Characterization

#### **Chapter 5**

## Temperature Dependence of the Stress Due to Additives in KCl Single Crystals

*Yohichi Kohzuki*

#### **Abstract**

The influence of the state of additive cations on the various deformation characteristics was studied for KCl:Sr2+ single crystal at room temperature. This result gives the heat treatment suitable for the crystal immediately before deformation tests, such as compression and tension. Four kinds of single crystals (KCl: Mg2+, Ca2+, Sr2+ or Ba2+) were plastically deformed by compression at 77 to room temperature. The plasticity of the crystal depends on dislocation motion from a microscopic viewpoint. When a dislocation breaks away from the defect around the additive cation with the help of thermal activation on the slip plane in the crystal, the variation of effective stress with the temperature was investigated by the combination method of strain-rate cycling tests and ultrasonic oscillations. Furthermore, the critical temperature *T*<sup>c</sup> at which the effective stress due to the additives is zero was estimated for each of the crystals. As a result,*T*<sup>c</sup> value tends to be larger with the divalent cationic size.

**Keywords:** dislocation, divalent cation, effective stress, yield stress, heat treatment

#### **1. Introduction**

In alkali halide crystals doped with divalent cations (divalent impurities), the additive cations are expected to be paired with vacancies to conserve the electrical neutrality. They are often formed a divalent impurity-vacancy (I-V) dipole for the impure crystals quenched from a high temperature. Then, the asymmetrical distortions are produced around the I-V dipoles. Mobile dislocations on a slip plane interact strongly only with these defects lying within one atom spacing of the glide plane [1]. The solution hardening is named "rapid hardening," which can be distinguished from "gradual hardening" due to the defects of cubic symmetry around the monovalent dopants in the crystals [2–4]. Effects of different defects on the hardness of some alkali halide crystals are listed in **Table 1**. The effects are expressed as an increase in flow stress per square root of concentration of point defects (i.e., Δ*τ* / Δ*c* 1/2) in terms of the shear modulus, *μ*. Despite the same matrix (see NaCl in **Table 1**), the hardening due to substitutional divalent additions is much larger than the case of monovalent ones. It has been well known for many years that aliovalent impurities (aliovalent cations) are a much more potent source of solution strengthening in ionic crystals than isovalent cations [5].


*Δ*c *represents the increment of the concentration of point defects and* μ *is the shear modulus. The measurements were made at room temperature unless otherwise noted. Δ*τ */Δ*c *in Refs. [2–4] is replaced by Δ*τ*/Δ*c*1/2.*

#### **Table 1.**

*Hardening due to various defects in alkali halide crystals. Defects concentration is below 10*�*<sup>4</sup> [2–4].*

In view of the different types of atomic defects, solution hardening may be divided into two classes: rapid hardening and gradual hardening by Fleischer and Hibbard [3] and Johnston et al. [4]. Roughly speaking, the value of Δ*τ* /Δ*c* 1/2 for rapid hardening is over several ten times as large as that for the gradual hardening as shown in **Table 1**.

It is well known that various characteristics of deformation are influenced by the state of impurities in a crystal. The following is concerned with it and presents the heat-treatment condition suitable for the deformation tests such as compression and tension for KCl:Sr2+ crystals. Furthermore, the influence of divalent cationic size on the deformation characteristics is reported analyzing the data obtained by the original method (strain-rate cycling tests associated with ultrasonic oscillation), which can separate the effective stress due to weak obstacles such as additive ions from that due to dislocation cuttings.

#### **2. Experimental procedure**

The initial dislocation density, the dielectric loss peak due to the I-V dipoles, and yield stress were measured for KCl:Sr2+ crystal as explained in Sections 2.2 to 2.4.

#### **2.1 Preparation of specimens**

KCl doped with SrCl2 was grown from the melt of reagent-grade powders by the Kyropoulos method in air. The specimens, which were cloven out of singlecrystalline ingots to the size of 5 � <sup>5</sup> � 15 mm<sup>3</sup> , were kept immediately at 973 K for 24 h, followed by cooling to room temperature at a rate of 40 Kh�<sup>1</sup> . This treatment is because the density of dislocations is reduced as much as possible. Owing to the gradual cooling, the additive ions (Sr2+) are expected to aggregate in the crystal. Accordingly, the specimens were further kept at 373 to 873 K for 30 min, followed by quenching to room temperature to disperse the additive ions (Sr2+) into them.

#### **2.2 Initial dislocation density (***ρ***)**

Using an etch pits technique, the initial density (*ρ*) of dislocations in KCl:Sr2+ (0.3 mol.% in the melt) was detected with a corrosive liquid (saturated solution of PbCl2 + ethyl alcohol added two drops of water). The etching was made at room

temperature for 30 min. The measurement of dislocation density in a crystal was carried out by the etch pit technique.

#### **2.3 Dielectric loss factor (tan** *δ***)**

The dielectric loss factor tan *δ* as a function of frequency was measured for KCl: Sr2+ (1.0 mol.% in the melt) in a thermostatic bath at 300 to 873 K by using an Andoh electricity TR-10C model.

### **2.4 Yield stress (***τ***y)**

The values of yield stress *τ*<sup>y</sup> were obtained at room temperature for KCl:Sr2+ (1.0 mol.% in the melt) compressed along the <100 > axis at the crosshead speed of 20 μm min<sup>1</sup> . The *τ*<sup>y</sup> values were determined by the intersection of the tangent to the easy glide region in the stress-strain curve and the straight line extrapolated from the elastic region of the curve.

#### **2.5 Combination method of strain-rate cycling tests and ultrasonic oscillation**

Four kinds of single crystals (KCl:Mg2+ (0.035 mol.% in the melt), Ca2+ (0.050, 0.065 mol.% in the melt), Sr2+ (0.035, 0.050, 0.065 mol.% in the melt) or Ba2+ (0.050, 0.065 mol.% in the melt)) were prepared by cleaving the single crystalline ingots to the size of 5 <sup>5</sup> 15 mm<sup>3</sup> . The test pieces were kept immediately below the melting point (1043 K) for 24 h and were gradually cooled to room temperature at a rate of 40 Kh<sup>1</sup> . Further, they were held at 673 K for 30 min and were rapidly cooled by water-quenching immediately before the following tests.

The heat-treated test pieces were compressed along the <100> axis at 77 K to the room temperature and the ultrasonic oscillatory stress (*τ*v) was intermittently superimposed in the same direction as the compression. The strain-rate cycling test off or on the ultrasonic oscillation (20 kHz) is illustrated in **Figure 1**. Superposing oscillatory stress, a stress drop (Δ*τ*) is caused during plastic deformation. The strain-rate cycling between strain-rates of *<sup>ε</sup>*\_ <sup>1</sup> (2.2 <sup>10</sup><sup>5</sup> <sup>s</sup> 1 ) and *<sup>ε</sup>*\_<sup>2</sup> (1.1 <sup>10</sup><sup>4</sup> <sup>s</sup> 1 ) was undertaken keeping the stress amplitude of *τ*<sup>v</sup> constant. This led to the increase Δ*τ*' in stress due to the strain-rate cycling. The strain-rate sensitivity (Δ*τ*'/Δln *ε*\_) of the flow stress, which is derived from Δ*τ*'/1.609, was used as a measurement of the strain-rate sensitivity.

#### **Figure 1.**

*Change in applied shear stress (τ*a*) for the strain-rate cycling tests between the two strain rates, ε*\_<sup>1</sup> *(2.2 <sup>10</sup><sup>5</sup> <sup>s</sup> 1 ) and <sup>ε</sup>*\_<sup>2</sup> *(1.1 <sup>10</sup><sup>4</sup> <sup>s</sup> 1 ), off or on the ultrasonic oscillatory stress (τv) due to the oscillation (20 kHz).*

#### **3. Results and discussion**

#### **3.1 Deformation characteristics influenced by different heat treatments for KCl:Sr2+ crystals**

**Figure 2** shows the optical micrograph of the etch pits for KCl:Sr2+ (0.05 mol.% in the melt) at room temperature after the annealing at 973 K for 24 h. The position of the dislocation after the annealed treatment is marked by a pyramidal pit, and the position where a dislocation slipped out of the crystal after the treatment is marked by a flat-bottom pit.

Although, it is difficult to resolve the individual etch pits for high dislocation density, the dislocation density on a (100) plane is found to be 1.27 � <sup>10</sup><sup>4</sup> cm�<sup>2</sup> from this micrograph for the annealed specimen (i.e., KCl:Sr2+ (0.05 mol.% in the melt)).

The height of the loss peak is related to the concentration of the isolated I-V dipole (see Eq. (1)). The details are explained about KCl:Sr2+ (0.05 mol.% in the melt) below.

Dielectric absorption of an I-V dipole causes a peak on the relative curve of tan *δ*


$$\tan \delta = \frac{2\pi e^2 c}{3\epsilon' akT} \,, \ (\text{maximum}) \tag{1}$$

where *e* is the elementary electric charge, *c* is the concentration of I-V dipoles, *ε*<sup>0</sup> is the dielectric constant in the matrix, *a* is the lattice constant, *k* is the Boltzmann constant, and *T* is the absolute temperature. **Figure 3** shows the tan *δ*-frequency curves for KCl:Sr2+ at 393 K. The solid and dotted curves correspond to the quenched KCl:Sr2+ (0.05 mol.% in the melt). That is to say, the crystals were held

**Figure 2.** *Dislocations on a (100) plane for KCl:Sr2+ (0.05 mol.%) after the heat treatment at 973 K for 24 h.*

*Temperature Dependence of the Stress Due to Additives in KCl Single Crystals DOI: http://dx.doi.org/10.5772/intechopen.104552*

#### **Figure 3.**

*Dielectric loss in KCl:Sr2+(0.05 mol.% in the melt) at 393 K. Dotted line (*� � *- -) shows the losses coming from the I-V dipoles.*

within 673 K for 30 min, followed by quenching to room temperature. The dotted line shows Debye peak obtained by subtracting the d.c. part which is obtained by extrapolating the linear part of the solid curve in the low-frequency region to the high-frequency region. By introducing the peak height of the dotted curve into Eq. (1), the concentration of the isolated I-V dipoles was determined to be 98.3 ppm for the quenched crystal by dielectric loss measurement.

As mentioned above, the dielectric loss factor tan *δ* is proportional to the concentration of the isolated I-V dipoles at a given temperature.

**Figure 4** shows the variations in the initial dislocation density (*ρ*), the dielectric loss peak due to the I-V dipoles (tan *δ*), and yield stress (*τ*y) as against the temperature quenched KCl:Sr2+ (0.3 and 1.0 mol.% in the melt) single crystals [7]. The *<sup>ρ</sup>* value is about 5 � <sup>1</sup> � <sup>10</sup>4cm�<sup>2</sup> independently of quenching temperature below 673 K, but it remarkably increases above 673 K. The *τ*<sup>y</sup> value also remarkably increases for the crystals quenched at the temperature above 673 K as the variation in dislocation density. And then it becomes a constant value 29 MPa above 723 K. While the tan *δ* value does not vary and is almost constant by quenching from the temperature below 573 K or above 673 K. Its value becomes 0.3 �10�<sup>2</sup> up to 0.9 �10�<sup>2</sup> between the two quenching temperatures (i.e., 573 K and 673 K). The variation in tan *δ* value is similar to it in the yield stress within the temperature.

The difference in dislocation density is slight and the tan *δ* obviously becomes larger with a higher quenching temperature between 573 and 673 K as shown in **Figure 4**. The concentration of isolated I-V dipoles, which is proportional to the tan *δ* (see Eq. (1)), affects the yield stress, as reported in the papers [8–11]. Therefore, the specimens are determined to be quenched from 673 K to room temperature immediately before deformation tests such as compression in this chapter.

#### **3.2 Temperature dependence of** *τ***p1,** *τ***p2, and yield stress (***τ***y)**

The variation of the strain-rate sensitivity and the stress decrement with the shear strain is shown in **Figure 5** for KCl:Sr2+ (0.050 mol.% in melt) single crystals at 200 K. Δ*τ*<sup>0</sup> /Δln *ε*\_ tends to increase with shear strain and decrease with stress amplitude in

**Figure 4.**

*Quenching temperature dependence of initial dislocation density (ρ), the dielectric loss peaks due to the I-V dipoles (tanδ), and yield stress (τy) for KCl:Sr2+ crystals (reproduced from Ref. [7]).*

#### **Figure 5.**

*Shear strain (*ε*) dependence of (*a*) the strain-rate sensitivity (Δτ* <sup>0</sup> */Δln ε*\_*) and (b) the stress decrement (Δτ) for KCl:Sr2+ (0.050 mol.%) at 200 K. <sup>τ</sup><sup>v</sup> (arb. units): (○) 0, (*●*) 10, (*▲*) 25, (*▽*) 35, (*▼*) 45, and (□) 50 (reproduced from Ref. [12] with permission from the publisher).*

*Temperature Dependence of the Stress Due to Additives in KCl Single Crystals DOI: http://dx.doi.org/10.5772/intechopen.104552*

#### **Figure 6.**

*Strain-rate sensitivity (Δτ* <sup>0</sup> */Δln ε*\_*) vs. the stress decrement (Δτ) at strain of 10% for KCl:Sr2+ (0.050 mol.%) at temperatures of (○) 103 K, (*Δ) 133 K, (□) 200 K, (⋄) 225 K *(reproduced from Ref. [13] with permission from the publisher)*.

**Figure 5(a)**. Δ*τ* does not change significantly with shear strain but increases with stress amplitude at a given temperature and shear strain in **Figure 5(b)**.

Δ*τ*<sup>0</sup> /Δln *ε*\_ vs. Δ*τ* curve is further obtained from **Figure 5** at a given strain, which provides the relative curve for a fixed internal structure of the crystal and is shown by open squares in **Figure 6** for KCl:Sr2+ (0.050 mol.% in melt) crystal at the shear strain of 10%. The details were described in the review article [14].

Relation between the strain-rate sensitivity and the stress decrement for KCl:Sr2+ (0.050 mol.% in melt) at the shear strain of 10% is shown by open symbols in **Figure 6**. The relative curve has a stair like shape. **Figure 6** shows the influence of temperature on the relation between the strain-rate sensitivity, Δ*τ*<sup>0</sup> /Δln *ε*\_, and the stress decrement, Δ*τ*, for KCl single crystals doped with Sr2+ as weak obstacles. As the temperature is high, the Δ*τ* value at first bending point, *τ*p1, shifts in the direction of low stress decrement and does not appear up to 225 K. The first plateau region indicates that the average length of the dislocation segment remains constant in that region. This is because the strain-rate sensitivity of effective stress (*τ* \* ) due to impurities is inversely proportional to the average length of the dislocation segment. That is to say, it is given by

$$\left(\frac{\Delta\tau^\*}{\Delta\ln\dot{e}}\right)\_T = \frac{kT}{bLd} \tag{2}$$

where *b* is the magnitude of the Burgers vector, *L* is the average length of dislocation segments, and *d* is the activation distance. Therefore, the application of

#### **Figure 7.**

*Temperature dependence of (○) τp1, (*Δ) τp2, and (□) τ<sup>y</sup> for KCl:Sr2+ ((a) 0.065, (b) 0.050, (c) 0.035 mol.% in the melt) *(reproduced from Ref. [12])*.

oscillation with low-stress amplitude cannot influence the average length of the dislocation segment at low temperature, but even a low-stress amplitude can do so at a temperature of 225 K. Such a phenomenon was also observed for the other specimens: KCl doped with Mg2+, Ca2+ or Ba2+ separately.

**Figure 7(a)**–**(c)** shows the dependence of *τ*p1, *τ*p2, and yield stress (*τ*y) on temperature for KCl:Sr2+ ((a) 0.065, (b) 0.050 and (c) 0.035 mol.%, respectively) crystals. *τ*p2 is the Δ*τ* value at the second bending point on the plots of Δ*τ* vs. (Δ*τ*<sup>0</sup> /Δln *ε*\_). It is clear from the figure that both *τ*p1 and *τ*p2 tend to increase with decreasing temperature as well as *τ*<sup>y</sup> for the three crystals and the *τ*<sup>y</sup> curve seems to approach a constant stress at high temperature. Two values of *τ*p1 and *τ*p2 increase with increasing Sr2+ concentration at a given temperature as shown in the figure. Similar results as the case of KCl:Sr2+ are also observed for the other crystals (i.e., KCl: Mg2+, Ca2+ or Ba2+).

#### **3.3 Critical temperature (***Tc***)**

**Figure 8(a)**–**(d)** shows the dependence of *τ*p1 on the temperature for KCl doped with Mg2+ (0.035 mol.% in melt), Ca2+ (0.065 mol.% in melt), Sr2+ (0.050 mol.% in melt) or Ba2+ (0.065 mol.% in melt). *τ*p1 is considered the effective stress due to the weak obstacles (Mg2+, Ca2+, Sr2+ or Ba2+ ions in this Section 3.2) on the mobile dislocation during plastic deformation [13]. *τ*p1 decreases with increasing temperature for the four kinds of crystals in the figure. The critical temperature (*T*c) at which *τ*p1 is zero can be determined from the

*Temperature Dependence of the Stress Due to Additives in KCl Single Crystals DOI: http://dx.doi.org/10.5772/intechopen.104552*

**Figure 8.**

*Temperature dependence of τp1 for various crystals: (*a*) KCl:Mg2+ (0.035 mol.% in the melt), (b) KCl:Ca2+ (0.065 mol.% in the melt), (c) KCl:Sr2+ (0.050 mol.% in the melt), and (d) KCl:Ba2+ (0.065 mol.% in the melt). (Reproduced from Ref. [13] with permission from the publisher).*

intersection with the abscissa and is around 180, 220, 230 and 260 K for KCl:Mg2+, KCl:Ca2+, KCl:Sr2+, and KCl:Ba2+, respectively.

The tetragonal distortion resulting from the introduction of the divalent cations into alkali halides is generally formed in the lattice. A dislocation moves on a single slip plane and interacts strongly only with those defects. Then, the relation between the effective stress and the temperature can be approximated as the linear relationship of *τ* <sup>∗</sup> <sup>1</sup>*=*<sup>2</sup> vs. *T*1/2 (i.e., the Fleischer's model [1]). *τ* <sup>∗</sup> is the effective stress due to the divalent cations. The critical temperature can be also determined from *τ*p1 1/2 vs. *T*1/2 for each specimen. The values of *T*<sup>c</sup> are given in **Table 2**. When the divalent ionic size becomes closer to it of K<sup>+</sup> from the small divalent cationic size side,*T*<sup>c</sup> tends to increase. *T*<sup>c</sup> is not influenced by the concentration of additives (i.e., Mg2+, Ca2+, Sr2+or Ba2+ here) [12, 15, 16] and is expressed by


**Table 2.**

*Tc and ionic radius values for various crystals.*

$$T\_c = \left(\Delta \mathbf{G}\_0/\mathbf{k}\right) \ln\left[\dot{\varepsilon}/\left\{\rho\_\mathbf{m} b^2 \nu\_\mathbf{D} \left(\mathbf{L}\_0/\mathbf{L}\right)^2\right\}\right],\tag{3}$$

where Δ*G*<sup>0</sup> is the Gibbs free energy for the breakaway of a dislocation from an impurity, *ρ*<sup>m</sup> is the density of mobile dislocations, the *ν<sup>D</sup>* is the Debye frequency, and *L*<sup>0</sup> is the average spacing of divalent cations on a slip plane. At the temperature of *T*c, thermal fluctuations can provide the entire energy for breaking through the impurity. The additive ions act no longer as obstacles to dislocation motion.

#### **4. Conclusions**

The concentration of isolated I-V dipoles affects the *τ*<sup>y</sup> values. The values of *τ*<sup>y</sup> and *ρ* remarkably increase with the quenching temperature above 673 K. As for tan *δ*, it does not vary by quenching from the temperature below 573 K or above 673 K. Within 573 to 673 K, the difference in *ρ* is slight and the values of tan *δ* and *τ*<sup>y</sup> obviously become larger with a higher quenching temperature. Based on these results, KCl:Sr2+ single crystals are determined to be quenched from 673 K to room temperature immediately before deformation tests such as compression.

The following two points were mainly mentioned from the experimental results and the discussion based on the data *τ*p1 of the first bending point on the plots of Δ*τ* vs. (Δ*τ*'/Δln *ε*\_).


#### **Conflict of interest**

The author declares no conflict of interest.

*Temperature Dependence of the Stress Due to Additives in KCl Single Crystals DOI: http://dx.doi.org/10.5772/intechopen.104552*

### **Author details**

Yohichi Kohzuki

Department of Mechanical Engineering, Saitama Institute of Technology, Fukaya, Japan

\*Address all correspondence to: kohzuki@sit.ac.jp

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **References**

[1] Fleischer RL. Rapid solution hardening, dislocation mobility, and the flow stress of crystals. Journal of Applied Physics. 1962;**33**:3504-3508. DOI: 10.1063/1.1702437

[2] Fleischer RL. Solution hardening by tetragonal distortions: Application to irradiation hardening in F.C.C. crystals. Acta Metallurgica. 1962;**10**:835-842. DOI: 10.1016/0001-6160(62)90098-6

[3] Fleischer RL, Hibbard WR. The Relation between Structure and Mechanical Properties of Metal. London: Her Majesty's Stationary Office; 1963. p. 261

[4] Johnston WG, Nadeau JS, Fleischer RL. The hardening of alkalihalide crystals by point defects. Journal of the Physical Society of Japan. 1963; **(Suppl. I)18**:7-15

[5] Sprackling MT. The plastic deformation of simple ionic crystals. In: Alper AM, Margrave JL, Nowick AS, editors. Materials Science and Technology. London New York San Francisco: Academic Press; 1976. p. 93

[6] Lidiard AB. Handbuch der Physik. Vol. 20. Berlin: Springer; 1957. p. 246

[7] Katoh S. Influence of Impurity Concentration on the Blaha Effect (Thesis). Kanazawa: Kanazawa Univ; 1987. pp. 16-21 (in Japanese)

[8] Dryden JS, Morimoto S, Cook JS. The hardness of alkali halide crystals containing divalent ion impurities. Philosophical Magazine. 1965;**12**: 379-391. DOI: 10.1080/ 14786436508218880

[9] Orozco ME, Mendoza AA, Soullard J, Rubio OJ. Changes in yield stress of NaCl:Pb2+ crystals during dissolution and precipitation of solid solutions. Japanese Journal of Applied Physics.

1982;**21**:249-254. DOI: 10.1143/ JJAP.21.249

[10] Zaldo C, Solé JG, Agulló-López F. Mechanical strengthening and impurity precipitation behaviour for divalent cation-doped alkali halides. Journal of Materials Science. 1982;**17**:1465-1473. DOI: 10.1007/BF00752261

[11] Reddy BK. Annealing and ageing studies in quenched KBr:Ba2+ single crystals. Physica Status Solidi A: Applications and Materials Science. 1987;**99**:K7-K10. DOI: 10.1002/ pssa.2210990140

[12] Kohzuki Y, Ohgaku T, Takeuchi N. Interaction between a dislocation and impurities in KCl single crystals. Journal of Materials Science. 1993;**28**:3612-3616. DOI: 10.1007/BF01159844

[13] Kohzuki Y, Ohgaku T, Takeuchi N. Interaction between a dislocation and various divalent impurities in KCl single crystals. Journal of Materials Science. 1995;**30**:101-104. DOI: 10.1007/ BF00352137

[14] Kohzuki Y. Study on dislocationdopant ions interaction in ionic crystals by the strain-rate cycling test during the Blaha effect. Crystals. 2018;**8**:31-54. DOI: 10.3390/cryst8010031

[15] Hikata A, Johnson R. A, Elbaum C. Interaction of dislocations with electrons and with phonons. Physical Review 1970;B2:4856–4863. DOI: 10.1103/PhysRevB.2.4856

[16] Ohgaku T, Takeuchi N. Interaction between a dislocation and monovalent impurities in KCl single crystals. Physica Status Solidi A. 1992;**134**:397-404. DOI: 10.1002/pssa.2211340210

## **Chapter 6** Elasticity of Auxetic Materials

*Jeremiah Rushchitsky*

### **Abstract**

The auxeticity of elastic materials is described and explained by the use of the linear and nonlinear models of elastic deformations for a wide range of strain values up to moderate levels. This chapter consists of three parts – general information on auxetic materials, description of auxetics by the model of the linear theory of elasticity, description of auxetics by the models of the nonlinear theory of elasticity. The analytical expressions are offered that corresponds to three kinds of universal deformations (simple shear, uniaxial tension, omniaxial tension) within the framework of three well-known in the nonlinear theory of elasticity models – twoconstant Neo-Hookean model, three-constant Mooney-Rivlin model, five-constant Murna-ghan model. A most interesting novelty consists in that the sample from elastic material is deformed as the conventional material for small values of strains whereas as the auxetic with increasing to moderate values of strains.

**Keywords:** auxetic material, elastic deformation of auxetics, three main effects of auxeticity, linear elastic model (Hookean model), nonlinear elastic models (twoconstant Neo-Hookean model, three-constant Mooney-Rivlin model, five-constant Murnaghan model, new mechanical effects

#### **1. Introduction**

To begin with, let us recall the definition of elasticity of deformation of the material. So, the property of elasticity consists in that the body practically simultaneously takes the initial configuration after removing the deformation causes. In other words, if deformations are elastic, then they simultaneously vanish after removing the action of forces, caused the deformations.

This property, as also other properties, though, is displayed seldom in the pure form, that is, it is accompanied in real solid materials by several other properties. But in most cases, elasticity is the main and pre-vailing property.

It is worthy to note at beginning of this chapter that the analysis of auxetic materials as the deforming elastically materials is dominating over other types of deformation (thermoelastic, viscoelastic, elastoplastic, magnetoelastic, etc). Therefore, the theme "Elasticity of Auxetic Materials" is related to the main part of studies of auxetics.

At present, the auxetic materials are thought of as some subclass of nontraditional (nonconventional) ma-terials which are known as metamaterials. The metamaterials include the mechanical metamaterials, which in turn include the auxetic materials. At present, a sufficiently big group of scientists work in the area of auxetic materials. It includes mainly specialists from material science, to the lesser extent from statistical physics, and even to the lesser extent from experimental mechanics. The state-of-the-art in science on auxe tic materials is shown in the monographs [1–3] and the review articles [4–23].

The auxetic materials were discovered and identified as a novel class of materials about forty years ago. Usually, two publications of Gibson L.J., Ashby M.F. et al [19, 20] are shown as the pioneer ones.

The term "auxetic material" was introduced by Evans in 1991 [21] for a new range of materials, which he defined them "the materials with negative Poisson ratio (NPR)". This needs some scientific comments relative to the term and definition.

*Comment 1* (to the term "auxetic material"). At present, Wikipedia and other sources propose for such materials the name "auxetics". Both names come from the Greek word *αυξητικοζ* (that which tends to increase). But this does not explain why just "auxetic". The next comment on definition gives some clearness.

*Comment* 2 (to the definition of auxetic material). Auxetic materials are deformed elastically exhibiting the unconventional property of increasing the crosssection (growing swollen) of cylindrical or prismatic samp-le under uniaxial *tension*, whereas in the conventional materials this cross-section decreases (grows thin). Just this is reflected in the name "auxetic" and shown in **Figure 1** [4].

The point is that the property of the decrease is described in the linear theory of elasticity by the use of the Poisson ratio as the elastic constant. A change of the decrease of cross-section on the increase of one means a change of positive values of the Poisson ratio on the negative ones.

The presented short information on auxetics shows that their definition is based on the secondary fact – the negativity of the Poisson ratio, which corresponds to the model of the linearly elastic body. The primary fact consists in observation in the standard for mechanics of materials (which does not depend on the model of deformation) experiment of longitudinal tension of a prism when the transverse deformation of the prism is positive (a material as if swells) in contrast to the classical materials, where it is negative.

The adherence of researchers of auxetic materials to the foams can be seen in the often used (described verbally or by the picture) demonstration of auxeticity of the foam as increasing the volume of sample from the foam under tension. It is shown in **Figure 2** [[10] (left), [20] (right)].

These pictures are really very demonstrative because they show two basic features.

*Feature 1.* The sample length is possibly not sufficient to create the classical conditions of the test on the universal deformation of uniaxial tension-compression.

**Figure 1.**

*Test on uniaxial tension for conventional and non-conventional materials.*

**Figure 2.** *Usually used test-demonstration of auxeticity.*

#### *Elasticity of Auxetic Materials DOI: http://dx.doi.org/10.5772/intechopen.99543*

*Feature 2.* The longitudinal and transverse strains are *seemingly* not sufficiently small in this test to be described by the linear theory of elasticity.

Starting with the first works on auxetics, the discussed real materials were the different kinds of foams. It is considered that the first observed auxetic materials were the foams which are characterized by the small value of density and the porous internal structure (see Lakes [24] and Wojciechowski [25]). In the next studies, the new auxetics were revealed, the density of which was also small and which have a porous structure. But it was shown later that small density is not the defining property of auxetics, because the significant part of foams has not the property of auxeticity. The defining characteristics of auxetics are new three mechanical phenomena which will be described below.

The common concept was adopted almost at the initial part of studies that the auxeticity of materials is caused by the internal structure of these materials. This corresponds to the general concept of mechanics (which is clearly shown in mechanics of composite materials) that the specificities of deformation of materials can be explained by the existence of some specific internal structure. Only the answer should be found which concrete specificity is characteristic for the auxetic materials. Therefore, an essential part of studies of auxetics consists in the finding of diverse variants of internal structure that are further studied by methods of molecular physics and computational simulations. The most popular is a so-called hexagonal system (it is shown in **Figure 3** [12]; left – before stretching, right – after stretching). Just this structure shows the swelling of the sample and is given by different authors to illustrate the auxeticity.

**Figure 3.** *The most known interpretation of the internal structure of auxetic material.*

It should be noted that the mechanics of materials works with the continuum models. This means that any discrete models of the internal structure must be transformed into the continuum one (here the different ways of averaging are usually applied). In mechanics, the internal structure of materials can appear on two different stages of modeling the materials. First, on the stage of changing the discrete structure of a material by the continuous one (that is when the notion of the continuum is introduced according to the principle of continualization). Second, on the stage of modeling the piece-wise inhomogeneous continuum by the homo-geneous continuum (that is when the principle of homogenization is applied). The first stage is usually asso-ciated with methods of molecular physics, whereas the second stage is a standard one in mechanics of composite materials. This is peculiar to all the materials that are studied in mechanics and refers also to the theory of elasticity within the framework of which the elastic deformation of aux-etic materials is studied.

For the presence in the material property of auxeticity, its internal structure has to change under defor mation by the special way exhibiting the unusual (nontraditional) mechanical effects. Note that mechanics of materials studied traditionally first the elastic deformation and this concern both traditional (non-auxe-tic), and nontraditional (auxetic) materials.

As far as the number of known nonauxetic materials exceeds the number of auxetic ones on many or-ders, then the term "unusual effect" is looking

#### **Figure 4.** *Sample from the polyurethane foam (left – traditional structure, right – auxetic structure).*

appropriate. In contrast to the traditional effects that count tens, the effects of auxeticity are observed as now in the identical mechanical problems in three types of such problems that are realized experimentally and described theoretically. An identity consists in that the samples from material must be compared when the internal structure of a material in cases "auxetic non-auxetic" is differing by the only geometrical shape of pores. This case is shown in **Figure 4** for the sample from the polyurethane foam (left – traditional structure, right – auxetic structure) [2].

Now, some facts from this theory should be shown concerning the phenomenon of auxeticity. But first three specific appearances of auxeticity must be described and commented on.

### **2. Three specific properties of auxetic materials**

Only one of these specificities is well known – the swelling under the tension of the standard sample (standard mechanical test). This test is described above and shown in **Figure 1**.

But the fact is known that the auxeticity is generated by the special kind of internal structure of material and appears in three basic mechanical tests on deformation of material

1.*Swelling under tension.*

2.*Hardening under indentation (impact).*

3.*Synclastic* and *anticlastic deformation of thin flexible plate.*

Test 2 on indentation (statical Hertz problem, problem on hardness by Rockwell-Brinell-Wikkers) and impact (dynamical Hertz problem) shows the effect of hardness of auxetics in the contact zone. Within the framework of the theory of elasticity, this problem is solving numerically with the given exactness. A scheme of test that exhibits the essential difference in the degree of indentation of the spherical indentor into the traditional (left) and auxetic (right) materials is shown in **Figure 5** [12].

**Figure 5.** *Test for hardness material (left – traditional structure, right – auxetic structure).*

#### *Elasticity of Auxetic Materials DOI: http://dx.doi.org/10.5772/intechopen.99543*

**Figure 6.** *Test on synclastic (left) and anticlastic (right) deformation.*

Test 3 on synclastic and anticlastic deformation of flexible elastic plate is stated within the assumption that the plate is quadratic in plan and is loaded by the balanced system of three forces – one force is applied at the center of a plate and directed upward, whereas two other identical forces are applied at the centers of two opposite ends of the plate and directed downward. Within the framework of the theory of flexible elastic plates, this problem is solving numerically with the given exactness. The simple experiment that exhibits the essential difference in deformation of the plate from the traditional and auxetic materials is shown in **Figure 6** [4] (left traditional material, right – auxetic material).

Note that these basic phenomena of deformation of auxetics can be described only in the terms of the theory of elasticity.

#### **3. Some facts from the linear theory of elasticity necessary for describing the auxetic materials**

Because the elastic deformation is described in mechanics only by the theory of elasticity, then some facts from this theory should be recalled before the discussion of the specificities of elastic deformation of auxetics. At that, the division of the theory of elasticity on the linear and nonlinear theories should be taken into account. It is important to remember that the linear theory is based on the one (Hookean) model, whereas the nonlinear uses many different models.

#### **3.1 Universal deformations**

Universal deformations (uniform deformations, universal states) occupy a special place in the theory of elasticity just owing to their universality [26]. This universality consists in that the theoretically and experi-mentally determining elastic constants of material in samples, in which the universal deformation is created purposely, are valid also for all other deformed states both samples and any different products made of this material. It is considered therefore that the particular importance of universal deformation (their fundamen tality) consists in the possibility to use them in the determination of properties of materials from tests [26–31]. To realize the universal deformation, two conditions have to be fulfilled: 1. Uniformity of deformation must not depend on the choice of material. 2. Deformation of material has to occur by using only the surface loads.

In the theory of infinitesimal deformations, the next kinds of universal deformations are studied more of-ten and in detail: simple shear, simple (uniaxial) tension-compression, uniform volume (omniaxial) tension-compression. In the linear theory of elasticity, the experiment with a sample, in which the simple shear is realized, allows determining the elastic shear modulus *μ*. The experiment with a

sample, in which the uniaxial tension is realized, allows determining Young elastic modulus *E* and Poisson ratio *ν*. The experiment with a sample, in which the uniform compression is realized, allows determining the elastic bulk modulus *k*.

While being passed from the linear model, which is valid for only the very small deformations to the mo-dels of non-small (moderate or large) ones, that is, from the linear mechanics of materials to nonlinear me-chanics of materials, the universal states permit to describe theoretically and experimentally many nonlinear phenomena. The history of mechanics testifies to the experimental observation in the XIX century of the non linear effects that arose under the simple shear and were named later by the names of Poynting and Kelvin [27–31]. After about a hundred years in the XX century, these effects were described theoretically within the framework of the nonlinear Mooney-Rivlin model [31–35].

The mechanics of composite materials is one more area of application of universal deformations. The mo-del of averaged (effective, reduced) moduli is in this case the simplest and most used model. In the theory of effective moduli, the composite materials of the complex internal structure with internal links are treated usually as homogeneous elastic media. A possibility to create in such media the states with universal deformations was used in the evaluation of effective moduli by different authors and different methods. It was found that it is sufficient for isotropic (granular) composites to study the energy stored in the elementary volumes of composites under only two kinds of universal deformations: simple shear and omniaxial compression. In the case of transversely isotropic (fibrous or layered) composites, the different directions need analysis of universal deformations for each direction separately.

#### **3.2 Classical procedures of estimating the values of elastic moduli in the linear theory of elasticity**

Perhaps, the eldest and exhausting procedures are shown in the classical Love's book [36]. Let us save the Love's notations and write according to [36] the internal energy of deformation of the linearly elastic isotropic body *W* in the form

$$\mathcal{W} = \lambda \left(\varepsilon\_{\text{xx}} + \varepsilon\_{\text{yy}} + \varepsilon\_{\text{xx}}\right)^2 + 2\mu \left(\varepsilon\_{\text{xx}}^2 + \varepsilon\_{\text{yy}}^2 + \varepsilon\_{\text{xx}}^2\right) + \mu \left(\varepsilon\_{\text{xy}}^2 + \varepsilon\_{\text{xx}}^2 + \varepsilon\_{\text{yx}}^2\right), \tag{1}$$

where *λ*, *μ* are the Lame moduli, *εxx*, … , *εyz* are the components of the strain tensor. The Hooke law has the form

$$\begin{aligned} X\_{\rm x} &= \lambda \Delta + 2\mu \varepsilon\_{\rm xx}, Y\_{\rm y} = \lambda \Delta + 2\mu \varepsilon\_{\rm yy}, Z\_{\rm z} = \lambda \Delta + 2\mu \varepsilon\_{\rm xx}, \\ X\_{\rm y} &= 2\mu \varepsilon\_{\rm xy}, Z\_{\rm x} = 2\mu \varepsilon\_{\rm xx}, Y\_{\rm x} = 2\mu \varepsilon\_{\rm yx}. \end{aligned} \tag{2}$$

Here Δ ¼ *εxx* þ *εyy* þ *εzz* is the dilatation.

The classical procedure of introducing the Young modulus and Poisson ratio is as follows: the cylinder or prism of any shape is considered, then the axis of the cylinder is chosen in direction *Ox* and the prism is stretched at the ends by a uniform tension *T*. Because the lateral surface of the prism is assumed to be free of stresses, then the stress state of a prism is uniform and is characterized by only one stress *Xx* ¼ *T*. In this case, the Hooke law becomes simpler

$$T = \lambda \Delta + 2\mu \varepsilon\_{xx}, \mathbf{0} = \lambda \Delta + 2\mu \varepsilon\_{\mathcal{\mathcal{V}}}, \mathbf{0} = \lambda \Delta + 2\mu \varepsilon\_{xx}.\tag{3}$$

An expression for dilatation follows from equalities (3) *T* ¼ ð Þ 3*λ* þ 2*μ* Δ ! Δ ¼ *T=*ð Þ 3*λ* þ 2*μ* .

*Elasticity of Auxetic Materials DOI: http://dx.doi.org/10.5772/intechopen.99543*

The substitution of the last expression for dilatation into the first equality (2) gives relations

$$T = \frac{\lambda}{3\lambda + 2\mu} T + 2\mu \varepsilon\_{\text{xx}} \to T = \frac{\mu(3\lambda + 2\mu)}{\lambda + \mu} \varepsilon\_{\text{xx}}.\tag{4}$$

The expression (4) represents the elementary law *T* ¼ *Eεxx* of link between tension and deformation of the prism, in which the Young modulus *E* is used

$$E = \frac{\mu(\mathfrak{Z}\lambda + \mathfrak{Z}\mu)}{\lambda + \mu}. \tag{5}$$

The substitution of expression for dilatation into the second and third equalities (2) gives relations

$$-\varepsilon\_{\mathcal{Y}} = -\varepsilon\_{\text{xx}} = \frac{\lambda}{2(\lambda + \mu)} \varepsilon\_{\text{xx}},\tag{6}$$

which express the classical Poisson law on the transverse compression under the longitudinal extension and permit to introduce of the Poisson ratio

$$
\sigma = \frac{-\varepsilon\_{yy}}{\varepsilon\_{\text{xx}}} = \frac{-\varepsilon\_{\text{xx}}}{\varepsilon\_{\text{xx}}} = \frac{\lambda}{2(\lambda + \mu)}.\tag{7}
$$

Let us repeat now the procedure associated with introducing the universal (uniform) deformation – the uniform compression. Thus, the body of arbitrary shape is considered, to all points of which the constant pressure �*p* is applied. In this body, the uniform stress state arises which is characterized by stresses *Xx* ¼ *Yy* ¼ *Zz* ¼ �*p*, *Xy* ¼ *Yz* ¼ *Zx* ¼ 0. The Hooke law becomes simpler

$$-p = \lambda \Delta + 2\mu \varepsilon\_{xx}, \ -p = \lambda \Delta + 2\mu \varepsilon\_{\eta \gamma}, \ -p = \lambda \Delta + 2\mu \varepsilon\_{xx}, \varepsilon\_{\chi \gamma} = \varepsilon\_{\chi x} = \varepsilon\_{xx} = 0. \tag{8}$$

The relations (8) can be transformed to �3*<sup>p</sup>* <sup>¼</sup> ð Þ <sup>3</sup>*<sup>λ</sup>* <sup>þ</sup> <sup>2</sup>*<sup>μ</sup> <sup>ε</sup>xx* <sup>þ</sup> *<sup>ε</sup>yy* <sup>þ</sup> *<sup>ε</sup>zz* ! �*p* ¼ ½ � *λ* þ ð Þ 2*=*3 *μ* Δ.

In this way, the modulus of compression *k* is defined

$$
\lambda = \lambda + (2/3)\mu. \tag{9}
$$

The classical Love's reasoning, which is repeated in most books on the linear theory of elasticity, is based on the representation of moduli *λ*, *μ*, *k* through moduli *E*, *σ*

$$\lambda = \frac{E\sigma}{(1+\sigma)(1-2\sigma)}, \ \mu = \frac{E}{2(1+\sigma)}, \ \ k = \frac{E}{3(1-2\sigma)}.\tag{10}$$

The formulas (10) are commented in ([36], p. 104) as follows: "If *σ* were > 1*=*2, *k* would be negative, or the material expands under pressure. If *σ* were < � 1, *μ* would be negative, and the function *W* would not be a positive quadratic function. We may show that this would also be the case if *k* were negative. Negative values *σ* are not excluded by the condition of stability, but such values have not been found for any isotro-pic material."

Because the comments of negativity of Poisson ratio is found in the books on the theory of elasticity very seldom, therefore a few sentences from Lurie's book ([29],

p. 117) are worthy to be cited: "A tension of the rod with negative *ν* (but the more than �1) would be accompanied by increasing of transverse sizes. Ener-getically, the existence of such elastic materials is not excluded." "In hypothetic material with *ν*< � 1, the hy-drostatic compression of the cube would be accompanied by increasing its volume."

Note that the Poisson ratio is denoted in the theory of elasticity by *σ* "sigma" and *ν* "nu". Love uses *σ*, whereas Lurie uses *ν*.

It should be also noted that not all authors of books on the linear isotropic theory of elasticity discuss the restrictions on changing the Poisson ratio (for example, Germain, Nowacki, Hahn do not made this in their well-known books [37–39]). The constitutive relations and classical restrictions on elastic constants are discussed in the most comprehensive and modern treatment of the theory of elasticity [28] (Subsection 3.3 "Constitutive relations").

But in some books, the discussion is presented and all authors start with one and the same postulate: in the procedure of restrictions in changing the Poisson ratio, the primary requirement is a positiveness of internal energy *W* (1). The representation of energy can be different for different elastic moduli. For example, Leibensohn [40], Love [36], Lurie [29] choose the pair *λ*, *μ* and use the representation (1). Landau and Lifshits [41] use the pair *k*, *μ*. In all the cases, *W* has a form of a quadratic function with coefficients composed of elastic moduli.

Thus, in most cases, the expression (1) is analyzed. It is assumed that the sufficient and being in line with experimental observations condition is the condition of positiveness of Lame moduli

$$
\lambda > 0, \mu > 0. \tag{11}
$$

Further, the formulas (10) are considered, in which without controversy the Young modulus is assumed positive *E*>0. Then positiveness of expressions 1 þ *σ* >0, 1 � 2*σ* >0 provides validity of formula (11), from which the well-known restriction on the Poisson ratio follows

$$-1 < \sigma < 1/2. \tag{12}$$

Let us recall that all the elastic moduli in the classical linear isotropic theory of elasticity are always posi-tive. The obvious contradiction between the assumption of negativity of the Poisson ratio and the primary statement on the positivity of Lame moduli (11) in condition when the Poisson ratio is defined by formula (7) is commented in the classical theory of elasticity anybody. To all appearances, this situation is occurred owing to the incredibility of negative values if only one of the elastic moduli *λ*, *μ*, *E*, *k*.

Note finally that two experimental approaches to determine the value of Poisson ratio for concrete material are used at present time ([27], subsections 2.18, 3.27, 3.28). The first approach is the older one. It is based on the experimental determination of Young, shear, and compression moduli and subsequent calculation of Poisson ratio by formulas (10) *σ* ¼ ð Þ� *E=*2*μ* 1, *σ* ¼ ð Þ 1*=*2 ½ � ð Þ� *E=*3*k* 1 . Here, the problem of the exactness of calculation arises. Let us cite Bell's book ([27], subsection 3.28): "Remind of the Grüneisen's conclusion that the errors of �1% in values *E* and *μ* result in the error of 10% in the value of Poisson ratio." Therefore, the second approach seems to be more preferable. It is associated with Kirchhoff's experiments (1859), in which the Poisson ratio is determined from the direct experiment on simultaneous bending and torsion.

Let us recall that the primary phenomenon in the determination of the Poisson ratio is the contraction of a sample (transverse deformation of a sample) under its elongation (its longitudinal deformation).

#### **3.3 Refinement of procedures of estimating the values of elastic moduli**

Let us save the initial postulate that the primary requirement is the positivity of internal energy *W* (1) and reject the sufficient (and not necessary) condition of positivity of *W* when the positivity of Lame moduli *λ*, *μ* is assumed and suppose the general condition of positivity of *W*.

Because the Lame modulus *μ* has a physical sense of the shear modulus and until now the facts of observation its negativity (a shear in the direction opposite to the direction of shear force) are not reported, then we can agree to its positivity. This condition of positivity can be also substantiated theoretically based on ana-lysis of universal deformation of simple shear. To describe the simple shear, the coordinate plane (for example, *xOy*) should be chosen and only one non-zero component *ux*,*<sup>y</sup>* of the displacement gradient should be given. This can be commented geometrically as deformation of the elementary rectangle *ABCD* with sides *dx*, *dy* parallel to the coordinate axes into the parallelogram *AB*<sup>0</sup> *C*0 *D*, which results from the longitudinal shift of the rectangle side *BC*. Then the shear angle ∡*BAB*<sup>0</sup> ¼ *γ* is linked with *ux*,*<sup>y</sup>* in a next way *ux*,*<sup>y</sup>* ¼ tan *γ* ¼ *τ* and *εxy* ¼ ð Þ 1*=*2 *τ*. The Hooke law becomes the simplest form *σxy* ¼ 2*μεxy* and the corresponding representation of internal energy is as follows *<sup>W</sup>* <sup>¼</sup> ð Þ <sup>1</sup>*=*<sup>2</sup> *μτ*2. Then the positivity of shear modulus (14) follows from the positivity of energy *W*.

Now, the next refinement can be formulated.

*Refinement 1.* The Lame modulus *λ* can be negative if the Poisson ratio *σ* ¼ *λ=*½ � 2ð Þ *λ* þ *μ* can be assumed possible negative.

*Refinement 2.* If the Poisson ratio *σ* is assumed to be possible negative and the shear modulus *μ* is positive, then according to definition (7) the negative Lame modulus *λ* can not exceed by its absolute value the shear modulus

$$|\lambda| < \mu. \tag{13}$$

Let us return to the primary definition of the Poisson ratio (7), which is found from the solution of the problem of unilateral tension. In this case, the internal energy has the form

$$\begin{split} \mathcal{W} &= \lambda(\epsilon\_{11} + \epsilon\_{22} + \epsilon\_{33})^2 + 2\mu \left(\epsilon\_{11}^2 + \epsilon\_{22}^2 + \epsilon\_{33}^2\right) = \lambda(\epsilon\_{11} + 2\sigma\epsilon\_{11})^2 + 2\mu \left(\epsilon\_{11}^2 + (\sigma\epsilon\_{11})^2 + (\sigma\epsilon\_{11})^2\right) \\ &= \left[\lambda(\mathbf{1} + 2\sigma)^2 + 2\mu \left(\mathbf{1} + 2\sigma^2\right)\right] \epsilon\_{11} \mathbf{1}^2 > 0. \end{split} \tag{14}$$

Then *<sup>λ</sup>* <sup>þ</sup> 2 1 <sup>þ</sup> <sup>2</sup>*σ*<sup>2</sup> ð Þ*=*ð Þ <sup>1</sup> <sup>þ</sup> <sup>2</sup>*<sup>σ</sup>* <sup>2</sup> h i*μ*>0*:* permits to the formulation of some new refinements.

*Refinement 3.* If the Poisson ratio *σ* is assumed to be possible negative and the shear modulus *μ* is positive, then the condition of positivity of internal energy admits arbitrary negative values of the Poisson ratio.

(because the coefficient ahead of *μ* is always positive). The case 1 þ 2*σ* ¼ 0 ! *σ* ¼ �0, 5 is the peculiar one – the value of modulus *λ* is practically not restricted at its neighborhood.

*Refinement 4.* The Lame modulus *λ* is already restricted from below according to (14), but also the additional condition (16) exists

$$|\lambda| < \left[ 2\left( \mathbf{1} + 2\sigma^2 \right) / \left( \mathbf{1} + 2\sigma \right)^2 \right] \mu. \tag{15}$$

The condition (15) is less strong: the coefficient ahead of *μ* exceeds 1 for all negative *σ* (in condition (13), the coefficient ahead of *μ* is equal to 1). Therefore, the condition (13) remains.

Let us turn to formula (9), which expresses the compression modulus *k* through the Lame moduli *λ*, *μ*. It follows from (9) that the modulus *k* will be negative if only the negative Lame modulus *λ* exceeds 2ð Þ *=*3 *μ* by absolute value

$$
\lambda = \lambda + (\mathfrak{I}/\mathfrak{J})\mu < 0 \to |\lambda| > (\mathfrak{I}/\mathfrak{J})\mu = \mathfrak{0}, \mathfrak{G}\mathfrak{G}\mathfrak{I}\mu.\tag{16}
$$

Comparison with restrictions (13) and (15) on the absolute values of negative Lame modulus *λ* in the case of negative values of Poisson ratio *σ* shows that (16) does not conflict with (13) and (15).

*Refinement 5.* If the Poisson ratio *σ* is assumed to be possible negative and the shear modulus *μ* is positive, then the compression modulus *k* can be negative.

The situation with refinements becomes clearer if the moduli *λ*, *E* and *k* are written through *μ* and *σ*

$$
\lambda = \frac{2\sigma}{1 - 2\sigma} \mu, E = 2(1 + \sigma)\mu, k = \frac{2}{3} \frac{1 + \sigma}{1 - 2\sigma} \mu. \tag{17}
$$

A few statements can be formulated at the end of this subchapter.

*Statement 1.* The classical restrictions of positivity of the elastic moduli in the isotropic theory of elasticity should be refined for auxetic materials: most elastic moduli can be negative.

*Statement 2.* Seemingly, the auxetics should be defined by the primary physical phenomenon of positivity of transverse deformation of a prism, which is observed in the standard in mechanics of materials experiment of longitudinal tension of a prism. In this case, the auxetics will be associated not only with the isotropic elastic materials.

*Statement 3.* In the case of auxetic materials, the Lame modulus *λ* is always negative and the Young *E* and compression *k* moduli are negative when the negative Poisson ratio is less than �1: *σ* < � 1.

*Statement 4.* When the problems of the linear isotropic theory of elasticity being studied for auxetic materials, then at least two elastic moduli for these materials should be determined from the direct experiments (unilateral tension, omnilateral compression, simple shear, torsion).

#### **4. Specificities of describing the auxetic materials by the nonlinear theory of elasticity**

#### **4.1 Essentials of nonlinear theory of elasticity**

While being studied the auxetics from the position of the nonlinear theory of elasticity, some essential differences between the linear and nonlinear descriptions should be taken into account. Therefore, the basic notions of the nonlinear approach seem to be worthy to show here very shortly [31, 34, 35, 42, 43].

A body is termed some area *V* of 3D space R<sup>3</sup> , in each point of which the density of mass *ρ* is given (the area occupied by the material continuum). In this way, a real body, the shape of which coincides with *V*, is changed on a fictitious body. This fictitious body is the basic notion of mechanics. The Lagrangian f g *xk* or Eulerian f g *Xk* coordinate systems can be given in R3 . In the theory of deformation of a body as a change of its initial shape, the notions are utilized that are associated with a geometry of body (kinematic notions) and with the forces acting on the body from outside and inside (kinetic notions). The notions of the configuration *χ*, the vector

#### *Elasticity of Auxetic Materials DOI: http://dx.doi.org/10.5772/intechopen.99543*

of displacement *u* ! <sup>¼</sup> f g *uk* , the principal extensions *<sup>λ</sup>k*, the strain tensor *<sup>ε</sup>ik* are referred to as the notions of kinematics. The external and internal forces, as well as the tensors of internal stresses, refer to the notions of kinetics,

The configuration of the body at a moment *t* is called the actual one, whereas the configuration of the body at arbitrarily chosen initial moment *t <sup>o</sup>* is called the reference one. The coordinates of the body point before deformation are denoted by *xk*. It is assumed that after deformation this point is displaced on the va-lue *uk*ð Þ *x*1, *x*2, *x*3, *t :* Then the vector with components *uk* is called the displacement vector and the coordinates of the point after deformation are presented in the form *ξ<sup>k</sup>* ¼ *xk* þ *uk*ð Þ *x*1, *x*2, *x*3, *t* . The frequently used Cauchy-Green strain tensor is given by the known displacement vector *u* !ð Þ *xk*, *<sup>t</sup>* in the Lagrangian coordinates f g *xk* and the reference configuration

$$
\varepsilon\_{nm}(\mathbf{x}\_k, t) = (\mathbf{1}/2)(u\_{n,m} + u\_{m,n} + u\_{n,i}u\_{m,i}).\tag{18}
$$

As a result, the deformation of the body is given by nine components of displacement gradients *ui*,*<sup>k</sup>*. Such a description of deformation is used in most models of the nonlinear theory of elasticity. But the process of deformation can be described also by other parameters of the geometry change of the body. It seems meaning ful to use often the first three algebraic invariants of tensor (18) *A*<sup>1</sup> ¼ *<sup>ε</sup>mnδmn*, *<sup>A</sup>*<sup>2</sup> <sup>¼</sup> ð Þ <sup>1</sup>*=*<sup>2</sup> ð Þ *<sup>ε</sup>mnδmn* <sup>2</sup> � *<sup>ε</sup>ikεik* h i, *<sup>A</sup>*<sup>3</sup> <sup>¼</sup> det *<sup>ε</sup>mn*, which can be rewritten through the principal values of tensor (18) *ε<sup>k</sup>* by the formulas *A*<sup>1</sup> ¼ *ε*<sup>1</sup> þ *ε*<sup>2</sup> þ *ε*3, *A*<sup>2</sup> ¼ *ε*1*ε*<sup>2</sup> þ *ε*1*ε*<sup>3</sup> þ *ε*2*ε*3, *A*<sup>3</sup> ¼ *ε*1*ε*2*ε*3*:* The often used invariants *I*1,*I*2,*I*<sup>3</sup> of tensor *εik* are linked with the algebraic invariants of the same tensor by relations

$$\begin{aligned} I\_1 &= 3 + 2\varepsilon\_{nn} = 3 + 2A\_1, & I\_2 &= 3 + 4\varepsilon\_{nn} + 2(\varepsilon\_{nn}\varepsilon\_{mm} - \varepsilon\_{nm}\varepsilon\_{mm}) = 3 + 4A\_1 + 2\left(A\_1^2 - A\_2\right), \\ I\_3 &= \det\left|\left|\delta\_{pq} + 2\varepsilon\_{pq}\right|\right| = 1 + 2A\_1 + 2\left(A\_1^2 - A\_2\right) + (4/3)\left(2A\_3 - 3A\_2A\_1 + A\_1^3\right). \end{aligned}$$

In several models of nonlinear deformation of materials, the elongation coefficients (principal extensions) defined as a change of length of the conditional linear elements (the infinitesimal segments that are directed arbitrarily) are used

$$
\lambda\_k = \sqrt{\mathbf{1} + 2\varepsilon\_k}. \tag{19}
$$

A simpler formula *λ<sup>k</sup>* � 1≈ *ε<sup>k</sup>* is valid for the case of linear theory. Additionally to three parameters (19), three parameters should be introduced that characterize a change of the angles between linear elements and areas of elements of coordinate surfaces.

It seems to be necessary to show the very often used notation of the displacement gradient

> **F** ¼ 1 þ *u*1,1 *u*1,2 *u*1,3 *u*2,1 1 þ *u*2,2 *u*2,3 *u*3,1 *u*3,2 1 þ *u*3,3 2 6 4 3 7 5

and notation of the left Cauchy-Green strain tensor **B** = **F F**<sup>T</sup> associated with it. The most used are two ten-sors of internal stresses: the symmetric Cauchy-Lagrange tensor *σik*, which is measured on the unit of area of the deformed body, and the nonsymmetric Kirchhoff tensor*tik*, which is measured on the unit area of the undeformed body.

#### **4.2 Universal deformation of simple shear in the nonlinear approach**

The simple shear is described in subsubsection 3.3, where the basic formula *u*1,2 ¼ tan *γ* ¼ *τ* > 0 is shown.

In the linear theory, the shear angle is assumed to be small and then *γ* ≈ tan *γ* ¼ *τ*. The nonlinear app-roach introduces some complications. The Cauchy-Green strain tensor is characterized by only three non-zero components

$$\begin{split} \varepsilon\_{11} &= (\mathbf{1}/2)(u\_{1,1} + u\_{1,1} + u\_{1,k}u\_{1,k}) = (\mathbf{1}/2)(u\_{1,2}u\_{1,2} + u\_{1,3}u\_{1,3}) = \tau^2; \varepsilon\_{12} = \varepsilon\_{21} \\ &= (\mathbf{1}/2)(u\_{1,2} + u\_{2,1} + u\_{1,k}u\_{2,k}) = (\mathbf{1}/2)\tau, \end{split} \tag{20}$$

The principal extensions are written through the shear angle by formulas *λ*<sup>1</sup> ¼ 1, *λ*<sup>2</sup> ¼ *λ*<sup>3</sup> ¼ *τ*.

#### **4.3 Universal deformation of uniaxial tension in the nonlinear approach**

This kind of deformation is also described above. It is characterized in the nonlinear approach by only one nonzero component *σ*<sup>11</sup> of the stress tensor and two nonzero components *ε*11, *ε*<sup>22</sup> ¼ *ε*<sup>33</sup> of the strain tensor (or two principal extensions *λ*1, *λ*<sup>2</sup> ¼ *λ*3).

#### **4.4 Universal deformation of uniform (omniaxial) compression-tension**

A sample has the shape of a cube, to sides of which the uniform surface load (hydrostatic compression) is applied. Then the uniform stress state is formed in the cube. The normal stresses are equal to each other *σ*<sup>11</sup> ¼ *σ*<sup>22</sup> ¼ *σ*33, and the shear stresses *σik*ð Þ *i* 6¼ *k* are absent. This type of deformation is defined as follows

$$u\_{1,1} = u\_{2,2} = u\_{3,3} = \varepsilon > 0,\\ u\_{1,1} + u\_{2,2} + u\_{3,3} = \mathfrak{Z}\varepsilon = \varepsilon,\\ u\_{k,m} = (\partial u\_k / \partial \mathfrak{x}\_m) = \mathbf{0} \ \ (k \neq m). \tag{21}$$

The Cauchy-Green strain tensor is simplified *ε*<sup>11</sup> ¼ *ε*<sup>22</sup> ¼ *ε*<sup>33</sup> ¼ *ε* þ ð Þ <sup>1</sup>*=*<sup>2</sup> *<sup>ε</sup>*2, *<sup>ε</sup>ik* <sup>¼</sup> <sup>0</sup> ð Þ *<sup>i</sup>* 6¼ *<sup>k</sup>* and the algebraic invariants are written in the form

$$I\_1 = \varepsilon\_{11} + \varepsilon\_{22} + \varepsilon\_{33} = \varepsilon,\\ I\_2 = \left(\varepsilon\_{11}\right)^2 + \left(\varepsilon\_{22}\right)^2 + \left(\varepsilon\_{33}\right)^2,\\ I\_3 = \left(\varepsilon\_{11}\right)^3 + \left(\varepsilon\_{22}\right)^3 + \left(\varepsilon\_{33}\right)^3. \tag{22}$$

The principal extensions are equal to each other

$$
\lambda\_1 = \lambda\_2 = \lambda\_3. \tag{23}
$$

#### **4.5 Three nonlinear models of hyperelastic deformation**

These models are related to the models of hyperelastic materials. This class of materials is characterized by the way of introduction of constitutive equations. First, the function of kinematic parameters (elastic potential, internal energy) is defined, from which later the constitutive equations are derived mathematically and sub-stantiated physically. Model 1 is chosen as the simplest one. Model 2 is wellworking for the not-small (large or finite) deformations. Model 3 belongs to the most used in the nonlinear mechanics of materials.

#### *4.5.1 Two-constant Neo-Hookean model (model 1)*

The elastic potential of this model is defined as follows [31, 34, 35, 42, 43]

$$W = \mathbf{C}\_1(\tilde{I}\_1 - \mathbf{3}) + D\_1(f - \mathbf{1})^2, \quad \tilde{I}\_1 = f^{-2/3} I\_1, \ f = \text{det}\boldsymbol{u}\_{i,k},$$

$$W(\boldsymbol{\lambda}\_1, \boldsymbol{\lambda}\_2, \boldsymbol{\lambda}\_3) = \mathbf{C}\_1 \Big[ \left(\boldsymbol{\lambda}\_1 \boldsymbol{\lambda}\_2 \boldsymbol{\lambda}\_3\right)^{-2/3} \left(\boldsymbol{\lambda}\_1^2 + \boldsymbol{\lambda}\_2^2 + \boldsymbol{\lambda}\_3^2\right) - \mathbf{3} \Big] + D\_1(\boldsymbol{\lambda}\_1 \boldsymbol{\lambda}\_2 \boldsymbol{\lambda}\_3 - \mathbf{1})^2. \tag{24}$$

#### *Elasticity of Auxetic Materials DOI: http://dx.doi.org/10.5772/intechopen.99543*

Here the elastic constants of the model are linked with the classical elastic constants by relation 2*C*<sup>1</sup> ¼ *μ*; 2*D*<sup>1</sup> ¼ *k*.

The constitutive equations have the form

$$
\sigma\_{nm} = 2C\_1 J^{-5/3} [B\_{nm} - (\mathbf{1}/3)I\_1 \delta\_{nm}] + 2D\_1 (f - \mathbf{1}) \delta\_{nm} \tag{25}
$$

$$
\sigma\_{nn} = 2C\_1 J^{-5/3} (\lambda\_n - (\mathbf{1}/3)I\_1) + 2D\_1 (f - \mathbf{1}).
$$

It is considered that this model describes well the deformation of rubber under the principal extensions up to 20% from the initial state. Since these extensions are linked with the principal values of the strain ten-sor by relation *<sup>λ</sup><sup>k</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2*ε<sup>k</sup>* <sup>p</sup> , then it is assumed *λ<sup>k</sup>* � 1≈*εkk* approximately with exactness to ≤ 1% in the cases of universal deformations for Neo-Hookean model, what is true in the case of linear theory too. Because the extensions in the linear theory are two orders less, then this observation testifies the fact that the Neo-Hookean model extends essentially the area of allowable values of strains as compared with the Hookean model.

#### *4.5.2 Three-constant Mooney-Rivlin model (model 2)*

The elastic potential of the Mooney -Rivlin model is defined as follows [31–35, 42, 43]

$$\mathcal{W} = \mathbf{C}\_{10} \left(\overline{I}\_1 - 3\right) + \mathbf{C}\_{01} \left(\overline{I}\_2 - 3\right) + D\_1 (f - 1)^2, \ \overline{I}\_2 = J^{-4/3} I\_2,\tag{26}$$

$$\mathcal{W}(\lambda\_1, \lambda\_2, \lambda\_3) = \mathbf{C}\_{10} \left[ \left(\lambda\_1 \lambda\_2 \lambda\_3\right)^{-2/3} \left(\lambda\_1^2 + \lambda\_2^2 + \lambda\_3^2\right) - 3 \right]$$

$$+ \mathbf{C}\_{01} \left[ \left(\lambda\_1 \lambda\_2 \lambda\_3\right)^{-4/3} \left(\lambda\_1^2 \lambda\_2^2 + \lambda\_1^2 \lambda\_2^2 + \lambda\_2^2 \lambda\_3^2\right) - 3 \right] + D\_1 (\lambda\_1 \lambda\_2 \lambda\_3 - 1)^2,$$

where the elastic constants are linked with the classical constants by relations 2ð Þ¼ *C*<sup>10</sup> þ *C*<sup>01</sup> *μ*; 2*D*<sup>1</sup> ¼ *k*.

The stresses are determined by formulas

$$\begin{split} \sigma &= 2J^{-5/3} \left( \mathbf{C}\_{10} + \mathbf{C}\_{01} J^{-2/3} I\_1 \right) \mathbf{B} - 2J^{-7/3} \mathbf{C}\_{01} \mathbf{B} \mathbf{B} \\ &+ \left[ 2D\_1 (J - 1) - (2/3) J^{-5/3} \left( \mathbf{C}\_{10} I\_1 + 2 \mathbf{C}\_{01} J^{-2/3} I\_2 \right) \right] \mathbf{1}, \end{split} \tag{27}$$

$$\begin{split} \sigma\_{kk} &= \lambda\_k \frac{\partial W}{\partial \lambda\_k} \\ &= 2 \mathcal{C}\_{10} (\lambda\_1 \lambda\_2 \lambda\_3)^{-5/3} \left[ \lambda\_k^2 - (1/3) \left( \lambda\_1^2 + \lambda\_2^2 + \lambda\_3^2 \right) \right] \\ &+ 2 \mathcal{C}\_{01} (\lambda\_1 \lambda\_2 \lambda\_3)^{-7/3} \left[ \lambda\_k^2 \left( \lambda\_n^2 + \lambda\_m^2 \right) - (2/3) \lambda\_k \left( \lambda\_1^2 \lambda\_2^2 + \lambda\_1^2 \lambda\_2^2 + \lambda\_2^2 \lambda\_3^2 \right) \right] + D\_1(\lambda\_1 \lambda\_2 \lambda\_3 - 1). \end{split} \tag{28}$$

Here the indexes *knm* form the cyclic permutation from numbers 123.

The Mooney-Rivlin model is the classical one. This can be seen from the next historical information.

*Information.* An effect of nonlinear dependence of decreasing the shear stresses when the torsion angle (de-formation) to the level of nonsmall values is called "the Poynting effect" owing to his publication of 1909, where this effect was described. At that, Poynting does not mention the results of Coloumb (1784), Wert-heim (1857), Kelvin (1865), Bauschinger (1881), Tomlinson (1883), where this effect was also described in one way or another. But only within the framework of finite elastic deformations, which was developed in 20 century, this effect was satisfactorily

explained by Rivlin in 1951. He used the model of nonlinear defor-mation which now is termed "the Mooney-Rivlin model".

#### *4.5.3 Five-constant Murnaghan model (model 3)*

The elastic potential in the Murnaghan model has the form [31, 34, 35, 42–45]

$$W(\varepsilon\_{ik}) = (\mathbf{1}/2)\boldsymbol{\lambda}(\varepsilon\_{mm})^2 + \mu(\varepsilon\_{ik})^2 + (\mathbf{1}/3)A\varepsilon\_{ik}\varepsilon\_{im}e\_{km} + B(\varepsilon\_{ik})^2e\_{mm} + (\mathbf{1}/3)C(\varepsilon\_{mm})^3,\tag{29}$$

$$W(I\_1, I\_2, I\_3) = (\mathbf{1}/2)\boldsymbol{I}\_1^2 + \mu I\_2 + (\mathbf{1}/3)A\mathbf{1}\_3 + B\mathbf{1}\_1\mathbf{1}\_2 + (\mathbf{1}/3)\mathbf{1}\_1^3.$$

The Cauchy-Green strain tensor *εik* and five elastic constants (two Lame elastic constants *λ*, *μ* and three Murnaghan elastic constants *A*, *B*,*C*) are used in this potential.

The Murnaghan model can be considered as the classical one in the nonlinear theory of hyperelastic ma-terials. It takes into account all the quadratic and cubic summands from the expansion of the internal energy and describes the deformation of a big class of engineering and other materials. If to unite the data on the constants of Murnaghan model, shown in books [21, 42, 44], then the sufficiently full information can be ob-tained on many tens of materials.

#### **5. Description of deformations of the auxetic materials by the models 1–3**

#### **5.1 Universal deformation of simple shear**

This kind of deformation of the auxetics needs some preliminary discussion. First, mechanics distinguishes the simple and pure shears. The state of such deformations is standard in the test for the determination of the shear modulus. Second, it is a common position in mechanics that this modulus is always positive. This means that new effects relative to auxetic materials will most likely not be found. Third, owing to the written above comments, the one only positive result can be reached: the degree of the description of the classical nonlinear effects the Poynting and Kelvin effects – can be considered for the chosen three nonlinear models.

The following materials are used in the numerical evaluations below (elastic constants are shown): 1. Rubber - *μ* ¼ 20 MPa, *k* ¼ 2*:*0 GPa. 2. Foam - *λ* ¼ <sup>0</sup>*:*<sup>58</sup> � <sup>10</sup>9, *<sup>μ</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>39</sup> � <sup>10</sup>9, *<sup>k</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>84</sup> � <sup>10</sup>9. 3. Foam - *<sup>λ</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>58</sup> � <sup>10</sup>9, *<sup>μ</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>39</sup> � <sup>10</sup>9, *<sup>A</sup>* ¼ �1*:*<sup>0</sup> � <sup>10</sup>10, *<sup>B</sup>* ¼ �0*:*<sup>9</sup> � <sup>10</sup>10, *<sup>C</sup>* ¼ �1*:*<sup>1</sup> � <sup>10</sup>10. 4. Polystyrene - *<sup>λ</sup>* <sup>¼</sup> <sup>3</sup>*:*<sup>7</sup> � <sup>10</sup>9, *<sup>μ</sup>* <sup>¼</sup> <sup>1</sup>*:*<sup>14</sup> � <sup>10</sup>9, *<sup>A</sup>* ¼ �1*:*<sup>1</sup> � <sup>10</sup>10, *<sup>B</sup>* ¼ �0*:*<sup>79</sup> � 1010, *<sup>C</sup>* ¼ �0*:*<sup>98</sup> � <sup>10</sup>10.

#### *5.1.1 Simple shear in model 1*

In this case *<sup>J</sup>* <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>τ</sup>* <sup>2</sup> , *<sup>I</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>*τ*2. Then expressions for displacement gradients **F** and components of tensor **В** are simplified

$$\mathbf{F} = \begin{bmatrix} \mathbf{1} & \boldsymbol{\tau} & \boldsymbol{\tau} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{1} \end{bmatrix}, \mathbf{B} = \begin{bmatrix} \mathbf{1} + 2\boldsymbol{\tau}^2 & \boldsymbol{\tau} & \boldsymbol{\tau} \\ \boldsymbol{\tau} & \mathbf{1} & \mathbf{0} \\ \boldsymbol{\tau} & \mathbf{0} & \mathbf{1} \end{bmatrix}.$$

As a result, the components of stress tensor have the form

$$\begin{aligned} \sigma\_{12} &= \sigma\_{21} = \sigma\_{13} = \sigma\_{31} = 2\mathbf{C}\_{1}(\mathbf{1} + \boldsymbol{\tau})^{-10/3}\boldsymbol{\tau}, \quad \sigma\_{32} = \sigma\_{23} = \mathbf{0}, \\ \sigma\_{11} &= (8/3)\mathbf{C}\_{1}(\mathbf{1} + \boldsymbol{\tau})^{-10/3}(\boldsymbol{\tau} - \mathbf{1})\boldsymbol{\tau} + 2D\_{1}\boldsymbol{\tau}(\boldsymbol{\tau} + \mathbf{2}), \\ \sigma\_{22} &= \sigma\_{33} = -(4/3)\mathbf{C}\_{1}(\mathbf{1} + \boldsymbol{\tau})^{-10/3}(\mathbf{1} + 2\boldsymbol{\tau})\boldsymbol{\tau} + 2D\_{1}\boldsymbol{\tau}(\boldsymbol{\tau} + \mathbf{2}). \end{aligned} \tag{30}$$

The formulas (30) show that the Poynting effect (when the values of shear angle increase from the sufficiently small values to the moderate ones, then the shear stress depends nonlinearly on the shear angle) is described by the Neo-Hookean model, because Eq. (30) demonstrates just this nonlinear dependence for the moderate values of shear angle.

**Figure 7** shows the dependence of the shear stress on the shear angle *σ*<sup>12</sup> � *τ* for the silicon rubber (here and in all next plots, stress is measured by MPa).

#### *5.1.2 Simple shear in model 2*

The expressions for gradient **F** and components of tensor **В** are the same as for the Neo-Hookean model. As a result, the expressions from formula (30) are simplified *<sup>λ</sup>*<sup>1</sup> <sup>¼</sup> 1, *<sup>λ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>3</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>τ</sup>*, *<sup>J</sup>* <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>τ</sup>* <sup>2</sup> , *<sup>I</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>*τ*2, *<sup>I</sup>*<sup>2</sup> <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>τ</sup>* <sup>2</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>τ</sup>* <sup>2</sup> h i and components of the stress tensor have the form

$$
\sigma\_{12} = \sigma\_{21} = 2C\_{10}(1+\tau)^{-10/3}\tau - 2C\_{01}(1+\tau)^{-14/3}(1+4\tau)\tau,\tag{31}
$$

$$
\sigma\_{23} = \sigma\_{32} = -2\mathcal{C}\_{01}(1+\tau)^{-14/3}\tau^2,\tag{32}
$$

$$\begin{split} \sigma\_{11} &= 2\mathbf{C}\_{10}(\mathbf{1} + \boldsymbol{\tau})^{-10/3}(\mathbf{4}/\mathbf{3}) \left(\mathbf{1} + \boldsymbol{\tau} + 2\boldsymbol{\tau}^{2}\right) + 2D\_{1}\boldsymbol{\tau}(\mathbf{1} + 2\boldsymbol{\tau}) + \\ &+ 2C\_{01}(\mathbf{1} + \boldsymbol{\tau})^{-14/3}(\mathbf{4}/\mathbf{3}) \left(\mathbf{3} + 5\boldsymbol{\tau} + 5\boldsymbol{\tau}^{2} + 4\boldsymbol{\tau}^{3} - 2\boldsymbol{\tau}^{4}\right), \end{split} \tag{33}$$

*σ*<sup>22</sup> ¼ *σ*<sup>33</sup>

$$=2\mathbf{C}\_{1}(\mathbf{1}+\boldsymbol{\tau})^{-2}\left[\left(\mathbf{1}+\boldsymbol{\tau}\right)^{-4/3}+\left(\mathbf{1}+2\boldsymbol{\tau}^{2}\right)\left(\mathbf{1}-\left(\mathbf{1}+\boldsymbol{\tau}\right)^{-4/3}\right)-\mathbf{1}\right]+2D\_{1}\boldsymbol{\tau}(\mathbf{1}+2\boldsymbol{\tau}).\tag{34}$$

Thus, the Mooney-Rivlin model (that is, more complicated as compared with the Neo-Hookean model) describes the more complicated stress state, which is characterized by six components of the stress tensor. This model describes well-known nonlinear effects. The Poynting effect follows from the representation of the shear stresses by formula (31). The Kelvin effect follows from formulas (33) and (34).

**Figure 7.** *Dependence of the shear stress on the shear angle σ*<sup>12</sup> � *τ.*

**Figure 8.** *Dependence of shear stress σ*<sup>12</sup> *on the shear strain τ.*

Also, formula (32) describes one more nonlinear effect: an initiation of shear stresses *σ*<sup>23</sup> ¼ *σ*32. **Figure 8** shows the nonlinear dependence of shear stress *σ*<sup>12</sup> on the shear strain *τ*, that is built for the silicon rubber. Comparison with **Figure 7**, which corresponds to the Neo-Hookean model, shows that the Mooney-Rivlin model describes the more essential deviation from the linear Hookean description of simple shear.

#### *5.1.3 Simple shear in model 3*

The Cauchy-Green strain tensor is characterized by three components

$$\varepsilon\_{22} = (\mathbf{1}/2)(\boldsymbol{\mu}\_{2,2} + \boldsymbol{\mu}\_{2,2} + \boldsymbol{\mu}\_{k,2}\boldsymbol{\mu}\_{k,2}) = (\mathbf{1}/2)\boldsymbol{\tau}^2,\tag{35}$$

$$
\mathfrak{e}\_{12} = \mathfrak{e}\_{21} = (\mathbf{1}/2)(\mathfrak{u}\_{1,2} + \mathfrak{u}\_{2,1} + \mathfrak{u}\_{k,1}\mathfrak{u}\_{k,2}) = (\mathbf{1}/2)\mathfrak{r}.\tag{36}
$$

To calculate the stresses, it is necessary to write the potential (29) concerning the formulas (35) and (36)

$$\begin{aligned} W(\varepsilon\_{ik}) &= (\mathbf{1}/2)\boldsymbol{\lambda}(\varepsilon\_{22})^2 + \mu \left[ \left( \varepsilon\_{22} \right)^2 + \left( \varepsilon\_{12} \right)^2 + \left( \varepsilon\_{21} \right)^2 \right] \\ &+ (\mathbf{1}/3)\boldsymbol{\lambda} \left[ \varepsilon\_{22} \left( \varepsilon\_{12}\varepsilon\_{12} + \varepsilon\_{21}\varepsilon\_{21} + \varepsilon\_{12}\varepsilon\_{21} \right) + \left( \varepsilon\_{22} \right)^3 \right] \\ &+ B \left[ \left( \varepsilon\_{22} \right)^2 + \left( \varepsilon\_{12} \right)^2 + \left( \varepsilon\_{21} \right)^2 \right] \varepsilon\_{22} + (\mathbf{1}/3)\mathbf{C} \left( \varepsilon\_{22} \right)^3, \end{aligned} \tag{37}$$
 
$$W(\tau) = (\mathbf{1}/2)\mu \tau^2 + (\mathbf{1}/8)[\left( \lambda + 2\mu \right) + A + B] \tau^4 + (\mathbf{1}/24)[A + 3B + C] \tau^6 \tag{38}$$

The Lagrange stress tensor is determined by the formula *<sup>σ</sup>ik*ð Þ¼ *xn*, *<sup>t</sup> <sup>∂</sup>W=∂εik* and has two nonlinear com-ponents

$$\begin{split} \sigma\_{22} &= (\lambda + 2\mu)\varepsilon\_{22} + A \left[ (\varepsilon\_{22})^2 + (\mathbf{1}/3)(\varepsilon\_{12}\varepsilon\_{12} + \varepsilon\_{21}\varepsilon\_{21} + \varepsilon\_{12}\varepsilon\_{21}) \right] \\ &+ B \left[ 3(\varepsilon\_{22})^2 + (\varepsilon\_{12})^2 + (\varepsilon\_{21})^2 \right] + C(\varepsilon\_{22})^2 \\ &= (\mathbf{1}/4) [2(\lambda + 2\mu) + (\mathbf{A} + 2\mathbf{B})] \mathbf{r}^2 + (\mathbf{1}/4)(\mathbf{A} + 3\mathbf{B} + \mathbf{C}) \mathbf{r}^4, \end{split} \tag{39}$$
 
$$\sigma\_{12} = \sigma\_{21} = 2\mu\varepsilon\_{12} + [(\mathbf{1}/3)\mathbf{A}(\varepsilon\_{12} + \varepsilon\_{21}) + 2\mathbf{B}\varepsilon\_{12}] \varepsilon\_{22} = \mu\tau + (\mathbf{1}/6)(\mathbf{A} + 3\mathbf{B})\mathbf{r}^3. \tag{40}$$

The shear stress contains the linear and nonlinear summands and describes the simple shear. The normal stress describes the change of volume under deformation and testifies the break of the state of simple shear in the nonlinear description of

#### *Elasticity of Auxetic Materials DOI: http://dx.doi.org/10.5772/intechopen.99543*

deformation. To build the plots of dependence (40) choose two nonstandard for the Murnaghan model materials – foam and polystyrene – which can experience not only the small by values strains but also the moderate ones. **Figures 9** and **10** show the dependence of the shear stress *σ*<sup>12</sup> on the shear angle *τ* for the foam and polystyrene.

The dependences *σ*<sup>12</sup> � *τ* for models 1–3 can be commented in the following way: these models describe well the nonlinear Poynting effect. At the same time, many scientists working with auxetic materials report the experimental dependences that coincide quantitatively with the shown here theoretical (and based on them numerical) dependences (for example, [46–48]). Also, some conclusions to dependence *σ*<sup>12</sup> � *τ* for models 1–3 can be formulated: the developed in mechanics of materials nonlinear models of deformation of elastic materials can be recommended for the description of auxetic materials.

#### **5.2 Universal deformation of uniaxial tension**

This kind of deformation is fundamental for the auxetics because just in tests on the uniaxial tension-compression the phenomenon of auxeticity was first observed.

#### *5.2.1 Uniaxial tension in model 1*

The formulas for the principal extensions *<sup>λ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>λ</sup>*3, *<sup>J</sup>* <sup>¼</sup> *<sup>λ</sup>*1*λ*<sup>2</sup> 2, *<sup>I</sup>*<sup>1</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>*λ*<sup>2</sup> <sup>2</sup> are valid in this model and the normal stresses (the shear stresses are absent in this state of deformation) are given by the formulas

$$
\sigma\_{11} = (\mathbf{2}/\mathbf{3})\mu \left(\lambda\_1 \lambda\_2^2\right)^{-5/3} \left(\lambda\_1^2 - \lambda\_2^2\right) + k \left(\lambda\_1 \lambda\_2^2 - \mathbf{1}\right),
\tag{41}
$$

$$
\sigma\_{22} = \sigma\_{33} = - (\mathbf{1}/3) \mu \left(\boldsymbol{\lambda}\_1 \boldsymbol{\lambda}\_2^2\right)^{-5/3} \left(\boldsymbol{\lambda}\_1^2 - \boldsymbol{\lambda}\_2^2\right) + k \left(\boldsymbol{\lambda}\_1 \boldsymbol{\lambda}\_2^2 - \mathbf{1}\right). \tag{42}
$$

Note that the stresses are depending in model 1 on two principal extensions – longitudinal and transverse.

**Figure 9.**

*Dependence of the shear stress on the shear angle (foam).*

**Figure 10.** *Dependence of the shear stress on the shear angle (polystyrene).*

If to assume that all three normal stresses on the lateral surface of the sample are absent (the surface is free of stresses), then

$$
\sigma\_{11} = \mathfrak{B} \left( \mathbb{A}\_1 \mathbb{A}\_2^2 - \mathbf{1} \right). \tag{43}
$$

It follows from (43) that the Poynting-type effect (when the principal extensions increase from the sufficiently small values to the moderate ones, then the normal stress in the direction of tension depends nonlinearly on these extensions) is described by the Neo-Hookean model.

**Figure 11** shows the dependence of the longitudinal stress on principal extensions and is built for the rubber with allowance for that the value ð Þ¼ *μ=*3*k* 0, 00334 is very small compared to the unit (the bulk mo-dulus is essentially more of the shear one). Then the dependence is valid

$$
\epsilon\_{22} = (\mathbf{1}/2)(\mathbf{1} + 2\epsilon\_{11})^{-2} - \mathbf{1}/2. \tag{44}
$$

**Figure 12** corresponds to formulas (41) and (42). and shows a dependence of the longitudinal principal extension on the transverse principal extension. Note that the silicon rubber is characterized by the big difference between values of shear and bulk moduli that can reach hundred times. Therefore, the new material is chosen further for the numerical analysis – the foam, which values of elastic constants is characterized by about equal by the order. **Figure 12** shows also that with an increase of extension *λ*<sup>1</sup> the increase of extension *λ*<sup>2</sup> slows.

It looks, in this case, to be illogical to neglect the first summand in (41) and (42). Note here that the ratio ð Þ *λ*2*=λ*<sup>1</sup> corresponds in the linear theory to the Poisson's ratio.

#### *5.2.2 Uniaxial tension in model 2*

The uniaxial tension in direction of the abscissa axis is characterized by parameters: *<sup>λ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>λ</sup>*3, *<sup>J</sup>* <sup>¼</sup> *<sup>λ</sup>*1*λ*<sup>2</sup> 2, *<sup>I</sup>*<sup>1</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>*λ*<sup>2</sup> 2, *<sup>I</sup>*<sup>2</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>4</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>*λ*<sup>2</sup> 1*λ*2 2, *<sup>B</sup>*<sup>11</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>2</sup> 1, ð Þ *BB* <sup>11</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>4</sup> <sup>1</sup> . The normal stresses are given by the formulas

**Figure 11.** *Dependence of the longitudinal stress on the principal extensions.*

**Figure 12.** *Dependence of the longitudinal principal extension on the transverse principal extension.*

*Elasticity of Auxetic Materials DOI: http://dx.doi.org/10.5772/intechopen.99543*

$$\begin{split} \sigma\_{11} &= 2\mathbf{C}\_{10} (\mathbf{2}/\mathbf{3}) \left(\lambda\_1 \lambda\_2^2\right)^{-5/3} \left(\lambda\_1^2 - \lambda\_2^2\right) + 2\mathbf{C}\_{01} \left(\lambda\_1 \lambda\_2^2\right)^{-7/3} \left(\lambda\_1^4 + (2/3)\lambda\_1^2 \lambda\_2^2 - (5/3)\lambda\_2^4\right) \\ &+ 2\mathbf{D}\_1 \left(\lambda\_1 \lambda\_2^2 - 1\right), \end{split} \tag{45}$$

$$
\sigma\_{22} = \sigma\_{33} = (2/3)C\_{10} \left(\lambda\_1 \lambda\_2^2\right)^{-5/3} \left(\lambda\_2^2 - \lambda\_1^2\right) + 2C\_{01} \left(\lambda\_1 \lambda\_2^2\right)^{-7/3} (1/3) \lambda\_2^2 \left(\lambda\_2^2 - \lambda\_1^2\right) + 2D\_1 \left(\lambda\_1 \lambda\_2^2 - 1\right). \tag{46}
$$

Assume that all three normal stresses over the sample lateral surface are absent. Then Eq. (45) is simplified to the form

$$
\sigma\_{11} = 2\mathbf{C}\_{01} \left(\boldsymbol{\lambda}\_1 \boldsymbol{\lambda}\_2^2\right)^{-7/3} \left(\boldsymbol{\lambda}\_1^4 - \boldsymbol{\lambda}\_2^4\right) + \mathbf{6} \mathbf{D}\_1 \left(\boldsymbol{\lambda}\_1 \boldsymbol{\lambda}\_2^2 - \mathbf{1}\right). \tag{47}
$$

The last formula testifies: the Mooney-Rivlin model describes the Poynting-type effect.

Two elastic constants are presented in (47) in contrast to the Neo-Hookean model, where the shear modulus was absent. It should be noted that in both models – Neo -Hookean and Moo-ney-Rivlin –the tension in the longitudinal direction stress *σ*<sup>11</sup> depends already on two principal extensions. **Figure 13** shows a dependence of the longitudinal stress on principal extensions is built for the silicon rubber. It coincides practically with **Figure 11** (Neo-Hookean model) and shows that the constant *C*<sup>01</sup> of the Mooney-Rivlin model effects not essentially on the stress *σ*<sup>11</sup> and the dependence (45) rests weakly nonlinear within the accepted restrictions.

The Eq. (46) can be transformed into the form

$$
\lambda\_1^6 - \lambda\_1^3/\sigma^2 + \left[ (2\text{C}\_{10}/\text{}\not{\text{G}}D\_1)\stackrel{\text{-}\gtr{\text{Q}}}{\sqrt[3]{\sigma^4}} + (2\text{C}\_{01}/\text{}\not{\text{G}}D\_1)\stackrel{\text{-}\gtr{\text{Q}}}{\sqrt[3]{\sigma^2}} \right] \sigma^{-4} \left(\sigma^2 - 1\right) = 0,
$$

$$
\sigma = (\lambda\_2/\lambda\_1), \quad \lambda\_1^3 = 1/2\sigma^2 \pm 1/2\sigma^2 \sqrt{1 - \begin{bmatrix} (2\text{C}\_{10}/\text{}\not{\text{G}}D\_1)\stackrel{\text{-}\gtr{\text{Q}}}{\sqrt[3]{\sigma^4}} \\ + (2\text{C}\_{01}/\text{}\not{\text{G}}D\_1)\stackrel{\text{-}\gtr{\text{Q}}}{\sqrt[3]{\sigma^2}} \end{bmatrix}},\tag{48}
$$

The corresponding to the model 1 plot from **Figure 11** is practically identical with the plot from **Figure 13** corresponding to model 2.

#### *5.2.3 Uniaxial tension in model 3*

The uniaxial tension in this model is characterized by three nonzero components of the strain tensor *εkk* and one non-zero component of the stress tensor *σ*11. Then the constitutive equations are somewhat simp-lified.

$$\begin{aligned} \sigma\_{11} &= \lambda I\_1 + 2\mu \varepsilon\_{11} + A \left( \varepsilon\_{11} \right)^2 + B \left( \mathbf{E} + 2\varepsilon\_{11} I\_1 \right) + C \left( \mathbf{E} + 2\varepsilon\_{22} \varepsilon\_{11} + 2\varepsilon\_{33} \varepsilon\_{11} \right) \\\ I\_1 &= \varepsilon\_{11} + \varepsilon\_{22} + \varepsilon\_{33}, \ \mathbf{E} = \left( \varepsilon\_{11} \right)^2 + \left( \varepsilon\_{22} \right)^2 + \left( \varepsilon\_{33} \right)^2. \end{aligned} \tag{49}$$

**Figure 13.** *Dependence of the longitudinal stress on principal extensions.*

$$\mathbf{0} = \lambda I\_1 + 2\mu \mathbf{e}\_{22} + A(\mathbf{e}\_{22})^2 + B(\mathbf{E} + 2\mathbf{e}\_{22}I\_1) + \mathbf{C}(\mathbf{E} + 2\mathbf{e}\_{22}\mathbf{e}\_{33} + 2\mathbf{e}\_{22}\mathbf{e}\_{11}),\tag{50}$$

$$\mathbf{0} = \lambda \mathbf{I}\_1 + 2\mu \mathbf{e}\_{33} + A(\mathbf{e}\_{33})^2 + B(\mathbf{E} + 2\mathbf{e}\_{33}\mathbf{I}\_1) + \mathbf{C}(\mathbf{E} + 2\mathbf{e}\_{22}\mathbf{e}\_{11} + 2\mathbf{e}\_{22}\mathbf{e}\_{33}).\tag{51}$$

Let us remind that in the linear theory of elasticity, corresponding to the Hookean model, the constitutive equations are significantly simpler

$$
\sigma\_{11} = \lambda I\_1 + 2\mu \varepsilon\_{11}, \mathbf{0} = \lambda I\_1 + 2\mu \varepsilon\_{22}, \mathbf{0} = \lambda I\_1 + 2\mu \varepsilon\_{33}.\tag{52}
$$

Apply further to the nonlinear Eqs. (49)–(51) the procedure of analysis of the state of uniaxial tension that is used in the linear theory of elasticity as applied to Eqs. (52). Subtraction of Eq. (51) from Eq. (50) gives the formula

$$\mathbf{0} = \mathbf{2}\mu(\varepsilon\_{22} - \varepsilon\_{33}) + A\left((\varepsilon\_{22})^2 - (\varepsilon\_{33})^2\right) + \mathbf{2}\mathcal{B}(\varepsilon\_{22} - \varepsilon\_{33})(\varepsilon\_{11} + \varepsilon\_{22} + \varepsilon\_{33}),$$

from which the equality of components of transverse strains *ε*<sup>22</sup> ¼ *ε*<sup>33</sup> follows. The addition of formulas (36)–(38) results in the following formula

$$\begin{aligned} \left[\sigma\text{n}/(3\lambda+2\mu)-[(\mathbf{A}+3\mathbf{B}+\mathbf{C})/3\lambda+2\mu]\right](\epsilon\mathbf{e}\mathbf{n})^2+2(\epsilon\mathbf{e}\mathbf{n})^2 &= \left[2\mathbf{B}/(3\lambda+2\mu)\right](\epsilon\mathbf{e}\mathbf{n}+2\epsilon\mathbf{e}\mathbf{n})^2 \\ -\left[4\mathbf{C}/(3\lambda+2\mu)\right]\left[\left(\epsilon\mathbf{e}\mathbf{n}\right)^2+2\epsilon\mathbf{e}\_{22}\epsilon\_{11}\right] &= \epsilon\mathbf{e}\_{11}+2\epsilon\mathbf{e}\_{22} \\ \mathbf{n} &\tag{53} \end{aligned} \tag{53}$$

Substitution of formula (53) into the relation (49) gives new relation

$$\begin{split} \sigma\_{11} &= Ee\_{11} + \left( A + \frac{2\lambda + 3\mu}{\lambda + \mu} B + C \right) (e\_{11})^2 - \frac{\lambda}{\lambda + \mu} \left( A + \frac{4\lambda - 2\mu}{\lambda} B - \frac{2\mu}{\lambda} C \right) (e\_{22})^2 \\ &+ \frac{2(\lambda + 2\mu)}{\lambda + \mu} (B + C) e\_{11} e\_{22} .\end{split} \tag{54}$$

The relation (54) shows that model 3, like models 1 and 2, describes the Poynting-type effect.

**Figures 14** and **15** show the dependence *σ*<sup>11</sup> ¼ *σ*11ð Þ *ε*11, *ε*<sup>22</sup> among the longitudinal stress *σ*<sup>11</sup> and strains *ε*11, *ε*<sup>22</sup> for the foam and polystyrene and the moderate values of strains. Both plots demonstrate an essential nonlinearity under moderate strains. This new nonlinear effect will be true for auxetic materials too.

Write now the constitutive Eq. (49) with allowance for equality *ε*<sup>22</sup> ¼ *ε*<sup>33</sup> and transform it to the form of a quadratic equation relative to the ratio *ε*22*=ε*<sup>11</sup>

**Figure 14.** *Dependence σ*<sup>11</sup> ¼ *σ*11ð Þ *ε*11, *ε*<sup>22</sup> *(foam).*

*Elasticity of Auxetic Materials DOI: http://dx.doi.org/10.5772/intechopen.99543*

**Figure 15.** *Dependenceσ*<sup>11</sup> ¼ *σ*11ð Þ *ε*11, *ε*<sup>22</sup> *(polystyrene).*

$$2\left(\frac{\varepsilon\_{22}}{\varepsilon\_{11}}\right)^2 + 2\frac{[((\lambda+\mu)/\varepsilon\_{11}) + (B+C)]}{(A+6B+4C)}\frac{\varepsilon\_{22}}{\varepsilon\_{11}} + \frac{(\lambda/\varepsilon\_{11}) + (B+C)}{(A+6B+4C)} = \mathbf{0}.$$

The solution of this equation has the form

$$\left(\epsilon\mathbf{e}\_{2}/\epsilon\mathbf{n}\right) = -\left\{ \left[ (\lambda+\mu)/\epsilon\mathbf{n} + (\mathbf{B}+\mathbf{C}) \right]/(\mathbf{A}+\mathbf{6B}+\mathbf{4C}) \right\} \left[ \mathbf{1} \pm \sqrt{\mathbf{1} - \frac{(\mathbf{A}+\mathbf{6B}+\mathbf{4C})[\lambda/\epsilon\_{11} + (\mathbf{B}+\mathbf{C})]}{\left[ (\lambda+\mu)/\epsilon\_{11} + (\mathbf{B}+\mathbf{C}) \right]^{2}}} \right]. \tag{55}$$

Thus, Eq. (55) shows that the ratio ð Þ �*ε*22*=ε*<sup>11</sup> is not constant in the Murnaghan nonlinear model. This can be treated as the new mechanical nonlinear effect which is looking very promising for the auxetic materials.

**Figures 16** and **17** show a dependence of the ratio ð Þ �*ε*22*=ε*<sup>11</sup> on the strain *ε*<sup>11</sup> and are built for the foam and polystyrene for the moderate strains. The plot's main features are as follows: the ratio ð Þ �*ε*22*=ε*<sup>11</sup> is de-creased essentially from the initial value, which corresponds to the Poisson ratio for small strain in the con-ventional materials, to the negative values under the moderate values of longitudinal strain that is observed in the auxetic materials. So, the ratio, that is, treated as the Poisson's ratio for small strain, in the case of mo-derate strain becomes the characteristics of transition of the material from the category of conventional ma-terials into the

**Figure 16.** *Dependence of the ratio* ð Þ �*ε*22*=ε*<sup>11</sup> *on the strain ε*<sup>11</sup> *(foam).*

**Figure 17.** *Dependence of the ratio* ð Þ �*ε*22*=ε*<sup>11</sup> *on the strain ε*11*(polystyrene).*

**Figure 18.** *Experimental dependence of the ratio* ð Þ �*ε*22*=ε*<sup>11</sup> *on the strain ε*11*.*

category of nonconventional materials. This can be considered as the newly revealed theore-tically nonlinear effect.

Thus, an analysis of universal deformation of uniaxial tension for model 3 revealed the new property: the material with conventional properties under small strains is transformed under moderate strains into the nonconventional (auxetic) material. The uncommonness of this observation consists in that usually the material is considered either the conventional or the nonconventional during all the processes of deformation.

Let us compare the plots from **Figures 16** and **17** with the experimental data from ([49], Figure 4) shown here as **Figure 18** (dependence of the ratio ð Þ �*ε*22*=ε*<sup>11</sup> on the strain *ε*11), where the deformation of the foams was studied for the finite strains with increasing the longitudinal strain *ε*<sup>11</sup> from 0.1 to 1.4. Note that the theoretical plots are constructed for the range from *ε*<sup>11</sup> ¼ 0 to the moderate values 0.23 (foam) and 0.33 (polystyrene). This comparison shows that ð Þ *ε*22*=ε*<sup>11</sup> increases within the range *ε*<sup>11</sup> ∈ð Þ 0, 0; 0, 3 *:* Thus, model 3 describes some experimental observations of the foam.

**Figures 19** and **20** show the dependence of longitudinal and transverse strains. Three stages can be marked out: 1. A decrease of transverse strain becomes slower under transition to the moderate strains. 2. The strain *ε*<sup>22</sup> reaches the local minimum

**Figure 19.**

*Dependence of longitudinal and transverse strains (foam).*

**Figure 20.** *Dependence of longitudinal and transverse strains (polystyrene).*

#### *Elasticity of Auxetic Materials DOI: http://dx.doi.org/10.5772/intechopen.99543*

and further increases. 3. When the strain *ε*<sup>11</sup> continues to increase, the strain *ε*<sup>22</sup> possesses zero value and further increases possessing already positive values.

The shown feature confirms once again the new mechanical effect – a transition of the material under its deformation to the level of moderate values of the longitudinal stretching from the class of conventional ma-terials into the class of the auxetic materials. In other words, the standard sample in conditions of universal deformation of uniaxial tension is deformed for small strains as if it is made of the conventional material (its cross-section is decreased) and with increasing the values of longitudinal stretching to the moderate values the sample cross-section starts to increase, what is the characteristic just for auxetic materials.

The plots from **Figures 19** and **20** can be compared with the plot, obtained experimentally in [48]. This article reports that the new metamaterials were created from soft silicon rubber. The samples we-re deformed in conditions of uniaxial compression up to moderate values of longitudinal strain 0,35. The shown in the **Figure 21** plot corresponds to **Figure 2a** in [48] and shows a dependence of longitudinal and transverse strains. Comparison of plots from **Figure 11** (uniaxial stretching) and **Figure 12** (uniaxial compression) demonstrates the common property of forming the hump in the area of negative values of transverse strain, which is transformed with the increasing values of longitudinal strain roughly into the straight line in the area of positive values of transverse strain.

Thus, the nonlinear Murnaghan model describes within conditions of uniaxial tension some nonlinear phenomena of deformation, which can be linked with the properties of deformation of auxetic materials. Note that the shown feature is clearly visible only within the framework of the Murnaghan model, but the Neo-Hookean and Mooney -Rivlin models also describe the hump formation, as can be seen in **Figure 6**.

#### **5.3 Universal deformation of omniaxial tension**

#### *5.3.1 Omniaxial tension in model 1*

In this case *<sup>λ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>λ</sup>*3, *<sup>J</sup>* <sup>¼</sup> *<sup>λ</sup>*<sup>3</sup> 1, *<sup>I</sup>*<sup>1</sup> <sup>¼</sup> <sup>3</sup>*λ*<sup>2</sup> <sup>1</sup> and the normal stress is equal

$$
\sigma\_{11} = 2D\_1(\lambda\_1^3 - 1). \tag{56}
$$

The formula (56) describes the Poynting-type effect relative to the bulk modulus (the dependence *σ*<sup>11</sup> on the extension *λ*<sup>1</sup> is evidently nonlinear).

**Figure 22** shows a dependence of the stress on the principal extension and is built for the silicon rubber. The plot testifies that model 1 describes the nonlinear

**Figure 21.** *Experimental dependence of longitudinal and transverse strains.*

**Figure 22.** *Dependence of stress on principal extension.*

change of the sample volume while being subjected to the universal deformation of uniform compression-tension.

#### *5.3.2 Omniaxial tension in model 2*

In this case *<sup>λ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>λ</sup>*3, *<sup>J</sup>* <sup>¼</sup> *<sup>λ</sup>*<sup>3</sup> 1, *<sup>I</sup>*<sup>1</sup> <sup>¼</sup> <sup>3</sup>*λ*<sup>2</sup> 1, *<sup>I</sup>*<sup>2</sup> <sup>¼</sup> <sup>3</sup>*λ*<sup>4</sup> <sup>1</sup> (note that they are identical for any nonlinear model). The formula for normal stress coincides with the analogous formula for model 1 (56) and verifies the non-linear dependence of the tension stress on the principal extension.

#### *5.3.3 Omniaxial tension in model 3*

The components of displacement gradients and Cauchy-Green strain tensor are as follows

$$\begin{aligned} \mu\_{1,1} = \mu\_{2,2} = \mu\_{3,3} = \varepsilon > 0, \mu\_{1,1} + \mu\_{2,2} + \mu\_{3,3} = \mathfrak{Z}\varepsilon = \varepsilon, \mu\_{k,m} = (\partial u\_k / \partial \mathfrak{x}\_m) \\ \quad = \mathbf{0} \quad (k \neq m); \varepsilon\_{11} = \varepsilon\_{22} = \varepsilon\_{33} = \varepsilon + (1/2)\varepsilon^2, \qquad \varepsilon\_{ik} = \mathbf{0} \quad (i \neq k) \end{aligned} \tag{57}$$

The corresponding algebraic invariants of the Cauchy-Green tensor are written in the form

$$\begin{split} I\_1 &= \mathbf{e}\_{11} + \mathbf{e}\_{22} + \mathbf{e}\_{33} = \mathbf{e}, I\_2 = \left(\mathbf{e}\_{11}\right)^2 + \left(\mathbf{e}\_{22}\right)^2 + \left(\mathbf{e}\_{33}\right)^2 = (\mathbf{1}/3)\mathbf{e}^2, I\_3 \\ &= \left(\mathbf{e}\_{11}\right)^3 + \left(\mathbf{e}\_{22}\right)^3 + \left(\mathbf{e}\_{33}\right)^3 = (\mathbf{1}/9)\mathbf{e}^3. \end{split} \tag{58}$$

The formulas for invariants (58) allow writing the potential in the simpler form

$$\begin{aligned} \mathcal{W}(\varepsilon) &= (3/2)(3\lambda + 2\mu)\varepsilon^2 + ((9/2)\lambda + 3\mu + A + 9B + 9C)\varepsilon^3 + \\ &+ (3/2)(4(3\lambda + 2\mu) + (A + 9B + 9C))\varepsilon^4 + (3/4)(A + 9B + 9C)\varepsilon^5 + (1/8)(A + 9B + 9C)\varepsilon^6. \end{aligned} \tag{59}$$

The stresses are evaluated by the formulas (the normal stresses only are nonzero)

$$\sigma\_{11} = \sigma\_{22} = \sigma\_{33} = (3\lambda + 2\mu)\varepsilon + [(3/2)(3\lambda + 2\mu) + (A + 9B + 7C)] \ \ e^2$$

$$\sigma + (A + 9B + 7C)\left(\varepsilon^3 + (1/4)\varepsilon^4\right), \quad \sigma\_{12} = \sigma\_{23} = \sigma\_{31} = \mathbf{0}.$$

Thus, the stresses contain linear and nonlinear summands.

The interdependence between the first invariant of the stress tensor *σkk* and the parameter of the omni-axial tension *e* has the form

*Elasticity of Auxetic Materials DOI: http://dx.doi.org/10.5772/intechopen.99543*

$$\begin{split} \sigma\_{kk} &= (3\lambda + 2\mu)e + [(1/2)(3\lambda + 2\mu) + (1/3)(A + 9B + 7C)]e^2 \\ &+ (A + 9B + 7C)[(1/9)e^3 + (1/108)e^4]. \end{split} \tag{60}$$

The plots in **Figures 23** and **24** show a dependence *σkk*ð Þ*e* for the foam and polystyrene evaluated formula (60). It follows from them that they are similar to the parabola with a vertex in a positive half of the plane *σkkOe*. The parabola's right branch then passes into the negative half of the plane. Both plots have "the hump" in the positive branch of the plane.

A presence of "the hump" testifies that the nonlinear Murnaghan model describes the transition of the material of the sample-cube from the class of conventional materials into the class of auxetic materials. The fact is that the sample is compressed for the small values of uniform tension and in the following increase of the tension the strain the sample swells. But this phenomenon is characteristic of only auxetic materials.

Thus, three nonlinear models which are used in the analysis describe the nonlinear Poynting-type effects in conditions of three used above universal deformations and the moderate strains. This agrees quantitatively with experimental observations of nonlinear dependences *σ* � *ε* (stress versus strain) in auxetic materials for the moderate strains.

The main new effects are revealed: the nonlinear Murnaghan model describes in the case of uniaxial and omniaxial tension the transition of the material from the class of conventional materials into the class of the auxetic materials. This occurs when the material is deformed to the level of moderate values of the longitudinal stretching. In other words, the shown experiments and proposed theoretical analysis testify that the stan-dard sample in conditions of the mentioned universal deformation of uniaxial tension is deformed for small strains as if it is made of the conventional material (its cross-section is decreased) and with increasing the values of longitudinal stretching to the moderate values the sample cross-section starts to increase, what is the characteristic just for auxetic materials.

**Figure 23.** *Dependence σkk*ð Þ*e (foam).*

**Figure 24.** *Dependence σkk*ð Þ*e (polystyrene).*

### **6. Final conclusions**

The elasticity is the property of auxetic materials, which is especially characteristic and most studied for these materials. Historically, the auxetics were treated from the point of view of the linear theory of elasticity what was not quite adequate in some cases.

As the part of classical mechanics of elastic materials, the mechanics of auxetic materials needs at present more and more experimental studies (the level of such studies as compared with the classical ones can be seen from the famous Bell's book [27]).

The nonlinear theory of elasticity is seemingly quite prospective for a description of elastic deformation of the auxetic materials but it is essentially more complicated in the mathematical apparatus and concrete investigations.

#### **Author details**

Jeremiah Rushchitsky S.P. Timoshenko Institute of Mechanics, Kyiv, Ukraine

\*Address all correspondence to: rushch@inmech.kiev.ua

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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#### **Chapter 7**

## Perspective Chapter: Improvement of Elastomer Elongation and Output for Dielectric Elastomers

*Seiki Chiba, Mikio Waki, Shijie Zhu, Tonghuan Qu and Kazuhiro Ohyama*

#### **Abstract**

The need for light, high-strength, and artificial muscles is growing rapidly. A well-known type of artificial muscle meeting these requirements is the dielectric elastic (DE) type, which uses electrostatic force between electrodes. In hopes of utilizing, it practically for a variety of purposes, research and development is rapidly progressing all over the world as a technology for practical use. Much of the market demand is dominated by more output-focused applications such as DE power suits, DE motors, DE muscles for robots, and larger DE power systems. To meet these demands, the elasticity of the elastomer is very important. In this paper, we discussed what the important factors are for SS curves, viscoelasticity tests, etc. of the dielectric elastomer materials. Recent attempts have been also made to use new carbon foam materials such as SWCNTs and MWCNTs as electrodes for DEs. These electrodes bring the elastomers to a higher level of performance.

**Keywords:** Dielectric elastomer, Actuator, Sensor, Generator, Large deformation, High efficiency, Artificial Muscle

#### **1. Introduction**

The creation of artificial muscle has long been a scientific aspiration. It is well known that Wilhelm Conrad Röntgen, who discovered X-rays, conducted experiments using rubber strings as artificial muscles [1]. In the 1950s, artificial muscles using EPA (Electro Active Polymer) became mainstream. Since then, the need for light, high-strength, and artificial muscles has been growing rapidly.

EAP type artificial muscles which drive a polymer membrane by applying electrical stimulation, are actuators that realize movements similar to living muscles by electrical control. Because they move softly, they are also called soft actuators. **Figure 1** shows the following types are of EAPs: **1)** DEs (dielectric elastomers) which are driven by the generated coulomb force, made by sandwiching an elastomer between flexible electrodes. [2, 3], **2)** IPMCs (Ionic polymer-metal composites), which owe their power to the movement of ions and water molecules in the polymer film (combination of an electrolyte film and a thin metal electrode) [4], **3)** CPs (conductive polymer) which use a drive force moving ions by applying a voltage between the conductive polymers [5], **4)** ionic polymer gel Ion polymer gels, which utilize the movement of ions due to chemical changes (e.g., Ph changes)

**Figure 1.**

*Typical electroactive polymers (EAP).*

within the gel [6], and **5)** CNT (carbon nanotube) actuators, which are ideally for nanomachines and do not require ion intercalation [7]. In addition, the piezopolymer utilizes a piezoelectric phenomenon [8], and some are driven by heat, air, light, etc. [9–13].

The most promising candidate from technologies above is the DE [2, 14]. In 1990, Chiba and Pelrine began research and development of dielectric elastomers for the first time in the world. [2], but now, research and development for their practical uses are rapidly progressing all over the world as a technology for practical use [2, 3, 14–34].

Most of the current market demand is for DE power suits, DE motors, DE muscles for robots, and systems that drive them in reverse to generate electricity efficiently. To meet these demands, the elasticity of the elastomer is extremely important. In this paper, we discussed important factors (including cross-linking agents and double bond breaks) through SS (strain stress) curves, viscoelasticity tests, etc. of DE materials. In addition, recent attempts have been made to use new carbon foam materials such as SWCNTs and MWCNTs as electrodes for DEs. These electrodes bring the above-improved elastomers to a higher performance. They will also be discussed in this paper.

### **2. Background of DEs**

The structure of a DE is very simple and consists of a polymer film (elastomer), which is the main material, and two electrodes that sandwich it [2, 3]. When a potential difference is applied between the electrodes, the Coulomb force causes the polymer film to contract in the thickness direction and expand in the plane direction (see **Figure 2**).

#### **Figure 2.**

*DE artificial muscle actuator structure and operating principle: (a) The black sheet is the flexible and stretchable electrode, and (b) The yellow part is the elastomer.*

*Perspective Chapter: Improvement of Elastomer Elongation and Output for Dielectric Elastomers DOI: http://dx.doi.org/10.5772/intechopen.99713*

At the material level, a DE actuator has a fast response speed (over 100 kHz), with a high strain rate (up to 680%) [15], high pressure (up to 8 MPa), and power density of 1 W/g [16]. A DE actuator having only 0.15 g of DE can lift the weight of 8 kg easily by 1 mm or more with the actuation speed of 88 msec, using Single-wall carbon nanotubes as electrodes [17]. Since the elongation and the output are in inverse proportion to each other, it is possible to suppress the output and increase the elongation. In addition, as mentioned above, power generation is possible by reversing the movement of the DE actuator. Its efficiency is excellent, at over 70% [18].

A mathematical model of the DE actuator can be described as follows: The strain (deformation) observed in the elastomer membrane is mainly caused by the interaction of electrostatic charges between the electrodes [19]. Opposite charges on the two electrodes attract each other, and the same charges repel each other. This phenomenon can be derived by using a simple electrostatic model to derive the effective pressure generated by the electrodes of the elastomer membrane as a function of the applied voltage [19]. Pressure ρ is

$$
\rho = \varepsilon\_r \varepsilon\_o E^2 = \varepsilon\_r \varepsilon\_o \left(\mathbf{V} / \mathbf{t}\right)^2 \tag{1}
$$

Here, εr and εo are the permittivity and the relative permittivity (dielectric constant) of the polymer in the free space, respectively, E is the electric field strength, V is the applied voltage, and t is the film thickness. The responsiveness of this polymer is similar to that of conventional electrostrictive polymers, and the pressure is proportional to the square of the electric field strength. For small strains with free boundary conditions, the actuator energy density, ea, of the material can be written as

$$\mathbf{e\_z} = \mathbf{P} \mathbf{s\_z} = \mathbf{Y} \mathbf{s\_z}^2 = \left(\boldsymbol{\varepsilon\_r} \boldsymbol{\varepsilon\_o}\right)^2 \left(\mathbf{V} / \mathbf{t}\right)^4 / \mathbf{Y} \tag{2}$$

where Y is the modulus of elasticity and sz is the polymer thickness strain [14]. Conventionally, the elastic energy density ea = 1/2 Ysz 2 is often used (see **Table 1**).


#### **Table 1.**

*The result of performance measurements of eight polymers (elastomers).*

**Figure 3.**

*Operating principle of DE power generation: (a) Thick lines are compliant electrodes, and (b) the yellow line between the thick lines is the dielectric elastomer.*

As described above, when the movement of the dielectric elastomer actuator is reversed, the power generation mode is set. This field of power generation research has become more active around the world since [14, 15, 18, 19, 32–56] it was first tested with a DE generator on a buoy [34]. The principle is simple and utilizes the increase in electrostatic energy generated by changing the shape of the dielectric elastomer actuator with an external force (see **Figure 3**). That is, when some mechanical energy is applied to the dielectric elastomer to extend it, the thickness direction becomes thinner and the area expands (Increase in capacitance). At this time, electrostatic energy is generated on the polymer and stored as an electric charge. When the mechanical energy decreases, the elasticity of the dielectric itself increases the thickness in the thickness direction and reduces the area (Reduction of capacitance). At this time, the electric charge is pushed out toward the electrode. Such changes in charge increase the voltage difference, resulting in increased electrostatic energy [19]. The capacitance of the DE film "C" is given as follows:

$$\mathbf{C} = \varepsilon\_0 \varepsilon \mathbf{A} / \mathbf{t} = \varepsilon\_0 \varepsilon \mathbf{b} / \mathbf{t}^2 \tag{3}$$

where ε0 is the dielectric permittivity of free space, ε is the dielectric constant of the polymer film, A is the active polymer area, and t and b are the thickness and the volume of the polymer. The second equality in Eq. (3) can be written because the volume of the elastomer is essentially constant, i.e., At = b = constant. The energy output of a DE generator per cycle of stretching and contraction is

$$\mathbf{E} = \mathbf{0}.\mathbf{5C}\_1\mathbf{V}\_\mathbf{b}^{\text{-2}} \left(\mathbf{C}\_1/\mathbf{C}\_2 - \mathbf{1}\right) \tag{4}$$

where C1 and C2 are the total capacitances of the DE films in the stretched and contracted states, respectively, and Vb is the bias voltage.

#### **3. Materials for DEs**

The main parameters that improve the performance of the DE are the withstand voltage of the elastomer film, the dielectric constant (including the improvement of the dielectric constant due to additives), Young's modulus, the type of electrode used, use of a cross-linking agent, and the elastomer structure Improvements (such as the addition of monomers or cutting one of the double bonds).

**Table 1** shows the measurement performance of some polymers [3, 20]. This table shows measurements of strain, electric field, modulus of elasticity, and permittivity. The pressure is calculated from Eq. (1) and the elastic energy density is estimated using the strain (measured value) and the pressure calculated from Eq. (2).

*Perspective Chapter: Improvement of Elastomer Elongation and Output for Dielectric Elastomers DOI: http://dx.doi.org/10.5772/intechopen.99713*

Note:


As shown in **Table 1**, the DE polymers (elastomers) that can obtain a value with a large strain has a large value of any one of elastic energy density, breakdown electric field, Young's modulus, or permittivity, or some of them are combined thereof. However, increasing these parameters will stiffen the elastomer and will not significantly deform the DE. In other words, the power obtained will not increase unless the elastomer is hardened and deformed (thickness) significantly.

A large deformation is important not only for actuators but also for power generation elements. That is, a large deformation produces more power (see Eq. (4)).

#### **3.1 Elastomer properties obtained from SS curves/dynamic viscoelasticity tests**

The SS curve and dynamic viscoelasticity were measured using Silicon 1 and Acrylic 1 [21]. The results are shown in **Figures 4** and **5**. First of all, we would like to point out that the research target is artificial muscles, and it is recommended to consider the tensile speed of the SS curve and the viscoelasticity test from the operating speed required for robots and power assist devices. In **Figure 4**, the SS curve was measured by changing the measurement speed in 4 steps, and the curve changed depending on the tensile speed. Similarly, in **Figure 5**, the curve of dynamic viscoelasticity changed depending on the measurement speed [14].

What is interesting here is that acrylic has higher viscoelasticity, so it depends more on tensile speed than silicon. This indicates that it is important to test it with the response required for the artificial muscle. In other words, until now, researchers have overlooked the importance of viscoelasticity. This meaning is easier to understand by looking at the results of the dynamic viscoelasticity test in **Figure 5**.

As the **Figure 5**, clearly shows, the acrylic is more affected by dynamic viscoelasticity than the silicon. As a result, when the driving voltage is increased and each elastomer is stretched, the silicon DE becomes harder, and the amount of stretching

**Figure 4.** *Relationship of stress-strain for tensile tests: (a) Acrylic 1, (b) Silicone 1.*

**Figure 5.**

*The frequency dependence of G', G'' and tan δ of (a) the acrylic 1 and (b) the silicone 1.*

is smaller. Of course, it is also a fact that the difference in the dielectric constant and Young's modulus of both films is the cause (see Eqs. (1) and (2)). Nevertheless, the above behavior can be explained using the SS curves in **Figure 4**. The silicon curve stands up more. This does not mean that silicon has poor performance. It is a proposed that it is better to change the material depending on the application. Silicon has a faster drive speed and a higher rate than acrylic. Therefore, it is advisable to select the type of elastomer depending on where it is used, for example, for applications such as robots or power suits. Illustrating this point, in human muscles, there are slow muscles and fast muscles, each of which has an important mission. Moreover, Silicon can be used from a relatively high temperature to a considerably low temperature. Compared to acrylic, it could have a considerable advantage in devices that are used at higher or lower temperatures [21].

In terms of artificial muscles, e.g., for human uses, it seems that a flat hill-like shape with a gentle rise, like acrylic, is preferable.

#### **3.2 Attempts to increase the dielectric constant of elastomers**

As an attempt to increase the dielectric constant of the elastomer, there is a method of adding a monomer to change the structure or adding a substance having a very high dielectric constant such as Barium titanate. However, in general, these methodss make the elastomer harder and less stretchable. Examples of adding Barium titanate to our synthetic acrylic are shown below. Here, the acrylic we have synthesized is called a base acrylic.

As a method for dispersing barium titanate, a predetermined amount of barium titanate was added to the polymer-containing liquid and crushed with a homogenizer. As a result of particle size measurement by SEM, the median diameter was about 450 nm [57]. It was also confirmed by using SEM that the barium titanate was uniformly mixed.

Elastomer sheets (thickness: 400 μm) were prepared by a) adding 1 wt% of Barium titanate to the base acrylic, and b) adding a 2 wt% of Barium titanate to the acrylic. The SS curves of those elastomers were measured as shown **Figure 6**. The acrylics, which are the base for the Barium titanate were slightly crosslinked. The permittivity of the acrylic was measured using the parallel plate capacitance method. The withstand voltage was measured using a general dielectric breakdown tester. The relationship between the withstand voltage and the capacitance of these films is shown in **Figure 7**.

From **Figures 6** and **7**, as we initially expected, the withstand voltage and the amount of capacitance of the film containing a large amount of Barium titanate increased, and the film became harder and less stretchable by that amount. Circular *Perspective Chapter: Improvement of Elastomer Elongation and Output for Dielectric Elastomers DOI: http://dx.doi.org/10.5772/intechopen.99713*

#### **Figure 6.**

*The SS curve of the elastomer sheet with a small amount of a) 1wt% and b) 2wt% of Barium titanate added to the base acrylics.*

**Figure 7.** *Relationship between the breakdown of the electric field and the capacitances of these elastomer films.*

actuators were produced using either the base acrylic film without adding the barium titanate, or the films containing 1 wt% or the barium titanate and 2 wt%, and the elongations of each were compared. As a result, the actuator using the base film showed the largest elongation. In fact, the film that was hardened by adding the barium titanate was superior in increasing the withstand voltage. From those results, it was found that even if a substance with a high dielectric constant was added, it did not give a significantly good effect. The dynamic viscoelasticity of the base acrylic +2% of Barium titanate is shown in **Figure 8** (**Figure 8** is in Section 3.3).

#### **3.3 Adjustment of cross-linking agents/reduction of double bonds**

**Figure 8** shows the SS curve when the amount of cross-linking agent added to the above base acrylic is changed. Assuming that the amount of the cross-linking agent added to the base acrylic (blue) is 1, red, green, black, and orange are added at rations of 2:1, 1.5:1, 0.8:1, and 0.5:1, respectively. Not surprisingly, the less cross-linking agent we add, the better the elongation. Due to that reduced strength, circular actuators need to be manufactured and evaluated to determine how appropriate they are.

#### **Figure 8.**

*The SS curve when the amount of cross-linking agent added to the above base acrylic changed. The case where the amount of the cross-linking agent added was changed and the case where the amount of the double bond was reduced by using HNBR (Hydrogenated acrylonitrile butadiene rubber).*

#### **Figure 9.**

*The SS curve when the amount of cross-linking agent added to the above base acrylic changed: The case where the amount of the cross-linking agent added was changed and the case where the amount of the double bond was reduced by using HNBR (Hydrogenated acrylonitrile butadiene rubber).*

In this experiment as well, the tensile speeds were set to 100 mm / min and 400 mm / min, but as mentioned above, such an evaluation is important for artificial muscles. That is, the test should be performed according to the actual running speed of the muscle. In this case, 100 mm / min / sec is clearly affected by viscoelasticity. In other words, even if the elongation increases, the stress does not increase so much (the inclination angle is gentle), and as a result, it could become easy to deform as the DE.

Next, attempts were made to not only change the amount of additive added, but also to reduce the amount of double bonds. **Table 2** shows the conditions of the case where the amount of the cross-linking agent added was changed and the case where the amount of the double bond was reduced by using HNBR (Hydrogenated acrylonitrile butadiene rubber). HNBR is a hard material with a dielectric constant of 15, but as shown in **Table 2**, when the ratio of double bonds is reduced, the slope of the SS curve becomes gentle (see **Figure 9**). HNBR Ver.3 has a dramatically reduced slope because the cross-linking agent has also been reduced. The capacitance was also 11.

*Perspective Chapter: Improvement of Elastomer Elongation and Output for Dielectric Elastomers DOI: http://dx.doi.org/10.5772/intechopen.99713*


**Table 2.**

*The case where the amount of the cross-linking agent added was changed and the case where the amount of the double bond was reduced by using HNBR (hydrogenated acrylonitrile butadiene rubber).*

#### **Figure 10.**

*The SS curves of HNBER (base material), HNBER ver.1, HNBR ver.2 and HNBER ver.3. Note: The silicon for this test was used the silicon German-made. This is because the silicon 1 was tested long time ago there is no remaining stock.*

With these membranes, it is necessary to make a circular actuator and measure the elongation, but unfortunately it has not been done yet. Perhaps HNBR ver.3 is a little too soft and it could be difficult to make a DE. Or, because it is soft, the Coulomb force might be dispersed and it might not be possible to drive it well. Further studies are desired on the proportion of double bonds and the amount of cross-linking agent.

The dynamic viscoelasticity of HNBR was also measured (see **Figure 10**). Silicon and acrylic are also shown in this figure for comparison.

**Figure 10** shows the frequency dependence of the storage modulus of five kinds of materials. From 0.032 Hz, it can be seen that the storage elastic modulus of the acrylic 1 (see **Table 1**) increases. It can be seen that the storage elastic modulus of HNBR ver.3 gradually increases, but the storage elastic modulus of the silicon (made in Germany) basically does not change as the frequency increases. Again, silicon could be a bit difficult to get the most out of as an artificial muscle. One of the reasons might be that the structure of silicon is generally a chain structure. Of course, silicon has excellent temperature characteristics and DE responsiveness, and can be driven efficiently. As for our recommendation, it is a good idea to use both fast (silicon) and slow (acrylic) muscles well, like human muscles. Since HNBR is rubber, it is resistant to humidity and can withstand temperature changes. In addition, the results of dynamic

viscoelastic research show that it is somewhat suitable for driving a DE. In particular, it seems that the amount of cross-linking agent added should be selected appropriately. We believe it is particularly suitable for ocean power generation. Since ocean power generation is exposed to a harsh natural environment, it is desirable to use a material that is tough and suitable for DEs. Another point that greatly contributes to power generation efficiency is that there are many changes in thickness (see Eq. (3)), and in that respect, acrylic is most suitable, but acrylic is not very suitable for harsh natural environments. We hope that moisture resistant acrylics will be developed. The film with 2% barium titanate added to the base acrylic is considerably harder than the other films as mentioned above, but the withstand voltage of this film is high (see **Figure 6**) and the elastic modulus is also increased. Therefore, if it could withstand a higher voltage, it might be used as a high-power DE in the future.

#### **3.4 Pre-strain**

Pre-strain will increase the performance of the DE. This is because when repeated tests were performed to know the SS curve of the elastomer sample, the film stretched and it could not return to its original length, so there was no choice but to stretch the film a little in advance and evaluate it [14, 20]. After that, if more pre-strain is applied, it will be advantageous because the strain is applied in advance compared to the case where it is not stretched, and the performance will be further improved [34]. In order to utilize the pre-distortion, it is advantageous to use a material having a gentle SS curve, such as acrylic (see **Figure 4**). As described above, as the degree of pulling increases, the film becomes harder and harder to stretch. However, since the curve of acrylic is gentle, it is harder to harden than silicon. In the dynamic viscoelasticity test, acrylic is more frequency dependent and has more storage modulus than silicon (see **Figures 5** and **8**). This means that even if the film becomes hard, it can function as a DE because the modulus increases. Again, this frequency dependence is also important for use as an artificial muscle.

#### **3.5 Adopted CNTs as electrodes**

Recently, attempts have been made to use new carbon foam materials such as SWCNTs and MWCNTs as electrodes for DEs. These electrodes bring the elastomers to a positively improved higher performance. **Table 3** shows how much weight can be lifted with a stroke of 5 mm due to the difference in electrodes. The elastomer used is acrylic 1, and its weight is 0.1 g [10]. Diaphragm actuators with a diameter of 8 cm were manufactured and those elongations were measured.

Since these electrodes are not optimized, it seems possible to lift heavier weights while having sufficient elongation in the very near future. In addition, these are single layers of DEs and are very light, so it is possible to have multiple layers of DEs, which is close enough to the range applicable to robots and power suits.

On the other hand, these electrodes are also promising as power generation elements. A power generation experiment was conducted using a drape type DE having a height of 120 mm and a diameter of 260 mm. The amount of power generation when the DE was pulled by about 60 mm is summarized in **Table 4** [15]. The drape weighs 4.6 g and uses acrylic 1. Carbon grease, Carbon black, MWCNTs (multi-walled carbon nanotubes), and SWCNTs (single-walled carbon nanotubes) were used as electrode materials.

Using MWCNTs or SWCNTs makes it possible to obtain more power, as shown in **Table 4**.

*Perspective Chapter: Improvement of Elastomer Elongation and Output for Dielectric Elastomers DOI: http://dx.doi.org/10.5772/intechopen.99713*


#### **Table 3.**

*Types of electrodes and weight that can be lifted.*


#### **Table 4.**

*Differences in power obtained when changing the electrode materials.*

This is because the conductivities of MWCNTs and SWCNTs are much higher than that of carbon black or Carbon grease.

In this way, the highly conductive material significantly increases the elongation of the DE actuator, resulting in greater elongation and also increasing the amount of power generated by the DE element.

#### **4. Conclusion**

From the above experimental results and those discussions, the following was found:


### **Acknowledgements**

We thank Zeon Corporation for supplying SWCNT ZEONANO®-SG101 and HNBR.

### **Author details**

Seiki Chiba1 \*, Mikio Waki2 , Shijie Zhu3 , Tonghuan Qu3 and Kazuhiro Ohyama3

1 Chiba Science Institute, Yagumo, Meguro ward, Tokyo, Japan

2 Wits Inc., Oshiage, Sakura, Tochigi, Japan

3 Fukuoka Institute of Technology, Wajirohigashi, Higashi-ward, Fukuoka, Japan

\*Address all correspondence to: epam@hyperdrive-web.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Perspective Chapter: Improvement of Elastomer Elongation and Output for Dielectric Elastomers DOI: http://dx.doi.org/10.5772/intechopen.99713*

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#### **Chapter 8**

## Compression and Recovery Functional Application for the Sportswear Fabric

*Ramratan Guru, Rajeev Kumar Varshney and Rohit Kumar*

#### **Abstract**

A sportswear fabric should have good stretch and recovery behaviour. This study facilitates an effective design and development of high-stretch sportswear using different knitted structure. Nine types of knitted fabrics were produced by varying the type of fibre and type of structure. An experiment work is done to study the fabric size, stretch and elastic recovery properties. The statistical analysis showed that type of fibre and type of knitted structure significantly influence the fabric stretch. Plain structure fabric showed higher stretch value than rib and interlock-knitted fabric. The high stitch density caused by reduce stretch value in the course- and wale-wise due to yarn floating rather than overlapping influenced the weight and thickness of knitted fabrics. The elastic recovery analysis indicated that the recovery value of plain-knitted structure with polyester-spandex blend is higher among studied fabrics. However, the recovery value decreased over time in comparison with stretch value.

**Keywords:** sportswear fabric, stretch and recovery performance, polyester, micro-polyester, polyester-spandex

#### **1. Introduction**

The stretchable knitted structures play an important role in body comfort and fit. The knitted structures allow wearer the freedom of movement with least resistance due to their stretchability and elasticity [1, 2]. Regular physical activity is important to maintain consistency in human health. To achieve comfort and functional support during various activities such as walking, stretching, jogging, athletes and sports persons use sports clothing [3, 4]. Stretch properties represent a significant mechanical property of clothing material that influences clothing pressure. Stretch properties are measured as the percentage of fabric stretch and fabric growth, and recovery [5–7].

Basically, two types of category are normally available in sportswear stretch cloths: First is comfort stretch range of about 20–30% and other power stretch cloth range approximately 30–50%. The basic designs are used for high-active sportswear garments in elasticity and compression cloths.

The power stretch cloths need to have more extensibility and quicker recovery performance [8–12].

The high-compression clothes are more utilized medical compression garments and sportswear cloths sectors. Study on the evaluation of elastic recovery of cotton-knitted fabrics was conducted [13–16]. It is found that length of these

cloths is different to a more extent value. As per information, fabric elongation is different from single-structure knit fabric in lengthwise ranging 3–6%, with double cloth knit having lengthwise 3–50%. Elongation in single-structure knit fabric is of widthwise 3–180% and double cloth knit elongation in widthwise of 6–155%. It is elastic recovery different from single-structure knit lengthwise to other structures. According to author, single-knit lengthwise elastic recovery is of range 100–56%, double cloth knit lengthwise elastic recovery is of range 100–57%, and single-structure knit elastic recovery widthwise is found 100–56%, double cloth knit widthwise elasticity recovery is of range 100–30% that is basically found in knit cloths [13–17].

Plain knit had more elongation and growth as compared with double knit. The growth after 30 seconds or relaxation was observed to be 36% and plain knit stretched more under load and after the load was released that exhibited more growth than the double knit. The stretch of knit fabrics is affected more by the cover factor than by the yarn diameter, loop length, loop density or the shape of the loop [18]. Spandex is widely used in sportswear for its superior stretch and recovery properties. Dynamic elastic repossession can assess the immediate apparel response due to body movement; the elastic bare-plaited fabric is found to have higher dynamic elastic recovery than cloth knitted from lycra core spun. The basic phenomena are essential use in stretch and recovery of the cloth to pressure generated by compression apparel. It is found that knitted fabric in normal stretch and recovery performance as compared to compression sportswear garment. Therefore, Lycra is used in knit cloths in blend with other fibres for proper utilization of stretchability and elasticity recovery properties in sportswear garments [19, 20].

The objective of this study was to investigate the effect of the stretch, growth fabric and recovery properties of polyester-spandex-blended, micro-polyester and 100% polyester-knitted fabric. These works could facilitate the design and development of sportswear with the required stretch and recovery properties.

#### **2. Materials and methods**

#### **2.1 Materials used**

In this study, three different filament yarns—polyester, micro-polyester, blend of polyester-spandex and non-circular cross section were used to prepare samples. The knitted structures—single jersey, interlock and rib fabrics were produced on weft-circular knitting machine (**Table 1**).

#### **2.2 Testing methods**

The knitted fabric was conditioned in standard atmospheric condition of 65+/−2% RH and 27+/−2°C temperature and the samples are in condition for 24 hours before testing. The stretch and recovery property tester was using the ASTM D 2594-2004 (2008) standard.

#### *2.2.1 Statistical analysis*

One-way ANOVA (Minitab 17 statistics software) tests were used to determine the significant difference between the stretch and elastic recovery properties of fabrics. In order to infer whether the parameters were significant or not, *p* values were examined. If the '*p*' value of a parameter is greater than 0.05 (*p* > 0.05), the parameter was not significant and should not be investigated.


*Compression and Recovery Functional Application for the Sportswear Fabric DOI: http://dx.doi.org/10.5772/intechopen.101316*

*\*Wpcm—wale per centimetre, Cpcm—course per centimetre, \* PET—polyester, \* all three fabrics are made from 100 deniers polyester tex 11.11.*

#### **Table 1.**

*Geometrical properties of knitted fabrics.*

#### *2.2.2 Fabric particulars*

The fabric details measured were as follows: wales per inch (wpi), course per inch (cpi), linear density of yarn (denier), fabric mass per unit area (g/m2 ) and fabric thickness (mm). The wpi and cpi were measured according to the ASTM D-3887 Standard. Yarn linear density and fabric mass per unit area were measured according to ASTM D 1059-01 and ASTM D 3776 M-09a standard respectively by using an electronic weighing balance. Thickness testing was carried out as per BS EN ISO 9073-2 using the electronic thickness tester at 0.25-KPa pressure. For each sample, 30 readings were taken to get the result at 95% confidence level.

#### *2.2.3 Stretch and recovery property tester*

ASTM D 2594-2004 (2008) standard test method for stretch properties of knitted fabric was applied under the form-fitting standards as sample size 18.3 cm in wale direction and 21.5 cm in course direction positions and 4.54 kgf weight apply in the Fabric specimen of the lower bench marks were calculated after 60 s and 1 h. Stretch percentage points to test the stretch and elastic recovery of the experimental samples are as shown in **Figures 1** and **2**.

Stretch, growth and recovery percentages were calculated by Eqs. (1)–(3) given below:

$$\text{Fabric structures} \\ \text{ch} \%= \frac{\text{B} - \text{A}}{\text{A}} \\ \text{X100} \\ \tag{1}$$

$$\text{Fabric growth}\% = \frac{\text{C} - A}{\text{A}} \text{X100} \tag{2}$$

$$\text{Fabric recovery} \%= \frac{\text{B} - \text{C}}{\text{B} - \text{A}} X \mathbf{100} \tag{3}$$

where A: the distance marked between the upper and bottom parts of the fabric; B: the distance between the marked points after holding the sample for 5 min with 4.54 Kgf load; C: the distance between the marked points after 5-min relaxation.

**Figure 1.** *Stretch and recovery setup assembly.*

**Figure 2.** *Fabric stretch equipment.*

#### **3. Results and discussion**

**Figures 3** and **4** and show knit specimen changes in weight and thickness. The heavier weight of knit specimen was, the thicker its thickness was in descending order 'interlock structure polyester-spandex, micro-polyester and 100% polyester-knitted fabric', 'rib structure polyester-spandex, micro-polyester and 100% polyester-knitted fabric', 'plain structure polyester-spandex, micro-polyester and 100% polyesterknitted fabric'. Thickness and weight of specimen were influenced by density change caused by reducing and increasing fabric size. Thus, high density caused by floating in course-wise causes more knitted fabric weight gain than by loop overlapping.

*Compression and Recovery Functional Application for the Sportswear Fabric DOI: http://dx.doi.org/10.5772/intechopen.101316*

**Figure 4.**

*Fabric thickness comparison on knit structure.*

#### **3.1 Stretch properties**

**Figure 5** and **Tables 2** and **3** indicate that the stretch value decreased in rib and interlock structure-knitted fabrics and direction except for wale-wise and coursewise as compared with plain structure fabrics. The plain structure fabrics have higher-stretch (%) polyester-spandex blend because lycra filament yarn have more stretch properties compared with other polyester and micro-polyester yarn [1].

The interlock structure three-knitted fabric showed a sharp decrease, while rib interlock structure three-knitted fabric had relatively small decrease. It seems that the material effect by stretch properties added to the reducing cause by yarn floating in the fabric structure, which held the loops reduced the stretch value of the fabric. The stretch value in course-wise is influenced by yarn floating rather than loop overlapping, while stretch value in wale-wise is caused by loop overlapping versus yarn floating [10].

#### **Figure 5.**

*Stretch comparison on knit structure.*


**Note: A** *—Course-wise stretch percentage,* **B** *—course-wise recovery after 60 sec %,* **C***—course-wise recovery after 1 hr. %,* **D***—Wale-wise stretch percentage,* **E***—wale-wise recovery after 60 sec %,* **F***—wale-wise recovery after 1 hr. %.*

#### **Table 2.**

*Mean value of stretch and recovery test results.*


**Note: A***—Course-wise stretch percentage,* **B***—course-wise recovery after 60 sec %,* **C***—course-wise recovery after 1 hr. %,* **D***—wale-wise stretch percentage,* **E***—wale-wise recovery after 60 sec %,* **F***—wale-wise recovery after 1 hr. %.*

#### **Table 3.**

*One-way ANOVA of stretch and recovery properties of sportswear-knitted fabric structures.*

*Compression and Recovery Functional Application for the Sportswear Fabric DOI: http://dx.doi.org/10.5772/intechopen.101316*

**Table 3** shows the ANOVA statistical analysis results at 5% significance level. Stretch and elastic recovery properties of the sportswear-knitted fabrics show significant difference between them (course-wise stretch (%): Factual = 5.432 and wale-wise stretch (%): Factual = 2.813 in comparison with Fcritical = 2.26) at degree of freedom 9.

#### **3.2 Elastic recovery properties**

There was significant value change on knit structure and direction in elastic recovery as shown in **Figures 6** and **7**. The recovery value gap among knitted specimen was lower at 1 h than at 60 sec. The stretch loops bent and restricted by the external force loop of stretch take on a form of stability and shape retention in cover time [11–13].

**Figure 6.** *Elastic recovery 60 sec on knit structure fabric.*

**Figure 7.** *Elastic recovery 1 h on knit structure fabric.*

The ANOVA results show in **Table 3** that with respect to stretch properties after 60 sec %, there is a significant difference between the knitted fabric coursewise recovery after 60 sec %, degree of freedom 9. [Factual = 3.16 > Fcritical = 2.26 (*p* < 0.05)]. And for wale-wise recovery after 60 sec %, there is a significant difference between the structures [Factual = 2.44 > Fcritical = 2.26 (*p* < 0.05)].

It was found from **Table 3**, ANOVA results show that there is a significant difference between the course-wise recovery after 1 h %, value of knitted fabrics [Factual = 2.91 > Fcritical = 2.26 (*p* < 0.05)]. Also, it is noticed that there is a significant difference in wale-wise recovery after 1 h %, between the knitted fabrics [Factual = 5.53 > Fcritical = 2.26 (*p* < 0.05)].

### **4. Conclusion**

The followings conclusions are derived from the above experimental work and given below:


*Compression and Recovery Functional Application for the Sportswear Fabric DOI: http://dx.doi.org/10.5772/intechopen.101316*

### **Author details**

Ramratan Guru1 \*, Rajeev Kumar Varshney2 and Rohit Kumar2

1 Department of Handloom and Textile Technology, Indian Institute of Handloom Technology, Varanasi, UP, India

2 Department of Textile Engineering, Giani Zail Singh Campus College of Engineering and Technology, MRSPTU, Bathinda, Punjab, India

\*Address all correspondence to: ramratan333@gmail.com

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **References**

[1] Senthilkumar M, Anbumani N. Dynamics of elastic knitted fabrics for sportswear. Journal of Industrial Textiles. 2011;**41**(1):13-24

[2] Rhie J. Fundamental relationship between extensibility of stretch fabric and its pressure. Family and Environment Research. 1992;**30**(1):1-2

[3] Chakraborty JN, Deora D. Functional and Interactive Sportswear. Asian Textile Journal. 2013;**22**(9):69

[4] Ramratan, Choudhary AK. Thermophysiological study of active knitted sportswear: A critical review. Asian Textile Journal. 2018;**27**(7):53

[5] Ramratan, Choudhary AK. Influence of functional finishes on characteristics of knitted sportswear fabrics. Asian Textile Journal. 2018;**27**(8):43

[6] Yamada T, Matsuo M. Clothing pressure of knitted fabrics estimated in relation to tensile load under extension and recovery processes by simultaneous measurements. Textile Research Journal. 2009;**79**(11):1033

[7] Sang JS, Park MJ. Knit structure and properties of high stretch compression garments. Textile Science and Engineering. 2013;**50**(6):359-365

[8] Lyle D. Performance of Textiles. New York: Wiley and Sons, Inc.; 1977. pp. 168-169

[9] Senthilkumar R, Sundaresan S. Textiles in Sports and Leisure. The Indian Textile Journal. 2013;**123**(5):89-95

[10] Ladumor HC, Manish B, Vaishali DS. Elastic recovery characteristics of waist band using high stretch polyester in place of Lycra a technical review. International Journal for Scientific Research & Development. 2015;**3**(4):104-107

[11] Kentaro K, Takayuki O. Stretch properties of weft knitted fabrics. Journal of the Textile Machinery Society of Japan. 1996;**19**(4):112-117

[12] Senthilkumar M, Anbumanl N. Effect on laundering on dynamic elastic behavior of cotton and cotton spandex knitted fabrics. Journal of Textile And Apparel Technology and Management. 2012;**7**(4):1-10

[13] Ashayeri E, Alam FMS. Factors influencing the effectiveness of compression garments used in sports. Procedia Engineering. 2010;**2**(1): 2823-2829

[14] Su CI, Yang HY. Structure and elasticity of fine electrometric yarns. Textile Research Journal. 2004;**74**(12): 1041

[15] Song G. Improving Comfort in Clothing. Cambridge: Woodhead Publishing Limited; 2011. p. 114

[16] Shishoo R. Textiles in Sport. Cambridge, England: Woodhead Publishing in Textiles; 2005. pp. 1-8

[17] Robert SH, Fletcher HM. Elastic properties of plain and double-knit cotton fabrics. Textile Research Journal. 1964;**649**

[18] Saricam C. Absorption, wicking and drying characteristics of compression garments. Journal of Engineered Fibres and Fabrics. 2015;**10**(30):146-154

[19] Manshahia M, Das A. High active sportswear a critical review. Indian Journal of Fiber and Textile Research. 2014;**39**(2):441-449

[20] Morton WE, Hearle JWS. Physical Properties of Textiles Fibres. England: The Textile Institute Woodhead Publishing Limited; 1993

#### **Chapter 9**

## Characterizing Stress-Strain Behavior of Materials through Nanoindentation

*Indrani Sen and S. Sujith Kumar*

#### **Abstract**

Nanoindentation is a widely used state of the art facility to precisely and conveniently evaluate the mechanical properties of a wide group of materials. Along with the determination of elastic modulus and hardness of materials, this chapter particularly aims to explore the possibilities to assess the corresponding stress–strain characteristics of elastic–plastic materials and most importantly unique pseudoelastic materials. The suitability of continuous stiffness measurement (CSM) based nanoindenter systems along with the adaptability of the instrument without CSM for precisely evaluating the deformation behavior of specialized materials is discussed in details. In this regard, the roll of indenter tip geometry and size is greatly emphasized. The recent research in the field is reviewed thoroughly and the updated protocol generated is illustrated.

**Keywords:** nanoindentation, stress-strain curve, small-scale, plasticity, NiTi

#### **1. Introduction**

Since the early 19th century, indentation technique has been extensively used for characterizing the mechanical properties of vast range of materials. In general, the indentation test is known to measure the *hardness* of materials. In conventional techniques, the mean contact pressure (*MCP*) upon indenting a specimen surface is evaluated. This is done on the basis of the residual area measured from the image of the indent impression and the known value of the applied load. The quantitative parameter, thus evaluated, represents the material's response against deformation. In fact, *MCP* measured at the fully developed plastic zone is known as hardness [1]. With the progress in the technology and its incorporation in the experimental setup, instrumented indentation technique, particularly '*nanoindentation*' has been evolved to assess various mechanical as well as metallurgical properties of a range of materials [2–4]. This includes characterizing elastic moduli, residual stress, creep properties, dislocation density, strain rate sensitivity etc. [5–12]. Among all these developments, the potential of the nanoindentation technique in generating the indention stress (*σind*) – indentation strain (*εind*) curve is the most recent one and it is explained in detail in this present chapter [2, 3].

In nanoindentation, the associated high-resolution depth sensing technique aids to estimate the depth or size of the deformation zone. The process records the continuous response of indentation load (*P*) in the range of *μN* vs. indentation

depth (*h*) in the magnitude of *nm*. The *P-h* curve obtained therein helps to assess the various properties of the studied materials. Unlike the conventional technique, in instrumented nanoindentation, hardness is estimated by using indirect measurement of projected contact area from *P-h* curve and the known geometry of the indenter tip. Similarly, elastic modulus of the material is estimated using the slope of the unloading segment in the *P-h* response of materials [13]. This method of analysis has been used for various scientific studies to characterize the localized mechanical properties of the samples in sub-micron scale. In fact, this revolutionary modification in the assessment methodology through nanoindentation has opened up a wide range of studies to extract the different relevant mechanical properties of materials on a small-scale.

One of the breakthroughs is the capability of this technique in generating the *σind - εind* response of a material of interest [3, 14–16]. This novel and recent development plays a significant role in understanding the localized deformation capability of materials system. This is particularly because stress – strain characteristics can provide an insight into the elastic – plastic mechanisms of the materials, as per the conventional notion. In fact, estimation of localized stress – strain characteristics of a material through nanoindentation can even be a substitute for typically used small-scale characterization techniques for instance, micro-pillar compression [17, 18]. Nevertheless, nanoindentation is further beneficial owing to its easier sample preparation, simplicity in experimental execution, and non-destructive nature. This technique therefore has enormous potential for evaluation of smallscale mechanical properties of materials with minimal effort.

Considering this, the present chapter is dedicated to provide a reasonable understanding for generating *σind - εind* data from the *P-h* curve of nanoindentation. To develop a more conceptual idea for a new reader, the importance of indenter tip geometry in activating different deformation modes within the indented volume are discussed at the first hand. Subsequently, the basic relationships for the indentation, the method of analysis and generation of protocol for obtaining the *σind - εind* curve will be discussed.

#### **2. Role of indenter configuration**

It is noteworthy that both the uni-axial tensile/compression test as well as the indentation technique are capable to assess the stress–strain characteristics of a material, however, with usually different size-scale of samples along with varying stress-states. The former provides an understanding for the degree of bond stretching induced elastic deformation and dislocation mediated plastic/permanent deformation in the material. To obtain such desired information, the strain-induced into the material should be controlled in such a way that, the material's response reflects the gradual activation and transition from the elastic to the plastic deformation. This is realized in uni-axial deformation without any strain gradient in the specimen, at least macroscopically.

In contrary to that, upon indentation, presence multi-axial state of stress exists beneath the indenter tip. Moreover, the constraint nature of deformation induces strain gradient within the deformation volume. Hence, for assessing the elastic– plastic activity within the deformation zone, the indentation tests need to be specially designed to produce a smooth strain distribution (or gradient) along with its gradual increment. To maintain that, indenter tip geometry needs to be carefully chosen to reflect the *σind - εind* characteristics from the localized region. In this regard, the most suitable indenter configuration is spherical tip (or sphero-conical indenter).

Before getting into the details about the configuration of the spherical indenter tip and its importance for *σind - εind* generation, the reader needs to develop a comprehensive idea about the different type of indenter tips that are used in general. For the same, the geometrical aspect of indenter configuration is briefed here. From a geometrical point of view, indenters are classified into two: (i) geometrically similar indenters (*GSI*) and (ii) non-geometrically similar indenter (*N-GSI*) [4].

The most commonly used sharp pyramidal indenter such as four-sided Vickers (for micro-and macro-indentation) and three-sided Berkovich indenters (for nanoindentation) comes under the category of *GSI*. On the other hand, the spherical indenter falls under the category of *N-GSI*. The major difference in deformation characteristics experienced by a specimen surface, by indenting with any of these two categories of indenter tips can better be appreciated from **Figure 1**. Schematic representations in **Figure 1(a-c)** show the deformation modes activated in traditional elastic–plastic material while increasing the indentation load/depth, using *GSI*. The mathematical relation for geometrical similarity originates from the ratio of the contact radius (*ac*) to the maximum depth of indentation (*hmax*). For *GSI*, *ac-i/hmax-i* = *ac-j/hmax-j* = *ac-k/hmax-k =* constant. The subscript, *i*, *j*, *k* signify increasing level of *h*. Nevertheless, this constant ratio of *ac/hmax* ensures that the size of the deformation zone of indentation varies uniformly irrespective of the depth of penetration. This helps to estimate the property of the subjected material independent of the applied indentation load/depth. Nevertheless, owing to the sharp nature of the Vickers and Berkovich indenter, the strain-induced within the indentation volume is large enough to generate significant plastic deformation [1]. In that case, dislocation activity is always the dominant mechanism within the deformation volume beneath the indenter tip, irrespective of the change in depth of indentation, as apparent from **Figure 1**. This assists to precisely measure the hardness of a material independent of the indentation load, in the theoretical sense. However, it is realized that *GSI* is not adequate to assess the elastic deformation response of the indented material. In fact, while using conventional Vickers and Berkovich

#### **Figure 1.**

*Schematic illustration of the indentation behavior associated with traditional elastic-plastic metallic using (a-c) sharp geometrically similar indenter and (d-e) spherical non-geometrically similar indenter at various indentation depths.*

indenters, occurrence of prominent dislocation activity within the deformation volume negates any influence of elastic activity therein. This acts as the limitation of the most commonly using Vickers and Berkovich indenter for generating the stress–strain curve.

On the other hand, a completely different deformation response is being experienced, while the specimen surface is indented using *N-GSI* (spherical tip) with increasing indentation load/depth. **Figure 1(d-f)**, illustrate the deformation scenario within the indentation volume, in such case. It is evident from the figure that, nature of deformation is entirely different in comparison to that for *GSI*. This difference originates from the non-geometrical similarity of the indenter. In case, the specimen surface is indented with a *N-GSI*, *ac-i/hmax-i* < *ac-j/hmax-j* < *ac-k/hmax-k.* This essentially means with the progress of the indentation, increment in the contact radius becomes more pronounced with respect to the depth of penetration. Such movement of indenter within the material surface gradually increases the induced strain/stress into the material. Also, the blunt nature of the indenter assists in generating a smooth stress field within the indentation volume, specifically as compared to *GSI*. As a net effect, spherical indenter facilitates a gradual activation of elastic to the plastic deformation mechanism. This potential for gradual instigation of the deformation mechanism similar to that observed in case of uni-axial test, is exploited for *σind - εind* generation from nanoindentation.

Nevertheless, the most crucial part in this regard is the data analysis procedure that is necessary to convert the indentation *P-h* response into a reliable *σind - εind* curve. There have been numerous attempts to obtain a stress–strain curve from traditional indentation as well as instrumented one. In the process, the protocol for generation of indentation stress–strain curve has undergone various alterations, to precisely correlate the materials' property. In the next section, we have briefed the different approaches adopted to appreciate the *σind - εind* behavior of a material. This will help to understand the scientific developments that has been materialized on this particular topic, so far.

#### **3. Evolution of** *σind - εind* **generation protocols**

The concept for the generation of *σind - εind* curve from indentation is introduced by Tabor in the 1950s. Tabor has measured the *MCP* on the specimen indented with a spherical tip to estimate the stress that is induced in the process [1]. The most crucial part, however, is the estimation of *εind*. Tabor defined *εind* by the relation (*d/D*), where *d* is the diameter of the residual impression and *D* is the diameter of the indenter tip. Here *d* is measured using the traditional approach, i.e., by imaging of residual impression after unloading. The general trend of *σind - εind* characteristics of materials, generated following Tabor's protocol, resembles well with that evaluated through traditional uniaxial compression test [1]. However, this method of analysis accounts for only single *σind - εind* data from an indentation. So, it means that several indentation tests with different indentation parameters are necessary to be pursued, to obtain a continuous *σind - εind* curve for a material, making the process cumbersome.

Nevertheless, Tabor's approach revealed the potential of the indentation technique and instigated more studies to develop a state-of-the-art protocol for generating *σind - εind* curve of a material. In this regard, automation through the instrumented indentation has opened up enormous possibilities to generate the *σind - εind* curve using a single indentation. In turn, the localized deformation behavior of a material can be precisely obtained. First among all is the Field and Swan approach [19]. They have proposed to incorporate multiple partial unload

#### *Characterizing Stress-Strain Behavior of Materials through Nanoindentation DOI: http://dx.doi.org/10.5772/intechopen.98495*

segments during each indentation. Here, the *P-h* responses obtained for each particular segments are used to measure the corresponding *σind* and *εind* values*.* The strain, on the other hand, is estimated using the relation *a/Ri*, where *Ri* represents the radius of the indenter tip. As per Field and Swan approach, the deformation associated in each unloading segment is assumed to be purely elastic. Correspondingly, the classical Hertzian elastic relationship (explained in the next section by Eq. (1)) is applied on those *P-h* responses to assess the contact radius, *a*. From the measured *a* value, contact area (*Ac*) is estimated instead of residual impressionbased analysis in Tabor's protocol.

The Field and Swan approach has much significance in the present scenario, owing to its implementation of the Hertzian contact mechanics theory. Nevertheless, interpretation of indentation strain as per both Tabor's as well as Field and Swan approaches has been questioned for its integrity with the fundamental concept of strain. In general, strain is defined as the ratio of change in length to the initial length in a region of deformation considered. However, this fundamental relationship is not met in both these above-mentioned approaches.

In order to overcome this fundamental lacking, various studies have been conducted to formulate an adequate relationship for the *εind*. Among those attempts, the protocol developed by *Kalidindi* and *Pathak* has succeeded in defining *εind* as per the most basic concept of strain [16]. The present chapter is extensively covering the formulation and implementation of *Kalidindi* and *Pathak* protocol for the generation of *σind - εind* curve for a material subjected to nanoindentation. This protocol is essentially formulated based on classical Hertzian theory, which is explained below.

#### **4. Contact mechanics for spherical tip-based indentation**

Contact mechanics theory introduced by Hertz has provided a fundamental basis for the indentation technique [20]. Classical Hertzian theory predicts the elastic responses of frictionless contact between two different bodies of dissimilar geometries (with varying properties) in contact. This theory is formulated based on the assumption that material is homogenous and isotropic. In the present scenario of indentation using spherical indenter, the Hertzian theory for elastic contact between the sphere (indenter) and elastic half-space (specimen surface) is used for the formulation of *σind - εind* generation. In the indentation aspect, the material of interest is considered as an elastic half-space by following the criteria that indenter tip radius (*Ri*) should be at least ten times smaller than the horizontal dimensions of the sample [21].

As explained in previous Section 2 (see **Figure 1(d-f)**), indentation using spherical indenter tip facilitates the gradual activation of elastic to plastic mechanisms in the material. Therefore, for the sake of understanding, the overall deformation scenario can be categorized into (i) fully elastic and (ii) plastic following the initial elastic section. The schematic representation of these two modes of deformation and their corresponding *P-h* response is showed in **Figure 2**. In the first case, material recovers all the depth it penetrated upon the indentation (see **Figure 2(a)** and **(c)**). In the second case, some amount of permanent deformation is existing within the indentation volume (see **Figure 2(b)** and **(d)**). Hertz has provided the basis for the elastic deformation associated in two former cases using the relation below,

$$P = \frac{4}{3} E\_{\epsilon\emptyset} R\_{\epsilon\emptyset}^{1\_2} h\_r^{3\_2} \tag{1}$$

$$\frac{1}{E\_{\rm eff}} = \frac{1-\nu\_s^2}{E\_s} + \frac{1-\nu\_i^2}{E\_i}, \frac{1}{R\_{\rm eff}} = \frac{1}{R\_i} - \frac{1}{R\_s} \tag{2}$$

#### **Figure 2.**

*Schematic representation of indentation of behavior of material in (a) fully elastic condition and in the pressure of (b) plastic deformation. Corresponding indentation load vs. indentation depth responses of materials are shown in (c) and (d).*

Here *P* is the applied load, *hr* is the recoverable depth, *Reff* is the effective tip radius and *Eeff* is the effective elastic moduli. All the characteristic terms mentioned here can be appreciated from **Figure 2(b)**. The terms *hmax* and *hp* in **Figure 2** represent the maximum depth of indentation at *Pmax* and recurring plastic depth of indentation post-unloading (*P* is zero), respectively. In the Hertzian relation, the role of elastic deformation on the two mating parts is assessed using *Eeff.* The value of *Eeff* accommodates the elastic deformation associated with the hard indenter and soft sample. *Eeff* during the indentation is estimated using the relation (2). Similarly, *Reff* takes into account the influence of plastic activity on the overall deformation. It is related to the indenter tip radius (*Ri*) and the radius of curvature of the sample (*Rs*) upon the indentation. *Reff* of the sample is estimated using the relation (2).

All these relations derived by Hertz has laid the foundation for the formulation *of σind - εind* data from the nanoindentation *P-h* response. This is explained in details in the following section.

#### **5. Defining the indentation stress and indentation strain**

It is well understood from Section 3 that Tabor's and Field and Swan's protocols do not suffice to define the *εind* precisely. Nevertheless, *Kalidindi* and *Pathak* have defined the *σind* and *εind* by considering the size of the deformation zone formed beneath the indenter and correlated it with the fundamental Hertzian relationship [16]. This protocol has succeeded in producing comprehensive *σind* - *εind* data from the nanoindentation experiments (explained in Section 6).

As per this novel approach, eq. (1) is rearranged by incorporating the following relations:

$$
\sigma\_{ind} = \frac{P}{\pi \, d^2}; \sigma\_{ind} = \quad E\_{\text{eff}} \varepsilon\_{ind}; \varepsilon\_{ind} = \frac{4}{3\pi} \, \frac{h\_r}{a} \approx \frac{h\_r}{2.4 \, a} \tag{3}
$$

$$a = \sqrt{\mathcal{R}\_{\mathcal{eff}} h\_r} \tag{4}$$

*Characterizing Stress-Strain Behavior of Materials through Nanoindentation DOI: http://dx.doi.org/10.5772/intechopen.98495*

**Figure 3.**

*(a) Schematic representation of the deformation behaviour associated with indentation. Figure highlights the actual deformation zone of indentation and the idealized deformation zone of indentation. (b) Schematic representation of surface irregularities on a sample.*

The indentation strain defined using the above relationship satisfies the general definition of strain. This can be better appreciated from **Figure 3(a)**. In the figure, the dashed spherical shaped region beneath the indenter tip schematically shows the actual size of the deformation zone upon indentation. Based on the *εind* defined from Hertzian relation, the length of the deformation zone beneath the indenter tip at *Pmax* is noted to be *2.4a*. Interestingly, a simulative study on the prediction of indentation behavior strongly agrees with this relation for tungsten and aluminium [16]. This has validated the new definition of *εind*, which is derived without any alteration of the fundamental Hertzian relation. This novel protocol is remarkably different yet comprehensive with respect to the other discussed approaches. This is primarily because it basically takes into account the actual size of the deformation zone during the indentation, rather than simply estimating the *εind* data using the concept of variation in indent impression.

Furthermore, this novel protocol has provided a reasonable basis for the analogical comparison of indentation behavior using spherical indenter and uniaxial compression test. The overall nature of the material response upon nanoindentation can be considered as the replication of compressing up to a depth of *hmax* on a cylindrical sample of height *2.4a* and radius *a.* To visualize it clearly, the idealized deformation zone of indentation and actual deformation of indentation is schematically shown in **Figure 3(a)**. The shape of the actual deformation zone formed is schematically showed as spherical. The reader should be aware that, in reality, owing to the anisotropy in material's properties, the actual shape of the deformation zone of indentation can be slightly different from this schematic representation. It is also noteworthy that with slight alternation in relation (4), *hmax* can be used instead of *hr* in the numerator to accommodate the plastic activity [15]. This whole theoretical concept has paved the way for generating *σind - εind* curve from the *P-h* signal in nanoindentation. To realize it in a practical scenario, the reader has to understand the necessary steps to follow for obtaining a reliable output.

#### **6. Theoretical conceptualization to experimental execution**

As mentioned in Section 1, nanoindentation typically generates a *P-h* response and its characteristics define the mechanical property of the material indented. Compared to any other characterization technique, particularly, the most commonly used uni-axial tests, the size of the active deformation region for

nanoindentation is extremely small. Therefore, proper measures are necessary at every steps right from the precise sample preparation to the careful data analysis to obtain reliable data.

#### **6.1 Sample preparation**

The existence of an artefacts such as scratches or the presence of foreign particles on the surface can influence the *P-h* signal and thereby the generated *σind - εind* data. The poorly polished samples create a scratch on the surface, the depth of which can be in hundreds of nanometres. Data recorded from such a region will certainly influence the overall *σind - εind* characteristics and consequently alter the assessment of the true properties of the material. This can be visualized and understood from the schematic representation in **Figure 3(b)**. In the figure, red coloured triangular shape and yellow coloured circular shape reveal the presence of sample surface roughness and foreign particles respectively. As per the indentation sequence, the indenter will first acquire the data from those artefacts and move to the bulk of the sample. So, actual material which is supposed to show the pure elastic response initially, is now influenced by the presence of sample surface artefacts. As a net effect, the *P-h* response from the bulk sample is influenced by the surface roughness/foreign particle. Hence, the assessed properties are certainly different from the true ones [1]. In case of conventional uniaxial tests, such misinterpretation of results can be obtained in case a specimen slips upon loading, or even when elastic properties are estimated from a tensile experiment, without attaching an extensometer to the test specimen.

To avoid such issues, well-polished, smooth, flat and plane-parallel specimen should be subjected to nanoindentation. The necessary steps to achieve such artefact free surface vary with the material of interest. However, colloidal silica polish for few hours (minimum 3 h) after the conventional polish using silicon carbide paper with decreasing mesh size and diamond polish is prescribed for metallic specimens, to attain a reasonably good surface condition for the *σind - εind* generation. Depending on the surface characteristics of the material, electropolishing may also appear to be a better option to minimize the artefacts on the sample surface.

#### **6.2 Conversion of experimental** *P***-***h* **data to effective** *P***-***h* **data**

It is noted that theoretical predictions and the experimental outcome may result to some disparities in case of the nanoindentation test. In this regard, it is noteworthy that proper data analysis plays a key role in the generation of *σind - εind* curve. It is highlighted in the previous section (Section 6.1) that nanoindentation experiments mandate extremely good quality surface finish. Nevertheless, obtaining the required surface finish is difficult in practice. A proper data correction route on the experimentally obtained *P-h* curve, on the other hand, can negate the role of artefacts on the *σind - εind* analysis. This step is crucial to compute a reliable stress– strain curve. For the same, effective initial contact point between the indenter tip and the specimen surface is estimated following the "zero-point correction" (*ZPC*). In fact, *ZPC* deals with discarding the data points which are influenced by unavoidable surface irregularities. In turn, the effective contact point is determined on the basis of Hertzian theory which reciprocates the material behavior. According to the type of nanoindentation instrument used, *Kalidindi* and *Pathak* have proposed two different approaches for the data correction using *ZPC*. One is for nanoindenter with (a) Contact Stiffness Mode, *CSM* (or Dynamic Mechanical Analysis, *DMA*) and another for (b) Non-Contact Stiffness Mode, *N-CSM* [22]. These two modes are slightly different in the method of experimentation.

*Characterizing Stress-Strain Behavior of Materials through Nanoindentation DOI: http://dx.doi.org/10.5772/intechopen.98495*

#### *6.2.1* CSM *mode or* DMA *mode*

In *CSM* or *DMA* mode, harmonic force is imposed in the loading and unloading segment during the indentation. This is highlighted at the inset (a) of **Figure 4**. It can be hypothetically viewed as if the specimen undergoes multiple indentations with minimal depth scale (2 to 4 nm) while conducting a single indentation. Displacement responses corresponding to these harmonic forces are recorded throughout the indentation. These assist in assessing the variation in contact stiffness, *S* (*or dP dh*) upon the indentation. Precise determination of *S* from each steps of *CSM* leads to estimate the continuous variation in the related properties of materials with increasing *h*, for example, hardness and elastic modulus changes [22].

In the present scenario, the continuously varying *S, hr,* and *P* are obtained from the *CSM* mode of the nanoindenter and these signals are used for *ZPC.* For the same, the Hertzian relation (Eq. (1)) for elastic contact is rearranged into the following relationship,

$$P - \frac{2}{3}h\_r \mathbb{S} = -\frac{2}{3}h^\* \mathbb{S} + P^\* \tag{5}$$

Here *P\** and *h\** denote the effective indentation load and depth respectively. A linear regression analysis on relation (5) helps to trace the *P\** and *h\** values through the slope (�<sup>2</sup> <sup>3</sup> *<sup>h</sup>*<sup>∗</sup> ) and y-intercept (*P\** ). Once the *P\** and *h\** are established, the experimentally generated *P-h* signal has to be corrected for obtaining an '*effective P-h curve',* which is devoid of any influence from the surface artefacts [16].

#### *6.2.2 Non-*CSM *mode*

In *N-CSM* mode, indentation is performed without harmonic force. This is also highlighted in inset (b) of **Figure 4**. In this particular case, *ZPC* is performed by recasting the Hertzian equation as per the relation below (derived from Eq. (1),

$$k(h\_r - h^\*) = k\ (P - P^\*),\\k = \frac{3}{4} \frac{1}{E\_{\sharp f}} \frac{1}{\sqrt{R\_{\sharp f}}} \tag{6}$$

**Figure 4.**

*Indentation load vs. indentation depth response generated using nanoindentation. Insets in the figure highlight the method of indentation in CSM mode and N-CSM mode.*

In the above relationship, the *k* value is constant in the elastic segment [15]. It is worth reiterating here that within the elastic segment, continuously varying *h* equates with *hr* whereas *Reff* to *Ri* (explained in the subsequent Section 6.3). Also, prior understanding of elastic moduli of the material makes the calculation much easier. Essentially, regression analysis on the initial elastic segment of experimentally obtained data helps to calculate the values of *P\** and *h\** and thereby the effective *P-h* data is estimated.

#### *6.2.3 Selection of data segment*

The above-mentioned data correction procedures for nanoindenter with *CSM* or *N-CSM* mode, ideally has to be performed on the initial elastic segment of the *P-h* segment. Such elastic segment dwells within few nanometers, in reality. The exact value of this elastic segment however varies with the sharpness (or bluntness) of the indenter tip and the associated variation in strain gradient [2]. The question here is how to precisely choose a segment in the *P-h* curve which can be used for the data correction using Eqs. (5) and (6). This can be realized through the iteration process on the initial segment with a different depth limit. For instance, for nanoindentation with *hmax* of 250 nm, regression analysis has to be performed in initial segments with indentation depth of 10 nm (or any other limit) to higher. By doing so, the accurate point of transition from elastic to plastic (data limit) can be approximately finalized based on the continuity nature observed in the effective *P-h* curve as well as the corresponding *σind - εind* curve (explained in next Section 6.3).

#### **6.3 Conversion of effective** *P***-***h* **curve to** *σind* **-** *εind* **curve**

As explained in Section 5, the Hertzian relation has provided a basis to obtain *σind* - *εind* curve from the *P-h* response. Once the effective *P-h* response is computed using the steps mentioned in Section 6.2, Eqs. (3) and (4) are used for obtaining the corresponding *σind* - *εind* values. In this conversion process, estimating the continuously varying *a* is important for calculating the continuous evolution in the *σind* and *εind* values. It is particularly evident from Eq. (4) that, *a* is the main characterizing parameter to obtain the *σind* and *εind* values.

Prior to going through further details, the physical significance of *a* and the mechanisms behind its alteration during indentation are explained through **Figure 5**. The figure schematically shows the indentation behavior of different materials with different extents of elastic–plastic activities. Sample-1 with green color indicates the material with full elastic recovery. Sample-2 (orange) and sample-3 (blue) exhibit the indentation behavior of two materials with different degrees of plastic activities along with elastic deformation. In a fully elastic material (sample-1), the indented surface recovers the whole depth upon the complete removal of load. Thereby *Rs* attains infinity in this case (see **Figure 5**). So, *Reff* = *Ri* for material with full depth recovery (see equation (2)). Similarly, owing to the full recovery, continuously recording *h* signal can equate with the depth recovery (*hr*). In short, *Reff* = *Ri* and *h* = *hr* within the elastic regime of material upon indentation.

But, once the dislocation mediated plastic activity is instigated, *Rs* attains a finite value. The orange and light blue colors in **Figure 5** reveal the formation of finite values of *Rs* in the materials due to the occurrence of plastic deformation. In these two cases, *Reff* is no longer equal to *Ri.* It is reported that *Reff* is significantly larger than *Ri* once plastic deformation initiates in the material. Almost a 100-fold increment in the *Reff* is reported with presence of plastic activity in aluminum sample [3].

*Characterizing Stress-Strain Behavior of Materials through Nanoindentation DOI: http://dx.doi.org/10.5772/intechopen.98495*

#### **Figure 5.**

*Schematic representation of the nature of deformation volume beneath the indenter tip for materials with three different degree of elastic-plastic property. Green line shows the sample with full depth recovery. The orange and light blue colors reveal the indentation behaviour of samples with different shares of elastic and plastic activities.*

All these physical changes are also related to *hr* after the unloading. While comparing sample-2 and sample-3, depth recovery is noted to be higher for the former. Correspondingly, *Rs* in the material also changes. It is apparent from the **Figure 5** that *Rs-1* > *Rs-2* and *hr-1 > hr-2.* As a net effect of change in *Reff* and *hr*, contact between the indenter and sample deflects. This is reciprocated in the changes in *a* (*a1 > a2*). In conclusion, all three parameters are correlated which are primarily controlled by the share of elastic–plastic activities within the material of interest. Eq. (4) derived by Hertz relates all these physical phenomena and in the present scenario, it is utilized to estimate *σind*-*εind* curve using relation (3).

#### *6.3.1* σind*-*εind from CSM *nanoindenter*

Estimation of *a* from nanoindentation using *CSM* mode is straight forward. The interrelation between *S* and *a* are derived from Eqs. (1) and (4) according to the Hertzian theory, as shown below:

$$\frac{d\mathcal{P}}{dh} = 2 \, E\_{\epsilon \mathcal{U}} \mathcal{R}\_{\epsilon \mathcal{U}}^{1/2} h\_{\epsilon}^{1/2} = 2 \, E\_{\epsilon \mathcal{U}} a$$
 
$$a = \frac{\mathcal{S}}{2 \, E\_{\epsilon \mathcal{U}}}$$

The analytical significance of this mathematical derivation lies in the fact that unlike Tabor's approach, this expression (eq. (7)) enables to assess the nature of deformation inside the material without a visual inspection of residual impression.

In the data analysis, once the evolving values of *a* are established using eq. (7), the final *σind* - *εind* curve is generated from Eq. (3). **Figure 6(a)** shows *σind* - *εind* curves obtained before and after *ZPC* on experimental *P-h* data. Interestingly, in this novel protocol, elastic moduli measured from the loading and unloading segments of the *σind* - *εind* curve are noted to resemble each other [16]. This observation has validated the new definition for *εind* as well as the novel protocol for reliably assessing the mechanical property via *σind* - *εind* curve.

**Figure 6.**

*(a) σind* � *εind curve obtained before and after the zero-point correction. (b) Schematic representation of the P-h responses with multiple unloading segments for generating σind* � *εind curve in N-CSM measurement nanoindenter.*

#### *6.3.2* σind*-*εind from *non-*CSM *nanoindenter*

As compared to the *CSM* mode, experimentation and method of analysis is different in case of *N-CSM* mode of nanoindentation. In *N-CSM* mode, multiple unloading segments are introduced into the indentation test for measuring the evolution in *a* and thereby the continuous variation in *σind* and *εind* values. This is similar to Field and Swan approach in terms of experimentation. **Figure 6(b)** schematically shows the *P-h* curve obtained after the multiple unloading. Once the effective *P-h* curve is generated by employing *ZPC*, following Section 6.2, the evolving values of *a* are estimated from each segment. For the same, *Reff* value is estimated by fitting the unloading response using the modified Hertz relation as mentioned below,

$$h\_r = h\_{\text{max}} - h\_p = k \, P^{2/3} \tag{8}$$

here *k* is a function of *Reff* and *Eeff* (see relation (6)). *Eeff* can be traced from the prior understanding of elastic moduli of sample or from the initial elastic segment in the *P-h* curve [15]. So, from the understanding of *k* value of the respective alloys and the recorded value of *hp* with reduction in indentation load in the unloading segments, *Reff* is estimated by fitting using the relation (8). Once *Reff* is established, *a* can be determined from relation (4) and in turn *σind* - *εind* curve can be generated using Eq. (3). It is also important to note here that, number of data points in the resultant *σind* - *εind* curve depends on the number of unloading segments provided in the experiment.

#### **7. Protocol for** *σind* **-** *εind* **generation in pseudoelastic shape memory alloys**

Previous sections have elaborated the potential of the nanoindentation technique in appreciating the *σind* - *εind* characteristics of traditional elastic–plastic metallic systems. In a further extension, *Sujith* and *Sen* have revealed the capability of nanoindentation in assessing the unique pseudoelastic (or superelastic) properties of shape memory alloys (*SMA*) *via σind* - *εind* curve [2, 6]. This recent development has succeeded in the producing the specialized stress - strain characteristics of the pseudoelastic NiTi system using most commonly used *N-CSM* nanoindenter.

It is noteworthy at this point that as compared to the traditional elastic–plastic metallic alloys, pseudoelastic system is different owing to the occurrence of reversible stress-induced martensitic transformation (*SIMT*). In pseudoelastic alloys

*Characterizing Stress-Strain Behavior of Materials through Nanoindentation DOI: http://dx.doi.org/10.5772/intechopen.98495*

(some examples of metallic systems are NiTi, Cu-Al-Zn, Cu-Al-Ni, Ni-Ti-Fe, Fe-Mn-Si, Fe-Mn-Si-Co-Ni), parent austenitic phase transforms to product martensitic phase upon the application of stress and it reverts to the previous austenite with the release of stress. Owing to this reversible *SIMT* along with usual elastic deformation in the parent and product phase, the NiTi system in pseudoelastic state shows (8– 10) % of recoverable strain. This is also reflected as a unique characteristic in the conventional uni-axial stress - strain curve. Hence, evaluating such unique property using nanoindentation requires special attention in terms of (a) optimizing indentation parameters as well as (b) tailored *σind* - *εind* generation protocol. This investigation by *Sujith* and *Sen* is the first of its kind to consider spherical indenter tips with varying *Ri* as well as *Pmax* levels with the aim to identify the optimum combination to precisely evaluate localized pseudoelasticity in *SMA* through nanoindentation. Following steps are briefed:

#### **7.1 Optimizing indentation parameters**

For optimizing the indentation parameters, a detailed analysis is performed on the *P-h* curve obtained from various indenter configuration (*Ri* of 10 μm, 20 μm and 50 μm) as well as *Pmax* (1 mN to 7 mN). Details of the experiments and analysis procedures are reported elsewhere [2]*.* However, the key observations in this method of analysis are mentioned here.

Optimization of indentation parameters is performed based on the close scrutiny of the experimentally generated *P-h* curve using Hertzian theoretical prediction and the understanding of the pseudoelastic behavior in the alloy system. **Figure 7(a)** shows the method of analysis performed on the *P-h* curve. The black solid and the red dashed curves in **Figure 7(a)** show the experimental results and Hertzian theoretical predication of indentation response, respectively. Using this comparison, overall deformation mode in the indentation is parted into different sections. Correspondingly, the depth of indentation, specifically influenced by pseudoelasticity is assessed. This can be even better appreciated from **Figure 7(b)**. Physical variation associated with indentation volume of NiTi sample using *Ri* of 10 μm and 20 μm are schematically (in two halves) shown in **Figure 7(b)**. The region influenced by reversible *SIMT* is highlighted as green color in the schematics. This novel method of analysis is performed using a range of combination of indentation parameter. The most adequate combination to assess the pseudoelasticity is identified based on the share of reversible *SIMT* activity and the overall depth recoverability (minimum 90% depth recovery). Based on this systematic analysis, spherical indenter with *Ri* of 20 μm and *Pmax* of 5 mN is noted to be most suitable combination for appreciating pseudoelasticity devoid of the influence of dominant plasticity, in NiTi system.

#### **7.2** *σind* **-** *εind* **protocol**

Considering the extremely high depth recoverability (≥ 90%) of pseudoelastic NiTi system, following assumption is used while generating the corresponding *σind* - *εind* curves,

$$R\_{\it eff} = R\_i \text{ and } h\_r = h \tag{9}$$

Section 6.3 has already mentioned about the validity of this assumption when material shows full depth recovery. In the present scenario, same assumption is used with depth recovery limit of 90% of the *hmax*. This assists in converting the *P-h* response into *σind - εind* curve using Eq. (3), while employing relations (9) in it. Essentially, this new protocol defined the *εind* and *a* using following relation,

#### **Figure 7.**

*(a) The P-h response of pseudoelastic NiTi system at its optimized nanoindentation parameter condition (Ri* ¼ 20 *μm and Pmax* ¼ 5 *mN). Red dotted curve shows the theoretical prediction of P-h response and the green dotted region infers the region that is dominantly influenced by reversible SIMT in the NiTi alloy. Ptr in the graph highlights the indentation load at which SIMT initiates in the material. (b) Schematic representation of share of different deformation mechanisms within the nanoindentation volume for pseudoelastic NiTi system indented using Ri of 10 μm and 20 μm. (c) σind* � *εind curve corresponding to the P-h response (Figure 7(a)). Here, σtr is the transformation stress to initiate SIMT [2].*

$$
\varepsilon\_{ind} \approx \frac{h}{2.4 \, a} \, \left. a \right| = \sqrt{R\_i \, h} \, \tag{10}
$$

Prior to conversation of *P-h* results into *σind* - *εind* curve, *ZPC* is performed following Section 6.2.2. **Figure 7(c)** shows the *σind* - *εind* curve that is generated from the *P-h* response of NiTi system. Interestingly the curve has shown the signature trends of pseudoelastic system like sudden changes in the transformation strength, plateau strain, significant recovery etc. The transformation strength (*σtr*) of the NiTi system estimated from the nanoindentation resembles reasonably well with that derived from uni-axial compression test [2]. This has validated the present protocol for future analysis on smart characteristics of NiTi based shape memory alloys.

#### **8. Closure**

The present chapter elucidates the vast potential of nanoindentation technique to develop insights about the localized stress–strain characteristics of materials. Nevertheless, to achieve the *σind* - *εind* curve, experiments need to be carefully designed. Also, post indention analysis should be meticulously performed to obtain

#### *Characterizing Stress-Strain Behavior of Materials through Nanoindentation DOI: http://dx.doi.org/10.5772/intechopen.98495*

the reliable data. Blunt spherical indenter tip is primarily necessary to activate elastic and plastic mechanisms sequentially in the material and thereby to estimate the *σind* - *εind* curve. On the other hand, different post-indentation analysis has to be adopted based on the mode of nanoindenter and the material of interest to compute the *indentation stress–strain* data. Validation of the protocols are also discussed for pseudoelastic material systems. The detailed explanation provided in the present chapter based on the physical mechanism associated with different alloy system upon indentation and further data analysis can pave the way for future usage of this method of analysis in various studies.

### **Author details**

Indrani Sen1 \* and S. Sujith Kumar1,2

1 Department of Metallurgical and Materials Engineering, Indian Institute of Technology, Kharagpur, India

2 Department of Metallurgical Engineering and Materials Science, Indian Institute of Technology, Bombay, India

\*Address all correspondence to: indrani.sen@metal.iitkgp.ac.in

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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### **Chapter 10**
