**3.1. Hysteresis and biasing effects in ferroelastic materials**

Ferroelastic materials undergo a phase transition from a high temperature paraelastic phase to a low temperature ferroelastic phase at the Curie temperature, *T*c. In the ferroelastic phase, there are two or more equilibrium and switchable orientation states, characterized by a different spontaneous strain. A shift from one state to another can occur during the application of a mechanical stress, and in most cases it gives rise to hysteretic stress-strain curves (Fig. 12).[69] In tension-compression loading, the stress at which the strain is zero is called *the coercive stress* (Fig. 12a). The intersection of the two branches of the hysteresis curve is defined as the "*critical stress*", where the system experiences significant changes in the domain pattern.

**Figure 12.** (a) Tension-compression stress-strain curve of a ferroelastic material. Reproduced with permission from Sal‐ je [67]. Copyright 1993, Cambridge University Press. (b) Compressive stress–strain curve of a ferroelastic material.

A number of complex oxides with a perovskite structure, and shape memory alloys exhibiting the martensitic transformation represent two important families of ferroelastic materials. In ferroelastic ceramics, it is difficult to generate the tension-compression hysteresis loops due to their brittleness, and thus ferroelasticity in these systems is preferably investigated by studying compression test curves.[70] A typical compressive stress–strain curve of a ferroelastic material is shown in Fig. 12b. At the beginning of loading, the strain increases almost linearly with stress. Above a critical stress, the stress-strain curve shows a clear nonlinearity due to ferroe‐ lastic domain switching. When most of the switchable domains have been reoriented, the strain will again increase linearly with the stress. The coercive stress is commonly identified with the stress corresponding to the minimum value of the local slope of the stress-strain curve. It should be mentioned that in polar ferroelastic systems, where the spontaneous strain reorien‐ tation is also accompanied by polarization changes, the mechanical response depends on the electrical boundary conditions. In particular, higher stiffness is usually observed in open circuit conditions.

### **3.2. Perovskite oxides**

Perovskite oxides may exhibit asymmetric deformation under external tensile and compres‐ sive mechanical loading. Fett *et al.* [71] performed bending tests on unpoled and poled soft PZT ceramics and reported that in both cases the region of the samples under tension and compression exhibit different deformation behaviour (Fig. 13).

The compression stress-strain curves of soft and hard PZT ceramics show large differences. Acceptor-doped PZTs, which behave as "hard" ferroelectrics, present an analogous "hard" behaviour under mechanical loading, characterized by a low elastic compliance, a high coercive stress and a large mechanical factor of quality. Cao *et al.* [72] systematically studied the strain and polarization variations of soft and hard PZT ceramics upon unipolar compres‐ sive stress cycles applied parallel and perpendicularly to the poling direction. They found that both hard and soft PZT exhibit a nonlinear stress-strain curve when the stress exceeds a critical value at which ferroelastic domain switching begins to occur (Fig. 14). From the figure, one can see that the coercive stress of hard PZT is larger than that of the soft composition. In

**Figure 13.** Stress–strain curves for the (a) unpoled and (b) poled PZT ceramics. After [71].

**Figure 12.** (a) Tension-compression stress-strain curve of a ferroelastic material. Reproduced with permission from Sal‐ je [67]. Copyright 1993, Cambridge University Press. (b) Compressive stress–strain curve of a ferroelastic material.

A number of complex oxides with a perovskite structure, and shape memory alloys exhibiting the martensitic transformation represent two important families of ferroelastic materials. In ferroelastic ceramics, it is difficult to generate the tension-compression hysteresis loops due to their brittleness, and thus ferroelasticity in these systems is preferably investigated by studying compression test curves.[70] A typical compressive stress–strain curve of a ferroelastic material is shown in Fig. 12b. At the beginning of loading, the strain increases almost linearly with stress. Above a critical stress, the stress-strain curve shows a clear nonlinearity due to ferroe‐ lastic domain switching. When most of the switchable domains have been reoriented, the strain will again increase linearly with the stress. The coercive stress is commonly identified with the stress corresponding to the minimum value of the local slope of the stress-strain curve. It should be mentioned that in polar ferroelastic systems, where the spontaneous strain reorien‐ tation is also accompanied by polarization changes, the mechanical response depends on the electrical boundary conditions. In particular, higher stiffness is usually observed in open circuit

Perovskite oxides may exhibit asymmetric deformation under external tensile and compres‐ sive mechanical loading. Fett *et al.* [71] performed bending tests on unpoled and poled soft PZT ceramics and reported that in both cases the region of the samples under tension and

The compression stress-strain curves of soft and hard PZT ceramics show large differences. Acceptor-doped PZTs, which behave as "hard" ferroelectrics, present an analogous "hard" behaviour under mechanical loading, characterized by a low elastic compliance, a high coercive stress and a large mechanical factor of quality. Cao *et al.* [72] systematically studied the strain and polarization variations of soft and hard PZT ceramics upon unipolar compres‐ sive stress cycles applied parallel and perpendicularly to the poling direction. They found that both hard and soft PZT exhibit a nonlinear stress-strain curve when the stress exceeds a critical value at which ferroelastic domain switching begins to occur (Fig. 14). From the figure, one can see that the coercive stress of hard PZT is larger than that of the soft composition. In

compression exhibit different deformation behaviour (Fig. 13).

conditions.

**3.2. Perovskite oxides**

224 Ferroelectric Materials – Synthesis and Characterization

addition, hard PZT can recover most of their nonlinear strain upon unloading, whereas the nonlinear strain of soft PZT is mostly unrecoverable (Fig. 14a). Similar behaviour can be also observed in the stress-depolarization curves (Fig. 14b). It is generally accepted that the main variations of both strain and polarization in ferroelectric/ferroelastic materials under mechan‐ ical stress result from the switching of 90° domains. When the stress is perpendicular to the poling direction, the stress-strain curves show lower strain values (Fig. 14c) due to a smaller amount of domain reorientation compared to the parallel case. Hard PZT has a more stable domain structure and thus the strain and the polarization changes, induced by the applied mechanical stress, are smaller than those of the soft PZT ceramics (Fig. 14c). Marsilius *et al.* [73] studied the stress-strain behaviour of the soft and hard ferroelectric/ferroelastic ceramics at different temperatures and found that hard ceramics have a larger coercive stress and a larger coercive field than soft ceramics. In addition, they demonstrated that the difference between the coercive mechanical stresses of soft and hard materials is much larger than the difference between their coercive electric fields. This suggests that the effect of doping on stress-induced ferroelastic switching is greater than the effect of doping on the domain switching induced by an external electrical field. For mechanical stresses above 200 MPa, both the soft and hard ceramics showed similar mechanical behaviour, indicating that above a certain threshold the hardening mechanisms of doping can be eliminated by the switching of the defect dipoles in the direction perpendicular to the loading direction. Similar hardening effects were also observed in the ferroelastic non-polar LaCoO3 ceramics after a partial substitution of La with the acceptor ions of Ca, which determined a suppression of domain wall movement within the sub-coercive stress region [74] and an increase of the coercive stress [75].

Picht *et al.* [76] measured the stress-strain curves of unpoled BaTiO3 in short circuit conditions at different temperatures. As shown in Fig. 15, at room temperature BaTiO3 displays a compressive stress-strain curve typical of a ferroelastic material. The remanent strain and the hysteresis area decrease with increasing temperature. When the stress-strain curves were measured at the Curie point (T = 127 °C) and at a slightly higher temperature (e.g., T = 130 °C), a double stress-strain loop was observed (Fig. 15). The hysteresis disappeared when the

**Figure 14.** (a) Stress-strain curves of the soft and hard PZT ceramics under compression with the applied stress parallel to the polarization. (b) Depolarization vs. compressive stress curves of the hard and soft PZT ceramics. (c) Stress-strain curves of the soft and hard PZT ceramics under compression with the stress applied perpendicular to the polarization. Reproduced with permission from Cao and Evans [72]. Copyright 1993, Willey & Sons, Inc.

temperature was increased above 138 °C. The presence of the double loop above the Curie point was ascribed to a stress-induced paraelastic-to-ferroelastic phase transition.

**Figure 15.** Stress-strain curves of BaTiO3 ceramics at different temperatures. Reproduced with permission from Picht *et al*. [76]. Copyright 2012, AIP Publishing LLC.

### **3.3. Shape memory alloys**

Shape memory effect refers to a shape recovery of ferroelastics during heating at high tem‐ peratures after a deformation process induced at a lower temperature. The most important class of materials exhibiting a strong shape memory effect is represented by Shape Memory Alloys (SMAs). Plietsch *et al.* [77] have studied the stress-strain curves of the NiTi alloys subjected to either tension or compressive stress at different temperatures. They found that the pseudoelastic and martensitic NiTi phases show a pronounced asymmetry in stress-strain behaviour (Fig. 16). The asymmetry in the martensitic phase was ascribed to the different strength under tension and compression, which is likely caused by the presence of internal cracks, residual stresses or due to the Bauschinger effect.

**Figure 16.** Tension/compression hysteresis for NiTi: (a) austenitic, (b) pseudoelastic, and (c) martensitic alloy. These images were published in Plietsch and Ehrlich [77]. Copyright 1997, Elsevier.

Liu *et al.* [78] have systematically studied tension-compression behaviour of martensitic NiTi under both monotonic and cyclic tensile/compressive loading. They found that the deforma‐ tion mechanisms of the martensite phase are different under tension and compression, as evidenced by the asymmetric stress-strain curves (Fig. 17). Transmission electron microscopy (TEM) observations revealed that in the non-deformed specimen the martensite variants are well self-accommodated through the martensite twinning. Under tension, the interfaces between two variants are mobile and can migrate under stress. On the other hand, no migration of the junction planes between the neighbouring martensite plates was observed in samples compressed up to 4% strain. Instead, a high density of lattice defects, mainly dislocations, was observed inside both the martensite twin bands and twin boundaries. It was suggested that in the martensite phase the deformation mechanism under tension up to 4% may be predomi‐ nantly related to the migration of variant interfaces, while under compression up to 4%, the strain is likely caused by the generation and movement of lattice defects, mostly dislocations.

### **3.4. The Bauschinger effect**

temperature was increased above 138 °C. The presence of the double loop above the Curie

**Figure 14.** (a) Stress-strain curves of the soft and hard PZT ceramics under compression with the applied stress parallel to the polarization. (b) Depolarization vs. compressive stress curves of the hard and soft PZT ceramics. (c) Stress-strain curves of the soft and hard PZT ceramics under compression with the stress applied perpendicular to the polarization.

**Figure 15.** Stress-strain curves of BaTiO3 ceramics at different temperatures. Reproduced with permission from Picht *et*

Shape memory effect refers to a shape recovery of ferroelastics during heating at high tem‐ peratures after a deformation process induced at a lower temperature. The most important class of materials exhibiting a strong shape memory effect is represented by Shape Memory

*al*. [76]. Copyright 2012, AIP Publishing LLC.

226 Ferroelectric Materials – Synthesis and Characterization

**3.3. Shape memory alloys**

point was ascribed to a stress-induced paraelastic-to-ferroelastic phase transition.

Reproduced with permission from Cao and Evans [72]. Copyright 1993, Willey & Sons, Inc.

The Bauschinger effect refers to the phenomenon by which the yield stress of metals reduces in the direction opposite to that of the very first stress applied.[79] This effect is believed to play a dominant role in the asymmetrical strain-stress behaviour of both monotonically and cyclic deformed metals. The effect is closely related to the presence of a long range internal stress (LRIS). The concept of LRIS assumes that the variations of the local stresses due to the ons.

have systematica and cyclic tensi e phase are diffe ves (Fig. 17). Tr med specimen t ning. Under tens On the other han es was observed mainly dislocati was suggested th

ally studied tens ile/compressive erent under tens ransmission elec the martensite sion, the interfa nd, no migratio d in samples com ions, was obser hat in the marten

sion-compression loading. They fo ion and compres ctron microscopy variants are w ces between two on of the juncti mpressed up to rved inside both nsite phase the d o the migration ed by the genera

n behaviour of m ound that the def ssion, as evidenc y (TEM) observ well self-accom o variants are m ion planes betw 4% strain. Inste h the martensite deformation mec n of variant in ation and movem

martensitic NiTi formation mech ced by the asym vations revealed mmodated throug mobile and can m ween the neighb ead, a high den e twin bands an chanism under t nterfaces, while ment of lattice d

i under hanisms mmetric that in gh the migrate bouring nsity of nd twin tension under defects,

L b o s t m u m l b

Liu *et al.* [78] h both monotonic of the martensit stress-strain curv the non-deform martensite twinn under stress. O martensite plate lattice defects, m boundaries. It w

m

mostly dislocati

t **3** tension-compres **3.4 The Bausch** ssion cyclic load **hinger effect**  ding within 4% s strain. After [78] . **Figure 17.** Stress-strain curves of a NiTi SMAs under (a) a monotonic tension and compression and (b) tension-com‐ pression cyclic loading within 4% strain. After [78].

applied stress occur over the long length scales. After plastic deformation have been produced, high dislocation density regions, such as sub-grain boundaries, and low dislocation density regions, such as cell interiors, characterized by a different yield stress are formed in the system. Under mechanical loading, the internal stress of the high-dislocation density regions is higher than that in low-dislocation density regions. Upon unloading, the average internal stress is zero; at the same time, the stress in the high dislocation density regions is positive, while it is negative in the low dislocation density regions. As a consequence, during stress reversal (applied stress changes sign), plasticity occurs in the low dislocation density regions at a lower stress and gives rise to the observed lower yield stress. T d r T L l s d h u d The Bauschinge direction opposi role in the asym The effect is clo LRIS assumes t length scales. A sub-grain bound different yield s high-dislocation unloading, the a density regions er effect refers to ite to that of the mmetrical strainosely related to that the variation fter plastic defor daries, and low stress are formed n density region average internal is positive, w o the phenomen very first stress -stress behaviou the presence of ns of the local s rmation have be dislocation dens d in the system. ns is higher th l stress is zero; while it is nega non by which th applied.[79] Thi ur of both mono f a long range i stresses due to t een produced, hi sity regions, suc . Under mechan han that in low at the same tim ative in the low e yield stress of is effect is believ tonically and cy internal stress (L the applied stres gh dislocation d ch as cell interio nical loading, the w-dislocation d me, the stress in w dislocation d f metals reduces ved to play a do yclic deformed m LRIS). The conc ss occur over th density regions, s ors, characterize e internal stress density regions. n the high dislo density regions. s in the ominant metals. cept of he long such as ed by a s of the Upon ocation . As a

> stress changes gives rise to the

 sign), plasticit e observed lower

ty occurs in th r yield stress.

he low

versal (applied

#### **3.5. Microscopic mechanisms** d dislocation dens sity regions at a l lower stress and

during stress rev

c

consequence, d

In SMAs, the defects that might be present in the system, such as vacancies and solute atoms, tend to distribute following a short-range order symmetry, which will comply with the crystal symmetry after aging treatments.[80] In the aged martensite (Fig. 18a), the short-range order symmetry of defects conforms with the crystal symmetry of the martensite phase. During heating, the short-range ordering of defects does not change abruptly into a cubic symmetry at temperatures above the diffusionless martensitic transformation, because the diffusion of defects is a slow process (Fig. 18b). After aging, the short-range order symmetry of defects becomes conformed to the cubic crystal symmetry of the austenite parent phase (Fig. 18c). This phenomenon is referred to as the symmetry-conforming property of point defects in ferroe‐ lastic systems.[80] It should be mentioned that the short-range order of defects with martensitic symmetry can slightly distort the cubic lattice towards the martensitic symmetry, when the martensite is quickly heated up to the cubic phase. Such a lattice difference will produce a short range order-induced domain pattern in the cubic phase identical to the martensitic domain pattern. Thus, the symmetry property of point defects can give rise to the aginginduced two-way shape memory effect, namely the one observed in the aged martensitic phase of Au51Cd49 SMA.[80] **3** I d a c o **3.5 Microscopic** In SMAs, the de distribute follow aging treatment conforms with th of defects does **c mechanisms**  efects that might wing a short-rang s.[80] In the ag he crystal symm not change abr t be present in th ge order symmet ged martensite (F metry of the mart uptly into a cub 20 he system, such try, which will c Fig. 18a), the sh tensite phase. Du bic symmetry at as vacancies and comply with the hort-range order uring heating, th t temperatures a nd solute atoms, crystal symmetr r symmetry of d he short-range or above the diffus tend to ry after defects rdering sionless

**Figure 18.** Predicted microstructure and short range order changes during reverse martensitic transformation from symmetry conforming short range order principle. (a) Aged martensite in which short range order symmetry conforms to the crystal symmetry of martensite. (b) Parent phase immediately after diffusionless transformation from (a). (c) Pa‐ rent phase after aging that allows for the short range order symmetry of defects to conform to the cubic symmetry of the parent phase. After [80].
