FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric Surfaces

Duc-Nam Nguyen

### Abstract

The elastic deformation machining method is suitable for fabricating aspheric surfaces that have excellent physical properties of elastic materials. The machining process is carried out with the deformation model without mold. When vacuum pressure is supplied to the workpiece, the top surfaces of workpiece are deformed into aspheric shape. After machining process, the bottom surface will be formed into the aspheric shape and the top surface returns to its original flat surface form due to internal force and bending moments of the material. However, the accuracy will decrease due to the reduced thickness while the vacuum pressure keeps unchanged during machining process. Therefore, it is necessary to carry out the finite element analysis (FEA) to determine the vacuum pressure with corresponding to the reduced thickness. In addition, the mold with its surface approximates and the desired surface form of the lens is also presented. When uniform vacuum pressure is supplied to the workpiece through small holes of the mold, the workpiece will be deformed into aspheric profile as similar to the mold surfaces. In order to improving the form accuracy, the FEA and the experiment are studied for modifying the mold profile to correspond with bending strength of workpiece material.

Keywords: elastic deformation, aspheric surface, glass lapping, glass molding, vacuum pressure, experimental study, finite element analysis

#### 1. Introduction

Nowadays, for managing laser light in sophisticated and compact laser systems, aspheric lenses are the most powerful lenses. In these systems, it is generally accepted that spherical aberration is the most common performance detractor. From the use of spherical surfaces, it is found that they artificially limit focusing and collimating accuracy. In spite of the fact that spherical geometry is not optimal for refracting light that has been known for centuries, the high cost and difficulty of fabricating nonspherical (aspheric) surfaces has inhibited them from a wider use.

Because aspheric surfaces offer advantages such as high resolution, light weight, and low cost, they are widely used in the opto-electronics industry. As aspheric surfaces are more effective in shaping the light than spherical surfaces, they have

recently been used in measurement instruments, astronomy, and optical lens [1, 2]. Figure 1 shows some applications which employ aspheric surfaces.

In most general terms, an optical lens can be determined as a refracting device that reconfigures the light wave front incident upon it. The phase, direction of propagation, intensity, and polarization state are the properties of the incident light beam which are influenced by a lens. Surface form and roughness, diameter, subsurface defects generated during the fabrication process, shape accuracy, physical and mechanical properties of the optical material, and other optical conditions, such as the angle of incidence of light beam, absorption, reflection, and environmental influences, are some of the major characteristics that govern the performance of an optical lens [3].

To overcome the aberration problems of spherical lenses, a number of spherical surfaces with different signs of aberrations have to be utilized to balance and minimize the final aberration to obtain high quality images. In principle, the optical system designer can always use enough spherical lenses to simultaneously correct for all of the common optical aberrations in a lens system if the number of elements used in an optical system is not limited. The number of surfaces required to do this may be so large that the resulting lens assembly is excessively large in size and weight, and expensive to produce. In addition, the transmission of the assembly lens may be unduly reduced due to the residual reflections from each surface, and the bulk absorption in each lens.

The usage of aspheric surfaces, both with and without the incorporation of diffractive elements, allows the design and construction of assembly lens with the same or even better optical performance than an equivalent all-spherical system. However, in most cases, with a significant reduction in the number of elements required, there is a significant improvement in the overall lens assembly size, weight, cost, and optical transmission. In many cases, in an optical system, each aspheric surface can be applied to replace at least two other spherical surfaces. Hence, aspheric lenses are more efficient because additional error-correcting lenses are not required. Figure 2 is an illustration of spherical and aspheric lens systems.

Cutting techniques such as turning and milling processes are usually utilized for the production of aspheric glass lenses as shown in Figure 3.

The machining processes, which usually consist of computer numerically controlled (CNC) generators, are employed to machine an aspheric shape on a lens to generate the desired shape. In glass machining, the roughness on a cutting edge has a larger effect on surface finish than that of metal machining. Glass workpiece can be machined without brittle fractures with an undeformed chip thickness less than 1 μm in milling and turning processes [4–7].

> Thereafter, the optical lenses are fine machined by grinding, and then followed by polishing to achieve the good surfaces. In the grinding process, if the depth of cut is below a certain value, the material removal mode is ductile flow which is characterized by low surface roughness and subsurface damage [8–12]. Figure 4 is an

> FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric…

"Precessions" polishing is an automated polishing method that uses a 7-axis CNC machine tool for polishing spherical and aspheric surfaces [13, 14]. Based on contact between the workpiece surface and polishing tool, the polishing spot of desired size is generated by controlling the load cell in polishing process. The polishing tool then moves in angular steps around the local normal to the part surface during machining process. The 7-axis CNC capability of the machine also makes the generation of free-form surfaces possible. Figure 5 shows a schematic illustration of a "Preces-

illustration of a precision grinding process.

Schematic illustration of milling and turning processes.

sions" polishing process.

35

Figure 2.

Figure 3.

Spherical vs. aspheric lens systems.

DOI: http://dx.doi.org/10.5772/intechopen.79402

Figure 1. Application of aspheric surfaces: (a) measurement instruments, (b) astronomy, and (c) optical lens.

FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric… DOI: http://dx.doi.org/10.5772/intechopen.79402

Figure 2. Spherical vs. aspheric lens systems.

recently been used in measurement instruments, astronomy, and optical lens [1, 2].

In most general terms, an optical lens can be determined as a refracting device that reconfigures the light wave front incident upon it. The phase, direction of propagation, intensity, and polarization state are the properties of the incident light beam which are influenced by a lens. Surface form and roughness, diameter, subsurface defects generated during the fabrication process, shape accuracy, physical and mechanical properties of the optical material, and other optical conditions, such as the angle of incidence of light beam, absorption, reflection, and environmental influences, are some of the major characteristics that govern the performance of an

To overcome the aberration problems of spherical lenses, a number of spherical

surfaces with different signs of aberrations have to be utilized to balance and minimize the final aberration to obtain high quality images. In principle, the optical system designer can always use enough spherical lenses to simultaneously correct for all of the common optical aberrations in a lens system if the number of elements used in an optical system is not limited. The number of surfaces required to do this may be so large that the resulting lens assembly is excessively large in size and weight, and expensive to produce. In addition, the transmission of the assembly lens may be unduly reduced due to the residual reflections from each surface, and

The usage of aspheric surfaces, both with and without the incorporation of diffractive elements, allows the design and construction of assembly lens with the same or even better optical performance than an equivalent all-spherical system. However, in most cases, with a significant reduction in the number of elements required, there is a significant improvement in the overall lens assembly size, weight, cost, and optical transmission. In many cases, in an optical system, each aspheric surface can be applied to replace at least two other spherical surfaces. Hence, aspheric lenses are more efficient because additional error-correcting lenses are not required. Figure 2 is an illustration of spherical and aspheric lens systems. Cutting techniques such as turning and milling processes are usually utilized for

The machining processes, which usually consist of computer numerically controlled (CNC) generators, are employed to machine an aspheric shape on a lens to generate the desired shape. In glass machining, the roughness on a cutting edge has a larger effect on surface finish than that of metal machining. Glass workpiece can be machined without brittle fractures with an undeformed chip thickness less than

Application of aspheric surfaces: (a) measurement instruments, (b) astronomy, and (c) optical lens.

the production of aspheric glass lenses as shown in Figure 3.

1 μm in milling and turning processes [4–7].

Figure 1.

34

Figure 1 shows some applications which employ aspheric surfaces.

Elasticity of Materials ‐ Basic Principles and Design of Structures

optical lens [3].

the bulk absorption in each lens.

Figure 3. Schematic illustration of milling and turning processes.

Thereafter, the optical lenses are fine machined by grinding, and then followed by polishing to achieve the good surfaces. In the grinding process, if the depth of cut is below a certain value, the material removal mode is ductile flow which is characterized by low surface roughness and subsurface damage [8–12]. Figure 4 is an illustration of a precision grinding process.

"Precessions" polishing is an automated polishing method that uses a 7-axis CNC machine tool for polishing spherical and aspheric surfaces [13, 14]. Based on contact between the workpiece surface and polishing tool, the polishing spot of desired size is generated by controlling the load cell in polishing process. The polishing tool then moves in angular steps around the local normal to the part surface during machining process. The 7-axis CNC capability of the machine also makes the generation of free-form surfaces possible. Figure 5 shows a schematic illustration of a "Precessions" polishing process.

#### Figure 4.

Schematic illustration of a precision grinding process.

the pressed load for a short time at a slow cooling rate, the stress in the glass lens is relaxed. Lastly, the formed glass lens is rapidly cooled to ambient temperature and released from the molds (Figure 6d). A BK7 glass fabricated using this molding process has surface roughness of approximately 5 nm Ra, and form accuracy of

Schematic illustration of a lens molding process: (a) Molds and glass gob, (b) Heating, (c) Heating and

FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric…

In spite of the obvious advantages, there are serious drawbacks that currently limit the application of injection molding and glass molding technologies to smaller size aspheric lens fabrications. A typical drawback is the altering of optical properties such as refractive index, due to heating and annealing of the glass material, and the uneven shrinking due to the cooling process that causes error in lens profile [18]. In contrast, the elastic deformation machining method is a good technique that the workpiece will be deformed into aspheric shape prior to the lapping process under vacuum pressure. While the vacuum pressure is remained, the opposite side is polished to optical flatness by the lapping wheel. When the vacuum pressure is released, the bottom surface of the workpiece will be shaped into an aspheric shape and the top surface will restore to its flat surface form by internal force and bending moments. Consequently, for machining materials which have excellent physical properties due to their perfect crystal structures, the elastic deformation method is

Based on the elasticity of the material, the circular flat plate is deformed to an aspheric surface by applying the pressure in the elastic deformation machining method. The deflection of the circular plate can be calculated by using appropriate plate theory. There are two types of edge support for circular plate, such as fixed (or clamped) edge and simply supported edge which are considered in this section.

0.2 μm P-V.

Figure 6.

pressing, and (d) Cooling and release.

DOI: http://dx.doi.org/10.5772/intechopen.79402

appropriate [19].

37

2. Theory of elastic deformation

#### Figure 5. Schematic illustration of a "Precessions" polishing process.

To fabricate aspheric surfaces, the movement of the tool must be constrained in the machining process. A sub-aperture tool (smaller in size than the lens) on a modified polishing machine is then utilized, and by controlling the amount of time the tool spends working at a given lens location, a desired aspheric surface can be fabricated. In addition to the complexity of the machining processes, conventional aspheric fabrication is highly sensitive to the manufacturing conditions, which strongly depend on the positioning accuracy of the machine, the condition of the grinding wheel, and the vibrations in the system. These factors result in an expensive manufacturing cost and a low production yield.

Compared to traditional cold-working methods, glass molding and precision injection molding have greatly advanced the fabrication technologies for aspheric lens industry because of their unique advantages such as excellent compatibility, high efficiency, great flexibility, and high consistency [15–17]. The mass production of aspheric glass lenses is fabricated by applying the technologies. In the glass molding technique, a glass lens is fabricated by compressing glass melting at a high temperature and replicating the shapes of the mold without any need of further machining. Figure 6a shows the process begins by putting a glass gob on top of a lower mold. Both the glass gob and the mold are heated to a molding temperature above the transition temperature of glass (Figure 6b). After the glass and the mold temperature have reached a steady state molding temperature, the mold is closed by moving the lower mold (Figure 6c). The temperature is maintained during the molding step. All steps are performed in vacuum environment. Then, by holding

FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric… DOI: http://dx.doi.org/10.5772/intechopen.79402

Figure 6.

To fabricate aspheric surfaces, the movement of the tool must be constrained in

the machining process. A sub-aperture tool (smaller in size than the lens) on a modified polishing machine is then utilized, and by controlling the amount of time the tool spends working at a given lens location, a desired aspheric surface can be fabricated. In addition to the complexity of the machining processes, conventional aspheric fabrication is highly sensitive to the manufacturing conditions, which strongly depend on the positioning accuracy of the machine, the condition of the grinding wheel, and the vibrations in the system. These factors result in an expen-

Compared to traditional cold-working methods, glass molding and precision injection molding have greatly advanced the fabrication technologies for aspheric lens industry because of their unique advantages such as excellent compatibility, high efficiency, great flexibility, and high consistency [15–17]. The mass production of aspheric glass lenses is fabricated by applying the technologies. In the glass molding technique, a glass lens is fabricated by compressing glass melting at a high temperature and replicating the shapes of the mold without any need of further machining. Figure 6a shows the process begins by putting a glass gob on top of a lower mold. Both the glass gob and the mold are heated to a molding temperature above the transition temperature of glass (Figure 6b). After the glass and the mold temperature have reached a steady state molding temperature, the mold is closed by moving the lower mold (Figure 6c). The temperature is maintained during the molding step. All steps are performed in vacuum environment. Then, by holding

sive manufacturing cost and a low production yield.

Figure 4.

Figure 5.

36

Schematic illustration of a precision grinding process.

Elasticity of Materials ‐ Basic Principles and Design of Structures

Schematic illustration of a "Precessions" polishing process.

Schematic illustration of a lens molding process: (a) Molds and glass gob, (b) Heating, (c) Heating and pressing, and (d) Cooling and release.

the pressed load for a short time at a slow cooling rate, the stress in the glass lens is relaxed. Lastly, the formed glass lens is rapidly cooled to ambient temperature and released from the molds (Figure 6d). A BK7 glass fabricated using this molding process has surface roughness of approximately 5 nm Ra, and form accuracy of 0.2 μm P-V.

In spite of the obvious advantages, there are serious drawbacks that currently limit the application of injection molding and glass molding technologies to smaller size aspheric lens fabrications. A typical drawback is the altering of optical properties such as refractive index, due to heating and annealing of the glass material, and the uneven shrinking due to the cooling process that causes error in lens profile [18].

In contrast, the elastic deformation machining method is a good technique that the workpiece will be deformed into aspheric shape prior to the lapping process under vacuum pressure. While the vacuum pressure is remained, the opposite side is polished to optical flatness by the lapping wheel. When the vacuum pressure is released, the bottom surface of the workpiece will be shaped into an aspheric shape and the top surface will restore to its flat surface form by internal force and bending moments. Consequently, for machining materials which have excellent physical properties due to their perfect crystal structures, the elastic deformation method is appropriate [19].

#### 2. Theory of elastic deformation

Based on the elasticity of the material, the circular flat plate is deformed to an aspheric surface by applying the pressure in the elastic deformation machining method. The deflection of the circular plate can be calculated by using appropriate plate theory. There are two types of edge support for circular plate, such as fixed (or clamped) edge and simply supported edge which are considered in this section.

#### 2.1 Basic equations for circular plate in elastic deformation

The amount of deflection of circular plate can be determined by solving the differential equations of an appropriate plate theory [20]. Two types of edge support include clamped and simply supported edge which are used in the elastic deformation method. In the case of simple bending of circular plate, the amount of deflection w is assumed to be very small in comparison with plate thickness. According to the small deflection theory of thin homogenous elastic plates, the deformation in the middle plane of the plate can be neglected and the straight line initially normal to the middle surface to the plate remains straight. In addition, the stress (i.e., transverse normal stress) is small when compared to other stress components and should be neglected in stress-strain relationship. Under these conditions, the three dimensional plate problem can be reduced to two dimensions. The linear theory of elasticity can be used to derive the governing differential equation for a plate subject to uniform transverse loads. The equation for small deformation w of a thin circular plate of constant thickness h is:

$$D.\nabla^2(\nabla^2 \mathbf{w}) - p = 0; D = Eh^3/\mathbf{1} \mathbf{2} \left(\mathbf{1} - \nu^2\right) \tag{1}$$

Multiply both sides of Eq. (6) by r and then integrate to obtain,

By successive integrations, the deflection can arrive finally at

2

The shears for the symmetrically loaded plate can be given as follow,

d2 w dr<sup>2</sup> <sup>þ</sup>

pr 2D þ

wr <sup>¼</sup> pr<sup>4</sup>

The boundary conditions are w ¼ 0 and Mr ¼ 0 at r = a.

pa<sup>2</sup>

<sup>C</sup><sup>2</sup> ¼ � pa<sup>2</sup> 32D

pa<sup>4</sup>

<sup>64</sup> ð Þþ <sup>12</sup> <sup>þ</sup> <sup>4</sup><sup>ν</sup>

3 þ ν 1 þ ν C2

; C<sup>4</sup> <sup>¼</sup> pa<sup>4</sup> 64D

ln r þ C2r

1 r dw dr ! ¼ � <sup>Q</sup>

> C1 <sup>r</sup> ¼ � <sup>Q</sup>

This equation indicates that Q would approach infinity as r approaches zero. To prevent this from happening, we make C<sup>1</sup> = 0. From Eq. (10), it can be seen that w becomes infinity at r = 0. To avoid this, the constant C<sup>3</sup> must be zero. Thus,

<sup>64</sup><sup>D</sup> <sup>þ</sup> <sup>C</sup>2<sup>r</sup>

From Eq. (13), the amount of deflection wr is the function of r in cylindrical

<sup>64</sup><sup>D</sup> <sup>þ</sup> <sup>C</sup>1<sup>r</sup>

d dr

dw dr � � � � <sup>¼</sup> pr<sup>2</sup>

FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric…

dw dr � � � � <sup>¼</sup> pr

2D þ

C1 r

<sup>2</sup><sup>D</sup> <sup>þ</sup> <sup>C</sup><sup>1</sup> (8)

<sup>2</sup> <sup>þ</sup> <sup>C</sup>3ln <sup>r</sup> <sup>þ</sup> <sup>C</sup><sup>4</sup> (10)

<sup>D</sup> (11)

<sup>D</sup> (12)

<sup>2</sup> <sup>þ</sup> <sup>C</sup><sup>4</sup> (13)

<sup>64</sup><sup>D</sup> <sup>þ</sup> <sup>C</sup>2a<sup>2</sup> <sup>þ</sup> <sup>C</sup><sup>4</sup> <sup>¼</sup> <sup>0</sup> (14)

5 þ ν 1 þ ν

<sup>2</sup> ð Þ¼ <sup>1</sup> <sup>þ</sup> <sup>ν</sup> <sup>0</sup> (15)

(16)

(9)

r d dr 1 r d dr <sup>r</sup>

DOI: http://dx.doi.org/10.5772/intechopen.79402

d dr 1 r d dr <sup>r</sup>

wr <sup>¼</sup> pr<sup>4</sup>

2.2 Circular plate with simply supported edge

and from Eq. (9),

coordinate system.

and

39

Eq. (13) can be written,

From the equations, we can find

or,

E and v are the Young's modulus and Poisson's coefficient. D is the rigidity constant of the plate, and p is the load on the plate. Because of the rotational symmetry, the Laplacian operator ∇<sup>2</sup> in polar coordinates r and θ can be written as:

$$\nabla^4 w = \left(\frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + \frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}\right) \left(\frac{\partial^2 w}{\partial r^2} + \frac{1}{r}\frac{\partial w}{\partial r} + \frac{1}{r^2}\frac{\partial^2 w}{\partial \theta^2}\right) \tag{2}$$

The moment can be written in the form,

$$M\_r = -D\left[\frac{\partial^2 w}{\partial r^2} + \nu \left(\frac{\mathbf{1}}{r^2} \frac{\partial^2 w}{\partial \theta^2} + \frac{\mathbf{1}}{r} \frac{\partial w}{\partial r}\right)\right] \tag{3}$$

$$M\_t = -D\left[\frac{1}{r}\frac{\partial w}{\partial r} + \frac{1}{r^2}\frac{\partial^2 w}{\partial \theta^2} + \nu \frac{\partial^2 w}{\partial r^2}\right] \tag{4}$$

where Mr and Mt are radial moment and tangential moment.

If the load acting on the plate is symmetrically distributed about the axis perpendicular to the middle plane of the plate, the deflection w is independent of θ, when Eqs. (3) and (4) becomes:

$$
\left(\frac{d}{dr^2} + \frac{1}{r}\frac{d}{dr}\right)\left(\frac{d^2w}{dr^2} + \frac{1}{r}\frac{dw}{dr}\right) = \frac{p}{D} \tag{5}
$$

In other form, it shown as

$$\left(\frac{d^2w}{dr^2} + \frac{1}{r}\frac{dw}{dr}\right) = \frac{1}{r}\frac{d}{dr}\left(r\frac{dw}{dr}\right) \tag{6}$$

Eq. (5) can be written as

$$\frac{1}{r}\frac{d}{dr}\left\{r\frac{d}{dr}\left[\frac{1}{r}\frac{d}{dr}\left(r\frac{dw}{dr}\right)\right]\right\}=\frac{p}{D}\tag{7}$$

FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric… DOI: http://dx.doi.org/10.5772/intechopen.79402

Multiply both sides of Eq. (6) by r and then integrate to obtain,

$$r\frac{d}{dr}\left[\frac{1}{r}\frac{d}{dr}\left(r\frac{dw}{dr}\right)\right] = \frac{pr^2}{2D} + C\_1 \tag{8}$$

or,

2.1 Basic equations for circular plate in elastic deformation

Elasticity of Materials ‐ Basic Principles and Design of Structures

w of a thin circular plate of constant thickness h is:

D:∇<sup>2</sup> ∇<sup>2</sup>

∂r<sup>2</sup> þ 1 r ∂ ∂r þ 1 r2 ∂2 ∂θ<sup>2</sup>

Mr ¼ �<sup>D</sup> <sup>∂</sup><sup>2</sup>

Mt ¼ �<sup>D</sup> <sup>1</sup>

where Mr and Mt are radial moment and tangential moment.

d dr<sup>2</sup> <sup>þ</sup>

> d2 w dr<sup>2</sup> <sup>þ</sup>

1 r d dr <sup>r</sup>

1 r d dr � � d<sup>2</sup>

> 1 r dw dr

!

d dr 1 r d dr <sup>r</sup>

<sup>∇</sup><sup>4</sup><sup>w</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup>

The moment can be written in the form,

when Eqs. (3) and (4) becomes:

In other form, it shown as

Eq. (5) can be written as

38

The amount of deflection of circular plate can be determined by solving the differential equations of an appropriate plate theory [20]. Two types of edge support include clamped and simply supported edge which are used in the elastic deformation method. In the case of simple bending of circular plate, the amount of deflection w is assumed to be very small in comparison with plate thickness. According to the small deflection theory of thin homogenous elastic plates, the deformation in the middle plane of the plate can be neglected and the straight line initially normal to the middle surface to the plate remains straight. In addition, the stress (i.e., transverse normal stress) is small when compared to other stress components and should be neglected in stress-strain relationship. Under these conditions, the three dimensional plate problem can be reduced to two dimensions. The linear theory of elasticity can be used to derive the governing differential equation for a plate subject to uniform transverse loads. The equation for small deformation

<sup>w</sup> � � � <sup>p</sup> <sup>¼</sup> <sup>0</sup>; D <sup>¼</sup> Eh<sup>3</sup>

� � ∂<sup>2</sup>

w <sup>∂</sup>r<sup>2</sup> <sup>þ</sup> <sup>ν</sup>

> r ∂w ∂r þ 1 r2 ∂2 w <sup>∂</sup>θ<sup>2</sup> <sup>þ</sup> <sup>ν</sup>

E and v are the Young's modulus and Poisson's coefficient. D is the rigidity constant of the plate, and p is the load on the plate. Because of the rotational symmetry, the Laplacian operator ∇<sup>2</sup> in polar coordinates r and θ can be written as:

> w ∂r<sup>2</sup> þ

1 r2 ∂2 w ∂θ<sup>2</sup> þ

If the load acting on the plate is symmetrically distributed about the axis perpendicular to the middle plane of the plate, the deflection w is independent of θ,

> w dr<sup>2</sup> <sup>þ</sup>

> > ¼ 1 r d dr <sup>r</sup>

> > > dw dr

� � � � � �

1 r dw dr

!

� � � �

� �

1 r ∂w ∂r þ 1 r2 ∂2 w ∂θ<sup>2</sup>

1 r ∂w ∂r

> ∂2 w ∂r2

> > ¼ p

dw dr � �

¼ p

<sup>D</sup> (5)

<sup>D</sup> (7)

� �

<sup>=</sup>12 1 � <sup>ν</sup><sup>2</sup> � � (1)

(2)

(3)

(4)

(6)

$$\frac{d}{dr}\left[\frac{1}{r}\frac{d}{dr}\left(r\frac{dw}{dr}\right)\right] = \frac{pr}{2D} + \frac{C\_1}{r} \tag{9}$$

By successive integrations, the deflection can arrive finally at

$$
\Delta w\_r = \frac{pr^4}{64D} + C\_1 r^2 \ln r + C\_2 r^2 + C\_3 \ln r + C\_4 \tag{10}
$$

The shears for the symmetrically loaded plate can be given as follow,

$$\frac{d}{dr}\left(\frac{d^2w}{dr^2} + \frac{1}{r}\frac{dw}{dr}\right) = -\frac{Q}{D} \tag{11}$$

and from Eq. (9),

$$\frac{pr}{2D} + \frac{C\_1}{r} = -\frac{Q}{D} \tag{12}$$

This equation indicates that Q would approach infinity as r approaches zero. To prevent this from happening, we make C<sup>1</sup> = 0. From Eq. (10), it can be seen that w becomes infinity at r = 0. To avoid this, the constant C<sup>3</sup> must be zero. Thus,

$$
\Delta w\_r = \frac{pr^4}{64D} + C\_2r^2 + C\_4 \tag{13}
$$

From Eq. (13), the amount of deflection wr is the function of r in cylindrical coordinate system.

#### 2.2 Circular plate with simply supported edge

The boundary conditions are w ¼ 0 and Mr ¼ 0 at r = a. Eq. (13) can be written,

$$\frac{p a^4}{64D} + \mathcal{C}\_2 a^2 + \mathcal{C}\_4 = 0\tag{14}$$

and

$$\frac{p a^2}{64}(12+4\nu) + \frac{C\_2}{2}(1+\nu) = 0\tag{15}$$

From the equations, we can find

$$\mathbf{C}\_{2} = -\frac{pa^{2}}{32D} \frac{\mathbf{\hat{3}} + \nu}{\mathbf{1} + \nu}; \mathbf{C}\_{4} = \frac{pa^{4}}{64D} \frac{\mathbf{\hat{5}} + \nu}{\mathbf{1} + \nu} \tag{16}$$

so that the deflection of every radial location can be calculated using,

$$aw\_r = \frac{p(a^2 - r^2)}{64D} \left[ \frac{5+\nu}{1+\nu} a^2 - r^2 \right] \tag{17}$$

The maximum deflection which occurs at r = 0, is given by,

$$w\_0 = \frac{pa^4}{64D} \left(\frac{5+\nu}{1+\nu}\right) \tag{18}$$

a hole, the workpiece is deformed in the middle. The edge of the workpiece is supported by the holder; therefore, it will be not moved. This makes the workpiece become a formed aspheric shape as presented in Figure 7b. The deflection of the workpiece can be calculated by using theoretical equation in boundary conditions of circular plate with simply supported edge. While the vacuum pressure is still remained, the workpiece and the holder start rotating and moving downward in contact with the lapping plate. Its opposite side will be polished to optical flatness as illustrated in Figure 7c and d. Then, the vacuum pressure is not supplied and the workpiece is also released from the holder as shown in Figure 7e. According to the Figure 7f, the bottom surface will be formed into the aspheric shape and the top surface returns to its original flat surface form due to material elasticity. It can be seen that the deformed workpiece surfaces can be restored by internal force and bending moments which are created from the vacuum pressure during machining

FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric…

3.1 Finite element analysis for elastic deformation machining process

The vacuum pressure affects an amount of elastic deformation of the workpiece; hence, the accuracy of manufactured profile will also be highly dependent on the vacuum pressure as well. Figure 8a and b illustrates that the workpiece is lapped and polished to a flat surface while the vacuum pressure stays it at the initial

The manufactured workpiece accuracy can be improved by adjusting the vacuum pressure during the machining process because of the changed workpiece thickness [21]. The vacuum pressure is defined by finite element analysis (FEA) results because theoretical calculation for complex surface is more difficult. In simulation process, a circular plate B270 optical glass with the edge supported by

All elements of modeling were created by meshing with A20-node quadratic brick elements in reduced integration (C3D20R). Figure 9 demonstrates the finite

(a and b) The deforming and lapping processes of glass plate. (a) The glass plate is deformed before lapping and

) Young's modulus (GPa) Knoop hardness HK100 (kg/mm<sup>2</sup>

2550 71.5 542 0.22

) Poisson ratio

process.

deformed state.

Figure 8.

Table 1.

41

the holding device is listed in Table 1.

DOI: http://dx.doi.org/10.5772/intechopen.79402

element model as follows.

(b) the glass plate is deformed in lapping.

Material properties of B270 optical glass [22].

Density (kg/m<sup>3</sup>

Substitute D = Eh<sup>3</sup> /12(1 � <sup>v</sup><sup>2</sup> ) into Eq. (18), we have:

$$w\_0 = \frac{3p(1-\nu)(5+\nu)a}{16E} \left(\frac{a}{h}\right)^3 \tag{19}$$

From Eq. (19), we can see that the maximum deflection of the circular plate is relative of the diameter a and the ratio between diameter a and thickness h.

#### 3. Elastic deformation machining method without mold

Figure 7 shows a schematic illustration of a lens elastic deformation process without mold.

Two surfaces of the workpiece are polished to certain flatness before fabricating as shown in Figure 7a. When vacuum pressure is supplied to the workpiece through

Figure 7. (a–f) Schematic illustration of an aspheric surface elastic deformation process.

#### FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric… DOI: http://dx.doi.org/10.5772/intechopen.79402

a hole, the workpiece is deformed in the middle. The edge of the workpiece is supported by the holder; therefore, it will be not moved. This makes the workpiece become a formed aspheric shape as presented in Figure 7b. The deflection of the workpiece can be calculated by using theoretical equation in boundary conditions of circular plate with simply supported edge. While the vacuum pressure is still remained, the workpiece and the holder start rotating and moving downward in contact with the lapping plate. Its opposite side will be polished to optical flatness as illustrated in Figure 7c and d. Then, the vacuum pressure is not supplied and the workpiece is also released from the holder as shown in Figure 7e. According to the Figure 7f, the bottom surface will be formed into the aspheric shape and the top surface returns to its original flat surface form due to material elasticity. It can be seen that the deformed workpiece surfaces can be restored by internal force and bending moments which are created from the vacuum pressure during machining process.

#### 3.1 Finite element analysis for elastic deformation machining process

The vacuum pressure affects an amount of elastic deformation of the workpiece; hence, the accuracy of manufactured profile will also be highly dependent on the vacuum pressure as well. Figure 8a and b illustrates that the workpiece is lapped and polished to a flat surface while the vacuum pressure stays it at the initial deformed state.

The manufactured workpiece accuracy can be improved by adjusting the vacuum pressure during the machining process because of the changed workpiece thickness [21]. The vacuum pressure is defined by finite element analysis (FEA) results because theoretical calculation for complex surface is more difficult. In simulation process, a circular plate B270 optical glass with the edge supported by the holding device is listed in Table 1.

All elements of modeling were created by meshing with A20-node quadratic brick elements in reduced integration (C3D20R). Figure 9 demonstrates the finite element model as follows.

#### Figure 8.

so that the deflection of every radial location can be calculated using,

wr <sup>¼</sup> p a<sup>2</sup> � <sup>r</sup><sup>2</sup> ð Þ 64D

The maximum deflection which occurs at r = 0, is given by,

/12(1 � <sup>v</sup><sup>2</sup>

Elasticity of Materials ‐ Basic Principles and Design of Structures

Substitute D = Eh<sup>3</sup>

without mold.

Figure 7.

40

<sup>w</sup><sup>0</sup> <sup>¼</sup> pa<sup>4</sup> 64D

<sup>w</sup><sup>0</sup> <sup>¼</sup> <sup>3</sup>pð Þ <sup>1</sup> � <sup>ν</sup> ð Þ <sup>5</sup> <sup>þ</sup> <sup>ν</sup> <sup>a</sup> 16E

relative of the diameter a and the ratio between diameter a and thickness h.

3. Elastic deformation machining method without mold

(a–f) Schematic illustration of an aspheric surface elastic deformation process.

5 þ ν 1 þ ν

5 þ ν 1 þ ν 

) into Eq. (18), we have:

From Eq. (19), we can see that the maximum deflection of the circular plate is

Figure 7 shows a schematic illustration of a lens elastic deformation process

Two surfaces of the workpiece are polished to certain flatness before fabricating as shown in Figure 7a. When vacuum pressure is supplied to the workpiece through

<sup>a</sup><sup>2</sup> � <sup>r</sup> 2

> a h <sup>3</sup>

(17)

(18)

(19)

(a and b) The deforming and lapping processes of glass plate. (a) The glass plate is deformed before lapping and (b) the glass plate is deformed in lapping.


#### Table 1.

Material properties of B270 optical glass [22].

Figure 9. Simulation model of workpiece.

When the vacuum pressure keeps unchanged, the workpiece thickness is reduced during the lapping process. Therefore, the surface form of a glass plate will have some errors compared to desired surface form at the end of the machining process. The results of the FEA indicate that the deflection of workpiece is greater than desired curve. In order to enhance its accuracy, the vacuum pressure should be fixed at 42 kPa as shown in Figure 10.

#### 3.2 Experimental setup

The B270 optical glass which is a clear, high transmission and high purity raw materials is chosen in this experiment. The workpiece sides are lapped and polished to flat surfaces. The lapping process is through the relative motion between the lapping plate and the workpiece, affected by abrasive slurry under distribution load. The silicon carbide (SiC) and cerium oxide (CeO2) abrasive grain slurry are used in the experiment. The principle of lapping process can be seen in Figure 11.

3.3 Experimental results

Lapping and polishing conditions.

Principle of lapping and polishing process.

DOI: http://dx.doi.org/10.5772/intechopen.79402

Figure 11.

Table 2.

Figure 12.

43

The component accuracy can be improved by adjusting the vacuum pressure values to compensate for its lost thickness during the lapping step. The vacuum pressure is defined through FEA results. Figure 10 shows that the deformation curve of the workpiece is close to the desired curve when the vacuum pressure is fixed at 42 kPa. Therefore, the vacuum pressure should be reduced from 50 to 42 kPa in the experiment. Figure 12 illustrates the deflection and deviation results of

Items Lapping Polishing Abrasive #1000 SiC #10,000 CeO2 Abrasive concentration in slurry (wt%) 10% 10% Machining load (N) 20 15 Rotating speed of lapping plate (rpm) 60 40 Machining time (min) 240 60

FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric…

Depending on reducing pressure from 50 to 42 kPa and keeping stable through

the entire lapping step, the experimental results agree greatly with theoretical

the experimental results and the theoretical calculations.

calculations. The peak-valley value is reached at 1.6 μm.

The experimental results of deflection and deviation against theoretical results.

In lapping process, a rigid iron surface covered by a flannelette plate is moved under the load on the glass surface, with abrasive particles suspended in water between them. Table 2 demonstrates parameters for the machining process. To remove microcrack layer and trace after the lapping process, a polishing step is required. This step is also carried out with the Nanopoli-100 precision polishing machine. The polishing parameters fixed unchanged as that in the initial lapping step, except that the abrasive is changed from SiC to CeO2 as a fine polishing step.

Figure 10. Finite element analysis results and analytical results: (a) P = 50 kPa and (b) P = 42 kPa.

FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric… DOI: http://dx.doi.org/10.5772/intechopen.79402

#### Figure 11.

When the vacuum pressure keeps unchanged, the workpiece thickness is reduced during the lapping process. Therefore, the surface form of a glass plate will have some errors compared to desired surface form at the end of the machining process. The results of the FEA indicate that the deflection of workpiece is greater than desired curve. In order to enhance its accuracy, the vacuum pressure should be

The B270 optical glass which is a clear, high transmission and high purity raw materials is chosen in this experiment. The workpiece sides are lapped and polished to flat surfaces. The lapping process is through the relative motion between the lapping plate and the workpiece, affected by abrasive slurry under distribution load. The silicon carbide (SiC) and cerium oxide (CeO2) abrasive grain slurry are used in

In lapping process, a rigid iron surface covered by a flannelette plate is moved under the load on the glass surface, with abrasive particles suspended in water between them. Table 2 demonstrates parameters for the machining process. To remove microcrack layer and trace after the lapping process, a polishing step is required. This step is also carried out with the Nanopoli-100 precision polishing machine. The polishing parameters fixed unchanged as that in the initial lapping step, except that the abrasive is changed from SiC to CeO2 as a fine polishing step.

the experiment. The principle of lapping process can be seen in Figure 11.

Finite element analysis results and analytical results: (a) P = 50 kPa and (b) P = 42 kPa.

fixed at 42 kPa as shown in Figure 10.

Elasticity of Materials ‐ Basic Principles and Design of Structures

3.2 Experimental setup

Simulation model of workpiece.

Figure 9.

Figure 10.

42

Principle of lapping and polishing process.


#### Table 2.

Lapping and polishing conditions.

#### 3.3 Experimental results

The component accuracy can be improved by adjusting the vacuum pressure values to compensate for its lost thickness during the lapping step. The vacuum pressure is defined through FEA results. Figure 10 shows that the deformation curve of the workpiece is close to the desired curve when the vacuum pressure is fixed at 42 kPa. Therefore, the vacuum pressure should be reduced from 50 to 42 kPa in the experiment. Figure 12 illustrates the deflection and deviation results of the experimental results and the theoretical calculations.

Depending on reducing pressure from 50 to 42 kPa and keeping stable through the entire lapping step, the experimental results agree greatly with theoretical calculations. The peak-valley value is reached at 1.6 μm.

Figure 12. The experimental results of deflection and deviation against theoretical results.

#### 4. Elastic deformation machining method with mold

In the elastic deformation machining process without mold, the thickness of the plate is reduced while the vacuum pressure remains unchanged. Thus, the workpiece deformation to increase as lapping progresses. This will cause large deviation in surface form between finished workpiece and theoretical calculation. The mold with its surface approximates the desired surface form of the lens which is used for improving the machining precision. When vacuum pressure is supplied, the top surface of the workpiece will be deformed and then contacts the molded surface. Figure 13 shows the basic concept of elastic deformation molding process [23].

The mold and workpiece surfaces are polished to flatness before fabricating as shown in Figure 13a. When uniform vacuum pressure is supplied to the workpiece through small holes of the mold, the workpiece will be deformed and then contacted with the aspheric surface of the mold as presented in Figure 13b. While deformed workpiece is kept stable under vacuum pressure, the bottom side of the workpiece is polished to flatness as illustrated in Figure 13c and d. Then, the vacuum pressure is not supplied; hence, the lapped side of workpiece will be formed into the mold surface while the opposite surface returns to its original flatness surface due to material elasticity as shown in Figure 13e.

#### 4.1 Finite element analysis for elastic deformation machining process with mold

The standard aspheric formula is:

n

where Z, depth or "Sag" of the curve; r, distance from the center; c, curvature (=1/

FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric…

the largest useful diameter [24]. Figure 14 illustrates the aspheric lens sag. An aspherical surface is built by using spherical surface combined with the higher order terms. Most optical designers use only the even-order terms from A2 to A20. The conic constant K has been used to design the initial aspheric, simple

simulation of contacting process between workpiece and mold surface.

Conic constant Surface type K = 0 Spherical K = 1 Paraboloid K < 1 Hyperboloid 1 < K < 0 Ellipsoid K > 0 Oblate ellipsoid

The radius (R) is used for determining the aspheric terms such as their shallow or depth. The closest spherical surface is the radius which reaches the aspheric sag at

In elastic deformation machining method, the accuracy of aspheric lens depends on the ability of elastic deformation and completely contacting the mold surfaces. The mold surface is defined by choosing the closest spherical surface (as shown in Figure 15). The FEA is designed for establishing the spherical surface through a

radius); K, conic constant; and A2i, higher order terms.

DOI: http://dx.doi.org/10.5772/intechopen.79402

paraboloid and hyperboloid (as shown in Table 3).

Figure 14.

Table 3.

Figure 15.

45

The aspheric surface from best fit sphere.

The relationship between conic constants and surface types.

The sag of aspheric lens.

Figure 13. Basic principle of elastic deformation molding process.

#### FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric… DOI: http://dx.doi.org/10.5772/intechopen.79402

where Z, depth or "Sag" of the curve; r, distance from the center; c, curvature (=1/ radius); K, conic constant; and A2i, higher order terms.

The radius (R) is used for determining the aspheric terms such as their shallow or depth. The closest spherical surface is the radius which reaches the aspheric sag at the largest useful diameter [24]. Figure 14 illustrates the aspheric lens sag.

An aspherical surface is built by using spherical surface combined with the higher order terms. Most optical designers use only the even-order terms from A2 to A20. The conic constant K has been used to design the initial aspheric, simple paraboloid and hyperboloid (as shown in Table 3).

In elastic deformation machining method, the accuracy of aspheric lens depends on the ability of elastic deformation and completely contacting the mold surfaces. The mold surface is defined by choosing the closest spherical surface (as shown in Figure 15). The FEA is designed for establishing the spherical surface through a simulation of contacting process between workpiece and mold surface.

Figure 14. The sag of aspheric lens.

4. Elastic deformation machining method with mold

Elasticity of Materials ‐ Basic Principles and Design of Structures

In the elastic deformation machining process without mold, the thickness of the plate is reduced while the vacuum pressure remains unchanged. Thus, the workpiece deformation to increase as lapping progresses. This will cause large deviation in surface form between finished workpiece and theoretical calculation. The mold with its surface approximates the desired surface form of the lens which is used for improving the machining precision. When vacuum pressure is supplied, the top surface of the workpiece will be deformed and then contacts the molded surface. Figure 13 shows the basic concept of elastic deformation molding process [23]. The mold and workpiece surfaces are polished to flatness before fabricating as shown in Figure 13a. When uniform vacuum pressure is supplied to the workpiece

through small holes of the mold, the workpiece will be deformed and then

flatness surface due to material elasticity as shown in Figure 13e.

<sup>Z</sup> <sup>¼</sup> cr<sup>2</sup>

mold

Figure 13.

44

The standard aspheric formula is:

Basic principle of elastic deformation molding process.

contacted with the aspheric surface of the mold as presented in Figure 13b. While deformed workpiece is kept stable under vacuum pressure, the bottom side of the workpiece is polished to flatness as illustrated in Figure 13c and d. Then, the vacuum pressure is not supplied; hence, the lapped side of workpiece will be formed into the mold surface while the opposite surface returns to its original

4.1 Finite element analysis for elastic deformation machining process with

<sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � ð Þ <sup>1</sup> <sup>þ</sup> <sup>k</sup> <sup>c</sup><sup>2</sup>r<sup>2</sup> <sup>p</sup> <sup>þ</sup> <sup>∑</sup>

n i¼2 A2ir

<sup>2</sup><sup>i</sup> (20)


#### Table 3.

The relationship between conic constants and surface types.

Figure 15. The aspheric surface from best fit sphere.

In the simulation process, the thickness (h = 1.0 mm) and diameter (D = 50 mm) of the workpiece are suggested. In addition, the radius (R = 2500 mm) of spherical surface is chosen. The parameters of FEA model can be seen in Figure 16a.

The axisymmetric model is selected in this simulation process. The mold is chosen as an analytical rigid shell and the workpiece is a deformable shell. The analytical step of model is "Dynamic, Explicit". The interaction and the contact property are "Surface to surface contact" and "Penalty contact method," respectively. All elements of the workpiece are divided in meshing with A4-node bilinear axisymmetric quadrilateral elements in reduced integration. The workpiece mesh and boundary conditions are described in Figure 16b. The values of uniform vacuum pressure are opted in range of 80 to 100 kPa. The conic constant K = 0.25 is selected for the simulation process.

It is clear to see that Figure 17 shows the deflection and deviation of workpiece under different vacuum pressures with the conic constant K = 3.

According to the results, the model with the conic constant K = 3, gives the best one and the deviation between the workpiece and the mold is the smallest. The workpiece and the mold can reach the best when the vacuum pressure approximates 95 kPa. However, when the vacuum pressure is larger than 95 kPa, the deviation results are still stable. Therefore, the conic constant K = 3 recommends for defining the aspheric surface of the mold.

The accuracy one can be innovated by modifying the mold profile to adopt with bending stress of workpiece material. This mold profile is redesigned by using the profile of workpiece after the deformed stage. An axisymmetric FEM model is established, and it consists of the new mold and workpiece. The uniform vacuum pressure is chosen as 95 kPa. The mold surface is redesigned with the conic constant K = 3 as displayed in Figure 18.

It can be noted that Figure 19a and b presents the deflection and deviation results between the workpiece and the new mold under supplied vacuum pressures, P = 95 kPa and K = 3.

The form accuracy of workpiece is enhanced by using the new mold surface. The

FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric…

Figure 20 presents that the experiment was conducted to a precision polishing machine Preci-Polish 300. The B270 glass with a diameter of 50 mm and a thickness

Experimental set-up in the lapping processes. 1-Lapping machine; 2-digital pressure switch; 3-regulator; 4 vacuum pump. 5-accumulator; 6-vacuum pipeline; 7-condition ring; 8-load; 9-mold; 10-lapping plate; and

maximum deviation is less than P-V 0.35 μm while the former mold is about

(a and b) Deflection and deviation results between the workpiece and modified mold.

15.02 μm.

Figure 20.

47

11-slurry pipeline.

Figure 19.

Figure 18.

The modified mold is chosen.

DOI: http://dx.doi.org/10.5772/intechopen.79402

4.2 Experimental setup

Figure 16. (a) The simulation model and (b) FEM simulation model.

Figure 17. (a and b) Deflection and deviation under different vacuum pressures (K = 3).

FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric… DOI: http://dx.doi.org/10.5772/intechopen.79402

Figure 18. The modified mold is chosen.

In the simulation process, the thickness (h = 1.0 mm) and diameter (D = 50 mm) of the workpiece are suggested. In addition, the radius (R = 2500 mm) of spherical

It is clear to see that Figure 17 shows the deflection and deviation of workpiece

According to the results, the model with the conic constant K = 3, gives the best one and the deviation between the workpiece and the mold is the smallest. The workpiece and the mold can reach the best when the vacuum pressure approximates 95 kPa. However, when the vacuum pressure is larger than 95 kPa, the deviation results are still stable. Therefore, the conic constant K = 3 recommends

The accuracy one can be innovated by modifying the mold profile to adopt with bending stress of workpiece material. This mold profile is redesigned by using the profile of workpiece after the deformed stage. An axisymmetric FEM model is established, and it consists of the new mold and workpiece. The uniform vacuum pressure is chosen as 95 kPa. The mold surface is redesigned with the conic

It can be noted that Figure 19a and b presents the deflection and deviation results between the workpiece and the new mold under supplied vacuum pressures,

under different vacuum pressures with the conic constant K = 3.

surface is chosen. The parameters of FEA model can be seen in Figure 16a. The axisymmetric model is selected in this simulation process. The mold is chosen as an analytical rigid shell and the workpiece is a deformable shell. The analytical step of model is "Dynamic, Explicit". The interaction and the contact property are "Surface to surface contact" and "Penalty contact method," respectively. All elements of the workpiece are divided in meshing with A4-node bilinear axisymmetric quadrilateral elements in reduced integration. The workpiece mesh and boundary conditions are described in Figure 16b. The values of uniform vacuum pressure are opted in range of 80 to 100 kPa. The conic constant K = 0.25 is

Elasticity of Materials ‐ Basic Principles and Design of Structures

selected for the simulation process.

for defining the aspheric surface of the mold.

constant K = 3 as displayed in Figure 18.

(a) The simulation model and (b) FEM simulation model.

(a and b) Deflection and deviation under different vacuum pressures (K = 3).

P = 95 kPa and K = 3.

Figure 16.

Figure 17.

46

Figure 19.

(a and b) Deflection and deviation results between the workpiece and modified mold.

The form accuracy of workpiece is enhanced by using the new mold surface. The maximum deviation is less than P-V 0.35 μm while the former mold is about 15.02 μm.

#### 4.2 Experimental setup

Figure 20 presents that the experiment was conducted to a precision polishing machine Preci-Polish 300. The B270 glass with a diameter of 50 mm and a thickness

#### Figure 20.

Experimental set-up in the lapping processes. 1-Lapping machine; 2-digital pressure switch; 3-regulator; 4 vacuum pump. 5-accumulator; 6-vacuum pipeline; 7-condition ring; 8-load; 9-mold; 10-lapping plate; and 11-slurry pipeline.

of 1.0 mm is utilized in the experiment process. In addition, Table 4 points the parameters for the machining process, in which the vacuum pressure is fixed as 95 kPa.

flatness by the lapping wheel. Then, the vacuum pressure is not supplied and hence, the bottom surface will be formed into the aspheric shape and the top surface will

FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric…

manufacturing of optical lens with large aperture and low thickness glass materials. In the elastic deformation machining process without mold, the manufactured workpiece accuracy can be increased by adjusting the vacuum pressure during the machining process because of the changed workpiece thickness. The vacuum pressure is defined through FEA results. According to the FEA, the deformation curve of the workpiece is reached to the desired curve when the vacuum pressure is fixed at 42 kPa. Depending on reducing the vacuum pressure from 50 to 42 kPa and keeping stable through the entire machining process, the experimental results agree greatly with theoretical calculations. The best peak-valley value P-V 1.6 μm was

In order to achieve form accuracy of the workpiece in the elastic deformation machining process with mold, the mold with its surface approximates the desired surface form of the lens which is used for improving the machining precision. The accuracy one can be innovated by modifying the mold profile to adopt with bending stress of workpiece material. This mold profile is redesigned by using the profile of workpiece after the deformed stage. According to the simulation results, the new mold surface with the conic constant K = 3 and vacuum pressure P = 95 kPa are used for the experimental process. In this case, the deviation of workpiece is less than P-V 0.01 μm within the radius of about 12 mm. The maximum deviation is P-V 0.6 μm; however, the former mold is about 18.93 μm. It is clear to see that the

Faculty of Mechanical Engineering, Industrial University of Ho Chi Minh City,

© 2018 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,

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

provided the original work is properly cited.

be restored to its flat surface form. Therefore, the method is suitable for

achieved in this method.

DOI: http://dx.doi.org/10.5772/intechopen.79402

Author details

Duc-Nam Nguyen

Vietnam

49

experimental results agree greatly with FEA results.

According to the simulation results, the new mold surface with the conic constant K = 3 is chosen. The multi-small holes of the modified mold surface are fabricated to fix the workpiece during the machining process. The new mold is machined on a precision CNC machining center.

#### 4.3 Experimental results

Figure 21a and b shows that experimental results are compared to FEA with new mold surface under applied vacuum pressure P = 95 kPa.

Based on the experimental and FEA results, the deviation of workpiece is less than P-V 0.01 μm within the radius of about 12 mm. The maximum deviation is P-V 0.6 μm; however, the former mold is about 18.93 μm. It is clear to see that the experimental results agree greatly with FEA results. Therefore, the form accuracy of the workpiece is significantly improved when the new mold profile is redesigned according to the FEA results with P = 95 kPa and K = 3.


#### Table 4.

Lapping and polishing parameters.

Figure 21. (a and b) Experimental and FEA results with modified mold surface.

#### 5. Conclusions

Based on the elasticity of the material, the elastic deformation machining is a method in which the vacuum pressure is used for fabricating complex aspheric surfaces. The amount of deflection of circular plate can be determined by solving the differential equations of an appropriate plate theory. The workpiece will be deformed into aspheric shape prior to the lapping process under the vacuum pressure. While the vacuum pressure is remained, the opposite side is polished to optical FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric… DOI: http://dx.doi.org/10.5772/intechopen.79402

flatness by the lapping wheel. Then, the vacuum pressure is not supplied and hence, the bottom surface will be formed into the aspheric shape and the top surface will be restored to its flat surface form. Therefore, the method is suitable for manufacturing of optical lens with large aperture and low thickness glass materials.

In the elastic deformation machining process without mold, the manufactured workpiece accuracy can be increased by adjusting the vacuum pressure during the machining process because of the changed workpiece thickness. The vacuum pressure is defined through FEA results. According to the FEA, the deformation curve of the workpiece is reached to the desired curve when the vacuum pressure is fixed at 42 kPa. Depending on reducing the vacuum pressure from 50 to 42 kPa and keeping stable through the entire machining process, the experimental results agree greatly with theoretical calculations. The best peak-valley value P-V 1.6 μm was achieved in this method.

In order to achieve form accuracy of the workpiece in the elastic deformation machining process with mold, the mold with its surface approximates the desired surface form of the lens which is used for improving the machining precision. The accuracy one can be innovated by modifying the mold profile to adopt with bending stress of workpiece material. This mold profile is redesigned by using the profile of workpiece after the deformed stage. According to the simulation results, the new mold surface with the conic constant K = 3 and vacuum pressure P = 95 kPa are used for the experimental process. In this case, the deviation of workpiece is less than P-V 0.01 μm within the radius of about 12 mm. The maximum deviation is P-V 0.6 μm; however, the former mold is about 18.93 μm. It is clear to see that the experimental results agree greatly with FEA results.

#### Author details

of 1.0 mm is utilized in the experiment process. In addition, Table 4 points the parameters for the machining process, in which the vacuum pressure is fixed as

machined on a precision CNC machining center.

Elasticity of Materials ‐ Basic Principles and Design of Structures

According to the simulation results, the new mold surface with the conic constant K = 3 is chosen. The multi-small holes of the modified mold surface are fabricated to fix the workpiece during the machining process. The new mold is

Figure 21a and b shows that experimental results are compared to FEA with

Based on the experimental and FEA results, the deviation of workpiece is less than P-V 0.01 μm within the radius of about 12 mm. The maximum deviation is P-V 0.6 μm; however, the former mold is about 18.93 μm. It is clear to see that the experimental results agree greatly with FEA results. Therefore, the form accuracy of the workpiece is significantly improved when the new mold profile is redesigned

Items Lapping Polishing Abrasive #1000 SiC #10,000 CeO2 Abrasive concentration in slurry (wt%) 10% 10% Machining load (N) 30 20 Rotating speed of lapping plate (rpm) 60 40 Machining time (min) 120 30

Based on the elasticity of the material, the elastic deformation machining is a method in which the vacuum pressure is used for fabricating complex aspheric surfaces. The amount of deflection of circular plate can be determined by solving the differential equations of an appropriate plate theory. The workpiece will be deformed into aspheric shape prior to the lapping process under the vacuum pressure. While the vacuum pressure is remained, the opposite side is polished to optical

new mold surface under applied vacuum pressure P = 95 kPa.

according to the FEA results with P = 95 kPa and K = 3.

(a and b) Experimental and FEA results with modified mold surface.

95 kPa.

4.3 Experimental results

5. Conclusions

Figure 21.

48

Table 4.

Lapping and polishing parameters.

Duc-Nam Nguyen Faculty of Mechanical Engineering, Industrial University of Ho Chi Minh City, Vietnam

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

© 2018 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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FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric…

[16] Mahajan P, Dora PT, Sandeep TS. Optimized design of optical surface of the mold in precision glass molding using the deviation approach.

International Journal for Computational Methods in Engineering Science and Mechanics. 2015;16:53-64. DOI: 10.1080/15502287.2014.976677

[17] Yi AY, Tao B, Klocke F. Residual stresses in glass after molding and its influence on optical properties. Procedia Engineering. 2011;19:402-406. DOI: 10.1016/j.proeng.2011.11.132

[18] Kunz A. Aspheric freedoms of glassprecision glass moulding allows costeffective fabrication of glass aspheres. Optik & Photonik. 2009;4:46-48. DOI:

10.1002/opph.201190063

CRC Press; 2001. p. 688

[19] Mori Y, Yamamura K, Endo K, Yamauchi K, Yasutake K, Goto H, Kakiuchi H, Sano Y, Mimura H. Creation of perfect surfaces. Journal of Crystal Growth. 2005;275:39-50. DOI: 10.1016/j.jcrysgro.2004.10.097

[20] Ventsel E, Krauthammer T, editors. Thin Plates and Shells: Theory, Analysis, and Applications. 1st ed. Boca Raton:

[21] Nguyen DN, Lv BH, Yuan JL, Wu Z, Lu HZ. Experimental study on elastic deformation machining process for aspheric surface glass. International Journal of Advanced Manufacturing Technology. 2013;65:525-531. DOI: 10.1007/s00170-012-4191-3

[22] Schott. Schott B270 Super-white Properties [Internet]. 2015. Available from: https://psec.uchicago.edu/glass/ SchottB270Properties-KnightOptical.

[23] Nguyen DN. Study on improving the precision of form surface produced

pdf [Accessed: 24-05-2018]

51

10.1117/12.2195764

[9] Lambropoulus JC, Fang T, Funkenbusch PD, Jacobs SD, Cumbo MJ, Golini D. Surface micro-roughness of optical glasses under deterministic micro-grinding. Applied Optics. 1996; 35:4448-4462. DOI: 10.1364/ AO.35.004448

[10] Namba Y, Abe M, Kobayashi A. Ultra-precision grinding of optical glasses to produce super-smooth surfaces. Annals of the CIRP. 1993;42: 417-420. DOI: 10.1016/S0007-8506(07) 62475-5

[11] Chen WK, Kuriyagawa T, Huang H, Yoshihara N. Machining of micro aspherical mould inserts. Journal of Precision Engineering. 2005;29:315-323. DOI: 10.1016/j.precisioneng.2004.11.002

[12] Brinksmeier E, Mutlugunes Y, Klocke F. Ultra-precision grinding. CIRP Annals-Manufacturing Technology. 2010;59:652-671. DOI: 10.1016/j.cirp.2010.05.001

[13] Bingham R, Walker D, Kim D. Novel automated process for aspheric surfaces. In: Proceedings of SPIE – The International Society for Optical Engineering. 2000;4093:445-448. DOI: 10.1117/12.405237

[14] Walker D, Brooks D, King A. The "Precessions" tooling for polishing and figuring flat, spherical and aspheric surfaces. Optics Express. 2003;11: 958-964:958-964. DOI: 1364/ OE.11.000958

[15] Nelson J, Scordato M, Schwertz K. Precision lens molding of asphero

FEA and Experimentally Determination of Applied Elasticity Problem for Fabricating Aspheric… DOI: http://dx.doi.org/10.5772/intechopen.79402

diffractive surfaces in chalcogenide materials. Optifab. 2015:96331L. DOI: 10.1117/12.2195764

References

pp. 844-847

2007.05.033

[1] Aono Y, Negishi M, Takano J. Development of large-aperture aspherical lens with glass molding. Advanced Optical Manufacturing and Testing Technology. 2000;4231:16-23.

Elasticity of Materials ‐ Basic Principles and Design of Structures

[8] Bifano TG, Dow TA, Scattergood RO.

Ductile-regime grinding: A new technology for machining brittle materials. Journal of Engineering for Industry. Transactions of the ASME. 1991;113:184-189. DOI: 10.1016/

0141-6359(92)90161-O

[9] Lambropoulus JC, Fang T, Funkenbusch PD, Jacobs SD, Cumbo MJ, Golini D. Surface micro-roughness of optical glasses under deterministic micro-grinding. Applied Optics. 1996;

35:4448-4462. DOI: 10.1364/

[10] Namba Y, Abe M, Kobayashi A. Ultra-precision grinding of optical glasses to produce super-smooth surfaces. Annals of the CIRP. 1993;42: 417-420. DOI: 10.1016/S0007-8506(07)

[11] Chen WK, Kuriyagawa T, Huang H, Yoshihara N. Machining of micro aspherical mould inserts. Journal of Precision Engineering. 2005;29:315-323. DOI: 10.1016/j.precisioneng.2004.11.002

[12] Brinksmeier E, Mutlugunes Y, Klocke F. Ultra-precision grinding. CIRP Annals-Manufacturing Technology. 2010;59:652-671. DOI:

[13] Bingham R, Walker D, Kim D. Novel automated process for aspheric surfaces. In: Proceedings of SPIE – The International Society for Optical Engineering. 2000;4093:445-448. DOI:

[14] Walker D, Brooks D, King A. The "Precessions" tooling for polishing and figuring flat, spherical and aspheric surfaces. Optics Express. 2003;11: 958-964:958-964. DOI: 1364/

[15] Nelson J, Scordato M, Schwertz K. Precision lens molding of asphero

10.1016/j.cirp.2010.05.001

10.1117/12.405237

OE.11.000958

AO.35.004448

62475-5

[2] Suzuki H, Highuchi O, Huriuchi H, Shibutani H. Precision cutting of microaxis-symmetric spherical surface with 3-axes controlled diamond tool. In: Proceedings of the Second Euspen International Conference on European Society for Precision Engineering and Nanotechnology; 27-31 May 2001; Italy.

[3] Anurag J. Experimental study and numerical analysis of compression molding process for manufacturing precision aspherical glass lenses [thesis]. Columbus: Ohio State University; 2006

[4] Takashi M, Takenori O. Cutting process of glass with inclined ball end mill. Journal of Materials Processing Technology. 2008;200:356-363. DOI: 10.1016/j.jmatprotec.2007.08.067

[5] Suzuki H, Moriwaki T, Yamamoto Y, Goto Y. Precision cutting of aspherical ceramic molds with micro PCD milling tool. Annals of the CIRP. 2007;56: 131-134. DOI: 10.1016/j.cirp.

[6] Kim HS, Kim EJ, Song BS. Diamond turning of large off-axis aspheric mirrors using a fast tool servo with on machine measurement. Journal of Materials Processing Technology. 2004;

[7] Li L, Yi AY, Huang C, Grewell DA, Benatar A, Chen Y. Fabrication of diffractive optics by use of slow tool servo diamond turning process. Optical Engineering. 2006;45:113401. DOI:

146:349-355. DOI: 10.1016/j. jmatprotec.2003.11.028

10.1117/1.2387142

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DOI: 10.1117/12.402759

[16] Mahajan P, Dora PT, Sandeep TS. Optimized design of optical surface of the mold in precision glass molding using the deviation approach. International Journal for Computational Methods in Engineering Science and Mechanics. 2015;16:53-64. DOI: 10.1080/15502287.2014.976677

[17] Yi AY, Tao B, Klocke F. Residual stresses in glass after molding and its influence on optical properties. Procedia Engineering. 2011;19:402-406. DOI: 10.1016/j.proeng.2011.11.132

[18] Kunz A. Aspheric freedoms of glassprecision glass moulding allows costeffective fabrication of glass aspheres. Optik & Photonik. 2009;4:46-48. DOI: 10.1002/opph.201190063

[19] Mori Y, Yamamura K, Endo K, Yamauchi K, Yasutake K, Goto H, Kakiuchi H, Sano Y, Mimura H. Creation of perfect surfaces. Journal of Crystal Growth. 2005;275:39-50. DOI: 10.1016/j.jcrysgro.2004.10.097

[20] Ventsel E, Krauthammer T, editors. Thin Plates and Shells: Theory, Analysis, and Applications. 1st ed. Boca Raton: CRC Press; 2001. p. 688

[21] Nguyen DN, Lv BH, Yuan JL, Wu Z, Lu HZ. Experimental study on elastic deformation machining process for aspheric surface glass. International Journal of Advanced Manufacturing Technology. 2013;65:525-531. DOI: 10.1007/s00170-012-4191-3

[22] Schott. Schott B270 Super-white Properties [Internet]. 2015. Available from: https://psec.uchicago.edu/glass/ SchottB270Properties-KnightOptical. pdf [Accessed: 24-05-2018]

[23] Nguyen DN. Study on improving the precision of form surface produced in elastic deformation molding process. International Journal of Advanced Manufacturing Technology. 2017;93: 3473-3484. DOI: 10.1007/s00170-017- 0766-3

[24] Kweon G, Kim CH. Aspherical lens design by using a numerical analysis. Journal of the Korean Physical Society. 2007;51:93-103. DOI: 10.3938/jkps.51.93

Chapter 4

Abstract

1. Introduction

bonding.

53

Concept of Phase Transition Based

The use of elastic constants systematics to describe fundamental properties of engineering materials has made materials science education and its related subjects increasingly important not only for manufacturing engineers but also for mankind at large. In this chapter, we present actual scaling of phase transition-driven considerations, such as martensitic transformation and transformable shape memory formation via elastic constant systematics. The scaling in terms of the simple and polycrystals mechanical stability criteria based on the elastic moduli and an acoustic anisotropy is in good agreement with novel experimental data from the literatures, and further, a long-standing concern in predicting polycrystalline elastic constants

Keywords: elastic, elastic modulus, martensitic transformation, shape memory

The ingenuity and the art required to tailor precisely the desired physical and structural properties in materials have been the main goal of the material scientists and engineers. Elastic response (i.e. elastic constant) to an applied load is one of such basic properties of all solids and originates from the distortion of atomic bonds. Simply put, elastic constants are a reflection of the fundamental thermodynamic properties that take place in the crystal lattice of solids. Complementary to this, the otherwise inaccessible essential information can be revealed from their temperature and stress dependencies of these important constants. For instance, the crystal structures of the three long periods of transition elements change more or less systematically from hcp through bcc to fcc as their group numbers increase from IV to VIII as does their elastic properties. Thus, the knowledge of microscopic elasticity can provide a fruitful ground for the exploration of the material behaviour yet uncommon to our knowledge about the relationship between crystal structure and

The earliest foundation of elastic theory dates back to seventeenth century (around 1821), when Navier first gave the equation for the equilibrium and motion of elastic solids [1], but modern foundation of microscopic elastic theory was established by the work of Born and Huang [2], followed by other excellent treatments [3]. It is well known that crystalline solids are by no means ideal and invariably contain some lattice defects such as vacancies, solute atoms or some extent of disorder. These point defects strongly affect almost all properties of

on Elastic Systematics

Paul S. Nnamchi and Camillus S. Obayi

was considered beyond the commonly encountered criteria.

effect, elastic constant, ductility criterion, mechanical properties

#### Chapter 4

## Concept of Phase Transition Based on Elastic Systematics

Paul S. Nnamchi and Camillus S. Obayi

#### Abstract

The use of elastic constants systematics to describe fundamental properties of engineering materials has made materials science education and its related subjects increasingly important not only for manufacturing engineers but also for mankind at large. In this chapter, we present actual scaling of phase transition-driven considerations, such as martensitic transformation and transformable shape memory formation via elastic constant systematics. The scaling in terms of the simple and polycrystals mechanical stability criteria based on the elastic moduli and an acoustic anisotropy is in good agreement with novel experimental data from the literatures, and further, a long-standing concern in predicting polycrystalline elastic constants was considered beyond the commonly encountered criteria.

Keywords: elastic, elastic modulus, martensitic transformation, shape memory effect, elastic constant, ductility criterion, mechanical properties

#### 1. Introduction

The ingenuity and the art required to tailor precisely the desired physical and structural properties in materials have been the main goal of the material scientists and engineers. Elastic response (i.e. elastic constant) to an applied load is one of such basic properties of all solids and originates from the distortion of atomic bonds. Simply put, elastic constants are a reflection of the fundamental thermodynamic properties that take place in the crystal lattice of solids. Complementary to this, the otherwise inaccessible essential information can be revealed from their temperature and stress dependencies of these important constants. For instance, the crystal structures of the three long periods of transition elements change more or less systematically from hcp through bcc to fcc as their group numbers increase from IV to VIII as does their elastic properties. Thus, the knowledge of microscopic elasticity can provide a fruitful ground for the exploration of the material behaviour yet uncommon to our knowledge about the relationship between crystal structure and bonding.

The earliest foundation of elastic theory dates back to seventeenth century (around 1821), when Navier first gave the equation for the equilibrium and motion of elastic solids [1], but modern foundation of microscopic elastic theory was established by the work of Born and Huang [2], followed by other excellent treatments [3]. It is well known that crystalline solids are by no means ideal and invariably contain some lattice defects such as vacancies, solute atoms or some extent of disorder. These point defects strongly affect almost all properties of

materials, including elastic behaviours. In effect, the early investigators of these phenomena were motivated by the response of naturally occurring anisotropic materials such as wood and other crystalline solids. On that premise, of interest here is the relationship between crystal structure and elastic properties, mainly because of the important information they provide about nature of binding forces in solids.

Here, Sijkl represents the elastic compliance of the crystal. From symmetry or equilibrium principles, the state of stress in an elastic body can be approximated by six independent stress and strain components. And as such the stress and strain

Here, exx, eyy and ezz are tensile strains, exy, eyz and ezx are shear strains. The experimental values of elastic constants, Cijkl, were originally determined by considering the response of crystals to small strains or unstressed lattice using Eq. (1). Beyond using Eq. (1) based on measured stress-strain relations, there are now methods of determining elastic constants from the first principles often referred to as ab initio methods. There are many methods of evaluating elastic coefficients such as the one based on expanding the internal strain energy of the crystal [7]. Thus, we

> 1 2 V ∑ i ∑ i

where U is the energy of the crystal, is a quadratic function of the strains, in the approximation of Hooke's law (recall the expression for the energy of a stretched spring). V0 is its equilibrium volume and e denotes an elastic strain. If the material

is a crystal, the number of independent elastic constants is reduced further

Elastic coefficients and elastic moduli have significant effect of mechanical response of crystals. Elastic constants, Cij(C11, C12, C44) and elastic moduli such as bulk modulus (B), shear modulus (G), Young's modulus (E) influence mechanical response of crystals. For instance, the bulk modulus (B) is associated with the hardness of materials which is of extreme importance in high-temperature and pressure applications, while elastic constants could provide essential information about bonding between adjacent atomic planes, anisotropic character of bonding and structural stability [7]. By far, the most widely reported elastic properties are E, G and B, corresponding to tensile, shear and hydrostatic loading, respectively. Since B signifies the compressibility of a substance, it can be calculated from the partial derivative of volume (V) and pressure (P) at constant temperature (T), as

It is worth pointing out that other definitions of elastic constant are possible. Elastic modulus E measures the resistance to a change in atomic separation distance within the plane of the bond and so can be determined from the linear portion of the interatomic potential. G quantifies the resistance to shear loading and B, since it corresponds to a volumetric dilatation, is dependent on the electronic properties of

ð3Þ

Cijeiej þ :… (4)

B ¼ <sup>δ</sup><sup>V</sup> ð Þ =<sup>δ</sup><sup>P</sup> <sup>T</sup> (5)

components in Eq. (1) can be expressed in three orthogonal axes as:

Concept of Phase Transition Based on Elastic Systematics

DOI: http://dx.doi.org/10.5772/intechopen.81340

U ¼ U<sup>0</sup> þ V0∑σiei þ

may write as Eq. (4),

per Eq. (5).

55

depending on the crystal system.

Over the past three decades, elastic constants of some simple crystals have been a subject of numerous researches and have been investigated both theoretically and experimentally. Some of the outcomes have revealed that fundamental elastic properties of a martensitic crystal are fully determined by the elastic constants Cij. All macroscopic elastic moduli (Young's and shear modulus, Poisson ratio, etc.) can be derived from the Cij at least within certain upper and lower bounds [4]. There is considerable evidence that the magnitude of <sup>C</sup>' <sup>¼</sup> ð Þ <sup>C</sup><sup>11</sup> � <sup>C</sup><sup>12</sup> <sup>=</sup>2 elastic shear modulus in metallic bcc structures is closely related to the occurrence of martensitic phase transformations and is thus a useful parameter for estimating bcc structures [5]. Similarly, whether a structural material shows plastic flow or brittle fracture on loading is of clear practical significance. Brittleness in polycrystalline metals can be intrinsic or induced. The basic question is: Do these two general properties (i.e. phase stability and elastic properties) of crystals correlate to each other?

#### 2. Analytical criterion of elastic constants of perfect crystals

The elastic properties are among the most important physical properties of materials and the importance of studying elastic properties of materials cannot be overemphasised. The knowledge of elastic properties is essential for both structural design and experimental mechanics [6]. It also enables the assessment of the sufficiency of strength, stiffness and stability of newly developed materials. Although the crystals are assumed to free from lattice imperfections and difficult to produce, their study had always been the building block for a better understanding of the behaviour of bulk materials. Usually, the determination of elastic properties of crystalline solids is based on its single or perfect crystal configuration under special loading conditions. The elastic moduli are the material constants that connect stress with strain and are therefore crucial to engineering applications. A crystal subjected to external load undergoes dimensional change. If the eternal load is a stress tensor denoted by σij, then the deformation per unit length in three-dimensional space, can be described by a strain tensor, eij. Within the elastic limit or for sufficiently small deformations, the stress tensor is a linear function of the strain tensor and the generalised delta notation of Hooke's law can be used to express the relationship between these two quantities [7] as:

$$
\sigma\_{ij} = \mathbf{C}\_{ijkl} \mathbf{e}\_{kl} \tag{1}
$$

where Cijkl is the proportionality constant that characterises the crystal's resistance to elastic shape change; often referred to as the elastic coefficients or elastic constants or elastic moduli or stress-strain coefficients [8].

The inverse relation between the strain and the stress can be determined by taking the inverse of stress-strain relation to get:

$$e\_{ij} = S\_{ijkl} \sigma\_{kl} \tag{2}$$

Concept of Phase Transition Based on Elastic Systematics DOI: http://dx.doi.org/10.5772/intechopen.81340

materials, including elastic behaviours. In effect, the early investigators of these phenomena were motivated by the response of naturally occurring anisotropic materials such as wood and other crystalline solids. On that premise, of interest here is the relationship between crystal structure and elastic properties, mainly because of the important information they provide about nature of binding forces

Elasticity of Materials ‐ Basic Principles and Design of Structures

Over the past three decades, elastic constants of some simple crystals have been a subject of numerous researches and have been investigated both theoretically and experimentally. Some of the outcomes have revealed that fundamental elastic properties of a martensitic crystal are fully determined by the elastic constants Cij. All macroscopic elastic moduli (Young's and shear modulus, Poisson ratio, etc.) can be derived from the Cij at least within certain upper and lower bounds [4]. There is considerable evidence that the magnitude of <sup>C</sup>' <sup>¼</sup> ð Þ <sup>C</sup><sup>11</sup> � <sup>C</sup><sup>12</sup> <sup>=</sup>2 elastic shear modulus in metallic bcc structures is closely related to the occurrence of martensitic phase transformations and is thus a useful parameter for estimating bcc structures [5]. Similarly, whether a structural material shows plastic flow or brittle fracture on loading is of clear practical significance. Brittleness in polycrystalline metals can be intrinsic or induced. The basic question is: Do these two general properties (i.e. phase

stability and elastic properties) of crystals correlate to each other?

2. Analytical criterion of elastic constants of perfect crystals

The elastic properties are among the most important physical properties of materials and the importance of studying elastic properties of materials cannot be overemphasised. The knowledge of elastic properties is essential for both structural design and experimental mechanics [6]. It also enables the assessment of the sufficiency of strength, stiffness and stability of newly developed materials. Although the crystals are assumed to free from lattice imperfections and difficult to produce, their study had always been the building block for a better understanding of the behaviour of bulk materials. Usually, the determination of elastic properties of crystalline solids is based on its single or perfect crystal configuration under special loading conditions. The elastic moduli are the material constants that connect stress with strain and are therefore crucial to engineering applications. A crystal subjected to external load undergoes dimensional change. If the eternal load is a stress tensor denoted by σij, then the deformation per unit length in three-dimensional space, can be described by a strain tensor, eij. Within the elastic limit or for sufficiently small deformations, the stress tensor is a linear function of the strain tensor and the generalised delta notation of Hooke's law can be used to express the relationship between

where Cijkl is the proportionality constant that characterises the crystal's resistance to elastic shape change; often referred to as the elastic coefficients or elastic

The inverse relation between the strain and the stress can be determined by

constants or elastic moduli or stress-strain coefficients [8].

taking the inverse of stress-strain relation to get:

σij ¼ Cijklekl (1)

ð2Þ

in solids.

these two quantities [7] as:

54

Here, Sijkl represents the elastic compliance of the crystal. From symmetry or equilibrium principles, the state of stress in an elastic body can be approximated by six independent stress and strain components. And as such the stress and strain components in Eq. (1) can be expressed in three orthogonal axes as:

ð3Þ

Here, exx, eyy and ezz are tensile strains, exy, eyz and ezx are shear strains. The experimental values of elastic constants, Cijkl, were originally determined by considering the response of crystals to small strains or unstressed lattice using Eq. (1). Beyond using Eq. (1) based on measured stress-strain relations, there are now methods of determining elastic constants from the first principles often referred to as ab initio methods. There are many methods of evaluating elastic coefficients such as the one based on expanding the internal strain energy of the crystal [7]. Thus, we may write as Eq. (4),

$$U = U\_0 + V\_0 \Sigma \sigma\_i \mathbf{e}\_i + \frac{1}{2} V \sum\_i \sum\_i \mathbf{C}\_{\vec{\eta}} \mathbf{e}\_i \mathbf{e}\_j + \dots \tag{4}$$

where U is the energy of the crystal, is a quadratic function of the strains, in the approximation of Hooke's law (recall the expression for the energy of a stretched spring). V0 is its equilibrium volume and e denotes an elastic strain. If the material is a crystal, the number of independent elastic constants is reduced further depending on the crystal system.

Elastic coefficients and elastic moduli have significant effect of mechanical response of crystals. Elastic constants, Cij(C11, C12, C44) and elastic moduli such as bulk modulus (B), shear modulus (G), Young's modulus (E) influence mechanical response of crystals. For instance, the bulk modulus (B) is associated with the hardness of materials which is of extreme importance in high-temperature and pressure applications, while elastic constants could provide essential information about bonding between adjacent atomic planes, anisotropic character of bonding and structural stability [7]. By far, the most widely reported elastic properties are E, G and B, corresponding to tensile, shear and hydrostatic loading, respectively. Since B signifies the compressibility of a substance, it can be calculated from the partial derivative of volume (V) and pressure (P) at constant temperature (T), as per Eq. (5).

$$B = \begin{pmatrix} \delta V / \delta p \end{pmatrix}\_T \tag{5}$$

It is worth pointing out that other definitions of elastic constant are possible. Elastic modulus E measures the resistance to a change in atomic separation distance within the plane of the bond and so can be determined from the linear portion of the interatomic potential. G quantifies the resistance to shear loading and B, since it corresponds to a volumetric dilatation, is dependent on the electronic properties of

a solid, i.e. the compressibility of the electron gas. Elastic moduli are therefore controlled by interatomic interactions and so may be considered a fundamental property of condensed matter. By excitation of longitudinal and transverse phone modes, E and G can, respectively, be calculated if the density (ρ) of the material is known. This is done via an ultrasonic probe which emits and measures the longitudinal (vl) and transverse (vt) sound wave velocities, from which E and G can be calculated via Eqs. (6) and (7):

$$E = \rho v\_l^2 \tag{6}$$

deformed homogeneously by infinitesimal strain as shown in Eqs. (14) and (15)

∂2 U ∂2 ei∂<sup>2</sup> ej

where U is the elastic energy, VO is the volume of unstressed sample, Cij (I, j = 1–6)

Most real materials (cubic and non-cubic polycrystalline structures) have some types of symmetry, which further reduces the required number of independent elastic moduli. In the case of cubic systems, such as bcc, fcc, NaCl type, or CsCl type) structures, in particular, number of independent elastic moduli is reduced from 36 to 9, as Cij = Cji and there being strong symmetry in the two lattices. Therefore, the conditions for stability reduced to a very simple form using three different elastic constants: C11, C<sup>22</sup> and C44. The mechanical stability criteria are

> C<sup>11</sup> � j j C12; >0 C<sup>11</sup> þ 2 C12>0 C44>0 C12>C<sup>11</sup>

Although hexagonal and tetragonal systems have the same form for the elastic matrix, the hexagonal has five, while tetragonal has six independent elastic constants. By direct calculation of the Eigen values of the stiffness matrix, according to

Similarly, for the orthorhombic system, there are nine independent elastic constants: C11, C22, C33, C44, C12C55, C66, C23 and C13. The mechanical stability of the structure at each concentration can be judged by calculated elastic stiffness. According to Born's criteria [3], the requirement of mechanical stability in an

> C11>0;C22>0;C44>0;C33>0;C55>0;C66>0; C<sup>11</sup> þ C22>2 C12;C<sup>11</sup> þ C33>2 C13; C<sup>11</sup> þ C<sup>22</sup> C<sup>33</sup> þ 2 C<sup>12</sup> þ 2 C<sup>23</sup> þ 2 C13>0;

Further to this, conditions for stability for some high-symmetry crystal classes have been studied. However, there is still some confusion about the form of stability

<sup>13</sup> <C33ð Þ C<sup>11</sup> þ C<sup>12</sup>

<sup>C</sup>44>0;C66>0 (17)

The condition when B < 0 is referred to as spinodal instability.

[11], four conditions can be derived for elastic stability in both classes:

<sup>C</sup>11>j j <sup>C</sup>12; ;2C<sup>2</sup>

orthorhombic system leads to the following equations [12].

criteria for other crystal systems and classes [8].

is the elastic constant and ei and ej are the applied strains [2]. In Eq. (15), O (e3) denotes the terms of numerical error in the order e3 or higher. A crystal lattice is dynamically said to be stable only if elastic energy U is positive for any small deformation [9], which implies that principal minors of the determinant with elements Cij

!

Cijeiej <sup>þ</sup> O e<sup>3</sup> � � (15)

(14)

(16)

(18)

Cij <sup>¼</sup> <sup>1</sup> V<sup>0</sup>

<sup>U</sup> <sup>¼</sup> <sup>1</sup> 2 V<sup>0</sup> ∑ 6 i,j¼<sup>1</sup>

Concept of Phase Transition Based on Elastic Systematics

DOI: http://dx.doi.org/10.5772/intechopen.81340

[3], respectively:

are all positive [3].

given by [10]:

57

$$\mathbf{G} = \rho v\_t^2 \tag{7}$$

E, G and B can also be calculated from Cij elastic constants. For a material with cubic structure, the number of Cij in the elastic tensor can be reduced from 36 to just 9, due to Cij = Cji and there being strong symmetry in a cubic lattice. The resulting relevant Cij are C11, C<sup>12</sup> and C44.

$$C\_{12} = B + \frac{4G}{3} \tag{8}$$

$$\mathbf{C\_{11}} = \mathbf{3B} - \frac{\mathbf{C\_{11}}}{2} \tag{9}$$

$$\mathbf{C\_{11} = G} \tag{10}$$

$$\stackrel{\circ}{C} = \mathbf{C}\_{11} - \frac{\mathbf{C}\_{12}}{2} \tag{11}$$

The tetragonal shear modulus, C , corresponds to a specific phonon vibration mode in the atomic structure, and is thus directional in nature. In comparison, B is non-directional as it relates to a volumetric effect.

$$B = \frac{\mathbf{C\_{11}} + \mathbf{2C\_{12}}}{3} \tag{12}$$

$$G = \frac{C\_{11} - C\_{12}}{3} \tag{13}$$

#### 3. Elastic and lattice stability criteria

#### 3.1 Lattice stability in perfect crystal

Elastic properties of a material are very important because they check the mechanical stability, ductile or brittle behaviour based on the analysis of elastic constants, Cij, bulk modulus B and shear modulus G. For example, the bulk modulus measures the resistance of the volume variation in a solid and provides an estimation of the elastic response of the materials under hydrostatic pressure. The shear modulus describes the resistance of a material to shape change.

The fundamental understanding of the conditions of mechanical stability of unstressed crystal structure was laid by the work of Max-Born and co-authors in the 1940s [3], and consolidated later in 1954 [3]. This and other text books gave the generic requirements for elastic stability of crystal lattices in terms of elastic constants [3] and offers simplified equivalents of the generic conditions for some high-symmetry classes. The general stability condition can be stated by considering the second-order elastic matrix and the elastic energy of the crystal a solid, i.e. the compressibility of the electron gas. Elastic moduli are therefore controlled by interatomic interactions and so may be considered a fundamental property of condensed matter. By excitation of longitudinal and transverse phone modes, E and G can, respectively, be calculated if the density (ρ) of the material is known. This is done via an ultrasonic probe which emits and measures the longitudinal (vl) and transverse (vt) sound wave velocities, from which E and G can be

Elasticity of Materials ‐ Basic Principles and Design of Structures

<sup>E</sup> <sup>¼</sup> <sup>ρ</sup>v<sup>2</sup>

<sup>G</sup> <sup>¼</sup> <sup>ρ</sup>v<sup>2</sup>

E, G and B can also be calculated from Cij elastic constants. For a material with cubic structure, the number of Cij in the elastic tensor can be reduced from 36 to just 9, due to Cij = Cji and there being strong symmetry in a cubic lattice. The

4G

C<sup>12</sup> ¼ B þ

<sup>C</sup><sup>11</sup> <sup>¼</sup> <sup>3</sup><sup>B</sup> � <sup>C</sup><sup>11</sup>

<sup>C</sup><sup>&#</sup>x27; <sup>¼</sup> <sup>C</sup><sup>11</sup> � <sup>C</sup><sup>12</sup>

The tetragonal shear modulus, C , corresponds to a specific phonon vibration mode in the atomic structure, and is thus directional in nature. In comparison, B is

<sup>B</sup> <sup>¼</sup> <sup>C</sup><sup>11</sup> <sup>þ</sup> <sup>2</sup>C<sup>12</sup>

Elastic properties of a material are very important because they check the mechanical stability, ductile or brittle behaviour based on the analysis of elastic constants, Cij, bulk modulus B and shear modulus G. For example, the bulk modulus measures the resistance of the volume variation in a solid and provides an estimation of the elastic response of the materials under hydrostatic pressure. The shear

The fundamental understanding of the conditions of mechanical stability of unstressed crystal structure was laid by the work of Max-Born and co-authors in the 1940s [3], and consolidated later in 1954 [3]. This and other text books gave the generic requirements for elastic stability of crystal lattices in terms of elastic constants [3] and offers simplified equivalents of the generic conditions for some high-symmetry classes. The general stability condition can be stated by considering the second-order elastic matrix and the elastic energy of the crystal

modulus describes the resistance of a material to shape change.

<sup>l</sup> (6)

<sup>t</sup> (7)

<sup>3</sup> (8)

<sup>2</sup> (9)

<sup>2</sup> (11)

<sup>3</sup> (12)

ð13Þ

C<sup>11</sup> ¼ G (10)

calculated via Eqs. (6) and (7):

resulting relevant Cij are C11, C<sup>12</sup> and C44.

non-directional as it relates to a volumetric effect.

3. Elastic and lattice stability criteria

3.1 Lattice stability in perfect crystal

56

deformed homogeneously by infinitesimal strain as shown in Eqs. (14) and (15) [3], respectively:

$$\mathbf{C}\_{\vec{\eta}} = \frac{\mathbf{1}}{V\_0} \left( \frac{\partial^2 U}{\partial^2 e\_i \partial^2 e\_j} \right) \tag{14}$$

$$U = \frac{1}{2} V\_0 \sum\_{i,j=1}^6 \mathbf{C}\_{ij} \mathbf{e}\_i \mathbf{e}\_j + \mathcal{O}\left(e^3\right) \tag{15}$$

where U is the elastic energy, VO is the volume of unstressed sample, Cij (I, j = 1–6) is the elastic constant and ei and ej are the applied strains [2]. In Eq. (15), O (e3) denotes the terms of numerical error in the order e3 or higher. A crystal lattice is dynamically said to be stable only if elastic energy U is positive for any small deformation [9], which implies that principal minors of the determinant with elements Cij are all positive [3].

Most real materials (cubic and non-cubic polycrystalline structures) have some types of symmetry, which further reduces the required number of independent elastic moduli. In the case of cubic systems, such as bcc, fcc, NaCl type, or CsCl type) structures, in particular, number of independent elastic moduli is reduced from 36 to 9, as Cij = Cji and there being strong symmetry in the two lattices. Therefore, the conditions for stability reduced to a very simple form using three different elastic constants: C11, C<sup>22</sup> and C44. The mechanical stability criteria are given by [10]:

$$\begin{aligned} \mathbf{C\_{11}} &- |\mathbf{C\_{12}}| \succ \mathbf{0} \\ \mathbf{C\_{11}} &+ 2 \, \mathbf{C\_{12}} \succeq \mathbf{0} \\ \mathbf{C\_{44}} &\succeq \mathbf{0} \end{aligned} \tag{16}$$
 
$$\mathbf{C\_{12}} \succeq \mathbf{C\_{11}}$$

The condition when B < 0 is referred to as spinodal instability.

Although hexagonal and tetragonal systems have the same form for the elastic matrix, the hexagonal has five, while tetragonal has six independent elastic constants. By direct calculation of the Eigen values of the stiffness matrix, according to [11], four conditions can be derived for elastic stability in both classes:

$$\begin{aligned} \text{C}\_{11} & \text{\$\bf C}\_{12}, \text{\$\bf 2C}\_{13}^2 \prec \text{C}\_{33} (\text{\$\bf C}\_{11} + \text{\$\bf C}\_{12})\\ & \text{\$\bf C}\_{44} \succ \text{0}; \text{C}\_{66} \succ \text{0} \end{aligned} \tag{17}$$

Similarly, for the orthorhombic system, there are nine independent elastic constants: C11, C22, C33, C44, C12C55, C66, C23 and C13. The mechanical stability of the structure at each concentration can be judged by calculated elastic stiffness. According to Born's criteria [3], the requirement of mechanical stability in an orthorhombic system leads to the following equations [12].

$$\begin{aligned} \text{C}\_{11} \rhd \text{0;} \text{C}\_{22} \rhd \text{0;} \text{C}\_{44} \rhd \text{0;} \text{C}\_{33} \rhd \text{0;} \text{C}\_{55} \rhd \text{0;} \text{C}\_{66} \rhd \text{0;}\\ \text{C}\_{11} + \text{C}\_{22} \rhd \text{2} \text{C}\_{12} \text{C}\_{11} + \text{C}\_{33} \rhd \text{2} \text{C}\_{13};\\ \text{C}\_{11} + \text{C}\_{22} \text{C}\_{33} + 2 \text{C}\_{12} + 2 \text{C}\_{23} + 2 \text{C}\_{13} \rhd \text{0;} \end{aligned} \tag{18}$$

Further to this, conditions for stability for some high-symmetry crystal classes have been studied. However, there is still some confusion about the form of stability criteria for other crystal systems and classes [8].

A crystal lattice is said to be stable in the absence of external load (unstressed condition) and in the harmonic approximation [13] if and only if it has both dynamic and elastic stability. Dynamic stability implies that its phonon modes have positive frequencies for all wave vectors, while its elastic stability is dependent on elastic energy given by Eq. (15) being always positive (U > 0,∀ε 6¼ 0). Elastic stability criterion is mathematically equivalent to the following necessary and sufficient conditions: the elastic matrix C is definite exactly positive and all Eigen values of matrix C are positive; all the leading principal and arbitrary minors of matrix C are all positive. The closed form expressions for necessary and sufficient elastic stability criteria for other crystal lattices have been studied. While the stability criterion is linear for some crystal lattices, it is quadratic and even polynomial for others. Thus, the mechanical stability of a crystal is combination of the elastic constant and Born's stability criteria. The elastic constant of a stable crystal must satisfy the Born's criteria to prove its mechanical stability.

<sup>ϵ</sup> <sup>¼</sup> <sup>ϵ</sup><sup>0</sup> <sup>þ</sup> GTϵ0, (22)

<sup>T</sup> <sup>¼</sup> <sup>δ</sup>Cð Þ <sup>1</sup> � <sup>G</sup>δ<sup>C</sup> �<sup>1</sup> (23)

t<sup>α</sup> ¼ τ: (25)

, finding the exact solution

<sup>C</sup><sup>∗</sup> <sup>¼</sup> <sup>C</sup><sup>0</sup> <sup>þ</sup> h i <sup>T</sup> <sup>=</sup>ð Þ <sup>1</sup> <sup>þ</sup> h i GT �<sup>1</sup> (24)

<sup>t</sup><sup>α</sup> <sup>¼</sup> <sup>δ</sup>C<sup>α</sup> <sup>þ</sup> <sup>δ</sup>CαGt<sup>α</sup> <sup>¼</sup> <sup>δ</sup>Cαð Þ <sup>1</sup> � <sup>G</sup>δC<sup>α</sup> �<sup>1</sup> (26)

<sup>δ</sup>C<sup>α</sup> <sup>¼</sup> <sup>δ</sup><sup>C</sup> <sup>¼</sup> <sup>C</sup> � <sup>C</sup><sup>0</sup> (27)

<sup>C</sup><sup>∗</sup> <sup>¼</sup> <sup>C</sup><sup>0</sup> <sup>þ</sup> h i<sup>τ</sup> ð Þ <sup>1</sup> <sup>þ</sup> h i <sup>G</sup><sup>τ</sup> �<sup>1</sup> (28)

h iτ ¼ 0 (29)

¼ 0 (30)

<sup>11</sup> <sup>þ</sup> <sup>2</sup>C<sup>O</sup> <sup>12</sup> =3, the

¼ 0 (31)

Here, ϵ<sup>0</sup> and G are the strain and modified Green's function of the medium

Here, І is equivalent to the unit tensor. Combining Eqs. (21) and (22), we get:

of h i T and h i GT for realistic cases is impossible. By neglecting the intergranular scattering that may occur in some cases in the form of a grain-to-grain positionorientation correlation function however, the T-matrix can be written in terms of

> T ≈ ∑α

For single-phase polycrystal, the self-consistent solution of Eq. (11) can be

For a multi-phase polycrystals, a solution to Eq. (4) can be found by evaluating

The application of the method to both single-phase aggregates and multi-phase composites is relevant to many multi-component alloys. For a single-phase polycrystal with cubic symmetry [16, 20] to the following expression for B<sup>∗</sup> and

<sup>μ</sup><sup>∗</sup><sup>2</sup> � 3C44 BO <sup>þ</sup> 4C<sup>00</sup> <sup>μ</sup><sup>∗</sup> � 6BOC44C<sup>00</sup>

the volume fraction and <sup>τ</sup> of each phase i u<sup>2</sup> and <sup>τ</sup><sup>2</sup> ð Þ, respectively [19], via:

∑ i v2 τ2 

In Eq. (31), three independent single-crystal elastic constants (C11, C12, C44<sup>Þ</sup> define the single-crystal bulk modulus <sup>B</sup><sup>o</sup> <sup>¼</sup> <sup>C</sup><sup>0</sup>

tetragonal shear modulus C<sup>0</sup> ¼ ð Þ C<sup>11</sup>�C<sup>12</sup> =2 and trigonal shear modulus, C44,

defined by CO, and the T-matrix given by:

DOI: http://dx.doi.org/10.5772/intechopen.81340

Concept of Phase Transition Based on Elastic Systematics

single-grain T-matrix (tα) for each grain α

Inserting Eq. (21) into (22) leads to:

obtained by choosing a C<sup>∗</sup> that satisfies:

<sup>8</sup>μ<sup>∗</sup><sup>3</sup> <sup>þ</sup> 9B0 <sup>þ</sup> 4C<sup>00</sup> <sup>0</sup>

<sup>44</sup> <sup>¼</sup> <sup>μ</sup><sup>0</sup> <sup>þ</sup> h i <sup>τ</sup><sup>44</sup>

<sup>1</sup>þG44τ44.

where

<sup>μ</sup><sup>∗</sup> : <sup>B</sup><sup>∗</sup> <sup>¼</sup> Bo

<sup>μ</sup><sup>∗</sup> <sup>¼</sup> <sup>C</sup><sup>∗</sup>

59

Although Eq. (21) constitutes an exact solution for C<sup>∗</sup>

∑ α

#### 3.2 Relative stability of polycrystalline materials

In the case of multi-phase stability, multi-phase composites can be obtained based on multiple scattering theory. For example, polycrystalline materials consisting of two phases, namely cubic and orthorhombic phases can be obtained by homogenising the integral elastic response of the multi-phase polycrystalline samples, following the effective medium approach originally applied by Zeller and Dederichs [13] to determine elastic properties of single-phase polycrystals with cubic symmetry. This type of concept was generalised by Middya and Basu [14] and further extended by Middya [15] and by Raabe et al. [16] to multi-phase composites to determine: (i) the elastic single constants and (ii) the volume fraction of the components within a self-consistent T-matrix solution for the effective medium elastic properties of hexagonal, and orthorhombic polycrystals.

The subset of supercells or cubic and orthorhombic symmetries consisting of three (C11, C12, C44) and nine (C11, C12, C13, C22, C23, C33, C44, C55, C66) elastic constants, respectively, was calculated by employing the methodology explained in [16–18] for the elastic properties of the multi-component alloys. This can be viewed as a macroscopic homogeneous effective medium consisting of microscopic fluctuations and characterised by an effective stiffness of Cijkl defined by:

$$
\langle \sigma\_{\vec{\eta}}(\mathbf{r}) \rangle = \mathbf{C}\_{\vec{\eta}kl} \langle c\_{kl}(\mathbf{r}) \rangle \tag{19}
$$

Here, Cijkl is the local elastic constant tensor with <sup>σ</sup>ijð Þ<sup>r</sup> and h i <sup>ϵ</sup>klð Þ<sup>r</sup> as the local stress and strain field at a point r, respectively, and the angular brackets denoting ensemble averages. A repeated index implies the usual summation convention. The effective stiffness of Cijkl is defined by:

$$
\langle \sigma\_{\vec{\eta}}(\mathbf{r}) \rangle = \mathcal{C}^\*\_{\vec{\eta}kl} \langle \varepsilon\_{kl}(\mathbf{r}) \rangle \tag{20}
$$

Since the aggregate represents a body in equilibrium, σijð Þr ∣j ¼ 0, where <sup>∣</sup><sup>j</sup> <sup>¼</sup> <sup>∂</sup>=∂rj and the local elastic constant tensor can now be decomposed into an arbitrary constants part (Co ijkl) and a fluctuating part—δCð Þr .

$$\mathbf{C}\_{ijkl}(r) = \mathbf{C}\_{ijkl}^{\circ} + \delta \mathbf{C}\_{ijkl}(r) \tag{21}$$

As shown in [16], an integral part of Eq. (19) is the interactive equivalent solution representing the resulting local strain ϵ distribution (in a short notation) as: Concept of Phase Transition Based on Elastic Systematics DOI: http://dx.doi.org/10.5772/intechopen.81340

$$
\epsilon = \epsilon^0 + GT\epsilon^0,\tag{22}
$$

Here, ϵ<sup>0</sup> and G are the strain and modified Green's function of the medium defined by CO, and the T-matrix given by:

$$T = \delta \mathbf{C} (\mathbf{1} - \mathbf{G} \delta \mathbf{C})^{-1} \tag{23}$$

Here, І is equivalent to the unit tensor. Combining Eqs. (21) and (22), we get:

$$\mathbf{C}^\* = \mathbf{C}^0 + \langle T \rangle / (\mathbf{1} + \langle GT \rangle)^{-1} \tag{24}$$

Although Eq. (21) constitutes an exact solution for C<sup>∗</sup> , finding the exact solution of h i T and h i GT for realistic cases is impossible. By neglecting the intergranular scattering that may occur in some cases in the form of a grain-to-grain positionorientation correlation function however, the T-matrix can be written in terms of single-grain T-matrix (tα) for each grain α

$$T \approx \sum\_{a} t\_a = \text{\textquotedblleft}.\tag{25}$$

where

A crystal lattice is said to be stable in the absence of external load (unstressed

In the case of multi-phase stability, multi-phase composites can be obtained

consisting of two phases, namely cubic and orthorhombic phases can be obtained by homogenising the integral elastic response of the multi-phase polycrystalline samples, following the effective medium approach originally applied by Zeller and Dederichs [13] to determine elastic properties of single-phase polycrystals with cubic symmetry. This type of concept was generalised by Middya and Basu [14] and further extended by Middya [15] and by Raabe et al. [16] to multi-phase composites to determine: (i) the elastic single constants and (ii) the volume fraction of the components within a self-consistent T-matrix solution for the effective medium

The subset of supercells or cubic and orthorhombic symmetries consisting of three (C11, C12, C44) and nine (C11, C12, C13, C22, C23, C33, C44, C55, C66) elastic constants, respectively, was calculated by employing the methodology explained in [16–18] for the elastic properties of the multi-component alloys. This can be viewed as a macroscopic homogeneous effective medium consisting of microscopic fluctu-

Here, Cijkl is the local elastic constant tensor with <sup>σ</sup>ijð Þ<sup>r</sup> and h i <sup>ϵ</sup>klð Þ<sup>r</sup> as the local stress and strain field at a point r, respectively, and the angular brackets denoting ensemble averages. A repeated index implies the usual summation convention. The

<sup>σ</sup>ijð Þ<sup>r</sup> <sup>¼</sup> <sup>C</sup><sup>∗</sup>

Cijklð Þ¼ <sup>r</sup> Co

Since the aggregate represents a body in equilibrium, σijð Þr ∣j ¼ 0, where <sup>∣</sup><sup>j</sup> <sup>¼</sup> <sup>∂</sup>=∂rj and the local elastic constant tensor can now be decomposed into an

As shown in [16], an integral part of Eq. (19) is the interactive equivalent solution representing the resulting local strain ϵ distribution (in a short notation) as:

ijkl) and a fluctuating part—δCð Þr .

<sup>σ</sup>ijð Þ<sup>r</sup> <sup>¼</sup> Cijklh i <sup>ϵ</sup>klð Þ<sup>r</sup> (19)

ijklh i ϵklð Þr (20)

ijkl þ δCijklð Þr (21)

based on multiple scattering theory. For example, polycrystalline materials

condition) and in the harmonic approximation [13] if and only if it has both dynamic and elastic stability. Dynamic stability implies that its phonon modes have positive frequencies for all wave vectors, while its elastic stability is dependent on elastic energy given by Eq. (15) being always positive (U > 0,∀ε 6¼ 0). Elastic stability criterion is mathematically equivalent to the following necessary and sufficient conditions: the elastic matrix C is definite exactly positive and all Eigen values of matrix C are positive; all the leading principal and arbitrary minors of matrix C are all positive. The closed form expressions for necessary and sufficient elastic stability criteria for other crystal lattices have been studied. While the stability criterion is linear for some crystal lattices, it is quadratic and even polynomial for others. Thus, the mechanical stability of a crystal is combination of the elastic constant and Born's stability criteria. The elastic constant of a stable crystal must

satisfy the Born's criteria to prove its mechanical stability.

Elasticity of Materials ‐ Basic Principles and Design of Structures

elastic properties of hexagonal, and orthorhombic polycrystals.

ations and characterised by an effective stiffness of Cijkl defined by:

effective stiffness of Cijkl is defined by:

arbitrary constants part (Co

58

3.2 Relative stability of polycrystalline materials

$$\mathbf{t}\_a = \delta \mathbf{C}\_a + \delta \mathbf{C}\_a \mathbf{G} \mathbf{t}\_a = \delta \mathbf{C}\_a (\mathbf{1} - \mathbf{G} \delta \mathbf{C}\_a)^{-1} \tag{26}$$

$$\sum\_{a} \delta \mathbf{C}\_{a} = \delta \mathbf{C} = \mathbf{C} - \mathbf{C}^{0} \tag{27}$$

Inserting Eq. (21) into (22) leads to:

$$\mathbf{C}^\* = \mathbf{C}^0 + \langle \mathbf{r} \rangle (\mathbf{1} + \langle G\mathbf{r} \rangle)^{-1} \tag{28}$$

For single-phase polycrystal, the self-consistent solution of Eq. (11) can be obtained by choosing a C<sup>∗</sup> that satisfies:

$$
\langle \pi \rangle = \mathbf{0} \tag{29}
$$

For a multi-phase polycrystals, a solution to Eq. (4) can be found by evaluating the volume fraction and <sup>τ</sup> of each phase i u<sup>2</sup> and <sup>τ</sup><sup>2</sup> ð Þ, respectively [19], via:

$$
\left\langle \sum\_{i} v^{2} \tau^{2} \right\rangle = 0 \tag{30}
$$

The application of the method to both single-phase aggregates and multi-phase composites is relevant to many multi-component alloys. For a single-phase polycrystal with cubic symmetry [16, 20] to the following expression for B<sup>∗</sup> and <sup>μ</sup><sup>∗</sup> : <sup>B</sup><sup>∗</sup> <sup>¼</sup> Bo

$$\mathbf{8}\boldsymbol{\mu}^{\*3} + \left(\mathbf{9B}\_0 + \mathbf{4C}^{\boldsymbol{\prime}^{\prime}}\right)\boldsymbol{\mu}^{\*2} - \mathbf{3C}\_{44}\left(\mathbf{B}\_0 + \mathbf{4C}^{\boldsymbol{\prime}^{\prime}}\right)\boldsymbol{\mu}^{\*} - \mathbf{6B}\_0\mathbf{C}\_{44}\mathbf{C}^{\boldsymbol{\prime}^{\prime}} = \mathbf{0} \tag{31}$$

In Eq. (31), three independent single-crystal elastic constants (C11, C12, C44<sup>Þ</sup> define the single-crystal bulk modulus <sup>B</sup><sup>o</sup> <sup>¼</sup> <sup>C</sup><sup>0</sup> <sup>11</sup> <sup>þ</sup> <sup>2</sup>C<sup>O</sup> <sup>12</sup> =3, the tetragonal shear modulus C<sup>0</sup> ¼ ð Þ C<sup>11</sup>�C<sup>12</sup> =2 and trigonal shear modulus, C44, <sup>μ</sup><sup>∗</sup> <sup>¼</sup> <sup>C</sup><sup>∗</sup> <sup>44</sup> <sup>¼</sup> <sup>μ</sup><sup>0</sup> <sup>þ</sup> h i <sup>τ</sup><sup>44</sup> <sup>1</sup>þG44τ44.

Here, C<sup>∗</sup> <sup>44</sup> is the homogenised bulk modulus. The details of the equation for calculating the elastic constants of polycrystals alloy with hexagonal symmetry have been explained elsewhere by [20], and the details here concern polycrystals with orthorhombic symmetry. Eqs. (29) and (30) are reduced to a set of coupled equations for B<sup>∗</sup> and μ<sup>∗</sup>:

$$\mathbf{0} = \Re(K\_V - B^\*) + \mathbf{2}\beta(d - c + e) + \mathbf{3}\boldsymbol{\beta}^2 \boldsymbol{\Delta}' \tag{32}$$

<sup>15</sup>τ<sup>44</sup> <sup>¼</sup> <sup>a</sup> � <sup>b</sup> <sup>þ</sup> <sup>β</sup>ð Þþ <sup>2</sup><sup>d</sup> � <sup>2</sup><sup>c</sup> � <sup>e</sup> <sup>3</sup>γð Þþ <sup>d</sup> � <sup>c</sup> <sup>þ</sup> <sup>e</sup> υβΔ<sup>0</sup>

� � <sup>þ</sup> β βð Þ <sup>þ</sup> <sup>2</sup><sup>γ</sup> ð Þ� <sup>c</sup> � <sup>d</sup> <sup>2</sup>eβγ � <sup>1</sup>

<sup>C</sup><sup>55</sup> � <sup>G</sup><sup>e</sup> <sup>O</sup> <sup>1</sup> � <sup>2</sup>k C<sup>55</sup> � <sup>G</sup><sup>e</sup> <sup>O</sup> � � <sup>þ</sup>

<sup>9</sup> Kv � <sup>B</sup>e<sup>0</sup> � � <sup>þ</sup> <sup>2</sup>βð Þþ <sup>d</sup> � <sup>c</sup> <sup>þ</sup> <sup>e</sup> <sup>3</sup>β<sup>2</sup>

3 1 � αβ � <sup>9</sup><sup>γ</sup> KV � <sup>B</sup>e<sup>0</sup> � � <sup>þ</sup> β βð Þ <sup>þ</sup> <sup>2</sup><sup>γ</sup> ð Þ� <sup>c</sup> � <sup>d</sup> <sup>2</sup>eβγ � <sup>1</sup>ηβ2Δ<sup>0</sup>

Here, β is defined in Eq. (28), η is defined in Eq. (29), and Δ<sup>0</sup> in Eq. (23). Here,

replaces G<sup>∗</sup> and B<sup>∗</sup> in the equations for β, υ and Δ<sup>0</sup>

Poisson's ratio ð Þ<sup>υ</sup> <sup>∗</sup> for (an elastically isotropic) polycrystal can be determined using standard elasticity relationships. The homogenised polycrystalline Young's modulus

G∗

Be∗

<sup>E</sup><sup>∗</sup> <sup>¼</sup> <sup>9</sup>Be<sup>∗</sup>

<sup>G</sup><sup>∗</sup> <sup>¼</sup> <sup>3</sup>Ee<sup>∗</sup>

4. Correlation of elastic constants with properties of polycrystalline

materials with respect to their dependency on specific crystal structure.

Strength and ductility have always been one of the crucial issues to study for metal materials. The tendency of materials to be ductile or brittle is being predicted using models based on elastic constants. Some of these include that of Pugh criterion [23] and Cauchy pressure as defined by Pettifor [24]. Pugh proposed an empirical relationship between the plasticity and fracture properties showing the ratio G/B indicates the intrinsic ability of a crystalline metal to resist fracture and deform plastically [25]. This represents a competition between plasticity and fracture considering that B and G represent resistance to fracture and plastic deformation, respectively. Thus, the force required to propagate a dislocation is proportional

3Be∗

9Be∗

In many problem relating to polycrystalline or anisotropic materials, it is customary to make use of the properties in an elastically isotropic materials. Most of the common metals and engineering alloys, however, exhibit a marked degree of anisotropy in their single-crystal elastic behaviour and it is therefore more desirable to obtain same on the bases of anisotropic elastic property. The fundamental factors determining the intrinsic plasticity or brittleness behaviour in solids have great link with interatomic potentials, for instance, there is a correlation with the ratio of the elastic shear modulus μ to the bulk modulus B. It is evident, elastic moduli show trends with a range of properties, including hardness, yield strength, toughness and fragility [21, 22]. In this section, for limitation of space, we will, in particular, consider elastic aspect of polycrystals

have been determined, the homogenised Young's modulus Ee

h i (50)

<sup>3</sup> υβ<sup>2</sup> Δ00

<sup>C</sup><sup>66</sup> � <sup>e</sup><sup>μ</sup>

Δ0

<sup>þ</sup> <sup>G</sup><sup>∗</sup> (51)

� <sup>E</sup>e<sup>∗</sup> (52)

<sup>1</sup> � <sup>2</sup><sup>β</sup> <sup>C</sup><sup>66</sup> � <sup>G</sup><sup>e</sup> <sup>O</sup> � �

O

1

(49)

CA

. As soon as G<sup>∗</sup>

and

3

� �<sup>∗</sup>

1 � αβ � 9γ Kv � Be<sup>0</sup>

DOI: http://dx.doi.org/10.5772/intechopen.81340

<sup>1</sup> � <sup>2</sup>k C<sup>44</sup> � <sup>G</sup><sup>e</sup> <sup>O</sup> � � <sup>þ</sup>

Concept of Phase Transition Based on Elastic Systematics

<sup>þ</sup><sup>3</sup> <sup>C</sup><sup>44</sup> � <sup>G</sup><sup>e</sup> <sup>O</sup>

0

B@

τ<sup>11</sup> þ 2τ<sup>12</sup> ¼

again <sup>G</sup><sup>∗</sup> and <sup>B</sup>e<sup>∗</sup>

is calculated using:

materials

61

4.1 Elasticity and ductility criteria

and <sup>B</sup>e<sup>∗</sup>

$$\begin{aligned} \text{RO} &= \frac{a - b + \beta(2d - 2c - e) + 3\gamma(d - c + e) + \eta\beta\Delta'}{\mathbf{1} - a\beta - 9\gamma(k\_v - B\_0) + \beta(\beta + 2\gamma)(c - d) - 2e\beta\gamma - \frac{1}{3}\eta\beta^2\Delta'} + \\ &\quad \text{3} \left(\frac{\mathbf{C}\_{44} - \mu^O}{\mathbf{1} - 2\beta(\mathbf{C}\_{44} - \mu^O)} + \frac{\mathbf{C}\_{55} - \mu^O}{\mathbf{1} - 2\beta(\mathbf{C}\_{55} - \mu^O)} + \frac{\mathbf{C}\_{66} - \mu^O}{\mathbf{1} - 2\beta(\mathbf{C}\_{66} - \mu^O)}\right) \end{aligned} \tag{33}$$

where

$$\Re \mathbf{K\_{v}} = \mathbf{C\_{11}} + \mathbf{C\_{22}} + \mathbf{C\_{33}} + \mathcal{Z}(\mathbf{C\_{12}} + \mathbf{C\_{13}} + \mathbf{C\_{23}}),\tag{34}$$

$$B = \mathfrak{H}\_3^\!(\mathbb{C}\_{11} + \mathbb{Z}\mathbb{C}\_{12})\mu^\* = \mathbb{C}\_{44} \tag{35}$$

$$\chi = \mathfrak{P}(\mathfrak{q} - \mathfrak{B}\mathfrak{k}) \tag{36}$$

$$
\sigma = \delta \mathbf{C}\_{11} + \delta \mathbf{C}\_{22} + \delta \mathbf{C}\_{33} \\
\mathbf{b} = \delta \mathbf{C}\_{12} + \delta \mathbf{C}\_{13} + \delta \mathbf{C}\_{23} \tag{37}
$$

$$
\omega = \delta \mathbf{C}\_{11} \delta \mathbf{C}\_{22} + \delta \mathbf{C}\_{11} \delta \mathbf{C}\_{33} + \delta \mathbf{C}\_{22} \delta \mathbf{C}\_{33} \\
\mu d = \delta \mathbf{C}\_{12}^2 + \delta \mathbf{C}\_{13}^2 + \delta \mathbf{C}\_{23}^2 \tag{38}
$$

$$\mathbf{e} = \delta \mathbf{C}\_{12} \delta \mathbf{C}\_{13} + \delta \mathbf{C}\_{12} \delta \mathbf{C}\_{23} + \delta \mathbf{C}\_{13} \delta \mathbf{C}\_{23} - \delta \mathbf{C}\_{11} \delta \mathbf{C}\_{23} - \delta \mathbf{C}\_{22} \delta \mathbf{C}\_{13} - \delta \mathbf{C}\_{33} \delta \mathbf{C}\_{12} \tag{39}$$

$$
\Delta' = \delta \mathbf{C}\_{11} \delta \mathbf{C}\_{22} \delta \mathbf{C}\_{33} + 2 \delta \mathbf{C}\_{12} \delta \mathbf{C}\_{13} \delta \mathbf{C}\_{23} - \delta \mathbf{C}\_{11} \delta \mathbf{C}\_{23}^2 - \delta \mathbf{C}\_{22} \delta \mathbf{C}\_{13}^2 - \delta \mathbf{C}\_{33} \delta \mathbf{C}\_{12}^2 \tag{40}
$$

$$\delta \mathbf{C}\_{11} = \mathbf{C}\_{11} - \mathbf{C}\_{11}^{0} = \mathbf{C}\_{11} - K^{0} - \frac{4}{3} \mu^{0};\\ \delta \mathbf{C}\_{22} = \mathbf{C}\_{22} = \mathbf{C}\_{22} - K^{0} - \frac{4}{3} \mu^{0} \tag{41}$$

$$\delta \mathbf{C}\_{33} = \mathbf{C}\_{33} - K^0 - \frac{4}{3} \mu^0 ; \delta \mathbf{C}\_{12} = \mathbf{C}\_{12} - \mathbf{C}\_{12}^0 = \mathbf{C}\_{12} - K^0 + \frac{4}{3} \mu^0 \tag{42}$$

$$\delta \mathbf{C}\_{13} = \mathbf{C}\_{13} - K^0 + \frac{4}{3} \mu^0 ; \delta \mathbf{C}\_{23} = \mathbf{C}\_{23} - K^0 + \frac{4}{3} \mu^0 \tag{43}$$

$$\beta = \frac{-\Im(B^\* + 2\mu^\*)}{5\mu^\*(3B^\* + 4\mu^\*)},\tag{44}$$

$$
\mathfrak{n}/\mathfrak{3} = -\mathbf{1}/\mathfrak{3}\mathfrak{B}^\* + 4\mu^\*,\tag{45}
$$

$$\mathbf{C\_{66}} = (\mathbf{1}/\mathbf{2})(\mathbf{C\_{11}} - \mathbf{C\_{12}}) \tag{46}$$

and orthorhombic symmetry has nine of the single crystal elastic constants, namely: C11, C22,C33, C44C55, C66, C12, C23 and C13.

The elastic constants of a multi-phase polycrystals were determined directly by coupling Eq. (13) for τ<sup>44</sup> and the ð Þ τ<sup>11</sup> þ 2τ<sup>12</sup> components of the T-matrix. For materials with cubic symmetry, the equation is defined as:

$$\mathfrak{Tr}\_{44} = \left(\frac{1}{\mathcal{C}\_{11} - \mathcal{C}\_{12} - \widetilde{\mathcal{G}}}^{-\beta}\right)^{-1} + \mathfrak{Z}\left(\frac{1}{\mathcal{C}\_{44} - \widetilde{\mathcal{G}}}^{-2\beta}\right)^{-1} \tag{47}$$

$$
\pi\_{11} + 2\pi\_{12} = \frac{\Im(\mathbf{C}\_{11} + \mathbf{2C}\_{12}) - \mathfrak{B}\widetilde{\boldsymbol{B}}^\*}{\Im - (\mathbf{C}\_{11} + \mathbf{2C}\_{12}) - \mathfrak{B}\widetilde{\boldsymbol{B}}^\*} \tag{48}
$$

This is where <sup>β</sup> is defined in Eq. (31) with <sup>G</sup><sup>e</sup> <sup>∗</sup> and <sup>B</sup>e<sup>∗</sup> replacing G<sup>∗</sup> and B<sup>∗</sup>. For materials with orthorhombic symmetry, the equation reads:

Concept of Phase Transition Based on Elastic Systematics DOI: http://dx.doi.org/10.5772/intechopen.81340

Here, C<sup>∗</sup>

tions for B<sup>∗</sup> and μ<sup>∗</sup>:

where

<sup>3</sup> <sup>C</sup><sup>44</sup> � <sup>μ</sup><sup>O</sup>

<sup>1</sup> � <sup>2</sup><sup>β</sup> <sup>C</sup><sup>44</sup> � <sup>μ</sup><sup>O</sup> ð Þ <sup>þ</sup>

Elasticity of Materials ‐ Basic Principles and Design of Structures

<sup>44</sup> is the homogenised bulk modulus. The details of the equation for calculating the elastic constants of polycrystals alloy with hexagonal symmetry have been explained elsewhere by [20], and the details here concern polycrystals with orthorhombic symmetry. Eqs. (29) and (30) are reduced to a set of coupled equa-

Δ<sup>0</sup> (32)

(33)

<sup>3</sup> <sup>η</sup>β<sup>2</sup> Δ00 þ

<sup>C</sup><sup>66</sup> � <sup>μ</sup><sup>O</sup> 1 � 2β C<sup>66</sup> � μ<sup>O</sup> ð Þ

<sup>0</sup> <sup>¼</sup> <sup>9</sup> KV � <sup>B</sup><sup>∗</sup> ð Þþ <sup>2</sup>βð Þþ <sup>d</sup> � <sup>c</sup> <sup>þ</sup> <sup>e</sup> <sup>3</sup>β<sup>2</sup>

<sup>C</sup><sup>55</sup> � <sup>μ</sup><sup>O</sup> <sup>1</sup> � <sup>2</sup><sup>β</sup> <sup>C</sup><sup>55</sup> � <sup>μ</sup><sup>O</sup> ð Þ <sup>þ</sup>

e ¼ δC12δC13 þ δC12δC23 þ δC13δC23 � δC11δC23 � δC22δC13 � δC33δC12 (39)

<sup>3</sup> <sup>μ</sup>0;δC<sup>12</sup> <sup>¼</sup> <sup>C</sup><sup>12</sup> � <sup>C</sup><sup>O</sup>

4

<sup>β</sup> <sup>¼</sup> �<sup>3</sup> <sup>B</sup><sup>∗</sup> <sup>þ</sup> <sup>2</sup>μ<sup>∗</sup> ð Þ

<sup>η</sup>=<sup>3</sup> ¼ �1=3B<sup>∗</sup> <sup>þ</sup> <sup>4</sup>μ<sup>∗</sup>

and orthorhombic symmetry has nine of the single crystal elastic constants,

<sup>τ</sup><sup>11</sup> <sup>þ</sup> <sup>2</sup>τ<sup>12</sup> <sup>¼</sup> <sup>3</sup>ð Þ� <sup>C</sup><sup>11</sup> <sup>þ</sup> <sup>2</sup>C<sup>12</sup> <sup>9</sup>Be<sup>∗</sup>

coupling Eq. (13) for τ<sup>44</sup> and the ð Þ τ<sup>11</sup> þ 2τ<sup>12</sup> components of the T-matrix. For

The elastic constants of a multi-phase polycrystals were determined directly by

9Kv ¼ C11 þ C22 þ C33 þ 2 Cð Þ <sup>12</sup> þ C13 þ C23 , (34)

a ¼ δC<sup>11</sup> þ δC<sup>22</sup> þ δC33;b ¼ δC<sup>12</sup> þ δC<sup>13</sup> þ δC<sup>23</sup> (37)

<sup>=</sup>3ð Þ <sup>C</sup><sup>11</sup> <sup>þ</sup> <sup>2</sup>C<sup>12</sup> <sup>μ</sup><sup>∗</sup> <sup>¼</sup> <sup>C</sup><sup>44</sup> (35)

<sup>12</sup> <sup>þ</sup> <sup>δ</sup>C<sup>2</sup>

<sup>23</sup> � <sup>δ</sup>C22δC<sup>2</sup>

<sup>12</sup> <sup>¼</sup> <sup>C</sup><sup>12</sup> � <sup>K</sup><sup>0</sup> <sup>þ</sup>

<sup>3</sup> <sup>μ</sup>0;δC<sup>22</sup> <sup>¼</sup> <sup>C</sup><sup>22</sup> <sup>¼</sup> <sup>C</sup><sup>22</sup> � <sup>K</sup><sup>0</sup> � <sup>4</sup>

<sup>3</sup> <sup>μ</sup>0;δC<sup>23</sup> <sup>¼</sup> <sup>C</sup><sup>23</sup> � <sup>K</sup><sup>0</sup> <sup>þ</sup>

=9ð Þ η � 3β (36)

<sup>13</sup> <sup>þ</sup> <sup>δ</sup>C<sup>2</sup>

<sup>13</sup> � <sup>δ</sup>C33δC<sup>2</sup>

4

, (45)

4

<sup>5</sup>μ<sup>∗</sup> <sup>3</sup>B<sup>∗</sup> <sup>þ</sup> <sup>4</sup>μ<sup>∗</sup> ð Þ, (44)

C<sup>66</sup> ¼ ð Þ 1=2 ð Þ C<sup>11</sup> � C<sup>12</sup> (46)

<sup>þ</sup> <sup>3</sup> <sup>1</sup>

and <sup>B</sup>e<sup>∗</sup>

<sup>C</sup><sup>44</sup> � <sup>G</sup><sup>e</sup> <sup>∗</sup>

�2<sup>β</sup> !�<sup>1</sup>

<sup>3</sup> � ð Þ� <sup>C</sup><sup>11</sup> <sup>þ</sup> <sup>2</sup>C<sup>12</sup> <sup>3</sup>Be<sup>∗</sup> (48)

replacing G<sup>∗</sup> and B<sup>∗</sup>. For

<sup>23</sup> (38)

<sup>12</sup> (40)

(47)

<sup>3</sup> <sup>μ</sup><sup>0</sup> (41)

<sup>3</sup> <sup>μ</sup><sup>0</sup> (42)

<sup>3</sup> <sup>μ</sup><sup>0</sup> (43)

� �

<sup>0</sup> <sup>¼</sup> <sup>a</sup> � <sup>b</sup> <sup>þ</sup> <sup>β</sup>ð Þþ <sup>2</sup><sup>d</sup> � <sup>2</sup><sup>c</sup> � <sup>e</sup> <sup>3</sup>γð Þþ <sup>d</sup> � <sup>c</sup> <sup>þ</sup> <sup>e</sup> <sup>η</sup>βΔ<sup>0</sup> <sup>1</sup> � αβ � <sup>9</sup>γð Þþ kv � <sup>B</sup><sup>0</sup> β βð Þ <sup>þ</sup> <sup>2</sup><sup>γ</sup> ð Þ� <sup>c</sup> � <sup>d</sup> <sup>2</sup>eβγ � <sup>1</sup>

B ¼ <sup>1</sup>

<sup>c</sup> <sup>¼</sup> <sup>δ</sup>C11δC<sup>22</sup> <sup>þ</sup> <sup>δ</sup>C11δC<sup>33</sup> <sup>þ</sup> <sup>δ</sup>C22δC33;<sup>d</sup> <sup>¼</sup> <sup>δ</sup>C<sup>2</sup>

<sup>11</sup> <sup>¼</sup> <sup>C</sup><sup>11</sup> � <sup>K</sup><sup>0</sup> � <sup>4</sup>

<sup>δ</sup>C<sup>13</sup> <sup>¼</sup> <sup>C</sup><sup>13</sup> � <sup>K</sup><sup>0</sup> <sup>þ</sup>

namely: C11, C22,C33, C44C55, C66, C12, C23 and C13.

<sup>5</sup>τ<sup>44</sup> <sup>¼</sup> <sup>1</sup>

This is where <sup>β</sup> is defined in Eq. (31) with <sup>G</sup><sup>e</sup> <sup>∗</sup>

60

materials with orthorhombic symmetry, the equation reads:

materials with cubic symmetry, the equation is defined as:

<sup>C</sup><sup>11</sup> � <sup>C</sup><sup>12</sup> � <sup>G</sup><sup>e</sup> <sup>∗</sup>

�<sup>β</sup> !�<sup>1</sup>

<sup>Δ</sup><sup>0</sup> <sup>¼</sup> <sup>δ</sup>C11δC22δC<sup>33</sup> <sup>þ</sup> <sup>2</sup>δC12δC13δC<sup>23</sup> � <sup>δ</sup>C11δC<sup>2</sup>

<sup>δ</sup>C<sup>11</sup> <sup>¼</sup> <sup>C</sup><sup>11</sup> � <sup>C</sup><sup>O</sup>

<sup>δ</sup>C<sup>33</sup> <sup>¼</sup> <sup>C</sup><sup>33</sup> � <sup>K</sup><sup>0</sup> � <sup>4</sup>

γ ¼ <sup>1</sup>

$$\begin{aligned} \text{15\tau}\_{44} &= \frac{a - b + \beta(2d - 2c - e) + 3\gamma(d - c + e) + v\beta\Delta'}{1 - a\beta - 9\gamma\left(K\_v - \tilde{B}\_0\right) + \beta(\beta + 2\gamma)(c - d) - 2e\beta\gamma - \frac{1}{3}v\vartheta^2\Delta''} \\ &+ 3\left(\frac{C\_{44} - \tilde{G}^O}{1 - 2k\left(C\_{44} - \tilde{G}^O\right)} + \frac{C\_{55} - \tilde{G}^O}{1 - 2k\left(C\_{55} - \tilde{G}^O\right)} + \frac{C\_{66} - \tilde{\mu}^O}{1 - 2\beta\left(C\_{66} - \tilde{G}^O\right)}\right) \end{aligned} \tag{49}$$
 
$$\tau\_{11} + 2\tau\_{12} = \frac{9\left(K\_v - \tilde{B}^O\right) + 2\beta(d - c + e) + 3\beta^2\Delta'}{3\left[1 - a\beta - 9\gamma\left(K\_V - \tilde{B}^O\right) + \beta(\beta + 2\gamma)(c - d) - 2e\beta\gamma - \frac{10\beta^2\Delta'}{3}\right]} \tag{50}$$

Here, β is defined in Eq. (28), η is defined in Eq. (29), and Δ<sup>0</sup> in Eq. (23). Here, again <sup>G</sup><sup>∗</sup> and <sup>B</sup>e<sup>∗</sup> replaces G<sup>∗</sup> and B<sup>∗</sup> in the equations for β, υ and Δ<sup>0</sup> . As soon as G<sup>∗</sup> and <sup>B</sup>e<sup>∗</sup> have been determined, the homogenised Young's modulus Ee � �<sup>∗</sup> and Poisson's ratio ð Þ<sup>υ</sup> <sup>∗</sup> for (an elastically isotropic) polycrystal can be determined using standard elasticity relationships. The homogenised polycrystalline Young's modulus is calculated using:

$$E^\* = \frac{\mathfrak{B}\tilde{\boldsymbol{B}}^\* G^\*}{\mathfrak{B}\tilde{\boldsymbol{B}}^\* + G^\*} \tag{51}$$

$$G^\* = \frac{\mathbf{3}\widetilde{E}^\*\widetilde{B}^\*}{\mathbf{9}\widetilde{B}^\* - \widetilde{E}^\*} \tag{52}$$

#### 4. Correlation of elastic constants with properties of polycrystalline materials

In many problem relating to polycrystalline or anisotropic materials, it is customary to make use of the properties in an elastically isotropic materials. Most of the common metals and engineering alloys, however, exhibit a marked degree of anisotropy in their single-crystal elastic behaviour and it is therefore more desirable to obtain same on the bases of anisotropic elastic property. The fundamental factors determining the intrinsic plasticity or brittleness behaviour in solids have great link with interatomic potentials, for instance, there is a correlation with the ratio of the elastic shear modulus μ to the bulk modulus B. It is evident, elastic moduli show trends with a range of properties, including hardness, yield strength, toughness and fragility [21, 22]. In this section, for limitation of space, we will, in particular, consider elastic aspect of polycrystals materials with respect to their dependency on specific crystal structure.

#### 4.1 Elasticity and ductility criteria

Strength and ductility have always been one of the crucial issues to study for metal materials. The tendency of materials to be ductile or brittle is being predicted using models based on elastic constants. Some of these include that of Pugh criterion [23] and Cauchy pressure as defined by Pettifor [24]. Pugh proposed an empirical relationship between the plasticity and fracture properties showing the ratio G/B indicates the intrinsic ability of a crystalline metal to resist fracture and deform plastically [25]. This represents a competition between plasticity and fracture considering that B and G represent resistance to fracture and plastic deformation, respectively. Thus, the force required to propagate a dislocation is proportional to Gb where b is the Burgers vector. This implies that a material with high value of the ratio tends to be brittle (fracture is easier and plasticity is much less), while a low value indicates ductility (plasticity is easier and fracture is not). Fracture strength is also proportional to Ba (a, is the lattice constant) since B is related to surface energy, which indicates brittle fracture strength.

These empirical observations implicate G/B as explaining well brittle or tough behaviour [19, 26]. Pugh's criterion is the most widely used model to predict plastic behaviour of materials [27]. Since yield strength and fracture stress scale with shear modulus and elastic constant, respectively, the Pugh's ratio determines the likelihood of material's failure. If the effect of crystal structure is neglected, high value of Pugh's ratio indicates that a material is prone to brittle failure, while low value of G/B implies ductile failure. The large data on polycrystalline pure metals collected by Pugh [2], when he provided a qualitative ranking from ductile (e.g. Ag, Au, Cd, Cu) to brittle (e.g. Be, Ir) behaviour as G/B increases. For cubic close-packed (ccp) metals, the critical ratio Gð Þ =B crit dividing the two regimes is in the range 0.43–0.56, and for hexagonal close-packed metals, it is 0.60–0.63. Cottrell [28] has estimated Gð Þ =B crit for transgranular fracture from measured surface energies: 0.32–0.57 for ccp metals and 0.35–0.68 for body-centred cubic metals. The spread in values for each structure type largely indicates the interrelationship between crystal structure and elastic constant. Each structure type, however, includes metals with widely differing degrees of elastic anisotropy. Detailed analysis requires knowledge of the relevant elastic constants.

On the other hand, the Cauchy pressure ductility criterion is associated with elastic constants of single cubic crystals such as C12–C44 and is useful in describing the nature of bonding in a material [27]. When a material has high resistance to bond bending as found in covalently bonded solids, it will have a negative Cauchy pressure (C44 > C12). This is in contrast with materials with metallic bonding which exhibit positive Cauchy pressure. When compared with Pugh's ductility criterion, ductile and brittle behaviours are considered to be indicated by a positive and a negative Cauchy pressure, respectively. Although Pugh's and Cauchy pressure criteria are adjured to be based on easily measurable properties of materials such as elastic constants, they do not give the critical value dividing brittle and ductile materials. It is proven in certain materials, including metallic glasses and composites, which religiously respect this dividing line [21]. The behaviour is shown graphically in Figure 1. A summary of the correlation between C12–C44 and Gð Þ =B crit

for a wide variety of aluminide group of materials is displayed in Figure 2. As can be seen, it is evident that an intrinsic correlation between strength and ductility of Albased materials. It has been observed the criteria indicate a trend in a class of materials with similar deformation mechanism, but is limited by the effects of

Several authors have studied elastic softening behaviour and recent evidence suggests elastic moduli manifest array of trends with a range of properties including mechanical such as hardness, yield strength, toughness and fragility [22, 30]. In early 1950s, Gilman and Cohen [31] made a historic revelations when they observed that there is a linear correlation between the hardness and elasticity in polycrystalline materials. Nevertheless, successive studies demonstrated that an uniformed linear correlation between hardness and bulk modulus does not really hold for a variety of materials [29] as illustrated in Figure 3(a). Following this, Tester [32] proposed a better empirical link between hardness and shear modulus (G), as illustrated in Figure 3(b). Although, the link between hardness and elastic shear modulus can be arguable, it is certain that he had demonstrated that the shear modulus, the resistance to reversible deformation under shear strain, can correctly provide a key assessment of hardness or ductility criteria for some materials. It is well known that some phase exhibits more hardness or ductility properties than others. Accordingly, it is fair to say that such descriptions could lead to further outlandish discovery in connections with regards to phase components in poly-

Martensitic transformation (MT) is a first-order phase type of transformation from a high-symmetry phase (austenite) at high temperature to a crystallographically low-symmetry phase (martensite) at low temperature. Martensitic behaviour has been extensively studied for decades because of its importance

specimen sizes and crystal structures on deformation processes.

Correlation between C12–C44 and G/B for 35 aluminides (culled from [29]).

Concept of Phase Transition Based on Elastic Systematics

DOI: http://dx.doi.org/10.5772/intechopen.81340

4.2 Elastic moduli and martensitic transformation

crystalline solids.

63

Figure 2.

#### Figure 1.

Ductile and brittle phase fields in metallic glasses, where G\* is the local modulus and 'G' the global modulus. Decreasing the fraction of low G sites reduces the need for a globally low ν (culled from [28]).

to Gb where b is the Burgers vector. This implies that a material with high value of the ratio tends to be brittle (fracture is easier and plasticity is much less), while a low value indicates ductility (plasticity is easier and fracture is not). Fracture strength is also proportional to Ba (a, is the lattice constant) since B is related to

These empirical observations implicate G/B as explaining well brittle or tough behaviour [19, 26]. Pugh's criterion is the most widely used model to predict plastic behaviour of materials [27]. Since yield strength and fracture stress scale with shear modulus and elastic constant, respectively, the Pugh's ratio determines the likelihood of material's failure. If the effect of crystal structure is neglected, high value of Pugh's ratio indicates that a material is prone to brittle failure, while low value of G/B implies ductile failure. The large data on polycrystalline pure metals collected by Pugh [2], when he provided a qualitative ranking from ductile (e.g. Ag, Au, Cd, Cu) to brittle (e.g. Be, Ir) behaviour as G/B increases. For cubic close-packed (ccp) metals, the critical ratio Gð Þ =B crit dividing the two regimes is in the range 0.43–0.56, and for hexagonal close-packed metals, it is 0.60–0.63. Cottrell [28] has estimated Gð Þ =B crit for transgranular fracture from measured surface energies: 0.32–0.57 for ccp metals and 0.35–0.68 for body-centred cubic metals. The spread in values for each structure type largely indicates the interrelationship between crystal structure and elastic constant. Each structure type, however, includes metals with widely differing degrees of elastic anisotropy. Detailed analysis requires knowledge of the relevant elastic constants. On the other hand, the Cauchy pressure ductility criterion is associated with elastic constants of single cubic crystals such as C12–C44 and is useful in describing the nature of bonding in a material [27]. When a material has high resistance to bond bending as found in covalently bonded solids, it will have a negative Cauchy pressure (C44 > C12). This is in contrast with materials with metallic bonding which exhibit positive Cauchy pressure. When compared with Pugh's ductility criterion, ductile and brittle behaviours are considered to be indicated by a positive and a negative Cauchy pressure, respectively. Although Pugh's and Cauchy pressure criteria are adjured to be based on easily measurable properties of materials such as elastic constants, they do not give the critical value dividing brittle and ductile materials. It is proven in certain materials, including metallic glasses and composites, which religiously respect this dividing line [21]. The behaviour is shown graphically in Figure 1. A summary of the correlation between C12–C44 and Gð Þ =B crit

Ductile and brittle phase fields in metallic glasses, where G\* is the local modulus and 'G' the global modulus.

Decreasing the fraction of low G sites reduces the need for a globally low ν (culled from [28]).

surface energy, which indicates brittle fracture strength.

Elasticity of Materials ‐ Basic Principles and Design of Structures

Figure 1.

62

Figure 2. Correlation between C12–C44 and G/B for 35 aluminides (culled from [29]).

for a wide variety of aluminide group of materials is displayed in Figure 2. As can be seen, it is evident that an intrinsic correlation between strength and ductility of Albased materials. It has been observed the criteria indicate a trend in a class of materials with similar deformation mechanism, but is limited by the effects of specimen sizes and crystal structures on deformation processes.

Several authors have studied elastic softening behaviour and recent evidence suggests elastic moduli manifest array of trends with a range of properties including mechanical such as hardness, yield strength, toughness and fragility [22, 30]. In early 1950s, Gilman and Cohen [31] made a historic revelations when they observed that there is a linear correlation between the hardness and elasticity in polycrystalline materials. Nevertheless, successive studies demonstrated that an uniformed linear correlation between hardness and bulk modulus does not really hold for a variety of materials [29] as illustrated in Figure 3(a). Following this, Tester [32] proposed a better empirical link between hardness and shear modulus (G), as illustrated in Figure 3(b). Although, the link between hardness and elastic shear modulus can be arguable, it is certain that he had demonstrated that the shear modulus, the resistance to reversible deformation under shear strain, can correctly provide a key assessment of hardness or ductility criteria for some materials. It is well known that some phase exhibits more hardness or ductility properties than others. Accordingly, it is fair to say that such descriptions could lead to further outlandish discovery in connections with regards to phase components in polycrystalline solids.

#### 4.2 Elastic moduli and martensitic transformation

Martensitic transformation (MT) is a first-order phase type of transformation from a high-symmetry phase (austenite) at high temperature to a crystallographically low-symmetry phase (martensite) at low temperature. Martensitic behaviour has been extensively studied for decades because of its importance

Figure 3.

Correlation of experimental Vickers hardness (HvÞ with (a) bulk modulus (B) and with (b) shear modulus (G) for 39 compounds [29].

in metallurgy and its key role in shape memory phenomenon. Shape memory alloys (SMA) are materials such as TiNi and TiNi-based alloys [33], Ti-Nb [34], Ti-Mo [20, 31] etc. that exhibit diffusion-less first-order martensitic phase transitions induced by the change of temperature and/or stress. The relation between softening of elastic constants and martensitic transformation has attracted considerable attention for many years and has been discussed by many researchers [35, 36]. This interesting feature of martensitic transformation in shape memory alloys is the existence of precursor phenomena [1, 2]. The relations between MT temperature and elastic constants were investigated by Ren et al. [36]. Experiments [37] indicate that martensitic transition occurs at almost constant values of C' . Slight change in composition would cause strong deviation in the critical temperature at which C' softens to a critical value and martensitic transition occurs. In some alloys exhibiting martensitic transformation, softening of elastic constants <sup>C</sup>' <sup>¼</sup> ð Þ <sup>C</sup><sup>11</sup> � <sup>C</sup><sup>12</sup> <sup>=</sup>2 and large elastic anisotropy, A <sup>¼</sup> ð Þ <sup>C</sup><sup>44</sup> <sup>=</sup>C' was observed in the parent phase, but the significance of the softening is largely different between the alloys. For example, Earlier Takashi Fukuda and co-workers [34] observed the value of C' near the transformation start temperature is approximately 0.01 GPa in In-27Ti (at %) alloy [38], 1 GPa in Au-30Cu-47Zn (at %) alloy [37], 5 GPa in Fe-30Pd (at %) alloy [39], 8 GPa in Cu-14Al-4Ni (at %) alloy [9], and 14 GPa in Ti-50.8Ni (at %) [33] and Al-63.2Ni (at %) alloys [40]. Because of such a large distribution of C' at the Ms temperature, the influence of softening of C' on martensitic transformation is expected to be significantly different between these alloys. Martensitic transformation in some alloys is probably strongly related to the softening of C' , while that in others is weakly related despite the fact that the softening appears before the transformation.

composition-dependence of MT temperature. As a result, the modulus softens abruptly within a narrow temperature window around martensitic start temperature, Ms. However, this is unsurprising since it is well known that they are a consequence of weak restoring forces in specific crystallographic directions that announce the possibility of a dynamical instability. The elastic constants are closely related to the acoustic lattice vibrations or even atomic bondings in crystals, and accordingly will be related to the transformation mechanism for not only the Martensitic alloys but also any other compounds which accompany

Correlations between parameters reduced elastic-stiffness coefficients ð Þ C<sup>12</sup> =ð Þ C<sup>11</sup> vs. ð Þ C<sup>44</sup> =ð Þ C<sup>11</sup> for several

Following the above, Nnamchi et al. [41] in a recent study considered the link between different groups of shape memory materials with elastic systematics found a clear delineated in a 2D plot of two dimensionless ratios of elastic constants or reduced elastic-stiffness coefficients, ð Þ C<sup>12</sup> =ð Þ C<sup>11</sup> vs. ð Þ C<sup>44</sup> =ð Þ C<sup>11</sup> formally popularised earlier by Blackman [42], It is only one table with different sections. (see Figure 4 and Tables 1 and 2). This reveals among others the elastic anisotropy, proximity to Born mechanical instability, elastic-constants (interatomic-bonding) changes caused by alloying, pressure, temperature, phase transformations and similarities in types of interatomic bonding. The significance of the softening is largely different between the alloys. Inspecting the diagram, we notice materials with similar chemical bonding tend to fall in the same region of

This work C11 C12 C12/C11 C44/C11 Ref.

 Ti-3Mo 159.3 115 0.72 1.21 [34] Ti-6Mo 111.3 69.07 0.62 0.93 [34] Ti-10Mo 167 19.6 0.12 0.081 [34] Ti-14Mo 179.2 17.9 0.10 0.074 [34]

shear-like or displacive transitions.

classes of shape memory materials (culled from [34]).

Concept of Phase Transition Based on Elastic Systematics

DOI: http://dx.doi.org/10.5772/intechopen.81340

Figure 4.

65

the diagrams. Such diagrams provide many uses.

Previously, Zener [5] established a correlation between the magnitudes of <sup>C</sup>' <sup>¼</sup> ð Þ <sup>C</sup><sup>11</sup> � <sup>C</sup><sup>12</sup> <sup>=</sup>2 elastic shear modulus in metallic bcc structures with the occurrence of martensitic phase transformations suggesting links with phase stability, via the atomic interactions. He observed that the large value suggests that C' is much smaller than C<sup>44</sup> and that MT temperature is dominated by C' [5]. Thus, independent elastic constants are needed to characterize the material response, such as Martensitic transformations (MTs), Shape memory etc. Martensitic transformations (MTs) are often accompanied by elastic modulus softening (acoustic phonon softening) [5]. This explains the strong

#### Figure 4.

in metallurgy and its key role in shape memory phenomenon. Shape memory alloys (SMA) are materials such as TiNi and TiNi-based alloys [33], Ti-Nb [34], Ti-Mo [20, 31] etc. that exhibit diffusion-less first-order martensitic phase transitions induced by the change of temperature and/or stress. The relation between softening of elastic constants and martensitic transformation has attracted considerable attention for many years and has been discussed by many researchers [35, 36]. This interesting feature of martensitic transformation in shape memory alloys is the existence of precursor phenomena [1, 2]. The relations between MT temperature and elastic constants were investigated by Ren et al. [36]. Experiments [37] indicate that martensitic transition occurs at

Correlation of experimental Vickers hardness (HvÞ with (a) bulk modulus (B) and with (b) shear modulus

Elasticity of Materials ‐ Basic Principles and Design of Structures

deviation in the critical temperature at which C' softens to a critical value and martensitic transition occurs. In some alloys exhibiting martensitic transformation, softening of elastic constants <sup>C</sup>' <sup>¼</sup> ð Þ <sup>C</sup><sup>11</sup> � <sup>C</sup><sup>12</sup> <sup>=</sup>2 and large elastic

significance of the softening is largely different between the alloys. For example, Earlier Takashi Fukuda and co-workers [34] observed the value of C' near the transformation start temperature is approximately 0.01 GPa in In-27Ti (at %) alloy [38], 1 GPa in Au-30Cu-47Zn (at %) alloy [37], 5 GPa in Fe-30Pd (at %) alloy [39], 8 GPa in Cu-14Al-4Ni (at %) alloy [9], and 14 GPa in Ti-50.8Ni (at %) [33] and Al-63.2Ni (at %) alloys [40]. Because of such a large distribution of C' at the Ms temperature, the influence of softening of C'

anisotropy, A <sup>¼</sup> ð Þ <sup>C</sup><sup>44</sup> <sup>=</sup>C' was observed in the parent phase, but the

on martensitic transformation is expected to be significantly different between these alloys. Martensitic transformation in some alloys is probably

despite the fact that the softening appears before the transformation.

[5]. Thus, independent elastic constants are needed to characterize the material response, such as Martensitic transformations (MTs), Shape memory etc. Martensitic transformations (MTs) are often accompanied by elastic modu-

lus softening (acoustic phonon softening) [5]. This explains the strong

Previously, Zener [5] established a correlation between the magnitudes of <sup>C</sup>' <sup>¼</sup> ð Þ <sup>C</sup><sup>11</sup> � <sup>C</sup><sup>12</sup> <sup>=</sup>2 elastic shear modulus in metallic bcc structures with the occurrence of martensitic phase transformations suggesting links with phase stability, via the atomic interactions. He observed that the large value suggests that C' is much smaller than C<sup>44</sup> and that MT temperature is dominated by C'

. Slight change in composition would cause strong

, while that in others is weakly related

almost constant values of C'

Figure 3.

(G) for 39 compounds [29].

strongly related to the softening of C'

64

Correlations between parameters reduced elastic-stiffness coefficients ð Þ C<sup>12</sup> =ð Þ C<sup>11</sup> vs. ð Þ C<sup>44</sup> =ð Þ C<sup>11</sup> for several classes of shape memory materials (culled from [34]).

composition-dependence of MT temperature. As a result, the modulus softens abruptly within a narrow temperature window around martensitic start temperature, Ms. However, this is unsurprising since it is well known that they are a consequence of weak restoring forces in specific crystallographic directions that announce the possibility of a dynamical instability. The elastic constants are closely related to the acoustic lattice vibrations or even atomic bondings in crystals, and accordingly will be related to the transformation mechanism for not only the Martensitic alloys but also any other compounds which accompany shear-like or displacive transitions.

Following the above, Nnamchi et al. [41] in a recent study considered the link between different groups of shape memory materials with elastic systematics found a clear delineated in a 2D plot of two dimensionless ratios of elastic constants or reduced elastic-stiffness coefficients, ð Þ C<sup>12</sup> =ð Þ C<sup>11</sup> vs. ð Þ C<sup>44</sup> =ð Þ C<sup>11</sup> formally popularised earlier by Blackman [42], It is only one table with different sections. (see Figure 4 and Tables 1 and 2). This reveals among others the elastic anisotropy, proximity to Born mechanical instability, elastic-constants (interatomic-bonding) changes caused by alloying, pressure, temperature, phase transformations and similarities in types of interatomic bonding. The significance of the softening is largely different between the alloys. Inspecting the diagram, we notice materials with similar chemical bonding tend to fall in the same region of the diagrams. Such diagrams provide many uses.



5. Summary and future challenges

Elastic constant of some bcc and fcc metals and alloys.

envisaged in new future.

community.

67

Table 2.

materials.

The following bullet points summarise some of the main challenges facing the

BCC elements C12/C11 C44/C114 Ref.

FCC elements [53]

 V 0.52 0.19 [53] Nb 0.59 0.13 [53] Ta 0.60 0.31 [53] Mo 0.38 0.28 [53] W 0.5 0.43 [53] Li 0.83 0.78 [53] Na 0.82 0.75 [53] K 0.79 0.73 [53] Ba 0.43 0.7 [53]

Concept of Phase Transition Based on Elastic Systematics

DOI: http://dx.doi.org/10.5772/intechopen.81340

 Au 0.83 0.22 [53] Pd 0.79 0.3 [53] Pt 0.74 0.2 [53] Ag 0.76 0.39 [53] Cu 0.76 0.43 [53] β-Co 0.69 0.6 [53] α-Sr 0.65 0.39 [53] γ-Fe 0.68 0.5 [53] Ni 0.62 0.51 [53] δ-Pu 0.78 0.96 [53]

• Some empirical elastic relationship such as a low G/B ratio (or high ν) favours toughness but also indicates a fragility in polycrystalline materials, though they can be typically difficult to vitrify in some polycrystalline

• Some empirical correlations exist in most of the metallic elements in the

stating the chemical species that should be present, and their rough proportions, and instead gives exact elastic relationship. However, a more rigorous that delineated the phase stability using systematics could be

periodic table have been found, and alloy development has moved beyond the bucket chemistry type approach used in the early days of elastic properties research. While a number of general guidelines exist for explaining elastic systematics property formation (such as Zener, and Burger's rules), Pugh and Pettifor's criterion [16, 17] in addition to Blackmans have gone beyond simply

#### Table 1.

Elastic constant of some bcc and fcc metals and alloys.


#### Concept of Phase Transition Based on Elastic Systematics DOI: http://dx.doi.org/10.5772/intechopen.81340

This work C11 C12 C12/C11 C44/C11 Ref.

5 Ti-18Mo 192.6 16.3 0.085 0.066 [34] 6 Ti-23Mo 197.5 16 0.081 0.051 [34]

 Ti50Ni30Cu20 209 183 0.88 0.17 [43] Ti-50Ni 165 140 0.85 0.21 [44] Ti-29Nb-13Ta-4.6Zr 67.1 39.9 0.87 0.19 [45] Ti-30Nb-10Ta-5Zr 128 92 0.86 0.24 [46] Ti-35Nb 163.5 142 0.87 0.22 [47] Ti-30Nb-5Ta-5Zr 70 30 0.87 0.185 [48] 7 Ti-32.7Nb-11.6Ta-4.49Zr-0.066O-0.052N 137 91.1 0.86 0.12 [49]

 Ag-75Au 230 161.5 0.702 0.33 [43] Cu-4.17Si 117 85.2 0.73 0.64 [43] α-Ag-2.4Zn 190 162 0.85 0.43 [43] α-Cu-9.98Al 199 179 0.89 0.50 [43] α-Cu-22.7Zn 158.9 136.2 0.86 0.43 [43]

 Ti-35.37Nb 130.2 52 0.40 0.078 [50] Ti-35Nb-2Zr-0.7Ta 183 31.4 0.17 0.15 [49] Ti-35.4Nb-1.9Ta-2.8Zr-0.37O 122 27 0.22 0.11 [49] Ti-24.1Nb-4Zr-8.06Sn-0.15O 140 26.3 0.19 0.16 [49] Ti-35Nb-10Ta-4.6Zr-0.16O 102.5 36 0.16 0.12 [51] 6 Ti-23.9Nb-3.75Zr-8.01Sn-0.04O 157.2 36 0.26 0.127 [51] Ti-24Nb-4Zr-7.9Sn-0.17O 0.23 0.22 [51] Ti-24Nb-4Zr-7.6Sn-0.07O 122 31.4 0.26 0.21 [49] 9 Ti-35.2Nb-10.5Ta-4.97Zr-0.091O-0.014N 140 27 0.19 0.1 [51] 10 Ti-23.9Nb-3.8Zr-7.61Sn-0.08O 102.5 26.3 0.12 0.13 [51] Ti-24Nb-4Zr-7.9Sn 157.2 46 0.29 0.27 [52]

 Cu44.9- 50Zn 125 80 0.64 0.6 [43] Au47.5-50Cd 142 96.77 0.68 0.53 [43] Ag45-50Zn 132.8 83.16 0.63 0.57 [43] γ-FeNi 209 183 0.65 0.54 [53] CuAlNi 142.8 93.7 0.66 0.59 [54] B2-NiTi 162 104 0.64 0.52 [55] Cu2.726A11.122Ni 0.152 137 89.2 0.65 0.59 [56] Cu2.742Al1.105Ni0.152 136 81.763 0.65 0.61 [56]

Non-SIM (BCC)alloys

Elasticity of Materials ‐ Basic Principles and Design of Structures

Non-SIM (FCC) alloys

SIM (BCC) alloys

SIM (FCC) alloys

Elastic constant of some bcc and fcc metals and alloys.

Table 1.

66

Table 2. Elastic constant of some bcc and fcc metals and alloys.

#### 5. Summary and future challenges

The following bullet points summarise some of the main challenges facing the community.


#### Glossary of symbols


Author details

69

Paul S. Nnamchi1,2\* and Camillus S. Obayi2

Concept of Phase Transition Based on Elastic Systematics

DOI: http://dx.doi.org/10.5772/intechopen.81340

University, Newcastle, United Kingdom

provided the original work is properly cited.

Nsukka, Enugu State, Nigeria

1 Department of Mechanical and Construction Engineering, Northumbria

\*Address all correspondence to: nnamchi.paul@gmail.com

2 Department of Metallurgical and Materials Engineering, University of Nigeria,

© 2018 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, Concept of Phase Transition Based on Elastic Systematics DOI: http://dx.doi.org/10.5772/intechopen.81340

Glossary of symbols

<sup>C</sup>44, <sup>μ</sup><sup>∗</sup> <sup>¼</sup> <sup>C</sup><sup>∗</sup>

68

Symbols used symbols derived from disambiguation (e.g. d for

<sup>C</sup><sup>44</sup> single crystal bulk modulus; Bo <sup>¼</sup> <sup>C</sup><sup>0</sup>

C<sup>0</sup> ¼ ð Þ C11� C<sup>12</sup> =2 tetragonal shear modulus

Elasticity of Materials ‐ Basic Principles and Design of Structures

ρ the density of the material

<sup>A</sup> <sup>¼</sup> ð Þ <sup>C</sup><sup>44</sup> <sup>=</sup>C' elastic anisotropy

V0 equilibrium volume e an elastic strain

<sup>σ</sup>ijð Þ<sup>r</sup> � � effective stiffness of Cijkl

І equivalent to the unit tensor

MF martensite finish temperature

SME shape memory effect

1þG44τ<sup>44</sup>

<sup>44</sup> <sup>¼</sup> <sup>μ</sup><sup>0</sup> <sup>þ</sup> h i <sup>τ</sup><sup>44</sup>

Cijkl the local elastic constant tensor with <sup>σ</sup>ijð Þ<sup>r</sup> � � and

ensemble averages

G the ratio of shearing stress τ to shearing strain γ

B bulk modulus, ratio between the fluid pressure

V<sup>L</sup> and V<sup>S</sup> the ultrasonic longitudinal and shear wave velocities respectively

U the energy of the crystal, and quadratic function of the strains

<sup>ϵ</sup> <sup>¼</sup> <sup>ϵ</sup><sup>0</sup> <sup>þ</sup> GTϵ<sup>0</sup> <sup>ϵ</sup><sup>0</sup> and GT are the strain and modify Green'<sup>s</sup> function <sup>T</sup> T-matrix is given by <sup>T</sup> <sup>¼</sup> <sup>δ</sup>Cð Þ <sup>1</sup> � <sup>G</sup>δ<sup>C</sup> �<sup>1</sup>

<sup>e</sup>μ<sup>∗</sup> homogenised polycrystalline Poisson's ratio MS martensite formation start temperature

<sup>Y</sup>e<sup>∗</sup> the homogenised polycrystalline Young's modulus

E modulus of elasticity or Young's modulus G modulus of rigidity or shear modulus

trigonal shear modulus

and the Volumetric Strain

h i ϵklð Þr as the local stress and strain field at a point r, respectively, and the angular brackets denote

within the proportional limit of a material

<sup>11</sup> <sup>þ</sup> <sup>2</sup>C<sup>O</sup> 12 � �=3

#### Author details

Paul S. Nnamchi1,2\* and Camillus S. Obayi2

1 Department of Mechanical and Construction Engineering, Northumbria University, Newcastle, United Kingdom

2 Department of Metallurgical and Materials Engineering, University of Nigeria, Nsukka, Enugu State, Nigeria

\*Address all correspondence to: nnamchi.paul@gmail.com

© 2018 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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[13] Zeller R, Dederichs PH. Physica Status Solidi B. 1973;55:831-842

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2015;50:52-58

70

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Chapter 5

Abstract

and Nobuhiro Okude

for estimating the effect of injection repair.

tomography, repair method

1. Introduction

73

Repair Inspection Technique

Deteriorated Concrete Structures

Katsufumi Hashimoto,Tomoki Shiotani,Takahiro Nishida

Applying elastic wave tomography as an innovative NDT method, the evaluation

of velocity distribution in three-dimensional (3D) before and after the repair is introduced in this study. The increase in the velocity with penetration of the repair material according to the repair effect is identified visually and quantitatively. The 3D tomography technique is newly proposed for one-side access inspection, using drill hammering to generate an elastic wave. Accordingly, the elastic wave velocity distribution result enables to visualize the internal quality of concrete after patch repair is successfully done. In addition, an attempt for reinforced concrete (RC) slab panels is made to confirm the effectiveness of the repair by comparing the velocity distribution of elastic waves obtained from acoustic emission (AE) tomography analysis, before and after the repair. Thus, the velocity recoveries due to injection are found in all the slab panels, and it is confirmed that the elastic wave velocities obtained using this technique can serve as an indicator for examining the state of crack and void filling with injected material. Further, a good correlation is found between the low-velocity region before repair and the amount of injection. These results show the potential of the AE tomography technique to be used as a method

Keywords: elastic wave, acoustic emission, wave velocity distribution,

It is highly demanded to establish sufficient management systems for the inspection of existing concrete infrastructures in order to manage and extend their service lives. As for aging infrastructure, severe deterioration is currently reported, where it is known as a critical issue in our society, and large budgets are required to repair damaged structures. Since budgetary restrictions are often imposed, preventive and proactive maintenance techniques of infrastructure are sufficiently needed with nondestructive testing (NDT) methods. In addition to conventional NDT, innovative methods must be established to appropriately assess and evaluate

Based on Elastic-Wave

Tomography Applied for

[52] Hao Y, Li S, Sun B, Sui M, Yang R. Physical Review Letters. 2008;98:1-4

[53] Paszkiewicz, Wolski. Journal de Physique(Conference series). 2008;104: 012038

[54] Mañosa L et al. Physical Review B. 1994;49:9969-9972

[55] Zhou L, Cornely P, Trivisonno J, Lahrman D. An ultrasonic study of the martensitic phase transformation in NiAl alloys. Honolulu, HI, USA: IEEE Symposium on Ultrasonics; 1990; 3:1389-1329

[56] Sedlak P, Seiner H, Landa M, Novak V, Sittner P, Li M. Acta Materialia. 2005; 53:3643-3661

#### Chapter 5

[52] Hao Y, Li S, Sun B, Sui M, Yang R. Physical Review Letters. 2008;98:1-4

Elasticity of Materials ‐ Basic Principles and Design of Structures

[53] Paszkiewicz, Wolski. Journal de Physique(Conference series). 2008;104:

[54] Mañosa L et al. Physical Review B.

[55] Zhou L, Cornely P, Trivisonno J, Lahrman D. An ultrasonic study of the martensitic phase transformation in NiAl alloys. Honolulu, HI, USA: IEEE Symposium on Ultrasonics; 1990;

[56] Sedlak P, Seiner H, Landa M, Novak V, Sittner P, Li M. Acta Materialia. 2005;

012038

1994;49:9969-9972

3:1389-1329

53:3643-3661

72
