Development of Uncoupled Anisotropic Ductile Fracture Criteria

Abstract

Two mainstreams of transformation methods from isotropic fracture criteria to anisotropic ones are to replace the Von Mises yield function with other yield function and linear transformation of strain or stress tensor. Several typical anisotropic criteria such as the transformation methods of Park, Luo, Lou and Yoon, Jia and Bai and a new transformation method are analyzed and discussed from the aspects of required experimental data, material constant amounts, stress state range, average predictive error (Err_avg), etc. The fitting results demonstrate that the prediction accuracy of the methods of Luo, Lou and Yoon, Jia and Bai and the new method are better than that of the isotropic criteria from the Err_avg aspect based on the experimental results of DP590. In addition, the new transformation method linearly transforms both stress and strain tensor without increasing too many parameters.

Appendices

Appendix 1: Rodrigues’ Rotation Formula

The strain tensor can be expressed by three principal strains according to the coordinate system along rolling direction, transverse direction and normal direction (RD, TD and ND) as shown in Eq 41.

$$\left[ {\begin{array}{*{20}l} {\varepsilon_{x} } \\ {\varepsilon_{y} } \\ {\varepsilon_{z} } \\ {\varepsilon_{xy} } \\ {\varepsilon_{xz} } \\ {\varepsilon_{yz} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\left( {tx^{2} + c} \right)^{2} } \hfill & {\left( {sz + txy} \right)^{2} } \hfill & {\left( {sy - txz} \right)^{2} } \hfill \\ {\left( {sz - txy} \right)^{2} } \hfill & {\left( {ty^{2} + c} \right)^{2} } \hfill & {\left( {sx + tyz} \right)^{2} } \hfill \\ {\left( {sy + txz} \right)^{2} } \hfill & {\left( {sx - tyz} \right)^{2} } \hfill & {\left( {tz^{2} + c} \right)^{2} } \hfill \\ {\left( {tx^{2} + c} \right)\left( { - sz + txy} \right)} \hfill & {\left( {ty^{2} + c} \right)\left( {sz + txy} \right)} \hfill & { - \,\left( {sy - txz} \right)\left( {sx + tyz} \right)} \hfill \\ {\left( {tx^{2} + c} \right)\left( {sy + txz} \right)} \hfill & { - \,\left( {sx - tyz} \right)\left( {sz + txy} \right)} \hfill & {\left( {tz^{2} + c} \right)\left( { - sy + txz} \right)} \hfill \\ { - \,\left( {sz - txy} \right)\left( {sy + txz} \right)} \hfill & {\left( {ty^{2} + c} \right)\left( { - sx + tyz} \right)} \hfill & {\left( {tz^{2} + c} \right)\left( {sx + tyz} \right)} \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {\varepsilon_{1} } \\ {\varepsilon_{2} } \\ {\varepsilon_{3} } \\ \end{array} } \right]$$

(41)

where $t = 1 - \cos \theta$, $c = \cos \theta$, $s = \sin \theta$, $\hat{u} = x\hat{i} + y\hat{j} + z\hat{k}$. $\hat{u}$ and θ denote a unit vector for a rotation axis and the rotation angle, respectively. For the application of the fracture surface to sheet metal forming, the unit vector of a rotation axis can be defined as $\hat{u} = x\hat{i} + y\hat{j} + z\hat{k} = \hat{k}$. Therefore, Eq 41 can be simplified to Eq 42.

$$\left[ {\begin{array}{*{20}l} {\varepsilon_{x} } \\ {\varepsilon_{y} } \\ {\varepsilon_{z} } \\ {\varepsilon_{xy} } \\ {\varepsilon_{xz} } \\ {\varepsilon_{yz} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {c^{2} } \hfill & {s^{2} } \hfill & 0 \hfill \\ {s^{2} } \hfill & {c^{2} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\left( {t + c} \right)^{2} } \hfill \\ { - \,cs} \hfill & {cs} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varepsilon_{1} } \\ {\varepsilon_{2} } \\ {\varepsilon_{3} } \\ \end{array} } \right]$$

(42)

Appendix 2: A Definition of the Strain Rate Potential for Plastically Deforming Metals

Due to the incompressibility during the plastic deformation, Barlat et al. (Ref 50) redefined the strain rate potential ψ based on the work-equivalent effective strain obtained by the TBH model.

$$\varPsi = \left| {\varepsilon_{1}^{d} } \right|^{b} + \left| {\varepsilon_{2}^{d} } \right|^{b} + \left| {\varepsilon_{3}^{d} } \right|^{b} = 2\left( {d\bar{\varepsilon }_{d} } \right)^{b}$$

(43)

Accordingly, the equivalent plastic strain can be expressed as follows:

$$\left( {d\bar{\varepsilon }_{d} } \right)^{b} = \frac{1}{2}\left( {\left| {d\varepsilon_{1}^{d} } \right|^{b} + \left| {d\varepsilon_{2}^{d} } \right|^{b} + \left| {d\varepsilon_{3}^{d} } \right|^{b} } \right)$$

(44)

Based on the procedure of transforming the isotropic yield function (Ref 54, 55) into the anisotropic yield function (Ref 56), the anisotropic equivalent plastic strain rate with six plastic strain rate components in the axes of orthotropic symmetry x, y and z is shown as follows.

$$\dot{\varepsilon }_{d} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\beta_{3} \left( {\dot{\varepsilon }_{x} - \dot{\varepsilon }_{y} } \right) - \beta_{2} \left( {\dot{\varepsilon }_{z} - \dot{\varepsilon }_{x} } \right)}}{3}} \\ {\beta_{6} \dot{\varepsilon }_{xy} } \\ {\beta_{5} \dot{\varepsilon }_{zx} } \\ \end{array} } & {\begin{array}{*{20}c} {\beta_{6} \dot{\varepsilon }_{xy} } \\ {\frac{{\beta_{1} \left( {\dot{\varepsilon }_{y} - \dot{\varepsilon }_{z} } \right) - \beta_{3} \left( {\dot{\varepsilon }_{x} - \dot{\varepsilon }_{y} } \right)}}{3}} \\ {\beta_{4} \dot{\varepsilon }_{yz} } \\ \end{array} } & {\begin{array}{*{20}c} {\beta_{5} \dot{\varepsilon }_{zx} } \\ {\beta_{4} \dot{\varepsilon }_{yz} } \\ {\frac{{\beta_{2} \left( {\dot{\varepsilon }_{z} - \dot{\varepsilon }_{x} } \right) - \beta_{1} \left( {\dot{\varepsilon }_{y} - \dot{\varepsilon }_{z} } \right)}}{3}} \\ \end{array} } \\ \end{array} } \right]$$

(45)

The six material constants of β_i (i = 1,2,3,4,5,6) indicate the anisotropy of alloys. The eigenvalues of $\dot{\varvec{\varepsilon }}_{\varvec{d}}$ including $\varepsilon_{1}^{d}$, $\varepsilon_{2}^{d}$ and $\varepsilon_{3}^{d}$ can be obtained by solving the eigenfunction.

$$\dot{\varepsilon }_{d}^{3} - 3I_{2} \dot{\varepsilon }_{d} - 2I_{3} = 0$$

(46)

where

$$\begin{aligned} I_{2} & = \frac{{\left[ {\beta_{3} \left( {\dot{\varepsilon }_{x} - \dot{\varepsilon }_{y} } \right) - \beta_{2} \left( {\dot{\varepsilon }_{z} - \dot{\varepsilon }_{x} } \right)} \right]^{2} }}{54} + \frac{{\left[ {\beta_{1} \left( {\dot{\varepsilon }_{y} - \dot{\varepsilon }_{z} } \right) - \beta_{3} \left( {\dot{\varepsilon }_{x} - \dot{\varepsilon }_{y} } \right)} \right]^{2} }}{54} \\ & \quad + \;\frac{{\left[ {\beta_{2} \left( {\dot{\varepsilon }_{z} - \dot{\varepsilon }_{x} } \right) - \beta_{1} \left( {\dot{\varepsilon }_{y} - \dot{\varepsilon }_{z} } \right)} \right]^{2} }}{54} + \frac{{\left( {\beta_{4} \dot{\varepsilon }_{yz} } \right)^{2} + \left( {\beta_{5} \dot{\varepsilon }_{zx} } \right)^{2} + \left( {\beta_{6} \dot{\varepsilon }_{xy} } \right)^{2} }}{3} \\ I_{3} & = \frac{{\left[ {\beta_{3} \left( {\dot{\varepsilon }_{x} - \dot{\varepsilon }_{y} } \right) - \beta_{2} \left( {\dot{\varepsilon }_{z} - \dot{\varepsilon }_{x} } \right)} \right]\left[ {\beta_{1} \left( {\dot{\varepsilon }_{y} - \dot{\varepsilon }_{z} } \right) - \beta_{3} \left( {\dot{\varepsilon }_{x} - \dot{\varepsilon }_{y} } \right)} \right]}}{54} - \frac{{\left[ {\beta_{3} \left( {\dot{\varepsilon }_{x} - \dot{\varepsilon }_{y} } \right) - \beta_{2} \left( {\dot{\varepsilon }_{z} - \dot{\varepsilon }_{x} } \right)} \right]\left( {\beta_{4} \dot{\varepsilon }_{yz} } \right)^{2} }}{6} \\ & \quad - \;\frac{{\left[ {\beta_{1} \left( {\dot{\varepsilon }_{y} - \dot{\varepsilon }_{z} } \right) - \beta_{3} \left( {\dot{\varepsilon }_{x} - \dot{\varepsilon }_{y} } \right)} \right]\left( {\beta_{5} \dot{\varepsilon }_{zx} } \right)^{2} }}{6} - \frac{{\left[ {\beta_{2} \left( {\dot{\varepsilon }_{z} - \dot{\varepsilon }_{x} } \right) - \beta_{1} \left( {\dot{\varepsilon }_{y} - \dot{\varepsilon }_{z} } \right)} \right]\left( {\beta_{6} \dot{\varepsilon }_{xy} } \right)^{2} }}{6} \\ & \quad + \;\beta_{4} \beta_{5} \beta_{6} \dot{\varepsilon }_{yz} \dot{\varepsilon }_{zx} \dot{\varepsilon }_{xy} \\ \end{aligned}$$

(47)

$$\theta = \cos^{ - 1} \left( {\frac{{I_{3} }}{{I_{2}^{3/2} }}} \right),\quad 0 \le \theta \le \pi$$

(48)

The expressions of eigenvalues of $\dot{\varvec{\varepsilon }}_{\varvec{d}}$ are shown as follows:

$$\begin{aligned} \dot{\varepsilon }_{1}^{d} & = 2\sqrt {I_{2} } \cos \left( {\frac{\theta }{3}} \right) \\ \dot{\varepsilon }_{2}^{d} & = 2\sqrt {I_{2} } \cos \left( {\frac{\theta - 2\pi }{3}} \right) \\ \dot{\varepsilon }_{3}^{d} & = 2\sqrt {I_{2} } \cos \left( {\frac{\theta + 2\pi }{3}} \right) \\ \end{aligned}$$

(49)

$\dot{\varvec{\varepsilon }}_{\varvec{d}}$ is reduced to $\dot{\varvec{\varepsilon }}_{\varvec{p}}$ if all the material constants are equal to 1. Therefore, above method induces a new method for explicitly expressing the principal strain by strain components. Moreover, the expression of $\dot{\varvec{\varepsilon }}_{\varvec{d}}$ meets the constant-volume principle in the metal forming processes regardless of the anisotropic constant values. The transformation method of Lou and Yoon involves the anisotropy. However, it is not based on the constant-volume principle in the process of introducing anisotropic parameters.

Appendix 3: Explicit Function of the Principal Strain Through Strain Components

The process of calculating the principal strain according to strain components is similar to that of getting eigenvalues of the third-order square matrix. Next, the general process of obtaining them is introduced.

$$\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {m_{11} } \\ {m_{21} } \\ {m_{31} } \\ \end{array} } & {\begin{array}{*{20}c} {m_{12} } \\ {m_{22} } \\ {m_{32} } \\ \end{array} } & {\begin{array}{*{20}c} {m_{13} } \\ {m_{23} } \\ {m_{33} } \\ \end{array} } \\ \end{array} } \right]$$

(50)

The corresponding eigenfunction is shown in Eq 51.

$$A_{1} x^{3} + A_{2} x^{2} + A_{3} x + A_{4} = 0 \left( {A_{1} \ne 0} \right)$$

(51)

where the coefficients are listed as follows:

$$\left\{ {\begin{array}{*{20}l} {A_{1} = 1} \hfill \\ {A_{2} = - \;\left( {m_{11} + m_{22} + m_{33} } \right)} \hfill \\ {A_{3} = m_{11} m_{22} + m_{22} m_{33} + m_{33} m_{11} - m_{12} m_{21} - m_{23} m_{32} - m_{31} m_{13} } \hfill \\ {A_{4} = m_{11} m_{23} m_{32} + m_{22} m_{13} m_{31} + m_{33} m_{12} m_{21} - m_{11} m_{22} m_{33} - m_{12} m_{23} m_{31} - m_{13} m_{21} m_{32} } \hfill \\ \end{array} } \right.$$

(52)

Let

$$y = x + \frac{{A_{2} }}{{3A_{1} }}$$

(53)

Equation 51 is simplified to Eq 54 as follows:

$$y^{3} + py + q = 0$$

(54)

where the coefficients are listed as follows:

$$\left\{ {\begin{array}{*{20}l} {p = \frac{{A_{3} }}{{A_{1} }} - \frac{1}{3}\left( {\frac{{A_{2} }}{{A_{1} }}} \right)^{2} } \hfill \\ {q = \frac{2}{27}\left( {\frac{{A_{2} }}{{A_{1} }}} \right)^{3} - \frac{1}{3}\frac{{A_{2} }}{{A_{1} }}\frac{{A_{3} }}{{A_{1} }} + \frac{{A_{4} }}{{A_{1} }}} \hfill \\ \end{array} } \right.$$

(55)

The calculating results are shown as follows:

$$\left\{ {\begin{array}{*{20}c} {y_{1} = \sqrt[3]{{ - \frac{q}{2} + \sqrt {\left( {\frac{q}{2}} \right)^{2} + \left( {\frac{p}{3}} \right)^{3} } }} + \sqrt[3]{{ - \frac{q}{2} - \sqrt {\left( {\frac{q}{2}} \right)^{2} + \left( {\frac{p}{3}} \right)^{3} } }}} \\ {y_{2} = \omega_{1} \sqrt[3]{{ - \frac{q}{2} + \sqrt {\left( {\frac{q}{2}} \right)^{2} + \left( {\frac{p}{3}} \right)^{3} } }} + \omega_{2} \sqrt[3]{{ - \frac{q}{2} - \sqrt {\left( {\frac{q}{2}} \right)^{2} + \left( {\frac{p}{3}} \right)^{3} } }}} \\ {y_{3} = \omega_{2} \sqrt[3]{{ - \frac{q}{2} + \sqrt {\left( {\frac{q}{2}} \right)^{2} + \left( {\frac{p}{3}} \right)^{3} } }} + \omega_{1} \sqrt[3]{{ - \frac{q}{2} - \sqrt {\left( {\frac{q}{2}} \right)^{2} + \left( {\frac{p}{3}} \right)^{3} } }}} \\ \end{array} } \right.$$

(56)

where ω₁ and ω₂ are expressed in Eq 57.

$$\left\{ {\begin{array}{*{20}c} {\omega_{1} = \frac{ - 1 + \sqrt 3 i}{2}} \\ {\omega_{2} = \frac{ - 1 - \sqrt 3 i}{2}} \\ \end{array} } \right.$$

(57)

Eigenvalues of the third-order square matrix are revealed as below:

$$\left\{ {\begin{array}{*{20}c} {x_{1} = y_{1} - \frac{{A_{2} }}{{3A_{1} }}} \\ {x_{2} = y_{2} - \frac{{A_{2} }}{{3A_{1} }}} \\ {x_{3} = y_{3} - \frac{{A_{2} }}{{3A_{1} }}} \\ \end{array} } \right.$$

(58)

According to the above method of the strain, the principal strain is shown as follows.

$$\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\varepsilon_{x} } \\ {\varepsilon_{yx} } \\ {\varepsilon_{zx} } \\ \end{array} } & {\begin{array}{*{20}c} {\varepsilon_{xy} } \\ {\varepsilon_{y} } \\ {\varepsilon_{zy} } \\ \end{array} } & {\begin{array}{*{20}c} {\varepsilon_{xz} } \\ {\varepsilon_{yz} } \\ {\varepsilon_{z} } \\ \end{array} } \\ \end{array} } \right]$$

(59)

The corresponding eigenfunction is shown in Eq 60:

$$A_{1} \varepsilon^{3} + A_{2} \varepsilon^{2} + A_{3} \varepsilon + A_{4} = 0 \left( {A_{1} \ne 0} \right)$$

(60)

where the coefficients are listed as follows:

$$\left\{ {\begin{array}{*{20}l} {A_{1} = 1} \hfill \\ {A_{2} = - \left( {\varepsilon_{x} + \varepsilon_{y} + \varepsilon_{z} } \right)} \hfill \\ {A_{3} = \varepsilon_{x} \varepsilon_{y} + \varepsilon_{y} \varepsilon_{z} + \varepsilon_{z} \varepsilon_{x} - \left( {\varepsilon_{xy} } \right)^{2} - \left( {\varepsilon_{yz} } \right)^{2} - \left( {\varepsilon_{zx} } \right)^{2} } \hfill \\ {A_{4} = \varepsilon_{x} \left( {\varepsilon_{yz} } \right)^{2} + \varepsilon_{y} \left( {\varepsilon_{zx} } \right)^{2} + \varepsilon_{z} \left( {\varepsilon_{xy} } \right)^{2} - \varepsilon_{x} \varepsilon_{y} \varepsilon_{z} - 2\varepsilon_{xy} \varepsilon_{yz} \varepsilon_{zx} } \hfill \\ \end{array} } \right.$$

(61)

Because ε_p is a symmetric matrix and ε_yz= ε_zx= 0 under plane stress conditions, Eq 61 can be simplified to Eq 62.

$$\left\{ {\begin{array}{*{20}l} {A_{1} = 1} \hfill \\ {A_{2} = - \left( {\varepsilon_{x} + \varepsilon_{y} + \varepsilon_{z} } \right) = 0} \hfill \\ {A_{3} = \varepsilon_{x} \varepsilon_{y} + \varepsilon_{y} \varepsilon_{z} + \varepsilon_{z} \varepsilon_{x} - \left( {\varepsilon_{xy} } \right)^{2} } \hfill \\ {A_{4} = \varepsilon_{z} \left( {\varepsilon_{xy} } \right)^{2} - \varepsilon_{x} \varepsilon_{y} \varepsilon_{z} } \hfill \\ \end{array} } \right.$$

(62)

$$\varepsilon^{3} + A_{3} \varepsilon + A_{4} = 0$$

(63)

Let

$$\left\{ {\begin{array}{*{20}c} {M = \frac{{\varepsilon_{x} \varepsilon_{y} \varepsilon_{z} - \varepsilon_{z} \left( {\varepsilon_{xy} } \right)^{2} }}{2}} \\ {N = \sqrt {\left( {\frac{{\varepsilon_{z} \left( {\varepsilon_{xy} } \right)^{2} - \varepsilon_{x} \varepsilon_{y} \varepsilon_{z} }}{2}} \right)^{2} + \left( {\frac{{\varepsilon_{x} \varepsilon_{y} + \varepsilon_{y} \varepsilon_{z} + \varepsilon_{z} \varepsilon_{x} - \left( {\varepsilon_{xy} } \right)^{2} }}{3}} \right)^{3} } } \\ \end{array} } \right.$$

(64)

The three principal strains are expressed as follows:

$$\left\{ {\begin{array}{*{20}c} {\varepsilon_{1} = \sqrt[3]{M + N} + \sqrt[3]{M - N}} \\ {\varepsilon_{2} = \omega_{1} \sqrt[3]{M + N} + \omega_{2} \sqrt[3]{M - N}} \\ {\varepsilon_{3} = \omega_{2} \sqrt[3]{M + N} + \omega_{1} \sqrt[3]{M - N}} \\ \end{array} } \right.$$

(65)

The above method is used to the transformation method of Lou and Yoon. According to Eq 25, the equations of M and N are shown as follows.

$$\begin{aligned} M & = \frac{1}{54}\left[ {\left( {d_{13} d_{21} d_{31} - d_{13} d_{23} d_{31} } \right)\left( {\varepsilon_{x} } \right)^{3} + \left( {d_{12} d_{23} d_{32} - d_{13} d_{23} d_{32} } \right)\left( {\varepsilon_{y} } \right)^{3} } \right. \\ & \quad +\, \left( {d_{13} d_{21} d_{32} - d_{13} d_{23} d_{32} - 2d_{13} d_{23} d_{31} + d_{13} d_{21} d_{31} + d_{12} d_{23} d_{31} - d_{12} d_{21} d_{31} } \right)\left( {\varepsilon_{x} } \right)^{2} \varepsilon_{y} \\ & \quad +\, \left( {d_{12} d_{23} d_{31} - d_{13} d_{23} d_{31} - 2d_{13} d_{23} d_{32} + d_{13} d_{21} d_{32} + d_{12} d_{23} d_{32} - d_{12} d_{21} d_{32} } \right)\varepsilon_{x} \left( {\varepsilon_{y} } \right)^{2} \\ & \quad \left. { +\, d_{31} \left( {d_{44} } \right)^{2} \varepsilon_{x} \left( {\varepsilon_{xy} } \right)^{2} + d_{32} \left( {d_{44} } \right)^{2} \varepsilon_{y} \left( {\varepsilon_{xy} } \right)^{2} } \right] \\ N & = \sqrt {\begin{aligned} \left\{ {\frac{1}{54}\left[ {\begin{array}{*{20}l} {\left( {d_{13} d_{21} d_{31} - d_{13} d_{23} d_{31} } \right)\left( {\varepsilon_{x} } \right)^{3} + \left( {d_{12} d_{23} d_{32} - d_{13} d_{23} d_{32} } \right)\left( {\varepsilon_{y} } \right)^{3} } \hfill \\ { + \left( {d_{13} d_{21} d_{32} - d_{13} d_{23} d_{32} - 2d_{13} d_{23} d_{31} + d_{13} d_{21} d_{31} + d_{12} d_{23} d_{31} - d_{12} d_{21} d_{31} } \right)\left( {\varepsilon_{x} } \right)^{2} \varepsilon_{y} } \hfill \\ { + d_{12} d_{23} d_{31} - d_{13} d_{23} d_{31} - 2d_{13} d_{23} d_{32} + d_{13} d_{21} d_{32} + d_{12} d_{23} d_{32} - d_{12} d_{21} d_{32} \varepsilon_{x} \left( {\varepsilon_{y} } \right)^{2} } \hfill \\ { + d_{31} \left( {d_{44} } \right)^{2} \varepsilon_{x} \left( {\varepsilon_{xy} } \right)^{2} + d_{32} \left( {d_{44} } \right)^{2} \varepsilon_{y} \left( {\varepsilon_{xy} } \right)^{2} } \hfill \\ \end{array} } \right]} \right\} \hfill \\ + \left\{ {\frac{1}{27}\left[ {\begin{array}{*{20}l} {\left( {d_{13} d_{23} - d_{13} d_{21} - d_{21} d_{31} - d_{23} d_{31} - d_{13} d_{31} } \right)\left( {\varepsilon_{x} } \right)^{2} } \hfill \\ { + \left( {d_{13} d_{23} - d_{12} d_{23} - d_{12} d_{32} + d_{23} d_{32} + d_{13} d_{32} } \right)\left( {\varepsilon_{y} } \right)^{2} } \hfill \\ { + \left( \begin{aligned} d_{21} d_{32} - d_{23} d_{32} - d_{23} d_{31} + 2d_{13} d_{23} + d_{13} d_{21} - d_{12} d_{23} + d_{12} d_{21} \hfill \\ d_{13} d_{32} - d_{13} d_{31} - d_{12} d_{31} \hfill \\ \end{aligned} \right)\varepsilon_{x} \varepsilon_{y} } \hfill \\ { - (d_{44} )^{2} \left( {\varepsilon_{xy} } \right)^{2} } \hfill \\ \end{array} } \right]} \right\}^{3} \hfill \\ \end{aligned}^{2}} \\ \end{aligned} .$$

(66)