#### 期刊菜单

General Solution of Stress and Deformation in Cracked Body for Composite Materials
DOI: 10.12677/IJM.2021.104025, PDF, HTML, XML, 下载: 75  浏览: 113

Abstract: The investigation of the mechanical behaviour for composite materials has even more importance to the applications in new engineering structures. Particularly, the crack problem of anisotropic materials must be discussed for the key points as in mechanical study. This paper is mainly to solve the boundary problem of the centre-cracked plate with composite materials under mixed loading. The general solution about the elastic mechanics had been achieved by way of establishing relational equations and using pan-complex variable method. The stress functions had been selected rationally to meet the needs of the plane stress boundary conditions. The whole solutions on stress field and deformation field for the anisotropic plate have been derived. The real function expressions of crack displacement fields and the singular stress fields near the crack-tip region have been determined.

1. 引言

2. 弹性力学基本方程及复变函数方法

2.1. 弹性力学基本方程

$\begin{array}{l}{\epsilon }_{x}={a}_{11}{\sigma }_{x}+{a}_{12}{\sigma }_{y}+{a}_{16}{\tau }_{xy}\\ {\epsilon }_{y}={a}_{12}{\sigma }_{x}+{a}_{22}{\sigma }_{y}+{a}_{26}{\tau }_{xy}\\ {\gamma }_{xy}={a}_{16}{\sigma }_{x}+{a}_{26}{\sigma }_{y}+{a}_{66}{\tau }_{xy}\end{array}\right\}$ (1)

${\epsilon }_{x}=\frac{\partial u}{\partial x},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\epsilon }_{y}=\frac{\partial v}{\partial y},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\gamma }_{xy}=\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}$ (2)

${\sigma }_{x}=\frac{{\partial }^{2}F}{\partial {y}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{y}=\frac{{\partial }^{2}F}{\partial {x}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\tau }_{xy}=-\frac{{\partial }^{2}F}{\partial x\partial y}$ (3)

$\begin{array}{l}\frac{\partial u}{\partial x}={a}_{11}\frac{{\partial }^{2}F}{\partial {y}^{2}}+{a}_{12}\frac{{\partial }^{2}F}{\partial {x}^{2}}-{a}_{16}\frac{{\partial }^{2}F}{\partial x\partial y}\\ \frac{\partial v}{\partial y}={a}_{12}\frac{{\partial }^{2}F}{\partial {y}^{2}}+{a}_{22}\frac{{\partial }^{2}F}{\partial {x}^{2}}-{a}_{26}\frac{{\partial }^{2}F}{\partial x\partial y}\\ \frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}={a}_{16}\frac{{\partial }^{2}F}{\partial {y}^{2}}+{a}_{26}\frac{{\partial }^{2}F}{\partial {x}^{2}}-{a}_{66}\frac{{\partial }^{2}F}{\partial x\partial y}\end{array}\right\}$ (4)

$\frac{{\partial }^{4}F}{\partial {y}^{4}}+{A}_{1}\frac{{\partial }^{4}F}{\partial x\partial {y}^{3}}+{A}_{2}\frac{{\partial }^{4}F}{\partial {x}^{2}\partial {y}^{2}}+{A}_{3}\frac{{\partial }^{4}F}{\partial {x}^{3}\partial y}+{A}_{4}\frac{{\partial }^{4}F}{\partial {x}^{4}}=0$ (5)

2.2. 复变函数方法

$w=x+qy=x+gy+ihy,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{¯}{w}=x+\stackrel{¯}{q}y=x+gy-ihy$ (6)

$q\stackrel{¯}{q}={g}^{2}+{h}^{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}w\stackrel{¯}{w}={\left(x+gy\right)}^{2}+{h}^{2}{y}^{2}$

$L=|w|=|\stackrel{¯}{w}|=\sqrt{{\left(x+gy\right)}^{2}+{h}^{2}{y}^{2}}$ (7)

$\Psi$ 是全纯泛复函， $\Psi =\Psi \left(w\right)$，其共轭复函为： $\stackrel{¯}{\Psi }=\stackrel{¯}{\Psi \left(w\right)}=\stackrel{¯}{\Psi }\left(\stackrel{¯}{w}\right)$。关于泛复变函数对坐标变量 $\left(x,y\right)$ 的偏导数确定如下：

$\frac{\partial \Psi }{\partial x}=\frac{\text{d}\Psi }{\text{d}w}\frac{\partial w}{\partial x}=\frac{\text{d}\Psi }{\text{d}w}={\Psi }^{\prime },\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \Psi }{\partial y}=\frac{\text{d}\Psi }{\text{d}w}\frac{\partial w}{\partial y}=q\frac{\text{d}\Psi }{\text{d}w}=q{\Psi }^{\prime }$

$\frac{\partial \stackrel{¯}{\Psi }}{\partial x}=\frac{\text{d}\stackrel{¯}{\Psi }}{\text{d}\stackrel{¯}{w}}\frac{\partial \stackrel{¯}{w}}{\partial x}=\frac{\text{d}\stackrel{¯}{\Psi }}{\text{d}\stackrel{¯}{w}}=\stackrel{¯}{{\Psi }^{\prime }},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \stackrel{¯}{\Psi }}{\partial y}=\frac{\text{d}\stackrel{¯}{\Psi }}{\text{d}\stackrel{¯}{w}}\frac{\partial \stackrel{¯}{w}}{\partial y}=\stackrel{¯}{q}\frac{\text{d}\stackrel{¯}{\Psi }}{\text{d}\stackrel{¯}{w}}=\stackrel{¯}{q}\stackrel{¯}{{\Psi }^{\prime }}$

$F=B\Psi \left(w\right)+\stackrel{¯}{B}\stackrel{¯}{\Psi }\left(\stackrel{¯}{w}\right)+\frac{T}{2}{y}^{2}=2\mathrm{Re}\left(B\Psi \right)+\frac{T}{2}{y}^{2}$ (8)

$\frac{\partial F}{\partial x}=B{\Psi }^{\prime }+\stackrel{¯}{B}{\stackrel{¯}{\Psi }}^{\prime }=2\mathrm{Re}\left(B{\Psi }^{\prime }\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial F}{\partial y}=qB{\Psi }^{\prime }+\stackrel{¯}{q}\stackrel{¯}{B}{\stackrel{¯}{\Psi }}^{\prime }+Ty=2\mathrm{Re}\left(qB{\Psi }^{\prime }\right)+Ty$

$\frac{{\partial }^{2}F}{\partial {x}^{2}}=B{\Psi }^{″}+\stackrel{¯}{B}\stackrel{¯}{{\Psi }^{″}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\partial }^{2}F}{\partial {y}^{2}}={q}^{2}B{\Psi }^{″}+{\stackrel{¯}{q}}^{2}\stackrel{¯}{B}{\stackrel{¯}{\Psi }}^{″}+T,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\partial }^{2}F}{\partial x\partial y}=qB{\Psi }^{″}+\stackrel{¯}{q}\stackrel{¯}{B}{\stackrel{¯}{\Psi }}^{″}$

$\begin{array}{l}{\sigma }_{x}=\frac{{\partial }^{2}F}{\partial {y}^{2}}={q}^{2}B{\Psi }^{″}+{\stackrel{¯}{q}}^{2}\stackrel{¯}{B}{\stackrel{¯}{\Psi }}^{″}+T=2\mathrm{Re}\left({q}^{2}B\frac{{\text{d}}^{2}\Psi }{\text{d}{w}^{2}}\right)+T\\ {\sigma }_{y}=\frac{{\partial }^{2}F}{\partial {x}^{2}}=B{\Psi }^{″}+\stackrel{¯}{B}{\stackrel{¯}{\Psi }}^{″}=2\mathrm{Re}\left(B\frac{{\text{d}}^{2}\Psi }{\text{d}{w}^{2}}\right)\\ {\tau }_{xy}=-\frac{{\partial }^{2}F}{\partial x\partial y}=-qB{\Psi }^{″}-\stackrel{¯}{q}\stackrel{¯}{B}{\stackrel{¯}{\Psi }}^{″}=-2\mathrm{Re}\left(qB\frac{{\text{d}}^{2}\Psi }{\text{d}{w}^{2}}\right)\end{array}\right\}$ (9)

$Re\left[\left({q}^{4}+{A}_{1}{q}^{3}+{A}_{2}{q}^{2}+{A}_{3}q+{A}_{4}\right)B\frac{{\text{d}}^{4}\Psi }{\text{d}{w}^{4}}\right]=0$

${q}^{4}+{A}_{1}{q}^{3}+{A}_{2}{q}^{2}+{A}_{3}q+{A}_{4}=0$ (10)

${w}_{1}=x+{q}_{1}y=x+{g}_{1}y+i{h}_{1}y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{w}_{2}=x+{q}_{2}y=x+{g}_{2}y+i{h}_{2}y$ (11)

$F={B}_{1}{\Psi }_{1}\left({w}_{1}\right)+{B}_{2}{\Psi }_{2}\left({w}_{2}\right)+{\stackrel{¯}{B}}_{1}{\stackrel{¯}{\Psi }}_{1}\left({\stackrel{¯}{w}}_{1}\right)+{\stackrel{¯}{B}}_{2}{\stackrel{¯}{\Psi }}_{2}\left({\stackrel{¯}{w}}_{2}\right)+\frac{T}{2}{y}^{2}$ (12)

$\frac{\partial F}{\partial x}=2\mathrm{Re}\left({B}_{1}\frac{\text{d}{\Psi }_{1}}{\text{d}{w}_{1}}+{B}_{2}\frac{\text{d}{\Psi }_{2}}{\text{d}{w}_{2}}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial F}{\partial y}=2\mathrm{Re}\left({q}_{1}{B}_{1}\frac{\text{d}{\Psi }_{1}}{\text{d}{w}_{1}}+{q}_{2}{B}_{2}\frac{\text{d}{\Psi }_{2}}{\text{d}{w}_{2}}\right)+Ty$

$\begin{array}{l}\frac{{\partial }^{2}F}{\partial {x}^{2}}=2\mathrm{Re}\left({B}_{1}\frac{{\text{d}}^{2}{\Psi }_{1}}{\text{d}{w}_{1}^{2}}+{B}_{2}\frac{{\text{d}}^{2}{\Psi }_{2}}{\text{d}{w}_{2}^{2}}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\partial }^{2}F}{\partial {y}^{2}}=2\mathrm{Re}\left({q}_{1}^{2}{B}_{1}\frac{{\text{d}}^{2}{\Psi }_{1}}{\text{d}{w}_{1}^{2}}+{q}_{2}^{2}{B}_{2}\frac{{\text{d}}^{2}{\Psi }_{2}}{\text{d}{w}_{2}^{2}}\right)+T\\ \frac{{\partial }^{2}F}{\partial x\partial y}=2\mathrm{Re}\left({q}_{1}{B}_{1}\frac{{\text{d}}^{2}{\Psi }_{1}}{\text{d}{w}_{1}^{2}}+{q}_{2}{B}_{2}\frac{{\text{d}}^{2}{\Psi }_{2}}{\text{d}{w}_{2}^{2}}\right)\end{array}$

$\begin{array}{l}{\sigma }_{x}=2\mathrm{Re}\left({q}_{1}^{2}{B}_{1}\frac{{\text{d}}^{2}{\Psi }_{1}}{\text{d}{w}_{1}^{2}}+{q}_{2}^{2}{B}_{2}\frac{{\text{d}}^{2}{\Psi }_{2}}{\text{d}{w}_{2}^{2}}\right)+T\\ {\sigma }_{y}=2\mathrm{Re}\left({B}_{1}\frac{{\text{d}}^{2}{\Psi }_{1}}{\text{d}{w}_{1}^{2}}+{B}_{2}\frac{{\text{d}}^{2}{\Psi }_{2}}{\text{d}{w}_{2}^{2}}\right)\\ {\tau }_{xy}=-2\mathrm{Re}\left({q}_{1}{B}_{1}\frac{{\text{d}}^{2}{\Psi }_{1}}{\text{d}{w}_{1}^{2}}+{q}_{2}{B}_{2}\frac{{\text{d}}^{2}{\Psi }_{2}}{\text{d}{w}_{2}^{2}}\right)\end{array}\right\}$ (13)

$\begin{array}{l}\frac{\partial u}{\partial x}=2\mathrm{Re}\left[{B}_{3}\frac{{\text{d}}^{2}{\Psi }_{1}}{\text{d}{w}_{1}^{2}}+{B}_{4}\frac{{\text{d}}^{2}{\Psi }_{2}}{\text{d}{w}_{2}^{2}}\right]+{a}_{11}T\\ \frac{\partial v}{\partial y}=2\mathrm{Re}\left[{B}_{5}\frac{{\text{d}}^{2}{\Psi }_{1}}{\text{d}{w}_{1}^{2}}+{B}_{6}\frac{{\text{d}}^{2}{\Psi }_{2}}{\text{d}{w}_{2}^{2}}\right]+{a}_{12}T\end{array}\right\}$ (14a)

$\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}=2\mathrm{Re}\left[{B}_{7}\frac{{\text{d}}^{2}{\Psi }_{1}}{\text{d}{w}_{1}^{2}}+{B}_{8}\frac{{\text{d}}^{2}{\Psi }_{2}}{\text{d}{w}_{2}^{2}}\right]+{a}_{16}T$ (14b)

$\begin{array}{l}{B}_{3}=\left({a}_{11}{q}_{1}^{2}+{a}_{12}-{a}_{16}{q}_{1}\right){B}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{4}=\left({a}_{11}{q}_{2}^{2}+{a}_{12}-{a}_{16}{q}_{2}\right){B}_{2}\\ {B}_{5}=\left({a}_{12}{q}_{1}^{2}+{a}_{22}-{a}_{26}{q}_{1}\right){B}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{6}=\left({a}_{12}{q}_{2}^{2}+{a}_{22}-{a}_{26}{q}_{2}\right){B}_{2}\\ {B}_{7}=\left({a}_{16}{q}_{1}^{2}+{a}_{26}-{a}_{66}{q}_{1}\right){B}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{8}=\left({a}_{16}{q}_{2}^{2}+{a}_{26}-{a}_{66}{q}_{2}\right){B}_{2}\end{array}$ (14c)

$\begin{array}{l}u=2\mathrm{Re}\left({B}_{3}\frac{\text{d}{\Psi }_{1}}{\text{d}{w}_{1}}+{B}_{4}\frac{\text{d}{\Psi }_{2}}{\text{d}{w}_{2}}\right)+{a}_{11}Tx+{a}_{16}Ty+{u}_{0}\\ v=2\mathrm{Re}\left(\frac{{B}_{5}}{{q}_{1}}\frac{\text{d}{\Psi }_{1}}{\text{d}{w}_{1}}+\frac{{B}_{6}}{{q}_{2}}\frac{\text{d}{\Psi }_{2}}{\text{d}{w}_{2}}\right)+{a}_{12}Ty+{v}_{0}\end{array}\right\}$ (15)

${B}_{3}{q}_{1}+\frac{{B}_{5}}{{q}_{1}}={B}_{7},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{4}{q}_{2}+\frac{{B}_{6}}{{q}_{2}}={B}_{8}$

3. 含裂纹各向异性板的应力与变形解法

3.1. 全场解答及坐标变换

$\begin{array}{l}{\Phi }_{1}={\Phi }_{1}\left({w}_{1}\right)=\sqrt{{w}_{1}^{2}-{a}^{2}}\\ {\Phi }_{2}={\Phi }_{2}\left({w}_{2}\right)=\sqrt{{w}_{2}^{2}-{a}^{2}}\end{array}\right\}$ (16)

${w}_{1}=x+{q}_{1}y=x+{g}_{1}y+i{h}_{1}y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{w}_{2}=x+{q}_{2}y=x+{g}_{2}y+i{h}_{2}y$

$\begin{array}{l}{\Psi }_{1}=\frac{{w}_{1}}{2}{\Phi }_{1}-\frac{{a}^{2}}{2}\mathrm{ln}\left({w}_{1}+{\Phi }_{1}\right)\\ {\Psi }_{2}=\frac{{w}_{2}}{2}{\Phi }_{2}-\frac{{a}^{2}}{2}\mathrm{ln}\left({w}_{2}+{\Phi }_{2}\right)\end{array}\right\}$ (17)

$\frac{\text{d}{\Psi }_{1}}{\text{d}{w}_{1}}={\Phi }_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\text{d}}^{2}{\Psi }_{1}}{\text{d}{w}_{1}^{2}}=\frac{{w}_{1}}{{\Phi }_{1}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\text{d}{\Psi }_{2}}{\text{d}{w}_{2}}={\Phi }_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\text{d}}^{2}{\Psi }_{2}}{\text{d}{w}_{2}^{2}}=\frac{{w}_{2}}{{\Phi }_{2}}$

$\begin{array}{l}{\sigma }_{x}=2\mathrm{Re}\left({q}_{1}^{2}{B}_{1}\frac{{w}_{1}}{{\Phi }_{1}}+{q}_{2}^{2}{B}_{2}\frac{{w}_{2}}{{\Phi }_{2}}\right)+T\\ {\sigma }_{y}=2\mathrm{Re}\left({B}_{1}\frac{{w}_{1}}{{\Phi }_{1}}+{B}_{2}\frac{{w}_{2}}{{\Phi }_{2}}\right)\\ {\tau }_{xy}=-2\mathrm{Re}\left({q}_{1}{B}_{1}\frac{{w}_{1}}{{\Phi }_{1}}+{q}_{2}{B}_{2}\frac{{w}_{2}}{{\Phi }_{2}}\right)\end{array}\right\}$ (18)

$\begin{array}{l}u=2\mathrm{Re}\left({B}_{3}{\Phi }_{1}+{B}_{4}{\Phi }_{2}\right)+{a}_{11}Tx+{a}_{16}Ty+{u}_{0}\\ v=2\mathrm{Re}\left(\frac{{B}_{5}}{{q}_{1}}{\Phi }_{1}+\frac{{B}_{6}}{{q}_{2}}{\Phi }_{2}\right)+{a}_{12}Ty+{v}_{0}\end{array}\right\}$ (19)

${\sigma }_{x}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{y}=\sigma ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\tau }_{xy}=\tau \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(|w|\to \infty \right)$

$2Re\left({q}_{1}^{2}{B}_{1}+{q}_{2}^{2}{B}_{2}\right)+T=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2\mathrm{Re}\left({B}_{1}+{B}_{2}\right)=\sigma ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\mathrm{Re}\left({q}_{1}{B}_{1}+{q}_{2}{B}_{2}\right)=\tau$

$Im\left({B}_{1}+{B}_{2}\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Im\left({q}_{1}{B}_{1}+{q}_{2}{B}_{2}\right)=0$

${B}_{1}=-\frac{\sigma {q}_{2}+\tau }{2\left({q}_{1}-{q}_{2}\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{2}=\frac{\sigma {q}_{1}+\tau }{2\left({q}_{1}-{q}_{2}\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}T=-2Re\left({q}_{1}^{2}{B}_{1}+{q}_{2}^{2}{B}_{2}\right)$

$\begin{array}{l}{B}_{1}=\frac{\sigma }{2}\left(\frac{{D}_{2}}{{D}_{12}}+i\frac{{D}_{3}}{{D}_{12}}\right)+\frac{\tau }{2}\left(\frac{{g}_{2}-{g}_{1}}{{D}_{1}{}_{2}}+i\frac{{h}_{1}-{h}_{2}}{{D}_{1}{}_{2}}\right)\\ {B}_{2}=\frac{\sigma }{2}\left(\frac{{D}_{1}}{{D}_{12}}-i\frac{{D}_{3}}{{D}_{12}}\right)-\frac{\tau }{2}\left(\frac{{g}_{2}-{g}_{1}}{{D}_{12}}+i\frac{{h}_{1}-{h}_{2}}{{D}_{12}}\right)\end{array}\right\}$ (20-a)

$\begin{array}{l}{D}_{1}={g}_{1}\left({g}_{1}-{g}_{2}\right)+{h}_{1}\left({h}_{1}-{h}_{2}\right)\\ {D}_{2}={g}_{2}\left({g}_{2}-{g}_{1}\right)+{h}_{2}\left({h}_{2}-{h}_{1}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{D}_{3}={g}_{2}{h}_{1}-{g}_{1}{h}_{2}\\ {D}_{12}=\left({q}_{1}-{q}_{2}\right)\left({\stackrel{¯}{q}}_{1}-{\stackrel{¯}{q}}_{2}\right)={\left({g}_{1}-{g}_{2}\right)}^{2}+{\left({h}_{1}-{h}_{2}\right)}^{2}={D}_{1}+{D}_{2}\end{array}\right\}$ (20-b)

$\begin{array}{l}{D}_{4}={g}_{2}{D}_{1}+{h}_{2}{D}_{3}=-{g}_{1}{D}_{2}+{h}_{1}{D}_{3}={g}_{1}{g}_{2}\left({g}_{1}-{g}_{2}\right)+{g}_{2}{h}_{1}^{2}-{g}_{1}{h}_{2}^{2}\\ {D}_{5}={h}_{1}{D}_{2}+{g}_{1}{D}_{3}=-{h}_{2}{D}_{1}+{g}_{2}{D}_{3}={g}_{2}^{2}{h}_{1}-{g}_{1}^{2}{h}_{2}-{h}_{1}{h}_{2}\left({h}_{1}-{h}_{2}\right)\end{array}\right\}$ (21)

${q}_{1}{B}_{1}=-\frac{\sigma }{2}\frac{{D}_{4}-i{D}_{5}}{{D}_{12}}-\frac{\tau }{2}\frac{{D}_{1}-i{D}_{3}}{{D}_{1}{}_{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{2}{B}_{2}=\frac{\sigma }{2}\frac{{D}_{4}-i{D}_{5}}{{D}_{12}}-\frac{\tau }{2}\frac{{D}_{2}+i{D}_{3}}{{D}_{1}{}_{2}}$

${q}_{1}^{2}{B}_{1}=-\frac{\sigma }{2}\frac{{C}_{1}+i{C}_{2}}{{D}_{12}}-\frac{\tau }{2}\frac{{C}_{3}+i{C}_{4}}{{D}_{12}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{2}^{2}{B}_{2}=\frac{\sigma }{2}\frac{{C}_{5}+i{C}_{6}}{{D}_{12}}-\frac{\tau }{2}\frac{{C}_{7}+i{C}_{8}}{{D}_{12}}$

$\begin{array}{l}{C}_{1}={g}_{1}{D}_{4}+{h}_{1}{D}_{5},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{2}={h}_{1}{D}_{4}-{g}_{1}{D}_{5},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{3}={g}_{1}{D}_{1}+{h}_{1}{D}_{3}\\ {C}_{4}={h}_{1}{D}_{1}-{g}_{1}{D}_{3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{5}={g}_{2}{D}_{4}+{h}_{2}{D}_{5},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{6}={h}_{2}{D}_{4}-{g}_{2}{D}_{5}\\ {C}_{7}={g}_{2}{D}_{2}-{h}_{2}{D}_{3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{8}={h}_{2}{D}_{2}+{g}_{2}{D}_{3}\end{array}\right\}$ (22)

${q}_{1}^{2}{B}_{1}+{q}_{2}^{2}{B}_{2}=-\frac{\sigma }{2}\left[{g}_{1}{g}_{2}-{h}_{1}{h}_{2}+i\left({g}_{2}{h}_{1}+{g}_{1}{h}_{2}\right)\right]-\frac{\tau }{2}\left[{g}_{1}+{g}_{2}+i\left({h}_{1}+{h}_{2}\right)\right]$

$\begin{array}{l}2Im\left({q}_{1}^{2}{B}_{1}+{q}_{2}^{2}{B}_{2}\right)=-\sigma \left({g}_{2}{h}_{1}+{g}_{1}{h}_{2}\right)-\tau \left({h}_{1}+{h}_{2}\right)\\ T=-2Re\left({q}_{1}^{2}{B}_{1}+{q}_{2}^{2}{B}_{2}\right)=\sigma \left({g}_{1}{g}_{2}-{h}_{1}{h}_{2}\right)+\tau \left({g}_{1}+{g}_{2}\right)\end{array}\right\}$ (23)

$\begin{array}{l}{B}_{3}=\sigma {a}_{11}\frac{{B}_{31}+i{B}_{32}}{2{D}_{12}}+\tau {a}_{11}\frac{{B}_{33}+i{B}_{34}}{2{D}_{12}}\\ {B}_{4}=\sigma {a}_{11}\frac{{B}_{41}-i{B}_{42}}{2{D}_{12}}-\tau {a}_{11}\frac{{B}_{43}+i{B}_{44}}{2{D}_{12}}\\ \frac{{B}_{5}}{{q}_{1}}=\sigma {a}_{22}\frac{{B}_{51}+i{B}_{52}}{2{D}_{12}}+\tau {a}_{22}\frac{{B}_{53}+i{B}_{54}}{2{D}_{12}}\\ \frac{{B}_{6}}{{q}_{2}}=\sigma {a}_{22}\frac{{B}_{61}-i{B}_{62}}{2{D}_{12}}-\tau {a}_{22}\frac{{B}_{63}+i{B}_{64}}{2{D}_{12}}\end{array}\right\}$ (24)

$\begin{array}{l}{B}_{31}=\left({g}_{1}^{2}-{h}_{1}^{2}+\frac{{a}_{12}}{{a}_{11}}-{g}_{1}\frac{{a}_{16}}{{a}_{11}}\right){D}_{2}-\left(2{g}_{1}-\frac{{a}_{16}}{{a}_{11}}\right){h}_{1}{D}_{3}\\ {B}_{32}=\left({g}_{1}^{2}-{h}_{1}^{2}+\frac{{a}_{12}}{{a}_{11}}-{g}_{1}\frac{{a}_{16}}{{a}_{11}}\right){D}_{3}+\left(2{g}_{1}-\frac{{a}_{16}}{{a}_{11}}\right){h}_{1}{D}_{2}\\ {B}_{33}=\left({g}_{1}^{2}-{h}_{1}^{2}+\frac{{a}_{12}}{{a}_{11}}-{g}_{1}\frac{{a}_{16}}{{a}_{11}}\right)\left({g}_{2}-{g}_{1}\right)-\left(2{g}_{1}-\frac{{a}_{16}}{{a}_{11}}\right){h}_{1}\left({h}_{1}-{h}_{2}\right)\\ {B}_{34}=\left({g}_{1}^{2}-{h}_{1}^{2}+\frac{{a}_{12}}{{a}_{11}}-{g}_{1}\frac{{a}_{16}}{{a}_{11}}\right)\left({h}_{1}-{h}_{2}\right)+\left(2{g}_{1}-\frac{{a}_{16}}{{a}_{11}}\right){h}_{1}\left({g}_{2}-{g}_{1}\right)\end{array}$

$\begin{array}{l}{B}_{41}=\left({g}_{2}^{2}-{h}_{2}^{2}+\frac{{a}_{12}}{{a}_{11}}-{g}_{2}\frac{{a}_{16}}{{a}_{11}}\right){D}_{1}+\left(2{g}_{2}-\frac{{a}_{16}}{{a}_{11}}\right){h}_{2}{D}_{3}\\ {B}_{42}=\left({g}_{2}^{2}-{h}_{2}^{2}+\frac{{a}_{12}}{{a}_{11}}-{g}_{2}\frac{{a}_{16}}{{a}_{11}}\right){D}_{3}-\left(2{g}_{2}-\frac{{a}_{16}}{{a}_{11}}\right){h}_{2}{D}_{1}\\ {B}_{43}=\left({g}_{2}^{2}-{h}_{2}^{2}+\frac{{a}_{12}}{{a}_{11}}-{g}_{2}\frac{{a}_{16}}{{a}_{11}}\right)\left({g}_{2}-{g}_{1}\right)-\left(2{g}_{2}-\frac{{a}_{16}}{{a}_{11}}\right){h}_{2}\left({h}_{1}-{h}_{2}\right)\\ {B}_{44}=\left({g}_{2}^{2}-{h}_{2}^{2}+\frac{{a}_{12}}{{a}_{11}}-{g}_{2}\frac{{a}_{16}}{{a}_{11}}\right)\left({h}_{1}-{h}_{2}\right)+\left(2{g}_{2}-\frac{{a}_{16}}{{a}_{11}}\right){h}_{2}\left({g}_{2}-{g}_{1}\right)\end{array}$

$\begin{array}{l}{B}_{51}=\left({g}_{1}\frac{{a}_{12}}{{a}_{22}}+\frac{{g}_{1}}{{g}_{1}^{2}+{h}_{1}^{2}}-\frac{{a}_{26}}{{a}_{22}}\right){D}_{2}-\left(\frac{{h}_{1}{a}_{12}}{{a}_{22}}-\frac{{h}_{1}}{{g}_{1}^{2}+{h}_{1}^{2}}\right){D}_{3}\\ {B}_{52}=\left({g}_{1}\frac{{a}_{12}}{{a}_{22}}+\frac{{g}_{1}}{{g}_{1}^{2}+{h}_{1}^{2}}-\frac{{a}_{26}}{{a}_{22}}\right){D}_{3}+\left(\frac{{h}_{1}{a}_{12}}{{a}_{22}}-\frac{{h}_{1}}{{g}_{1}^{2}+{h}_{1}^{2}}\right){D}_{2}\\ {B}_{53}=\left({g}_{1}\frac{{a}_{12}}{{a}_{22}}+\frac{{g}_{1}}{{g}_{1}^{2}+{h}_{1}^{2}}-\frac{{a}_{26}}{{a}_{22}}\right)\left({g}_{2}-{g}_{1}\right)-\left(\frac{{h}_{1}{a}_{12}}{{a}_{22}}-\frac{{h}_{1}}{{g}_{1}^{2}+{h}_{1}^{2}}\right)\left({h}_{1}-{h}_{2}\right)\\ {B}_{54}=\left({g}_{1}\frac{{a}_{12}}{{a}_{22}}+\frac{{g}_{1}}{{g}_{1}^{2}+{h}_{1}^{2}}-\frac{{a}_{26}}{{a}_{22}}\right)\left({h}_{1}-{h}_{2}\right)+\left(\frac{{h}_{1}{a}_{12}}{{a}_{22}}-\frac{{h}_{1}}{{g}_{1}^{2}+{h}_{1}^{2}}\right)\left({g}_{2}-{g}_{1}\right)\end{array}$

$\begin{array}{l}{B}_{61}=\left(\frac{{g}_{2}{a}_{12}}{{a}_{22}}+\frac{{g}_{2}}{{g}_{2}^{2}+{h}_{2}^{2}}-\frac{{a}_{26}}{{a}_{22}}\right){D}_{1}+\left(\frac{{h}_{2}{a}_{12}}{{a}_{22}}-\frac{{h}_{2}}{{g}_{2}^{2}+{h}_{2}^{2}}\right){D}_{3}\\ {B}_{62}=\left(\frac{{g}_{2}{a}_{12}}{{a}_{22}}+\frac{{g}_{2}}{{g}_{2}^{2}+{h}_{2}^{2}}-\frac{{a}_{26}}{{a}_{22}}\right){D}_{3}-\left(\frac{{h}_{2}{a}_{12}}{{a}_{22}}-\frac{{h}_{2}}{{g}_{2}^{2}+{h}_{2}^{2}}\right){D}_{1}\\ {B}_{63}=\left(\frac{{g}_{2}{a}_{12}}{{a}_{22}}+\frac{{g}_{2}}{{g}_{2}^{2}+{h}_{2}^{2}}-\frac{{a}_{26}}{{a}_{22}}\right)\left({g}_{2}-{g}_{1}\right)-\left(\frac{{h}_{2}{a}_{12}}{{a}_{22}}-\frac{{h}_{2}}{{g}_{2}^{2}+{h}_{2}^{2}}\right)\left({h}_{1}-{h}_{2}\right)\\ {B}_{64}=\left(\frac{{g}_{2}{a}_{12}}{{a}_{22}}+\frac{{g}_{2}}{{g}_{2}^{2}+{h}_{2}^{2}}-\frac{{a}_{26}}{{a}_{22}}\right)\left({h}_{1}-{h}_{2}\right)+\left(\frac{{h}_{2}{a}_{12}}{{a}_{22}}-\frac{{h}_{2}}{{g}_{2}^{2}+{h}_{2}^{2}}\right)\left({g}_{2}-{g}_{1}\right)\end{array}$

$x=a+r\mathrm{cos}\theta ={r}_{\omega }\mathrm{cos}\omega ={r}_{\alpha }\mathrm{cos}\alpha -a,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=r\mathrm{sin}\theta ={r}_{\omega }\mathrm{sin}\omega ={r}_{\alpha }\mathrm{sin}\alpha$

$\begin{array}{l}w={r}_{\omega }\left(\mathrm{cos}\omega +g\mathrm{sin}\omega +ih\mathrm{sin}\omega \right)={r}_{\omega }{J}_{\omega }\left(\mathrm{cos}\stackrel{^}{\omega }+i\mathrm{sin}\stackrel{^}{\omega }\right)={r}_{\omega }{J}_{\omega }{\text{e}}^{i\stackrel{^}{\omega }}\\ w+a={r}_{\alpha }\left(\mathrm{cos}\alpha +g\mathrm{sin}\alpha +ih\mathrm{sin}\alpha \right)={r}_{\alpha }{J}_{\alpha }\left(\mathrm{cos}\stackrel{^}{\alpha }+i\mathrm{sin}\stackrel{^}{\alpha }\right)={r}_{\alpha }{J}_{\alpha }{\text{e}}^{i\stackrel{^}{\alpha }}\\ w-a=r\left(\mathrm{cos}\theta +g\mathrm{sin}\theta +ih\mathrm{sin}\theta \right)=rJ\left(cos\beta +i\mathrm{sin}\beta \right)=rJ{\text{e}}^{i\beta }\end{array}\right\}$ (25)

$\mathrm{cos}\omega +g\mathrm{sin}\omega ={J}_{\omega }\mathrm{cos}\stackrel{^}{\omega },\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}h\mathrm{sin}\omega ={J}_{\omega }\mathrm{sin}\stackrel{^}{\omega }$

$\mathrm{cos}\alpha +g\mathrm{sin}\alpha ={J}_{\alpha }\mathrm{cos}\stackrel{^}{\alpha },\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}h\mathrm{sin}\alpha ={J}_{\alpha }\mathrm{sin}\stackrel{^}{\alpha }$

$\mathrm{cos}\theta +g\mathrm{sin}\theta =Jcos\beta ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}h\mathrm{sin}\theta =J\mathrm{sin}\beta$

${J}_{\omega }=\sqrt{{\left(\mathrm{cos}\omega +g\mathrm{sin}\omega \right)}^{2}+{\left(h\mathrm{sin}\omega \right)}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{J}_{\alpha }=\sqrt{{\left(\mathrm{cos}\alpha +g\mathrm{sin}\alpha \right)}^{2}+{\left(h\mathrm{sin}\alpha \right)}^{2}}$

$\mathrm{tan}\stackrel{^}{\omega }=\frac{h\mathrm{sin}\omega }{\mathrm{cos}\omega +g\mathrm{sin}\omega },\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{tan}\stackrel{^}{\alpha }=\frac{h\mathrm{sin}\alpha }{\mathrm{cos}\alpha +g\mathrm{sin}\alpha }$

$J=\sqrt{{\left(\mathrm{cos}\theta +g\mathrm{sin}\theta \right)}^{2}+{\left(h\mathrm{sin}\theta \right)}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{tan}\beta =\frac{h\mathrm{sin}\theta }{\mathrm{cos}\theta +g\mathrm{sin}\theta }$

${w}_{1}={r}_{\omega }\left(\mathrm{cos}\omega +{g}_{1}\mathrm{sin}\omega +i{h}_{1}\mathrm{sin}\omega \right)={r}_{\omega }{J}_{\omega 1}\left(\mathrm{cos}{\stackrel{^}{\omega }}_{1}+i\mathrm{sin}{\stackrel{^}{\omega }}_{1}\right)={r}_{\omega }{J}_{\omega 1}{\text{e}}^{i{\stackrel{^}{\omega }}_{1}}$

${w}_{1}+a={r}_{\alpha }\left(\mathrm{cos}\alpha +{g}_{1}\mathrm{sin}\alpha +i{h}_{1}\mathrm{sin}\alpha \right)={r}_{\alpha }{J}_{\alpha 1}\left(\mathrm{cos}{\stackrel{^}{\alpha }}_{1}+i\mathrm{sin}{\stackrel{^}{\alpha }}_{1}\right)={r}_{\alpha }{J}_{\alpha 1}{\text{e}}^{i{\stackrel{^}{\alpha }}_{1}}$

${w}_{1}-a=r\left(\mathrm{cos}\theta +{g}_{1}\mathrm{sin}\theta +i{h}_{1}\mathrm{sin}\theta \right)=r{J}_{1}\left(cos{\beta }_{1}+i\mathrm{sin}{\beta }_{1}\right)=r{J}_{1}{\text{e}}^{i{\beta }_{1}}$

${w}_{2}={r}_{\omega }\left(\mathrm{cos}\omega +{g}_{2}\mathrm{sin}\omega +i{h}_{2}\mathrm{sin}\omega \right)={r}_{\omega }{J}_{\omega 2}\left(\mathrm{cos}{\stackrel{^}{\omega }}_{2}+i\mathrm{sin}{\stackrel{^}{\omega }}_{2}\right)={r}_{\omega }{J}_{\omega 2}{\text{e}}^{i{\stackrel{^}{\omega }}_{2}}$

${w}_{2}+a={r}_{\alpha }\left(\mathrm{cos}\alpha +{g}_{2}\mathrm{sin}\alpha +i{h}_{2}\mathrm{sin}\alpha \right)={r}_{\alpha }{J}_{\alpha 2}\left(\mathrm{cos}{\stackrel{^}{\alpha }}_{2}+i\mathrm{sin}{\stackrel{^}{\alpha }}_{2}\right)={r}_{\alpha }{J}_{\alpha 2}{\text{e}}^{i{\stackrel{^}{\alpha }}_{2}}$

${w}_{2}-a=r\left(\mathrm{cos}\theta +{g}_{2}\mathrm{sin}\theta +i{h}_{2}\mathrm{sin}\theta \right)=r{J}_{2}\left(cos{\beta }_{2}+i\mathrm{sin}{\beta }_{2}\right)=r{J}_{2}{\text{e}}^{i{\beta }_{2}}$

$\begin{array}{l}{\Phi }_{1}=\sqrt{{w}_{1}^{2}-{a}^{2}}=\sqrt{\left({w}_{1}+a\right)\left({w}_{1}-a\right)}=\sqrt{{r}_{\alpha }{J}_{\alpha 1}}\sqrt{r{J}_{1}}\mathrm{exp}\left(i\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}\right)\\ {\Phi }_{2}=\sqrt{{w}_{2}^{2}-{a}^{2}}=\sqrt{\left({w}_{2}+a\right)\left({w}_{2}-a\right)}=\sqrt{{r}_{\alpha }{J}_{\alpha 2}}\sqrt{r{J}_{2}}\mathrm{exp}\left(i\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}\right)\end{array}\right\}$ (26)

$\begin{array}{l}{\sigma }_{x}=2\mathrm{Re}\left[{q}_{1}^{2}{B}_{1}\frac{{r}_{\omega }{J}_{\omega 1}\left(cos{\lambda }_{1}+i\mathrm{sin}{\lambda }_{1}\right)}{\sqrt{{r}_{\alpha }{J}_{\alpha 1}}\sqrt{r{J}_{1}}}+{q}_{2}^{2}{B}_{2}\frac{{r}_{\omega }{J}_{\omega 2}\left(cos{\lambda }_{2}+i\mathrm{sin}{\lambda }_{2}\right)}{\sqrt{{r}_{\alpha }{J}_{\alpha 2}}\sqrt{r{J}_{2}}}\right]+T\\ {\sigma }_{y}=2\mathrm{Re}\left[{B}_{1}\frac{{r}_{\omega }{J}_{\omega 1}\left(cos{\lambda }_{1}+i\mathrm{sin}{\lambda }_{1}\right)}{\sqrt{{r}_{\alpha }{J}_{\alpha 1}}\sqrt{r{J}_{1}}}+{B}_{2}\frac{{r}_{\omega }{J}_{\omega 2}\left(cos{\lambda }_{2}+i\mathrm{sin}{\lambda }_{2}\right)}{\sqrt{{r}_{\alpha }{J}_{\alpha 2}}\sqrt{r{J}_{2}}}\right]\\ {\tau }_{xy}=-2\mathrm{Re}\left[{q}_{1}{B}_{1}\frac{{r}_{\omega }{J}_{\omega 1}\left(cos{\lambda }_{1}+i\mathrm{sin}{\lambda }_{1}\right)}{\sqrt{{r}_{\alpha }{J}_{\alpha 1}}\sqrt{r{J}_{1}}}+{q}_{2}{B}_{2}\frac{{r}_{\omega }{J}_{\omega 2}\left(cos{\lambda }_{2}+i\mathrm{sin}{\lambda }_{2}\right)}{\sqrt{{r}_{\alpha }{J}_{\alpha 2}}\sqrt{r{J}_{2}}}\right]\end{array}\right\}$ (27)

${\lambda }_{1}={\stackrel{^}{\omega }}_{1}-\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\lambda }_{2}={\stackrel{^}{\omega }}_{2}-\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}$

$\begin{array}{l}u=2\mathrm{Re}\left[{B}_{3}\sqrt{{r}_{\alpha }{J}_{\alpha 1}}\sqrt{r{J}_{1}}\left(cos\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}+isin\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{B}_{4}\sqrt{{r}_{\alpha }{J}_{\alpha 2}}\sqrt{r{J}_{2}}\left(cos\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}+isin\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}\right)\right]+{a}_{11}Tx+{a}_{16}Ty+{u}_{0}\\ v=2\mathrm{Re}\left[\frac{{B}_{5}}{{q}_{1}}\sqrt{{r}_{\alpha }{J}_{\alpha 1}}\sqrt{r{J}_{1}}\left(cos\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}+isin\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{B}_{6}}{{q}_{2}}\sqrt{{r}_{\alpha }{J}_{\alpha 2}}\sqrt{r{J}_{2}}\left(cos\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}+isin\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}\right)\right]+{a}_{12}Ty+{v}_{0}\end{array}\right\}$ (28)

$\begin{array}{l}{\sigma }_{x}=-\frac{{r}_{\omega }{J}_{\omega 1}}{\sqrt{{r}_{\alpha }{J}_{\alpha 1}}\sqrt{r{J}_{1}}}\left[\sigma \left(\frac{{C}_{1}}{{D}_{12}}cos{\lambda }_{1}-\frac{{C}_{2}}{{D}_{12}}\mathrm{sin}{\lambda }_{1}\right)+\tau \left(\frac{{C}_{3}}{{D}_{12}}cos{\lambda }_{1}-\frac{{C}_{4}}{{D}_{12}}\mathrm{sin}{\lambda }_{1}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{r}_{\omega }{J}_{\omega 2}}{\sqrt{{r}_{\alpha }{J}_{\alpha 2}}\sqrt{r{J}_{2}}}\left[\sigma \left(\frac{{C}_{5}}{{D}_{12}}cos{\lambda }_{2}-\frac{{C}_{6}}{{D}_{12}}\mathrm{sin}{\lambda }_{2}\right)-\tau \left(\frac{{C}_{7}}{{D}_{12}}cos{\lambda }_{2}-\frac{{C}_{8}}{{D}_{12}}\mathrm{sin}{\lambda }_{2}\right)\right]+T\end{array}$ (29-a)

$\begin{array}{l}{\sigma }_{y}=\frac{{r}_{\omega }{J}_{\omega 1}}{\sqrt{{r}_{\alpha }{J}_{\alpha 1}}\sqrt{r{J}_{1}}}\left[\sigma \left(\frac{{D}_{2}}{{D}_{12}}cos{\lambda }_{1}-\frac{{D}_{3}}{{D}_{12}}\mathrm{sin}{\lambda }_{1}\right)+\tau \left(\frac{{g}_{2}-{g}_{1}}{{D}_{12}}cos{\lambda }_{1}-\frac{{h}_{1}-{h}_{2}}{{D}_{12}}\mathrm{sin}{\lambda }_{1}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{r}_{\omega }{J}_{\omega 2}}{\sqrt{{r}_{\alpha }{J}_{\alpha 2}}\sqrt{r{J}_{2}}}\left[\sigma \left(\frac{{D}_{1}}{{D}_{12}}cos{\lambda }_{2}+\frac{{D}_{3}}{{D}_{12}}\mathrm{sin}{\lambda }_{2}\right)-\tau \left(\frac{{g}_{2}-{g}_{1}}{{D}_{12}}cos{\lambda }_{2}-\frac{{h}_{1}-{h}_{2}}{{D}_{12}}\mathrm{sin}{\lambda }_{2}\right)\right]\end{array}$ (29-b)

$\begin{array}{l}{\tau }_{xy}=\frac{{r}_{\omega }{J}_{\omega 1}}{\sqrt{{r}_{\alpha }{J}_{\alpha 1}}\sqrt{r{J}_{1}}}\left[\sigma \left(\frac{{D}_{4}}{{D}_{12}}cos{\lambda }_{1}+\frac{{D}_{5}}{{D}_{12}}\mathrm{sin}{\lambda }_{1}\right)+\tau \left(\frac{{D}_{1}}{{D}_{1}{}_{2}}cos{\lambda }_{1}+\frac{{D}_{3}}{{D}_{1}{}_{2}}\mathrm{sin}{\lambda }_{1}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-\frac{{r}_{\omega }{J}_{\omega 2}}{\sqrt{{r}_{\alpha }{J}_{\alpha 2}}\sqrt{r{J}_{2}}}\left[\sigma \left(\frac{{D}_{4}}{{D}_{12}}cos{\lambda }_{2}+\frac{{D}_{5}}{{D}_{12}}\mathrm{sin}{\lambda }_{2}\right)-\tau \left(\frac{{D}_{2}}{{D}_{12}}cos{\lambda }_{2}-\frac{{D}_{3}}{{D}_{12}}\mathrm{sin}{\lambda }_{2}\right)\right]\end{array}$ (29-c)

$\begin{array}{c}u=\sqrt{{r}_{\alpha }{J}_{\alpha 1}}\sqrt{r{J}_{1}}\left[\sigma {a}_{11}\left(\frac{{B}_{31}}{{D}_{12}}cos\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}-\frac{{B}_{32}}{{D}_{12}}sin\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}\right)\\ \text{\hspace{0.17em}}+\tau {a}_{11}\left(\frac{{B}_{33}}{{D}_{12}}cos\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}-\frac{{B}_{34}}{{D}_{12}}sin\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}\right)\right]\\ \text{\hspace{0.17em}}+\sqrt{{r}_{\alpha }{J}_{\alpha 2}}\sqrt{r{J}_{2}}\left[\sigma {a}_{11}\left(\frac{{B}_{41}}{{D}_{12}}cos\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}+\frac{{B}_{42}}{{D}_{12}}sin\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}\right)\\ \text{\hspace{0.17em}}-\tau {a}_{11}\left(\frac{{B}_{43}}{{D}_{12}}cos\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}-\frac{{B}_{44}}{{D}_{12}}sin\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}\right)\right]+{a}_{11}Tx+{a}_{16}Ty+{u}_{0}\end{array}$ (30-a)

$\begin{array}{c}v=\sqrt{{r}_{\alpha }{J}_{\alpha 1}}\sqrt{r{J}_{1}}\left[\sigma {a}_{22}\left(\frac{{B}_{51}}{{D}_{12}}cos\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}-\frac{{B}_{52}}{{D}_{12}}sin\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}\right)\\ \text{\hspace{0.17em}}+\tau {a}_{22}\left(\frac{{B}_{53}}{{D}_{12}}cos\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}-\frac{{B}_{54}}{{D}_{12}}sin\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}\right)\right]\\ \text{\hspace{0.17em}}+\sqrt{{r}_{\alpha }{J}_{\alpha 2}}\sqrt{r{J}_{2}}\left[\sigma {a}_{22}\left(\frac{{B}_{61}}{{D}_{12}}cos\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}+\frac{{B}_{62}}{{D}_{12}}sin\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}\right)\\ \text{\hspace{0.17em}}-\tau {a}_{22}\left(\frac{{B}_{63}}{{D}_{12}}cos\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}-\frac{{B}_{64}}{{D}_{12}}sin\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}\right)\right]+{a}_{12}Ty+{v}_{0}\end{array}$ (30-b)

3.2. 裂纹端部应力场

$\alpha \to 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\omega \to 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{^}{\alpha }\to 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{^}{\omega }\to 0$

${r}_{\alpha }\to 2a,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{r}_{\omega }\to a,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{J}_{\alpha 1}\to 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{J}_{\alpha 2}\to 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{J}_{\omega 1}\to 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{J}_{\omega 2}\to 1$

${\lambda }_{1}={\stackrel{^}{\omega }}_{1}-\frac{{\stackrel{^}{\alpha }}_{1}+{\beta }_{1}}{2}\to -\frac{{\beta }_{1}}{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\lambda }_{2}={\stackrel{^}{\omega }}_{2}-\frac{{\stackrel{^}{\alpha }}_{2}+{\beta }_{2}}{2}\to -\frac{{\beta }_{2}}{2}$

$\begin{array}{l}{\sigma }_{x}=-\frac{\sqrt{a}}{\sqrt{2r}}\left[\frac{\sigma }{{D}_{12}}\left(\frac{{C}_{1}}{\sqrt{{J}_{1}}}cos\frac{{\beta }_{1}}{2}+\frac{{C}_{2}}{\sqrt{{J}_{1}}}\mathrm{sin}\frac{{\beta }_{1}}{2}-\frac{{C}_{5}}{\sqrt{{J}_{2}}}cos\frac{{\beta }_{2}}{2}-\frac{{C}_{6}}{\sqrt{{J}_{2}}}\mathrm{sin}\frac{{\beta }_{2}}{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\tau }{{D}_{12}}\left(\frac{{C}_{3}}{\sqrt{{J}_{1}}}cos\frac{{\beta }_{1}}{2}+\frac{{C}_{4}}{\sqrt{{J}_{1}}}\mathrm{sin}\frac{{\beta }_{1}}{2}+\frac{{C}_{7}}{\sqrt{{J}_{2}}}cos\frac{{\beta }_{2}}{2}+\frac{{C}_{8}}{\sqrt{{J}_{2}}}\mathrm{sin}\frac{{\beta }_{2}}{2}\right)\right]+T\end{array}$ (31-a)

$\begin{array}{l}{\sigma }_{y}=\frac{\sqrt{a}}{\sqrt{2r}}\left[\frac{\sigma }{{D}_{12}}\left(\frac{{D}_{2}}{\sqrt{{J}_{1}}}cos\frac{{\beta }_{1}}{2}+\frac{{D}_{3}}{\sqrt{{J}_{1}}}\mathrm{sin}\frac{{\beta }_{1}}{2}+\frac{{D}_{1}}{\sqrt{{J}_{2}}}cos\frac{{\beta }_{2}}{2}-\frac{{D}_{3}}{\sqrt{{J}_{2}}}\mathrm{sin}\frac{{\beta }_{2}}{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\tau }{{D}_{12}}\left(\frac{{g}_{2}-{g}_{1}}{\sqrt{{J}_{1}}}cos\frac{{\beta }_{1}}{2}+\frac{{h}_{1}-{h}_{2}}{\sqrt{{J}_{1}}}\mathrm{sin}\frac{{\beta }_{1}}{2}-\frac{{g}_{2}-{g}_{1}}{\sqrt{{J}_{2}}}cos\frac{{\beta }_{2}}{2}-\frac{{h}_{1}-{h}_{2}}{\sqrt{{J}_{2}}}\mathrm{sin}\frac{{\beta }_{2}}{2}\right)\right]\end{array}$ (31-b)

$\begin{array}{l}{\tau }_{xy}=\frac{\sqrt{a}}{\sqrt{2r}}\left[\frac{\sigma }{{D}_{12}}\left(\frac{{D}_{4}}{\sqrt{{J}_{1}}}cos\frac{{\beta }_{1}}{2}-\frac{{D}_{5}}{\sqrt{{J}_{1}}}\mathrm{sin}\frac{{\beta }_{1}}{2}-\frac{{D}_{4}}{\sqrt{{J}_{2}}}cos\frac{{\beta }_{2}}{2}+\frac{{D}_{5}}{\sqrt{{J}_{2}}}\mathrm{sin}\frac{{\beta }_{2}}{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\tau }{{D}_{12}}\left(\frac{{D}_{1}}{\sqrt{{J}_{1}}}cos\frac{{\beta }_{1}}{2}-\frac{{D}_{3}}{\sqrt{{J}_{1}}}\mathrm{sin}\frac{{\beta }_{1}}{2}+\frac{{D}_{2}}{\sqrt{{J}_{2}}}cos\frac{{\beta }_{2}}{2}+\frac{{D}_{3}}{\sqrt{{J}_{2}}}\mathrm{sin}\frac{{\beta }_{2}}{2}\right)\right]\end{array}$ (31-c)

${{\sigma }_{y}|}_{\theta =0}=\frac{\sigma \sqrt{a}}{\sqrt{2r}}=\frac{{K}_{I}}{\sqrt{2\pi r}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{{\tau }_{xy}|}_{\theta =0}=\frac{\tau \sqrt{a}}{\sqrt{2r}}=\frac{{K}_{II}}{\sqrt{2\pi r}}$ (32)

3.3. 裂纹变形分析

$-a (裂纹上面)

$-a (裂纹下面)

$-a

$\begin{array}{l}{u}^{+}=\sqrt{{a}^{2}-{x}^{2}}\left(\sigma {a}_{11}\frac{{B}_{42}-{B}_{32}}{{D}_{12}}+\tau {a}_{11}\frac{{B}_{44}-{B}_{34}}{{D}_{12}}\right)+{a}_{11}Tx+{u}_{0}\\ {u}^{-}=-\sqrt{{a}^{2}-{x}^{2}}\left(\sigma {a}_{11}\frac{{B}_{42}-{B}_{32}}{{D}_{12}}+\tau {a}_{11}\frac{{B}_{44}-{B}_{34}}{{D}_{12}}\right)+{a}_{11}Tx+{u}_{0}\end{array}$

$\begin{array}{l}{v}^{+}=\sqrt{{a}^{2}-{x}^{2}}\left(\sigma {a}_{22}\frac{{B}_{62}-{B}_{52}}{{D}_{12}}+\tau {a}_{22}\frac{{B}_{64}-{B}_{54}}{{D}_{12}}\right)+{v}_{0}\\ {v}^{-}=-\sqrt{{a}^{2}-{x}^{2}}\left(\sigma {a}_{22}\frac{{B}_{62}-{B}_{52}}{{D}_{12}}+\tau {a}_{22}\frac{{B}_{64}-{B}_{54}}{{D}_{12}}\right)+{v}_{0}\end{array}$

$\begin{array}{l}M=\sigma {a}_{11}\frac{{B}_{42}-{B}_{32}}{{D}_{12}}+\tau {a}_{11}\frac{{B}_{44}-{B}_{34}}{{D}_{12}}\\ N=\sigma {a}_{22}\frac{{B}_{62}-{B}_{52}}{{D}_{12}}+\tau {a}_{22}\frac{{B}_{64}-{B}_{54}}{{D}_{12}}\end{array}\right\}$ (33)

$\begin{array}{l}{\Delta }_{1}={u}^{+}-{u}^{-}=2M\sqrt{{a}^{2}-{x}^{2}}\\ {\Delta }_{2}={v}^{+}-{v}^{-}=2N\sqrt{{a}^{2}-{x}^{2}}\end{array}\right\}$ (34)

${\left(\frac{{\Delta }_{1}}{2M}\right)}^{2}+{x}^{2}={a}^{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(\frac{{\Delta }_{2}}{2N}\right)}^{2}+{x}^{2}={a}^{2}$ (35)

${\Delta }_{1}=2M\sqrt{2ar},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Delta }_{2}=2N\sqrt{2ar}$ (36)

4. 计算举例

${a}_{11}=50{\left(TPa\right)}^{-1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{22}=75{\left(TPa\right)}^{-1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{66}=150{\left(TPa\right)}^{-1}$

${a}_{12}=-35{\left(TPa\right)}^{-1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{16}=-60{\left(TPa\right)}^{-1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{26}=-20{\left(TPa\right)}^{-1}$

${A}_{1}=-\frac{2{a}_{16}}{{a}_{11}}=2.4,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{A}_{2}=\frac{{a}_{66}+2{a}_{12}}{{a}_{11}}=1.6,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{A}_{3}=-\frac{2{a}_{26}}{{a}_{11}}=0.8,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{A}_{4}=\frac{{a}_{22}}{{a}_{11}}=1.5$

${q}^{4}+{A}_{1}{q}^{3}+{A}_{2}{q}^{2}+{A}_{3}q+{A}_{4}={q}^{4}+2.4{q}^{3}+1.6{q}^{2}+0.8q+1.5=0$

${q}_{1}=0.235+0.774i,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{2}=-1.435+0.485i$

${q}_{3}=0.235-0.774i,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{4}=-1.435-0.485i$

${g}_{1}=0.235,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}_{1}=0.774,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{g}_{2}=-1.435,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}_{2}=0.485$

$\begin{array}{l}{D}_{1}={g}_{1}\left({g}_{1}-{g}_{2}\right)+{h}_{1}\left({h}_{1}-{h}_{2}\right)=0.616\\ {D}_{2}={g}_{2}\left({g}_{2}-{g}_{1}\right)+{h}_{2}\left({h}_{2}-{h}_{1}\right)=2.256\\ {D}_{3}={g}_{2}{h}_{1}-{g}_{1}{h}_{2}=-1.225,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{D}_{12}={D}_{1}+{D}_{2}=2.872\end{array}$

$\begin{array}{l}{D}_{4}={g}_{1}{g}_{2}\left({g}_{1}-{g}_{2}\right)+{g}_{2}{h}_{1}^{2}-{g}_{1}{h}_{2}^{2}=-1.478\\ {D}_{5}={g}_{2}^{2}{h}_{1}-{g}_{1}^{2}{h}_{2}-{h}_{1}{h}_{2}\left({h}_{1}-{h}_{2}\right)=1.459\end{array}$

$\begin{array}{l}{C}_{1}={g}_{1}{D}_{4}+{h}_{1}{D}_{5}=0.782,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{2}={h}_{1}{D}_{4}-{g}_{1}{D}_{5}=-1.487\\ {C}_{3}={g}_{1}{D}_{1}+{h}_{1}{D}_{3}=-0.8034,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{4}={h}_{1}{D}_{1}-{g}_{1}{D}_{3}=0.7747\\ {C}_{5}={g}_{2}{D}_{4}+{h}_{2}{D}_{5}=2.8285,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{6}={h}_{2}{D}_{4}-{g}_{2}{D}_{5}=1.3768\\ {C}_{7}={g}_{2}{D}_{2}-{h}_{2}{D}_{3}=-2.6432,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{8}={h}_{2}{D}_{2}+{g}_{2}{D}_{3}=2.852\end{array}$

$\begin{array}{l}2{B}_{1}=\sigma \left(\frac{{D}_{2}}{{D}_{12}}+i\frac{{D}_{3}}{{D}_{12}}\right)+\tau \left(\frac{{g}_{2}-{g}_{1}}{{D}_{1}{}_{2}}+i\frac{{h}_{1}-{h}_{2}}{{D}_{1}{}_{2}}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }=\sigma \left(0.7855-0.4265i\right)-\tau \left(0.5815-0.1006i\right)\\ 2{B}_{2}=\sigma \left(\frac{{D}_{1}}{{D}_{12}}-i\frac{{D}_{3}}{{D}_{12}}\right)-\tau \left(\frac{{g}_{2}-{g}_{1}}{{D}_{12}}+i\frac{{h}_{1}-{h}_{2}}{{D}_{12}}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\sigma \left(0.2145+0.4265i\right)+\tau \left(0.5815-0.1006i\right)\end{array}$

$T=-2Re\left({q}_{1}^{2}{B}_{1}+{q}_{2}^{2}{B}_{2}\right)=\sigma \left({g}_{1}{g}_{2}-{h}_{1}{h}_{2}\right)+\tau \left({g}_{1}+{g}_{2}\right)=-0.7126\sigma -1.2\tau$

${B}_{31}=-0.587,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{32}=4.095,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{33}=1.233,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{34}=-2.437$

${B}_{41}=0.624,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{42}=1.232,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{43}=1.233,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{44}=1.18$

${B}_{51}=-0.727,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{52}=-4.116,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{53}=-0.416,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{54}=2.728$

${B}_{61}=0.728,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{62}=-0.111,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{63}=-0.393,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{64}=0.821$

$\begin{array}{l}M=\sigma {a}_{11}\frac{{B}_{42}-{B}_{32}}{{D}_{12}}+\tau {a}_{11}\frac{{B}_{44}-{B}_{34}}{{D}_{12}}=-0.997{a}_{11}\sigma +1.259{a}_{11}\tau \\ N=\sigma {a}_{22}\frac{{B}_{62}-{B}_{52}}{{D}_{12}}+\tau {a}_{22}\frac{{B}_{64}-{B}_{54}}{{D}_{12}}=1.394{a}_{22}\sigma -0.661{a}_{22}\tau \end{array}$

$\begin{array}{l}{\Delta }_{1}=2M\sqrt{2ar}=\left(-1.994{a}_{11}\sigma +2.518{a}_{11}\tau \right)\sqrt{2ar}\\ {\Delta }_{2}=2N\sqrt{2ar}=\left(2.788{a}_{22}\sigma -1.322{a}_{22}\tau \right)\sqrt{2ar}\end{array}$

$\begin{array}{l}{\sigma }_{x}=-\frac{{K}_{I}}{\sqrt{2\pi r}}\left(\frac{0.2723}{\sqrt{{J}_{1}}}cos\frac{{\beta }_{1}}{2}-\frac{0.5178}{\sqrt{{J}_{1}}}\mathrm{sin}\frac{{\beta }_{1}}{2}-\frac{0.9849}{\sqrt{{J}_{2}}}cos\frac{{\beta }_{2}}{2}-\frac{0.4794}{\sqrt{{J}_{2}}}\mathrm{sin}\frac{{\beta }_{2}}{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{K}_{II}}{\sqrt{2\pi r}}\left(-\frac{0.2797}{\sqrt{{J}_{1}}}cos\frac{{\beta }_{1}}{2}+\frac{0.2697}{\sqrt{{J}_{1}}}\mathrm{sin}\frac{{\beta }_{1}}{2}-\frac{0.9203}{\sqrt{{J}_{2}}}cos\frac{{\beta }_{2}}{2}+\frac{0.993}{\sqrt{{J}_{2}}}\mathrm{sin}\frac{{\beta }_{2}}{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-0.7126\sigma -1.2\tau \end{array}$

$\begin{array}{l}{\sigma }_{y}=\frac{{K}_{I}}{\sqrt{2\pi r}}\left(\frac{0.7855}{\sqrt{{J}_{1}}}cos\frac{{\beta }_{1}}{2}-\frac{0.4265}{\sqrt{{J}_{1}}}\mathrm{sin}\frac{{\beta }_{1}}{2}+\frac{0.2145}{\sqrt{{J}_{2}}}cos\frac{{\beta }_{2}}{2}+\frac{0.4265}{\sqrt{{J}_{2}}}\mathrm{sin}\frac{{\beta }_{2}}{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{K}_{II}}{\sqrt{2\pi r}}\left(-\frac{0.5815}{\sqrt{{J}_{1}}}cos\frac{{\beta }_{1}}{2}+\frac{0.1006}{\sqrt{{J}_{1}}}\mathrm{sin}\frac{{\beta }_{1}}{2}+\frac{0.5815}{\sqrt{{J}_{2}}}cos\frac{{\beta }_{2}}{2}-\frac{0.1006}{\sqrt{{J}_{2}}}\mathrm{sin}\frac{{\beta }_{2}}{2}\right)\end{array}$

$\begin{array}{l}{\tau }_{xy}=\frac{{K}_{I}}{\sqrt{2\pi r}}\left(-\frac{0.5146}{\sqrt{{J}_{1}}}cos\frac{{\beta }_{1}}{2}-\frac{0.508}{\sqrt{{J}_{1}}}\mathrm{sin}\frac{{\beta }_{1}}{2}+\frac{0.5146}{\sqrt{{J}_{2}}}cos\frac{{\beta }_{2}}{2}+\frac{0.508}{\sqrt{{J}_{2}}}\mathrm{sin}\frac{{\beta }_{2}}{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{K}_{II}}{\sqrt{2\pi r}}\left(\frac{0.2145}{\sqrt{{J}_{1}}}cos\frac{{\beta }_{1}}{2}+\frac{0.4265}{\sqrt{{J}_{1}}}\mathrm{sin}\frac{{\beta }_{1}}{2}+\frac{0.7855}{\sqrt{{J}_{2}}}cos\frac{{\beta }_{2}}{2}-\frac{0.4265}{\sqrt{{J}_{2}}}\mathrm{sin}\frac{{\beta }_{2}}{2}\right)\end{array}$

$\begin{array}{l}{J}_{1}=\sqrt{{\left(\mathrm{cos}\theta +0.235\mathrm{sin}\theta \right)}^{2}+{\left(0.774\mathrm{sin}\theta \right)}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{1}=\mathrm{arctan}\frac{0.774\mathrm{sin}\theta }{\mathrm{cos}\theta +0.235\mathrm{sin}\theta }\\ {J}_{2}=\sqrt{{\left(\mathrm{cos}\theta -1.435\mathrm{sin}\theta \right)}^{2}+{\left(0.485\mathrm{sin}\theta \right)}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{2}=\mathrm{arctan}\frac{0.485\mathrm{sin}\theta }{\mathrm{cos}\theta -1.435\mathrm{sin}\theta }\end{array}$

 [1] 张行. 断裂与损伤力学[M]. 北京: 北京航空航天大学出版社, 2009: 1-37. [2] 李群, 欧卓成, 陈宜亨. 高等断裂力学[M]. 北京: 科学出版社, 2017: 16-71. [3] Sun, C.T. and Jin, Z.-H. (2012) Fracture Mechanics. Elsevier Inc., United States. [4] Sih, G.C. (1991) Mechanics of Fracture Initiation and Propagation. Kluwer Academic Publishers, Netherlands. https://doi.org/10.1007/978-94-011-3734-8 [5] 李星, 路见可. 双周期弹性断裂理论[M]. 北京: 科学出版社, 2015: 25-68. [6] Smith, D.J., Ayatollahi, M.R. and Pavier, M.J. (2001) The Role of T-Stress in Brittle Fracture for Linear Elastic Materials under Mixed-Mode Loading. Fatigue & Fracture of Engineering Materials & Structures, 24, 137-150. https://doi.org/10.1046/j.1460-2695.2001.00377.x [7] Friedrich, K. (1989) Application of Fracture Mechanics to Composite Materials. Elsevier Science Pub. Co., Netherlands. [8] 杨维阳, 李俊林, 张雪霞. 复合材料断裂复变方法[M]. 北京: 科学出版社, 2005: 12-72. [9] Zhang, H. and Qiao, P. (2019) A State-Based Peridynamic Model for Quantitative Elastic and Fracture Analysis of Orthotropic Materials. Engineering Fracture Mechanics, 206, 147-171. https://doi.org/10.1016/j.engfracmech.2018.10.003 [10] 贾普荣, 锁永永. 正交异性材料平面裂纹尖端应力场[J]. 应用力学学报, 2020, 37(1): 78-85. [11] 贾普荣. 正交异性板裂纹端部应力及变形通解[J]. 力学研究, 2020, 9(2): 70-76. [12] 贾普荣. 正交异性材料I+II+III混合型裂纹尖端应力分析[J]. 力学研究, 2020, 9(4): 123-134. [13] 贾普荣. 基于泛复函的各向异性板裂纹尖端应力场解法[J]. 力学研究, 2021, 10(2): 90-98.