New approach to interface crack problems in transversely isotropic materials

Abstract

The novelty of the approach contains several aspects: the method of derivation of the governing equations, the form and number of equations, the optimal choice of elastic constants. While majority of published articles derives the governing equations from the Green’s function for the compound space and the use of the reciprocal theorem, we use the combination of Green’s functions for 2 different half-spaces and Fourier transform. The usual approach leads to 3 hypersingular integral equations, we arrive at 2 integro-differential equations (one of them being complex). While the coefficients of the governing equations in existing publications are presented in terms of the results of the solution of a set of linear algebraic equations and are too cumbersome to be written explicitly in terms of the basic elastic constants, our choice of the basic constants leads to quite elegant explicit expressions for these coefficients and reveals certain symmetry, which was noticed only numerically in previous publications. The new form of the governing equations allows us to obtain an exact closed form solution for an axisymmetric interface crack problem by elementary means. The same method can be applied to obtain the exact solution for a non-axisymmetric problem. A comparison is made with the existing exact solutions and some are shown to be incorrect.

Appendices

Appendix 1

We present below the table of Fourier integral transforms, used in this article.

$$\begin{aligned}&\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty } \frac{1}{\sqrt{\xi _{1}^{2}+\xi _{2}^{2}}}\exp (-\zeta z)\exp [-i(x\xi _{1}+y\xi _{2})]\mathrm{d}\xi _{1}\mathrm{d}\xi _{2}=\frac{2\pi }{R_{0}}, \end{aligned}$$

(151)

$$\begin{aligned}&\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }\frac{i\xi _{2}}{(\xi _{1}^{2}+\xi _{2}^{2})} \exp (-\zeta z)\exp [-i(x\xi _{1}+y\xi _{2})]\mathrm{d}\xi _{1}\mathrm{d}\xi _{2}= \frac{2\pi y}{R_{0}(R_{0}+z_{g})}, \end{aligned}$$

(152)

$$\begin{aligned}&\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }\frac{i\xi _{1}}{(\xi _{1}^{2} +\xi _{2}^{2})}\exp (-\zeta z)\exp [-i(x\xi _{1}+y\xi _{2})]\mathrm{d}\xi _{1}\mathrm{d}\xi _{2}=\frac{2\pi x}{R_{0}(R_{0}+z_{g})}, \end{aligned}$$

(153)

$$\begin{aligned}&\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }\frac{\xi _{1}^{2}}{(\xi _{1}^{2} +\xi _{2}^{2})^{3/2}}\exp (-\zeta z) \exp [-i(x\xi _{1}+y\xi _{2})]\mathrm{d}\xi _{1}\mathrm{d}\xi _{2} =\pi \left[ \frac{1}{R_{0}}+\frac{y^{2}-x^{2}}{R_{0}(R_{0}+z_{g})^{2}}\right] , \end{aligned}$$

(154)

$$\begin{aligned}&\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty } \frac{\xi _{2}^{2}}{(\xi _{1}^{2}+\xi _{2}^{2})^{3/2}}\exp (-\zeta z)\exp [-i(x\xi _{1}+y\xi _{2})]\mathrm{d}\xi _{1}\mathrm{d}\xi _{2} =\pi \left[ \frac{1}{R_{0}}-\frac{y^{2}-x^{2}}{R_{0}(R_{0}+z_{g})^{2}}\right] , \end{aligned}$$

(155)

$$\begin{aligned}&\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }\frac{\xi _{1} \xi _{2}}{(\xi _{1}^{2}+\xi _{2}^{2})^{3/2}}\exp (-\zeta z) \exp [-i(x\xi _{1}+y\xi _{2})]\mathrm{d}\xi _{1}\mathrm{d}\xi _{2}=-\frac{2\pi xy}{R_{0}(R_{0}+z_{g})^{2}}, \end{aligned}$$

(156)

The following notation was introduced in the formulas above:

$$\begin{aligned} \zeta =\frac{1}{\gamma }\sqrt{\xi _{1}^{2}+\xi _{2}^{2}},\qquad R_{0} =\sqrt{z_{g}^{2}+x^{2}+y^{2}},\qquad z_{g}=\frac{z}{\gamma }. \end{aligned}$$

(157)

Appendix 2

We present below the solutions of the set of linear algebraic equations (58–63)

$$\begin{aligned} A_{11}^{+}= & {} \frac{A_{1}^{+}}{D}[-(c_{44}^{-})^{2}(1+m_{1}^{-}) (1+m_{2}^{-})(m_{2}^{+}\gamma _{1}^{+}+m_{1}^{+}\gamma _{2}^{+}) (\gamma _{1}^{-}-\gamma _{2}^{-})\nonumber \\&-(c_{44}^{+})^{2}(1+m_{1}^{+}) (1+m_{2}^{+})(m_{2}^{-}\gamma _{1}^{-}-m_{1}^{-}\gamma _{2}^{-}) (\gamma _{1}^{+}+\gamma _{2}^{+}) \nonumber \\&+c_{44}^{+}c_{44}^{-}[(m_{1}^{+}-m_{2}^{+})(m_{1}^{-}-m_{2}^{-}) (\gamma _{1}^{+}\gamma _{2}^{+}-\gamma _{1}^{-}\gamma _{2}^{-})\nonumber \\&+2(1+m_{1}^{+}) (1+m_{1}^{-})(m_{2}^{+}\gamma _{1}^{+}+\gamma _{2}^{+})(m_{2}^{-}\gamma _{1}^{-} -\gamma _{2}^{-})]], \end{aligned}$$

(158)

$$\begin{aligned} A_{21}^{+}= & {} 2\gamma _{2}^{+}\frac{A_{1}^{+}}{D}[(c_{44}^{-})^{2}m_{1}^{+} (1+m_{1}^{-})(1+m_{2}^{-})(\gamma _{1}^{-}-\gamma _{2}^{-})+(c_{44}^{+})^{2} (1+m_{1}^{+})^{2}(m_{2}^{-}\gamma _{1}^{-}-m_{1}^{-}\gamma _{2}^{-}) \nonumber \\&+c_{44}^{+}c_{44}^{-}(1+m_{1}^{+})^{2}(1+m_{2}^{-})(m_{1}^{-}\gamma _{2}^{-} -\gamma _{1}^{-})], \end{aligned}$$

(159)

$$\begin{aligned} A_{12}^{+}= & {} 2\gamma _{1}^{+}\frac{A_{2}^{+}}{D}[-(c_{44}^{-})^{2}m_{2}^{+}(1+m_{1}^{-}) (1+m_{2}^{-})(\gamma _{1}^{-}-\gamma _{2}^{-})\nonumber \\&-(c_{44}^{+})^{2}(1+m_{2}^{+})^{2}(m_{2}^{-} \gamma _{1}^{-}-m_{1}^{-}\gamma _{2}^{-})\nonumber \\&+c_{44}^{+}c_{44}^{-}(1+m_{2}^{+})^{2}(1+m_{2}^{-})(\gamma _{1}^{-}-m_{1}^{-} \gamma _{2}^{-})], \end{aligned}$$

(160)

$$\begin{aligned} A_{22}^{+}= & {} \frac{A_{2}^{+}}{D}[(c_{44}^{-})^{2}(1+m_{1}^{-})(1+m_{2}^{-})(m_{2}^{+} \gamma _{1}^{+}+m_{1}^{+}\gamma _{2}^{+})(\gamma _{1}^{-}-\gamma _{2}^{-})\nonumber \\&+(c_{44}^{+})^{2} (1+m_{1}^{+})(1+m_{2}^{+})(m_{2}^{-}\gamma _{1}^{-}-m_{1}^{-} \gamma _{2}^{-}) (\gamma _{1}^{+}+\gamma _{2}^{+}) \nonumber \\&+c_{44}^{+}c_{44}^{-}[(m_{1}^{+}-m_{2}^{+})(m_{1}^{-}-m_{2}^{-}) (\gamma _{1}^{+}\gamma _{2}^{+}-\gamma _{1}^{-}\gamma _{2}^{-})\nonumber \\&-2(m_{1}^{+}+1) (m_{1}^{-}+1)(m_{2}^{+}\gamma _{1}^{+}+\gamma _{2}^{+}) (m_{2}^{-}\gamma _{1}^{-} -\gamma _{2}^{-})]], \end{aligned}$$

(161)

$$\begin{aligned} A_{11}^{-}= & {} -2c_{44}^{+}(m_{1}^{+}-m_{2}^{+})\gamma _{1}^{-}\frac{A_{1}^{+}}{D} [c_{44}^{-}(m_{2}^{-}+1)(m_{1}^{+}\gamma _{2}^{+}-\gamma _{2}^{-})-c_{44}^{+} (m_{1}^{+}+1)(m_{2}^{-}\gamma _{2}^{+}-\gamma _{2}^{-})], \end{aligned}$$

(162)

$$\begin{aligned} A_{21}^{-}= & {} 2c_{44}^{+}(m_{1}^{+}-m_{2}^{+})\gamma _{2}^{-}\frac{A_{1}^{+}}{D} [c_{44}^{-}(m_{1}^{-}+1)(m_{1}^{+}\gamma _{2}^{+}-\gamma _{1}^{-})-c_{44}^{+} (m_{1}^{+}+1)(m_{1}^{-}\gamma _{2}^{+}-\gamma _{1}^{-})], \end{aligned}$$

(163)

$$\begin{aligned} A_{12}^{-}= & {} -2c_{44}^{+}(m_{1}^{+}-m_{2}^{+})\gamma _{1}^{-}\frac{A_{2}^{+}}{D} [c_{44}^{-}(m_{2}^{-}+1)(m_{2}^{+}\gamma _{1}^{+}-\gamma _{2}^{-})-c_{44}^{+} (m_{2}^{+}+1)(m_{2}^{-}\gamma _{1}^{+}-\gamma _{2}^{-})], \end{aligned}$$

(164)

$$\begin{aligned} A_{22}^{-}= & {} 2c_{44}^{+}(m_{1}^{+}-m_{2}^{+})\gamma _{2}^{-}\frac{A_{2}^{+}}{D} [c_{44}^{-}(m_{1}^{-}+1)(m_{2}^{+}\gamma _{1}^{+}-\gamma _{1}^{-})-c_{44}^{+} (m_{2}^{+}+1)(m_{1}^{-}\gamma _{1}^{+}-\gamma _{1}^{-})], \end{aligned}$$

(165)

$$\begin{aligned} D= & {} -(c_{44}^{-})^{2}(1+m_{1}^{-})(1+m_{2}^{-})(m_{2}^{+}\gamma _{1}^{+}-m_{1}^{+} \gamma _{2}^{+})(\gamma _{1}^{-}-\gamma _{2}^{-})\nonumber \\&-(c_{44}^{+})^{2}(1+m_{1}^{+}) (1+m_{2}^{+})(m_{2}^{-}\gamma _{1}^{-}-m_{1}^{-}\gamma _{2}^{-})(\gamma _{1}^{+}-\gamma _{2}^{+}) \nonumber \\&+c_{44}^{+}c_{44}^{-}[(m_{1}^{+}-m_{2}^{+})(m_{1}^{-}-m_{2}^{-})(\gamma _{1}^{+} \gamma _{2}^{+}+\gamma _{1}^{-}\gamma _{2}^{-})\nonumber \\&+2(1+m_{1}^{+})(1+m_{1}^{-})(m_{2}^{+} \gamma _{1}^{+}-\gamma _{2}^{+})(m_{2}^{-}\gamma _{1}^{-}-\gamma _{2}^{-})],\nonumber \\ \end{aligned}$$

(166)

The solutions for $B_{kj}$ are exactly the same as for $A_{kj}$ given above, except that the coefficients $A_{k}^{+}$ should be replaced by $-B_{k}^{+}$ for $\{j,k\}=1,2$. The solutions for $B_{3}$ are

$$\begin{aligned} B_{3}^{-}=\frac{i\gamma _{3}^{+}\gamma _{3}^{-}}{4\pi (c_{44}^{-}\gamma _{3}^{+} +c_{44}^{+}\gamma _{3}^{-})},\qquad B_{3p}^{+}=\frac{i\gamma _{3}^{+}(c_{44}^{+} \gamma _{3}^{-}-c_{44}^{-}\gamma _{3}^{+})}{8\pi c_{44}^{+}(c_{44}^{-}\gamma _{3}^{+} +c_{44}^{+}\gamma _{3}^{-})}. \end{aligned}$$

(167)