An efficient numerical method for quasi-static crack propagation in heterogeneous media

Abstract

The paper is devoted to the problem of slow crack growth in heterogeneous media. The crack is subjected to arbitrary pressure distribution on the crack surface. The problem relates to construction of the so-called equilibrium crack. For such a crack, stress intensity factors are equal to the material fracture toughness at each point of the crack contour. The crack shape and size depend on spatial distributions of the elastic properties and fracture toughness of the medium, and the type of loading. In the paper, attention is focused on the case of layered elastic media when a planar crack propagates orthogonally to the layers. The problem is reduced to a system of surface integral equations for the crack opening vector and volume integral equations for stresses in the medium. For discretization of these equations, a regular node grid and Gaussian approximating functions are used. For iterative solution of the discretized equations, fast Fourier transform technique is employed. An iteration process is proposed for the construction of the crack shape in the process of crack growth. Examples of crack evolution for various properties of medium and types of loading are presented.

Appendix

The integral ${\varGamma }_{ijkl}(x)$ in Eq. (16) can be calculated in explicit analytical form. In the basis of six linearly independent four rank tensors $E_{ijkl}^{\left( p\right) }(m)$

$$\begin{aligned} E_{ijkl}^{1}= & {} \delta _{ik}\delta _{jl}|_{(ij)(kl)}\,, \;\;E_{ijkl}^{2}=\delta _{ij}\delta _{kl}\,,\nonumber \\ E_{ijkl}^{3}= & {} \delta _{ij}m_{k}m_{l}\,, \end{aligned}$$

(A1)

$$\begin{aligned} E_{ijkl}^{4}= & {} m_{i}m_{j}\delta _{kl}\,,\;\; E_{ijkl}^{5}=m_{i}m_{k}\delta _{jl}|_{(ij)(kl)}\,,\nonumber \\ E_{ijkl}^{6}= & {} m_{i}m_{j}m_{k}m_{l},\, \end{aligned}$$

(A2)

the tensor ${\varGamma }_{ijkl}(x)$ takes the form

$$\begin{aligned} {\varGamma }_{ijkl}(x)=-\,2\mu _{0}\sum \limits _{p=1}^{6}g^{\left( p\right) }\left( \frac{\left| x\right| }{h}\right) E_{ijkl}^{p}(m), \quad m= \frac{x}{\left| x\right| }. \end{aligned}$$

(A3)

Scalar functions $g^{\left( p\right) }(z)$ ($p=1,2,\ldots ,6$) in this equation are expressed in terms of three scalar functions $\psi _{0}(z),\psi _{1}(z),\psi _{2}(z)$

$$\begin{aligned} g^{1}= & {} \left( \psi _{0}-2\psi _{1}\right) +4\kappa _{0}\psi _{2}, \nonumber \\ g^{2}= & {} \left( 2\kappa _{0}-1\right) \left( \psi _{0}-2\psi _{1}\right) +2\kappa _{0}\psi _{2}, \end{aligned}$$

(A4)

$$\begin{aligned} g^{3}= & {} g^{4}=\left( 1-2\kappa _{0}\right) \phi _{0}+2\kappa _{0}\phi _{1}, \nonumber \\ g^{5}= & {} -\frac{1}{2}\left( \phi _{0}-16\kappa _{0}\phi _{1}\right) , { \ \ }g^{6}=2\kappa _{0}\phi _{2}. \end{aligned}$$

(A5)

$$\begin{aligned} \phi _{0}= & {} \psi _{0}-3\psi _{1}, \ \phi _{1}=\psi _{0}-5\psi _{1},\nonumber \\ \phi _{2}= & {} \psi _{0}-10\psi _{1}+35\psi _{2}, \end{aligned}$$

(A6)

$$\begin{aligned} \psi _{0}\left( z\right)= & {} \frac{1}{\left( \pi H\right) ^{3/2}}\exp \left( -\, \frac{z^{2}}{H}\right) ,\end{aligned}$$

(A7)

$$\begin{aligned} \psi _{1}\left( z\right)= & {} \frac{1}{4\pi ^{3/2}z^{3}\sqrt{H}}\left[ -\,2z\exp \left( -\,\frac{z^{2}}{H}\right) \right. \nonumber \\&\left. + \,\sqrt{\pi H}erf\left( \frac{z}{\sqrt{H}}\right) \right] ,\end{aligned}$$

(A8)

$$\begin{aligned} \psi _{2}\left( z\right)= & {} \frac{1}{16\pi ^{2}z^{5}}\left[ 6\sqrt{\pi H}z\exp \left( -\frac{z^{2}}{H}\right) \right. \nonumber \\&\left. +\,\pi \left( -3H+2z^{2}\right) erf\left( \frac{z}{\sqrt{H}} \right) \right] . \end{aligned}$$

(A9)

Here $erf\left( z\right)$ is the probability integral

$$\begin{aligned} erf(z)=\frac{2}{\sqrt{\pi }}\int \nolimits _{0}^{z}\exp ( -\,t^{2})dt. \end{aligned}$$

(A10)

The integral $I_{ijk}(x_{1},x_{2},x_{3})$ in Eq. (25) is presented in the form

$$\begin{aligned} I_{ijk}(x_{1},x_{2},x_{3})= & {} \iint \limits _{-\infty }^{\ \ \infty }S_{ijk3}(x_{1}-x_{1}^{\prime },x_{2}-x_{2}^{\prime },x_{3})\nonumber \\&\cdot \exp \left[ -\frac{(x_{1}^{\prime 2}+x_{2}^{\prime 2})}{h^{2}H}\right] dx_{1}^{\prime }x_{2}^{\prime }.\nonumber \\ \end{aligned}$$

(A11)

Calculation of the double integral in eq (29) yields the following equation for $I_{ijk}(\varsigma _{1},\varsigma _{2},\varsigma _{3})$:

$$\begin{aligned}&I_{ijk}(x_{1},x_{2},x_{3}) \nonumber \\&\quad =I_{ijk}(r,x_{3})=-\,s_{1}q_{(i}\theta _{j)k}+s_{2}\theta _{ij}q_{k} \nonumber \\&\qquad +\, s_{3}n_{(i}\theta _{j)k}-s_{4}\theta _{ij}n_{k}-s_{5}q_{(i}n_{j)}q_{k}+s_{6}q_{i}q_{j}n_{k}\nonumber \\&\qquad +\,s_{7}(2q_{(i}n_{j)}n_{k}+n_{i}n_{j}q_{k}) \nonumber \\&\qquad -\,s_{8}n_{i}n_{j}n_{k}-s_{9}q_{i}q_{j}q_{k}, \end{aligned}$$

(A12)

$$\begin{aligned} q_{i}= & {} \frac{x_{i}}{r}\quad \text { }\left( i=1,2\right) ;\quad r=\sqrt{ x_{1}^{2}+x_{2}^{2}},\quad q_{3}=0;\nonumber \\ \theta _{ij}= & {} \delta _{ij}-n_{i}n_{j}. \end{aligned}$$

(A13)

Here $n_{i}=n_{i}^{(s)}$ is the normal to ${\varOmega }$ at the sth node, scalar coefficients $s_{\alpha }=s_{\alpha }(r,x_{3})$ are

$$\begin{aligned} s_{1}= & {} 2\mu _{0}(g_{1}-4\kappa _{0}g_{2}), \nonumber \\ s_{2}= & {} 2\mu _{0}\left[ (1-2\kappa _{0})g_{1}+2\kappa _{0}g_{2}\right] , \end{aligned}$$

(A14)

$$\begin{aligned} s_{3}= & {} 2\mu _{0}(g_{3}-4\kappa _{0}g_{4}), \nonumber \\ s_{4}= & {} 2\mu _{0}\left[ (1-2\kappa _{0})g_{3}+2\kappa _{0}g_{4}\right] , \end{aligned}$$

(A15)

$$\begin{aligned} s_{5}= & {} 2\mu _{0}\left( g_{5}-4\kappa _{0}g_{6}\right) , \nonumber \\ s_{6}= & {} 2\mu _{0} \left[ (1-2\kappa _{0})g_{5}+2\kappa _{0}g_{6}\right] , \end{aligned}$$

(A16)

$$\begin{aligned} s_{7}= & {} 4\mu _{0}\kappa _{0}g_{7}, \ s_{8}=4\mu _{0}\kappa _{0}g_{8}, \ s_{9}=4\mu _{0}\kappa _{0}g_{9}. \end{aligned}$$

(A17)

Functions $g_{1},g_{2},\ldots ,g_{9}$ in these equations are expressed in terms of five linearly independent integrals $F_{\alpha }(r,x_{3}):$

$$\begin{aligned} g_{1}= & {} \frac{r}{2h^{2}}\mathrm {sign}(x_{3})(F_{3}+F_{4}),\quad g_{2} = \frac{x_{3}}{ 2hr}F_{2},\nonumber \\ g_{3}= & {} \frac{1}{2h}(F_{2}-F_{1}), \end{aligned}$$

(A18)

$$\begin{aligned} g_{4}= & {} \frac{1}{4h}\left[ F_{1}+F_{2}-\frac{|x_{3}|}{h}(F_{3}+F_{4}) \right] , \nonumber \\ g_{5}= & {} \frac{1}{h}F_{2}, \end{aligned}$$

(A19)

$$\begin{aligned} g_{6}= & {} \frac{1}{2h}\left( F_{2}-\frac{|x_{3}|}{h}F_{4}\right) , \nonumber \\ g_{7}= & {} \frac{x_{3}}{2h^{2}}\left( F_{5}-4\frac{h}{r}F_{2}\right) , \end{aligned}$$

(A20)

$$\begin{aligned} g_{8}= & {} \frac{1}{2h}\left( F_{1}+\frac{|x_{3}|}{h}F_{3}\right) , \quad g_{9}=\frac{x_{3}}{ 2h^{2}}F_{5}. \end{aligned}$$

(A21)

Here $F_{\alpha }=F_{\alpha }(r,x_{3})$ are 1D-absolutely converging integrals ($\rho =r/h$, $z=x_{3}/h$)

$$\begin{aligned} F_{1}(\rho ,z)= & {} \frac{1}{4\pi }\int \limits _{0}^{\infty }\exp \left( -\,k|z|- \frac{k^{2}H}{4}\right) J_{0}(k\rho )k^{2}dk, \end{aligned}$$

(A22)

$$\begin{aligned} F_{2}(\rho ,z)= & {} \frac{1}{4\pi }\int \limits _{0}^{\infty }\exp \left( -\,k|z|- \frac{k^{2}H}{4}\right) J_{2}(k\rho )k^{2}dk, \end{aligned}$$

(A23)

$$\begin{aligned} F_{3}(\rho ,z)= & {} \frac{1}{4\pi }\int \limits _{0}^{\infty }\exp \left( -\,k|z|- \frac{k^{2}H}{4}\right) J_{0}(k\rho )k^{3}dk, \end{aligned}$$

(A24)

$$\begin{aligned} F_{4}(\rho ,z)= & {} \frac{1}{4\pi }\int \limits _{0}^{\infty }\exp \left( -\,k|z|- \frac{k^{2}H}{4}\right) J_{2}(k\rho )k^{3}dk, \end{aligned}$$

(A25)

$$\begin{aligned} F_{5}(\rho ,z)= & {} \frac{1}{4\pi }\int \limits _{0}^{\infty }\exp \left( -\,k|z|- \frac{k^{2}H}{4}\right) J_{3}(k\rho )k^{3}dk.\nonumber \\ \end{aligned}$$

(A26)

In these equations, $J_{n}(k\rho )$ ($n=0,2,3$) are Bessel functions of the first kind. Note that these functions do not depend on the node grid step h but are functions of non-dimensional variables $\rho$ and z.

If $x_{3}=0$, the tensor $I_{ijk}(x_{1},x_{2},0)$ is calculated explicitly and takes the form

$$\begin{aligned} I_{ijk}(x_{1},x_{2},0)= & {} I_{ijk}(r,0) \nonumber \\= & {} s_{1}n_{(i}\theta _{j)k}-s_{2}\theta _{ij}n_{k}-s_{3}q_{(i} n_{j)}q_{k} \nonumber \\&+\, s_{4}q_{i}q_{j}n_{k}-s_{5}n_{i}n_{j}n_{k}, \end{aligned}$$

(A27)

$$\begin{aligned} q_{i}= & {} \frac{x_{i}}{r}\quad \text { }\left( i=1,2\right) , \nonumber \\ r= & {} \sqrt{x_{1}^{2}+x_{2}^{2}},\quad \text { }q_{3}=0,\nonumber \\ \theta _{ij}= & {} \delta _{ij}-n_{i}n_{j}. \end{aligned}$$

(A28)

Here the scalar coefficients $s_{\alpha }=s_{\alpha }(r,\varsigma _{3})$ are

$$\begin{aligned} s_{1}= & {} \mu _{0}\frac{1}{h}\left[ \left( -1-2\kappa _{0}\right) F_{1}+\left( 1-2\kappa _{0}\right) F_{2}\right] , \end{aligned}$$

(A29)

$$\begin{aligned} s_{2}= & {} \mu _{0}\frac{1}{h}\left[ \left( -1+3\kappa _{0}\right) F_{1}+\left( 1-\kappa _{0}\right) F_{2}\right] , \end{aligned}$$

(A30)

$$\begin{aligned} s_{3}= & {} 2\mu _{0}\frac{1}{h}\left( 1-2\kappa _{0}\right) F_{2},\text { } s_{4}=2\mu _{0}\frac{1}{h}\left[ (1-\kappa _{0})F_{2}\right] ,\nonumber \\ s_{5}= & {} \mu _{0}\kappa _{0}\frac{2}{h}F_{1}. \end{aligned}$$

(A31)

The functions $F_{\alpha }=F_{\alpha }(r)$ have the form

$$\begin{aligned} F_{1}(\rho )= & {} \frac{\exp (-\rho ^{2}/2H)}{2\sqrt{\pi H^{3}}} \nonumber \\&\cdot \left[ \left( 1-\frac{\rho ^{2}}{H}\right) I_{0}\left( \frac{\rho ^{2}}{2H} \right) +\frac{\rho ^{2}}{H}I_{1}\left( \frac{\rho ^{2}}{2H}\right) \right] , \end{aligned}$$

(A32)

$$\begin{aligned} F_{2}(\rho )= & {} \frac{\exp (-\rho ^{2}/2H)}{2\sqrt{\pi H^{3}}} \nonumber \\&\cdot \left[ \frac{\rho ^{2}}{H}I_{0}\left( \frac{\rho ^{2}}{2H}\right) -\left( 1+ \frac{\rho ^{2}}{H}\right) I_{1}\left( \frac{\rho ^{2}}{2H}\right) \right] .\nonumber \\ \end{aligned}$$

(A33)

In these equations, $I_{0}(z)$ and $I_{1}(z)$ are the first kind modified Bessel functions of order 0 and 1 correspondingly, $\rho =r/h$.