Singular perturbation solutions of a class of systems of singular integral equations (Q2706076)

The authors examine a new class of systems of strongly singular integrodifferential equations which emerges in the study of the bridged interface crack growth. Let \(u(x)=(u_1(x), u_2(x))^T\), \(-1<x<1\), a two-dimensional \(C^1\) vector function, \(\alpha_1, \alpha_2, \alpha_3, \alpha_4\) real numbers, NEWLINE\[NEWLINE\alpha_0= \sqrt{\alpha_1 \alpha_2-\alpha_3}\quad\text{and}\quad \Lambda= {1\over \alpha_0} \left(\begin{matrix} \alpha_1 & \alpha_3+i \alpha_4\\ \alpha_3-i\alpha_4 & \alpha_2\end{matrix} \right)NEWLINE\]NEWLINE be a complex \(2\times 2\) positive-definite matrix. If \(a\) and \(f\) are functions defined on the open interval \(]-1,1[\) define the convolution integration NEWLINE\[NEWLINE(a*f)(x)= \int^1_{-1}a (x-t)f(t)dt.NEWLINE\]NEWLINE The authors consider the system NEWLINE\[NEWLINE\varepsilon\left( {1\over\pi} \text{Re}\left( {1\over(x-i\alpha_0)^2} \Lambda\right) *u(x)+T(x) \right)+ f\bigl(u(x),x\bigr)=0 \tag{1}NEWLINE\]NEWLINE for \(\varepsilon\ll 1\), \(-1<x<1\), with auxiliary condition \(u(\pm 1)=0\) and where NEWLINE\[NEWLINE\left( \begin{matrix} a & b\\ c & d\end{matrix} \right)* {f\choose g}=\left( \begin{matrix} a*f & b*g \\ c*f & d*g \end{matrix} \right).NEWLINE\]NEWLINE Consider \(\Lambda= \Lambda_1-i \Lambda_2\) where NEWLINE\[NEWLINE\Lambda_1={1 \over\alpha_0} \left(\begin{matrix} \alpha_1 & \alpha_3 \\ \alpha_3 & \alpha_2 \end{matrix}\right), \quad\Lambda_2= {1\over\alpha_0} \left( \begin{matrix} 0 & -\alpha_4 \\ \alpha_4 & 0\end{matrix} \right)NEWLINE\]NEWLINE then the system (1) becomes NEWLINE\[NEWLINE\varepsilon\left( {1\over\pi} \Lambda_1\int^1_{-1} {1\over(x-\xi)^2} u(\xi)d\xi- \Lambda_2 {d\over dx}u(x)+ T(x)\right) +f\bigl(u(x), x\bigr) =0, \quad -1<x<1. \tag{2}NEWLINE\]NEWLINE Systems (1) or (2) represent an equilibrium of total forces acting on the surfaces of a bridged interface crack. The convolution term in (1) corresponds to the nondimensional crack resistence with the matrix \(\Lambda\) determined by the elastic parameters of the bimaterial. The parameter \(\varepsilon\) is the inverse of the nondimensional crack length \(\ell={K_0L \over \alpha_0}\) where \(K_0\) is the strength of the bridging force and \(L\) is the physical crack length.NEWLINENEWLINENEWLINEThe authors generalize the singular perturbation method to solve the problem (1) or (2). Explicit expressions for the asymptotic solutions up to the \(\varepsilon\) order are presented for two reasons. First, for the general case of a bridged interface crack problem in an anisotropic bimaterial where the bimaterial constant matrices \(\Lambda_1\) and \(\Lambda_2\) are unrestricted. Second, for a special case where in the matrix \(\Lambda_1\) the component \(\alpha_3=0\) which is the case of an isotropic bimaterial.