Uniform in-plane stresses and strains inside an incompressible nonlinear elastic elliptical inhomogeneity (Q6049491)

The authors consider an incompressible and nonlinear elastic elliptical inhomogeneity defined through \(L:\{x_{1}^{2}/a^{2}+x_{2}^{2}/b^{2}=1\}\), with \(a\geq b>0\), perfectly bonded to an infinite linear isotropic elastic matrix submitted to uniform remote in-plane stresses \((\sigma _{11}^{\infty },\sigma _{22}^{\infty },\sigma _{12}^{\infty })\). For the plane strain deformation of the linear isotropic elastic material in the matrix and according to Muskhelishvili's relationships, the authors write the three in-plane stresses \((\sigma _{11},\sigma _{22},\sigma _{12})\), the two in-plane displacements \((u_{1},u_{2})\) and the two stress functions \( (\varphi _{1},\varphi _{2})\) in terms of two analytic functions \(\phi (z)\) and \(\psi (z)\) as: \(\sigma _{11}+\sigma _{22}=2[\phi ^{\prime }(z)+ \overline{\phi ^{\prime }(z)}]\), \(\sigma _{22}-\sigma _{11}+2i\sigma _{12}=2[ \overline{z}\phi ^{\prime \prime }(z)+\psi ^{\prime }(z)]\), \(2\mu _{2}(u_{1}+iu_{2})=\kappa _{2}\phi (z)-z\overline{\phi ^{\prime }(z)}- \overline{\psi (z)}\), \(\varphi _{1}+i\varphi _{2}=i[\phi (z)+z\overline{\phi ^{\prime }(z)}+\overline{\psi (z)}]\), where \(\kappa _{2}=3-4\nu _{2}\), \(\nu _{2}\) being Poisson's ratio and \(\mu _{2}\) the shear modulus of the linear isotropic elastic material. The stresses are related to the two stress functions \(\varphi _{1}\) and \(\varphi _{2}\) through \(\sigma _{11}=-\varphi _{1,2}\), \(\sigma _{12}=\varphi _{1,1}\), \(\sigma _{21}=-\varphi _{2,2}\), \( \sigma _{22}=\varphi _{2,1}\). Inside the inhomogeneity, the effective stress \(\sigma _{e}\) of the nonlinear elastic material is linked to the effective strain \(\varepsilon _{e}\) through the power-law: \(\sigma _{e}=\mu _{1}\varepsilon _{e}^{p-2}\), where \(\mu _{1}>0\) and \(p>1\). The authors introduce the conformal mapping function for the matrix: \(z=\omega (\xi )=R(\xi +m\xi )\), \(\left\vert \xi \right\vert >1\), where \(R=\frac{a+b}{2}\), \( m=\frac{a-b}{a+b}\), \(0\leq m<1\), and they rewrite the above expressions in equations which involve two unknown complex numbers \(A\) and \(B\). They draw computations, also taking into account the boundary conditions, which finally lead to a nonlinear equation whose solution is \(\left\vert B\right\vert \) and which requires numerical tools for its resolution. From the value of \(\left\vert B\right\vert \), the authors derive that of \(A\) and \( B\). They finally prove that the internal in-plane elastic field of stresses and strains inside the incompressible nonlinear elastic elliptical inhomogeneity are uniform, the internal uniform elastic field being related to the remote loading in a nonlinear way.