Anisotropic elastic materials that uncouple all three displacement components, and existence of one-displacement Green's function (Q5925975)

An attempt is made to obtain a class of two-dimensional displacement of three-dimensional anisotropic elastostatics of the form \({\mathbf u}= [u_1, u_2, u_3]\), where \(u_1= u_1(x_1, x_2)\) and \(u_2= u_2(x_1, x_2)\) correspond to a plane-strain elastic state in \((x_1, x_2)\) plane, while \(u_3= u_3(x_1, x_2)\) corresponds to an antiplane elastic state in \((x_1,x_2)\) plane. By seeking \({\mathbf u}\) in the form \({\mathbf u}={\mathbf a}F(z)\), where \(z= x_1+ px_2\), \(F= F(z)\) is a prescribed function, and \({\mathbf a}\) and \(p\) stand for an eigenvector and eigenvalue, respectively, the problem is reduced to the study of the following eigenproblem: find a pair \(({\mathbf a},p)\) that satisfies the homogeneous algebraic equation \([{\mathbf Q}+ p({\mathbf R}+{\mathbf R}^T)+ p^2{\mathbf T}]{\mathbf a}= 0\), where \(3\times 3\) matrices \({\mathbf Q}\), \({\mathbf R}\), and \({\mathbf T}\) in components are defined by \((i,j,k,l= 1,2,3): Q_{ik}= C_{i1k1}\), \(R_{ik}= C_{i1k2}\), \(T_{ik}= C_{i2k2}\), and \(C_{ijkl}\) is the elasticity tensor. The author shows that, if suitable restrictions are imposed on matrices \({\mathbf Q}\), \({\mathbf R}\), and \({\mathbf T}\), the elastic state corresponding to \((u_1,u_2)\) can be separated from the elastic state corresponding to \(u_3\) as in the case of three-dimensional homogeneous isotropic elastostatics. Also, it is shown that for such a restrictive anisotropic material it is possible to obtain closed-form displacement fields corresponding to a concentrated line force (or a line dislocation) parallel to the \(x_3\) axis in an infinite space, in a semi-space with clamped boundary, and in an infinite space with rigid elliptic inclusion.