The task of this work is to study the scattering of SH waves by homogeneous lining structures in unbounded inhomogeneous medium. The shear modulus is assumed to be a function of coordinates (x,y). A two-dimensional scattering model is established. Based on the complex function theory, the expressions of wave field in the lining medium are derived. Meanwhile, a displacement auxiliary function and a pair of mapping functions are introduced to obtain the wave field expressions in inhomogeneous media. The unknown coefficients in the expression can be solved by boundary conditions on the lining structure. The stress concentration phenomenon on the lining structure is discussed in numerical examples. The distributions of dynamic stress concentration factor on the inner and outer boundary are analyzed under different influencing factors. Finally, it is found that the inhomogeneous parameters and reference wave numbers of the medium have an obvious influence on the distribution of dynamic stress concentration factor.