Constructions of the Method of Difference Potentials for the Computation of Stressed States of Elastic Compressible Materials

In this chapter we specialize the difference potential and other main constructions of the method of difference potentials to the Lamé system in a bounded domain D: (I)$$ L_{D\overline D } u_{\overline D } = \mu \Delta u_{\overline D } + (\lambda + \mu ){\text{grad div }}u_{\overline D } = f_D , $$ which describes the plane stressed state of isotropic elastic materials. The Lamé constants λ and μ are positive and characterize the elastic properties of the material. The Lamé system is strongly elliptic in the sense of Vishik. The vector function $$ u_{\overline D } \left( {x,y} \right) = \left\{ {u_{\overline D }^{(x)} ,u_{\overline D }^{(y)} } \right\} $$ describes displacement from the equilibrium position (x, y).