Mathematical modeling of the deformation of composite plane with interface crack for semi-linear material

The exact analytical solutions have been obtained for the nonlinear problems (plane-strain and plane-stress) for the bi-material plane with an interface crack. The plane is formed by joining of two half-planes made from different materials. Mechanical properties of halfplanes are described with the model of semi-linear material. The application of this model has allowed using the methods of the complex functions in the nonlinear boundary value problems. For this particular case the problem is solved for the plane with a free interface crack at given constant nominal (Piola) stresses at infinity. The expressions for nominal stresses, Cauchy stresses and displacements are obtained. From the general solutions the asymptotic expansions of these functions have been constructed in vicinities of crack tips. It is established that in the nonlinear problem of uniaxial extension of a plane with a free crack the formulas which give the crack disclosing differ by a constant factor from the formulas of linear elasticity. The stress intensity factors (SIF) of nonlinear and linear problems coincide. The nominal stresses have the root singularity at the tips of a crack; the Cauchy stresses have no singularity.