Mathematical modeling of the deformation of composite plane with interface crack for John’s harmonic material

The exact analytical solution of a nonlinear plane-strain problem for a bimaterial plane with an interface crack (cut) has been obtained. The plane is formed by joining of two half-planes made from different materials. Mechanical properties of half-planes are described with the model of a John's harmonic material. The application of this model has permitted using the methods of the complex functions in the nonlinear boundary value problems. As particular case, the problem is solved for the plane with a free interface crack at given constant nominal stresses at infinity. The expressions are obtained for nominal (Piola) stresses, Cauchy stresses and displacements. From the general solutions the asymptotic expansions of these functions have been constructed in vicinities of crack tips. In nonlinear problems of uniaxial extension of a plane with a free crack it is established that the formulas which give the crack disclosing and the stress intensity factors near the crack tips coincide completely with similar formulas derived from the equations of linear elasticity. The nominal stresses have root singularity at the tips of a crack; the Cauchy stresses have no singularity. Refs 16. Figs 3. Table 1.