Analysis of a cracked harmonic substrate under a rigid punch

The study of the mechanical action between a punch and a cracked substrate has some theoretical guidance for the material protection. So the coupling problem of a cracked semi-infinite harmonic substrate under the action of a rigid flat punch is studied. The mixed boundary value problem is transformed into the Riemann-Hilbert boundary value problem by applying the complex-variable method, and then converted into singular integral equation for a numerical solution. The stress intensity factors at the contact ends and crack tips and the Piola stresses of whole harmonic material can be expressed as complex functions. The results indicate that the stressed state of harmonic solid near the crack tip and contact ends have similar features as those in linear elastic solids. The crack causes an obvious impact on the stress distributions near the contact region. The study provides theoretical guidance for analyzing the damaged problems of some soft materials under small deformation.