Efficient Simulations of the Nonlinear Wave Modulation Induced by a Closed Crack Using Local Contact Modelling

Nonlinear wave modulation methods have been used for the detection of small defects such as closed cracks or delaminations. These methods rely on the nonlinear interaction of a low frequency pumping wave and a high frequency probing wave, which produces sideband frequencies equal to the sum and difference of the input frequencies. Since two very different frequencies are involved, numerical modelling approaches generally require to run the simulations for a long time in order to achieve a steady state regarding the low frequency excitation. In this work, we propose a numerical model where the low frequency excitation is accounted by a time dependent stress state at the contact interface. This local stress state is the prospective stress in the un-cracked structure due to the pumping wave. This is performed by implementing an additional stress parameter directly in the contact laws used to model the contact dynamics occurring at the crack. The high frequency excitation is generated in the solid by imposing displacements on the boundary of the domain, as done in previous work. This concept is first demonstrated through a 1D Finite Difference model with a contact interface between a semi-infinite solid and a rigid boundary. Secondly, this approach is implemented in a 2D Finite Element model where a closed crack of finite length is considered. The results demonstrate the interest of the method for investigating nonlinear wave modulation, and they provide a benchmark for understanding experimental results that may involve more complicated manifestations of contact acoustic nonlinearity.