Contact acoustic nonlinearity underlies modern nondestructive testing methods that use nonlinear ultrasound to detect cracks, delamination, debondings, welding defects, imperfect gluing, etc. Since the nonlinear response of a sample highly depends on the actual configuration of internal contacts (called cracks for brevity), numerical modeling is most suitable. In this communication, we present a numerical toolbox for modeling vibrations or acoustic wave propagation in solids containing cracks of known geometry. With the help of our numerical tool, the user can calculate the nonlinear time-dependent distributions of stress, strain, displacement, etc., in a sample with known excitation sources and compare the results to actual measurements. The final objective would be to estimate geometric and physical parameters of defects using this comparison.

Description of contact acoustic nonlinearity requires the junction of two classical disciplines: acoustics and contact mechanics. The numerical tool also contains two components: a unit for solving the elasticity equations in the bulk volume and a unit that provides the appropriate boundary conditions imposed at the internal defect boundaries in the material. The solid mechanics unit can be programmed using available finite element software that accepts internal boundaries and user-supplied boundary conditions. We have used COMSOL® whose features provide an interface to an external crack model programmed in MATLAB®. The crack model is created using physics-based theoretical considerations.

The presence of cracks invokes two major mechanisms of nonlinearity: asymmetric reaction of a crack on normal compression/tension, and friction-induced hysteresis activated by shearing action. On the other hand, an internal contact can evolve in one of several regimes, such as contact loss, stick, and sliding. The crack model has to take into account these phenomena to provide the load-displacement relationships for any value of the drive parameters. The basic friction model is Coulomb's friction law written for loads. Obviously, this model does not have the desired properties, since, for instance, in the sliding regime, when the tangential load exceeds the threshold defined by the normal one, the tangential displacement remains undefined. Therefore we use another concept, based on the Coulomb friction law as well, that includes the account for roughness and therefore results in appearance of an additional contact regime of partial slip, when some parts of the contact zone slip and some do not. This situation is successfully dealt with by using the previously developed method of memory diagrams. In this method, the hysteretic load-displacement solution is constructed with the help of an internal system function (memory diagram) that contains all memory information. This displacement-driven solution can be easily extended onto two other contact regimes (contact loss and total sliding) and is finally computed for any combination of normal and tangential displacement values and their time dependencies. Memory diagrams have to be maintained at each discretization point on the crack surface and updated following the applied displacement field.

In this communication, we present an example of wave propagation in a sample of known geometry and quantitatively illustrate its nonlinear behavior (e.g. compute the field of secondary nonlinear sources). Our results may be of interest for researchers working in experimental nondestructive testing, nonlinear acoustics, or dealing with vibrations of loaded contacts, nonlinear metamaterials, imaging techniques including nonlinear vibrometry, thermography, etc.