Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction – Part I: Theoretical background
Abstract
Abstract
Our study aims at the creation of a numerical toolbox that describes wave propagation in samples containing internal contacts (e.g. cracks, delaminations, debondings, imperfect intergranular joints) of
known geometry with postulated contact interaction laws including friction. The code consists of two
entities: the contact model and the solid mechanics module. Part I of the paper concerns the modeling
of internal contacts (called cracks for brevity), while part II is related to the integration of the developed
contact model into a solid mechanics module that allows the description of wave propagation processes.
The contact model is used to produce normal and tangential load-displacement relationships, which in
turn are used by the solid mechanics module as boundary conditions at the internal contacts. Due to friction, the tangential reaction curve is hysteretic and memory-dependent. In addition, it depends on the
normal reaction curve. An essential feature of the proposed contact model is that it takes into account
the roughness of the contact faces. On one hand, accounting for roughness makes the contact model more
complicated since it gives rise to a partial slip regime when some parts on the contact area experience slip
and some do not. On the other hand, as we will show, the concept of contact surfaces covered by asperities receding under load makes it possible to formulate a consistent contact model that provides nonlinear load-displacement relationships for any value of the drive displacements and their histories. This is a
strong advantage, since this way, the displacement-driven model allows for a simple explicit procedure of
data exchange with the solid mechanics module, while more traditional flat-surface contacts driven by
loads generate a complex iterative procedure. More specifically, the proposed contact model is based
on the previously developed method of memory diagrams that allows one to automatically obtain
memory-dependent solutions to frictional contact problems in the particular case of partial slip. Here
we extend the solution onto cases of total sliding and contact loss which is possible while using the
displacement-driven formulation. The method requires the knowledge of the normal contact response
obtained in our case as a result of statistical consideration of roughness of contact faces.