Absorbing Boundary Conditions for Elastic Waves in Functionally Graded Materials

Minimizing artificial reflections in wave propagation simulations from the edges of a finite domain is a persistent challenge. Absorbing boundary conditions (ABCs) are an effective solution to this problem. However, these conditions remain to be developed for Functionally Graded Materials (FGMs). Our work examines the propagation of longitudinal elastic waves in FGMs, focusing on boundary reflections. The classical D’Alembert solution for homogeneous materials results in forward and backward waves (distortionless displacement fields), often known as Riemann Invariants (RIs). In FGMs, however, these waves exhibit distortion, resulting in Riemann Variables (RVs) that are space- and time-dependent.

In our work, we derive closed-form solutions for the displacement field in a functionally graded bar subjected to a very high constant strain rate of the order of 10⁵/s to10⁷/s. We recast the second-order variable-coefficient governing equation into two first-order PDEs and determine the associated RVs. Further, we analyze the conditions under which the wave reflections can be controlled effectively by formulating the ABCs in terms of the derived RVs.