Efficient Steady-State Computation for Wear of Multimaterial Composites

Abstract

Traditionally, iterative schemes have been used to predict evolving material profiles under abrasive wear. In this work, more efficient continuous formulations are presented for predicting the wear of tribological systems. Following previous work, the formulation is based on a two parameter elastic Pasternak foundation model. It is considered as a simplified framework to analyze the wear of multimaterial surfaces. It is shown that the evolving wear profile is also the solution of a parabolic partial differential equation (PDE). The wearing profile is proven to converge to a steady-state that propagates with constant wear rate. A relationship between this velocity and the inverse rule of mixtures or harmonic mean for composites is derived. For cases where only the final steady-state profile is of interest, it is shown that the steady-state profile can be accurately and directly determined by solving a simple elliptic differential system—thus avoiding iterative schemes altogether. Stability analysis is performed to identify conditions under which an iterative scheme can provide accurate predictions and several comparisons between iterative and the proposed formulation are made. Prospects of the new continuous wear formulation and steady-state characterization are discussed for advanced optimization, design, manufacturing, and control applications.

Introducing a level-set based shape and topology optimization method for the wear of composite materials with geometric constraints

“…In Rowe et al (2014), Feppon et al (2015), Sidebottom et al (2015), a Pasternak elastic foundation model (Kerr (1964), Pasternak (1954) was assumed to characterize the pressure distribution, p, with a constant average pressure constraint ), < p >= 1 | | pdx = P 0 :…”

“…The time dependent equation describing the evolution of the profile, z, is obtained by combining (1) and (3) (see Feppon et al (2015)), and is written in the weak form Allaire (2007b) as:…”

“…Boundary conditions in addition to (6) are required to ensure the well-posedness of the model. In Rowe et al (2014), Feppon et al (2015), periodic boundary conditions were considered by tribologists because they were both conveniently implemented in iterative schemes and compatible with the study of multimaterial composites on a periodic, unit-cell basis. For these reasons, they will be used for the topology optimization results presented in Section 6.…”

“…In Feppon et al (2015), it has been shown that the wellestablished framework offered by elastic foundation models (Kerr (1964), Pasternak (1954)) allows one to propose a continuous and mathematically well-posed formulation for the abrasive sliding wear of composites or heterogeneous materials. Foundation models simplify the contact mechanics analysis by postulating explicit relationships between surface topography and pressure distributions (Fig.…”

“…The focus will be on linear abrasive sliding wear for which sliding direction is fixed. A generalization of the wear model employed in (Rowe et al (2014), Feppon et al (2015), Sawyer (2004)), that is mathematically suitable for topology optimization, is introduced in Section 2. Governing equations for the corresponding steady-state solution are explained.…”