Accretion–ablation mechanics

Abstract

In this paper, we formulate a geometric nonlinear theory of the mechanics of accreting–ablating bodies. This is a generalization of the theory of accretion mechanics of Sozio & Yavari (Sozio & Yavari 2019 J. Nonlinear Sci. 29 , 1813–1863 (doi:10.1007/s00332-019-09531-w)). More specifically, we are interested in large deformation analysis of bodies that undergo a continuous and simultaneous accretion and ablation on their boundaries while under external loads. In this formulation, the natural configuration of an accreting–ablating body is a time-dependent Riemannian 3 -manifold with a metric that is an unknown a priori and is determined after solving the accretion–ablation initial-boundary-value problem. In addition to the time of attachment map, we introduce a time of detachment map that along with the time of attachment map, and the accretion and ablation velocities, describes the time-dependent reference configuration of the body. The kinematics, material manifold, material metric, constitutive equations and the balance laws are discussed in detail. As a concrete example and application of the geometric theory, we analyse a thick hollow circular cylinder made of an arbitrary incompressible isotropic material that is under a finite time-dependent extension while undergoing continuous ablation on its inner cylinder boundary and accretion on its outer cylinder boundary. The state of deformation and stress during the accretion–ablation process, and the residual stretch and stress after the completion of the accretion–ablation process, are computed. This article is part of the theme issue ‘Foundational issues, analysis and geometry in continuum mechanics’.