A 2-D biharmonic equation for the displacement of an isotropic Kelvin–Voigt viscoelastic plate

Abstract

Cancellous bone can be regarded as a lattice of asymptotically small rods and plates. In this article, a thin plate of an isotropic Kelvin–Voigt viscoelastic material with a thickness ϵ ≪ 1 is considered. This thickness leads to asymptotic expansions of the displacement, stress and strain tensors, and of the change in the volumetric fraction of solid bone in the bone matrix. A rate equation describing the change in the volumetric fraction with respect to time governs the remodeling process of bone deposition and reabsorption. It has linear and quadratic terms of strain tensors and thus, its own asymptotic expansion. This expansion's leading term is used in some simple numerical simulations.

Keywords:

• isotropic Kelvin–Voigt viscoelasticity
• bone remodeling
• cancellous bone

AMS Subject Classifications:

• 74D10
• 74K20