On the geometric description of non linear elasticity via an energy approach using barycentric coordinates

Odysseas Kosmas, Pieter Boom, Andrey Jivkov

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Abstract

The deformation of a solid in Euclidean space, caused by the change of boundary conditions, can be fully described via a deformation gradient. For reversible (conservative) solid behaviour, the work done by changing boundary conditions is stored as a potential (elastic) energy, a function of the deformation gradient invariants. Using this, we present a discrete energy formulation that uses maps between nodal positions, which are linked via standard barycentric coordinates. A derivation is provided for domains tessellated by tetrahedrons, where the energy functionals are constrained with prescribed boundary conditions via Lagrange multipliers. Analysis of such domains is performed via energy minimisation, where the constraints and eliminated via premultiplication of the discrete equations by a discrete null-space matrix of the constraint gradients. Numerical examples are provided to verify the accuracy of the proposed technique. The definition of standard barycentric coordinates is restricted to three dimensional convex polygons. We show that for an explicit energy expression, applicable also to non-convex polytopes, general barycentric coordinates can be a fundamental tool. We define the discrete energy via a gradient for general polytopes, which is a natural extension of the definition for discrete domains tessellated by tetrahedra. We finally prove that the resulting expressions can fully define the deformation of solids.
Original languageEnglish
Article number1689
JournalMathematics
Volume9
DOIs
Publication statusPublished - 19 Jul 2021

Keywords

  • nonlinear elasticity
  • general barycentric coordinates
  • energy minimization
  • Lagrange multipliers
  • null-space method

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