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On geometric discretization of elasticity

### Abstract

This paper presents a geometric discretization of elasticity when the ambient space is Euclidean. This theory is built on ideas from algebraic topology, exterior calculus, and the recent developments of discrete exterior calculus. We first review some geometric ideas in continuum mechanics and show how constitutive equations of linearized elasticity, similar to those of electromagnetism, can be written in terms of a material Hodge star operator. In the discrete theory presented in this paper, instead of referring to continuum quantities, we postulate the existence of some discrete scalar-valued and vector-valued primal and dual differential forms on a discretized solid, which is assumed to be a triangulated domain. We find the discrete governing equations by requiring energy balance invariance under time-dependent rigid translations and rotations of the ambient space. There are several subtle differences between the discrete and continuous theories. For example, power of tractions in the discrete theory is written on a layer of cells with a nonzero volume. We obtain the compatibility equations of this discrete theory using tools from algebraic topology. We study a discrete Cosserat medium and obtain its governing equations. Finally, we study the geometric structure of linearized elasticity and write its governing equations in a matrix form. We show that, in addition to constitutive equations, balance of angular momentum is also metric dependent; all the other governing equations are topological.

© 2008 American Institute of Physics

Received 24 June 2007
Accepted 12 December 2007
Published online 05 February 2008

Acknowledgments:
The author benefitted from discussions with M. Arroyo, A. N. Hirani, J. E. Marsden, and A. Ozakin.

Article outline:

I. INTRODUCTION AND MOTIVATION
II. ALGEBRAIC TOPOLOGY, EXTERIOR CALCULUS, BUNDLE-VALUED DIFFERENTIAL FORMS, AND DISCRETE EXTERIOR CALCULUS
A. Discrete exterior calculus (DEC)
B. Discrete bundle-valued differential forms
III. GEOMETRIC CONTINUUM MECHANICS
A. Geometry of Maxwell’sequations
B. Nonlinear elasticity
C. Geometric character of stress
D. Linear elasticity as a geometric linearization of nonlinear elasticity
1. Material Hodge star in linear elasticity
IV. A DISCRETE THEORY OF ELASTICITY
A. Energy balance for a discretized solid
B. Invariance of energy balance
C. Discrete compatibility equations
D. Number of equations and unknowns
E. A discrete Cosserat elasticity
F. A geometric formulation of linear elasticity
V. CONCLUSIONS

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2008-02-05

2016-02-06

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