INTRODUCTION TO LINEAR ELASTICITY AND ITS FINITE ELEMENT APPROXIMATION

INTRODUCTION TO LINEAR ELASTICITY AND ITS FINITE ELEMENT APPROXIMATION LONG CHEN CONTENTS 1. Introduction 1 2. Physical Interpretation 4 3. Tensors 7 3.1. Forces 7 3.2. Stress Tensor 8 3.3. Strain Tensor 8 3.4. Relation between Trace, Strain and Stress 8 4. Variational Forms 9 4.1. Displacement Formulation 9 4.2. Mixed Formulation of Hellinger and Reissner 10 4.3. Mixed formulation of Hu and Washizu 12 5. Finite Element Methods 12 5.1. Locking and Nearly Incompressible Material 13 5.2. Symmetric stress and discontinuous displacement 13 References 13 The theory of elasticity is to study the deformation of elastic solid bodies under external load. A body is elastic if, when the external forces are removed, the bodies return to their original (undeformed) shape. 1. I NTRODUCTION How to describe the deformation of a solid body? Let us denoted a solid body by a bounded domain Ω ⊂ R 3 . Then the deformed domain can be described as the image of a vector-valued function Φ:Ω → R 3 , i.e. Φ(Ω), which is called a conﬁguration or a placement of a body. For most problems of interest, we can require that the map be 1-1 and differentiable. Since we are interested in the deformation, i.e., the change of the domain, we let Φ(x)= x + u(x) or equivalently u(x) = Φ(x) - x and call u the displacement. Here we deal with continuum mechanics, i.e., systems have properties deﬁned at all points in space, ignoring details in atom and molecules level. The displacement is not the deformation. For example, a translation or a rotation, so- called rigid body motions, of Ω will lead to a non-trivial displacement u 6=0, but the shape and volume of Ω does not change at all. How to describe the shape mathematically? Vectors. For example, a cube can be described by 3 orthogonal vectors. The change of the Date: Created Mar 19, 2011, Updated September 16, 2016. 1

INTRODUCTION TO LINEAR ELASTICITY AND ITS FINITE ELEMENT APPROXIMATION

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