Solid Mechanics
Solid mechanics is arguably one of the most important areas of application for finite elements. Indeed, finite element analysis is used together with computer aided design (CAD) to optimize and speed up the design and manufacturing process of practically
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Solid Mechanics
Abstract Solid mechanics is arguably one of the most important areas of application for finite elements. Indeed, finite element analysis is used together with computer aided design (CAD) to optimize and speed up the design and manufacturing process of practically all mechanical structures, ranging from bearings to airplanes. In this chapter we derive the equations of linear elasticity and formulate finite element approximations of them. We do this in the abstract setting of elliptic partial differential equations introduced before and prove existence and uniqueness of the solution using the Lax-Milgram lemma. A priori and a posteriori error estimates are also proved. Some effort is laid on explaining the implementation of the finite element method. We also touch upon thermal stress – and modal analysis.
11.1 Governing Equations 11.1.1 Cauchy’s Equilibrium Equation Consider a volume ˝ occupied by an elastic material and let ! denote an arbitrary subdomain of ˝ with boundary @! and exterior normal n. Two types of forces can act on !. First, there are forces acting on the whole volume, so called body forces. These are described by a force density f , which expresses force per unit volume. The most common body force is gravity with f D Œ0; 0; 9:82T . Second, there are forces acting on the boundary @!. These are assumed to have the form n, where isP the so called stress tensor n denotes the vector with components . n/i D 3j D1 ij nj . The stress tensor is a second order tensor with components ij , i; j D 1; 2; 3, where ij expresses the force per unit area in direction xi on a surface with unit normal in direction xj . It follows from conservation of angular momentum that the stress tensor is symmetric with six independent components.
M.G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Texts in Computational Science and Engineering 10, DOI 10.1007/978-3-642-33287-6__11, © Springer-Verlag Berlin Heidelberg 2013
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11 Solid Mechanics
Summing body forces and contact forces we obtain the total net force F on ˝ Z
Z f dx C
F D !
nds
(11.1)
@!
Using the divergence theorem on the surface integral we obtain Z .f C r / dx
F D
(11.2)
!
In equilibrium F D 0, and since ! is arbitrary, we conclude that f Cr D0
(11.3)
which is Cauchy’s equilibrium equation. It is a system of equations, given in component form by 11 12 13 C C D0 @x1 @x2 @x3 21 22 23 f2 C C C D0 @x1 @x2 @x3 31 32 33 C C D0 f3 C @x1 @x2 @x3 f1 C
(11.4a) (11.4b) (11.4c)
As there are six independent stress components, the above system with three equations needs to be closed by a material specific so-called constitutive equation, which relates the stress in the material to its deformation.
11.1.2 Constitutive Equations and Hooke’s Law The displacement of a material particle is defined as the vector u D x x0 , where x is the current and x0 the initial position of the particle. Under assumption of small displacement gradients the measure of deformation
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