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Concept
Linear elasticity
Summary
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.
The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations (or strains) and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding.
These assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis.
Equations governing a linear elastic boundary value problem are based on three tensor partial differential equations for the balance of linear momentum and six infinitesimal strain-displacement relations. The system of differential equations is completed by a set of linear algebraic constitutive relations.
In direct tensor form that is independent of the choice of coordinate system, these governing equations are:
Equation of motion, which is an expression of Newton's second law:
Strain-displacement equations:
Constitutive equations. For elastic materials, Hooke's law represents the material behavior and relates the unknown stresses and strains. The general equation for Hooke's law is
where is the Cauchy stress tensor, is the infinitesimal strain tensor, is the displacement vector, is the fourth-order stiffness tensor, is the body force per unit volume, is the mass density, represents the nabla operator, represents a transpose, represents the second derivative with respect to time, and is the inner product of two second-order tensors (summation over repeated indices is implied).
Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are:
Equation of motion: where the subscript is a shorthand for and indicates , is the Cauchy stress tensor, is the body force density, is the mass density, and is the displacement.
Official source
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