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Linear elasticity is a mathematical model as to how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.
Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886. [1] In the continuum description of a solid body we imagine the body to be composed of a set of infinitesimal volumes or material points.
Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) [1] is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral ...
Linear elasticity is used widely in the design and analysis of structures such as beams, plates and shells, and sandwich composites. This theory is also the basis of much of fracture mechanics . Hyperelasticity is primarily used to determine the response of elastomer -based objects such as gaskets and of biological materials such as soft ...
In the linear theory of elasticity Clapeyron's theorem states that the potential energy of deformation of a body, which is in equilibrium under a given load, is equal to half the work done by the external forces computed assuming these forces had remained constant from the initial state to the final state.
In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces (or body forces that could be expressed as potentials) on the boundary are (using index notation) the equilibrium equation:
The elasticity tensor is a fourth-rank tensor describing the stress-strain relation in a linear elastic material. [ 1 ] [ 2 ] Other names are elastic modulus tensor and stiffness tensor . Common symbols include C {\displaystyle \mathbf {C} } and Y {\displaystyle \mathbf {Y} } .
A starting point for solving contact problems is to understand the effect of a "point-load" applied to an isotropic, homogeneous, and linear elastic half-plane, shown in the figure to the right. The problem may be either plane stress or plane strain. This is a boundary value problem of linear elasticity subject to the traction boundary conditions: