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Stress Functions
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: :\sigma_=0\, where \sigma is the stress tensor, and the Beltrami-Michell compatibility equations: :\sigma_+\frac\sigma_=0 A general solution of these equations may be expressed in terms of the Beltrami stress tensor. Stress functions are derived as special cases of this Beltrami stress tensor which, although less general, sometimes will yield a more tractable method of solution for the elastic equations. Beltrami stress functions It can be shown that a complete solution to the equilibrium equations may be written as :\sigma=\nabla \times \Phi \times \nabla Using index notation: :\sigma_=\varepsilon_\varepsilon_\Phi_ : where \Phi_ is an arbitrary second-rank tensor field that is at least twice differentiable, and is known as the ''Beltram ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Elasticity
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. Mathematical formulation 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
