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Two other elastic moduli are Lamé's first parameter, λ, and P-wave modulus, M, as used in table of modulus comparisons given below references. Homogeneous and isotropic (similar in all directions) materials (solids) have their (linear) elastic properties fully described by two elastic moduli, and one may choose any pair.
The most general linear relation between two second-rank tensors , is = where are the components of a fourth-rank tensor . [1] [note 1] The elasticity tensor is defined as for the case where and are the stress and strain tensors, respectively.
The two parameters together constitute a parameterization of the elastic moduli for homogeneous isotropic media, popular in mathematical literature, and are thus related to the other elastic moduli; for instance, the bulk modulus can be expressed as K = λ + 2 / 3 μ. Relations for other moduli are found in the (λ, G) row of the ...
Elastic properties describe the reversible deformation (elastic response) of a material to an applied stress. They are a subset of the material properties that provide a quantitative description of the characteristics of a material, like its strength .
Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are: [1]. 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.
Conversion formulae Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these; thus, given any two, any other of the elastic moduli can be calculated according to these formulas, provided both for 3D materials (first part of the table) and for 2D materials (second part).
The curve () describes the deflection of the beam in the direction at some position (recall that the beam is modeled as a one-dimensional object). is a distributed load, in other words a force per unit length (analogous to pressure being a force per area); it may be a function of , , or other variables.
The acoustoelastic effect is an effect of finite deformation of non-linear elastic materials. A modern comprehensive account of this can be found in. [1] This book treats the application of the non-linear elasticity theory and the analysis of the mechanical properties of solid materials capable of large elastic deformations.