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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 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} } .
Brillouin spectroscopy is an empirical spectroscopy technique which allows the determination of elastic moduli of materials. The technique uses inelastic scattering of light when it encounters acoustic phonons in a crystal, a process known as Brillouin scattering, to determine phonon energies and therefore interatomic potentials of a material. [1]
+, where is a fourth-order tensor of second-order elastic moduli, while is a sixth-order tensor of third-order elastic moduli. The symmetry of = together with the scalar strain energy density function implies that the second order moduli have the following symmetry: = =, which reduce the number of independent elastic constants from 81 to 36.
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.
Orthotropic materials have three planes/axes of symmetry. An isotropic material, in contrast, has the same properties in every direction. It can be proved that a material having two planes of symmetry must have a third one. Isotropic materials have an infinite number of planes of symmetry.
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.
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 .