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Concept
Elasticity tensor
Summary
The elasticity tensor is a fourth-rank tensor describing the stress-strain relation in
a linear elastic material. Other names are elastic modulus tensor and stiffness tensor. Common symbols include and .
The defining equation can be written as
where and are the components of the Cauchy stress tensor and infinitesimal strain tensor, and are the components of the elasticity tensor. Summation over repeated indices is implied. This relationship can be interpreted as a generalization of Hooke's law to a 3D continuum.
A general fourth-rank tensor in 3D has 34 = 81 independent components , but the elasticity tensor has at most 21 independent components. This fact follows from the symmetry of the stress and strain tensors, together with the requirement that the stress derives from an elastic energy potential. For isotropic materials, the elasticity tensor has just two independent components, which can be chosen to be the bulk modulus and shear modulus.
The most general linear relation between two second-rank tensors is
where are the components of a fourth-rank tensor . The elasticity tensor is defined as for the case where and are the stress and strain tensors, respectively.
The compliance tensor is defined from the inverse stress-strain relation:
The two are related by
where is the Kronecker delta.
Unless otherwise noted, this article assumes is defined from the stress-strain relation of a linear elastic material, in the limit of small strain.
For an isotropic material, simplifies to
where and are scalar functions of the material coordinates
and is the metric tensor in the reference frame of the material. In an orthonormal Cartesian coordinate basis, there is no distinction between upper and lower indices, and the metric tensor can be replaced with the Kronecker delta:
Substituting the first equation into the stress-strain relation and summing over repeated indices gives
where is the trace of .
In this form, and can be identified with the first and second Lamé parameters.
Official source
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