Elasticity tensor

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 ${\displaystyle \mathbf {C} }$ and ${\displaystyle \mathbf {Y} }$.

The defining equation can be written as

${\displaystyle T^{ij}=C^{ijkl}E_{kl}}$

where ${\displaystyle T^{ij}}$ and ${\displaystyle E_{kl}}$ are the components of the Cauchy stress tensor and infinitesimal strain tensor, and ${\displaystyle C^{ijkl}}$ are the components of the elasticity tensor. Summation over repeated indices is implied.^{[note 1]} This relationship can be interpreted as a generalization of Hooke's law to a 3D continuum.

A general fourth-rank tensor ${\displaystyle \mathbf {F} }$ in 3D has 3⁴ = 81 independent components ${\displaystyle F_{ijkl}}$, but the elasticity tensor has at most 21 independent components.^[3] 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.^[3]

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