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 C {\displaystyle \mathbf {C} } and Y {\displaystyle \mathbf {Y} } . The defining equation can be written as
T i j = C i j k l E k l {\displaystyle T^{ij}=C^{ijkl}E_{kl}}
where T i j {\displaystyle T^{ij}} and E k l {\displaystyle E_{kl}} are the components of the Cauchy stress tensor and infinitesimal strain tensor, and C i j k l {\displaystyle C^{ijkl}} 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 F {\displaystyle \mathbf {F} } in 3D has 34 = 81 independent components F i j k l {\displaystyle F_{ijkl}} , 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.
Definition The most general linear relation between two second-rank tensors T , E {\displaystyle \mathbf {T} ,\mathbf {E} } is
T i j = C i j k l E k l {\displaystyle T^{ij}=C^{ijkl}E_{kl}}
where C i j k l {\displaystyle C^{ijkl}} are the components of a fourth-rank tensor C {\displaystyle \mathbf {C} } . The elasticity tensor is defined as C {\displaystyle \mathbf {C} } for the case where T {\displaystyle \mathbf {T} } and E {\displaystyle \mathbf {E} } are the stress and strain tensors, respectively. The compliance tensor K {\displaystyle \mathbf {K} } is defined from the inverse stress-strain relation:
E i j = K i j k l T k l {\displaystyle E^{ij}=K^{ijkl}T_{kl}}
The two are related by
K i j p q C p q k l = 1 2 ( δ i k δ j l + δ i l δ j k ) {\displaystyle K_{ijpq}C^{pqkl}={\frac {1}{2}}\left(\delta _{i}^{k}\delta _{j}^{l}+\delta _{i}^{l}\delta _{j}^{k}\right)}
where δ n m {\displaystyle \delta _{n}^{m}} is the Kronecker delta. Unless otherwise noted, this article assumes C {\displaystyle \mathbf {C} } is defined from the stress-strain relation of a linear elastic material, in the limit of small strain.
Special cases
Isotropic For an isotropic material, C {\displaystyle \mathbf {C} } simplifies to
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