Anisotropic elasticity In
linear elasticity, the relation between
stress and
strain depend on the type of material under consideration. This relation is known as
Hooke's law. For anisotropic materials Hooke's law can be written as :\boldsymbol{\sigma} = \mathsf{c}\cdot\boldsymbol{\varepsilon} where \boldsymbol{\sigma} is the stress
tensor, \boldsymbol{\varepsilon} is the strain tensor, and \mathsf{c} is the elastic
stiffness tensor. If the tensors in the above expression are described in terms of components with respect to an
orthonormal coordinate system we can write :\sigma_{ij} = c_{ijk\ell}~ \varepsilon_{k\ell} where summation has been assumed over repeated indices. Since the stress and strain tensors are
symmetric, and since the stress-strain relation in linear elasticity can be derived from a
strain energy density function, the following symmetries hold for linear elastic materials :c_{ijk\ell} = c_{jik\ell} ~,~~c_{ijk\ell} = c_{ij\ell k} ~,~~ c_{ijk\ell} = c_{k\ell ij} ~. Because of the above symmetries, the stress-strain relation for linear elastic materials can be expressed in matrix form as : \begin{bmatrix}\sigma_{11}\\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{bmatrix} = \begin{bmatrix} c_{1111} & c_{1122} & c_{1133} & c_{1123} & c_{1131} & c_{1112} \\ c_{2211} & c_{2222} & c_{2233} & c_{2223} & c_{2231} & c_{2212} \\ c_{3311} & c_{3322} & c_{3333} & c_{3323} & c_{3331} & c_{3312} \\ c_{2311} & c_{2322} & c_{2333} & c_{2323} & c_{2331} & c_{2312} \\ c_{3111} & c_{3122} & c_{3133} & c_{3123} & c_{3131} & c_{3112} \\ c_{1211} & c_{1222} & c_{1233} & c_{1223} & c_{1231} & c_{1212} \end{bmatrix} \begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{31} \\ 2\varepsilon_{12} \end{bmatrix} An alternative representation in
Voigt notation is : \begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end{bmatrix} = \begin{bmatrix} C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\ C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\ C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\ C_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} \end{bmatrix} \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \varepsilon_4 \\ \varepsilon_5 \\ \varepsilon_6 \end{bmatrix} or : \underline{\underline{\boldsymbol{\sigma}}} = \underline{\underline{\mathsf{C}}}~\underline{\underline{\boldsymbol{\varepsilon}}} The
stiffness matrix \underline{\underline{\mathsf{C}}} in the above relation satisfies
point symmetry.
Condition for material symmetry The stiffness matrix \underline{\underline{\mathsf{C}}} satisfies a given symmetry condition if it does not change when subjected to the corresponding
orthogonal transformation. The orthogonal transformation may represent symmetry with respect to a
point, an
axis, or a
plane. Orthogonal transformations in linear elasticity include rotations and reflections, but not shape changing transformations and can be represented, in orthonormal coordinates, by a 3\times 3 matrix \underline{\underline{\mathbf{A}}} given by : \underline{\underline{\mathbf{A}}} = \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix}~. In Voigt notation, the transformation matrix for the
stress tensor can be expressed as a 6\times6 matrix \underline{\underline{\mathsf{A}_\sigma}} given by : \underline{\underline{\mathsf{S}}} = \begin{bmatrix} \tfrac{1}{E_{\rm 1}} & - \tfrac{\nu_{\rm 21}}{E_{\rm 2}} & - \tfrac{\nu_{\rm 31}}{E_{\rm 3}} & 0 & 0 & 0 \\ -\tfrac{\nu_{\rm 12}}{E_{\rm 1}} & \tfrac{1}{E_{\rm 2}} & - \tfrac{\nu_{\rm 32}}{E_{\rm 3}} & 0 & 0 & 0 \\ -\tfrac{\nu_{\rm 13}}{E_{\rm 1}} & - \tfrac{\nu_{\rm 23}}{E_{\rm 2}} & \tfrac{1}{E_{\rm 3}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \tfrac{1}{G_{\rm 23}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm 31}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm 12}} \\ \end{bmatrix} where {E}_{\rm i}\, is the
Young's modulus along axis i, G_{\rm ij}\, is the
shear modulus in direction j on the plane whose normal is in direction i, and \nu_{\rm ij}\, is the
Poisson's ratio that corresponds to a contraction in direction j when an extension is applied in direction i. Only nine (9) from these twelve (12) elastic constants are independent.
Bounds on the moduli of orthotropic elastic materials The strain-stress relation for orthotropic linear elastic materials can be written in Voigt notation as : \underline{\underline{\boldsymbol{\varepsilon}}} = \underline{\underline{\mathsf{S}}}~\underline{\underline{\boldsymbol{\sigma}}} where the compliance matrix \underline{\underline{\mathsf{S}}} is given by : \underline{\underline{\mathsf{S}}} = \begin{bmatrix} S_{11} & S_{12} & S_{13} & 0 & 0 & 0 \\ S_{12} & S_{22} & S_{23} & 0 & 0 & 0 \\ S_{13} & S_{23} & S_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & S_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & S_{55} & 0\\ 0 & 0 & 0 & 0 & 0 & S_{66} \end{bmatrix} The compliance matrix is
symmetric and must be
positive definite for the
strain energy density to be positive. This implies from
Sylvester's criterion that all the principal
minors of the matrix are positive, i.e., : \Delta_k := \det(\underline{\underline{\mathsf{S}_k}}) > 0 where \underline{\underline{\mathsf{S}_k}} is the k\times k principal
submatrix of \underline{\underline{\mathsf{S}}}. Then, : \begin{align} \Delta_1 > 0 & \implies \quad S_{11} > 0 \\ \Delta_2 > 0 & \implies \quad S_{11}S_{22} - S_{12}^2 > 0 \\ \Delta_3 > 0 & \implies \quad (S_{11}S_{22}-S_{12}^2)S_{33}-S_{11}S_{23}^2+2S_{12}S_{23}S_{13}-S_{22}S_{13}^2 >0 \\ \Delta_4 > 0 & \implies \quad S_{44}\Delta_3 > 0 \implies S_{44} > 0\\ \Delta_5 > 0 & \implies \quad S_{44}S_{55}\Delta_3 > 0 \implies S_{55} > 0 \\ \Delta_6 > 0 & \implies \quad S_{44}S_{55}S_{66}\Delta_3 > 0 \implies S_{66} > 0 \end{align} We can show that this set of conditions implies that : S_{11} > 0 ~,~~ S_{22} > 0 ~,~~ S_{33} > 0 ~,~~ S_{44} > 0 ~,~~ S_{55} > 0 ~,~~ S_{66} > 0 or : E_1 > 0 , E_2 > 0, E_3 > 0, G_{12} > 0 , G_{23} > 0, G_{13} > 0 However, no similar lower bounds can be placed on the values of the Poisson's ratios \nu_{ij}. ==See also==