Isotropic materials Isotropic materials are characterized by properties which are independent of direction in space. Physical equations involving isotropic materials must therefore be independent of the coordinate system chosen to represent them. The strain tensor is a symmetric tensor. Since the
trace of any tensor is independent of any coordinate system, the most complete coordinate-free decomposition of a symmetric tensor is to represent it as the sum of a constant tensor and a traceless symmetric tensor. Thus in
index notation: \varepsilon_{ij} = \left(\tfrac13\varepsilon_{kk}\delta_{ij}\right) + \left(\varepsilon_{ij}-\tfrac13\varepsilon_{kk}\delta_{ij}\right) where is the
Kronecker delta. In direct tensor notation: \boldsymbol{\varepsilon} = \operatorname{vol}(\boldsymbol{\varepsilon}) + \operatorname{dev}(\boldsymbol{\varepsilon}) \,; \qquad \operatorname{vol}(\boldsymbol{\varepsilon}) = \tfrac13\operatorname{tr}(\boldsymbol{\varepsilon})~\mathbf{I} \,; \qquad \operatorname{dev}(\boldsymbol{\varepsilon}) = \boldsymbol{\varepsilon} - \operatorname{vol}(\boldsymbol{\varepsilon}) where is the second-order identity tensor. The first term on the right is the constant tensor, also known as the
volumetric strain tensor, and the second term is the traceless symmetric tensor, also known as the
deviatoric strain tensor or shear tensor. The most general form of Hooke's law for isotropic materials may now be written as a linear combination of these two tensors: \sigma_{ij}=3K\left(\tfrac{1}{3}\varepsilon_{kk}\delta_{ij}\right) +2G\left(\varepsilon_{ij}-\tfrac{1}{3}\varepsilon_{kk}\delta_{ij}\right)\,; \qquad \boldsymbol{\sigma} = 3K\operatorname{vol}(\boldsymbol{\varepsilon}) + 2G\operatorname{dev}(\boldsymbol{\varepsilon}) where is the
bulk modulus and is the
shear modulus. Using the relationships between the
elastic moduli, these equations may also be expressed in various other ways. A common form of Hooke's law for isotropic materials, expressed in direct tensor notation, is \boldsymbol{\sigma} = \lambda\operatorname{tr}(\boldsymbol{\varepsilon})\mathbf{I} + 2\mu\boldsymbol{\varepsilon} = \mathsf{c}:\boldsymbol{\varepsilon} \,; \qquad \mathsf{c} = \lambda\mathbf{I}\otimes\mathbf{I} + 2\mu\mathsf{I} where and are the
Lamé constants, is the second-rank identity tensor, and
I is the symmetric part of the fourth-rank identity tensor. In index notation: \sigma_{ij} = \lambda\varepsilon_{kk}~\delta_{ij} + 2\mu\varepsilon_{ij} = c_{ijkl}\varepsilon_{kl} \,;\qquad c_{ijkl} = \lambda\delta_{ij}\delta_{kl} + \mu\left(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}\right) The inverse relationship is \boldsymbol{\varepsilon} = \frac{1}{2\mu}\boldsymbol{\sigma} - \frac{\lambda}{2\mu(3\lambda+2\mu)}\operatorname{tr}(\boldsymbol{\sigma})\mathbf{I} = \frac{1}{2G} \boldsymbol{\sigma} + \left(\frac{1}{9K} - \frac{1}{6G}\right)\operatorname{tr}(\boldsymbol{\sigma})\mathbf{I} Therefore, the compliance tensor in the relation is \mathsf{s} = - \frac{\lambda}{2\mu(3\lambda+2\mu)}\mathbf{I}\otimes\mathbf{I} + \frac{1}{2\mu}\mathsf{I} = \left(\frac{1}{9K} - \frac{1}{6G}\right)\mathbf{I}\otimes\mathbf{I} + \frac{1}{2G}\mathsf{I} In terms of
Young's modulus and
Poisson's ratio, Hooke's law for isotropic materials can then be expressed as \varepsilon_{ij}=\frac{1}{E}\big(\sigma_{ij}-\nu(\sigma_{kk}\delta_{ij}-\sigma_{ij})\big) \,; \qquad \boldsymbol{\varepsilon} = \frac{1}{E} \big(\boldsymbol{\sigma} - \nu(\operatorname{tr}(\boldsymbol{\sigma})\mathbf{I} - \boldsymbol{\sigma})\big) = \frac{1+\nu}{E}\boldsymbol{\sigma} - \frac{\nu}{E}\operatorname{tr}(\boldsymbol{\sigma})\mathbf{I} This is the form in which the strain is expressed in terms of the stress tensor in engineering. The expression in expanded form is \begin{align} \varepsilon_{11} & = \frac{1}{E} \big(\sigma_{11} - \nu(\sigma_{22}+\sigma_{33}) \big) \\ \varepsilon_{22} & = \frac{1}{E} \big(\sigma_{22} - \nu(\sigma_{11}+\sigma_{33}) \big) \\ \varepsilon_{33} & = \frac{1}{E} \big(\sigma_{33} - \nu(\sigma_{11}+\sigma_{22}) \big) \\ \varepsilon_{12} & = \frac{1}{2G} \sigma_{12} \,;\qquad \varepsilon_{13} = \frac{1}{2G}\sigma_{13} \,;\qquad \varepsilon_{23} = \frac{1}{2G}\sigma_{23} \end{align} where is
Young's modulus and is
Poisson's ratio. (See
3-D elasticity). {{math proof Consider the strain and stress relation as a superposition of two effects: stretching in direction of the load (1) and shrinking (caused by the load) in perpendicular directions (2 and 3), \begin{align} \varepsilon_1' &= \frac{1}{E}\sigma_1 \,, \\ \varepsilon_2' &= -\frac{\nu}{E}\sigma_1 \,, \\ \varepsilon_3' &= -\frac{\nu}{E}\sigma_1 \,, \end{align} where is Poisson's ratio and is Young's modulus. We get similar equations to the loads in directions 2 and 3, \begin{align} \varepsilon_1'' &= -\frac{\nu}{E}\sigma_2 \,, \\ \varepsilon_2'' &= \frac{1}{E}\sigma_2 \,, \\ \varepsilon_3'' &= -\frac{\nu}{E}\sigma_2 \,, \end{align} and \begin{align} \varepsilon_1''' &= -\frac{\nu}{E}\sigma_3 \,, \\ \varepsilon_2''' &= -\frac{\nu}{E}\sigma_3 \,, \\ \varepsilon_3''' &= \frac{1}{E}\sigma_3 \,. \end{align} Summing the three cases together () we get \begin{align} \varepsilon_1 &= \frac{1}{E}\big(\sigma_1-\nu(\sigma_2+\sigma_3)\big) \,, \\ \varepsilon_2 &= \frac{1}{E}\big(\sigma_2-\nu(\sigma_1+\sigma_3)\big) \,, \\ \varepsilon_3 &= \frac{1}{E}\big(\sigma_3-\nu(\sigma_1+\sigma_2)\big) \,, \end{align} or by adding and subtracting one \begin{align} \varepsilon_1 &= \frac{1}{E}\big((1+\nu)\sigma_1-\nu(\sigma_1+\sigma_2+\sigma_3)\big) \,, \\ \varepsilon_2 &= \frac{1}{E}\big((1+\nu)\sigma_2-\nu(\sigma_1+\sigma_2+\sigma_3)\big) \,, \\ \varepsilon_3 &= \frac{1}{E}\big((1+\nu)\sigma_3-\nu(\sigma_1+\sigma_2+\sigma_3)\big) \,, \end{align} and further we get by solving \sigma_1 = \frac{E}{1+\nu}\varepsilon_1 + \frac{\nu}{1+\nu}(\sigma_1+\sigma_2+\sigma_3)\,. Calculating the sum \begin{align} \varepsilon_1 + \varepsilon_2 + \varepsilon_3 &= \frac{1}{E}\big((1+\nu)(\sigma_1 + \sigma_2 + \sigma_3) - 3\nu(\sigma_1 + \sigma_2 + \sigma_3)\big) = \frac{1-2\nu}{E}(\sigma_1 + \sigma_2 + \sigma_3) \\ \sigma_1 +\sigma_2+\sigma_3 &= \frac{E}{1-2\nu}(\varepsilon_1 + \varepsilon_2 +\varepsilon_3) \end{align} and substituting it to the equation solved for gives \begin{align} \sigma_1 &= \frac{E}{1+\nu}\varepsilon_1 + \frac{E\nu}{(1+\nu)(1-2\nu)}(\varepsilon_1 + \varepsilon_2 +\varepsilon_3) \\ &= 2\mu\varepsilon_1 + \lambda(\varepsilon_1 + \varepsilon_2 +\varepsilon_3)\,, \end{align} where and are the
Lamé parameters. Similar treatment of directions 2 and 3 gives the Hooke's law in three dimensions. }} In matrix form, Hooke's law for isotropic materials can be written as \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{13} \\ 2\varepsilon_{12} \end{bmatrix} \,=\, \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ \gamma_{23} \\ \gamma_{13} \\ \gamma_{12} \end{bmatrix} \,=\, \frac{1}{E} \begin{bmatrix} 1 & -\nu & -\nu & 0 & 0 & 0 \\ -\nu & 1 & -\nu & 0 & 0 & 0 \\ -\nu & -\nu & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2+2\nu & 0 & 0 \\ 0 & 0 & 0 & 0 & 2+2\nu & 0 \\ 0 & 0 & 0 & 0 & 0 & 2+2\nu \end{bmatrix} \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{13} \\ \sigma_{12} \end{bmatrix} where is the
engineering shear strain. The inverse relation may be written as \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{13} \\ \sigma_{12} \end{bmatrix} \,=\, \frac{E}{(1+\nu)(1-2\nu)} \begin{bmatrix} 1-\nu & \nu & \nu & 0 & 0 & 0 \\ \nu & 1-\nu & \nu & 0 & 0 & 0 \\ \nu & \nu & 1-\nu & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1-2\nu}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1-2\nu}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1-2\nu}{2} \end{bmatrix} \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{13} \\ 2\varepsilon_{12} \end{bmatrix} which can be simplified thanks to the Lamé constants: \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{13} \\ \sigma_{12} \end{bmatrix} \,=\, \begin{bmatrix} 2\mu+\lambda & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & 2\mu+\lambda & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & 2\mu+\lambda & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu \end{bmatrix} \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{13} \\ 2\varepsilon_{12} \end{bmatrix} In vector notation this becomes \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{12} & \sigma_{22} & \sigma_{23} \\ \sigma_{13} & \sigma_{23} & \sigma_{33} \end{bmatrix} \,=\, 2\mu \begin{bmatrix} \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ \varepsilon_{12} & \varepsilon_{22} & \varepsilon_{23} \\ \varepsilon_{13} & \varepsilon_{23} & \varepsilon_{33} \end{bmatrix} + \lambda \mathbf{I}\left(\varepsilon_{11} + \varepsilon_{22} + \varepsilon_{33} \right) where is the identity tensor.
Plane stress Under
plane stress conditions, . In that case Hooke's law takes the form \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix} \,=\, \frac{E}{1-\nu^2} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1-\nu}{2} \end{bmatrix} \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ 2\varepsilon_{12} \end{bmatrix} In vector notation this becomes \begin{bmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{12} & \sigma_{22} \end{bmatrix} \,=\, \frac{E}{1-\nu^2} \left((1-\nu) \begin{bmatrix} \varepsilon_{11} & \varepsilon_{12} \\ \varepsilon_{12} & \varepsilon_{22} \end{bmatrix} + \nu \mathbf{I} \left(\varepsilon_{11} + \varepsilon_{22} \right) \right) The inverse relation is usually written in the reduced form \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ 2\varepsilon_{12} \end{bmatrix} \,=\, \frac{1}{E} \begin{bmatrix} 1 & -\nu & 0 \\ -\nu & 1 & 0 \\ 0 & 0 & 2+2\nu \end{bmatrix} \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix}
Plane strain Under
plane strain conditions, . In this case Hooke's law takes the form \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix} \,=\, \frac{E}{(1 + \nu)(1 - 2\nu)} \begin{bmatrix} 1 - \nu & \nu & 0 \\ \nu & 1 - \nu & 0 \\ 0 & 0 & \frac{1 - 2\nu}{2} \end{bmatrix} \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ 2\varepsilon_{12} \end{bmatrix}
Anisotropic materials The symmetry of the
Cauchy stress tensor () and the generalized Hooke's laws () implies that . Similarly, the symmetry of the
infinitesimal strain tensor implies that . These symmetries are called the
minor symmetries of the stiffness tensor
c. This reduces the number of elastic constants from 81 to 36. If in addition, since the displacement gradient and the Cauchy stress are work conjugate, the stress–strain relation can be derived from a strain energy density functional (), then \sigma_{ij} = \frac{\partial U}{\partial \varepsilon_{ij}} \quad \implies \quad c_{ijkl} = \frac{\partial^2 U}{\partial \varepsilon_{ij}\partial \varepsilon_{kl}}\,. The arbitrariness of the order of differentiation implies that . These are called the
major symmetries of the stiffness tensor. This reduces the number of elastic constants from 36 to 21. The major and minor symmetries indicate that the stiffness tensor has only 21 independent components.
Matrix representation (stiffness tensor) It is often useful to express the anisotropic form of Hooke's law in matrix notation, also called
Voigt notation. To do this we take advantage of the symmetry of the stress and strain tensors and express them as six-dimensional vectors in an orthonormal coordinate system () as [\boldsymbol{\sigma}] \,=\, \begin{bmatrix}\sigma_{11}\\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{13} \\ \sigma_{12} \end{bmatrix} \,\equiv\, \begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end{bmatrix} \,;\qquad [\boldsymbol{\varepsilon}] \,=\, \begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{13} \\ 2\varepsilon_{12} \end{bmatrix} \,\equiv\, \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \varepsilon_4 \\ \varepsilon_5 \\ \varepsilon_6 \end{bmatrix} Then the stiffness tensor (
c) can be expressed as [\mathsf{c}] \,=\, \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} \,\equiv\, \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} and Hooke's law is written as [\boldsymbol{\sigma}] = [\mathsf{C}][\boldsymbol{\varepsilon}] \qquad \text{or} \qquad \sigma_i = C_{ij} \varepsilon_j \,. Similarly the compliance tensor (
s) can be written as [\mathsf{s}] \,=\, \begin{bmatrix} s_{1111} & s_{1122} & s_{1133} & 2s_{1123} & 2s_{1131} & 2s_{1112} \\ s_{2211} & s_{2222} & s_{2233} & 2s_{2223} & 2s_{2231} & 2s_{2212} \\ s_{3311} & s_{3322} & s_{3333} & 2s_{3323} & 2s_{3331} & 2s_{3312} \\ 2s_{2311} & 2s_{2322} & 2s_{2333} & 4s_{2323} & 4s_{2331} & 4s_{2312} \\ 2s_{3111} & 2s_{3122} & 2s_{3133} & 4s_{3123} & 4s_{3131} & 4s_{3112} \\ 2s_{1211} & 2s_{1222} & 2s_{1233} & 4s_{1223} & 4s_{1231} & 4s_{1212} \end{bmatrix} \,\equiv\, \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16} \\ S_{12} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26} \\ S_{13} & S_{23} & S_{33} & S_{34} & S_{35} & S_{36} \\ S_{14} & S_{24} & S_{34} & S_{44} & S_{45} & S_{46} \\ S_{15} & S_{25} & S_{35} & S_{45} & S_{55} & S_{56} \\ S_{16} & S_{26} & S_{36} & S_{46} & S_{56} & S_{66} \end{bmatrix}
Change of coordinate system If a linear elastic material is rotated from a reference configuration to another, then the material is symmetric with respect to the rotation if the components of the stiffness tensor in the rotated configuration are related to the components in the reference configuration by the relation c_{pqrs} = l_{pi}l_{qj}l_{rk}l_{sl}c_{ijkl} where are the components of an
orthogonal rotation matrix . The same relation also holds for inversions. In matrix notation, if the transformed basis (rotated or inverted) is related to the reference basis by [\mathbf{e}_i'] = [L][\mathbf{e}_i] then C_{ij}\varepsilon_i\varepsilon_j = C_{ij}'\varepsilon'_i\varepsilon'_j \,. In addition, if the material is symmetric with respect to the transformation then C_{ij} = C'_{ij} \quad \implies \quad C_{ij}(\varepsilon_i\varepsilon_j - \varepsilon'_i\varepsilon'_j) = 0 \,.
Orthotropic materials Orthotropic materials have three
orthogonal planes of symmetry. If the basis vectors () are normals to the planes of symmetry then the coordinate transformation relations imply that \begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end{bmatrix} \,=\, \begin{bmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{22} & C_{23} & 0 & 0 & 0 \\ C_{13} & C_{23} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66} \end{bmatrix} \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \varepsilon_4 \\ \varepsilon_5 \\ \varepsilon_6 \end{bmatrix} The inverse of this relation is commonly written as \begin{bmatrix} \varepsilon_{xx} \\ \varepsilon_{yy} \\ \varepsilon_{zz} \\ 2\varepsilon_{yz} \\ 2\varepsilon_{zx} \\ 2\varepsilon_{xy} \end{bmatrix} \,=\, \begin{bmatrix} \frac{1}{E_{x}} & - \frac{\nu_{yx}}{E_{y}} & - \frac{\nu_{zx}}{E_{z}} & 0 & 0 & 0 \\ -\frac{\nu_{xy}}{E_{x}} & \frac{1}{E_{y}} & - \frac{\nu_{zy}}{E_{z}} & 0 & 0 & 0 \\ -\frac{\nu_{xz}}{E_{x}} & - \frac{\nu_{yz}}{E_{y}} & \frac{1}{E_{z}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G_{yz}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G_{zx}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G_{xy}} \\ \end{bmatrix} \begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{zz} \\ \sigma_{yz} \\ \sigma_{zx} \\ \sigma_{xy} \end{bmatrix} where • is the
Young's modulus along axis • is the
shear modulus in direction on the plane whose normal is in direction • is the
Poisson's ratio that corresponds to a contraction in direction when an extension is applied in direction . Under
plane stress conditions, , Hooke's law for an orthotropic material takes the form \begin{bmatrix}\varepsilon_{xx} \\ \varepsilon_{yy} \\ 2\varepsilon_{xy} \end{bmatrix} \,=\, \begin{bmatrix} \frac{1}{E_{x}} & -\frac{\nu_{yx}}{E_{y}} & 0 \\ -\frac{\nu_{xy}}{E_{x}} & \frac{1}{E_{y}} & 0 \\ 0 & 0 & \frac{1}{G_{xy}} \end{bmatrix} \begin{bmatrix}\sigma_{xx} \\ \sigma_{yy} \\ \sigma_{xy} \end{bmatrix} \,. The inverse relation is \begin{bmatrix}\sigma_{xx} \\ \sigma_{yy} \\ \sigma_{xy} \end{bmatrix} \,=\, \frac{1}{1-\nu_{xy}\nu_{yx}} \begin{bmatrix} E_{x} & \nu_{yx}E_{x} & 0 \\ \nu_{xy}E_{y} & E_{y} & 0 \\ 0 & 0 & G_{xy}(1-\nu_{xy}\nu_{yx}) \end{bmatrix} \begin{bmatrix}\varepsilon_{xx} \\ \varepsilon_{yy} \\ 2\varepsilon_{xy} \end{bmatrix} \,. The transposed form of the above stiffness matrix is also often used.
Transversely isotropic materials A
transversely isotropic material is symmetric with respect to a rotation about an
axis of symmetry. For such a material, if is the axis of symmetry, Hooke's law can be expressed as \begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end{bmatrix} \,=\, \begin{bmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{13} & 0 & 0 & 0 \\ C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{C_{11}-C_{12}}{2} \end{bmatrix} \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \varepsilon_4 \\ \varepsilon_5 \\ \varepsilon_6 \end{bmatrix} More frequently, the axis is taken to be the axis of symmetry and the inverse Hooke's law is written as \begin{bmatrix} \varepsilon_{xx} \\ \varepsilon_{yy} \\ \varepsilon_{zz} \\ 2\varepsilon_{yz} \\ 2\varepsilon_{zx} \\ 2\varepsilon_{xy} \end{bmatrix} \,=\, \begin{bmatrix} \frac{1}{E_{x}} & - \frac{\nu_{yx}}{E_{y}} & - \frac{\nu_{zx}}{E_{z}} & 0 & 0 & 0 \\ -\frac{\nu_{xy}}{E_{x}} & \frac{1}{E_{y}} & - \frac{\nu_{zy}}{E_{z}} & 0 & 0 & 0 \\ -\frac{\nu_{xz}}{E_{x}} & - \frac{\nu_{yz}}{E_{y}} & \frac{1}{E_{z}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G_{yz}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G_{xz}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G_{xy}} \\ \end{bmatrix} \begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{zz} \\ \sigma_{yz} \\ \sigma_{zx} \\ \sigma_{xy} \end{bmatrix} :
Universal elastic anisotropy index To grasp the degree of anisotropy of any class, a
universal elastic anisotropy index (AU) was formulated. It replaces the
Zener ratio, which is suited for
cubic crystals. == Thermodynamic basis ==