Orthotropic material

Material behavior is represented in physical theories by constitutive relations. A large class of physical behaviors can be represented by linear material models that take the form of a second-order tensor. The material tensor provides a relation between two vectors and can be written as

${\displaystyle \mathbf {f} ={\boldsymbol {K}}\cdot \mathbf {d} }$

where ${\displaystyle \mathbf {d} ,\mathbf {f} }$ are two vectors representing physical quantities and ${\displaystyle {\boldsymbol {K}}}$ is the second-order material tensor. If we express the above equation in terms of components with respect to an orthonormal coordinate system, we can write

${\displaystyle f_{i}=K_{ij}~d_{j}~.}$

Summation over repeated indices has been assumed in the above relation. In matrix form we have

${\displaystyle {\underline {\mathbf {f} }}={\underline {\underline {\boldsymbol {K}}}}~{\underline {\mathbf {d} }}\implies {\begin{bmatrix}f_{1}\\f_{2}\\f_{3}\end{bmatrix}}={\begin{bmatrix}K_{11}&K_{12}&K_{13}\\K_{21}&K_{22}&K_{23}\\K_{31}&K_{32}&K_{33}\end{bmatrix}}{\begin{bmatrix}d_{1}\\d_{2}\\d_{3}\end{bmatrix}}}$

Examples of physical problems that fit the above template are listed in the table below.^[2]

Problem	${\displaystyle \mathbf {f} }$	${\displaystyle \mathbf {d} }$	${\displaystyle {\boldsymbol {K}}}$
Electrical conduction	Electrical current ${\displaystyle \mathbf {J} }$	Electric field ${\displaystyle \mathbf {E} }$	Electrical conductivity ${\displaystyle {\boldsymbol {\sigma }}}$
Dielectrics	Electrical displacement ${\displaystyle \mathbf {D} }$	Electric field ${\displaystyle \mathbf {E} }$	Electric permittivity ${\displaystyle {\boldsymbol {\varepsilon }}}$
Magnetism	Magnetic induction ${\displaystyle \mathbf {B} }$	Magnetic field ${\displaystyle \mathbf {H} }$	Magnetic permeability ${\displaystyle {\boldsymbol {\mu }}}$
Thermal conduction	Heat flux ${\displaystyle \mathbf {q} }$	Temperature gradient ${\displaystyle -{\boldsymbol {\nabla }}T}$	Thermal conductivity ${\displaystyle {\boldsymbol {\kappa }}}$
Diffusion	Particle flux ${\displaystyle \mathbf {J} }$	Concentration gradient ${\displaystyle -{\boldsymbol {\nabla }}c}$	Diffusivity ${\displaystyle {\boldsymbol {D}}}$
Flow in porous media	Weighted fluid velocity ${\displaystyle \eta _{\mu }\mathbf {v} }$	Pressure gradient ${\displaystyle {\boldsymbol {\nabla }}P}$	Fluid permeability ${\displaystyle {\boldsymbol {\kappa }}}$

The material matrix ${\displaystyle {\underline {\underline {\boldsymbol {K}}}}}$ has a symmetry with respect to a given orthogonal transformation (${\displaystyle {\boldsymbol {A}}}$) if it does not change when subjected to that transformation. For invariance of the material properties under such a transformation we require

${\displaystyle {\boldsymbol {A}}\cdot \mathbf {f} ={\boldsymbol {K}}\cdot ({\boldsymbol {A}}\cdot {\boldsymbol {d}})\implies \mathbf {f} =({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {K}}\cdot {\boldsymbol {A}})\cdot {\boldsymbol {d}}}$

Hence the condition for material symmetry is (using the definition of an orthogonal transformation)

${\displaystyle {\boldsymbol {K}}={\boldsymbol {A}}^{-1}\cdot {\boldsymbol {K}}\cdot {\boldsymbol {A}}={\boldsymbol {A}}^{T}\cdot {\boldsymbol {K}}\cdot {\boldsymbol {A}}}$

Orthogonal transformations can be represented in Cartesian coordinates by a ${\displaystyle 3\times 3}$ matrix ${\displaystyle {\underline {\underline {\boldsymbol {A}}}}}$ given by

${\displaystyle {\underline {\underline {\boldsymbol {A}}}}={\begin{bmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{bmatrix}}~.}$

Therefore, the symmetry condition can be written in matrix form as

${\displaystyle {\underline {\underline {\boldsymbol {K}}}}={\underline {\underline {{\boldsymbol {A}}^{T}}}}~{\underline {\underline {\boldsymbol {K}}}}~{\underline {\underline {\boldsymbol {A}}}}}$

An orthotropic material has three orthogonal symmetry planes. If we choose an orthonormal coordinate system such that the axes coincide with the normals to the three symmetry planes, the transformation matrices are

${\displaystyle {\underline {\underline {{\boldsymbol {A}}_{1}}}}={\begin{bmatrix}-1&0&0\\0&1&0\\0&0&1\end{bmatrix}}~;~~{\underline {\underline {{\boldsymbol {A}}_{2}}}}={\begin{bmatrix}1&0&0\\0&-1&0\\0&0&1\end{bmatrix}}~;~~{\underline {\underline {{\boldsymbol {A}}_{3}}}}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}}$

It can be shown that if the matrix ${\displaystyle {\underline {\underline {\boldsymbol {K}}}}}$ for a material is invariant under reflection about two orthogonal planes then it is also invariant under reflection about the third orthogonal plane.

Consider the reflection ${\displaystyle {\underline {\underline {{\boldsymbol {A}}_{3}}}}}$ about the ${\displaystyle 1-2\,}$ plane. Then we have

${\displaystyle {\underline {\underline {\boldsymbol {K}}}}={\underline {\underline {{\boldsymbol {A}}_{3}^{T}}}}~{\underline {\underline {\boldsymbol {K}}}}~{\underline {\underline {{\boldsymbol {A}}_{3}}}}={\begin{bmatrix}K_{11}&K_{12}&-K_{13}\\K_{21}&K_{22}&-K_{23}\\-K_{31}&-K_{32}&K_{33}\end{bmatrix}}}$

The above relation implies that ${\displaystyle K_{13}=K_{23}=K_{31}=K_{32}=0}$. Next consider a reflection ${\displaystyle {\underline {\underline {{\boldsymbol {A}}_{2}}}}}$ about the ${\displaystyle 1-3\,}$ plane. We then have

${\displaystyle {\underline {\underline {\boldsymbol {K}}}}={\underline {\underline {{\boldsymbol {A}}_{2}^{T}}}}~{\underline {\underline {\boldsymbol {K}}}}~{\underline {\underline {{\boldsymbol {A}}_{2}}}}={\begin{bmatrix}K_{11}&-K_{12}&0\\-K_{21}&K_{22}&0\\0&0&K_{33}\end{bmatrix}}}$

That implies that ${\displaystyle K_{12}=K_{21}=0}$. Therefore, the material properties of an orthotropic material are described by the matrix

${\displaystyle {\underline {\underline {\boldsymbol {K}}}}={\begin{bmatrix}K_{11}&0&0\\0&K_{22}&0\\0&0&K_{33}\end{bmatrix}}}$

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^[3]

${\displaystyle {\boldsymbol {\sigma }}={\mathsf {c}}\cdot {\boldsymbol {\varepsilon }}}$

where ${\displaystyle {\boldsymbol {\sigma }}}$ is the stress tensor, ${\displaystyle {\boldsymbol {\varepsilon }}}$ is the strain tensor, and ${\displaystyle {\mathsf {c}}}$ 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

${\displaystyle \sigma _{ij}=c_{ijk\ell }~\varepsilon _{k\ell }}$

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

${\displaystyle c_{ijk\ell }=c_{jik\ell }~,~~c_{ijk\ell }=c_{ij\ell k}~,~~c_{ijk\ell }=c_{k\ell ij}~.}$

Because of the above symmetries, the stress-strain relation for linear elastic materials can be expressed in matrix form as

${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{31}\\\sigma _{12}\end{bmatrix}}={\begin{bmatrix}c_{1111}&c_{1122}&c_{1133}&c_{1123}&c_{1131}&c_{1112}\\c_{2211}&c_{2222}&c_{2233}&c_{2223}&c_{2231}&c_{2212}\\c_{3311}&c_{3322}&c_{3333}&c_{3323}&c_{3331}&c_{3312}\\c_{2311}&c_{2322}&c_{2333}&c_{2323}&c_{2331}&c_{2312}\\c_{3111}&c_{3122}&c_{3133}&c_{3123}&c_{3131}&c_{3112}\\c_{1211}&c_{1222}&c_{1233}&c_{1223}&c_{1231}&c_{1212}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{31}\\2\varepsilon _{12}\end{bmatrix}}}$

An alternative representation in Voigt notation is

${\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}$

or

${\displaystyle {\underline {\underline {\boldsymbol {\sigma }}}}={\underline {\underline {\mathsf {C}}}}~{\underline {\underline {\boldsymbol {\varepsilon }}}}}$

The stiffness matrix ${\displaystyle {\underline {\underline {\mathsf {C}}}}}$ in the above relation satisfies point symmetry.^[4]

The stiffness matrix ${\displaystyle {\underline {\underline {\mathsf {C}}}}}$ 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 ${\displaystyle 3\times 3}$ matrix ${\displaystyle {\underline {\underline {\mathbf {A} }}}}$ given by

${\displaystyle {\underline {\underline {\mathbf {A} }}}={\begin{bmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{bmatrix}}~.}$

In Voigt notation, the transformation matrix for the stress tensor can be expressed as a ${\displaystyle 6\times 6}$ matrix ${\displaystyle {\underline {\underline {{\mathsf {A}}_{\sigma }}}}}$ given by^[4]

${\displaystyle {\underline {\underline {{\mathsf {A}}_{\sigma }}}}={\begin{bmatrix}A_{11}^{2}&A_{12}^{2}&A_{13}^{2}&2A_{12}A_{13}&2A_{11}A_{13}&2A_{11}A_{12}\\A_{21}^{2}&A_{22}^{2}&A_{23}^{2}&2A_{22}A_{23}&2A_{21}A_{23}&2A_{21}A_{22}\\A_{31}^{2}&A_{32}^{2}&A_{33}^{2}&2A_{32}A_{33}&2A_{31}A_{33}&2A_{31}A_{32}\\A_{21}A_{31}&A_{22}A_{32}&A_{23}A_{33}&A_{22}A_{33}+A_{23}A_{32}&A_{21}A_{33}+A_{23}A_{31}&A_{21}A_{32}+A_{22}A_{31}\\A_{11}A_{31}&A_{12}A_{32}&A_{13}A_{33}&A_{12}A_{33}+A_{13}A_{32}&A_{11}A_{33}+A_{13}A_{31}&A_{11}A_{32}+A_{12}A_{31}\\A_{11}A_{21}&A_{12}A_{22}&A_{13}A_{23}&A_{12}A_{23}+A_{13}A_{22}&A_{11}A_{23}+A_{13}A_{21}&A_{11}A_{22}+A_{12}A_{21}\end{bmatrix}}}$

The transformation for the strain tensor has a slightly different form because of the choice of notation. This transformation matrix is

${\displaystyle {\underline {\underline {{\mathsf {A}}_{\varepsilon }}}}={\begin{bmatrix}A_{11}^{2}&A_{12}^{2}&A_{13}^{2}&A_{12}A_{13}&A_{11}A_{13}&A_{11}A_{12}\\A_{21}^{2}&A_{22}^{2}&A_{23}^{2}&A_{22}A_{23}&A_{21}A_{23}&A_{21}A_{22}\\A_{31}^{2}&A_{32}^{2}&A_{33}^{2}&A_{32}A_{33}&A_{31}A_{33}&A_{31}A_{32}\\2A_{21}A_{31}&2A_{22}A_{32}&2A_{23}A_{33}&A_{22}A_{33}+A_{23}A_{32}&A_{21}A_{33}+A_{23}A_{31}&A_{21}A_{32}+A_{22}A_{31}\\2A_{11}A_{31}&2A_{12}A_{32}&2A_{13}A_{33}&A_{12}A_{33}+A_{13}A_{32}&A_{11}A_{33}+A_{13}A_{31}&A_{11}A_{32}+A_{12}A_{31}\\2A_{11}A_{21}&2A_{12}A_{22}&2A_{13}A_{23}&A_{12}A_{23}+A_{13}A_{22}&A_{11}A_{23}+A_{13}A_{21}&A_{11}A_{22}+A_{12}A_{21}\end{bmatrix}}}$

It can be shown that ${\displaystyle {\underline {\underline {{\mathsf {A}}_{\varepsilon }}}}^{T}={\underline {\underline {{\mathsf {A}}_{\sigma }}}}^{-1}}$.

The elastic properties of a continuum are invariant under an orthogonal transformation ${\displaystyle {\underline {\underline {\mathbf {A} }}}}$ if and only if^[4]

${\displaystyle {\underline {\underline {\mathsf {C}}}}={\underline {\underline {{\mathsf {A}}_{\varepsilon }}}}^{T}~{\underline {\underline {\mathsf {C}}}}~{\underline {\underline {{\mathsf {A}}_{\varepsilon }}}}}$

An orthotropic elastic material has three orthogonal symmetry planes. If we choose an orthonormal coordinate system such that the axes coincide with the normals to the three symmetry planes, the transformation matrices are

${\displaystyle {\underline {\underline {\mathbf {A} _{1}}}}={\begin{bmatrix}-1&0&0\\0&1&0\\0&0&1\end{bmatrix}}~;~~{\underline {\underline {\mathbf {A} _{2}}}}={\begin{bmatrix}1&0&0\\0&-1&0\\0&0&1\end{bmatrix}}~;~~{\underline {\underline {\mathbf {A} _{3}}}}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}}$

We can show that if the matrix ${\displaystyle {\underline {\underline {\mathsf {C}}}}}$ for a linear elastic material is invariant under reflection about two orthogonal planes then it is also invariant under reflection about the third orthogonal plane.

If we consider the reflection ${\displaystyle {\underline {\underline {\mathbf {A} _{3}}}}}$ about the ${\displaystyle 1-2\,}$ plane, then we have

${\displaystyle {\underline {\underline {{\mathsf {A}}_{\varepsilon }}}}={\begin{bmatrix}1&0&0&0&0&0\\0&1&0&0&0&0\\0&0&1&0&0&0\\0&0&0&-1&0&0\\0&0&0&0&-1&0\\0&0&0&0&0&1\end{bmatrix}}}$

Then the requirement ${\displaystyle {\underline {\underline {\mathsf {C}}}}={\underline {\underline {{\mathsf {A}}_{\varepsilon }}}}^{T}~{\underline {\underline {\mathsf {C}}}}~{\underline {\underline {{\mathsf {A}}_{\varepsilon }}}}}$ implies that^[4]

${\displaystyle {\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&-C_{14}&-C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&-C_{24}&-C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&-C_{34}&-C_{35}&C_{36}\\-C_{14}&-C_{24}&-C_{34}&C_{44}&C_{45}&-C_{46}\\-C_{15}&-C_{25}&-C_{35}&C_{45}&C_{55}&-C_{56}\\C_{16}&C_{26}&C_{36}&-C_{46}&-C_{56}&C_{66}\end{bmatrix}}}$

The above requirement can be satisfied only if

${\displaystyle C_{14}=C_{15}=C_{24}=C_{25}=C_{34}=C_{35}=C_{46}=C_{56}=0~.}$

Let us next consider the reflection ${\displaystyle {\underline {\underline {\mathbf {A} _{2}}}}}$ about the ${\displaystyle 1-3\,}$ plane. In that case

${\displaystyle {\underline {\underline {{\mathsf {A}}_{\varepsilon }}}}={\begin{bmatrix}1&0&0&0&0&0\\0&1&0&0&0&0\\0&0&1&0&0&0\\0&0&0&-1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&-1\end{bmatrix}}}$

Using the invariance condition again, we get the additional requirement that

${\displaystyle C_{16}=C_{26}=C_{36}=C_{45}=0~.}$

No further information can be obtained because the reflection about third symmetry plane is not independent of reflections about the planes that we have already considered. Therefore, the stiffness matrix of an orthotropic linear elastic material can be written as

${\displaystyle {\underline {\underline {\mathsf {C}}}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{22}&C_{23}&0&0&0\\C_{13}&C_{23}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{55}&0\\0&0&0&0&0&C_{66}\end{bmatrix}}}$

The inverse of this matrix is commonly written as^[5]

${\displaystyle {\underline {\underline {\mathsf {S}}}}={\begin{bmatrix}{\tfrac {1}{E_{\rm {1}}}}&-{\tfrac {\nu _{\rm {21}}}{E_{\rm {2}}}}&-{\tfrac {\nu _{\rm {31}}}{E_{\rm {3}}}}&0&0&0\\-{\tfrac {\nu _{\rm {12}}}{E_{\rm {1}}}}&{\tfrac {1}{E_{\rm {2}}}}&-{\tfrac {\nu _{\rm {32}}}{E_{\rm {3}}}}&0&0&0\\-{\tfrac {\nu _{\rm {13}}}{E_{\rm {1}}}}&-{\tfrac {\nu _{\rm {23}}}{E_{\rm {2}}}}&{\tfrac {1}{E_{\rm {3}}}}&0&0&0\\0&0&0&{\tfrac {1}{G_{\rm {23}}}}&0&0\\0&0&0&0&{\tfrac {1}{G_{\rm {31}}}}&0\\0&0&0&0&0&{\tfrac {1}{G_{\rm {12}}}}\\\end{bmatrix}}}$

where ${\displaystyle {E}_{\rm {i}}\,}$ is the Young's modulus along axis ${\displaystyle i}$, ${\displaystyle G_{\rm {ij}}\,}$ is the shear modulus in direction ${\displaystyle j}$ on the plane whose normal is in direction ${\displaystyle i}$, and ${\displaystyle \nu _{\rm {ij}}\,}$ is the Poisson's ratio that corresponds to a contraction in direction ${\displaystyle j}$ when an extension is applied in direction ${\displaystyle i}$. Only nine (9) from these twelve (12) elastic constants are independent.

The strain-stress relation for orthotropic linear elastic materials can be written in Voigt notation as

${\displaystyle {\underline {\underline {\boldsymbol {\varepsilon }}}}={\underline {\underline {\mathsf {S}}}}~{\underline {\underline {\boldsymbol {\sigma }}}}}$

where the compliance matrix ${\displaystyle {\underline {\underline {\mathsf {S}}}}}$ is given by

${\displaystyle {\underline {\underline {\mathsf {S}}}}={\begin{bmatrix}S_{11}&S_{12}&S_{13}&0&0&0\\S_{12}&S_{22}&S_{23}&0&0&0\\S_{13}&S_{23}&S_{33}&0&0&0\\0&0&0&S_{44}&0&0\\0&0&0&0&S_{55}&0\\0&0&0&0&0&S_{66}\end{bmatrix}}}$

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,^[6] i.e.,

${\displaystyle \Delta _{k}:=\det({\underline {\underline {{\mathsf {S}}_{k}}}})>0}$

where ${\displaystyle {\underline {\underline {{\mathsf {S}}_{k}}}}}$ is the ${\displaystyle k\times k}$ principal submatrix of ${\displaystyle {\underline {\underline {\mathsf {S}}}}}$.

Then,

${\displaystyle {\begin{aligned}\Delta _{1}>0&\implies \quad S_{11}>0\\\Delta _{2}>0&\implies \quad S_{11}S_{22}-S_{12}^{2}>0\\\Delta _{3}>0&\implies \quad (S_{11}S_{22}-S_{12}^{2})S_{33}-S_{11}S_{23}^{2}+2S_{12}S_{23}S_{13}-S_{22}S_{13}^{2}>0\\\Delta _{4}>0&\implies \quad S_{44}\Delta _{3}>0\implies S_{44}>0\\\Delta _{5}>0&\implies \quad S_{44}S_{55}\Delta _{3}>0\implies S_{55}>0\\\Delta _{6}>0&\implies \quad S_{44}S_{55}S_{66}\Delta _{3}>0\implies S_{66}>0\end{aligned}}}$

We can show that this set of conditions implies that^[7]

${\displaystyle S_{11}>0~,~~S_{22}>0~,~~S_{33}>0~,~~S_{44}>0~,~~S_{55}>0~,~~S_{66}>0}$

or

${\displaystyle E_{1}>0,E_{2}>0,E_{3}>0,G_{12}>0,G_{23}>0,G_{13}>0}$

However, no similar lower bounds can be placed on the values of the Poisson's ratios ${\displaystyle \nu _{ij}}$.^[6]

• Anisotropy
• Clinotropic material
• Stress (mechanics)
• Infinitesimal strain theory
• Finite strain theory
• Hooke's law