Orthotropic material
These directional differences in strength can be quantified with Hankinson's equation.
They are a subset of anisotropic materials, because their properties change when measured from different directions.
In wood, one can define three mutually perpendicular directions at each point in which the properties are different.
It is most stiff (and strong) along the grain (axial direction), because most cellulose fibrils are aligned that way.
This anisotropy was provided by evolution, as it best enables the tree to remain upright.
This method is used to advantage in structural steel beams, and in aluminium aircraft skins.
If orthotropic properties vary between points inside an object, it possesses both orthotropy and inhomogeneity.
Isotropic materials have an infinite number of planes of symmetry.
Transversely isotropic materials are special orthotropic materials that have one axis of symmetry (any other pair of axes that are perpendicular to the main one and orthogonal among themselves are also axes of symmetry).
One common example of transversely isotropic material with one axis of symmetry is a polymer reinforced by parallel glass or graphite fibers.
Orthotropic material properties have been shown to provide a more accurate representation of bone's elastic symmetry and can also give information about the three-dimensional directionality of bone's tissue-level material properties.
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 where
If we express the above equation in terms of components with respect to an orthonormal coordinate system, we can write Summation over repeated indices has been assumed in the above relation.
In matrix form we have Examples of physical problems that fit the above template are listed in the table below.
If we choose an orthonormal coordinate system such that the axes coincide with the normals to the three symmetry planes, the transformation matrices are It can be shown that if the matrix
If the tensors in the above expression are described in terms of components with respect to an orthonormal coordinate system we can write 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 Because of the above symmetries, the stress-strain relation for linear elastic materials can be expressed in matrix form as An alternative representation in Voigt notation is or The stiffness matrix
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.
given by In Voigt notation, the transformation matrix for the stress tensor can be expressed as a
given by[4] The transformation for the strain tensor has a slightly different form because of the choice of notation.
The elastic properties of a continuum are invariant under an orthogonal transformation
if and only if[4] 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 We can show that if the matrix
In that case Using the invariance condition again, we get the additional requirement that 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.
The strain-stress relation for orthotropic linear elastic materials can be written in Voigt notation as where the compliance matrix
This implies from Sylvester's criterion that all the principal minors of the matrix are positive,[6] i.e., where
Then, We can show that this set of conditions implies that[7] or However, no similar lower bounds can be placed on the values of the Poisson's ratios
