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In material science and solid mechanics, orthotropic materials have material properties that differ along three mutually-orthogonal twofold axes of rotational symmetry. They are a subset of anisotropic materials, because their properties change when measured from different directions.
Contents
- Anisotropic material relations
- Condition for material symmetry
- Orthotropic material properties
- Anisotropic elasticity
- Stiffness and compliance matrices in orthotropic elasticity
- Bounds on the moduli of orthotropic elastic materials
- References
A familiar example of an orthotropic material is wood. In wood, one can define three mutually perpendicular directions at each point in which the properties are different. These are the axial direction (along the grain), the radial direction, and the circumferential direction. Because the preferred coordinate system is cylindrical-polar, this type of orthotropy is also called polar orthotropy. Mechanical properties, such as strength and stiffness, measured axially (along the grain) are typically better than those measured in the radial and circumferential directions (across the grain). These directional differences in strength can be quantified with Hankinson's equation.
Another example of an orthotropic material is sheet metal formed by squeezing thick sections of metal between heavy rollers. This flattens and stretches its grain structure. As a result, the material becomes anisotropic—its properties differ between the direction it was rolled in and each of the two transverse directions.
If orthotropic properties vary between points inside an object, it possesses both orthotropy and inhomogeneity. This suggests that orthotropy is the property of a point within an object rather than for the object as a whole (unless the object is homogeneous). The associated planes of symmetry are also defined for a small region around a point and do not necessarily have to be identical to the planes of symmetry of the whole object.
Orthotropic materials are a subset of anisotropic materials; their properties depend on the direction in which they are measured. Orthotropic materials have three planes/axes of symmetry. An isotropic material, in contrast, has the same properties in every direction. It can be proved that a material having two planes of symmetry must have a third one. 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. The strength and stiffness of such a composite material will usually be greater in a direction parallel to the fibers than in the transverse direction, and the thickness direction usually has properties similar to the transverse direction. Another example would be a biological membrane, in which the properties in the plane of the membrane will be different from those in the perpendicular direction. 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.
It is important to keep in mind that a material which is anisotropic on one length scale may be isotropic on another (usually larger) length scale. For instance, most metals are polycrystalline with very small grains. Each of the individual grains may be anisotropic, but if the material as a whole comprises many randomly oriented grains, then its measured mechanical properties will be an average of the properties over all possible orientations of the individual grains.
Anisotropic material relations
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
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.
Condition for material symmetry
The material matrix
Hence the condition for material symmetry is (using the definition of an orthogonal transformation)
Orthogonal transformations can be represented in Cartesian coordinates by a
Therefore the symmetry condition can be written in matrix form as
Orthotropic material properties
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
It can be shown that if the matrix
Consider the reflection
The above relation implies that
That implies that
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
where
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
Condition for material symmetry
The stiffness matrix
In Voigt notation, the transformation matrix for the stress tensor can be expressed as a
The transformation for the strain tensor has a slightly different form because of the choice of notation. This transformation matrix is
It can be shown that
The elastic properties of a continuum are invariant under an orthogonal transformation
Stiffness and compliance matrices in orthotropic elasticity
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
If we consider the reflection
Then the requirement
The above requirement can be satisfied only if
Let us next consider the reflection
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. Therefore, the stiffness matrix of an orthotropic linear elastic material can be written as
The inverse of this matrix is commonly written as
where
Bounds on the moduli of orthotropic elastic materials
The strain-stress relation for orthotropic linear elastic materials can be written in Voigt notation as
where the compliance matrix
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.,
where
Then,
We can show that this set of conditions implies that
or
However, no similar lower bounds can be placed on the values of the Poisson's ratios