Orthotropic Material
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In
material science A material is a substance or mixture of substances that constitutes an object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical and chemical properties, or on their geol ...
and
solid mechanics Solid mechanics (also known as mechanics of solids) is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation (mechanics), deformation under the action of forces, temperature chang ...
, orthotropic materials have material properties at a particular point which differ along three
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
axes, where each axis has twofold
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...
. 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. A familiar example of an orthotropic material is
wood Wood is a structural tissue/material found as xylem in the stems and roots of trees and other woody plants. It is an organic materiala natural composite of cellulosic fibers that are strong in tension and embedded in a matrix of lignin t ...
. 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. It is usually least stiff in the radial direction (between the growth rings), and is intermediate in the circumferential direction. This anisotropy was provided by evolution, as it best enables the tree to remain upright. Because the preferred
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
is cylindrical-polar, this type of orthotropy is also called polar orthotropy. Another example of an orthotropic material is
sheet metal Sheet metal is metal formed into thin, flat pieces, usually by an industrial process. Thicknesses can vary significantly; extremely thin sheets are considered foil (metal), foil or Metal leaf, leaf, and pieces thicker than 6 mm (0.25  ...
formed by squeezing thick sections of metal between heavy rollers. This flattens and stretches its grain structure. As a result, the material becomes
anisotropic Anisotropy () is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit ver ...
— its properties differ between the direction it was rolled in and each of the two transverse directions. 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. 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 In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...
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.Geraldes DM et al, 2014, A comparative study of orthotropic and isotropic bone adaptation in the femur, International Journal for Numerical Methods in Biomedical Engineering, Volume 30, Issue 9, pages 873–889, DOI: 10.1002/cnm.2633, http://onlinelibrary.wiley.com/wol1/doi/10.1002/cnm.2633/full 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 A grain is a small, hard, dry fruit ( caryopsis) – with or without an attached hull layer – harvested for human or animal consumption. A grain crop is a grain-producing plant. The two main types of commercial grain crops are cereals and le ...
. 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.


Orthotropy in physics


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 In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
. The material tensor provides a relation between two
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
s and can be written as : \mathbf = \boldsymbol\cdot\mathbf where \mathbf,\mathbf are two vectors representing physical quantities and \boldsymbol is the second-order material tensor. If we express the above equation in terms of components with respect to an
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpe ...
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
, we can write : f_i = K_~d_j ~. Summation over repeated indices has been assumed in the above relation. In matrix form we have : \underline = \underline~\underline \implies \begin f_1\\f_2\\f_3 \end = \begin K_ & K_ & K_ \\ K_ & K_ & K_ \\ K_ & K_ & K_ \end \begin d_1\\d_2\\d_3 \end Examples of physical problems that fit the above template are listed in the table below.Milton, G. W., 2002, The Theory of Composites, Cambridge University Press.


Condition for material symmetry

The material matrix \underline has a symmetry with respect to a given
orthogonal transformation In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we hav ...
(\boldsymbol) if it does not change when subjected to that transformation. For invariance of the material properties under such a transformation we require : \boldsymbol\cdot\mathbf = \boldsymbol\cdot(\boldsymbol\cdot\boldsymbol) \implies \mathbf = (\boldsymbol^\cdot\boldsymbol\cdot\boldsymbol)\cdot\boldsymbol Hence the condition for material symmetry is (using the definition of an orthogonal transformation) : \boldsymbol = \boldsymbol^\cdot\boldsymbol\cdot\boldsymbol = \boldsymbol^\cdot\boldsymbol\cdot\boldsymbol Orthogonal transformations can be represented in Cartesian coordinates by a 3\times 3 matrix \underline given by : \underline = \begin A_ & A_ & A_ \\ A_ & A_ & A_ \\ A_ & A_ & A_ \end~. Therefore, the symmetry condition can be written in matrix form as : \underline = \underline~\underline~\underline


Orthotropic material properties

An orthotropic material has three
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
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 : \underline = \begin-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ~;~~ \underline = \begin1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end ~;~~ \underline = \begin1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end It can be shown that if the matrix \underline 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 \underline about the 1-2\, plane. Then we have : \underline = \underline~\underline~\underline = \begin K_ & K_ & -K_ \\ K_ & K_ & -K_ \\ -K_ & -K_ & K_ \end The above relation implies that K_ = K_ = K_ = K_ = 0. Next consider a reflection \underline about the 1-3\, plane. We then have : \underline = \underline~\underline~\underline = \begin K_ & -K_ & 0 \\ -K_ & K_ & 0 \\ 0 & 0 & K_ \end That implies that K_ = K_ = 0. Therefore, the material properties of an orthotropic material are described by the matrix
: \underline = \begin K_ & 0 & 0 \\ 0 & K_ & 0 \\ 0 & 0 & K_ \end


Orthotropy in linear elasticity


Anisotropic elasticity

In
linear elasticity Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechani ...
, the relation between stress and strain depend on the type of material under consideration. This relation is known as
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
. For anisotropic materials Hooke's law can be written asLekhnitskii, S. G., 1963, Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day Inc. :\boldsymbol = \mathsf\cdot\boldsymbol where \boldsymbol is the stress
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
, \boldsymbol is the strain tensor, and \mathsf is the elastic
stiffness tensor Stiffness is the extent to which an object resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. Calculations The stiffness, k, of a ...
. If the tensors in the above expression are described in terms of components with respect to an
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpe ...
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
we can write :\sigma_ = c_~ \varepsilon_ where summation has been assumed over repeated indices. Since the stress and strain tensors are
symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, and since the stress-strain relation in linear elasticity can be derived from a
strain energy density function A strain energy density function or stored energy density function is a scalar (mathematics), scalar-valued function (mathematics), function that relates the strain energy density of a material to the deformation gradient. : W = \hat(\boldsy ...
, the following symmetries hold for linear elastic materials :c_ = c_ ~,~~c_ = c_ ~,~~ c_ = c_ ~. Because of the above symmetries, the stress-strain relation for linear elastic materials can be expressed in matrix form as : \begin\sigma_\\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \end = \begin c_ & c_ & c_ & c_ & c_ & c_ \\ c_ & c_ & c_ & c_ & c_ & c_ \\ c_ & c_ & c_ & c_ & c_ & c_ \\ c_ & c_ & c_ & c_ & c_ & c_ \\ c_ & c_ & c_ & c_ & c_ & c_ \\ c_ & c_ & c_ & c_ & c_ & c_ \end \begin\varepsilon_\\ \varepsilon_ \\ \varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \\ 2\varepsilon_ \end An alternative representation in Voigt notation is : \begin \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end = \begin C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \end \begin \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \varepsilon_4 \\ \varepsilon_5 \\ \varepsilon_6 \end or : \underline = \underline~\underline The stiffness matrix \underline in the above relation satisfies point symmetry.Slawinski, M. A., 2010, Waves and Rays in Elastic Continua: 2nd Ed., World Scientific

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Condition for material symmetry

The stiffness matrix \underline satisfies a given symmetry condition if it does not change when subjected to the corresponding
orthogonal transformation In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we hav ...
. The orthogonal transformation may represent symmetry with respect to a point symmetry, point, an
axis An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names ...
, 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 given by : \underline = \begin A_ & A_ & A_ \\ A_ & A_ & A_ \\ A_ & A_ & A_ \end~. In Voigt notation, the transformation matrix for the stress tensor can be expressed as a 6\times6 matrix \underline given by : \underline = \begin A_^2 & A_^2 & A_^2 & 2A_A_ & 2A_A_ & 2A_A_ \\ A_^2 & A_^2 & A_^2 & 2A_A_ & 2A_A_ & 2A_A_ \\ A_^2 & A_^2 & A_^2 & 2A_A_ & 2A_A_ & 2A_A_ \\ A_A_ & A_A_ & A_A_ & A_A_+A_A_ & A_A_+A_A_ & A_A_+A_A_ \\ A_A_ & A_A_ & A_A_ & A_A_+A_A_ & A_A_+A_A_ & A_A_+A_A_ \\ A_A_ & A_A_ & A_A_ & A_A_+A_A_ & A_A_+A_A_ & A_A_+A_A_ \end The transformation for the strain tensor has a slightly different form because of the choice of notation. This transformation matrix is : \underline = \begin A_^2 & A_^2 & A_^2 & A_A_ & A_A_ & A_A_ \\ A_^2 & A_^2 & A_^2 & A_A_ & A_A_ & A_A_ \\ A_^2 & A_^2 & A_^2 & A_A_ & A_A_ & A_A_ \\ 2A_A_ & 2A_A_ & 2A_A_ & A_A_+A_A_ & A_A_+A_A_ & A_A_+A_A_ \\ 2A_A_ & 2A_A_ & 2A_A_ & A_A_+A_A_ & A_A_+A_A_ & A_A_+A_A_ \\ 2A_A_ & 2A_A_ & 2A_A_ & A_A_+A_A_ & A_A_+A_A_ & A_A_+A_A_ \end It can be shown that \underline^T = \underline^.
The elastic properties of a continuum are invariant under an orthogonal transformation \underline if and only if : \underline = \underline^T~\underline~\underline


Stiffness and compliance matrices in orthotropic elasticity

An orthotropic elastic material has three
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
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 : \underline = \begin-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ~;~~ \underline = \begin1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end ~;~~ \underline = \begin1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end We can show that if the matrix \underline 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 \underline about the 1-2\, plane, then we have : \underline = \begin 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 Then the requirement \underline = \underline^T~\underline~\underline implies that : \begin C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \end = \begin C_ & C_ & C_ & -C_ & -C_ & C_ \\ C_ & C_ & C_ & -C_ & -C_ & C_ \\ C_ & C_ & C_ & -C_ & -C_ & C_ \\ -C_ & -C_ & -C_ & C_ & C_ & -C_ \\ -C_ & -C_ & -C_ & C_ & C_ & -C_ \\ C_ & C_ & C_ & -C_ & -C_ & C_ \end The above requirement can be satisfied only if : C_ = C_ = C_ = C_ = C_ = C_ = C_ = C_ = 0 ~. Let us next consider the reflection \underline about the 1-3\, plane. In that case : \underline = \begin 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 Using the invariance condition again, we get the additional requirement that : C_ = C_ = C_ = C_ = 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
: \underline = \begin C_ & C_ & C_ & 0 & 0 & 0 \\ C_ & C_ & C_ & 0 & 0 & 0 \\ C_ & C_ & C_ & 0 & 0 & 0 \\ 0 & 0 & 0 & C_ & 0 & 0 \\ 0 & 0 & 0 & 0 & C_ & 0\\ 0 & 0 & 0 & 0 & 0 & C_ \end
The inverse of this matrix is commonly written asBoresi, A. P, Schmidt, R. J. and Sidebottom, O. M., 1993, ''Advanced Mechanics of Materials'', Wiley. : \underline = \begin \tfrac & - \tfrac & - \tfrac & 0 & 0 & 0 \\ -\tfrac & \tfrac & - \tfrac & 0 & 0 & 0 \\ -\tfrac & - \tfrac & \tfrac & 0 & 0 & 0 \\ 0 & 0 & 0 & \tfrac & 0 & 0 \\ 0 & 0 & 0 & 0 & \tfrac & 0 \\ 0 & 0 & 0 & 0 & 0 & \tfrac \\ \end where _\, is the
Young's modulus Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Youn ...
along axis i, G_\, is the
shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the Elasticity (physics), elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear s ...
in direction j on the plane whose normal is in direction i, and \nu_\, is the
Poisson's ratio In materials science and solid mechanics, Poisson's ratio (symbol: ( nu)) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value ...
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~\underline where the compliance matrix \underline is given by : \underline = \begin S_ & S_ & S_ & 0 & 0 & 0 \\ S_ & S_ & S_ & 0 & 0 & 0 \\ S_ & S_ & S_ & 0 & 0 & 0 \\ 0 & 0 & 0 & S_ & 0 & 0 \\ 0 & 0 & 0 & 0 & S_ & 0\\ 0 & 0 & 0 & 0 & 0 & S_ \end The compliance matrix is
symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
and must be positive definite for the strain energy density to be positive. This implies from
Sylvester's criterion In mathematics, Sylvester’s criterion is a necessary and sufficient condition, necessary and sufficient criterion to determine whether a Hermitian matrix is Definite matrix, positive-definite. Sylvester's criterion states that a ''n'' × ''n'' ...
that all the principal minors of the matrix are positive,Ting, T. C. T. and Chen, T., 2005, ''Poisson's ratio for anisotropic elastic materials can have no bounds,'', Q. J. Mech. Appl. Math., 58(1), pp. 73-82. i.e., : \Delta_k := \det(\underline) > 0 where \underline is the k\times k principal
submatrix In mathematics, a matrix (: matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. ...
of \underline. Then, : \begin \Delta_1 > 0 & \implies \quad S_ > 0 \\ \Delta_2 > 0 & \implies \quad S_S_ - S_^2 > 0 \\ \Delta_3 > 0 & \implies \quad (S_S_-S_^2)S_-S_S_^2+2S_S_S_-S_S_^2 >0 \\ \Delta_4 > 0 & \implies \quad S_\Delta_3 > 0 \implies S_ > 0\\ \Delta_5 > 0 & \implies \quad S_S_\Delta_3 > 0 \implies S_ > 0 \\ \Delta_6 > 0 & \implies \quad S_S_S_\Delta_3 > 0 \implies S_ > 0 \end We can show that this set of conditions implies that. : S_ > 0 ~,~~ S_ > 0 ~,~~ S_ > 0 ~,~~ S_ > 0 ~,~~ S_ > 0 ~,~~ S_ > 0 or : E_1 > 0 , E_2 > 0, E_3 > 0, G_ > 0 , G_ > 0, G_ > 0 However, no similar lower bounds can be placed on the values of the Poisson's ratios \nu_.


See also

*
Anisotropy Anisotropy () is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit ve ...
* Clinotropic material *
Stress (mechanics) In continuum mechanics, stress is a physical quantity that describes forces present during deformation. For example, an object being pulled apart, such as a stretched elastic band, is subject to ''tensile'' stress and may undergo elongati ...
*
Infinitesimal strain theory In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
*
Finite strain theory In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal str ...
*
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...


References


Further reading


Orthotropy modeling equations
from OOFEM Matlib manual section.
Hooke's law for orthotropic materials
{{Topics in continuum mechanics Continuum mechanics Elasticity (physics) models Materials Orientation (geometry)