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Wood is an example of an orthotropic material. Material properties in three perpendicular directions (axial, radial, and circumferential) are different.

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.

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.^[1]

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.

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. The material tensor provides a relation between two vectors and can be written as

mathbf f=boldsymbol Kcdot mathbf d

where $mathbf d,mathbf f$ are two vectors representing physical quantities and $boldsymbol K$ 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

f_i=K_ij~d_j~.

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

underline mathbf f=underline underline boldsymbol K~underline mathbf dimplies beginbmatrixf_1\f_2\f_3endbmatrix=beginbmatrixK_11&K_12&K_13\K_21&K_22&K_23\K_31&K_32&K_33endbmatrixbeginbmatrixd_1\d_2\d_3endbmatrix

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

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

Condition for material symmetry

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

boldsymbol Acdot mathbf f=boldsymbol Kcdot (boldsymbol Acdot boldsymbol d)implies mathbf f=(boldsymbol A^-1cdot boldsymbol Kcdot boldsymbol A)cdot boldsymbol d

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

boldsymbol K=boldsymbol A^-1cdot boldsymbol Kcdot boldsymbol A=boldsymbol A^Tcdot boldsymbol Kcdot boldsymbol A

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

underline underline boldsymbol A=beginbmatrixA_11&A_12&A_13\A_21&A_22&A_23\A_31&A_32&A_33endbmatrix~.

Therefore the symmetry condition can be written in matrix form as

underline underline boldsymbol K=underline underline boldsymbol A^T~underline underline boldsymbol K~underline underline boldsymbol A

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

underline underline boldsymbol A_1=beginbmatrix-1&0&0\0&1&0\0&0&1endbmatrix~;~~underline underline boldsymbol A_2=beginbmatrix1&0&0\0&-1&0\0&0&1endbmatrix~;~~underline underline boldsymbol A_3=beginbmatrix1&0&0\0&1&0\0&0&-1endbmatrix

It can be shown that if the matrix $underline underline boldsymbol K$ 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 $underline underline boldsymbol A_3$ about the $1-2,$ plane. Then we have

underline underline boldsymbol K=underline underline boldsymbol A_3^T~underline underline boldsymbol K~underline underline boldsymbol A_3=beginbmatrixK_11&K_12&-K_13\K_21&K_22&-K_23\-K_31&-K_32&K_33endbmatrix

The above relation implies that $K_13=K_23=K_31=K_32=0$ . Next consider a reflection $underline underline boldsymbol A_2$ about the $1-3,$ plane. We then have

underline underline boldsymbol K=underline underline boldsymbol A_2^T~underline underline boldsymbol K~underline underline boldsymbol A_2=beginbmatrixK_11&-K_12&0\-K_21&K_22&0\0&0&K_33endbmatrix

That implies that $K_12=K_21=0$ . Therefore the material properties of an orthotropic material are described by the matrix

$underline underline boldsymbol K=beginbmatrixK_11&0&0\0&K_22&0\0&0&K_33endbmatrix$

Orthotropy in linear elasticity

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^[3]

boldsymbol sigma =mathsf ccdot boldsymbol varepsilon

where $boldsymbol sigma$ is the stress tensor, $boldsymbolvarepsilon$ is the strain tensor, and $mathsfc$ 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

sigma _ij=c_ijkell ~varepsilon _kell

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

c_ijkell =c_jikell ~,~~c_ijkell =c_ijell k~,~~c_ijkell =c_kell ij~.

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

beginbmatrixsigma _11\sigma _22\sigma _33\sigma _23\sigma _31\sigma _12endbmatrix=beginbmatrixc_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_1212endbmatrixbeginbmatrixvarepsilon _11\varepsilon _22\varepsilon _33\2varepsilon _23\2varepsilon _31\2varepsilon _12endbmatrix

An alternative representation in Voigt notation is

beginbmatrixsigma _1\sigma _2\sigma _3\sigma _4\sigma _5\sigma _6endbmatrix=beginbmatrixC_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_66endbmatrixbeginbmatrixvarepsilon _1\varepsilon _2\varepsilon _3\varepsilon _4\varepsilon _5\varepsilon _6endbmatrix

or

underline underline boldsymbol sigma =underline underline mathsf C~underline underline boldsymbol varepsilon

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

Condition for material symmetry

The stiffness matrix $underline underline mathsf C$ 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 $3times 3$ matrix $underline underline mathbf A$ given by

underline underline mathbf A=beginbmatrixA_11&A_12&A_13\A_21&A_22&A_23\A_31&A_32&A_33endbmatrix~.

In Voigt notation, the transformation matrix for the stress tensor can be expressed as a $6times 6$ matrix $underline underline mathsf A_sigma$ given by^[4]

underline underline mathsf A_sigma =beginbmatrixA_11^2&A_12^2&A_13^2&2A_12A_13&2A_11A_13&2A_11A_12\A_21^2&A_22^2&A_23^2&2A_22A_23&2A_21A_23&2A_21A_22\A_31^2&A_32^2&A_33^2&2A_32A_33&2A_31A_33&2A_31A_32\A_21A_31&A_22A_32&A_23A_33&A_22A_33+A_23A_32&A_21A_33+A_23A_31&A_21A_32+A_22A_31\A_11A_31&A_12A_32&A_13A_33&A_12A_33+A_13A_32&A_11A_33+A_13A_31&A_11A_32+A_12A_31\A_11A_21&A_12A_22&A_13A_23&A_12A_23+A_13A_22&A_11A_23+A_13A_21&A_11A_22+A_12A_21endbmatrix

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

underline underline mathsf A_varepsilon =beginbmatrixA_11^2&A_12^2&A_13^2&A_12A_13&A_11A_13&A_11A_12\A_21^2&A_22^2&A_23^2&A_22A_23&A_21A_23&A_21A_22\A_31^2&A_32^2&A_33^2&A_32A_33&A_31A_33&A_31A_32\2A_21A_31&2A_22A_32&2A_23A_33&A_22A_33+A_23A_32&A_21A_33+A_23A_31&A_21A_32+A_22A_31\2A_11A_31&2A_12A_32&2A_13A_33&A_12A_33+A_13A_32&A_11A_33+A_13A_31&A_11A_32+A_12A_31\2A_11A_21&2A_12A_22&2A_13A_23&A_12A_23+A_13A_22&A_11A_23+A_13A_21&A_11A_22+A_12A_21endbmatrix

It can be shown that $underline underline mathsf A_varepsilon ^T=underline underline mathsf A_sigma ^-1$ .

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

$underline underline mathsf C=underline underline mathsf A_varepsilon ^T~underline underline mathsf C~underline underline mathsf A_varepsilon$

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

underline underline mathbf A_1=beginbmatrix-1&0&0\0&1&0\0&0&1endbmatrix~;~~underline underline mathbf A_2=beginbmatrix1&0&0\0&-1&0\0&0&1endbmatrix~;~~underline underline mathbf A_3=beginbmatrix1&0&0\0&1&0\0&0&-1endbmatrix

We can show that if the matrix $underline underline mathsf C$ 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 $underline underline mathbf A_3$ about the $1-2,$ plane, then we have

underline underline mathsf A_varepsilon =beginbmatrix1&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&1endbmatrix

Then the requirement $underline underline mathsf C=underline underline mathsf A_varepsilon ^T~underline underline mathsf C~underline underline mathsf A_varepsilon$ implies that^[4]

beginbmatrixC_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_66endbmatrix=beginbmatrixC_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_66endbmatrix

The above requirement can be satisfied only if

C_14=C_15=C_24=C_25=C_34=C_35=C_46=C_56=0~.

Let us next consider the reflection $underline underline mathbf A_2$ about the $1-3,$ plane. In that case

underline underline mathsf A_varepsilon =beginbmatrix1&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&-1endbmatrix

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

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

$underline underline mathsf C=beginbmatrixC_11&C_12&C_13&0&0&0\C_12&C_22&C_23&0&0&0\C_13&C_23&C_33&0&0&0\0&0&0&C_44&0&0\0&0&0&0&C_55&0\0&0&0&0&0&C_66endbmatrix$

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

underline underline mathsf S=beginbmatrixtfrac 1E_rm 1&-tfrac nu _rm 21E_rm 2&-tfrac nu _rm 31E_rm 3&0&0&0\-tfrac nu _rm 12E_rm 1&tfrac 1E_rm 2&-tfrac nu _rm 32E_rm 3&0&0&0\-tfrac nu _rm 13E_rm 1&-tfrac nu _rm 23E_rm 2&tfrac 1E_rm 3&0&0&0\0&0&0&tfrac 1G_rm 23&0&0\0&0&0&0&tfrac 1G_rm 31&0\0&0&0&0&0&tfrac 1G_rm 12\endbmatrix

where $E_rm i,$ is the Young's modulus along axis $i$ , $G_rm ij,$ is the shear modulus in direction $j$ on the plane whose normal is in direction $i$ , and $nu_rm ij,$ is the Poisson's ratio that corresponds to a contraction in direction $j$ when an extension is applied in direction $i$ .

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 boldsymbol varepsilon =underline underline mathsf S~underline underline boldsymbol sigma

where the compliance matrix $underline underline mathsf S$ is given by

underline underline mathsf S=beginbmatrixS_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_66endbmatrix

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.,

Delta _k:=det(underline underline mathsf S_k)>0

where $underline underline mathsf S_k$ is the $ktimes k$ principal submatrix of $underline underline mathsf S$ .

Then,

beginalignedDelta _1>0&implies quad S_11>0\Delta _2>0&implies quad S_11S_22-S_12^2>0\Delta _3>0&implies quad (S_11S_22-S_12^2)S_33-S_11S_23^2+2S_12S_23S_13-S_22S_13^2>0\Delta _4>0&implies quad S_44Delta _3>0implies S_44>0\Delta _5>0&implies quad S_44S_55Delta _3>0implies S_55>0\Delta _6>0&implies quad S_44S_55S_66Delta _3>0implies S_66>0endaligned

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

S_11>0~,~~S_22>0~,~~S_33>0~,~~S_44>0~,~~S_55>0~,~~S_66>0

or

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 $nu _ij$ .^[6]

References

^ 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

^ Milton, G. W., 2002, The Theory of Composites, Cambridge University Press.

^ Lekhnitskii, S. G., 1963, Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day Inc.

^ ^a^b^c^d Slawinski, M. A., 2010, Waves and Rays in Elastic Continua: 2nd Ed., World Scientific. http://samizdat.mines.edu/wavesandrays/WavesAndRays.pdf^{[permanent dead link]}

^ Boresi, A. P, Schmidt, R. J. and Sidebottom, O. M., 1993, Advanced Mechanics of Materials, Wiley.

^ ^a^b 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.

^ Ting, T. C. T. (1996), "Positive definiteness of anisotropic elastic constants", Mathematics & Mechanics of Solids, 1 (3): 301–314, doi:10.1177/108128659600100302.mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em.

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Orthotropic material Contents Orthotropy in physics Orthotropy in linear elasticity See also References Further reading Navigation menu10.1177/108128659600100302Orthotropy modeling equationsHooke's law for orthotropic materials

Contents

Orthotropy in physics

Anisotropic material relations

Condition for material symmetry

Orthotropic material properties

Orthotropy in linear elasticity

Anisotropic elasticity

Condition for material symmetry

Stiffness and compliance matrices in orthotropic elasticity

Bounds on the moduli of orthotropic elastic materials

See also

References

Further reading

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