Editing Anisotropy (section)

===Materials science and engineering===
Anisotropy, in [[materials science]], is a material's directional dependence of a [[physical property]]. This is a critical consideration for [[materials selection]] in engineering applications. A material with physical properties that are symmetric about an axis that is normal to a plane of isotropy is called a [[transverse isotropy|transversely isotropic material]]. [[Tensor]] descriptions of material properties can be used to determine the directional dependence of that property. For a [[monocrystalline]] material, anisotropy is associated with the crystal symmetry in the sense that more symmetric crystal types have fewer independent coefficients in the tensor description of a given property.<ref>{{cite book |last1=Newnham |first1=Robert E. |title=Properties of Materials: Anisotropy, Symmetry, Structure |publisher=Oxford University Press |isbn=978-0198520764 |edition=1st}}</ref><ref>{{cite book |last1=Nye |first1=J.F. |title=Physical Properties of Crystals |publisher=Clarendon Press |edition=1st}}</ref> When a material is [[polycrystalline]], the directional dependence on properties is often related to the processing techniques it has undergone. A material with randomly oriented grains will be isotropic, whereas materials with [[texture (crystalline)|texture]] will be often be anisotropic. Textured materials are often the result of processing techniques like [[cold rolling]], [[wire drawing]], and [[heat treatment]].

Mechanical properties of materials such as [[Young's modulus]], [[ductility]], [[yield strength]], and high-temperature [[creep (deformation)|creep rate]], are often dependent on the direction of measurement.<ref>{{cite book |last1=Courtney |first1=Thomas H. |title=Mechanical Behavior of Materials |publisher=Waveland Pr Inc |isbn=978-1577664253 |edition=2nd |year=2005}}</ref> Fourth-rank [[tensor]] properties, like the elastic constants, are anisotropic, even for materials with cubic symmetry. The Young's modulus relates stress and strain when an isotropic material is elastically deformed; to describe elasticity in an anisotropic material, [[stiffness]] (or compliance) tensors are used instead.

In metals, anisotropic elasticity behavior is present in all single crystals with three independent coefficients for cubic crystals, for example. For face-centered cubic materials such as nickel and copper, the stiffness is highest along the <111> direction, normal to the close-packed planes, and smallest parallel to <100>. Tungsten is so nearly isotropic at room temperature that it can be considered to have only two stiffness coefficients; aluminium is another metal that is nearly isotropic.

For an isotropic material, <math display="block">G = E/[2(1 + \nu)], </math> where <math> G </math> is the [[shear modulus]], <math> E </math> is the [[Young's modulus]], and <math> \nu </math> is the material's [[Poisson's ratio]]. Therefore, for cubic materials, we can think of anisotropy, <math> a_r </math>, as the ratio between the empirically determined shear modulus for the cubic material and its (isotropic) equivalent:
<math display="block">a_r = \frac{G}{E/[2(1 + \nu)]} = \frac{2(1+\nu)G}{E} \equiv \frac{2 C_{44}}{C_{11} - C_{12}}.</math>

The latter expression is known as the [[Zener ratio]], <math> a_r </math>, where <math>C_{ij}</math> refers to [[Hooke's Law|elastic constants]] in [[Voigt notation|Voigt (vector-matrix) notation]]. For an isotropic material, the ratio is one.

Limitation of the [[Zener ratio]] to cubic materials is waived in the Tensorial anisotropy index A<sup>T</sup> <ref>{{cite journal |last1=Sokołowski |first1=Damian |last2=Kamiński |first2=Marcin |date=2018-09-01 |title=Homogenization of carbon/polymer composites with anisotropic distribution of particles and stochastic interface defects |journal=Acta Mechanica |language=en |volume=229 |issue=9 |pages=3727–3765 |doi=10.1007/s00707-018-2174-7 |s2cid=126198766 |issn=1619-6937 |doi-access=free}}</ref> that takes into consideration all the 27 components of the fully anisotropic stiffness tensor. It is composed of two major parts <math>A^I</math>and <math>A^A </math>, the former referring to components existing in cubic tensor and the latter in anisotropic tensor so that <math>A^T = A^I+A^A .</math> This first component includes the modified Zener ratio and additionally accounts for directional differences in the material, which exist in [[Orthotropic material|orthotropic]] material, for instance. The second component of this index <math>A^A </math> covers the influence of stiffness coefficients that are nonzero only for non-cubic materials and remains zero otherwise.

Fiber-reinforced or layered [[composite material]]s exhibit anisotropic mechanical properties, due to orientation of the reinforcement material. In many fiber-reinforced composites like carbon fiber or glass fiber based composites, the weave of the material (e.g. unidirectional or plain weave) can determine the extent of the anisotropy of the bulk material.<ref>{{cite web |title=Fabric Weave Styles |url=https://compositeenvisions.com/fabric-weave-styles/ |website=Composite Envisions |access-date=23 May 2019}}</ref> The tunability of orientation of the fibers allows for application-based designs of composite materials, depending on the direction of stresses applied onto the material.

Amorphous materials such as glass and polymers are typically isotropic. Due to the highly randomized orientation of [[macromolecule]]s in polymeric materials, [[polymer]]s are in general described as isotropic. However, [[mechanically gradient polymers]] can be engineered to have directionally dependent properties through processing techniques or introduction of anisotropy-inducing elements. Researchers have built composite materials with aligned fibers and voids to generate anisotropic [[hydrogel]]s, in order to mimic hierarchically ordered biological soft matter.<ref>{{cite journal |last1=Sano |first1=Koki |last2=Ishida |first2=Yasuhiro |last3=Aida |first3=Tazuko |title=Synthesis of Anisotropic Hydrogels and Their Applications |journal=Angewandte Chemie International Edition |date=16 October 2017 |volume=57 |issue=10 |pages=2532–2543 |doi=10.1002/anie.201708196 |pmid=29034553}}</ref> 3D printing, especially Fused Deposition Modeling, can introduce anisotropy into printed parts. This is because FDM is designed to extrude and print layers of thermoplastic materials.<ref>{{cite journal |last1=Wang |first1=Xin |last2=Jiang |first2=Man |last3=Gou |first3=Jihua |last4=Hui |first4=David |title=3D printing of polymer matrix composites: A review and prospective |journal=Composites Part B: Engineering |date=1 February 2017 |volume=110 |pages=442–458 |doi=10.1016/j.compositesb.2016.11.034}}</ref> This creates materials that are strong when tensile stress is applied in parallel to the layers and weak when the material is perpendicular to the layers.