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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.
Glass and metals are examples of isotropic materials. [3] Common anisotropic materials include wood (because its material properties are different parallel to and perpendicular to the grain) and layered rocks such as slate. Isotropic materials are useful since they are easier to shape, and their behavior is easier to predict.
In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise it is a definite quadratic form . More explicitly, if q is a quadratic form on a vector space V over F , then a non-zero vector v in V is said to be isotropic if q ( v ) = 0 .
An important goal of micromechanics is predicting the anisotropic response of the heterogeneous material on the basis of the geometries and properties of the individual phases, a task known as homogenization. [3] Micromechanics allows predicting multi-axial responses that are often difficult to measure experimentally.
Each iteration of the Sierpinski triangle contains triangles related to the next iteration by a scale factor of 1/2. In affine geometry, uniform scaling (or isotropic scaling [1]) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions (isotropically).
The Tensorial Anisotropy Index A T [5] extends the Zener ratio for fully anisotropic materials and overcomes the limitation of the AU that is designed for materials exhibiting internal symmetries of elastic crystals, which is not always observed in multi-component composites. It takes into consideration all the 21 coefficients of the fully ...
Isotropic lines have been used in cosmological writing to carry light. For example, in a mathematical encyclopedia, light consists of photons: "The worldline of a zero rest mass (such as a non-quantum model of a photon and other elementary particles of mass zero) is an isotropic line."
Additionally, all crystal structures, including the cubic crystal system, are anisotropic with respect to certain properties, and isotropic to others (such as density). [ 4 ] The anisotropy of a crystal’s properties depends on the rank of the tensor used to describe the property, as well as the symmetries present within the crystal.