If we utilize Einstein's summation convention, we can leave out the summation symbol and get: This can be done by noting the angle between each axis of the new coordinate system and each axis of the new coordinate system altogether there will be 9 angles, three of which are illustrated in Figure 2: If we have a vector P with components p 1, p 2, p 3 along the coordinate axes X 1, X 2, X 3 and we want to write P in terms of p′ 1, p′ 2, p′ 3 along new coordinate axes Z 1, Z 2, Z 3, we first need to describe how the coordinate systems are related to each other. As an example we will consider the transformation of a first rank tensor which is a vector. The elasticity of single crystals is described by a fourth rank tensor.Īs mentioned above, it is often desirable to know the value of a tensor property in a new coordinate system, so the tensor needs to be "transformed" from the original coordinate system to the new one. A fourth rank tensor is a four-dimensional array of numbers. Piezoelectricity is described by a third rank tensor. A third rank tensor would look like a three-dimensional matrix a cube of numbers. Stress, strain, thermal conductivity, magnetic susceptibility and electrical permittivity are all second rank tensors. A second rank tensor looks like a typical square matrix.
A zero rank tensor is a scalar, a first rank tensor is a vector a one-dimensional array of numbers. Tensors are referred to by their "rank" which is a description of the tensor's dimension. So you can see that even if E 1 is the only non-zero value in the electric field, all the components of D i may be non-zero.
D i can be calculated from E j as follows:ĭ 1 = k oK 11E 1 + k oK 12E 2 + k oK 13E 3 D 2 = k oK 21E 1 + k oK 22E 2 + k oK 23E 3 D 3 = k oK 31E 1 + k oK 32E 2 + k oK 33E 3 The dielectric constants "maps" the electric field E j into the electric displacement D i : The refractive index in any given direction through the crystal is governed by the dielectric constant K ij which is a tensor. You are probably familiar with the optical indicatrix which is an ellipsoid constructed on the three principle refractive indices. Another familiar tensor property is electrical permittivity which gives rise to birefringence in polarized light microscopy. Tensor math allows you to calculate the stresses acting on the crystallographic planes by transforming the stress tensor from one coordinate system to another.
However you may be really interested in understanding the stresses acting on various crystallographic planes, which are best viewed in terms of the crystallographic coordinates. It is easy to calculate the values in the stress tensor in the coordinate system tied to the apparatus. For example, let's say you measure the forces imposed on a single crystal in a deformation apparatus. Tensor math allows us to solve problems that involve tensors. These components form a second rank tensor the stress tensor (Figure 1). Any arbitrary stress state can be decomposed into 9 components (labeled σ ij). By dividing by the surface area over which the forces are acting, the stresses on the cube can be obtained. Figure 1 below, illustrates a unit cube of material with forces acting on it in three dimensions. Pressure is isotropic, but if a material has finite strength, it can support different forces applied in different directions. Stress, like pressure is defined as force per unit area. A simple example of a geophysically relevant tensor is stress. We use tensors as a tool to deal with more this complex situation and because single crystal properties are important for understanding the bulk behavior of rocks (and Earth), we wind up dealing with tensors fairly often in mineral physics.Ī tensor is a multi-dimensional array of numerical values that can be used to describe the physical state or properties of a material. For this reason properties such as the elasticity and thermal expansivity cannot be expressed as scalars. If one heats a block of glass it will expand by the same amount in each direction, but the expansion of a crystal will differ depending on whether one is measuring parallel to the a-axis or the b-axis. Many physical properties of crystalline materials are direction dependent because the arrangement of the atoms in the crystal lattice are different in different directions.
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