Notation
Here are a few points on the notation used in the remainder of the document.
Vectors will be represented for the most part by lowercase bold-faced latin letters, e.g. $a$, $b$, ... except for out of departure from convention such as $\mathbf{E}$ for the electric field vector.
Second order tensors will be represented for the most part by uppercase bold-faced latin letters, e.g. $\mathbf{A}$, $\mathbf{B}$, ... except for out of departure form convention such as $\mathbf{\sigma}$ for the Cauchy stress.
Common Variables
| Name | Symbol | Type |
|---|---|---|
| Acoustic tensor | $\mathbf{A}$ | Second order tensor |
| Cauchy Stress | $\sigma$ | Second order tensor (spatial) |
| Deformation gradient | $\mathbf{F}$ | Two point tensor |
| Density in current frame | $\rho$ | Scalar |
| Density in reference frame | $\rho_R$ | Scalar |
| Displacement | $\mathbf{u}$ | Vector |
| Displacement Gradient | $\nabla\mathbf{u}$ | Second order Tensor |
| Heat flux | $\mathbf{q}$ | Vector |
| Helmholtz free energy | $\psi$ | Scalar |
| Material tangent | $\mathbb{A}$ | Fourth order tensor (two bases) |
| Normal vector | $\mathbf{n}$ | Vector |
| PK1 Stress | $\mathbf{P}$ | Two point tensor |
| PK2 Stress | $\mathbf{S}$ | Second order tensor (material) |
| State Variables | $\mathcal{Z}$ | Arbitrary |
| Temperature | $\theta$ | Scalar |
| Temperature Gradient | $\nabla\theta$ | Vector |