Denotation

Denotation

Material Properties

• E: Young's modulus
• lambda_: Lamé's first parameter
• mu: Lamé's second parameter
• nu: Poisson's ratio
• q: parameters (e.g. material properties)

Array Annotations

• C: number of cells / elements / faces / tetras / triangles
• D: number of Dirichlet boundary conditions
• F: number of DoF after applying Dirichlet boundary conditions
• N: number of DoF without boundary conditions, usually N = 3 * V
• Q: number of parameters
• V: number of points / vertices

Variables

• dV: volume per element
• F: deformation gradient
• h: shape function
• J: det(F)
• PK1: first Piola–Kirchhoff stress tensor
• Psi: energy density
• R, S: polar decomposition of F
• u: displacement
• W: energy
• x: position
• X: undeformed position

Basic Formulas

\[ \frac{\partial\Psi}{\partial x} = \frac{\partial\Psi}{\partial F} \frac{\partial F}{\partial x} \]

Useful Identities

\[ I_1 = \operatorname{tr}(S), \quad I_2 = \operatorname{tr}(F^T F), \quad I_3 = \det(F) \]