List of Physical Quantity Symbols¶

This page lists the symbols for physical quantities used in the FrontISTR theory manual. For notation conventions (bold vectors/tensors, Einstein summation convention, Voigt notation), see Tensor Notation and Mathematical Foundations.

Configurations and Coordinate Systems¶

Symbol	Description
\(\boldsymbol{X}\)	Position vector of a material point in the reference (initial) configuration (material coordinates)
\(\boldsymbol{x}\)	Position vector of a material point in the current configuration (spatial coordinates)
\(\phi(\boldsymbol{X}, t)\)	Motion mapping: \(\boldsymbol{x} = \phi(\boldsymbol{X}, t)\)
\(\Omega_0\)	Domain occupied by the body in the reference configuration
\(\Omega\)	Domain occupied by the body in the current configuration
\(\Gamma_0\)	Boundary of \(\Omega_0\)
\(\Gamma\)	Boundary of \(\Omega\)
\(\boldsymbol{n}\)	Outward unit normal vector on the surface in the current configuration
\(\boldsymbol{N}\)	Outward unit normal vector on the surface in the reference configuration
\(t\)	Time

Convention: Quantities in the reference configuration use uppercase letters or a subscript \(0\); quantities in the current configuration use lowercase letters.

Displacement, Velocity, Acceleration, and Body Force¶

Symbol	Description
\(\boldsymbol{u}\)	Displacement vector: \(\boldsymbol{u} = \boldsymbol{x} - \boldsymbol{X}\)
\(\boldsymbol{v}\)	Velocity vector: \(\boldsymbol{v} = \dot{\boldsymbol{u}}\)
\(\boldsymbol{a}\)	Acceleration vector: \(\boldsymbol{a} = \dot{\boldsymbol{v}}\)
\(\boldsymbol{g}\)	Body force per unit mass
\(\boldsymbol{t}\)	Surface traction vector (force per unit area): \(\boldsymbol{t} = \boldsymbol{\sigma}\boldsymbol{n}\)

Deformation Tensors¶

Symbol	Description
\(\boldsymbol{F}\)	Deformation gradient tensor: \(F_{ij} = \partial x_i / \partial X_j\)
\(J\)	Volume ratio (Jacobian): \(J = \det \boldsymbol{F}\)
\(\boldsymbol{C}\)	Right Cauchy-Green deformation tensor: \(\boldsymbol{C} = \boldsymbol{F}^T \boldsymbol{F}\)
\(\boldsymbol{b}\)	Left Cauchy-Green deformation tensor: \(\boldsymbol{b} = \boldsymbol{F}\boldsymbol{F}^T\)
\(\boldsymbol{L}\)	Velocity gradient tensor: \(L_{ij} = \partial v_i / \partial x_j = \dot{F}_{ik}F^{-1}_{kj}\)
\(\boldsymbol{D}\)	Rate of deformation tensor (symmetric part of \(\boldsymbol{L}\)): \(\boldsymbol{D} = \tfrac{1}{2}(\boldsymbol{L}+\boldsymbol{L}^T)\)
\(\boldsymbol{W}\)	Spin tensor (skew-symmetric part of \(\boldsymbol{L}\)): \(\boldsymbol{W} = \tfrac{1}{2}(\boldsymbol{L}-\boldsymbol{L}^T)\)

Strain Tensors¶

Symbol	Description
\(\boldsymbol{E}\)	Green-Lagrange strain tensor (reference configuration): \(\boldsymbol{E} = \tfrac{1}{2}(\boldsymbol{C}-\boldsymbol{I})\)
\(\boldsymbol{e}\)	Almansi strain tensor (current configuration): \(\boldsymbol{e} = \tfrac{1}{2}(\boldsymbol{I}-\boldsymbol{b}^{-1})\)
\(\boldsymbol{\varepsilon}\)	Infinitesimal strain tensor (linearized): \(\varepsilon_{ij} = \tfrac{1}{2}(\partial u_i/\partial x_j + \partial u_j/\partial x_i)\)

Note: \(\boldsymbol{E}\) is used primarily in the Total Lagrange formulation; \(\boldsymbol{D}\) is used in the Updated Lagrange formulation.

Stress Tensors¶

Symbol	Description
\(\boldsymbol{\sigma}\)	Cauchy stress tensor (true stress, current configuration): \(d\boldsymbol{f} = \boldsymbol{\sigma}\boldsymbol{n}\,d\Gamma\)
\(\boldsymbol{P}\)	First Piola-Kirchhoff stress tensor (nominal stress): \(d\boldsymbol{f} = \boldsymbol{P}\boldsymbol{N}\,d\Gamma_0\)
\(\boldsymbol{S}\)	Second Piola-Kirchhoff stress tensor (reference configuration, symmetric): \(\boldsymbol{F}^{-1}d\boldsymbol{f} = \boldsymbol{S}\boldsymbol{N}\,d\Gamma_0\)

Conversion relations between stress tensors:

\[ \boldsymbol{P} = \boldsymbol{F}\boldsymbol{S}, \qquad \boldsymbol{\sigma} = \frac{1}{J}\boldsymbol{F}\boldsymbol{S}\boldsymbol{F}^T = \frac{1}{J}\boldsymbol{P}\boldsymbol{F}^T \]

Convention: The Total Lagrange formulation uses the pair \((\boldsymbol{S}, \boldsymbol{E})\); the Updated Lagrange formulation uses \((\boldsymbol{\sigma}, \boldsymbol{D})\).

Density and Mass¶

Symbol	Description
\(\rho\)	Mass density in the current configuration
\(\rho_0\)	Mass density in the reference configuration

Mass conservation: \(\rho_0 = J\rho\).

Material Parameters (Linear Elasticity)¶

Symbol	Description
\(E\)	Young's modulus
\(\nu\)	Poisson's ratio
\(\lambda\)	First Lamé constant: \(\lambda = E\nu / [(1+\nu)(1-2\nu)]\)
\(\mu\)	Second Lamé constant (shear modulus): \(\mu = E / [2(1+\nu)]\)
\(\boldsymbol{\mathsf{C}}\)	Elasticity tensor (4th order): \(\boldsymbol{S} = \boldsymbol{\mathsf{C}}:\boldsymbol{E}\), components \(C_{ijkl}\)
\(D\) (or \(\hat{\tilde{C}}\))	Material matrix (Voigt notation, \(6\times6\) in 3D): \(\hat{\sigma} = D\,\hat{\varepsilon}\)

Isotropic linear elastic tensor components:

\[ C_{ijkl} = \lambda\,\delta_{ij}\delta_{kl} + \mu\,(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}) \]

Hyperelastic Materials¶

Symbol	Description
\(W(\boldsymbol{C})\)	Strain energy density function (elastic potential)
\(I_1, I_2, I_3\)	Principal invariants of the right Cauchy-Green tensor \(\boldsymbol{C}\)
\(\tilde{I}_1, \tilde{I}_2, \tilde{I}_3\)	Reduced invariants of \(\boldsymbol{C}\) (volumetric change separated)
\(C_1, C_2\)	Mooney-Rivlin model material constants
\(D\)	Material constant for volumetric elasticity

Elastoplastic Materials¶

Symbol	Description
\(\boldsymbol{D}^e\)	Elastic part of the rate of deformation tensor
\(\boldsymbol{D}^p\)	Plastic part of the rate of deformation tensor
\(F(\boldsymbol{\sigma}, \kappa)\)	Yield function
\(\kappa\)	Internal variable representing the plastic state (isotropic hardening variable)
\(\lambda^p\)	Plastic multiplier (plastic strain rate multiplier): \(\lambda^p \geq 0\)
\(\Theta\)	Plastic potential (equals \(F\) for associated flow rule)
\(\bar{\sigma}\)	Equivalent stress (e.g., von Mises stress)
\(\bar{\varepsilon}^p\)	Equivalent plastic strain
\(H(\bar{\varepsilon}^p)\)	Isotropic hardening function

Complementarity condition: \(\lambda^p F(\boldsymbol{\sigma}, \kappa) = 0\), \(\lambda^p \geq 0\), \(F \leq 0\).

Finite Element Method¶

Symbol	Description
\(\Omega^e, \Omega^e_0\)	Element domain in the current and reference configurations
\(\boldsymbol{r}\)	Natural coordinates (element-local coordinates)
\(N_\alpha(\boldsymbol{r})\)	Shape function associated with node \(\alpha\)
\(n_e\)	Number of nodes per element
\(n_g\)	Total number of global nodes
\(\boldsymbol{X}^e_\alpha, \boldsymbol{u}^e_\alpha\)	Coordinates and displacement of element node \(\alpha\)
\(\boldsymbol{u}^n\)	Global nodal displacement vector
\(\boldsymbol{B}\)	Strain-displacement matrix (B-matrix)
\(\boldsymbol{K}^e\)	Element stiffness matrix
\(\boldsymbol{Q}^e\) (TL), \(\boldsymbol{q}^e\) (UL)	Element internal force vector
\(\alpha, \beta, \gamma, \ldots\)	Indices for element nodes
\(i, j, k, l, \ldots\)	Indices for degrees of freedom (1, 2, 3 in 3D)