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List of Physical Quantity Symbols¶
This page lists the symbols for physical quantities used in the FrontISTR theory manual. For notation conventions (bold notation for vectors and tensors, Einstein summation convention, and 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 \(\Omega\) |
| \(\Gamma_B, \Gamma_{0B}\) | Geometric (displacement) boundary: boundary in the current/reference configuration on which \(\boldsymbol{u} = \bar{\boldsymbol{u}}\) is prescribed |
| \(\Gamma_t, \Gamma_{0t}\) | Mechanical boundary: boundary in the current/reference configuration on which \(\boldsymbol{\sigma}\boldsymbol{n} = \bar{\boldsymbol{t}}\) (or \(\boldsymbol{P}\boldsymbol{N} = \bar{\boldsymbol{t}}\)) is prescribed |
| \(\boldsymbol{n}\) | Outward unit normal vector to a surface in the current configuration |
| \(\boldsymbol{N}\) | Outward unit normal vector to a surface in the reference configuration |
| \(t\) | Time |
Convention: Physical quantities in the reference configuration are denoted by uppercase letters or a subscript \(0\), while quantities in the current configuration are denoted by lowercase letters.
Time and Increments (Incremental Analysis)¶
| Symbol | Description |
|---|---|
| \(\Delta t\) | Time increment: \(\Delta t = t_{n+1} - t_n\) |
| \(^{t}(\cdot)\) | Physical quantity at time \(t\) (left superscript). Example: \(^{t}\boldsymbol{\sigma}\) is the Cauchy stress at time \(t\) |
| \(^{t}\Omega\) | Domain of the current configuration at time \(t\) |
| \(^{t}\Gamma\) | Boundary of the current configuration at time \(t\) |
Convention: In incremental analysis, the state up to time \(t\) is treated as known, and the state at time \(t + \Delta t\) is solved as unknown. The left superscript is omitted when the time need not be stated explicitly. A formulation that uses the reference configuration \(\Omega_0\) as the reference is called the Total Lagrange formulation, while a formulation that uses the current configuration \(^{t}\Omega\) at the start of the increment as the reference is called the Updated Lagrange formulation.
Displacement, Velocity, Acceleration, and Body Force¶
| Symbol | Description |
|---|---|
| \(\boldsymbol{u}\) | Displacement vector: \(\boldsymbol{u} = \boldsymbol{x} - \boldsymbol{X}\) |
| \(\bar{\boldsymbol{u}}\) | Displacement prescribed on the geometric boundary \(\Gamma_B\) |
| \(\delta \boldsymbol{u}\) | Virtual displacement (test function): \(\delta \boldsymbol{u} = \boldsymbol{0}\) on \(\Gamma_B\) |
| \(\boldsymbol{v}\) | Velocity vector: \(\boldsymbol{v} = \dot{\boldsymbol{u}}\) |
| \(\boldsymbol{a}\) | Acceleration vector: \(\boldsymbol{a} = \dot{\boldsymbol{v}}\) |
| \(\boldsymbol{g}\) | Body force (per unit mass) |
| \(\boldsymbol{t}\) | Traction vector (per unit area): \(\boldsymbol{t} = \boldsymbol{\sigma}\boldsymbol{n}\) |
| \(\bar{\boldsymbol{t}}\) | Traction prescribed on the mechanical boundary \(\Gamma_t\) (Neumann data) |
Differential Operators¶
| Symbol | Description |
|---|---|
| \(\dot{(\cdot)}\) | Material time derivative (time derivative following the same material point): \(\dot{A} \equiv DA/Dt\) |
| \(\nabla_X\) | Reference-configuration gradient (gradient with respect to material coordinates \(\boldsymbol{X}\)): \(\nabla_X = \partial / \partial \boldsymbol{X}\) |
| \(\nabla_x\) | Current-configuration gradient (gradient with respect to spatial coordinates \(\boldsymbol{x}\)): \(\nabla_x = \partial / \partial \boldsymbol{x}\) |
| \(\nabla_S\) | Symmetric gradient operator: \(\nabla_S \boldsymbol{u} = \tfrac{1}{2}(\nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^T)\) |
Note: \(\nabla_X\) and \(\nabla_x\) are related by \(\nabla_X = \boldsymbol{F}^T \nabla_x\). A gradient without a subscript, \(\nabla\), is used when the configuration is clear from context or when the distinction disappears, as in small-deformation problems.
Deformation Tensors¶
| Symbol | Description |
|---|---|
| \(\boldsymbol{F}\) | Deformation gradient tensor: \(F_{ij} = \partial x_i / \partial X_j\) |
| \(J\) | Volume ratio: \(J = dv/dV = \det \boldsymbol{F}\) |
| \(\boldsymbol{C}\) | 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): \(\boldsymbol{e} = \tfrac{1}{2}(\boldsymbol{I}-\boldsymbol{b}^{-1})\) |
| \(\boldsymbol{A}_{(L)}\) | Linear part of the Almansi strain tensor (symmetric displacement gradient in the current configuration): \(\boldsymbol{A}_{(L)} = \tfrac{1}{2}(\nabla_x \boldsymbol{u} + (\nabla_x \boldsymbol{u})^T)\) |
| \(\boldsymbol{\varepsilon}\) | Infinitesimal strain tensor (linear approximation): \(\varepsilon_{ij} = \tfrac{1}{2}(\partial u_i/\partial x_j + \partial u_j/\partial x_i)\) |
Note: \(\boldsymbol{E}\) is used primarily in the Total Lagrange formulation, while \(\boldsymbol{D}\) is used primarily 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 among the 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 \((\boldsymbol{S}, \boldsymbol{E})\), while the Updated Lagrange formulation uses the pair \((\boldsymbol{\sigma}, \boldsymbol{D})\).
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 Constants (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 (fourth order): \(\boldsymbol{S} = \boldsymbol{\mathsf{C}}:\boldsymbol{E}\), components \(C_{ijkl}\) |
| \(D\) (or \(\hat{\tilde{C}}\)) | Material matrix (Voigt notation, \(6\times6\) in 3D): \(\hat{\sigma} = D\,\hat{\varepsilon}\) |
Components of the elasticity tensor for an isotropic linear elastic material:
\[
C_{ijkl} = \lambda\,\delta_{ij}\delta_{kl} + \mu\,(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk})
\]
Hyperelastic Materials¶
| Symbol | Description |
|---|---|
| \(W(\boldsymbol{C})\) | Elastic potential function (strain-energy density function) |
| \(I_1, I_2, I_3\) | 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) |
| \(C_1, C_2\) | Material constants of the Mooney-Rivlin model |
| \(D\) | Material constant related to volumetric elasticity |
Elastoplastic Materials¶
| Symbol | Description |
|---|---|
| \(\boldsymbol{D}^e\) | Elastic component of the rate-of-deformation tensor |
| \(\boldsymbol{D}^p\) | Plastic component 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 (\(\Theta = F\) for an 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 |
|---|---|
| \(\Omega^h, \Omega_0^h\) | Domains in the current and reference configurations approximated by finite element discretization: \(\Omega^h = \bigcup_e \Omega^e\) |
| \(\Omega^e, \Omega^e_0\) | Element domain in the current and reference configurations |
| \(\Gamma^e_t, \Gamma^e_{0t}\) | Portion of the element boundary belonging to the mechanical boundary |
| \(\boldsymbol{r}\) | Natural coordinates (element-local coordinates) |
| \(\boldsymbol{r}_\alpha\) | Point in natural coordinates corresponding to node \(\alpha\) |
| \(N_\alpha^e(\boldsymbol{r})\) | Shape function corresponding to node \(\alpha\) of element \(e\) |
| \(n_e\) | Number of nodes constituting an element |
| \(n_g\) | Total number of global nodes |
| \(\boldsymbol{X}^e_\alpha, \boldsymbol{u}^e_\alpha\) | Coordinates and displacement of element node \(\alpha\) |
| \(\boldsymbol{X}^e, \boldsymbol{u}^e\) | Element nodal coordinate and displacement vectors: \(\boldsymbol{X}^e = (\boldsymbol{X}^{eT}_1, \ldots, \boldsymbol{X}^{eT}_{n_e})^T\), etc. |
| \(\boldsymbol{X}^n, \boldsymbol{u}^n\) | Global nodal coordinate and displacement vectors (ordered by node number, then degree of freedom) |
| \(\boldsymbol{B}\) | Strain-displacement relation matrix (B-matrix) |
| \(\boldsymbol{N}_\alpha, \boldsymbol{N}\) | Shape-function matrix: \(\boldsymbol{N}_\alpha\) is a \(d \times d\) block with the shape function \(N_\alpha^e\) for node \(\alpha\) on its diagonal, and \(\boldsymbol{N} = [\boldsymbol{N}_1, \ldots, \boldsymbol{N}_{n_e}]\). \(\delta\boldsymbol{u} = \boldsymbol{N}\, \delta\boldsymbol{u}^e\) |
| \(\boldsymbol{K}^e\) | Element stiffness matrix |
| \(\boldsymbol{K}^e_X, \boldsymbol{K}^e_x\) | Integrands of the element stiffness matrix (expressed in the reference and current configurations): \(\boldsymbol{K}^e = \int_{\Omega^e_0} \boldsymbol{K}^e_X\, dV = \int_{\Omega^e} \boldsymbol{K}^e_x\, dv\) |
| \(\boldsymbol{Q}^e\) (TL formulation), \(\boldsymbol{q}^e\) (UL formulation) | Element internal force vector |
| \(\boldsymbol{F}^e_\alpha, \boldsymbol{F}^e\) | Nodal external force acting on element node \(\alpha\), and the element nodal external-force vector: \(\boldsymbol{F}^e = (\boldsymbol{F}^{eT}_1, \ldots, \boldsymbol{F}^{eT}_{n_e})^T\) |
| \(\boldsymbol{Q}, \boldsymbol{F}\) | Global internal-force vector and global external-force vector: \(\boldsymbol{Q} = (\boldsymbol{Q}^T_1, \ldots, \boldsymbol{Q}^T_{n_g})^T\); similarly for \(\boldsymbol{F}\) |
| \(\boldsymbol{Q}_{i_g}, \boldsymbol{F}_{i_g}\) | Nodal internal force and nodal external force at global node \(i_g\) (nodal blocks of \(\boldsymbol{Q}, \boldsymbol{F}\)) |
| \(\boldsymbol{K}\) | Global tangent stiffness matrix: \(\boldsymbol{K} = \partial \boldsymbol{Q}/\partial \boldsymbol{u}\). Matrix formed by placing the \(3\times 3\) block \(\boldsymbol{K}_{i_gi_h}\) at row \(i_g\), column \(i_h\) |
| \(\boldsymbol{R}_i\) | Residual vector in Newton-Raphson iteration: \(\boldsymbol{R}_i = \boldsymbol{F} - \boldsymbol{Q}(\boldsymbol{u}_n + \Delta\boldsymbol{u})\) |
| \(d\boldsymbol{u}_i\) | Correction in Newton-Raphson iteration (obtained by solving the linear equation \(\boldsymbol{K}_i\, d\boldsymbol{u}_i = \boldsymbol{R}_{i-1}\) at iteration \(i\)) |
| \(i_g\) | Global node number (\(1 \leq i_g \leq n_g\)) |
| \(\mathrm{gdx}(e, \alpha)\) | Global node number corresponding to local node number \(\alpha\) of element \(e\): \(\mathrm{gdx}(e, \alpha) = i_g\) |
| \(\mathcal{E}(i_g)\) | Set of \((e, \alpha)\) pairs corresponding to global node number \(i_g\): \(\mathcal{E}(i_g) = \{ (e, \alpha) \mid \mathrm{gdx}(e, \alpha) = i_g \}\). Used in assembly operations |
| \(\mathcal{E}^2(i_g, i_h)\) | Set of \((e, \alpha, \beta)\) tuples corresponding to the pair of global node numbers \((i_g, i_h)\); used for stiffness-matrix assembly |
| \(\alpha, \beta, \gamma, \ldots\) | Indices for nodes constituting an element |
| \(i, j, k, l, \ldots\) | Indices for degrees of freedom (\(1, 2, 3\) in 3D) |
Advanced Elements and Structural Elements¶
| Symbol | Description |
|---|---|
| \(\bar{\boldsymbol{B}}\) | Strain-displacement relation matrix with its volumetric part modified by the B-bar method (the volumetric component is replaced by the B-matrix evaluated at the element center) |
| \(\bar{\boldsymbol{F}}\) | Deformation gradient with its volumetric part modified by the F-bar method: \(\bar{\boldsymbol{F}} = (J_0/J)^{1/3} \boldsymbol{F}\) |
| \(J_0\) | Volume ratio of the deformation gradient evaluated at the element center \(\boldsymbol{r}=\boldsymbol{0}\): \(J_0 = \det \boldsymbol{F}(\boldsymbol{0})\) |
| \(\boldsymbol{\alpha}\) | Internal degree-of-freedom vector for an incompatible-mode element (coefficients of additional displacement modes that do not enforce continuity on the element boundary) |
| \(M_k(\boldsymbol{r})\) | Shape functions for incompatible modes: \(M_1=1-\xi^2\), \(M_2=1-\eta^2\), \(M_3=1-\zeta^2\) |
| \(h\) | Shell-element thickness |
| \(A\) | Cross-sectional area of a beam element |
| \(I\) | Second moment of area of a beam-element cross section |
| \(G\) | Shear modulus: \(G = E/[2(1+\nu)]\) |
Related Topics¶
- Tensor Notation and Mathematical Foundations — Rules for bold notation, Einstein summation convention, and Voigt notation
- Motion, Deformation, and Strain — Definitions of configurations, deformation gradient, and strain
- Motion, Deformation, and Strain — Derivation of the governing equations
- Linear Elasticity — Explicit components of material matrix \(D\)