19 Appendix A — Notation & Conventions
Master symbol list, conventions, and typography at a glance
This appendix consolidates the notation used throughout the course. Each symbol is listed with its meaning and the lecture in which it is first introduced (where students can find the definition in context). Conventions collected from chapter-level callouts are gathered in §A.4; the typography rules governing the whole course are summarised in §A.5.
This is a reference, not a lecture — skim once on first reading, then return when a symbol is unfamiliar.
19.1 A.1 Symbols
19.1.1 Kinematics
| Symbol | Meaning | First in |
|---|---|---|
| \(\mathbf{X}\) | Material (reference) position | L03 |
| \(\mathbf{x}\) | Spatial (current) position | L03 |
| \(\mathbf{u}\) | Displacement field, \(\mathbf{u} = \mathbf{x} - \mathbf{X}\) | L03 |
| \(\mathbf{F}\) | Deformation gradient, \(\mathbf{F} = \partial\mathbf{x}/\partial\mathbf{X}\) | L03 |
| \(J\) | Jacobian, \(J = \det\mathbf{F}\) | L03 |
| \(\mathbf{C}\) | Right Cauchy-Green tensor, \(\mathbf{C} = \mathbf{F}^T\mathbf{F}\) (reference) | L03 |
| \(\mathbf{b}\) | Left Cauchy-Green / Finger tensor, \(\mathbf{b} = \mathbf{F}\mathbf{F}^T\) (current) | L05 |
| \(\bar{\mathbf{C}}, \bar{\mathbf{b}}\) | Isochoric (unimodular) parts, \(\det\bar{\mathbf{C}} = \det\bar{\mathbf{b}} = 1\) | L05 |
| \(\mathbf{E}\) | Green-Lagrange strain (reference, finite) | L02 |
| \(\mathbf{e}\) | Almansi-Euler strain (current, finite) | L02 |
| \(\boldsymbol{\varepsilon}\) | Infinitesimal (small) strain, \(\tfrac{1}{2}(\nabla\mathbf{u} + \nabla\mathbf{u}^T)\) | L03 |
| \(\boldsymbol{\varepsilon}^e\) | Elastic part of \(\boldsymbol{\varepsilon}\) (additive split) | L07 |
| \(\boldsymbol{\varepsilon}^p\) | Plastic part of \(\boldsymbol{\varepsilon}\) | L07 |
| \(\bar{\varepsilon}^p\) | Equivalent (accumulated) plastic strain | L07 |
| \(\boldsymbol{\varepsilon}^\text{th}\) | Thermal strain | L04 |
| \(\mathbf{F}^e, \mathbf{F}^p\) | Multiplicative split, \(\mathbf{F} = \mathbf{F}^e\mathbf{F}^p\) (finite-strain plasticity) | L04 |
| \(\mathbf{C}^e\) | Elastic part of right Cauchy-Green, \((\mathbf{F}^e)^T\mathbf{F}^e\) | L04 |
| \(\mathbf{R}, \mathbf{U}, \mathbf{V}\) | Polar decomposition: rotation, right stretch (ref.), left stretch (current) | L03 |
| \(\lambda_i\) | Principal stretches (eigenvalues of \(\mathbf{U}\) or \(\mathbf{V}\)) | L03 |
| \(\mathbf{L}\) | Velocity gradient, \(\dot{\mathbf{F}}\mathbf{F}^{-1}\) | L03 |
| \(\mathbf{D}\) | Rate of deformation, \(\operatorname{sym}\mathbf{L}\) | L03 |
| \(\mathbf{W}\) | Spin tensor, \(\operatorname{skew}\mathbf{L}\) | L03 |
| \(\mathbf{N}, \mathbf{n}\) | Unit normals — reference (\(\mathbf{N}\)) and current (\(\mathbf{n}\)) | L03 |
| \(dV, dv\) | Volume elements — reference (\(dV\)) and current (\(dv\)) | L03 |
| \(dA, da\) | Area elements — reference (\(dA\)) and current (\(da\)) | L03 |
19.1.2 Stress
| Symbol | Meaning | First in |
|---|---|---|
| \(\boldsymbol{\sigma}\) | Cauchy stress tensor (current configuration) | L03 |
| \(\mathbf{s}\) | Deviatoric Cauchy stress, \(\mathbf{s} = \operatorname{dev}\boldsymbol{\sigma}\) | L07 |
| \(p\) | Hydrostatic pressure, \(p = -\tfrac{1}{3}\operatorname{tr}\boldsymbol{\sigma}\) | L07 |
| \(\sigma_\text{eq}\) | Equivalent (von Mises) stress, \(\sqrt{3J_2}\) | L07 |
| \(I_1, J_2, J_3\) | Stress invariants; \(I_1 = \operatorname{tr}\boldsymbol{\sigma}\), \(J_2 = \tfrac{1}{2}\mathbf{s}:\mathbf{s}\) | L07 |
| \(\mathbf{P}\) | First Piola-Kirchhoff stress (reference) | L02 |
| \(\mathbf{S}\) | Second Piola-Kirchhoff stress (reference) | L02 |
| \(\boldsymbol{\tau}\) | Kirchhoff stress, \(\boldsymbol{\tau} = J\boldsymbol{\sigma}\) | L04 |
| \(\boldsymbol{\sigma}^\text{tr}\) | Trial (elastic-predictor) stress | L08 |
| \(\boldsymbol{\alpha}\) | Backstress tensor (deviatoric, J2 kinematic hardening) | L07 |
| \(\boldsymbol{\xi}\) | Effective (backstress-shifted) stress, \(\boldsymbol{\xi} = \boldsymbol{\sigma} - \boldsymbol{\alpha}\) | L08 App. |
| \(\mathbf{t}, \bar{\mathbf{t}}\) | Traction vector and prescribed boundary traction | L03 |
19.1.3 Elastic moduli
| Symbol | Meaning | First in |
|---|---|---|
| \(E\) | Young’s modulus | L03 |
| \(\nu\) | Poisson’s ratio | L03 |
| \(\mu\) (= \(G\)) | Shear modulus (Lamé’s second parameter) | L03 |
| \(\lambda\) | Lamé’s first parameter | L05 |
| \(\kappa\) (= \(K\)) | Bulk modulus | L05 |
| \(\mathbb{C}^e\) | Fourth-order elasticity tensor (material, small-strain) | L07 |
| \(\mathbb{D}\) | Fourth-order material (Lagrangian) tangent, finite strain | L05 |
| \(\mathbb{c}\) | Fourth-order spatial (Eulerian) tangent, push-forward of \(\mathbb{D}\) | L05 |
| \(\mathbb{C}^{ep}\) | Continuum elasto-plastic tangent (rate form) | L07 |
| \(\mathbb{C}^\text{alg}\) | Consistent (algorithmic) tangent, \(d\boldsymbol{\sigma}_{n+1}/d\boldsymbol{\varepsilon}_{n+1}\) | L08 |
| \(\mathbb{I}, \mathbf{I}\) | Fourth-order and second-order identity tensors | L02 |
| \(\mathbb{P}_\text{dev}\) | Fourth-order deviatoric projector, \(\mathbb{I} - \tfrac{1}{3}\mathbf{I}\otimes\mathbf{I}\) | L08 |
19.1.4 Plasticity & hardening
| Symbol | Meaning | First in |
|---|---|---|
| \(f\) | Yield function (admissibility: \(f \le 0\)) | L07 |
| \(f^\text{tr}\) | Trial yield function value, \(f(\boldsymbol{\sigma}^\text{tr}, q_n)\) | L08 |
| \(g\) | Plastic potential (flow potential); associated rule ⇔ \(g = f\) | L07 |
| \(\mathbf{n}\) | Flow direction, \(\mathbf{n} = \partial f/\partial\boldsymbol{\sigma}\) (or \(\partial g/\partial\boldsymbol{\sigma}\)) | L07 |
| \(\mathbf{n}^\text{tr}_\xi\) | Trial flow direction in effective-stress space | L08 App. |
| \(\dot{\gamma}\) | Plastic multiplier rate (consistency parameter) | L07 |
| \(\Delta\gamma\) | Discrete plastic multiplier over a time step | L08 |
| \(\sigma_y\) | Current yield stress (scalar) | L07 |
| \(\sigma_{y0}\) | Initial yield stress | L07 |
| \(H\) | Generic plastic modulus (used in combined-hardening pedagogical form) | L07 |
| \(H_\text{iso}\) | Isotropic hardening modulus | L07 |
| \(H_\text{kin}\) | Kinematic hardening modulus (Prager) | L07 |
| \(\beta\) | Isotropic/kinematic mixing parameter, \(\beta \in [0,1]\) | L07 |
| \(\kappa, q\) | Generic internal / hardening state variables | L07 |
| \(R(\Delta\gamma)\) | Non-linear scalar residual for \(\Delta\gamma\) (combined hardening) | L08 App. |
| \(m\) | Isotropic hardening exponent (power law), \(m \in [0,1]\) | L08 App. |
| \(\alpha_\text{DP}\) | Drucker-Prager pressure-sensitivity coefficient | L07 |
| \(\eta, \bar{\eta}, \xi\) | Drucker-Prager parameters (friction, dilatancy, cohesion scaling) | L09 |
| \(c\) | Cohesion (Mohr-Coulomb / Drucker-Prager) | L07 |
| \(\phi\) | Internal friction angle | L07 |
19.1.5 Damage & coupled models
| Symbol | Meaning | First in |
|---|---|---|
| \(D\) | Scalar damage variable, \(D \in [0,1]\) | L09 |
| \(\tilde{\boldsymbol{\sigma}}\) | Effective (undamaged) stress, \(\tilde{\boldsymbol{\sigma}} = \boldsymbol{\sigma}/(1-D)\) | L09 |
| \(Y\) | Damage energy release rate | L09 |
19.1.6 Thermodynamics & viscoelasticity
| Symbol | Meaning | First in |
|---|---|---|
| \(\Psi\) | Helmholtz free energy density | L04 |
| \(W\) | Strain energy density (hyperelasticity) | L05 |
| \(\theta\) | Absolute temperature | L04 |
| \(s\) | Specific entropy | L04 |
| \(\rho\) | Mass density (current config) | L04 |
| \(\rho_0\) | Mass density (reference config) | L04 |
| \(\mathbf{q}\) | Heat flux vector | L04 |
| \(r\) | Specific heat source | L04 |
| \(\mathcal{D}\) | Dissipation rate | L04 |
| \(\tau_i\) | Relaxation time (Maxwell branch \(i\)) | L06 |
| \(\tilde{\mathbf{C}}_i\) | Internal viscous deformation variable (per Maxwell branch) | L06 |
| \(\Gamma_i\) | Branch free-energy contribution | L06 |
19.1.7 Operators & decorators
| Symbol | Meaning |
|---|---|
| \(\operatorname{tr}(\cdot)\) | Trace, \(\operatorname{tr}\mathbf{X} = X_{ii}\) |
| \(\operatorname{dev}(\cdot)\) | Deviatoric projection, \(\operatorname{dev}\mathbf{X} = \mathbf{X} - \tfrac{1}{3}(\operatorname{tr}\mathbf{X})\mathbf{I}\) |
| \(\operatorname{sym}, \operatorname{skew}\) | Symmetric and skew-symmetric parts |
| \(\|\mathbf{X}\|\) | Frobenius norm, \(\sqrt{\mathbf{X}:\mathbf{X}}\) |
| \(\mathbf{A}:\mathbf{B}\) | Double contraction, \(A_{ij}B_{ij}\) |
| \(\mathbf{a}\otimes\mathbf{b}\) | Dyadic (tensor) product, \((a\otimes b)_{ij} = a_i b_j\) |
| \(\overline{\otimes},\ \underline{\otimes}\) | Non-symmetric fourth-order products: \((\mathbf{A}\overline{\otimes}\mathbf{B})_{ijkl} = A_{ik}B_{jl}\), \((\mathbf{A}\underline{\otimes}\mathbf{B})_{ijkl} = A_{il}B_{jk}\) |
| \(\dot{(\cdot)}\) | Material time derivative, \(D(\cdot)/Dt\) |
| \(\langle x\rangle\) | Macaulay bracket, \(\max(0, x)\) |
| \((\cdot)_n,\ (\cdot)_{n+1}\) | Value at discrete time step \(t_n,\ t_{n+1}\) |
| \((\cdot)^\text{tr}\) | Trial (elastic-predictor) quantity |
| \((\cdot)^\text{alg}\) | Algorithmic (consistent with discrete integrator) quantity |
| \(\bar{(\cdot)}\) | Isochoric / unimodular part (kinematics); or running average |
| \(\tilde{(\cdot)}\) | Internal / effective quantity (context-dependent) |
19.2 A.2 Abbreviations
| Abbreviation | Expansion |
|---|---|
| CDM | Continuum Damage Mechanics |
| DP | Drucker-Prager (yield criterion) |
| FE, FEM | Finite Element, Finite Element Method |
| GSM | Generalised Standard Material |
| J2 | Second-invariant plasticity (von Mises class) |
| KKT | Karush-Kuhn-Tucker (loading/unloading conditions) |
| MC | Mohr-Coulomb (yield criterion) |
| NR, N-R | Newton-Raphson |
| PK1, PK2 | First and Second Piola-Kirchhoff stress |
| RRM | Radial Return Method |
| RVE | Representative Volume Element |
| SEF | Strain Energy Function |
| TIV | Thermodynamics of Irreversible Variables |
| AF | Armstrong-Frederick (non-linear kinematic hardening) |
19.3 A.3 Conventions
19.3.1 A.3.1 Reference vs current configuration
Uppercase Latin letters denote quantities defined on the reference (Lagrangian, material) configuration; lowercase Latin letters denote quantities defined on the current (Eulerian, spatial) configuration.
- Position: \(\mathbf{X}\) (ref.) / \(\mathbf{x}\) (curr.)
- Normals: \(\mathbf{N}\) (ref.) / \(\mathbf{n}\) (curr.)
- Volume / area elements: \(dV, dA\) (ref.) / \(dv, da\) (curr.)
- Density: \(\rho_0\) (ref.) / \(\rho\) (curr.)
- Heat flux: \(\mathbf{Q}\) (ref. — if used) / \(\mathbf{q}\) (curr.)
- Right vs left Cauchy-Green: \(\mathbf{C} = \mathbf{F}^T\mathbf{F}\) (ref.) / \(\mathbf{b} = \mathbf{F}\mathbf{F}^T\) (curr.)
- Green-Lagrange vs Almansi strain: \(\mathbf{E}\) (ref.) / \(\mathbf{e}\) (curr.)
- Stress measures: \(\mathbf{P}, \mathbf{S}\) (ref.) / \(\boldsymbol{\sigma}, \boldsymbol{\tau}\) (curr.)
- Elasticity tensors: \(\mathbb{D}\) (material tangent) / \(\mathbb{c}\) (spatial tangent)
The deformation gradient \(\mathbf{F}\) is the one map that spans both configurations; by convention it is uppercase.
19.3.2 A.3.2 Sign conventions
- Cauchy stress is positive in tension, negative in compression.
- Hydrostatic pressure \(p = -\tfrac{1}{3}\operatorname{tr}\boldsymbol{\sigma}\); positive \(p\) means compressive mean stress.
- Yield function: \(f \le 0\) in the elastic (admissible) region, \(f = 0\) on the yield surface, \(f > 0\) inadmissible. Return-mapping algorithms project back to \(f = 0\).
- Heat flux \(\mathbf{q}\) points in the direction of positive heat flow (outward from hot to cold). In the energy balance, \(-\nabla\cdot\mathbf{q}\) is heat supplied to the body.
- Plastic multiplier \(\Delta\gamma \ge 0\) (non-negative by KKT).
- Damage \(D \in [0, 1]\), monotonically non-decreasing (\(\dot{D} \ge 0\)) for rate-independent models.
19.3.3 A.3.3 Strain regime by chapter
- L01–L06 develop the general kinematic framework; both finite-strain and infinitesimal-strain measures appear, always clearly labelled.
- L07, L08, L08 Appendix, L09 assume the small-strain regime: \(\boldsymbol{\varepsilon} = \tfrac{1}{2}(\nabla\mathbf{u} + \nabla\mathbf{u}^T)\) with additive decomposition \(\boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}^e + \boldsymbol{\varepsilon}^p\). Finite-strain plasticity uses the multiplicative split \(\mathbf{F} = \mathbf{F}^e\mathbf{F}^p\) (out of scope).
- Hyperelasticity (L05) and viscoelasticity (L06) use finite-strain measures (\(\mathbf{C}, \mathbf{b}, \bar{\mathbf{C}}\)).
19.3.4 A.3.4 Voigt / Mandel ordering
Fourth-order tensors \(\mathbb{C}\) are stored as 6×6 matrices using Voigt ordering: the index pair \((i,j)\) maps to a single index as
\[[1,1] \to 1,\ [2,2] \to 2,\ [3,3] \to 3,\ [2,3]=[3,2] \to 4,\ [1,3]=[3,1] \to 5,\ [1,2]=[2,1] \to 6.\]
Stress and strain vectors use the same ordering. Note that strain components 4–6 are engineering shears (\(\gamma_{ij} = 2\varepsilon_{ij}\) for \(i \neq j\)), so Voigt is not an isometry. Use Mandel ordering (components 4–6 scaled by \(\sqrt{2}\)) when isometric inner products are required.
19.3.5 A.3.5 Rate form
The material time derivative is denoted \(\dot{(\cdot)} = D(\cdot)/Dt\). The overdot form is used throughout the course; the \(D/Dt\) form is used only where the distinction from a spatial derivative matters (L03).
19.4 A.4 Typography at a glance
| Typography | Denotes | Examples |
|---|---|---|
| Lowercase italic Latin | Scalar | \(p,\ J,\ \sigma_y,\ \Delta\gamma,\ \bar{\varepsilon}^p\) |
| Bold upright Latin (uppercase) | 2nd-order tensor on the reference configuration | \(\mathbf{F},\ \mathbf{C},\ \mathbf{E},\ \mathbf{P},\ \mathbf{S},\ \mathbf{N}\) |
| Bold upright Latin (lowercase) | 2nd-order tensor on the current configuration | \(\mathbf{b},\ \mathbf{e},\ \mathbf{s},\ \mathbf{n},\ \mathbf{q}\) |
| Bold Greek | 2nd-order tensor (configuration named explicitly) | \(\boldsymbol{\sigma},\ \boldsymbol{\varepsilon},\ \boldsymbol{\alpha},\ \boldsymbol{\tau},\ \boldsymbol{\xi}\) |
| Blackboard bold (uppercase) | 4th-order tensor — material / Lagrangian frame | \(\mathbb{C}^e,\ \mathbb{D},\ \mathbb{I},\ \mathbb{P}_\text{dev}\) |
| Blackboard bold (lowercase) | 4th-order tensor — spatial / Eulerian frame | \(\mathbb{c}\) |
| Overbar \(\bar{(\cdot)}\) | Isochoric / unimodular part; or running average | \(\bar{\mathbf{C}},\ \bar{\mathbf{b}},\ \bar{\varepsilon}^p,\ \bar{\eta}\) |
| Tilde \(\tilde{(\cdot)}\) | Internal state variable; or effective quantity | \(\tilde{\mathbf{C}}_i,\ \tilde{\boldsymbol{\sigma}}\) |
Superscript tr |
Trial (elastic-predictor) quantity | \(\boldsymbol{\sigma}^\text{tr},\ f^\text{tr},\ \mathbf{s}^\text{tr}\) |
Superscript alg |
Algorithmic / consistent-with-discrete-integrator | \(\mathbb{C}^\text{alg}\) |
Superscript e, p, th |
Elastic, plastic, thermal component | \(\boldsymbol{\varepsilon}^e,\ \boldsymbol{\varepsilon}^p,\ \boldsymbol{\varepsilon}^\text{th},\ \mathbf{F}^e,\ \mathbf{F}^p\) |
Subscript 0 |
Reference-configuration value (for densities, etc.) | \(\rho_0,\ \sigma_{y0}\) |
Subscript y |
Yield (uniaxial) quantity | \(\sigma_y,\ \sigma_{y0}\) |
Subscript iso, kin, DP |
Isotropic / kinematic / Drucker-Prager specific | \(H_\text{iso},\ H_\text{kin},\ f_\text{DP},\ \alpha_\text{DP}\) |
| Subscripts \(n,\ n+1\) | Values at discrete time steps \(t_n,\ t_{n+1}\) | \(\boldsymbol{\sigma}_n,\ \boldsymbol{\sigma}_{n+1},\ \bar{\varepsilon}^p_{n+1}\) |
19.4.1 Colliding symbols — where context matters
A few glyphs necessarily mean two different things in different chapters. The course disambiguates by context and, where needed, by explicit callouts:
- \(\alpha\) / \(\boldsymbol{\alpha}\). Bold \(\boldsymbol{\alpha}\) is the backstress tensor (plasticity, L07–L09). Scalar \(\alpha_\text{DP}\) is the Drucker-Prager pressure coefficient (always subscripted). Scalar \(\alpha\) in L09 line 114 is the coefficient of thermal expansion (standard thermoelasticity notation, within a thermal-strain equation).
- \(\beta\). In L07 hardening, \(\beta \in [0,1]\) is the iso/kin mixing parameter. Not to be confused with the backstress (which has always been \(\boldsymbol{\alpha}\) in this course — no \(\boldsymbol{\beta}\) notation is used for backstress here).
- \(\lambda\). With subscript \(\lambda_i\): principal stretches (L02–L05). Bare \(\lambda\): Lamé’s first parameter (L05). It is never used for the plastic multiplier in this course (that is \(\dot{\gamma}\) / \(\Delta\gamma\)).
- \(\mathbf{C}\) vs \(\mathbf{C}^e\) vs \(\mathbb{C}^e\). \(\mathbf{C}\) is the right Cauchy-Green tensor (2nd-order kinematic quantity). \(\mathbf{C}^e\) is its elastic part (2nd-order, from the multiplicative split, L04). \(\mathbb{C}^e\) (blackboard) is the 4th-order elasticity tensor (L07–L09). Typography (bold upright vs. blackboard) makes this visually unambiguous.
- \(\mathbf{b}\). In this course, \(\mathbf{b}\) always denotes the left Cauchy-Green / Finger tensor. Body-force vectors, when they appear, are written \(\mathbf{b}_0\) (reference) or spelled out contextually.
- \(\mathbf{D}\) / \(D\). Bold \(\mathbf{D}\) is the rate-of-deformation tensor (L03). Scalar \(D\) is the damage variable (L09).
- \(\kappa\). Bare \(\kappa\) is the bulk modulus. In some plasticity textbooks \(\kappa\) denotes a hardening state variable; this course uses \(q\) or \(\bar{\varepsilon}^p\) instead.
19.5 A.5 Where to look
| If you’re unsure about… | See |
|---|---|
| A kinematic quantity (\(\mathbf{F}, \mathbf{C}, \mathbf{b}, \mathbf{E}\)) | L03 |
| Small-strain setup and weak-form integrals | L03 (end) |
| Hyperelastic strain-energy functions | L05 |
| Finite-strain elasticity tensors \(\mathbb{D}, \mathbb{c}\) | L05 (notation callouts) |
| Viscoelastic internal variables \(\tilde{\mathbf{C}}_i\) | L06 |
| Yield functions, flow rules, hardening rules | L07 |
| \(f \le 0\) sign convention and KKT conditions | L07 |
| Return-mapping algorithms, \(\Delta\gamma\), consistent tangent | L08 |
| Full worked derivation with combined hardening | L08 Appendix |
| Damage \(D\) and pressure-dependent plasticity (DP, MC) | L09 |
| Parameter identification, \(H_\text{iso}^*\) | L10 |
| Voigt / Mandel ordering, Gauss-point state variables | L08 (Implementation Notes) |