9 L07 — Plasticity: Theory
Yield Criteria, Flow Rules, Hardening, and Loading/Unloading Conditions
9.1 Phenomenology of Plasticity
Characteristic features:
- Irreversible deformation upon load removal
- A yield threshold — elastic below, plastic above
- Hardening: yield stress increases with accumulated plastic strain
- Bauschinger effect: kinematic hardening under reversed loading
The additive strain decomposition (small strains): \[ \boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}^e + \boldsymbol{\varepsilon}^p \]
9.2 Deviatoric Stress and Pressure
Hydrostatic pressure \(p\) and deviatoric stress \(\mathbf{s}\): \[ p = -\tfrac{1}{3}\mathrm{tr}\,\boldsymbol{\sigma}, \qquad \mathbf{s} = \boldsymbol{\sigma} + p\mathbf{I}. \]
Von Mises equivalent stress: \[ \sigma_\text{eq} = \sqrt{\tfrac{3}{2}\mathbf{s}:\mathbf{s}} = \sqrt{3J_2}, \qquad J_2 = \tfrac{1}{2}\mathbf{s}:\mathbf{s}. \]
9.3 Yield Criteria
Von Mises (J2): pressure-independent, metallic materials: \[ f(\boldsymbol{\sigma}, \kappa) = \sigma_\text{eq} - \sigma_y(\kappa) \leq 0 \]
Tresca: maximum shear stress criterion — inscribes Von Mises in principal-stress space.
Drucker-Prager: pressure-dependent, cohesive-frictional materials: \[ f = \alpha_\text{DP} I_1 + \sqrt{J_2} - k \leq 0 \]
Here \(\alpha_\text{DP}\) is the Drucker-Prager pressure-sensitivity coefficient — distinct from the backstress \(\boldsymbol{\alpha}\) used later in this chapter and from the equivalent plastic strain used elsewhere.
Mohr-Coulomb: \(\tau_\text{max} = c - \sigma_n\tan\phi\) on the slip plane.
9.4 Flow Rule
The plastic strain rate is governed by a flow rule: \[ \dot{\boldsymbol{\varepsilon}}^p = \dot{\gamma}\,\mathbf{m}(\boldsymbol{\sigma}) \]
(overdot denotes the material time derivative \(\dot{(\cdot)} = D(\cdot)/Dt\) — see L03)
Associated (normality): \(\mathbf{m} = \partial f/\partial\boldsymbol{\sigma}\) — plastic flow normal to yield surface. Follows from maximum plastic dissipation principle; ensures convexity.
Non-associated: \(\mathbf{m} \neq \partial f/\partial\boldsymbol{\sigma}\) — used when dilatancy must be independently controlled (Drucker-Prager, soils).
9.5 Hardening Rules
Isotropic hardening: yield surface expands uniformly. \[ \sigma_y = \sigma_{y0} + H_\text{iso}\bar{\varepsilon}^p, \qquad \dot{\bar{\varepsilon}}^p = \sqrt{\tfrac{2}{3}\dot{\boldsymbol{\varepsilon}}^p:\dot{\boldsymbol{\varepsilon}}^p} \]
Kinematic hardening: yield surface translates — captures Bauschinger effect. \[ f = \sigma_\text{eq}(\boldsymbol{\sigma} - \boldsymbol{\alpha}) - \sigma_{y0} \leq 0, \qquad \dot{\boldsymbol{\alpha}} = \tfrac{2}{3}H_\text{kin}\,\dot{\boldsymbol{\varepsilon}}^p \]
Here \(\boldsymbol{\alpha}\) is the backstress tensor (deviatoric in J2; centre of the yield surface in deviatoric-stress space) and \(H_\text{kin}\) is the linear kinematic (Prager) hardening modulus. Same symbols carry through L08 and L08-appendix.
Mixed (combined) hardening: both effects simultaneously.
9.6 Kuhn-Tucker Loading/Unloading Conditions
The standard Karush-Kuhn-Tucker (KKT) conditions govern plastic flow: \[ f \leq 0, \qquad \dot{\gamma} \geq 0, \qquad \dot{\gamma}\,f = 0. \]
Consistency condition (during plastic loading, \(f = 0\) and \(\dot{f} = 0\)): \[ \dot{f} = \frac{\partial f}{\partial\boldsymbol{\sigma}}:\dot{\boldsymbol{\sigma}} + \frac{\partial f}{\partial\kappa}\dot{\kappa} = 0 \]
This determines \(\dot{\gamma}\).
9.7 Elasto-Plastic Tangent Modulus
The continuum elasto-plastic modulus \(\mathbb{C}^{ep}\) relates stress and strain rates in the plastic loading regime:
\[ \dot{\boldsymbol{\sigma}} = \mathbb{C}^{ep}:\dot{\boldsymbol{\varepsilon}}, \qquad \mathbb{C}^{ep} = \mathbb{C}^e - \frac{(\mathbb{C}^e:\mathbf{n})\otimes(\mathbf{n}:\mathbb{C}^e)} {h + \mathbf{n}:\mathbb{C}^e:\mathbf{n}} \]
where \(\mathbf{n} = \partial f/\partial\boldsymbol{\sigma}\) and \(h\) is the hardening modulus.
Note: \(\mathbb{C}^{ep}\) is symmetric for associated flow; non-symmetric for non-associated.
9.8 Material Stability
Drucker’s stability postulate: for stable materials, the plastic work increment is non-negative: \[ d\boldsymbol{\sigma}:d\boldsymbol{\varepsilon}^p \geq 0 \]
Consequences:
- Yield surface must be convex
- Flow must obey normality (associated rule)
- \(\mathbb{C}^{ep}\) is positive semi-definite
Loss of stability → localization (shear bands), which requires special regularization.
9.9 One-Dimensional Elastoplasticity
9.9.1 Fundamentals
Small strain additive decomposition: \[ \boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}^e + \boldsymbol{\varepsilon}^p \]
Stress is related to elastic strain only. Plastic strain is an internal variable evolving with plastic deformation.
9.9.2 Elastic predictor, plastic corrector procedure
Given trial (elastic) stress and current state variables: \[ \sigma^\text{tr} = \sigma^n + E\Delta\varepsilon, \quad \text{Yield check:} \quad f^\text{tr} = |\sigma^\text{tr}| - \sigma_y^n \]
If elastic (\(f^\text{tr} \leq 0\)): Accept trial; no plastic flow.
If plastic (\(f^\text{tr} > 0\)): Return to yield surface via plastic consistency: \[ |\sigma^\text{tr} - \text{sgn}(\sigma^\text{tr})E\Delta\varepsilon_p| = \sigma_y^n + H_\text{iso}\Delta\varepsilon_p \]
Solving: \(\Delta\varepsilon_p = \frac{f^\text{tr}}{E+H_\text{iso}}\) and \(\sigma^{n+1} = \sigma^\text{tr} - \text{sgn}(\sigma^\text{tr})E\Delta\varepsilon_p\).
9.9.3 Isotropic hardening
Yield stress evolves with accumulated plastic strain: \[ \sigma_y = \sigma_{y0} + H_\text{iso}\bar{\varepsilon}^p \]
The plastic modulus \(H_\text{iso}\) relates stress increment to plastic strain increment: \[ H_\text{iso} = \frac{EE_t}{E - E_t}, \quad \text{where } E_t = \frac{EH_\text{iso}}{E+H_\text{iso}} \]
9.9.4 Kinematic hardening
Yield surface translates (back stress \(\alpha\) evolves): \[ \text{Effective stress:} \quad \eta = \sigma - \alpha, \quad \text{Yield:} \quad |\eta| = \sigma_{y0} \]
Back stress evolution: \(\alpha^{n+1} = \alpha^n + \text{sgn}(\eta)H_\text{kin}\Delta\varepsilon_p\).
Captures Bauschinger effect (yield strength reduction upon load reversal).
9.9.5 Combined hardening
Mix isotropic and kinematic via parameter \(\beta \in [0,1]\): \[ \sigma_y^{n+1} = \sigma_y^n + (1-\beta)H\Delta\varepsilon_p, \quad \alpha^{n+1} = \alpha^n + \beta H\Delta\varepsilon_p \]
In this combined-hardening simplification, \(H\) is a single mixed modulus; compare L08-appendix where iso and kin moduli are split as \(H_\text{iso}\) and \(H_\text{kin}\).
\(\beta = 0\) → isotropic; \(\beta = 1\) → kinematic.
9.10 Multi-Dimensional Theory
9.10.1 Deviatoric stress and strain invariants
Hydrostatic component and deviator: \[ \boldsymbol{\sigma}_m = \tfrac{1}{3}\mathrm{tr}(\boldsymbol{\sigma})\mathbf{I}, \quad \mathbf{s} = \boldsymbol{\sigma} - \boldsymbol{\sigma}_m \]
Second invariant of deviatoric stress: \[ J_2 = \tfrac{1}{2}\mathbf{s}:\mathbf{s} \]
9.10.2 Von Mises yield criterion (J2 plasticity)
Pressure-insensitive (applies to metals): \[ f = \sigma_\text{eq} - \sigma_y(\bar{\varepsilon}^p) = \sqrt{3J_2} - \sigma_y = 0 \]
Where \(\sigma_\text{eq} = \sqrt{\tfrac{3}{2}\mathbf{s}:\mathbf{s}}\) is the equivalent (von Mises) stress.
Plastic flow is deviatoric (dilatation elastic, distortion plastic).
9.10.3 Effective plastic strain
Conjugate measure to equivalent stress: \[ \bar{\varepsilon}^p = \int_0^t \dot{\bar{\varepsilon}}^p \, d\tau, \quad \dot{\bar{\varepsilon}}^p = \sqrt{\tfrac{2}{3}\dot{\boldsymbol{\varepsilon}}^p:\dot{\boldsymbol{\varepsilon}}^p} \]
Plasticity relations in principal stress space often cleaner; spectral decomposition aligns principal stresses with flow direction.
9.10.4 Normality and plastic flow
For associated plasticity (most common): \[ \dot{\boldsymbol{\varepsilon}}^p = \dot{\gamma}\frac{\partial f}{\partial\boldsymbol{\sigma}} = \dot{\gamma}\frac{3}{2}\frac{\mathbf{s}}{\sigma_\text{eq}} \]
Direction normal to yield surface; ensures convexity and stability.
9.10.5 Multi-axial hardening models
Isotropic: yield surface expands uniformly in deviatoric space.
Kinematic: surface translates (back stress tensor \(\boldsymbol{\alpha}\) evolves). \[ f = \|\mathbf{s} - \boldsymbol{\alpha}\| - \sqrt{\tfrac{2}{3}}\sigma_{y0} \]
Combined: both mechanisms active simultaneously.
9.10.6 Pressure-dependent criteria
Drucker-Prager: \[ f = \alpha_\text{DP} I_1 + \sqrt{J_2} - k \leq 0 \]
Common in geomechanics and concrete; \(I_1 = \mathrm{tr}(\boldsymbol{\sigma})\) is the first invariant (hydrostatic stress).
Mohr-Coulomb: defined by friction angle \(\phi\) and cohesion \(c\); forms a hexagonal cone in principal stress space.
Non-associated flow rule often needed to control dilatancy.
9.11 Finite Deformations and Objectivity
9.11.1 Objective stress rates
Constitutive laws written in rate form must use objective rates to be frame-indifferent: \[ \dot{\boldsymbol{\sigma}}^\mathcal{J} = \dot{\boldsymbol{\sigma}} - \mathbf{w}\boldsymbol{\sigma} + \boldsymbol{\sigma}\mathbf{w} \]
Jaumann rate: uses spin tensor \(\mathbf{w} = \tfrac{1}{2}(\mathbf{L} - \mathbf{L}^T)\) (skew part of velocity gradient).
9.11.2 Finite rotation: midpoint configuration
For finite rotations, use intermediate (midpoint) configuration to avoid spurious stress rotation:
Rotate stress to midpoint configuration \(\rightarrow\) perform plasticity update \(\rightarrow\) rotate back: \[ \bar{\boldsymbol{\sigma}} = \mathbf{R}\boldsymbol{\sigma}^n\mathbf{R}^T, \quad \text{update with } \bar{\boldsymbol{\sigma}}, \quad \boldsymbol{\sigma}^{n+1} = \mathbf{R}^T\bar{\boldsymbol{\sigma}}^{n+1}\mathbf{R} \]
\(\mathbf{R}\) extracted from incremental deformation gradient via polar decomposition or exponential map.
9.11.3 Multiplicative decomposition (large strains)
Deformation gradient splits elastically and plastically: \[ \mathbf{F} = \mathbf{F}^e\mathbf{F}^p \]
Plastic part \(\mathbf{F}^p\) lies in an unobservable intermediate configuration (stress-free after plastic flow).
Enables consistent treatment of large elastic and large plastic strains via hyperelasticity + plasticity in principal stress space.