Yield Criteria, Flow Rules, Hardening, and Loading/Unloading Conditions
📽 Slides: Open presentation
Characteristic features:
The additive strain decomposition (small strains): \[ \boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}^e + \boldsymbol{\varepsilon}^p \]
Regime. Unless stated otherwise, this chapter (and L08, L09) assumes 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 a multiplicative split \(\mathbf{F} = \mathbf{F}^e\mathbf{F}^p\) (outside scope).
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: \[ \sigma_\text{eq} = \sqrt{\tfrac{3}{2}\mathbf{s}:\mathbf{s}} = \sqrt{3J_2}, \qquad J_2 = \tfrac{1}{2}\mathbf{s}:\mathbf{s}. \]
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.
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).
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.
Yield function sign convention. Throughout L07–L09, \(f \le 0\) is the elastic (admissible) region, \(f = 0\) is the yield surface, and \(f > 0\) is inadmissible (non-physical — the return-mapping algorithms of L08 project back to \(f = 0\)).
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 \(\dot{\gamma}\).
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.
Drucker’s stability postulate: for stable materials, the plastic work increment is non-negative: \[ d\boldsymbol{\sigma}:d\boldsymbol{\varepsilon}^p \geq 0 \]
Consequences:
Loss of stability → localization (shear bands), which requires special regularization.
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.
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 \(\sigma^{n+1} = \sigma^\text{tr} - \text{sgn}(\sigma^\text{tr})E\Delta\varepsilon_p\).
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: \[ H_\text{iso} = \frac{EE_t}{E - E_t}, \quad \text{where } E_t = \frac{EH_\text{iso}}{E+H_\text{iso}} \]
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).
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.
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: \[ J_2 = \tfrac{1}{2}\mathbf{s}:\mathbf{s} \]
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).
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.
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.
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.
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.
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).
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.
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.
📖 Worked examples: fully numerical derivations of the RRM for combined Prager + power-law isotropic hardening are in L08 Appendix — Worked Examples.
Constitutive Models — Technion | Website