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Local resolution of constitutive laws — Simvia web edition

A Modified Version of "Local resolution of constitutive laws", code_aster / salome_meca course material authored and published by EDF S.A. under the GNU Free Documentation License. The content and the figures are the original course, unchanged; only the layout is new.

Authors: EDF S.A. (the original course material); Simvia (the modifications).

Publisher of this Modified Version: Simvia.

Copyright © 2026 Simvia, for the modifications. The original course material carries no copyright notice.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

The course runs in four parts, in the order the original deck presents them:

Part What it covers
1 General concepts why a behaviour law is needed, and how a plasticity model is built
2 Solving behaviour laws the algorithm code_aster runs, and where it sits in the global loop
3 Behaviour laws in code_aster the catalogue of available laws
4 Syntax in code_aster the COMPORTEMENT keywords that drive the integration
— Conclusion where the documentation lives

Part 1 · General concepts

In this part Why a behaviour law · Example: a metal in a 1D tensile test · A general theoretical framework · A formalism for plasticity · Hardening · The final system to solve

Original deck: slides 2–17.

Why a behaviour law

Consider a solid $\Omega$ in equilibrium with external forces:

$$ \int_\Omega \underline{\underline{\sigma}}(\vec{u}) : \underline{\underline{\varepsilon}}(\delta\vec{u}),d\Omega = \int_\Omega \vec{f}(\vec{u}) : \delta\vec{u},d\Omega

  • \int_{\Gamma_g} g(\vec{u}) : \delta\vec{u},d\Gamma $$
In 3D Count
Unknowns — displacement field 3 components
Unknowns — stress field 6 components
Equations — (weak) equilibrium 3 equations

Six equations are missing ⟹ a behaviour law is required to close the system.

Building the behaviour law is a cycle, not a sequence — testing sends you back to identification:

  1. Identify from experimental: from try to complete structure (representativity).
  2. Using formalism (general prooves for convergence).
  3. Develop in non-linear framework.
  4. Test (verification and validation).

Example: a metal in a 1D tensile test

Behaviour law for metal from 1D tensile-test (plasticity).

1/ Elaborate a representative try

Left: applied force F increasing linearly with time t, labelled Loading monotone. Centre: a tensile specimen pulled at both ends by force F. Right: the measured response, F over S against relative elongation delta-L over L, rising along a blue elastic branch to a marked yield stress sigma-Y, curving in red up to a peak stress sigma-p, then unloading along a steep red line.
Slide 6 — monotone loading, the specimen, and the response it produces.

We can observe:

  • an elasticity domain with a Yield Stress $\sigma^Y$;
  • an irreversible strain $\varepsilon^P$;
  • a hardening caracterized by $\sigma^P > \sigma^Y$;
  • plasticity, a phenomenon that is independent of velocity.

2/ Identify to extend to the REV (Representative Elementary Volume)

First part: the elastic domain until $\sigma^Y$. The elasticity domain is defined by

$$ f(\underline{\underline{\sigma}}) < 0 $$

An elliptical elasticity domain in the sigma-11 versus sigma-12 plane, filled blue, with experimentally identified points marked around its boundary in red and a green looping loading path staying inside it.
Slide 7 — the elasticity domain in stress space.

Experimentaly:

  • $f$ is a function of $\underline{\underline{\sigma}}^D = \mathrm{dev}(\underline{\underline{\sigma}})$
  • more particulary $f(\underline{\underline{\sigma}}^D) = f(J_1, J_2)$

with

$$ J_1 = \frac{1}{2},\mathrm{dev}(\underline{\underline{\sigma}}) . \mathrm{dev}(\underline{\underline{\sigma}}), \qquad J_2 = \frac{1}{3},trac!\left(\mathrm{dev}(\underline{\underline{\sigma}}) . \mathrm{dev}(\underline{\underline{\sigma}}) . \mathrm{dev}(\underline{\underline{\sigma}})\right) $$

Second part: the plastic domain from $\sigma^Y$. The partition of strains:

$$ \underline{\underline{\varepsilon}} = \underline{\underline{\varepsilon}}^e + \underline{\underline{\varepsilon}}^p $$

Stress against strain: a blue curved loading path rising to a maximum stress sigma-S, then a red dashed unloading line of the same slope E as the initial elastic branch, splitting the total strain into a permanent part epsilon-p and a recovered part epsilon-e.
Slide 8 — unloading is parallel to the elastic branch, which splits the total strain into εp and εe.

For elastic case:

$$ \underline{\underline{\sigma}} = \underline{\underline{\underline{\underline{E}}}} : \underline{\underline{\varepsilon}}^e \quad\Longleftrightarrow\quad \underline{\underline{\sigma}} = \lambda,tr(\underline{\underline{\varepsilon}}^e) + 2\mu.\underline{\underline{\varepsilon}}^e $$

A general theoretical framework

Mechanics and thermodynamics. First law of thermodynamics:

$$ \frac{\partial E_{\mathrm{int}}}{\partial t} + \frac{\partial E_{cin}}{\partial t} = P_{ext} + Q_{ext} $$

With PPV, we can formulate a variational internal energy:

$$ \rho,\frac{\partial e}{\partial t} = \underline{\underline{\sigma}} : \dot{\underline{\underline{\varepsilon}}} + r - div(\underline{q}) $$

The second law of thermodynamics: the Inegality of Clausius-Duhem

$$ D = \underline{\underline{\sigma}} : \dot{\underline{\underline{\varepsilon}}}

  • \rho\left(\frac{d\Psi}{dt} - s\dot{T}\right)
  • \frac{\underline{q}}{T}.\nabla T ;\ge; 0 $$

with $\Psi$ the Helmholtz's free energy.

The method of local state

The thermodynamic state, at the point and the instant considered, is entirely defined at this instant by the state variables (observable $(T, \underline{\underline{\varepsilon}})$ and internal $(V_1 = \underline{\underline{\varepsilon}}^p, V_k)$).

$$ \frac{d\Psi}{dt} = \frac{\partial\Psi}{\partial\underline{\underline{\varepsilon}}^e},\dot{\underline{\underline{\varepsilon}}}^e

  • \frac{\partial\Psi}{\partial T},\dot{T}
  • \frac{\partial\Psi}{\partial V_k},\dot{V_k} $$

$$ D_{\mathrm{int}} = \left(\underline{\underline{\sigma}} - \rho\frac{d\Psi}{d\underline{\underline{\varepsilon}}^e}\right) : \dot{\underline{\underline{\varepsilon}}}^e

  • \underline{\underline{\sigma}} : \dot{\underline{\underline{\varepsilon}}}^p
  • \rho\left(s + \frac{d\Psi}{dt}\right)
  • \frac{d\Psi}{dV_k}.\dot{V_k} $$
If Then
$\dot{\underline{\underline{\varepsilon}}}^p = \dot{V}^k = \dot{T} = 0$ the first state law: $\underline{\underline{\sigma}} = \rho\dfrac{\partial\Psi}{\partial\underline{\underline{\varepsilon}}^e}$
$\dot{\underline{\underline{\varepsilon}}}^p = \dot{V}^k = 0,\ \forall,\dot{T}$ the second state law: $s = -\dfrac{\partial\Psi}{\partial T}$

Collecting the state variables, their associated forces and the state laws:

State Variables Associated
thermodynamic forces		 State laws
observable internal
T s s = −∂Ψ/∂T
εe σ σ = ρ ∂Ψ/∂εe
εp −σ σ = −ρ ∂Ψ/∂εp
Vk Ak Ak = ρ ∂Ψ/∂Vk

Evolution of internal state variables ⟹ irreversible dissipation formalism.

A formalism for plasticity

The maximum plastic work (Hill 1951). In short, the principle postulate two important ideas:

  • the yield surface must be convex function,
  • the plastic strain rate is normal to the yield surface

A circular yield surface filled blue where f is less than zero, outlined red where f equals zero. At a point on the boundary the outward normal n, drawn red, and the plastic strain rate, drawn black, point in the same radial direction, while the stress rate, drawn blue, points in a different more tangential direction.
Slide 12 — the plastic strain rate is normal to the yield surface.

Evolution law (or law of normality) Outward normal to the boundary of the domain
$\dot{\underline{\underline{\varepsilon}}}^p = \dot\lambda\dfrac{\partial f}{\partial\sigma}$ $\underline{n} = \dfrac{\partial f}{\partial\sigma}$

Intensity of the flow. The plasticity multiplier is determined by the consistency relation:

$$ df = 0 \quad\Longleftrightarrow\quad \frac{\partial f}{\partial A_k},dA_k = 0 \qquad\Longrightarrow\qquad \boxed{;\dot{f} \equiv 0;} $$

Summary, a plasticity theory:

  • Defining hardening
  • Defining yield surface
  • Defining flow direction (normal = associative law)
  • Défining flow intensity (plastic multiplicator)

Hardening

Isotropic hardening

Left: the initial yield surface S-zero drawn as a small closed curve inside a larger similarly shaped surface S, with a red radial arrow labelled R sigma showing the dilatation. Right: the cyclic stress-strain response in red and green, with the initial elastic range marked two sigma-y and the grown yield stresses marked sigma-y-prime in tension and compression.
Slide 15 — the elasticity domain dilates.

$$ f(\underline{\underline{\sigma}}, p) = \sqrt{\frac{3}{2}}\left|\underline{\underline{\sigma}}^D\right| - R(p) - \sigma^y $$

An isotropic extension of the elasticity domain is taking into account:

  • Dilatation of the elasticity domain
  • Evolution of criterion is governed by a single scalar (internal state variable: cumulated plastic strain)

Kinematic hardening

Left: the initial yield surface S-zero and an equally sized surface S displaced sideways, with a red arrow labelled X marking the translation between their centres. Right: the cyclic stress-strain response, where the elastic range stays two sigma-y in both directions and an annotation marks the Bauschinger effect on reversal.
Slide 16 — the elasticity domain translates, which produces the Bauschinger effect.

$$ f(\sigma, p) = \left|\underline{\underline{\sigma}}^D - \underline{X}\right| - \sigma_y $$

An translation of the elasticity domain is taking into account:

  • Translation of the elasticity domain
  • Evolution of criterion is governed by a tensor (internal state variable: centre of the elasticity domain)

The final system to solve

The first two rows are common to both models, as in the original: they span the full width. Below them, the left column is isotropic hardening and the right column kinematic.

Isotropic hardening Kinematic hardening
Partition of strains ε = εe + εp
Elastic strains σ = λ tr(εe) + 2μ·εe
Plasticity criterion f(σ, p) = √(3/2)·‖σD‖ − R(p) − σy f(σ, p) = ‖σD − X‖ − σy
Flow law (normality) ε̇p = √(3/2)·ṗ·σD ⁄ ‖σD‖ ε̇p = √(3/2)·ṗ·(σD − X) ⁄ ‖σD − X‖
ṗ > 0  if  f(σ, p) = 0   ·   ṗ = 0  if  f(σ, p) < 0
Material parameters R(p) C,  with X = C·ε̇p

▲ back to the four parts

Part 2 · Solving behaviour laws

In this part The functional and the global loop · Time discretisation · Example of algorithm · Integration of constitutive laws: sum up

Original deck: slides 18–24.

The functional and the global loop

Algorithm:

  1. Define functional
  2. Solve functional

Behaviour law is a functional for stress from strains, external state variables (temperature…) and internal state variables

$$ \underline{\underline{\sigma}} = F!\left(\underline{\underline{\varepsilon}}, T, V_k\right) $$

Newton's method: need jacobian too!

$$ \left(\frac{\partial\underline{\underline{\sigma}}}{\partial\underline{\underline{\varepsilon}}}\right) $$

Flowchart of the global Newton loop. A green box, load increment R of u equals F i minus F i minus one, leads to calculation of the tangent matrix K, then to the displacement update u equals u minus K inverse times R, then to a red highlighted box calculating the stress sigma and the tangent operator d-sigma over d-epsilon, then to residual calculation and a test of convergence.
Slide 20 — the red box is the local resolution: it runs once per Gauss point per iteration, and returns both the stress and the tangent operator.

Time discretisation

From ODE equations ⟹ time discretization.

  • Implicit choice: stability
  • The choice of the time step depends on the radial nature of the problem
  • Unknown variables at time step: incremental scheme for stress

$$ \left(\Delta\underline{\underline{\varepsilon}}, V_k^-\right) \xrightarrow{;F_\sigma;} \Delta\underline{\underline{\sigma}} \quad\Longrightarrow\quad \underline{\underline{\sigma}}^+ = \underline{\underline{\sigma}}^- + \Delta\underline{\underline{\sigma}} $$

  • Scheme for internal state variables

$$ \left(\Delta\underline{\underline{\varepsilon}}, V_k^-\right) \xrightarrow{;F_V;} V_k^+ $$

Incremental choice for stress update: only for small strains! $F_V$ and $F_\sigma$ are non-linear functionals to solve.

Example of algorithm

The radial return algorithm. A test compares the elastic prediction against R of p-minus plus sigma-y. The Oui branch sets delta-p and delta-epsilon-p to zero. The Non branch computes delta-p from the hardening slope and delta-epsilon-p along the normal n. Both branches update the plastic strain and the cumulated plastic strain, then a green box gives the new stress. On the right, a sketch in the deviatoric plane shows the elastic prediction overshooting the updated surface and the plastic term correcting it back along the normal.
Slide 22 — the test, its two branches, and the same step drawn in the deviatoric plane.

Depending of $F_V$ and $F_\sigma$:

  • (pseudo)-time integration: implicit or explicit (code_aster: mainly implicit)
  • Non-linear solving: Newton's method, line-search, …
  • Warning! Hypothesis to solve non-linear equation! Implicit algorithm ⟹ unconditional stability BUT when solve ODE using RADIAL hypothesis of loads
  • Parameters for non-linear solving of behaviours laws:
    • In the COMPORTEMENT keyword
    • Sometimes, you can choose local non-linear algorithm to solve

Integration of constitutive laws: sum up

General non-linear algorithm:

  • In practice, only a few laws in code_aster
  • Solving the NL local system of n equations
  • Explicit method (Runge-Kutta) or implicit method (Newton)

Specific non-linear algorithms:

  • For some laws (in fact, most of the laws in code_aster!)
  • The system is reduced to one single scalar equation
  • Solved by various methods (secant, Newton, Dekker, Brent)
  • Analytical solution for some laws (ex: Von Mises isotropic hardening and / or linear kinematic)

▲ back to the four parts

Part 3 · Behaviour laws in code_aster

In this part Constitutive laws available · 2D and 3D continuum media · Beyond continuum media

Original deck: slides 25–30.

Constitutive laws available

More than 160 laws in the 13 stable version.

Various fields of applications

  • Metals, polycrystalline metals
  • Concrete
  • Soils

Various phenomena

  • Irradiation
  • Damage or cracking
  • Metallurgical phases

Documentation

  • Synthesis of non-linear constitutive laws: U4.51.11
  • DEFI_MATERIAU syntax: U4.43.01

2D and 3D continuum media

Family Laws
Non linear elasticity — Von Mises isotropic Pseudo-hardening ELAS_VMIS_LINE, ELAS_VMIS_TRAC, ELAS_HYPER
Incremental elasto-plasticity — Von Mises isotropic hardening, kinematic linear, mixed VMIS_ISOT_TRAC, VMIS_ISOT_PUIS, VMIS_ISOT_LINE, VMIS_CINE_LINE, VMIS_ECMI_TRAC, VMIS_ECMI_LINE
Other elastoplastic models (metals) VMIS_CIN1_CHAB, VMIS_CIN2_CHAB, VMIS_CIN2_MEMO
Elasto-visco-plasticity (metals) LEMAITRE, LEMA_SEUIL, VISC_CIN1_CHAB, VISC_CIN2_CHAB, VISC_ISOT_LINE, VISC_ISOT_TRAC, VISC_TAHERI, VISCOCHAB
Limit loads NORTON_HOFF
Polycrystalline metals POLY_CFC, MONOCRISTAL, POLYCRISTAL
Elasto-visco-plasticity under irradiation LMARC, LEMAITRE_IRRA, GATT_MONNERIE, VISC_IRRA_LOG, GRAN_IRRA_LOG, IRRAD3M
Damage or cracking of metals ENDO_FRAGILE, VENDOCHAB, ROUSSELIER, ROUSS_PR, ROUSS_VISC, RUPT_FRAG, BARENBLATT
Concrete BETON_DOUBLE_DP, GRANGER_FP, GRANGER_FP_V, GRANGER_FP_INDT, BAZANT_FP, ENDO_ISOT_BETON, ENDO_ORTH_BETON, MAZARS, JOINT_BA, CORR_ACIER, KIT_DDI, BETON_REGLE_PR, BETON_UMLV_FP, BETON_BURGER_FP, BETON_RAG
Soils and geomaterials DRUCK_PRAGER(N_A), CAM_CLAY, BARCELONE, CJS, HUJEUX, LAIGLE, LETK, HOEK_BROWN, KIT_HM, KIT_HHM, KIT_THH, KIT_THM, KIT_THHM

Metallurgical phases (elasto-visco-plastic) for steel or zirconium: META_X_Y_Z where

  • X = P (plasticity) or V (viscosity)
  • Y = IL (linear isotropic) or INL (nonlinear isotropic) or CL (linear kinematic)
  • Z = RE (restoration) and/or PT (transformation plasticity)

Beyond continuum media

  • Plates, shells and pipes (local behaviour = plane stress): all 3D constitutive laws (thanks to the method if C_PLAN is not supported: ALGO_C_PLAN = 'DEBORST')
  • Bars, multi-fiber beams, grids: all the laws of 1D behaviour (thanks to the DeBorst method if 1D is not supported: ALGO_1D = 'DEBORST')
  • Discrete elements, shear connections, reinforcements

▲ back to the four parts

Part 4 · Syntax in code_aster

In this part General algorithm · Specific algorithms

Original deck: slides 31–34.

Choice of parameters for the integration: under the factor key word COMPORTEMENT.

General algorithm

Resolution of the local NL system of n equations, selected with ALGO_INTE:

Explicit resolution Implicit resolution by a local Newton, with the possibility of LInear REsearch for certain laws
RUNGE_KUTTA NEWTON NEWTON_RELI
VISCOCHAB, VENDOCHAB, POLYCRISTAL, MONOCRISTAL, VMIS_POU_FLEJOU, VMIS_POU_LINE VISCOCHAB, LMARC, MONOCRISTAL, IRRAD3M, CJS, HUJEUX… VISCOCHAB, LMARC, MONOCRISTAL, IRRAD3M

Convergence

Keyword Meaning Default
RESI_INTE_RELA Residue to achieve $10^{-6}$
ITER_INTE_MAXI Maximum number of iterations 20

Tips

For behaviour which are "difficult" to integrate, increase ITER_INTE_MAXI:

Test case Setting Law
ssnd105b ITER_INTE_MAXI = 250 VISCOCHAB
ssnv172a ITER_INTE_MAXI = 100 MONOCRISTAL
ssnl106i ITER_INTE_MAXI = 500 VMIS_POU_LINE

For certain behaviours, it is better to integrate finely the behaviour (ex: Hujeux) RESI_INTE_RELA = 10⁻⁸.

Specific algorithms

  • For some laws (in fact, most of the laws in code_aster!)
  • The system is reduced to one single scalar equation $\Delta\lambda : \gamma(\Delta\lambda) = 0$
  • Solved by various methods:
    • ALGO_INTE = 'SECANTE', 'DEKKER', 'NEWTON_1D', 'BRENT'
    • Convergence: RESI_INTE_RELA ($10^{-6}$), ITER_INTE_MAXI (20)
  • Analytical resolution
    • VMIS_ISOT_LINE, VMIS_ISOT_TRAC, VMIS_ISOT_PUIS, …
    • CZM_*, ENDO_SCALAIRE, …
    • No additional keyword is required! (except for plane stresses)
    • Ex: hsnv125a: VMIS_ISOT_LINE in 3D and ITER_INTE_MAXI = 100

▲ back to the four parts

Conclusion

Documentation — utilisation

Doc Content
U4.51.11 synthesis of non-linear behaviour
U4.43.01 syntax of the command DEFI_MATERIAU

Documentation — reference

Doc Content
R5.03.02 integration of isotropic hardening or kinematic linear laws
R5.03.03 taking into account the plane stresses
R5.03.XX integration of other behaviours
R5.03.14 implicit and explicit integration of nonlinear laws
R5.03.03 Hypothesis of plane stresses in non-linear behaviours
R3.06.08 Finite elements dealing with the quasi-incompressibility
R5.03.21 Elasto(visco)plastic modelling with isotropic hardening in large strains (SIMO_MIEHE)

All of these are in the code_aster documentation index.

Is something missing or unclear in this document? Or feeling happy to have read such a clear tutorial?

Please, we welcome any feedbacks about Code_Aster training materials. Do not hesitate to share with us your comments on the Code_Aster forum.

History

  • Local resolution of constitutive laws — code_aster / salome_meca course material, authored and published by EDF S.A. under the GNU Free Documentation License, distributed as a slide deck. Its Title Page states no year.
  • Local resolution of constitutive laws — Simvia web edition, 2026, modified and published by Simvia. Converted from the slide deck into a web page: text and figures unchanged, layout new. https://simvia-tech.github.io/tutorials-code_aster/

GNU Free Documentation License

The full text of the license is in LICENSE at the root of this repository.