Constitutive Material Modeling Formulary

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Constitutive Material Modeling

- Formulary-

HS 2014

Dr. Falk K. Wittel

Computational Physics of Engineering Materials

Institute for Building Materials

ETH Zürich

1.Preliminaries 3

1.1.Vectors and tensors 3

1.2.Stress tensors 3

1.3.Strain tensors 6

1.4.Elasticity 9

2.Failure / Yield surfaces 13

2.1.Invariant spaces 13

2.1.One-parameter models 14

2.2.Two-parameter models 16

2.3.Multiple parameter models 17

2.4.Anisotropic failure / yield surface 19

3.Non-linear elasticity 20

3.1. CAUCHY-elastic material law 21

3.2.GREEN-elastic material laws (hyper elastic) 21

3.3.Hypo-elastic material laws 23

3.4.Variable Moduli models 23

4.Plasticity 23

4.1.Approximation of material curves 23


1.1.Vectors and tensors

EINSTEIN‘s summation convention:

Further notations:



1.2.Stress tensors

CAUCHY’s equation:

BOLTZMANN’s axiom:

; VOIGT-notation:

Transformation relation:

Transformation matrix: with


Back transformation:

Principal axis transformation:

Characteristic equation:

Ii 1.,2.,3. Invariants


Principal stresses:, resp.

Transformation matrix:

with Eigen vectors:

Principal shear stress:

Hydrostatic stress tensor:

Deviatoric stress tensor:

Decomposition of the stress tensor:

Invariants of the hydrostatic stress tensor:

Invariants of the deviatoric stress tensor:

Equilibrium condition:

1.3.Strain tensors

Displacement point P:

Displacement infinitesimal line element dx:

Displacement gradient:

Decomposition of deformation gradient in symmetric and antisymmetric component:

Transformation relation:

Principal axis system with principal strains:

Directions of maximum shear deformation:


Volumetric strain (dilatation):

Decomposition of strain tensors:


Octahedral strains:

Compatibility conditions (6compliance condition):

file:polar decomposition of f.png

Push forward operation:

Pull back operation:

Polar decomposition: ; ;

V left stretch tensor; U right stretch tensor; R orthonormal rotation tensor (R-1=RT)

Deformation tensors:

Right CAUCHY-GREEN deformation tensor C:

Left CAUCHY-GREEN deformation tensor B:

GREENs deformation tensor E:

EULER-ALMANSI strain tensor e:

HENCKYs deformation tensor :



Aelotropic body (21 independent parameters):

Compliance tensor:

Monotropic Body (13 independent parameters): Symmetry with respect to one plane

Orthotropic body (9 independent parameters): Symmetry with respect to two planes

, resp..

Positive definite of DAB: 1.



Symmetry condition DAB=DBA:

With engineering constants Ei, Gij, nij follows


Transversal isotropic body (5 independent parameters): rotation symmetry with respect to one axis

Isotropic body (2 independent parameters): Rotation symmetry with respect to two axes



Typical elasticity laws:

LAMEs constants:

Relation of elastic moduli:

Shear modulus


Constrained modulus

Bulk modulus

Lamé Parameters

Poisson number

Specific strain energy / complementary energy:

2.Failure / Yield surfaces

    1. Invariant spaces

Principal stress space

Invariant space

-Invariant space

LODE-angle :

file:octahedral stress planes.svg

Figure: Octahedral plane, deviatoric plane, meridian plane

Mean stress:

p,q,rInvariant space:

 Invariant space:

2.1.One-parameter models

RANKINE criterion (tension cutoff):

TRESCA criterion:

Von MISES criterion:

HOSFORD criterion:

n=1: TRESCA; n=2: von MISES

2.2.Two-parameter models

MOHR-COULOMB criterion: c, cohesion, internal friction angle

MC-criterion in the MOHRs plane.


+ if DP encloses the MC, -if DP enclosed by MC.

2.3.Multiple parameter models

BRESLER und PISTER (Parabolic dependence of  and ):

a,b,c failure parameters.

WILLAM und WARNKE: (3-parameter model) elliptic shape by dependence 

A constant.

ARGYRIS et al.: a,b,c failure parameters.

OTTOSEN (4-parameter model):

a, b, k1, k2 constants; (cos

HSIEH-TING-CHEN criterion (4-parameter model):

a,b,c,d failure parameter.

WILLAM-WARNKE criterion (5-parameter model):

2.4.Anisotropic failure / yield surface

LOGAN-HOSFORD yield criterion:

F,G,H scaling parameters in principal directions; n exponent e.g. for metal lattice (BCC n=6; FCC n=8).

HILLs yield criterion:

Generalized HILLs yield criterion:

CADDEL-RAGHAVA-ATKINS (CRA) yield criterion:


3.Non-linear elasticity

Elastic total stress-strain relations:

Incremental stress-strain relations:

3.1. CAUCHY-elastic material law

with or

Non- linear elastic of J2 power law type: , with b, m as material parameters


with is most general form.

3.2.GREEN-elastic material laws (hyper elastic)

, resp.., with

with CAYLEY-HAMILTON theorem

Neo-HOOKEian material:





Polynomic approach for hyper elastic constitutive equations:

Polynomic row development after W(I1,I2):


Incompressible: Dk material constants

Polynomial row development after W(I1,I2,I3):

Stability conditions:


Stresses and strains have to be unique.

Stability (DRUCKERs stability postulate):

    • Additional external forces result in deformation and hence positive work.

    • The resulting work due to loading and unloading by external forces is non-negative.

    • Stability in small:

    • Cyclic stability:

Normality conditon:

    • Outward pointing normal at the surface of constant energy is normal vector N. The proportionality holds

Convexity condition:

    • Every tangential plane never intersect the surface and is located entirely outside of the surface.

3.3.Hypo-elastic material laws

with tangent stiffness and compliance tensors C/D.

with secant moduli

3.4.Variable Moduli models


4.1.Approximation of material curves

RAMBERG-OSGOOG: K,n material parameters.

LUDWIK curves:

Exponential curves:

Power law curves:

SARGIN curves:


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