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.Preliminaries

1.1.Vectors and tensors


EINSTEIN‘s summation convention:



Further notations:



KRONECKER-symbol



LEVI-CIVITÀ-tensor:


1.2.Stress tensors


CAUCHY’s equation:

BOLTZMANN’s axiom:



; VOIGT-notation:

Transformation relation:

Transformation matrix: with

with



Back transformation:

Principal axis transformation:

Characteristic equation:

Ii 1.,2.,3. Invariants

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:

Invariants:

Volumetric strain (dilatation):



Decomposition of strain tensors:

Invariants:





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 :


1.4.Elasticity


Notation:

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.

2.

3.

Symmetry condition DAB=DBA:

With engineering constants Ei, Gij, nij follows



with

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



with

with

Typical elasticity laws:



LAMEs constants:



Relation of elastic moduli:






Shear modulus



E-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.

DRUCKER-PRAGER criterion:

+ 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:



DESHPOANDE-FLECK-ASHBY (DFA) 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

with is most general form.

3.2.GREEN-elastic material laws (hyper elastic)


, resp.., with

with CAYLEY-HAMILTON theorem



Neo-HOOKEian material:

Incompressible:

Compressible:

MOONEY-RIVLIN material:

with

Polynomic approach for hyper elastic constitutive equations:

Polynomic row development after W(I1,I2):

Compressible:

Incompressible: Dk material constants

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





Stability conditions:

Uniqueness:

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.Plasticity



4.1.Approximation of material curves


RAMBERG-OSGOOG: K,n material parameters.

LUDWIK curves:



Exponential curves:



Power law curves:



SARGIN curves:

TO BE CONTINUED





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