Constitutive Material Modeling Formulary

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:

I_i 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):

Push forward operation:

Pull back operation:

Polar decomposition: ; ;

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

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 D_AB: 1.

2.

3.

Symmetry condition D_AB=D_BA:

With engineering constants E_i, G_ij, n_ij 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. 
   
   
2. 
   
   

   1. Invariant spaces

Principal stress space

Invariant space

-Invariant space

LODE-angle :

Figure: Octahedral plane, deviatoric plane, meridian plane

Mean stress:



p,q,rInvariant 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:

1. 
   
   

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: D_k 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

2. 
   
   

4.1.Approximation of material curves

RAMBERG-OSGOOG: K,n material parameters.

LUDWIK curves:

Exponential curves:

Power law curves:



SARGIN curves:

TO BE CONTINUED