Constitutive Material Mod-eling

- Formulary-

HS 2014

Dr. Falk K. Wittel

Computational Physics of Engineering Materials

Institute for Building Materials

ETH Zürich

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1. Preliminaries....................................................................................................31.1. Vektors and tensors................................................................................31.2. Stress tensors.........................................................................................31.3. Strain tensors.........................................................................................51.4. Elasticity.................................................................................................7

2. Failure / Yield surfaces...................................................................................102.1. Invariant spaces...................................................................................102.2. One-parameter models.........................................................................112.3. Two-parameter models.........................................................................132.4. Multiple parameter models...................................................................132.5. Anisotropic failure / yield surface.........................................................15

3. Non-linear elasticity.......................................................................................153.1. CAUCHY-elastic material law................................................................163.2. GREEN-elastic material laws (hyperelastic)..........................................163.3. Hypo-elastic material laws....................................................................183.4. Variable Moduli models........................................................................18

4. Plasticity.........................................................................................................184.1. Approximation of material curves.........................................................18

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1.Preliminaries

1.1. Vectors and tensorsEINSTEIN‘s summation convention:

Further notations:

KRONECKER-symbol

LEVI-CIVITÀ-tensor:

1.2. Stress tensors

CAUCHY’s equation:

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BOLTZMANN’s axiom:

; VOIGT-notation:

Transformation relation:

Transformation matrix: with

with

Back transformation: Principal axis transformation:

Characteristic equation:

Ii 1.,2.,3. Invari-ants

Invariants:

Principal stresses: , resp. Transformation matrix:

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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:

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1.3. Strain tensors

Displacement point P:

Displacement infinitesimal line element dx:

Displacement gradient:

Decomposition of deformation gradient in symmetric and antisymmetric compo-nent:

Transformation relation:

Principal axis system with principal strains: Directions of maximum shear deformation:

Invariants:

Volumetric strain (dilatation): Decomposition of strain tensors:

Invariants:

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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=RT)Deformation tensors: Right CAUCHY-GREEN deformation tensor C:

Left CAUCHY-GREEN deformation tensor B:

GREENs deformation tensor E:

EULER-ALMANSI strain tensor e:

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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..

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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

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with

with Typical elasticity laws:

LAMEs constants:

Relation of elastic moduli:Shear

modulusE-modu-

lusCon-

strained modulus

Bulk modu-

lus

Lamé Pa-rameters

Poisson number

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Specific strain energy / complementary energy:

2.Failure / Yield surfaces

2.1. Invariant spaces

Principal stress space

Invariant space

-Invariant space

LODE-angle :

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Figure: Octahedral plane, deviatoric plane, meridian plane

Mean stress:

p,q,r Invariant space:

Invariant space:

2.2. One-parameter modelsRANKINE criterion (tension cutoff):

TRESCA criterion:

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Von MISES criterion:

HOSFORD criterion:

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

2.3. Two-parameter modelsMOHR-COULOMB criterion: c, cohesion, internal friction angle

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MC-criterion in the MOHRs plane.

DRUCKER-PRAGER criterion:

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

2.4. Multiple parameter modelsBRESLER und PISTER (Parabolic dependence of and ):

a,b,c failure parame-ters.

WILLAM und WARNKE: (3-parameter model) elliptic shape by dependence

A constant.

ARGYRIS et al.: a,b,c failure parame-ters.

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OTTOSEN (4-parameter model):

a, b, k1, k2 constants; (cosHSIEH-TING-CHEN criterion (4-parameter model):

a,b,c,d failure parameter.WILLAM-WARNKE criterion (5-parameter model):

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2.5. Anisotropic failure / yield surface LOGAN-HOSFORD yield criterion:

F,G,H scaling parameters in principal directions; n exponent e.g. for metal lat-tice (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 elasticityElastic total stress-strain relations:

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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

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with CAYLEY-HAMILTON theoremNeo-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 constantsPolynomial 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: 18

• 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 pa-rameters.

LUDWIK curves:

Exponential curves:

Power law curves:

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SARGIN curves:

TO BE CONTINUED

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