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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