Solid Mechanics Application Constitutive Laws

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Contents

Constitutive Laws designed by the Solid Mechanics Application

The Constitutive Law located in the kratos base has a general usage explained in the next HowTo:

The constitutive law base class was designed basically to deal with the general requirements of continuum mechanics for solids.

Definition

In the Solid Mechanics application the constitutive laws are classified as

  1. Elastic
    1. Plane Strain
    2. Plane Stress
    3. Axi-Symmetric
    4. 3D
  2. HyperElastic
    1. Plane Strain
    2. Plane Stress
    3. Axi-Symmetric
    4. 3D

Elastic laws correspond to Kirchooff-Saint Venant models usually perfoming the relationship between the 2ndPiola-Kirchhoff stress tensor and the Green-Lagrange strain tensor ( S = C:E ) for Finite Strains or the relationship between the Cauchy stress tensor and the infinitessimal strain tensor ( sigma = C:epsilon) for Infinitessimal Strains. The hyperelastic laws correspond to a simpler Neo-Hookean model which depends only in the Deformation Gradient tensor F, and it's determinant.

An extension to these models to plasticity is available and classified as

  1. ElasticPlastic
    1. Plane Strain
    2. Plane Stress
    3. Axi-Symmetric
    4. 3D
  2. HyperElasticPlastic
    1. Plane Strain
    2. Plane Stress
    3. Axi-Symmetric
    4. 3D

Models concerning to the characterization of the fracture in a continuous domain are classified as

  1. Damage
    1. Plane Strain
    2. Plane Stress
    3. Axi-Symmetric
    4. 3D

The constitutive models and constitutive laws are mathematical formulas based on thermodynamic laws aimed to predict the behavior or the response of one or more materials.

The computation of internal variables and the mapping of tensor spaces are typical operations performed on them.

A general Constitutive Law is defined using different type of parameters

  • 1. Material Properties (characteristics of each material, given by the experimental tests) 
 YOUNG_MODULUS
 POISSON_COEFFICIENT
 DENSITY
 THERMAL_EXPANSION_COEFFICIENT
 YIELD_STRESS
 ...
  • 2. Internal Variables (which define behaviour given by the mathematical model) 
 EquivalentPlasticPotencial
 EquivalentPlasticStrain
 DamageVariable
 ...

In plasticity the model can be described with three different parts:

A yield criterion that defines the elastic region, the yield surface (where the plasticity occurs) and the impossible region for the stress state function.

A hardening law that defines the evolution of the yield surface in terms of hardening (increase of the elastic region) or softening (decrease of the elastic region)

A flow rule that solves the constitutive model integrating the state function which is defined by the yield criterion and the hardening law.

These parts describe the evolution of the model properties and sets the resultant stress response given a strain state and taking in account the strain history.

Therefore to define a Plasticity model these three characteristics of a model need to be defined. In the Solid Mechanics Application is done by three different object classes that define the particularities of each part. Every plasticity law must define these parts, in a C++ code this corresponds to the definition of three member pointer variables that will be done in the constructor of each "plastic" constitutive law. For example: 

HyperElasticPlasticJ23DLaw()
{
  mpHardeningLaw   = HardeningLaw::Pointer( new NonLinearIsotropicKinematicHardeningLaw() );
  mpYieldCriterion = YieldCriterion::Pointer( new MisesHuberYieldCriterion(mpHardeningLaw) );
  mpFlowRule       = FlowRule::Pointer( new LinearAssociativePlasticFlowRule(mpYieldCriterion) );
}


List of Constiutive Laws for the Solid Mechanics Application

  • Next we present the list of constitutive laws implemented in the Solid Mechanics Application for the standard elements based in DISPLACEMENT dof's:

Elastic and HyperElastic Laws

Constitutive Law Dimension Material Properties Input Variables Type
LinearElastic3DLaw() 3 YOUNG_MODULUS Infinitessimal Strain / Deformation Gradient F Finite Strains / Infinitesimal Strains
POISSON_RATIO
LinearElasticPlaneStrain2DLaw() 2 plane-state YOUNG_MODULUS Infinitessimal Strain / Deformation Gradient F Finite Strains / Infinitesimal Strains
POISSON_RATIO
LinearElasticPlaneStress2DLaw() 2 plane-state YOUNG_MODULUS Infinitessimal Strain / Deformation Gradient F Finite Strains / Infinitesimal Strains
POISSON_RATIO
LinearElasticAxisym2DLaw() 2 axi-symmetric YOUNG_MODULUS Infinitessimal Strain / Deformation Gradient F Finite Strains / Infinitesimal Strains
POISSON_RATIO
HyperElastic3DLaw() 3 YOUNG_MODULUS Deformation Gradient F Finite Strains
POISSON_RATIO
HyperElasticPlaneStrain2DLaw() 2 plane-state YOUNG_MODULUS Deformation Gradient F Finite Strains
POISSON_RATIO
HyperElasticAxisym2DLaw() 2 axi-symmetric YOUNG_MODULUS Deformation Gradient F Finite Strains
POISSON_RATIO

Plasticity Laws

Constitutive Law Dimension Material Properties Yield Criterion Hardening Law Flow Rule Input Variables Type
HyperElasticPlasticJ23DLaw() 3 Mises-Huber Non-Linear Isotropic-Kinematic Linear-Associative YOUNG_MODULUS Deformation Gradient F Finite Strains
POISSON_RATIO
YIELD_STRESS
KINEMATIC_HARDENING_MODULUS
REFERENCE_HARDENING_MODULUS
INFINITY_HARDENING_MODULUS
HARDENING_EXPONENT
HyperElasticPlasticJ2PlaneStrain2DLaw() 2 plane-state Mises-Huber Non-Linear Isotropic-Kinematic Linear-Associative YOUNG_MODULUS Deformation Gradient F Finite Strains
POISSON_RATIO
YIELD_STRESS
KINEMATIC_HARDENING_MODULUS
REFERENCE_HARDENING_MODULUS
INFINITY_HARDENING_MODULUS
HARDENING_EXPONENT
HyperElasticPlasticJ2Axisym2DLaw() 2 axi-symmetric Mises-Huber Non-Linear Isotropic-Kinematic Linear-Associative YOUNG_MODULUS Deformation Gradient F Finite Strains
POISSON_RATIO
YIELD_STRESS
KINEMATIC_HARDENING_MODULUS
REFERENCE_HARDENING_MODULUS
INFINITY_HARDENING_MODULUS
HARDENING_EXPONENT

Damage Laws

Constitutive Law Dimension Material Properties Yield Criterion Hardening Law Flow Rule Input Variables Type
IsotropicDamageSimoJu3DLaw() 3 Simo-Ju Exponential Isotropic Damage YOUNG_MODULUS Infinitessimal Strain / Deformation Gradient FDeformation Gradient F Infinitessimal Strains
POISSON_RATIO
DAMAGE_THRESHOLD
STRENGTH_RATIO
FRACTURE_ENERGY
IsotropicDamageSimoJuPlaneStrain2DLaw() 2 plane-state Simo-Ju Exponential Isotropic Damage YOUNG_MODULUS Infinitessimal Strain / Deformation Gradient FDeformation Gradient F Infinitessimal Strains
POISSON_RATIO
DAMAGE_THRESHOLD
STRENGTH_RATIO
FRACTURE_ENERGY
IsotropicDamageSimoJuPlaneStress2DLaw() 2 plane-state Simo-Ju Exponential Isotropic Damage YOUNG_MODULUS Infinitessimal Strain / Deformation Gradient FDeformation Gradient F Infinitessimal Strains
POISSON_RATIO
DAMAGE_THRESHOLD
STRENGTH_RATIO
FRACTURE_ENERGY


  • Next we present the list of constitutive laws implemented in the Solid Mechanics Application for the hybrid elements based in DISPLACEMENT and PRESSURE dof's:

Elastic and HyperElastic Laws

Constitutive Law Dimension Material Properties Input Variables Type
HyperElasticUP3DLaw() 3 YOUNG_MODULUS Deformation Gradient F Finite Strains
POISSON_RATIO
HyperElasticUPPlaneStrain2DLaw() 2 plane-state YOUNG_MODULUS Deformation Gradient F Finite Strains
POISSON_RATIO
HyperElasticUPAxisym2DLaw() 2 axi-symmetric YOUNG_MODULUS Deformation Gradient F Finite Strains
POISSON_RATIO

Plasticity Laws

Constitutive Law Dimension Material Properties Yield Criterion Hardening Law Flow Rule Input Variables Type
HyperElasticPlasticUPJ23DLaw() 3 Mises-Huber Non-Linear Isotropic-Kinematic Linear-Associative YOUNG_MODULUS Deformation Gradient F Finite Strains
POISSON_RATIO
YIELD_STRESS
KINEMATIC_HARDENING_MODULUS
REFERENCE_HARDENING_MODULUS
INFINITY_HARDENING_MODULUS
HARDENING_EXPONENT
HyperElasticPlasticUPJ2PlaneStrain2DLaw() 2 plane-state Mises-Huber Non-Linear Isotropic-Kinematic Linear-Associative YOUNG_MODULUS Deformation Gradient F Finite Strains
POISSON_RATIO
YIELD_STRESS
KINEMATIC_HARDENING_MODULUS
REFERENCE_HARDENING_MODULUS
INFINITY_HARDENING_MODULUS
HARDENING_EXPONENT
HyperElasticPlasticUPJ2Axisym2DLaw() 2 axi-symmetric Mises-Huber Non-Linear Isotropic-Kinematic Linear-Associative YOUNG_MODULUS Deformation Gradient F Finite Strains
POISSON_RATIO
YIELD_STRESS
KINEMATIC_HARDENING_MODULUS
REFERENCE_HARDENING_MODULUS
INFINITY_HARDENING_MODULUS
HARDENING_EXPONENT
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