# Structural Application Constitutive Laws

## List of constitutive laws in Structural Application

The constitutive models and constitutive laws are mathematical formulas based on thermodynamic laws aimed at predicting the behavior or response from one or more materials. The computation internal variables and mapping of tensor spaces are typical operations performed on them.

In general, a constitutive law is defined using various parameters
that define the evolution of internal variables used in the model. The most common
are:

** A yield function ** that lets us know the domain of elastic material.
A constitutive model may have one or more combinations of sueprficies
fluence, it depends both know the material. An example of the same
mohr is Coulomb's surface, which is expressed in six functions as defined inteseccion the yield strength of material.

** A function of softening or hardening ** . Functions that define the post-peak response of the material. They can be linear In general, a constitutive law is defined using various parameters
that define the evolution of internal variables used in the model. The most common
are:

The parameters to define a yield surface are: a) the yield function b) flow rate: Associate or Associate c) Status: Plane Stress, Plane Strain, Full TrID.

Shown below a Python script which creates the object yield function and softening function.

Fluency_1 = EnergyYieldFunction (myState.Plane_Stress). Fluency_2 = VonMissesYieldFunction (myState.Plane_Stress, myPotencialPlastic.Associated). Fluency_3 = DruckerPragerYieldFunction (myState.Plane_Stress). fluency_4 = ModifiedMorhCoulombYieldFunction (myState.Plane_Strain, myPotencialPlastic.Not_Associated). fluency_5 = RankineYieldFunction (myState.Plane_Strain). print fluency_1. # Function to use Softening. behavior_1 = ExponentialSoftening (). behavior_2 = LinearSoftening (). print behavior_1

Among the most common is Hook's law or theory of elasticity,
Theory of plasticity and damage models. Combinations of them can
be creating, doing more complex formulation.

**Isotropic:** Models of elasticity in 2 dimensions (Plane Stress) and three dimensions.
Being known as *Isotropic2D* and *Isotropic3D*. Only the parameters required are
Young's modulus and Poisson ratio.

**Simple Damage Model:** Models of Damage in 2 dimensions (Plane Stress) and three dimensions.
They are known as *Isotropic_Damage* and *Isotropic_Damage_3D*.
The properties of the materials it need are:

- Young Modulus,
- Poisson Ratio,
- Fracture Energy.
- Fluency Criteria (The Energy Yield Function),
- Softening Behavior (Linear or Exponential).

**Simple Damage Model:** Models of Damage in 2 dimensions (Plane Stress) and three dimensions.
They are known as *Isotropic_Damage* and *Isotropic_Damage_3D*.
The properties of the materials it need are:

- Young Modulus,
- Poisson Ratio,
- Fracture Energy.
- Fluency Criteria (The Energy Yield Function),
- Softening Behavior (Linear or Exponential).

**Plasticty:** Models of plasticity in 2 dimensions (Plane Stress and Plane Strain ) and three dimensions.
They are known as *Plasticity_2D* and *Plasticity_3D*.
The properties of the materials it need are:

- Young Modulus,
- Poisson Ratio,
- Yield Stress.
- Plastic Modulus.
- Fluency Criteria (The Energy Yield Function),
- Softening Behavior (Linear or Exponential).

**Hyperelastic:** Models of Hyperelasticity (Neo-Hookean quasi-incompressible) in 2 dimensions (Membrane Element) and three dimensions.
Being known as *Hyperelastic2D* and *Hyperelastic3D*. The parameters required are
Shear modulus and Bulk modulus for the 3D case and additional 2 parameters for 2D to consider or not retraction: Retraction Time and Alpha.
The properties of the materials it need are:

- Shear modulus,
- Bulk modulus,
- Retraction Time (only for 2D case),
- Alpha (only for 2D case).

**Viscoelastic:** Model of Viscoelasticity (Generalized Maxwell model considering 5 devices) in 2 dimensions (Membrane Element). Vicoelasticity combined with neo-Hookean Hyperelasticity.
Being known as *Viscoelastic2D*. The viscoelastic parameters required are
Relaxation Time and Beta for each Maxwell device for the 2D case, and additional 4 parameters for the Hyperelastic part: Shear modulus, Bulk modulus, Retraction Time and Alpha (see above: Hyperelastic).
Important: the Maxwell parameters are declared in the python script used to run the problem.
In short, the list of parameters needed:

- Relaxation Time (for each Maxwell device),
- Beta (for each Maxwell device),
- Shear modulus,
- Bulk modulus,
- Retraction Time,
- Alpha.

**Viscofibers:** Viscoelastic Model with Fibers (elastic matrix and fibers are considered, neo-Hookean hyperelasticity) in 2 dimensions (Membrane Element). Two classes of fibers are considered at the element level.
Being known as *Viscofibers2D*. The parameters required are
Fiber parameters: K1 and K2
Direction of the fibers at the element level: 6 components (x, y and z directions for each fiber).
Relaxation Time and Beta for each Maxwell device for the 2D case, and additional 4 parameters for the Hyperelastic part: Shear modulus, Bulk modulus, Retraction Time and Alpha (see above: Hyperelastic).
Important: Fiber parameters, fiber directions and Maxwell parameters are declared in the python script used to run the problem.
In short, the list of parameters needed:

- Fiber parameters,
- Direction of the fibers,
- Relaxation Time (for each Maxwell device),
- Beta (for each Maxwell device),
- Shear modulus,
- Bulk modulus,
- Retraction Time,
- Alpha.

**Viscofibers_Hypermatrix:** Composite model comprising a Hyperelastic Matrix (neo-Hookean hyperelasticity) and Viscoelastic Fibers (Generalized Maxwell model considering 5 devices) in 2 dimensions (Membrane Element). Two classes of fibers are considered at the element level.
Being known as *Viscofibers_Hypermatrix2D*. The parameters required are
Fiber parameters: K1 and K2
Direction of the fibers at the element level: 6 components (x, y and z directions for each fiber).
Relaxation Time and Beta for each Maxwell device for the 2D case, and additional 4 parameters for the Hyperelastic part: Shear modulus, Bulk modulus, Retraction Time and Alpha (see above: Hyperelastic).
Important: Fiber parameters, fiber directions and Maxwell parameters are declared in the python script used to run the problem.
In short, the list of parameters needed:

- Fiber parameters,
- Direction of the fibers,
- Relaxation Time (for each Maxwell device),
- Beta (for each Maxwell device),
- Shear modulus,
- Bulk modulus,
- Retraction Time,
- Alpha.

Summary

- Isotropic2D()
- Isotropic3D()
- Isotropic_Damage()
- Isotropic_Damage3D()
- Plasticity_2D()
- Plasticity_3D()
- Hyperelastic2D()
- Hyperelastic3D()
- Viscoelastic2D()
- Viscofibers2D()
- Viscofibers_Hypermatrixc2D()

## List of Constitutive Law

Constitutive Law | Properties | Properties Variables | Element | Type |
---|---|---|---|---|

Isotropic2d() | Young Modulus | YOUNG_MODULUS | Total_Lagrangian and Linear Element | |

Poisson Ratio | POISSON_RATIO | Total_Lagrangian and Linear Element | ||

Isotropic3d() | Young Modulus | YOUNG_MODULUS | Total_Lagrangian and Linear Element | |

Poisson Ratio | POISSON_RATIO | Total_Lagrangian and Linear Element | ||

Isotropic_Damage() | Young Modulus | YOUNG_MODULUS | Total_Lagrangian | |

Poisson Ratio | POISSON_RATIO | Total_Lagrangian | ||

Fracture Energy | FRACTURE_ENERGY | Total_Lagrangian | ||

Traction_Strength | FT | Total_Lagrangian | ||

Compresion_Strength | FC | Total_Lagrangian | ||

Fluency Criteria (Only Yield Energy Criterium) | N/A | Total_Lagrangian | ||

Softening Behavior (Only Exponencial Softening) | N/A | Total_Lagrangian | ||

Isotropic_Damage3D() | Young Modulus | YOUNG_MODULUS | Total_Lagrangian | |

Poisson Ratio | POISSON_RATIO | Total_Lagrangian | ||

Fracture Energy | FRACTURE_ENERGY | Total_Lagrangian | ||

Traction_Strength | FT | Total_Lagrangian | ||

Compresion_Strength | FC | Total_Lagrangian | ||

Fluency Criteria (Only Yield Energy Criterium) | N/A | Total_Lagrangian | ||

Softening Behavior (Only Exponencial Softening) | N/A | Total_Lagrangian | ||

Platicity_2d() (Von Misses and Tresca) | Young Modulus | YOUNG_MODULUS | Total_Lagrangian | |

Poisson Ratio | POISSON_RATIO | Total_Lagrangian | ||

Yield Stress | YIELD_STRESS | Total_Lagrangian | ||

Isotropic Hardening Modulus | ISOTROPIC_HARDENING_MODULUS | Total_Lagrangian | ||

Kinematic Hardening Modulus | KINEMATIC_HARDENING_MODULUS | Total_Lagrangian | ||

Fluency Criteria (Only Yield Energy Criterium) | N/A | Total_Lagrangian | ||

Softening Behavior (Only Exponencial Softening) | N/A | Total_Lagrangian | ||

Platicity_2d() (Morh Coulomb Model) | Young Modulus | YOUNG_MODULUS | Total_Lagrangian | |

Poisson Ratio | POISSON_RATIO | Total_Lagrangian | ||

Fracture Energy | FRACTURE_ENERGY | Total_Lagrangian | ||

Crushing Energy | CRUSHING_ENERGY | Total_Lagrangian | ||

Friction Internal Angle | MAX_FRICTION_INTERNAL_ANGLE | Total_Lagrangian | ||

Friction Internal Angle | MAX_DILATANCY_ANGLE | Total_Lagrangian | ||

Traction_Strength | FT | Total_Lagrangian | ||

Compresion_Strength | FC | Total_Lagrangian | ||

Fluency Criteria (Only Yield Energy Criterium) | N/A | Total_Lagrangian | ||

Softening Behavior (Only Exponencial Softening) | N/A | Total_Lagrangian | ||

Hyperelastic2d() | Shear Modulus | MU | Membrane Element | |

Bulk Modulus | BULK_MODULUS | Membrane Element | ||

Retraction Time | RETRACTION_TIME | Membrane Element | ||

Alpha | ALPHA | Membrane Element | ||

Hyperelastic3d() | Shear Modulus | MU | Total_Lagrangian | |

Bulk Modulus | BULK_MODULUS | Total_Lagrangian | ||

Viscoelastic2d() | Relaxation Time (5 parameters) | N/A | Membrane Element | |

Beta (5 parameters) | N/A | Membrane Element | ||

Shear Modulus | MU | Membrane Element | ||

Bulk Modulus | BULK_MODULUS | Membrane Element | ||

Retraction Time | RETRACTION_TIME | Membrane Element | ||

Alpha | ALPHA | Membrane Element | ||

Viscofibers2d() | Fiber parameters K1 and K2 (2 parameters) | N/A | Membrane Element | |

Fiber direction (6 components) | N/A | Membrane Element | ||

Relaxation Time (5 parameters) | N/A | Membrane Element | ||

Beta (5 parameters) | N/A | Membrane Element | ||

Shear Modulus | MU | Membrane Element | ||

Bulk Modulus | BULK_MODULUS | Membrane Element | ||

Retraction Time | RETRACTION_TIME | Membrane Element | ||

Alpha | ALPHA | Membrane Element | ||

Viscofibers_Hypermatrix2d() | Fiber parameters K1 and K2 (2 parameters) | N/A | Membrane Element | |

Fiber direction (6 components) | N/A | Membrane Element | ||

Relaxation Time (5 parameters) | N/A | Membrane Element | ||

Beta (5 parameters) | N/A | Membrane Element | ||

Shear Modulus | MU | Membrane Element | ||

Bulk Modulus | BULK_MODULUS | Membrane Element | ||

Retraction Time | RETRACTION_TIME | Membrane Element | ||

Alpha | ALPHA | Membrane Element | ||

## List of flux

Fluency | Flux | State |
---|---|---|

Von Misses | Asociated | Plane Stress, Plane Strain , TriD |

Energy | Asociated | Plane Stress, Plane Strain , TriD |

Tresca | Asociated | Plane Stress, Plane Strain , TriD |

Morh-Coulomb | Asociated and NonAsociated | Plane Stress, Plane Strain , TriD |

Rankine | Asociated | Plane Stress, Plane Strain , TriD |

Drucker Prager | Asociated | Plane Stress, Plane Strain , TriD |