Viscoplasticity

Building blocks for viscoplastic problems. More...

Classes
class	Kaskade::Elastomechanics::MaterialLaws::ViscoPlasticEnergy< HyperelasticEnergy >
	An adaptor for using hyperelastic stored energies for viscoplasticity. More...

Functions
double	Kaskade::Elastomechanics::yieldStrength (std::string const &name)
	Yield strengths of several materials in Pa. More...
template<int dim, class Scalar >
Scalar	Kaskade::Elastomechanics::MaterialLaws::vonMisesStress (Dune::FieldMatrix< Scalar, dim, dim > const &stress)
	Computes the von Mises equivalent stress \( \sigma_v \). More...
template<int dim, class Scalar = double>
Dune::FieldVector< Scalar, dim *(dim+1)/2 >	Kaskade::Elastomechanics::MaterialLaws::duvautLionsJ2Flow (Dune::FieldMatrix< Scalar, dim, dim > const &cauchyStress, Dune::FieldMatrix< Scalar, dim *(dim+1)/2, dim *(dim+1)/2 > const &C, double tau, double sigmaY)
	A simple Duvaut-Lions flow rule with \( J_2 \) (von Mises) yield surface. More...

Detailed Description

Building blocks for viscoplastic problems.

Function Documentation

duvautLionsJ2Flow()

template<int dim, class Scalar = double>

Dune::FieldVector< Scalar, dim *(dim+1)/2 > Kaskade::Elastomechanics::MaterialLaws::duvautLionsJ2Flow	(	Dune::FieldMatrix< Scalar, dim, dim > const &	cauchyStress,
		Dune::FieldMatrix< Scalar, dim *(dim+1)/2, dim *(dim+1)/2 > const &	C,
		double	tau,
		double	sigmaY
	)

A simple Duvaut-Lions flow rule with \( J_2 \) (von Mises) yield surface.

The Duvaut-Lions model of viscoplasticity defines the flow rate, i.e. the time derivative of the viscoplastic strain, as

\[ \dot \epsilon^{\rm vp} = \tau^{-1} C^{-1} (\sigma - P\sigma), \]

where \( P \) is the closest point projector onto the admissible stress state. In \( J_2 \) viscoplasticity, the admissible stresses are characterized by \( \|\sigma\| \le \sqrt{2/3}\,\sigma_Y \), and the boundary of that set is known as von Mises yield surface.

Parameters

cauchyStress	the true stress tensor
C	the strain to stress matrix in Voigt notation
tau	the relaxation time (>0)
sigmaY	the yield strength (>=0)

Returns

the flow rate \( \dot \epsilon^{\rm vp} \) in Voigt notation

vonMisesStress()

template<int dim, class Scalar >

Scalar Kaskade::Elastomechanics::MaterialLaws::vonMisesStress

(

Dune::FieldMatrix< Scalar, dim, dim > const &

stress

)

Computes the von Mises equivalent stress \( \sigma_v \).

The equivalent von Mises stress is defined as

\[ 2\sigma_v^2 = (\sigma_{11}-\sigma_{22})^2 + (\sigma_{22}-\sigma_{33})^2 + (\sigma_{33}-\sigma_{11})^2 + 6 (\sigma_{23}^2+\sigma_{31}^2 + \sigma_{12}^2) \]

in terms of the Cauchy stress \( \sigma \).

Template Parameters

dim	the spatial dimension. For d=2, plane stress is assumed.
Scalar	the scalar field type of the stress tensor, usually double

Assuming there are finite element functions lambda and mu giving the material parameters, and u the displacement, a scalar finite element function sv of von Mises equivalent stress values can be obtained as follows:

interpolateGlobally<PlainAverage>(sv,makeFunctionView(u.space(), [&] (auto const& evaluator)
{
  ElasticModulus modulus(lambda.value(evaluator.cell(),evaluator.xloc()),
                         mu.value(evaluator.cell(),evaluator.xloc()));
  HyperelasticVariationalFunctional<StVenantKirchhoff<dim>,GreenLagrangeTensor<double,dim>>  material(modulus);
  material.setLinearizationPoint(u.derivative(evaluator));
  return Dune::FieldVector<double,1>(vonMisesStress(material.cauchyStress()));
}));

yieldStrength()

double Kaskade::Elastomechanics::yieldStrength

(

std::string const &

name

)

Yield strengths of several materials in Pa.

Throws a LookupException if the material is not available in the data base.