kinematics.h

// ---------------------------------------------------------------------
//
// Copyright (C) 2016 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE.md at
// the top level directory of deal.II.
//
// ---------------------------------------------------------------------

#ifndef dealii_elasticity_kinematics_h
#define dealii_elasticity_kinematics_h


#include <deal.II/base/numbers.h>
#include <deal.II/base/symmetric_tensor.h>
#include <deal.II/base/tensor.h>

#include <deal.II/physics/elasticity/standard_tensors.h>

DEAL_II_NAMESPACE_OPEN

namespace Physics
{
  namespace Elasticity
  {
    namespace Kinematics
    {

      template <int dim, typename Number>
      Tensor<2, dim, Number>
      F(const Tensor<2, dim, Number> &Grad_u);

      template <int dim, typename Number>
      Tensor<2, dim, Number>
      F_iso(const Tensor<2, dim, Number> &F);

      template <int dim, typename Number>
      SymmetricTensor<2, dim, Number>
      F_vol(const Tensor<2, dim, Number> &F);

      template <int dim, typename Number>
      SymmetricTensor<2, dim, Number>
      C(const Tensor<2, dim, Number> &F);

      template <int dim, typename Number>
      SymmetricTensor<2, dim, Number>
      b(const Tensor<2, dim, Number> &F);



      template <int dim, typename Number>
      SymmetricTensor<2, dim, Number>
      E(const Tensor<2, dim, Number> &F);

      template <int dim, typename Number>
      SymmetricTensor<2, dim, Number>
      epsilon(const Tensor<2, dim, Number> &Grad_u);

      template <int dim, typename Number>
      SymmetricTensor<2, dim, Number>
      e(const Tensor<2, dim, Number> &F);



      template <int dim, typename Number>
      Tensor<2, dim, Number>
      l(const Tensor<2, dim, Number> &F, const Tensor<2, dim, Number> &dF_dt);

      template <int dim, typename Number>
      SymmetricTensor<2, dim, Number>
      d(const Tensor<2, dim, Number> &F, const Tensor<2, dim, Number> &dF_dt);

      template <int dim, typename Number>
      Tensor<2, dim, Number>
      w(const Tensor<2, dim, Number> &F, const Tensor<2, dim, Number> &dF_dt);

    } // namespace Kinematics
  }   // namespace Elasticity
} // namespace Physics



#ifndef DOXYGEN

// ------------------------- inline functions ------------------------



template <int dim, typename Number>
inline Tensor<2, dim, Number>
Physics::Elasticity::Kinematics::F(const Tensor<2, dim, Number> &Grad_u)
{
  return StandardTensors<dim>::I + Grad_u;
}



template <int dim, typename Number>
inline Tensor<2, dim, Number>
Physics::Elasticity::Kinematics::F_iso(const Tensor<2, dim, Number> &F)
{
  return std::pow(determinant(F), -1.0 / dim) * F;
}



template <int dim, typename Number>
inline SymmetricTensor<2, dim, Number>
Physics::Elasticity::Kinematics::F_vol(const Tensor<2, dim, Number> &F)
{
  return internal::NumberType<Number>::value(
           std::pow(determinant(F), 1.0 / dim)) *
         static_cast<SymmetricTensor<2, dim, Number>>(
           unit_symmetric_tensor<dim>());
}



template <int dim, typename Number>
inline SymmetricTensor<2, dim, Number>
Physics::Elasticity::Kinematics::C(const Tensor<2, dim, Number> &F)
{
  return symmetrize(transpose(F) * F);
}



template <int dim, typename Number>
inline SymmetricTensor<2, dim, Number>
Physics::Elasticity::Kinematics::b(const Tensor<2, dim, Number> &F)
{
  return symmetrize(F * transpose(F));
}



template <int dim, typename Number>
inline SymmetricTensor<2, dim, Number>
Physics::Elasticity::Kinematics::E(const Tensor<2, dim, Number> &F)
{
  return internal::NumberType<Number>::value(0.5) *
         (C(F) - static_cast<SymmetricTensor<2, dim, Number>>(
                   StandardTensors<dim>::I));
}



template <int dim, typename Number>
inline SymmetricTensor<2, dim, Number>
Physics::Elasticity::Kinematics::epsilon(const Tensor<2, dim, Number> &Grad_u)
{
  // This is the equivalent to 0.5*symmetrize(Grad_u + transpose(Grad_u));
  return symmetrize(Grad_u);
}



template <int dim, typename Number>
inline SymmetricTensor<2, dim, Number>
Physics::Elasticity::Kinematics::e(const Tensor<2, dim, Number> &F)
{
  const Tensor<2, dim, Number> F_inv = invert(F);
  return internal::NumberType<Number>::value(0.5) *
         symmetrize(static_cast<SymmetricTensor<2, dim, Number>>(
                      StandardTensors<dim>::I) -
                    transpose(F_inv) * F_inv);
}



template <int dim, typename Number>
inline Tensor<2, dim, Number>
Physics::Elasticity::Kinematics::l(const Tensor<2, dim, Number> &F,
                                   const Tensor<2, dim, Number> &dF_dt)
{
  return dF_dt * invert(F);
}



template <int dim, typename Number>
inline SymmetricTensor<2, dim, Number>
Physics::Elasticity::Kinematics::d(const Tensor<2, dim, Number> &F,
                                   const Tensor<2, dim, Number> &dF_dt)
{
  return symmetrize(l(F, dF_dt));
}



template <int dim, typename Number>
inline Tensor<2, dim, Number>
Physics::Elasticity::Kinematics::w(const Tensor<2, dim, Number> &F,
                                   const Tensor<2, dim, Number> &dF_dt)
{
  // This could be implemented as w = l-d, but that would mean computing "l"
  // a second time.
  const Tensor<2, dim> grad_v = l(F, dF_dt);
  return internal::NumberType<Number>::value(0.5) *
         (grad_v - transpose(grad_v));
}

#endif // DOXYGEN

DEAL_II_NAMESPACE_CLOSE

#endif