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/**
* @file materials_toolbox.hh
*
* @author Till Junge <till.junge@epfl.ch>
*
* @date 02 Nov 2017
*
* @brief collection of common continuum mechanics tools
*
* Copyright © 2017 Till Junge
*
* µSpectre is free software; you can redistribute it and/or
* modify it under the terms of the GNU General Public License as
* published by the Free Software Foundation, either version 3, or (at
* your option) any later version.
*
* µSpectre is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with µSpectre; see the file COPYING. If not, write to the
* Free Software Foundation, Inc., 59 Temple Place - Suite 330,
* Boston, MA 02111-1307, USA.
*/
#ifndef MATERIALS_TOOLBOX_H
#define MATERIALS_TOOLBOX_H
#include "common/common.hh"
#include "common/tensor_algebra.hh"
#include "common/eigen_tools.hh"
#include "common/T4_map_proxy.hh"
#include <Eigen/Dense>
#include <unsupported/Eigen/MatrixFunctions>
#include <exception>
#include <sstream>
#include <iostream>
#include <tuple>
#include <type_traits>
namespace muSpectre {
namespace MatTB {
/**
* thrown when generic materials-related runtime errors occur
* (mostly continuum mechanics problems)
*/
class MaterialsToolboxError:public std::runtime_error{
public:
//! constructor
explicit MaterialsToolboxError(const std::string& what)
:std::runtime_error(what){}
//! constructor
explicit MaterialsToolboxError(const char * what)
:std::runtime_error(what){}
};
/* ---------------------------------------------------------------------- */
/**
* Flag used to designate whether the material should compute both stress
* and tangent moduli or only stress
*/
enum class NeedTangent {
yes, //!< compute both stress and tangent moduli
no //!< compute only stress
};
/**
* struct used to determine the exact type of a tuple of references obtained
* when a bunch of iterators over fiel_maps are dereferenced and their
* results are concatenated into a tuple
*/
template <class... T>
struct ReferenceTuple {
//! use this type
using type = std::tuple<typename T::reference ...>;
};
/**
* specialisation for tuples
*/
//template <>
template <class... T>
struct ReferenceTuple<std::tuple<T...>> {
//! use this type
using type = typename ReferenceTuple<T...>::type;
};
/**
* helper type for ReferenceTuple
*/
template <class... T>
using ReferenceTuple_t = typename ReferenceTuple<T...>::type;
/* ---------------------------------------------------------------------- */
namespace internal {
/** Structure for functions returning one strain measure as a
* function of another
**/
template <StrainMeasure In, StrainMeasure Out = In>
struct ConvertStrain {
static_assert((In == StrainMeasure::Gradient) ||
(In == StrainMeasure::Infinitesimal),
"This situation makes me suspect that you are not using "
"MatTb as intended. Disable this assert only if you are "
"sure about what you are doing.");
//! returns the converted strain
template <class Strain_t>
inline static decltype(auto)
compute(Strain_t&& input) {
// transparent case, in which no conversion is required:
// just a perfect forwarding
static_assert
((In == Out),
"This particular strain conversion is not implemented");
return std::forward<Strain_t>(input);
}
};
/* ---------------------------------------------------------------------- */
/** Specialisation for getting Green-Lagrange strain from the
transformation gradient
E = ¹/₂ (C - I) = ¹/₂ (Fᵀ·F - I)
**/
template <>
struct ConvertStrain<StrainMeasure::Gradient, StrainMeasure::GreenLagrange> {
//! returns the converted strain
template <class Strain_t>
inline static decltype(auto)
compute(Strain_t&& F) {
return .5*(F.transpose()*F - Strain_t::PlainObject::Identity());
}
};
/* ---------------------------------------------------------------------- */
/** Specialisation for getting Left Cauchy-Green strain from the
transformation gradient
B = F·Fᵀ = V²
**/
template <>
struct ConvertStrain<StrainMeasure::Gradient, StrainMeasure::LCauchyGreen> {
//! returns the converted strain
template <class Strain_t>
inline static decltype(auto)
compute(Strain_t&& F) {
return F*F.transpose();
}
};
/* ---------------------------------------------------------------------- */
/** Specialisation for getting Right Cauchy-Green strain from the
transformation gradient
C = Fᵀ·F = U²
**/
template <>
struct ConvertStrain<StrainMeasure::Gradient, StrainMeasure::RCauchyGreen> {
//! returns the converted strain
template <class Strain_t>
inline static decltype(auto)
compute(Strain_t&& F) {
return F.transpose()*F;
}
};
/* ---------------------------------------------------------------------- */
/** Specialisation for getting logarithmic (Hencky) strain from the
transformation gradient
E₀ = ¹/₂ ln C = ¹/₂ ln (Fᵀ·F)
**/
template <>
struct ConvertStrain<StrainMeasure::Gradient, StrainMeasure::Log> {
//! returns the converted strain
template <class Strain_t>
inline static decltype(auto)
compute(Strain_t&& F) {
constexpr Dim_t dim{EigenCheck::tensor_dim<Strain_t>::value};
return (.5*logm(Eigen::Matrix<Real, dim, dim>{F.transpose()*F})).eval();
}
};
} // internal
/* ---------------------------------------------------------------------- */
//! set of functions returning one strain measure as a function of
//! another
template <StrainMeasure In, StrainMeasure Out,
class Strain_t>
decltype(auto) convert_strain(Strain_t && strain) {
return internal::ConvertStrain<In, Out>::compute(std::move(strain));
};
/* ---------------------------------------------------------------------- */
namespace internal {
/** Structure for functions returning PK1 stress from other stress measures
**/
template <Dim_t Dim,
StressMeasure StressM,
StrainMeasure StrainM>
struct PK1_stress {
//! returns the converted stress
template <class Strain_t, class Stress_t>
inline static decltype(auto)
compute(Strain_t && /*strain*/, Stress_t && /*stress*/) {
// the following test always fails to generate a compile-time error
static_assert((StressM == StressMeasure::Cauchy) &&
(StressM == StressMeasure::PK1),
"The requested Stress conversion is not implemented. "
"You either made a programming mistake or need to "
"implement it as a specialisation of this function. "
"See PK2stress<PK1,T1, T2> for an example.");
}
//! returns the converted stress and stiffness
template <class Strain_t, class Stress_t, class Tangent_t>
inline static decltype(auto)
compute(Strain_t && /*strain*/, Stress_t && /*stress*/,
Tangent_t && /*stiffness*/) {
// the following test always fails to generate a compile-time error
static_assert((StressM == StressMeasure::Cauchy) &&
(StressM == StressMeasure::PK1),
"The requested Stress conversion is not implemented. "
"You either made a programming mistake or need to "
"implement it as a specialisation of this function. "
"See PK2stress<PK1,T1, T2> for an example.");
}
};
/* ---------------------------------------------------------------------- */
/** Specialisation for the transparent case, where we already
have PK1 stress
**/
template <Dim_t Dim, StrainMeasure StrainM>
struct PK1_stress<Dim, StressMeasure::PK1, StrainM>:
public PK1_stress<Dim, StressMeasure::no_stress_,
StrainMeasure::no_strain_> {
//! returns the converted stress
template <class Strain_t, class Stress_t>
inline static decltype(auto)
compute(Strain_t && /*dummy*/, Stress_t && P) {
return std::forward<Stress_t>(P);
}
};
/* ---------------------------------------------------------------------- */
/** Specialisation for the transparent case, where we already have PK1
stress *and* stiffness is given with respect to the transformation
gradient
**/
template <Dim_t Dim>
struct PK1_stress<Dim, StressMeasure::PK1, StrainMeasure::Gradient>:
public PK1_stress<Dim, StressMeasure::PK1,
StrainMeasure::no_strain_> {
//! base class
using Parent = PK1_stress<Dim, StressMeasure::PK1,
StrainMeasure::no_strain_>;
using Parent::compute;
//! returns the converted stress and stiffness
template <class Strain_t, class Stress_t, class Tangent_t>
inline static decltype(auto)
compute(Strain_t && /*dummy*/, Stress_t && P, Tangent_t && K) {
return std::make_tuple(std::forward<Stress_t>(P),
std::forward<Tangent_t>(K));
}
};
/* ---------------------------------------------------------------------- */
/**
* Specialisation for the case where we get material stress (PK2)
*/
template <Dim_t Dim, StrainMeasure StrainM>
struct PK1_stress<Dim, StressMeasure::PK2, StrainM>:
public PK1_stress<Dim, StressMeasure::no_stress_,
StrainMeasure::no_strain_> {
//! returns the converted stress
template <class Strain_t, class Stress_t>
inline static decltype(auto)
compute(Strain_t && F, Stress_t && S) {
return F*S;
}
};
/* ---------------------------------------------------------------------- */
/**
* Specialisation for the case where we get material stress (PK2) derived
* with respect to Green-Lagrange strain
*/
template <Dim_t Dim>
struct PK1_stress<Dim, StressMeasure::PK2, StrainMeasure::GreenLagrange>:
public PK1_stress<Dim, StressMeasure::PK2,
StrainMeasure::no_strain_> {
//! base class
using Parent = PK1_stress<Dim, StressMeasure::PK2,
StrainMeasure::no_strain_>;
using Parent::compute;
//! returns the converted stress and stiffness
template <class Strain_t, class Stress_t, class Tangent_t>
inline static decltype(auto)
compute(Strain_t && F, Stress_t && S, Tangent_t && C) {
using T4 = typename std::remove_reference_t<Tangent_t>::PlainObject;
using Tmap = T4MatMap<Real, Dim>;
T4 K;
Tmap Kmap{K.data()};
K.setZero();
for (int i = 0; i < Dim; ++i) {
for (int m = 0; m < Dim; ++m) {
for (int n = 0; n < Dim; ++n) {
get(Kmap,i,m,i,n) += S(m,n);
for (int j = 0; j < Dim; ++j) {
for (int r = 0; r < Dim; ++r) {
for (int s = 0; s < Dim; ++s) {
get(Kmap,i,m,j,n) += F(i,r)*get(C,r,m,n,s)*(F(j,s));
}
}
}
}
}
}
auto && P = compute(std::forward<Strain_t>(F),
std::forward<Stress_t>(S));
return std::make_tuple(std::move(P), std::move(K));
}
};
/* ---------------------------------------------------------------------- */
/**
* Specialisation for the case where we get Kirchhoff stress (τ)
*/
template <Dim_t Dim, StrainMeasure StrainM>
struct PK1_stress<Dim, StressMeasure::Kirchhoff, StrainM>:
public PK1_stress<Dim, StressMeasure::no_stress_,
StrainMeasure::no_strain_> {
//! returns the converted stress
template <class Strain_t, class Stress_t>
inline static decltype(auto)
compute(Strain_t && F, Stress_t && tau) {
return tau*F.inverse().transpose();
}
};
/* ---------------------------------------------------------------------- */
/**
* Specialisation for the case where we get Kirchhoff stress (τ) derived
* with respect to Gradient
*/
template <Dim_t Dim>
struct PK1_stress<Dim, StressMeasure::Kirchhoff, StrainMeasure::Gradient>:
public PK1_stress<Dim, StressMeasure::Kirchhoff,
StrainMeasure::no_strain_> {
//! short-hand
using Parent = PK1_stress<Dim, StressMeasure::Kirchhoff,
StrainMeasure::no_strain_>;
using Parent::compute;
//! returns the converted stress and stiffness
template <class Strain_t, class Stress_t, class Tangent_t>
inline static decltype(auto)
compute(Strain_t && F, Stress_t && tau, Tangent_t && C) {
using T4 = typename std::remove_reference_t<Tangent_t>::PlainObject;
using Tmap = T4MatMap<Real, Dim>;
T4 K;
Tmap Kmap{K.data()};
K.setZero();
auto && F_inv{F.inverse()};
for (int i = 0; i < Dim; ++i) {
for (int m = 0; m < Dim; ++m) {
for (int n = 0; n < Dim; ++n) {
for (int j = 0; j < Dim; ++j) {
for (int r = 0; r < Dim; ++r) {
for (int s = 0; s < Dim; ++s) {
get(Kmap,i,m,j,n) += F_inv(i,r)*get(C,r,m,n,s);
}
}
}
}
}
}
auto && P = tau * F_inv.transpose();
return std::make_tuple(std::move(P), std::move(K));
}
};
} // internal
/* ---------------------------------------------------------------------- */
//! set of functions returning an expression for PK2 stress based on
template <StressMeasure StressM, StrainMeasure StrainM,
class Stress_t, class Strain_t>
decltype(auto) PK1_stress(Strain_t && strain, Stress_t && stress) {
constexpr Dim_t dim{EigenCheck::tensor_dim<Strain_t>::value};
static_assert((dim == EigenCheck::tensor_dim<Stress_t>::value),
"Stress and strain tensors have differing dimensions");
return internal::PK1_stress<dim, StressM, StrainM>::compute
(std::forward<Strain_t>(strain),
std::forward<Stress_t>(stress));
};
/* ---------------------------------------------------------------------- */
//! set of functions returning an expression for PK2 stress based on
template <StressMeasure StressM, StrainMeasure StrainM,
class Stress_t, class Strain_t, class Tangent_t>
decltype(auto) PK1_stress(Strain_t && strain,
Stress_t && stress,
Tangent_t && tangent) {
constexpr Dim_t dim{EigenCheck::tensor_dim<Strain_t>::value};
static_assert((dim == EigenCheck::tensor_dim<Stress_t>::value),
"Stress and strain tensors have differing dimensions");
static_assert((dim == EigenCheck::tensor_4_dim<Tangent_t>::value),
"Stress and tangent tensors have differing dimensions");
return internal::PK1_stress<dim, StressM, StrainM>::compute
(std::forward<Strain_t>(strain),
std::forward<Stress_t>(stress),
std::forward<Tangent_t>(tangent));
};
namespace internal {
//! Base template for elastic modulus conversion
template <ElasticModulus Out, ElasticModulus In1, ElasticModulus In2>
struct Converter {
//! wrapped function (raison d'être)
inline constexpr static Real compute(const Real& /*in1*/,
const Real& /*in2*/) {
// if no return has happened until now, the conversion is not
// implemented yet
static_assert((In1 == In2),
"This conversion has not been implemented yet, please add "
"it here below as a specialisation of this function "
"template. Check "
"https://en.wikipedia.org/wiki/Lam%C3%A9_parameters for "
"// TODO: he formula.");
return 0;
}
};
/**
* Spectialisation for when the output is the first input
*/
template <ElasticModulus Out, ElasticModulus In>
struct Converter<Out, Out, In> {
//! wrapped function (raison d'être)
inline constexpr static Real compute(const Real & A, const Real & /*B*/) {
return A;
}
};
/**
* Spectialisation for when the output is the second input
*/
template <ElasticModulus Out, ElasticModulus In>
struct Converter<Out, In, Out> {
//! wrapped function (raison d'être)
inline constexpr static Real compute(const Real & /*A*/, const Real & B) {
return B;
}
};
/**
* Specialisation μ(E, ν)
*/
template <>
struct Converter<ElasticModulus::Shear,
ElasticModulus::Young,
ElasticModulus::Poisson> {
//! wrapped function (raison d'être)
inline constexpr static Real compute(const Real & E, const Real & nu) {
return E/(2*(1+nu));
}
};
/**
* Specialisation λ(E, ν)
*/
template <>
struct Converter<ElasticModulus::lambda,
ElasticModulus::Young,
ElasticModulus::Poisson> {
//! wrapped function (raison d'être)
inline constexpr static Real compute(const Real & E, const Real & nu) {
return E*nu/((1+nu)*(1-2*nu));
}
};
/**
* Specialisation K(E, ν)
*/
template <>
struct Converter<ElasticModulus::Bulk,
ElasticModulus::Young,
ElasticModulus::Poisson> {
//! wrapped function (raison d'être)
inline constexpr static Real compute(const Real & E, const Real & nu) {
return E/(3*(1-2*nu));
}
};
/**
* Specialisation E(K, µ)
*/
template <>
struct Converter<ElasticModulus::Young,
ElasticModulus::Bulk,
ElasticModulus::Shear> {
//! wrapped function (raison d'être)
inline constexpr static Real compute(const Real & K, const Real & G) {
return 9*K*G/(3*K+G);
}
};
/**
* Specialisation ν(K, µ)
*/
template <>
struct Converter<ElasticModulus::Poisson,
ElasticModulus::Bulk,
ElasticModulus::Shear> {
//! wrapped function (raison d'être)
inline constexpr static Real compute(const Real & K, const Real & G) {
return (3*K - 2*G) /(2*(3*K + G));
}
};
/**
* Specialisation E(λ, µ)
*/
template <>
struct Converter<ElasticModulus::Young,
ElasticModulus::lambda,
ElasticModulus::Shear> {
//! wrapped function (raison d'être)
inline constexpr static Real compute(const Real & lambda, const Real & G) {
return G * (3*lambda + 2*G)/(lambda + G);
}
};
} // internal
/**
* allows the conversion from any two distinct input moduli to a
* chosen output modulus
*/
template <ElasticModulus Out, ElasticModulus In1, ElasticModulus In2>
inline constexpr Real convert_elastic_modulus(const Real& in1,
const Real& in2) {
// enforcing sanity
static_assert((In1 != In2),
"The input modulus types cannot be identical");
// enforcing independence from order in which moduli are supplied
constexpr bool inverted{In1 > In2};
using Converter = std::conditional_t<inverted,
internal::Converter<Out, In2, In1>,
internal::Converter<Out, In1, In2>>;
if (inverted) {
return Converter::compute(std::move(in2),
std::move(in1));
} else {
return Converter::compute(std::move(in1),
std::move(in2));
}
}
//! static inline implementation of Hooke's law
template <Dim_t Dim, class Strain_t, class Tangent_t>
struct Hooke {
/**
* compute Lamé's first constant
* @param young: Young's modulus
* @param poisson: Poisson's ratio
*/
inline static constexpr Real
compute_lambda(const Real & young, const Real & poisson) {
return convert_elastic_modulus<ElasticModulus::lambda,
ElasticModulus::Young,
ElasticModulus::Poisson>(young, poisson);
}
/**
* compute Lamé's second constant (i.e., shear modulus)
* @param young: Young's modulus
* @param poisson: Poisson's ratio
*/
inline static constexpr Real
compute_mu(const Real & young, const Real & poisson) {
return convert_elastic_modulus<ElasticModulus::Shear,
ElasticModulus::Young,
ElasticModulus::Poisson>(young, poisson);
}
/**
* compute the bulk modulus
* @param young: Young's modulus
* @param poisson: Poisson's ratio
*/
inline static constexpr Real
compute_K(const Real & young, const Real & poisson) {
return convert_elastic_modulus<ElasticModulus::Bulk,
ElasticModulus::Young,
ElasticModulus::Poisson>(young, poisson);
}
/**
* compute the stiffness tensor
* @param lambda: Lamé's first constant
* @param mu: Lamé's second constant (i.e., shear modulus)
*/
inline static Eigen::TensorFixedSize<Real, Eigen::Sizes<Dim, Dim, Dim, Dim>>
compute_C(const Real & lambda, const Real & mu) {
return lambda*Tensors::outer<Dim>(Tensors::I2<Dim>(),Tensors::I2<Dim>()) +
2*mu*Tensors::I4S<Dim>();
}
/**
* compute the stiffness tensor
* @param lambda: Lamé's first constant
* @param mu: Lamé's second constant (i.e., shear modulus)
*/
inline static T4Mat<Real, Dim>
compute_C_T4(const Real & lambda, const Real & mu) {
return lambda*Matrices::Itrac<Dim>() + 2*mu*Matrices::Isymm<Dim>();
}
/**
* return stress
* @param lambda: First Lamé's constant
* @param mu: Second Lamé's constant (i.e. shear modulus)
* @param E: Green-Lagrange or small strain tensor
*/
template <class s_t>
inline static decltype(auto)
evaluate_stress(const Real & lambda, const Real & mu, s_t && E) {
return E.trace()*lambda * Strain_t::Identity() + 2*mu*E;
}
/**
* return stress and tangent stiffness
* @param lambda: First Lamé's constant
* @param mu: Second Lamé's constant (i.e. shear modulus)
* @param E: Green-Lagrange or small strain tensor
* @param C: stiffness tensor (Piola-Kirchhoff 2 (or σ) w.r.t to `E`)
*/
template <class s_t>
inline static decltype(auto)
evaluate_stress(const Real & lambda, const Real & mu,
Tangent_t && C, s_t && E) {
return std::make_tuple
(std::move(evaluate_stress(lambda, mu, std::move(E))),
std::move(C));
}
};
} // MatTB
} // muSpectre
#endif /* MATERIALS_TOOLBOX_H */
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