Hi,
I am trying to use symbolic differentiation to obtain the Kirchhoff stress
tensor using the material model given in step 44. I have attached the
minimal example herewith, which reproduces the following error:
In line 63, the differentiation of the energy function (psi_sd) with
respect to left cauchy green tensor (b_sd) results in the Kirchhoff stress
tensor tau_sd = zero tensor. I believe this is due to the fact that tau_sd
is stored as a function of F_sd (symbolic deformation gradient) and is
therefore unable to recognise the dependence with b_sd (symbolic left
cauchy green tensor).
How can i resolve this issue? Thanks in advance.
Regards
Vinayak
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//-----------------------------------------------------------
//
// Copyright (C) 2020 by the deal.II authors
//
// This file is subject to LGPL and may not be distributed
// without copyright and license information. Please refer
// to the file deal.II/doc/license.html for the text and
// further information on this license.
//
//-----------------------------------------------------------
#include <deal.II/base/symmetric_tensor.h>
#include <deal.II/base/tensor.h>
#include <deal.II/differentiation/sd.h>
#include <deal.II/physics/elasticity/kinematics.h>
#include <deal.II/physics/elasticity/standard_tensors.h>
using namespace dealii;
int
main()
{
const Differentiation::SD::Expression nu_e_sd("nu_e");
const Differentiation::SD::Expression lambda_sd("lambda");
const Differentiation::SD::Expression mu_e_sd("mu_e");
const Differentiation::SD::Expression kappa_e_sd("kappa_e");
const Differentiation::SD::Expression c_1_e_sd("c_1_e");
const int dim = 3;
const int spacedim = 3;
const Tensor<2, dim, Differentiation::SD::Expression> F_sd(
Differentiation::SD::make_tensor_of_symbols<2, dim>("F"));
std::cout << F_sd << std::endl;
const Differentiation::SD::Expression det_F_sd = determinant(F_sd);
const Tensor<2, 3, dealii::Differentiation::SD::Expression> F_bar_sd =
std::pow(det_F_sd, -1 / 3) * F_sd;
SymmetricTensor<2, spacedim, Differentiation::SD::Expression> b_bar_sd =
symmetrize(F_bar_sd * transpose(F_bar_sd));
SymmetricTensor<2, spacedim, Differentiation::SD::Expression> b_sd =
std::pow(det_F_sd, 1 / 3) * b_bar_sd;
std::cout << "b_sd: " << b_sd << std::endl;
const Differentiation::SD::Expression psi_vol_sd =
(kappa_e_sd / 4.0) *
(det_F_sd * det_F_sd - 1.0 - 2.0 * Differentiation::SD::log(det_F_sd));
const Differentiation::SD::Expression psi_iso_sd =
c_1_e_sd * (trace(b_bar_sd) - dim);
Differentiation::SD::Expression psi_sd;
psi_sd = psi_vol_sd + psi_iso_sd;
SymmetricTensor<2, dim, Differentiation::SD::Expression> tau_sd;
tau_sd = 2 * b_sd * Differentiation::SD::differentiate(psi_sd, b_sd);
Tensor<2, dim, Differentiation::SD::Expression> first_PK_sd;
first_PK_sd = 2 * b_sd * Differentiation::SD::differentiate(psi_sd, F_sd);
std::cout << "psi_sd: " << psi_sd << std::endl;
std::cout << "first_PK_sd: " << first_PK_sd << std::endl;
std::cout << "tau_sd: " << tau_sd << std::endl;
}
