Beispiel #1
0
  void HookesLaw::compute_stress_imp( unsigned int dim,
                                      const libMesh::TensorValue<libMesh::Real>& g_contra,
                                      const libMesh::TensorValue<libMesh::Real>& g_cov,
                                      const libMesh::TensorValue<libMesh::Real>& /*G_contra*/,
                                      const libMesh::TensorValue<libMesh::Real>& G_cov,
                                      libMesh::TensorValue<libMesh::Real>& stress )
  {
    stress.zero();

    for( unsigned int i = 0; i < dim; i++ )
      {
        for( unsigned int j = 0; j < dim; j++ )
          {
            for( unsigned int k = 0; k < dim; k++ )
              {
                for( unsigned int l = 0; l < dim; l++ )
                  {
                    libMesh::Real strain_kl = 0.5*(G_cov(k,l) - g_cov(k,l));

                    _C(i,j,k,l) = _lambda*g_contra(i,j)*g_contra(k,l) +
                                  _mu*(g_contra(i,k)*g_contra(j,l) + g_contra(i,l)*g_contra(j,k));

                    stress(i,j) += _C(i,j,k,l)*strain_kl;
                  }
              }
          }
      }

    return;
  }
  void IncompressiblePlaneStressHyperelasticity<StrainEnergy>::compute_stress_imp( unsigned int /*dim*/,
                                                                                   const libMesh::TensorValue<libMesh::Real>& a_contra,
                                                                                   const libMesh::TensorValue<libMesh::Real>& a_cov,
                                                                                   const libMesh::TensorValue<libMesh::Real>& A_contra,
                                                                                   const libMesh::TensorValue<libMesh::Real>& A_cov,
                                                                                   libMesh::TensorValue<libMesh::Real>& stress )
  {
    // We're treating a_* and A_* as 2x2, but we're cheating to pick up lambda^2
    libMesh::Real lambda_sq = A_cov(2,2);

    libMesh::Real A_over_a = 1.0/lambda_sq; // We're incompressible

    libMesh::Real I1, I2;
    this->compute_I1_I2(a_contra,a_cov,A_contra,A_cov,lambda_sq,A_over_a,I1,I2);

    libMesh::Real a_term, A_term;
    this->compute_stress_terms( lambda_sq, A_over_a, I1, I2, a_term, A_term );

    // Now compute stress
    stress.zero();
    for( unsigned int alpha = 0; alpha < 2; alpha++ )
      {
        for( unsigned int beta = 0; beta < 2; beta++ )
          {
            stress(alpha,beta) = a_contra(alpha,beta)*a_term + A_contra(alpha,beta)*A_term;
          }
      }

    return;
  }
  void IncompressiblePlaneStressHyperelasticity<StrainEnergy>::compute_stress_deriv_terms( libMesh::Real lambda_sq, libMesh::Real A_over_a,
                                                                                           libMesh::Real I1, libMesh::Real I2,
                                                                                           const libMesh::TensorValue<libMesh::Real>& a_contra,
                                                                                           const libMesh::TensorValue<libMesh::Real>& A_contra,
                                                                                           libMesh::TensorValue<libMesh::Real>& daterm_dstrain,
                                                                                           libMesh::TensorValue<libMesh::Real>& dAterm_dstrain) const
  {
    daterm_dstrain.zero();
    dAterm_dstrain.zero();

    libMesh::Real dWdI1 = _W.dI1(I1,I2,1.0); // We're incompressible
    libMesh::Real dWdI2 = _W.dI2(I1,I2,1.0);

    // A = det(A_cov) = 1/det(A_contra)
    //libMesh::Real A = 1.0/( A_contra(0,0)*A_contra(1,1) - A_contra(0,1)*A_contra(1,0) );

    for( unsigned int alpha = 0; alpha < 2; alpha++ )
      {
        for( unsigned int beta = 0; beta < 2; beta++ )
          {
            // a_term = 2.0*(dWdI1 + dWdI2*lambda^2);
            // => da_dstrain = 2.0*dWdI2*dlambda^2_dstrain
            // dlambda_sq_dstrain = -2*lambda^2 A^{alpha,beta}
            const libMesh::Real dlamsq_dstrain = -2.0*lambda_sq*A_contra(alpha,beta);

            daterm_dstrain(alpha,beta) = 2.0*dWdI2*dlamsq_dstrain;

            // A_term = 2.0*dWdI2*A_over_a + p;
            // A = det(A_cov) ==> dA_dstrain = 2*A*A_contra(alpha,beta)
            // p = -2.0*lambda_sq*( dWdI1 + dWdI2*(I1-lambda_sq) );
            const libMesh::Real dI1_dstrain = 2.0*a_contra(alpha,beta) + dlamsq_dstrain;

            const libMesh::Real dp_dstrain = -2.0*dlamsq_dstrain*( dWdI1 + dWdI2*(I1-lambda_sq) )
                                             -2.0*lambda_sq*dWdI2*(dI1_dstrain - dlamsq_dstrain);

            dAterm_dstrain(alpha,beta) = 2.0*dWdI2*A_over_a*(2.0*A_contra(alpha,beta)) + dp_dstrain;
          }
      }

    return;
  }
Beispiel #4
0
  void Hyperelasticity<StrainEnergy>::compute_stress_imp( unsigned int dim,
                                                          const libMesh::TensorValue<libMesh::Real>& g_contra,
                                                          const libMesh::TensorValue<libMesh::Real>& g_cov,
                                                          const libMesh::TensorValue<libMesh::Real>& G_contra,
                                                          const libMesh::TensorValue<libMesh::Real>& G_cov,
                                                          libMesh::TensorValue<libMesh::Real>& stress )
  {
    stress.zero();

    // Compute strain invariants
    libMesh::Real I3 = (g_contra*G_cov).det();

    libMesh::Real I1 = 0.0;
    libMesh::Real I2 = 0.0;
    for( unsigned int i = 0; i < dim; i++ )
      {
        for( unsigned int j = 0; j < dim; j++ )
          {
            I1 += g_contra(i,j)*G_cov(i,j);
            I2 += G_contra(i,j)*g_cov(i,j);
          }
      }

    I2 *= I3;

    libMesh::Real dWdI1 = _W.dI1(I1,I2,I3);
    libMesh::Real dWdI2 = _W.dI2(I1,I2,I3);
    libMesh::Real dWdI3 = _W.dI3(I1,I2,I3);

    // Now compute stress
    for( unsigned int i = 0; i < dim; i++ )
      {
        for( unsigned int j = 0; j < dim; j++ )
          {
            for( unsigned int k = 0; k < dim; k++ )
              {
                for( unsigned int l = 0; l < dim; l++ )
                  {
                    stress(i,j) += 2.0*dWdI1*g_contra(i,j)
                      + 2.0*dWdI2*(I1*g_contra(i,j) - g_contra(i,k)*g_contra(j,l)*G_cov(k,l))
                      + 2.0*dWdI3*G_contra(i,j);
                  }
              }
          }
      }

    return;
  }
  void IncompressiblePlaneStressHyperelasticity<StrainEnergy>::compute_stress_and_elasticity_imp( unsigned int /*dim*/,
                                                                                                  const libMesh::TensorValue<libMesh::Real>& a_contra,
                                                                                                  const libMesh::TensorValue<libMesh::Real>& a_cov,
                                                                                                  const libMesh::TensorValue<libMesh::Real>& A_contra,
                                                                                                  const libMesh::TensorValue<libMesh::Real>& A_cov,
                                                                                                  libMesh::TensorValue<libMesh::Real>& stress,
                                                                                                  ElasticityTensor& C)
  {
    // We're treating a_* and A_* as 2x2, but we're cheating to pick up lambda^2
    libMesh::Real lambda_sq = A_cov(2,2);

    libMesh::Real A_over_a = 1.0/lambda_sq; // We're incompressible

    libMesh::Real I1, I2;
    this->compute_I1_I2(a_contra,a_cov,A_contra,A_cov,lambda_sq,A_over_a,I1,I2);

    libMesh::Real a_term, A_term;
    this->compute_stress_terms( lambda_sq, A_over_a, I1, I2, a_term, A_term );

    libMesh::TensorValue<libMesh::Real> daterm_dstrain, dAterm_dstrain;
    this->compute_stress_deriv_terms( lambda_sq, A_over_a, I1, I2, a_contra, A_contra, daterm_dstrain, dAterm_dstrain );

    ElasticityTensor dAcontra_dstrain;
    this->compute_Acontra_deriv( A_contra, dAcontra_dstrain );

    // Now compute stress
    stress.zero();
    for( unsigned int alpha = 0; alpha < 2; alpha++ )
      {
        for( unsigned int beta = 0; beta < 2; beta++ )
          {
            stress(alpha,beta) = a_contra(alpha,beta)*a_term + A_contra(alpha,beta)*A_term;

            for( unsigned int lambda = 0; lambda < 2; lambda++ )
              {
                for( unsigned int mu = 0; mu < 2; mu++ )
                  {
                    C(alpha,beta,lambda,mu) = a_contra(alpha,beta)*daterm_dstrain(lambda,mu)
                                            + dAcontra_dstrain(alpha,beta,lambda,mu)*A_term
                                            + A_contra(alpha,beta)*dAterm_dstrain(lambda,mu);
                  }
              }
          }
      }

    return;
  }