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KASKADE 7 development version
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A stored energy density class for Taylor-based shell models. More...
#include <shellKinematics.hh>
A stored energy density class for Taylor-based shell models.
Assume on the reference configuration \(\Omega = \omega\times\mathopen]-\tau,\tau\mathclose[ \) with \( \omega\subset\R^d \) is a deformation \( \Phi:\Omega\to\R^{d+1}\) defined and there is a stored energy density \( W: \R^{(d+1)\times(d+1)}_{\rm sym} \to \R \) formulated in terms of the Euler-Lagrange strain tensor \( E = \frac{1}{2}(C-I) \) with \( C = \Phi'^T \Phi' \). Integrating over \( t\in \mathopen]-\tau,\tau\mathclose[ \), we define the marginal stored energy density
\[ w:\omega\to\R, \quad w(x) = \int_{-\tau}^\tau W(E(x,t)) \, dt. \]
For Kirchhoff shells ( \( d=2\)) with \( \Phi(x,t) = \phi(x) + t(x)n(x) \) and \( n(x) = \phi_{x_0}(x) \times \phi_{x_1}(x) \), the derivative is of the form \( \Phi'(x,t) = A(x)+tB(x) \). Then, the marginal stored energy is
\[ w(x) = 2\tau W(E(x,0)) + \frac{\tau^3}{3} W'(E(x,0))[B^TB] + \frac{\tau^3}{12} W''(E(x,0))[B^TA+A^TB,B^TA+A^TB] + \mathcal{O}(\tau^5), \]
where \( E(x,0) = \frac{1}{2} (A(x)^TA(x)-I) \). The first order term represents in-plane strain energy contributions, and the third order term bending energy. If \( \tau \) is small, the remainder term of order \( \tau^5 \) can be neglected.
This class implements this marginal stored energy density and its derivatives wrt \( \phi \) and \( t \) (and their spatial derivatives).
| Energy | the type of stored energy density \( W \) (currently, only 3D material laws can be used) |
| Real | the scalar field type of real numbers |
Explicit instantiations are provided for dim=2, Real=double, and Energy=StVenantKirchhoff<3,double>. If you want to instantiate the ShellEnergy for different material laws, include fem/diffops/shellKinematics.hpp, where the template definition is provided.
Definition at line 197 of file shellKinematics.hh.
Public Member Functions | |
| ShellEnergy (Energy const &W, Real tau) | |
| ShellEnergy (Energy const &W, Real tau, Dune::FieldMatrix< Real, dim+1, dim > const &dphi_dx, Tensor< Real, dim+1, dim, dim > const &ddphi_ddx, Real t, Dune::FieldVector< Real, dim > const &dt_dx) | |
| Constructor defining a linearization point. More... | |
| void | setLinearizationPoint (Dune::FieldMatrix< Real, dim+1, dim > const &dphi_dx, Tensor< Real, dim+1, dim, dim > const &ddphi_ddx, Real t, Dune::FieldVector< Real, dim > const &dt_dx) |
| Defines a new evaluation/linearization point. More... | |
| Real | d0 () const |
| Evaluates the marginal stored energy density \( w(x) \). More... | |
| template<int row> | |
| Dune::FieldVector< Real, row==0?dim+1:1 > | d1 (VariationalArg< Real, dim, 1 > const &v) const |
| Evaulates the parametric directional derivative of the marginal stored energy density \( w \) wrt changes. in \( \phi\) and \( t \). More... | |
| template<int row, int col> | |
| Dune::FieldMatrix< Real, row==0?dim+1:1, col==0?dim+1:1 > | d2 (VariationalArg< Real, dim, 1 > const &v, VariationalArg< Real, dim, 1 > const &w) const |
| Evaulates the second parametric directional derivative of the marginal stored energy density \( w \) wrt changes. in \( \phi\) and \( t \). More... | |
Static Public Attributes | |
| static int const | dim = Energy::dim-1 |
| The dimension of the parameter domain. More... | |
| Kaskade::Elastomechanics::ShellEnergy< Energy, Real >::ShellEnergy | ( | Energy const & | W, |
| Real | tau | ||
| ) |
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inline |
Constructor defining a linearization point.
| W | the volume stored energy density |
| dphi_dx | the spatial derivative of deformation \( \phi \) |
| ddphi_ddx | the second spatial derivative of deformation \( \phi \) |
| t | the local shell thickness \( t \) |
| dt_dx | the spatial derivative of the thickness \( t \) |
Definition at line 215 of file shellKinematics.hh.
| Real Kaskade::Elastomechanics::ShellEnergy< Energy, Real >::d0 | ( | ) | const |
Evaluates the marginal stored energy density \( w(x) \).
| Dune::FieldVector< Real, row==0?dim+1:1 > Kaskade::Elastomechanics::ShellEnergy< Energy, Real >::d1 | ( | VariationalArg< Real, dim, 1 > const & | v | ) | const |
Evaulates the parametric directional derivative of the marginal stored energy density \( w \) wrt changes. in \( \phi\) and \( t \).
| row | either 0 or 1. For row=0, the derivative wrt \( \phi \) is computed, for row=1 the derivative wrt \( t \). |
| v | the direction in which to take the derivative |
| Dune::FieldMatrix< Real, row==0?dim+1:1, col==0?dim+1:1 > Kaskade::Elastomechanics::ShellEnergy< Energy, Real >::d2 | ( | VariationalArg< Real, dim, 1 > const & | v, |
| VariationalArg< Real, dim, 1 > const & | w | ||
| ) | const |
Evaulates the second parametric directional derivative of the marginal stored energy density \( w \) wrt changes. in \( \phi\) and \( t \).
| void Kaskade::Elastomechanics::ShellEnergy< Energy, Real >::setLinearizationPoint | ( | Dune::FieldMatrix< Real, dim+1, dim > const & | dphi_dx, |
| Tensor< Real, dim+1, dim, dim > const & | ddphi_ddx, | ||
| Real | t, | ||
| Dune::FieldVector< Real, dim > const & | dt_dx | ||
| ) |
Defines a new evaluation/linearization point.
Referenced by Kaskade::Elastomechanics::ShellEnergy< Energy, Real >::ShellEnergy().
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static |
The dimension of the parameter domain.
Definition at line 203 of file shellKinematics.hh.
1.9.4