.. _large_deformation-gen_yeoh_tl_up_interactive: large_deformation/gen_yeoh_tl_up_interactive.py =============================================== **Description** Incompressible generalized Yeoh hyperelastic material model. In this model, the deformation energy density per unit reference volume is given by .. math:: W = K_1 \, \left( \overline I_1 - 3 \right)^{m} +K_2 \, \left( \overline I_1 - 3 \right)^{p} +K_3 \, \left( \overline I_1 - 3 \right)^{q} where :math:`\overline I_1` is the first main invariant of the deviatoric part of the right Cauchy-Green deformation tensor :math:`\ull{C}`, the coefficients :math:`K_1, K_2, K_3` and exponents :math:`m, p, q` are material parameters. Only a single term (:class:`dw_tl_he_genyeoh `) is used in this example for the sake of simplicity. Components of the second Piola-Kirchhoff stress are in the case of an incompressible material .. math:: S_{ij} = 2 \, \pdiff{W}{C_{ij}} - p \, F^{-1}_{ik} \, F^{-T}_{kj} \;, where :math:`p` is the hydrostatic pressure. The large deformation is described using the total Lagrangian formulation in this example. The incompressibility is treated by mixed displacement-pressure formulation. The weak formulation is: Find the displacement field :math:`\ul{u}` and pressure field :math:`p` such that: .. math:: \intl{\Omega\suz}{} \ull{S}\eff(\ul{u}, p) : \ull{E}(\ul{v}) \difd{V} = 0 \;, \quad \forall \ul{v} \;, \intl{\Omega\suz}{} q\, (J(\ul{u})-1) \difd{V} = 0 \;, \quad \forall q \;. The following formula holds for the axial true (Cauchy) stress in the case of uniaxial stress: .. math:: \sigma(\lambda) = \frac{2}{3} \, m \, K_1 \, \left( \lambda^2 + \frac{2}{\lambda} - 3 \right)^{m-1} \, \left( \lambda - \frac{1}{\lambda^2} \right) \;, where :math:`\lambda = l/l_0` is the prescribed stretch (:math:`l_0` and :math:`l` being the original and deformed specimen length respectively). The boundary conditions are set so that a state of uniaxial stress is achieved, i.e. appropriate components of displacement are fixed on the "Left", "Bottom", and "Near" faces and a monotonously increasing displacement is prescribed on the "Right" face. This prescribed displacement is then used to calculate :math:`\lambda` and to convert the second Piola-Kirchhoff stress to the true (Cauchy) stress. Note on material parameters --------------------------- The three-term generalized Yeoh model is meant to be used for modelling of filled rubbers. The following choice of parameters is suggested [1] based on experimental data and stability considerations: :math:`K_1 > 0`, :math:`K_2 < 0`, :math:`K_3 > 0`, :math:`0.7 < m < 1`, :math:`m < p < q`. Usage Examples -------------- Default options:: python3 sfepy/examples/large_deformation/gen_yeoh_tl_up_interactive.py To show a comparison of stress against the analytic formula:: python3 sfepy/examples/large_deformation/gen_yeoh_tl_up_interactive.py -p Using different mesh fineness:: python3 sfepy/examples/large_deformation/gen_yeoh_tl_up_interactive.py \ --shape "5, 5, 5" Different dimensions of the computational domain:: python3 sfepy/examples/large_deformation/gen_yeoh_tl_up_interactive.py \ --dims "2, 1, 3" Different length of time interval and/or number of time steps:: python3 sfepy/examples/large_deformation/gen_yeoh_tl_up_interactive.py \ -t 0,15,21 Use higher approximation order (the ``-t`` option to decrease the time step is required for convergence here):: python3 sfepy/examples/large_deformation/gen_yeoh_tl_up_interactive.py \ --order 2 -t 0,2,21 Change material parameters:: python3 sfepy/examples/large_deformation/gen_yeoh_tl_up_interactive.py -m 2,1 View the results using ``sfepy-view`` ------------------------------------- Show pressure on deformed mesh (use PgDn/PgUp to jump forward/back):: sfepy-view --fields=p:f1:wu:p1 domain.??.vtk Show the axial component of stress (second Piola-Kirchhoff):: sfepy-view --fields=stress:c0 domain.??.vtk [1] Travis W. Hohenberger, Richard J. Windslow, Nicola M. Pugno, James J. C. Busfield. Aconstitutive Model For Both Lowand High Strain Nonlinearities In Highly Filled Elastomers And Implementation With User-Defined Material Subroutines In Abaqus. Rubber Chemistry And Technology, Vol. 92, No. 4, Pp. 653-686 (2019) .. image:: /../doc/images/gallery/large_deformation-gen_yeoh_tl_up_interactive-gen_yeoh_tl_up_comparison.png .. image:: /../doc/images/gallery/large_deformation-gen_yeoh_tl_up_interactive.png :download:`source code ` .. literalinclude:: /../sfepy/examples/large_deformation/gen_yeoh_tl_up_interactive.py