Reference problem ===================== Geometry --------- The aim is to test the 2 laws of viscoplastic behavior VISC_ISOT_TRAC and VISC_ISOT_LINE on a representative Elementary Volume :math:`R` of dimension :math:`\mathrm{1mm}`, i.e. a cube in :math:`\mathrm{3D}`, the equivalent of a bar in plane deformations or of a cylinder in axisymmetry. Material properties ---------------------- +-------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ |Isotropic elasticity |Young's modulus: | + + + | | | + + .. image:: images/Object_1.svg + | | :width: 115 | + + :height: 18 + | | | + + + | | | + + + | | | + + Poisson's ratio: :math:`\nu =0.3` + | | | +-------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ |Traction curve (mod. A, B and C) | | + + .. image:: images/100002010000023B000001A8225D4A854006DF71.png + | | :width: 161 | + + :height: 26 + | | | + + + | | | +-------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ |Linear work hardening (mod. D, E, and F) | | + + .. image:: images/Object_2.svg + | | :width: 115 | + + :height: 18 + | | | + + + | | .. image:: images/Object_3.svg | + + :width: 115 + | | :height: 18 | + + + | | | +-------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ |Coefficient for viscous law VISC_SINH | | + + .. image:: images/Object_4.svg + | | :width: 115 | + + :height: 18 + | | | + + + | | .. image:: images/Object_5.svg | + + :width: 115 + | | :height: 18 | + + + | | | + + .. image:: images/Object_6.svg + | | :width: 115 | + + :height: 18 + | | | + + + | | | +-------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ |Rousselier model coefficients used to obtain the reference solution (mod. A, B and C)| | + + .. image:: images/Object_7.svg + | | :width: 115 | + + :height: 18 + | | | + + + | | .. image:: images/Object_8.svg | + + :width: 115 + | | :height: 18 | + + + | | | + + .. image:: images/Object_9.svg + | | :width: 115 | + + :height: 18 + | | | + + + | | | +-------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ Boundary conditions and loads ------------------------------------- The volume element is subjected to a simple homogeneous tensile test. It is therefore stuck in :math:`x` on the face [:ref:`3,4 <3,4>`] and in :math:`y` on the face [:ref:`1,3 <1,3>`] (and possibly in the :math:`z` direction) and subject to a displacement :math:`u(t)` in the direction .. image:: images/Object_10.svg :width: 115 :height: 18 .. _RefImage_Object_10.svg: on the face [:ref:`1, 2 <1, 2>`]. 3 values of :math:`T` are used :math:`2000s`, :math:`0.2s` and :math:`0.002s`, corresponding to deformation rates :math:`\dot{\varepsilon }` of :math:`{10}^{-3}{s}^{-1}`, :math:`10{s}^{-1}` and :math:`{10}^{3}{s}^{-1}`. +-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+ | | | + .. image:: images/Object_12.svg + .. image:: images/Object_13.svg + | :width: 115 | :width: 161 | + :height: 18 + :height: 26 + | | | + + + | | | +-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+ Initial conditions -------------------- Zero stresses and deformations at :math:`t=0`.