Reference problem ===================== Device description ------------------------- The system studied is represented by the rheological model below. It is composed of a non-linear element affected by the behavior DIS_ECRO_TRAC (between the nodes N1 and N2), a mass assigned to the node N2, a linear element (between the nodes N2 and N3). .. image:: images/10000000000003B7000000B6FF9EAA41D84DD261.png :width: 5.2in :height: 0.9953in .. _RefImage_10000000000003B7000000B6FF9EAA41D84DD261.png: **Figure** 1.1-a: Device model. The equations governing the behavior of the nonlinear element are in [:ref:`R5.03.17 `]. Modelizations ------------- The models tested are on elements DIS_T, mesh SEG2. The characteristics of the discrete elements are of the type: K_T_D_L. *Note: The units of the parameters must agree with the unit of effort, the unit of lengths* [:ref:`R5.03.17 `]. For all models the units are homogeneous to [N], [mm]. *Note: Models A and B treat the same system, with the same boundary conditions, the same load.* Modeling A ~~~~~~~~~~~~~~~ This modeling makes it possible to test the nonlinear behavior of the isotropic work hardening type, in the local discrete x-direction, of the DIS_ECRO_TRAC law with the DYNA_NON_LINE operator. B modeling ~~~~~~~~~~~~~~~ This modeling makes it possible to test the nonlinear behavior of the isotropic work hardening type, in the local discrete x-direction, of the DIS_ECRO_TRAC law with the DYNA_VIBRA operator. C modeling ~~~~~~~~~~~~~~~ This modeling makes it possible to test the nonlinear behavior of the isotropic work hardening type, in the local tangent plane of the discrete, of the DIS_ECRO_TRAC law with the DYNA_NON_LINE operator. D modeling ~~~~~~~~~~~~~~~ This modeling makes it possible to test the nonlinear behavior of the isotropic work hardening type, in the local tangent plane of the discrete, of the DIS_ECRO_TRAC law with the DYNA_VIBRA operator. E modeling ~~~~~~~~~~~~~~~ This modeling makes it possible to test the nonlinear behavior, such as kinematic work hardening, in the local tangential plane of the discrete, of the DIS_ECRO_TRAC law with the DYNA_NON_LINE operator. F modeling ~~~~~~~~~~~~~~~ This modeling makes it possible to test the nonlinear behavior, such as kinematic work hardening, in the local tangential plane of the discrete, of the DIS_ECRO_TRAC law with the DYNA_VIBRA operator. Material properties -------------------- A and B models ~~~~~~~~~~~~~~~~~~~~~~ The stiffness of the linear device is :math:`{k}_{\mathit{lin}}\phantom{\rule{2em}{0ex}}=\phantom{\rule{2em}{0ex}}400N.\mathit{mm}`. The mass is :math:`M\phantom{\rule{2em}{0ex}}=\phantom{\rule{2em}{0ex}}200\mathit{kg}`. The non-linear behavior used in the test case is shown in the figure: • Elastic behavior up to point :math:`(\mathrm{0.5mm},\mathrm{200.0N})`. • Nonlinear behavior, governed by the following equation: .. math:: : label: EQ-None R (p) =\ frac {{K} _ {\ mathit {elas}} .p} {\ left [1+ {\ left (\ frac {{k} _ {\ mathit {elas}} .p} .p} {{F} _ {u} _ {u}} — {F} _ {elas}}} .p} {{F} _ {elas}}} .p} {{F} _ {elas}} .p} {{F} _ {u}} — {F} _ {y}}\ right)} ^ {n}\ right]} ^ {(1/n)}} .p} {{F} _ {(1/n)}} .. image:: images/100002010000041D00000355903461E673CA258E.png :width: 3.9366in :height: 3.1811in .. _RefImage_100002010000041D00000355903461E673CA258E.png: **Figure** 1.3.1-a **: Nonlinear behavior** C and D modeling ~~~~~~~~~~~~~~~~~~~~~ The stiffness of the linear device is :math:`{k}_{\mathit{lin}}=400\mathit{N.mm}`. The mass is :math:`M=200\mathit{kg}`. The behavior is of the "isotropic work hardening" type in the tangential plane local to the element. It is defined by the following function: fctsy2 **= DEFI_FONCTION (NOM_PARA = "DTAN"**, **VALE =** **(** 0.0, 0.0, 0.0, 0.1, 0.1, 100.0, 0.2, 120.0, 20.2, 370.0), **)** SE and F modeling ~~~~~~~~~~~~~~~~~~~~~ The stiffness of the linear device is :math:`{k}_{\mathit{lin}}=400\mathit{N.mm}`. The mass is :math:`M=200\mathit{kg}`. The behavior is of the "kinematic work hardening" type in the tangential plane local to the element. It is defined by the following function: fctsy2 **= DEFI_FONCTION (NOM_PARA = "DTAN"**, **VALE =** **(** 0.0, 0.0, 0.0, 0.1, 100.0, 20.1, 350.0), **)** .. _DdeLink__11456_1213890583: Boundary conditions and loads ------------------------------------- A and B models ~~~~~~~~~~~~~~~~~~~~~~ Nodes N1, N2, N3 are stuck in the Y and Z directions. For models A and B, the nodes N1 and N3 are subject to displacement :math:`U(t)` in the X direction. For modeling B, nodes N1 and N3 are subject to displacement :math:`U(t)`, speed :math:`V(t)`, and acceleration :math:`\mathrm{\gamma }(t)` in the X direction. The condition while traveling as a function of time: :math:`U(t)={U}_{0}.\mathrm{sin}(2\pi .\mathit{f.t})/(2\pi .f)` with :math:`f=0.5\mathit{Hz}` and :math:`{U}_{0}=\mathrm{6.0mm}` The speed condition as a function of time: :math:`V(t)\phantom{\rule{2em}{0ex}}=\phantom{\rule{2em}{0ex}}\dot{U}(t)` The accelerating condition as a function of time: :math:`\mathrm{\gamma }(t)\phantom{\rule{2em}{0ex}}=\phantom{\rule{2em}{0ex}}\ddot{U}(t)` C, D, E, and F models ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Nodes N1, N2, N3 are locked in the X direction. For C, D, E, and F models, the nodes N1 and N3 are subject to :math:`U(t)` displacement in the Y and Z directions. For modeling B, the nodes N1 and N3 are subject to displacement :math:`U(t)`, speed :math:`V(t)`, and acceleration :math:`\gamma (t)` in the Y and Z directions. Travel conditions are functions of time: :math:`\mathit{Depl}={U}_{1.}\mathrm{sin}(2\mathrm{\pi }.{f}_{1}.t)+{U}_{2.}\mathrm{sin}(2\mathrm{\pi }.{f}_{2}.t)+{U}_{3.}\mathrm{sin}(2\mathrm{\pi }.{f}_{3}.t)` Following the direction :math:`Y`: :math:`(u,f)=[(\mathrm{0.20,0}.80),(\mathrm{0.15,1}.50),(\mathrm{0.10,3}.00)]` Following the direction :math:`Z`: :math:`(u,f)=[(-\mathrm{0.20,0}.90),(\mathrm{0.15,2}.00),(-\mathrm{0.10,2}.80)]` The speed condition as a function of time: :math:`V(t)=\dot{U}(t)` The accelerating condition as a function of time: :math:`\gamma (t)=\ddot{U}(t)`