L modeling ============== Characteristics of modeling ----------------------------------- Distortion and pure shear in the plane. +-----------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------+ | | | + .. image:: images/1000020000000171000001169669FCCAEC11AD67.png + .. image:: images/10000200000001540000011D5218302EC2AD09C1.png + | :width: 2.9236in | :width: 2.5874in | + :height: 2.2075in + :height: 2.252in + | | | + + + | | | +-----------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------+ Figure 14.1-a: Meshing and boundary conditions Modeling: DKTG. :math:`L=1.0m`. Boundary conditions (see figure above on the right) so that the plate is subject to pure distortion: :math:`{\varepsilon }_{\text{xy}}` must be constant or to pure shear: forces are applied. Therefore, the following displacement field is applied to the edges of the plate for distortion: :math:`\mathrm{\{}\begin{array}{c}{u}_{x}\mathrm{=}{D}_{0}\mathrm{\cdot }y\\ {u}_{y}\mathrm{=}{D}_{0}\mathrm{\cdot }x\end{array}\mathrm{\Rightarrow }\varepsilon \mathrm{=}\frac{1}{2}({u}_{x,y}+{u}_{y,x})\mathrm{=}{D}_{0}` So: * we impose an embedding in :math:`{A}_{1}`, * :math:`{u}_{x}={D}_{0}\cdot y,{u}_{y}=0` on edge :math:`{A}_{1}-{A}_{3}`, :math:`{u}_{x}=0,{u}_{y}={D}_{0}\cdot x` on edge :math:`{A}_{1}-{A}_{2}`, * :math:`{u}_{x}={D}_{0}\cdot y,{u}_{y}={D}_{0}\cdot L` on edge :math:`{A}_{2}-{A}_{4}`, :math:`{u}_{x}={D}_{0}\cdot L,{u}_{y}={D}_{0}\cdot x` on edge :math:`{A}_{3}-{A}_{4}`, where :math:`{D}_{0}=3.3{10}^{-3}` and :math:`f(t)` represent the magnitude of cyclic loading as a function of the (pseudo-time) parameter :math:`t`, defined as: .. image:: images/10000000000001F8000001201AA96929C8D1ABF2.png :width: 3.7063in :height: 1.9055in .. _RefImage_10000000000001F8000001201AA96929C8D1ABF2.png: Figure 14.1-b: loading function Integration increment: :math:`0.05s`. For shearing, the following forces are applied: * we impose :math:`{F}_{y}={F}_{0}` on :math:`{A}_{2}{A}_{4}`, * we impose :math:`{F}_{x}={F}_{0}` on :math:`{A}_{4}{A}_{3}`, * we impose :math:`{F}_{y}=-{F}_{0}` on :math:`{A}_{3}{A}_{1}`, * we impose :math:`{F}_{x}=-{F}_{0}` on :math:`{A}_{1}{A}_{2}`, with :math:`{F}_{0}=5000000N` Characteristics of the mesh ---------------------------- Knots: 121. Stitches: 200 TRIA3; 40 SEG2. Modeling DHRC ----------------- Refer to modeling DHRCexpliquée for the H (§ :ref:`10 `) pure traction-compression and I (§ :ref:`11 `) pure flexure models. Tested sizes and results ------------------------------ For the distortion, the shear force :math:`{N}_{\mathrm{xy}}` and :math:`B` obtained by the two models are compared; the tolerances are taken in absolute values based on these relative differences: .. csv-table:: "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**" "DIST. POS .- CHAR. ELAS. :math:`t=\mathrm{0,25}` ", "", "", "" "Relative difference :math:`{N}_{\mathit{xy}}` - DHRC - GLRC_DM "," NON_REGRESSION ", "-0.171493969414", "1 10-6" "DIST. **POS. -** CHAR. **ENDO** :math:`t=\mathrm{1,0}` ", "", "", "" "Relative difference :math:`{N}_{\mathrm{xy}}` - DHRC - GLRC_DM "," NON_REGRESSION ", "-0.151040071429", "1 10-6" "DIST. **POS. - FROM** CHAR. ELAS. :math:`t=\mathrm{1,5}` ", "", "", "" "Relative difference :math:`{N}_{\mathrm{xy}}` - DHRC - GLRC_DM "," NON_REGRESSION ", "-0.151040071429", "1 10-6" "DIST. **NEG. -** CHAR. ELAS. :math:`t=\mathrm{3,0}` ", "", "", "" "Relative difference :math:`{N}_{\mathrm{xy}}` - DHRC - GLRC_DM "," NON_REGRESSION ", "-0.151040071429", "1 10-6" "**DIST. NEGNEG. ATIVE - DECHAR. ELAS .** :math:`t=\mathrm{3,5}` ", "", "", "" "Relative difference :math:`{N}_{\mathrm{xy}}` - DHRC - GLRC_DM "," NON_REGRESSION ", "-0.151040071429", "1 10-6" Shearing force diagram :math:`{N}_{\mathit{xy}}` **(in the plan)** as a function of time: .. image:: images/10000000000001F800000120F246050AAE32834E.png :width: 5.2508in :height: 3.0008in .. _RefImage_10000000000001F800000120F246050AAE32834E.png: **shear force graph** :math:`{N}_{\mathit{xy}}` **** (in the plan) ****based on**:math:`{D}_{0}`**imposed: ** .. image:: images/10000000000001F80000012099AEC8B782E26B04.png :width: 5.2508in :height: 3.0008in .. _RefImage_10000000000001F80000012099AEC8B782E26B04.png: For shear, we do non-regression tests on shear deformations :math:`{\epsilon }_{\mathit{xy}}` and :math:`B`: .. csv-table:: "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**" "CISAIL. **POS. - ELAS .** :math:`t=\mathrm{0,1}` ", "", "", "" ":math:`{\epsilon }_{\mathit{xy}}` *-* GLRC_DM "," NON_REGRESSION ", "4.7587148531429.10-4", "1 10-6" "CISAIL. **POS. - ENDO .** :math:`t=\mathrm{0,8}` ", "", "", "" ":math:`{\epsilon }_{\mathit{xy}}` *-* GLRC_DM "," NON_REGRESSION ", "9.09169470373.10-3", "1 10-6" "CISAIL. **POS. - ELAS .** :math:`t=\mathrm{0,1}` ", "", "", "" ":math:`{\epsilon }_{\mathit{xy}}` *-* DHRC "," NON_REGRESSION ", "1.78152989856.10-4", "1 10-6" "CISAIL. **POS. - ENDO .** :math:`t=\mathrm{0,8}` ", "", "", "" ":math:`{\epsilon }_{\mathit{xy}}` *-* DHRC "," NON_REGRESSION ", "1.93228925951.10-3", "1 10-6", "1 10-6" .. _RefNumPara__30393805: .. _refnumpara__30393805: