D modeling ============== Characteristics of modeling ----------------------------------- Pure in-plane distortion and shear. +-----------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------+ | | | + .. image:: images/1000020000000171000001169669FCCAEC11AD67.png + .. image:: images/10000200000001540000011D5218302EC2AD09C1.png + | :width: 2.9236in | :width: 2.5874in | + :height: 2.2075in + :height: 2.252in + | | | + + + | | | +-----------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------+ Figure 6.1-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}^{-4}` 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: 3.572in :height: 1.9898in .. _RefImage_10000000000001F8000001201AA96929C8D1ABF2.png: Figure 6.1-b: loading function Integration increment: :math:`0.05s`. For shearing, the following forces are applied: * we impose :math:`{F}_{y}={D}_{0}` on :math:`{A}_{2}{A}_{4}`, * we impose :math:`{F}_{x}={D}_{0}` on :math:`{A}_{4}{A}_{3}`, * we impose :math:`{F}_{y}=-{D}_{0}` on :math:`{A}_{3}{A}_{1}`, * we impose :math:`{F}_{x}=-{D}_{0}` on :math:`{A}_{1}{A}_{2}`, Characteristics of the mesh ---------------------------- Knots: 121. Stitches: 200 TRIA3; 40 SEG2. 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: .. _DdeLink__15507_23373940: .. csv-table:: "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**" "DIST. POS. - PHASE CHAR. ELAS. :math:`t=\mathrm{0,25}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xy}}` "," AUTRE_ASTER "," ", "0", "5 10-2" "DIST. POS. - PHASE CHAR. ENDO. :math:`t=\mathrm{1,0}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xy}}` "," AUTRE_ASTER "," ", "0", "7 10 -2" "DIST. POS. - PHASE DECHAR. ELAS. :math:`t=\mathrm{1,5}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xy}}` "," AUTRE_ASTER "," ", "0", "7 10 -2" "DIST. NEG. - PHASE CHAR. ELAS. :math:`t=\mathrm{3,0}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xy}}` "," AUTRE_ASTER "," ", "0", "7 10-2" "DIST. NEG. - PHASE DECHAR. ELAS. :math:`t=\mathrm{3,5}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xy}}` "," AUTRE_ASTER "," ", "0", "7 10-2" Shearing force diagram :math:`{N}_{\mathrm{xy}}` **(in the plan)** as a function of time: .. image:: images/10000000000001F800000120E6683A740D0FB920.png :width: 5.2508in :height: 3.0008in .. _RefImage_10000000000001F800000120E6683A740D0FB920.png: **shear force graph** :math:`{N}_{\mathit{xy}}` **** (in the plan) ****based on**:math:`{D}_{0}`**imposed: ** .. image:: images/10000000000001F8000001201279C44A6B935BC6.png :width: 5.2508in :height: 3.0008in .. _RefImage_10000000000001F8000001201279C44A6B935BC6.png: **Diagram** **of the evolution of model damage** **** GLRC_DM **** **(** :math:`{d}_{1}={d}_{2}` **) as a function of time:** .. image:: images/10000000000002010000012094775FA9B1166E24.png :width: 5.3445in :height: 3.0008in .. _RefImage_10000000000002010000012094775FA9B1166E24.png: **For shear, we do non-regression tests on shear deformations** :math:`{\varepsilon }_{\mathrm{xy}}` in :math:`B`: .. csv-table:: "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**" "CIS. POS. - PHASE CHAR. ELAS. :math:`t=\mathrm{0,1}` ", "", "", "" "*Shear Deformations* :math:`{\varepsilon }_{\mathrm{xy}}` "," NON_REGRESSION "," ", "3.013 10-15", "1 10-6" "CIS. POS. - PHASE CHAR. ENDO. :math:`t=\mathrm{0,8}` ", "", "", "" "*Shear deformations* :math:`{\varepsilon }_{\mathrm{xy}}` "," NON_REGRESSION "," ", "2,410 10-14", "7 10 -2" notes --------- In order to have a better agreement between model GLRC_DM and the reference (multilayer model) in pure distortion, it was necessary to modify the Young's modulus from :math:`E=35620\mathit{MPa}` to :math:`E=42500\mathit{MPa}` compared to the A, B, C models, knowing that in pure distortion the steels are not loaded. It is verified that the shear force obtained with *Code_Aster* at time :math:`t=\mathrm{0,37427}s`, just when the first damage occurs, produces the theoretical elastic value: :math:`{N}_{\mathrm{xy}}^{D}=2\frac{\sqrt{2{\mu }_{m}{k}_{0}}}{\sqrt{2-{\gamma }_{\mathrm{mc}}-{\gamma }_{\mathrm{mt}}}}=\frac{{N}_{D}}{1+{\nu }_{m}}\cdot \sqrt{\frac{(1-{\nu }_{m})(1+2{\nu }_{m})(1-{\gamma }_{\mathrm{mt}})+{\nu }_{m}^{2}(1-{\gamma }_{\mathrm{mc}})}{2-{\gamma }_{\mathrm{mc}}-{\gamma }_{\mathrm{mt}}}}` that is: :math:`{N}_{\mathrm{xy}}^{D}=331128N/m`.