6. D modeling#

6.1. Characteristics of modeling#

Pure in-plane distortion and shear.

_images/1000020000000171000001169669FCCAEC11AD67.png _images/10000200000001540000011D5218302EC2AD09C1.png

Figure 6.1-a: Meshing and boundary conditions.

Modeling: DKTG. \(L=1.0m\).

Boundary conditions (see figure above on the right) so that the plate is subject to pure distortion: \({\varepsilon }_{\text{xy}}\) 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:

\(\mathrm{\{}\begin{array}{c}{u}_{x}\mathrm{=}{D}_{0}\mathrm{\cdot }y\\ {u}_{y}\mathrm{=}{D}_{0}\mathrm{\cdot }x\end{array}\mathrm{\Rightarrow }\varepsilon \mathrm{=}\frac{1}{2}({u}_{x,y}+{u}_{y,x})\mathrm{=}{D}_{0}\)

So:

  • we impose an embedding in \({A}_{1}\),

  • \({u}_{x}={D}_{0}\cdot y,{u}_{y}=0\) on edge \({A}_{1}-{A}_{3}\), \({u}_{x}=0,{u}_{y}={D}_{0}\cdot x\) on edge \({A}_{1}-{A}_{2}\),

  • \({u}_{x}={D}_{0}\cdot y,{u}_{y}={D}_{0}\cdot L\) on edge \({A}_{2}-{A}_{4}\), \({u}_{x}={D}_{0}\cdot L,{u}_{y}={D}_{0}\cdot x\) on edge \({A}_{3}-{A}_{4}\),

where \({D}_{0}=3.3{10}^{-4}\) and \(f(t)\) represent the magnitude of cyclic loading as a function of the (pseudo-time) parameter \(t\), defined as:

_images/10000000000001F8000001201AA96929C8D1ABF2.png

Figure 6.1-b: loading function

Integration increment: \(0.05s\).

For shearing, the following forces are applied:

  • we impose \({F}_{y}={D}_{0}\) on \({A}_{2}{A}_{4}\),

  • we impose \({F}_{x}={D}_{0}\) on \({A}_{4}{A}_{3}\),

  • we impose \({F}_{y}=-{D}_{0}\) on \({A}_{3}{A}_{1}\),

  • we impose \({F}_{x}=-{D}_{0}\) on \({A}_{1}{A}_{2}\),

6.2. Characteristics of the mesh#

Knots: 121.

Stitches: 200 TRIA3; 40 SEG2.

6.3. Tested sizes and results#

For the distortion, the shear force \({N}_{\mathrm{xy}}\) and \(B\) obtained by the two models are compared; the tolerances are taken in absolute values based on these relative differences:

Shearing force diagram \({N}_{\mathrm{xy}}\) (in the plan) as a function of time:

_images/10000000000001F800000120E6683A740D0FB920.png

shear force graph \({N}_{\mathit{xy}}\) **** (in the plan) **based on:math:`{D}_{0}`**imposed: **

_images/10000000000001F8000001201279C44A6B935BC6.png

Diagram of the evolution of model damage **** GLRC_DM **** ( \({d}_{1}={d}_{2}\) ) as a function of time:

_images/10000000000002010000012094775FA9B1166E24.png

For shear, we do non-regression tests on shear deformations \({\varepsilon }_{\mathrm{xy}}\) in \(B\):

Identification

Reference type

Reference value

Tolerance

CIS. POS. - PHASE CHAR. ELAS. \(t=\mathrm{0,1}\)

Shear Deformations \({\varepsilon }_{\mathrm{xy}}\)

NON_REGRESSION

3.013 10-15

1 10-6

CIS. POS. - PHASE CHAR. ENDO. \(t=\mathrm{0,8}\)

Shear deformations \({\varepsilon }_{\mathrm{xy}}\)

NON_REGRESSION

2,410 10-14

7 10 -2

6.4. 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 \(E=35620\mathit{MPa}\) to \(E=42500\mathit{MPa}\) 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 \(t=\mathrm{0,37427}s\), just when the first damage occurs, produces the theoretical elastic value:

\({N}_{\mathrm{xy}}^{D}=2\frac{\sqrt{2{\mu }_{m}{k}_{0}}}{\sqrt{2-{\gamma }_{\mathrm{mc}}-{\gamma }_{\mathrm{mt}}}}=\frac{{N}_{D}}{1+{\nu }_{m}}\cdot \sqrt{\frac{(1-{\nu }_{m})(1+2{\nu }_{m})(1-{\gamma }_{\mathrm{mt}})+{\nu }_{m}^{2}(1-{\gamma }_{\mathrm{mc}})}{2-{\gamma }_{\mathrm{mc}}-{\gamma }_{\mathrm{mt}}}}\)

that is: \({N}_{\mathrm{xy}}^{D}=331128N/m\).