6. E modeling

6.1. Characteristics of modeling

It is a nonlinear elasticity calculation under the hypothesis of small displacements, in plane deformations.

6.2. Characteristics of the mesh

The mesh is identical to that of modeling A.

6.3. Tested sizes and results

The quantity tested is the reproduction rate \(G\), calculated for the first integration ring defined in modeling A and for ten values of the imposed horizontal displacement \(\delta\) (\(\delta\) varying from \(0\) to \(\mathrm{0,2}\mathrm{mm}\)).

Two distinct series of calculations are carried out:

first, a law of behavior with linear hardening is taken. Three calculations are carried out using the different methods available to define the same work-hardening curve (TRACTION, ECRO_LINE and ECRO_PUIS). The solutions obtained by these three calculations are compared with each other;

then we choose a law of behavior with hardening in power law and we compare the results obtained between ELAS_VMIS_TRAC and ELAS_VMIS_PUIS.

The test is of a non-regression type. It is verified that:

the three different calculations in linear work hardening lead to exactly the same result on \(G\);

the two power law hardening calculations lead to very similar results (difference less than 0.3%). The difference decreases if the calculation time step is reduced.

Imposed displacement \(\delta\) (\(\mathrm{mm}\))	\(G(N/\mathrm{mm})\) - linear work hardening: ELAS_VMIS_TRAC	\(G(N/\mathrm{mm})\) - linear work hardening: ELAS_VMIS_LINE	\(G(N/\mathrm{mm})\) - linear work hardening: ELAS_VMIS_PUIS
0.02	3.59	3.59	3.59
0.04	14.47	14.47	14.47
0.06	32.88	32.88	32.88
0.08	59.42	59.42	59.42	59.42
0.10	94.50	94.50	94.50
0.12	134.98	134.98	134.98	134.98
0.14	178.95	178.95	178.95
0.16	226.24	226.24	226.24
0.18	276.47	276.47	276.47
0.20	329.61	329.61	329.61	329.61

Imposed displacement \(\delta\) (\(\mathrm{mm}\))	\(G(N/\mathrm{mm})\) - work hardening power: ELAS_VMIS_TRAC	\(G(N/\mathrm{mm})\) - work hardening power: ELAS_VMIS_PUIS
0.02	3.59	3.59
0.04	14.44	14.44
0.06	32.76	32.76
0.08	58.99	58.98
0.10	93.76	93.77
0.12	136.14	136.14	136.18
0.14	184.04	184.04	184.09
0.16	236.31	236.31	236.38
0.18	292.38	292.46
0.20	351.82	351.90

This test validates the calculation of \(G\) in linear elasticity for the various methods of defining the work hardening curve.