2. Benchmark solution#

For each modeling, this test carries out an inter-comparison between the reference solution (obtained with a very fine time step), the solution with a moderately coarse discretization, the solution with the effect of temperature (or another control variable), the solution by changing the system of units (\(\mathrm{Pa}\) in \(\mathrm{MPa}\)), and the solution obtained after rotation or symmetry.

2.1. Definition of robustness test cases#

We propose 3 analysis angles to test the robustness of the integration of laws of behavior:

  • study of equivalent problems

  • tangent matrix check

  • study of the discretization of the time step

For each of them, we study the evolution of the relative differences between several calculations using the same law but presenting different parameters or calculation options. The exploitation focuses on the invariants of the stress tensor: tensor trace, Von-Mises constraint and scalar internal variables: generally it is the cumulative plasticity.

2.2. Study of equivalent problems#

For a rough discretization of the paths: 1 time step for each segment of the journey, the solution obtained for each law is compared to 3 strictly equivalent problems for the state of the material point:

  • \(\mathrm{Tpa}\), same path with a change of unit, we substitute the \(\mathrm{Pa}\) for the \(\mathrm{MPa}\) in the material data and the possible parameters of the law,

  • \(\mathrm{Trot}\), path by imposing the same tensor \(\stackrel{ˉ}{\varepsilon }\) after a rotation: \({}^{t}R\cdot \stackrel{ˉ}{\varepsilon }\cdot R\) where \(R\) is a rotation matrix defined from the following arbitrary Euler angles: {\(\Psi =0.9\mathrm{radian}\), \(\theta =0.7\mathrm{radian}\) and \(\varphi =0.4\mathrm{radian}\)},

  • \(\mathrm{Tsym}\), path by imposing the tensor \(\stackrel{ˉ}{\varepsilon }\) after symmetry: permutation from \(x\) to \(y\), \(y\) to \(z\) and \(z\) to \(x\) to \(\mathrm{3D}\).

For each of these problems, the solution (stress invariants, cumulative equivalent plastic deformation) must be identical to the basic solution, obtained with the same time discretization. The reference value of the difference is therefore 0. In practice, this means that the difference found must be of the order of machine precision, i.e. approximately 1.E-15.

2.3. Tangent matrix test#

For each behavior, the tangent matrix is also tested, by difference with the matrix obtained by disturbance. Again, the reference value is 0.

2.4. Study of the discretization of the time step#

We study the behavior of law integration as a function of discretization. For the same modeling, and therefore a given behavior, several discretizations are studied here at different times, by multiplying the number of steps in the loading path by 5.