2. Benchmark solution

2.1. Calculation method used for the reference solution

The analytical solution corresponding to pre-work hardening can be calculated (monotonic traction, modeling A):

The system of equations for the problem with memory effect is written (20 equations) [R5.03.04]:

Elasticity: \(\tilde{\sigma }\mathrm{=}2\mu (\tilde{\varepsilon }\mathrm{-}{\varepsilon }^{p})\)

Plasticity criterion

Plastic flow:

with

Isotropic work hardening:

Maximum work hardening memory:

where q is determined by:

• a domain

  

characterizing the maximum plastic deformations, whose radius q measures and

The center

• \(\xi\) is calculated according to a law of normality, i.e.:

  

, with

On the surface of the maximum work hardening range, we have \(F\mathrm{=}0\). Applying condition \(\mathit{dF}\mathrm{=}0\), we get the speed expression:

For a material point under uniaxial load, the components of the (uniform) fields are:

In this case, during the first uniaxial load in the x direction:

In this case,

, implies that

. In this case,

In addition, in the case of a symmetric tensile compression cycle (in plastic deformation), during the first symmetric discharge (with

):

which corresponds well to the expected result (cf. [bib2]): domain \(F\mathrm{=}0\) centered on the origin, and radius the half-amplitude of plastic deformation.

In the case of increasing traction, and if kinematic work hardening is neglected, the equations to be solved become:

So we have to calculate the function

, such as:

with

.

In addition, we consider that we are in charge, so

It is therefore necessary to integrate the differential equation:

which is integrated as follows:

=>

Method for varying the constant:

by integrating:

From where

The constant \(K\) is defined by the initial conditions: for \(p\mathrm{=}0\), \(R\mathrm{=}0\)

either

Finally:

So we’re in charge of: \(\sigma \mathrm{=}{R}_{0}+R(p)\)

2.2. Benchmark results

Modeling A:

Value of \(\mathit{SIXX}\) at the final moment: \(\sigma \mathrm{=}{R}_{0}+R(p)\)

with

\(t\mathrm{=}\mathrm{8s}\), we need to find \(\mathit{SIXX}\mathrm{=}120\mathit{Mpa}\).

To do this, we calculate \(R(p)\) from the value of \(p\) at the time \(t\mathrm{=}\mathrm{8s}\).

Modeling B:

The results obtained with VISC_CIN2_MEMO will be compared with those obtained with VISCOCHAB, at the end of pre-work hardening and after 10 cycles. The curves below highlight the memory effect (in comparison with VISC_CIN2_CHAB, which does not model it): after pre-work hardening, the imposed deformation cycles stabilize at a stress amplitude greater than that obtained without a memory effect:

2.3. Uncertainty about the solution

• Modeling A: analytical

• B modeling: inter-comparison between VISCOCHAB and VISC_CIN2_MEMO: precision of numerical integration, estimated at less than \(\text{1\%}\).

• C modeling: validation of behaviors in 2D AXIS; the results must be identical to those of modeling B.

2.4. Bibliographical references

1. R5.03.04 « Elasto-visco-plastic behaviors of J.L.Chaboche ».

2. J.M. PROIX « Viscoplastic behavior taking into account the non-proportionality of the load » EDF R&D-CR- AMA12 -284, 12/12/12