2. Benchmark solution#

2.1. Calculation method used for the reference solution [bib1]#

2.1.1. Isotropic work hardening: analytical#

\(t=\mathrm{1s}\) ( \(T=0°C\) ):

_images/Object_5.svg

with

_images/Object_6.svg

either

_images/Object_7.svg

Heating: Maximum plastic deformation at \(t=\mathrm{1.45222s}\) (\(T=45.222°C\)):

_images/Object_8.svg

either

_images/Object_9.svg

Then: the plastic deformation no longer evolves.

2.1.2. Linear kinematic work hardening: analytics#

\(t=\mathrm{1s}\) ( \(T=0°C\) ):

_images/Object_10.svg

Heating: Constant plastic deformation up to \(t=356/316=\mathrm{1.12658s}\) (\(T=12.658°C\)):

Then the plastic deformation decreases to reach \(t=\mathrm{2s}\):

_images/Object_11.svg

2.1.3. Nonlinear kinematic hardening I: analytical#

\(t=\mathrm{1s}\) ( \(T=0°C\) ):

_images/Object_12.svg

with \(A=\frac{C}{D}=100\) either

_images/Object_14.svg

Heating: Maximum plastic deformation at \(t=\mathrm{1.26011s}\) (\(T=26.011°C\)):

_images/Object_15.svg

either

_images/Object_16.svg

Plastic deformation no longer evolves until \({t}_{1}=\mathrm{1.98332s}\) (\({T}_{1}=98.332°C\)) where the other end of the elasticity domain is encountered.

Then: the plastic deformation decreases to reach \(t=\mathrm{2s}\):

_images/Object_17.svg

with

_images/Object_18.svg

either

_images/Object_19.svg

2.1.4. Nonlinear kinematic hardening II#

Comparison to the reference solution proposed on day \({\Phi }^{2}\mathrm{As}\). (numerical result obtained with 10 time steps), and comparison to the results obtained with Code_Aster with very fine time discretization

2.2. Precision on the reference results#

We have an analytical solution for the first three behaviors, so the uncertainty is zero. It is estimated at \(\text{4\%}\) for the fourth (difference between the result for 10 steps and the result for 200 steps, the solution strongly dependent on time discretization).

2.3. Bibliographical references#

  1. IPSI: Phi2as study day on 30 March 2000 on the nonlinear behaviors of materials.