1. Reference problem#

1.1. Geometry#

Volume element materialized by a cube with a unit side:

\(E={2.10}^{5}\mathrm{MPa}\), \(\nu =0.3\), \(\alpha ={10}^{-5}°{C}^{-1}\)

The material is elastoplastic with different types of behavior:

\(\mathrm{C1}\): Isotropic hardening: the traction curve is of the form:

\(\sigma ({\varepsilon }^{P},T)={\sigma }_{y}(T)+Q(T)(1-{e}^{-b(T){\varepsilon }^{P}})\)

\(\mathrm{C2}\): Linear kinematic hardening:

\(\sigma ({\varepsilon }^{P},T)=\pm {\sigma }_{y}(T)+C(T){\varepsilon }^{P}\)

\(\mathrm{SIGY}=200.-\mathrm{1.7.T}\)	(in \(\mathrm{MPa}\))
\(C(T)=1000+\mathrm{2990.T}\)	(in \(\mathrm{MPa}\))

\(\mathrm{C3}\): Nonlinear kinematic hardening (\(I\)):

\(\mathrm{C4}\): Nonlinear kinematic hardening (\(\mathrm{II}\)):

same characteristics as for behavior \(\mathrm{C3}\), except \(D(T)=50+\mathrm{2T}\)

Such that the state of stress and deformation are uniform in the volume element:

Point \(B\) stuck in \(x\), \(y\) and \(z\). Point \(A\) stuck in \(z\), \(\mathrm{DY}=0\) on the side \(\mathrm{ABFE}\)

Force distributed on the face \(\mathrm{CDHG}\): \(\mathrm{Fy}\)

Uniform temperature \(T(t)\) on the cube. The reference temperature is \(0°C\).

\(\mathrm{Fy}\) and \(T\) vary over time as follows: