Using the penalty method

Consider a creep analysis of the thick-wall tube shown. Compute the relaxation history of stress under plane strain conditions using
(a) the axisymmetric 8-node elements and
(b) the 3D elements with the symmetry condition enforced by the penalty method.
Compare the results.

$E=2\times10^5\text{ MPa},$ $\nu=0.3,$ $\alpha=10^{-5}\text{ 1/K}.$

Norton's law: $\dot\varepsilon_c=\gamma(\sigma_e/\sigma_0)^n$ with $\gamma=2\times10^{-28}\text{ 1/h},$ $n=3,$ $\sigma_0=1\text{ Pa}.$

Plane strain condition.

Internal pressure $p=100\text{ MPa},$ $T=600\text{ }^\circ\text{C}.$

• Axisymmetric solution, TUBE7C1
• Penalty solution, TUBE56C1

Material description is included in the input files as in Relaxation, i.e., $$\dot\varepsilon_c=a_1 +a_2\sigma_e+a_3\sigma_e^2+a_4\sigma_e^3$$ where $$a_1=a_2=a_3=0,\quad a_4=2\times10^{-28}\text{ 1/h}$$

The process of relaxation is studied under steady-state conditions $(p,T)$ for 10 hours with the elastic solution being the initial stress state. It should be noted that the automatic integration step control causes the time increments to increase as relaxation proceeds—see the .oL protocols.