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Finite Element Analysis in Structural Mechanics

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Long-term creep

by Dr. Jiří Plešek

Problem description

Consider the rod shown for a steady-state creep analysis with the temperature T and the uniaxial stress $\sigma_{xx}$ being constant. The effective creep strain rate $\dot\varepsilon_c$ is assumed to be a function of the cumulated creep strain $\varepsilon_c.$

Material properties

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

Creep curve: $\varepsilon_c=a+b\tan\left(ct-d\right)$ with $a=0.949,$ $b=0.6322435755,$ $c=0.013127841\text{ 1/h},$ $d=0.9831024372.$


Clamped at $x=0.$ Statically determinate.


$\sigma_{xx}=30\text{ MPa},$ $T=800\text{ $^\circ$C}.$


The strain rate, which description is required in the input file, is obtained by differentiating the creep curve $$\dot\varepsilon_c=\frac{bc}{\cos^2(ct-d)}.$$

In order to derive the strain hardening form in terms of $\varepsilon_c$ we must eliminate parameter $t$ from the simultaneous expressions $\varepsilon_c(t)$ and $\dot\varepsilon_c(t).$ To this end we use the indentity $$\tan x=\frac{\sin x}{\cos x}=\frac{\sqrt{1-\cos^2 x}}{\cos x}=\sqrt{\frac{1}{\cos^2 x}-1}$$ and $$\frac{1}{\cos^2(ct-d)}=\frac{\dot\varepsilon_c}{bc}.$$

Substituting for $\tan(ct-d)=\sqrt{\dot\varepsilon_c/bc-1}$ into $\varepsilon_c$ we arrive at $$\varepsilon_c=a+b\sqrt{\frac{\dot\varepsilon_c}{bc}-1}$$ and finally $$\dot\varepsilon_c=bc+a^2\frac{c}{b} -2a\frac{c}{b}\varepsilon_c+\frac{c}{b}\varepsilon_c^2.$$ This polynomial can be written in the PMD format as $$\dot\varepsilon_c=a_1+a_2\varepsilon_c+a_3\varepsilon_c^2$$ where $$\begin{align} a_1 &= 2.7\times10^{-2}\text{ 1/h}, \\ a_2 &= -3.94099\times10^{-2}\text{ 1/h}, \\ a_3 &= 2.07639\times10^{-2}\text{ 1/h}. \end{align}$$

en/example/creep/1/start.txt · Last modified: 2022-02-11 14:25 by Petr Pařík