Bounds and self-consistent estimates for creep of polycrystalline materials

Abstract

A study of steady creep of face centred cubic (f.c.c.) and ionic polycrystals as it relates to single crystal creep behaviour is made by using an upper bound technique and a self-consistent method. Creep on a crystallographic slip system is assumed to occur in proportion to the resolved shear stress to a power. For the identical systems of an f.c.c. crystal the slip-rate on any system is taken as $\gamma =\alpha (\tau /\tau _{0})^{n}$ where $\alpha $ is a reference strain-rate $\tau $ is the resolved shear stress and $\tau _{0}$ is the reference shear stress. The tensile behaviour of a polycrystal of randomly orientated single crystals can be expressed as $\overline{\epsilon}=\alpha (\overline{\sigma}/\overline{\sigma}_{0})^{n}$ where $\overline{\epsilon}$ and $\overline{\sigma}$ are the overall uniaxial strain-rate and stress and $\overline{\sigma}_{0}$ is the uniaxial reference stress. The central result for an f.c.c. polycrystal in tension can be expressed as $\overline{\sigma}_{0}=h(n)\tau _{0}$. Calculated bounds to $h(n)$ coincide at one extreme $(n=\infty)$ with the Taylor result for rigid/perfectly plastic behaviour and at the other $(n=1)$ with the Voigt bound for linear viscoelastic behaviour. The self-consistent results, which are shown to be highly accurate for $n=1$, agree closely with the upper bound for $n\geq 3$. Two types of glide systems are considered for ionic crystals: A-systems, {110} $\langle 110\rangle $, with $\gamma =\alpha (\tau /\tau _{\text{A}})^{n}$; and B-systems, {100} $\langle 110\rangle $, with $\gamma =\alpha (\tau /\tau _{\text{B}})^{n}$. The upper bound to the tensile reference stress $\overline{\sigma}_{0}$ is shown to have the simple form $\overline{\sigma}_{0}\leq A(n)\tau _{\text{A}}+B(n)\tau _{\text{B}};A(n)$ and $B(n)$ are computed for the entire range of $n$, including the limit $n=\infty $. Self-consistent predictions are again in good agreement with the bounds for high $n$. Upper bounds in pure shear are also calculated for both f.c.c. and ionic polycrystals. These results, together with those for tension, provide a basis for assessing the most commonly used stress creep potentials. The simplest potential based on the single effective stress invariant is found to give a reasonably accurate characterization of multiaxial stress dependence.