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# Bounds and Self-Consistent Estimates for Creep of Polycrystalline Materials

Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, Vol. 348, No. 1652. (10 February 1976), pp. 101-127, doi:10.1098/rspa.1976.0027  Key: citeulike:12143089

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### 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 $γ =α (τ /τ _0)^n$ where $α$ is a reference strain-rate $τ$ is the resolved shear stress and $τ _0$ is the reference shear stress. The tensile behaviour of a polycrystal of randomly orientated single crystals can be expressed as $overlineε=α (overlineσ/overlineσ_0)^n$ where $overlineε$ and $overlineσ$ are the overall uniaxial strain-rate and stress and $overlineσ_0$ is the uniaxial reference stress. The central result for an f.c.c. polycrystal in tension can be expressed as $overlineσ_0=h(n)τ _0$. Calculated bounds to $h(n)$ coincide at one extreme $(n=∞)$ 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 $n≥ 3$. Two types of glide systems are considered for ionic crystals: A-systems, 110 $\langle 110\rangle$, with $γ =α (τ /τ _\textA)^n$; and B-systems, 100 $\langle 110\rangle$, with $γ =α (τ /τ _\textB)^n$. The upper bound to the tensile reference stress $overlineσ_0$ is shown to have the simple form $overlineσ_0≤ A(n)τ _\textA+B(n)τ _\textB;A(n)$ and $B(n)$ are computed for the entire range of $n$, including the limit $n=∞$. 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.