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 γ = α(ז/ז0)n where α is a reference strain-rate, ז is the resolved shear stress and ז0 is the reference shear stress. The tensile behaviour of a polycrystal of randomly orientated single crystals can be expressed as ∊̄ = α(σ̄/σ̄0)n where ∊̄ are σ̄ the overall uniaxial strain-rate and stress and σ̄0 is the uniaxial reference stress. The central result for an f. c. c. polycrystal in tension can be expressed as σ̄0 = h(n) ז0. 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} <110>, with γ = α(ז/זA)n; and B-systems, {100} <110>, with γ = α(ז/זB)n. The upper bound to the tensile reference stress σ̄0 is shown to have the simple form σ̄0 ≼ A(n)זA+B(n)זB; 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.