The main objective of this work is to quantify the effect of the reinforcing phase on the stress and strain fields at crack tip regions and to enhance the current understanding of the factors controlling short crack behaviour in Ni-base superalloys at high temperatures. The structure of the paper is as follows. First, a brief outline of the constitutive approach is presented. Then, the crack tip fields in a typical compact tension (CT) specimen are presented as a function of the volume fraction of the precipitate population. This is followed by a discussion of the results and the implications for crack growth predictions in single crystals. MULTI-SCALE CRYSTALLOGRAPHIC FORMULATION The average macroscopic stress-strain behaviour of the superalloy single crystal (SC) of interest, viz. CMSX4, is described by the multi-scale rate dependent crystallographic formulation recently proposed by Busso [5]. The flow rule relies on a stress-dependent activation energy expressed in terms of two internal state variables per slip system, α: a macroscopically average slip resistance Sα, and a internal or back stress Bα. Thus, ˙γα = ˙γo exp − Fo kθ 1− | τα −Bα|−Sαµ/µ0 ˆτ0µ/µ0 p q sgn(τα) , (1) where τα is the resolved shear stress, θ the absolute temperature, µ, µ0 the shear moduli at θ and 0 K, respectively, and F0, ˆτo, p, q and ˙γ0 are material parameters. The evolutionary behaviour of the overall slip resistance is given by, ˙Sα = n β=1 δαβ S hs −dD(S β −Sβ 0) | ˙γ β| , (2) where Sα 0 is the initial value of Sα, dD is a dynamic recovery function and nis the total number of slip systems. InEq. 2, δαβ S is the latent hardening or interaction function. Here, self-hardening is assumed so that δ αβ S =δαβ, the Kroneker delta. The formulation contains an explicit link between the γ precipitate population at the microscale and the behaviour of the homogeneous equivalent material at the macroscale. This link is introduced through the dynamic recovery function dD and the initial microstructural state, Sα 0 in Eq. 2 which, in turn, depend on the characteristics of the current precipitate population. Here, dD = ˆdD{l/lm, vf} , (3) Sα 0 = ˆS0 {l/lm, vf} , (4) where l/lmis the precipitate size normalised by a mean reference value, and vf the precipitate volume fraction. Equations 3 and 4 have been calibrated from FE analyses of periodic unit cells at the microscale containing the individual precipitates [4]. Note that the effects of the precipitate aspect ratio will not be quantified in this paper. For details, see [3] and [4]. The back stress evolves according to , ˙Bα = hB˙γ α −rDBα| ˙γα|, (5) where hB is the hardening coefficient, and rD a dynamic recovery function expressed in terms of the current overall deformation resistance, rD = hBµo S(α) µ 0 fcλ − µ . (6) Here, fc and λ are statistical factors, and µ 0 is the shear modulus in the slip plane. Note that the dependency of the back stress on the characteristics of the precipitate population is implicitly incorporated through Sα in Eq. 6. The above crystallographic formulation has been implemented numerically into a material subroutine in a commercial FE code [6] using a large strain algorithm with an implicit time-integration procedure. It relies on the multiplicative decomposition of the total deformation gradient, F, into an inelastic component, Fp, 2
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