Further Analysis of Dynamic Strain Aging in Context of an Internal State Variable Constitutive Formalism
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INTRODUCTION
THE analysis of stress–strain curves in a several metals and alloys in the context of an internal state variable constitutive model has led to a common observation when dynamic strain aging (DSA) becomes active. DSA manifests itself as higher than expected stress levels, generally at elevated temperatures. DSA occurs in a temperature regime when solute mobility is sufficient to enable solutes to travel to dislocations, thus promoting additional restriction to the motion of dislocations, which results in additional hardening.[1] This is often accompanied by serrated yielding, although the effects of DSA can be present before serrated yielding is observed.[2,3] Figure 1 shows stress–strain curves in niobium published by Nemat-Nasser and Guo[4] at a strain rate of 0.001 s1 over the temperature range of 77 K to 700 K. The curious feature of these measurements is that the curve at 700 K lies above the curves at 400 K, 500 K, and 600 K. The internal state variable model described by Follansbee[5] demands that the stress uniformly decrease with increasing temperature. Thus, the behavior illustrated in Figure 1 is inconsistent with this constitutive formulation. Trends such as those shown in Figure 1 have been observed in several metals and alloys. A unique ‘‘signature’’ of DSA has been observed in austenitic stainless steels,[5–7] vanadium,[5] niobium,[5] AISI 1018 steel,[5] PAUL S. FOLLANSBEE is with the Saint Vincent College, Latrobe, PA. Contact e-mail: [email protected] Manuscript submitted August 7, 2019.
METALLURGICAL AND MATERIALS TRANSACTIONS A
titanium,[5] Inconel 600,[8] and the high-entropy alloy CoCrFeMnNi.[9] The common observations in these metals and alloys will first be reviewed. Then, the simple analyses proffered in [5] will be rejected, and in Section III an alternate model that considers transport of solutes to the mobile dislocations will be proposed. One of the objectives of the remainder of this section is to restate some of the earlier observations in a form that is more consistent with the proposed model. Predictions of the model will be demonstrated in an austenitic stainless steel, Inconel 600, and niobium. Comparison of model predictions of strain-rate sensitivity in Inconel 600 will call into question one of the underlying assumptions of the analyses to date. The governing equation of the internal state variable model is[5] ^1 ^2 ^e r ra r r r ¼ þ s1 ðe_ ; TÞ þ s2 ðe_ ; TÞ þ se ðe_ ; TÞ l l l0 l0 l0
½1
where r is the stress, ra is an athermal stress (e.g., due to grain boundary strengthening), e_ is the strain rate, T is ^1 is a threshold stress representing the temperature, r ^2 is dislocation interactions with obstacle population 1, r a threshold stress representing dislocation interactions ^e is a threshold stress with obstacle population 2, r representing dislocation interactions with the stored dislocation density, s1, s2, and se are kinetic factors that lie between 0 and unity, l is the temperature dependent shear modulus, and l0 is the shear modulus at 0 K. Equa
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