Analysis of the Temperature and Strain-Rate Dependences of Strain Hardening
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munication Analysis of the Temperature and Strain-Rate Dependences of Strain Hardening JOHANNES KREYCA and ERNST KOZESCHNIK A classical constitutive modeling-based Ansatz for the impact of thermal activation on the stress–strain response of metallic materials is compared with the state parameter-based Kocks–Mecking model. The predicted functional dependencies suggest that, in the first approach, only the dislocation storage mechanism is a thermally activated process, whereas, in the second approach, only the mechanism of dynamic recovery is. In contradiction to each of these individual approaches, our analysis and comparison with experimental evidence shows that thermal activation contributes both to dislocation generation and annihilation. DOI: 10.1007/s11661-017-4402-5 Ó The Minerals, Metals & Materials Society and ASM International 2017
Physical models describing the stress–strain evolution of a material during plastic deformation[1–6] are commonly founded on (i) the Taylor equation,[7,8] which relates the stress contribution due to forest hardening, i.e., the true stress, r, to the average dislocation density, q, as pffiffiffi pffiffiffi ½1 r ¼ aMGb q ¼ g1 q; where M is the Taylor factor, a is the strengthening coefficient, G is the shear modulus, and b is the Burgers vector, and g1 ¼ aMGb; and (ii) a differential equation for the average dislocation density evolution in the form: dq dqþ dq ¼ : de de de
½2
In Eq. [2], the generation of dislocations due to plastic deformation is accounted for in the dislocation storage term, dqþ =de, whereas the annihilation of dislocations
JOHANNES KREYCA and ERNST KOZESCHNIK are with the Institute of Materials Science and Technology, TU Wien, Getreidemarkt 9, 1060 Vienna, Austria. Contact e-mail: [email protected] Manuscript submitted August 16, 2017.
METALLURGICAL AND MATERIALS TRANSACTIONS A
dqþ M : ¼ bL de
½3
With the assumption that the mean free path is indirectly proportional to the square root of the dislocation density, A L ¼ pffiffiffi ; q
½4
Equation [2] delivers the well-known Kocks–Mecking equation[11] reading dq pffiffiffi ¼ k1 q k2 q: de
½5
Here, k1 ¼ M=bA; and A is the proportionality constant. Dynamic recovery was first associated with thermally activated cross-slip by Mott.[12] A detailed theory was then developed in References 13, 14 concluding that the decreasing strain hardening rate during stage III hardening and its temperature and strain-rate dependences are a result of dynamic recovery due to thermally activated cross-slip. For a detailed review, see References 15 and 16. In most dislocation density-based models,[1–6] k1 is assumed to be temperature independent (except for the temperature dependence of the shear modulus), whereas k2 is treated as a temperature- and strain-rate-dependent parameter. The assumption of an athermal k1 is equivalent to stating that dislocation storage is not a thermally activated process. Importantly, these two assumptions have severe implications on the shape of the modeled stress–strain curves. Within the Kocks–Mecking fr
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