The Determination of Evolving Microstructure Using Constitutive Relationships
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ABSTRACT The evolution of the constitutive parameter, the mean slip distance, X, is monitored during tensile deformation of 3.2 gm grain size aluminum at 200 K. Transmission electron microscopy (TEM) confirms that the grain size, D, sets an upper limit to X. INTRODUCTION An ideal constitutive relation describing plastic deformation should integrally manifest the microplastic processes using microstructural parameters giving rise to the macroscopic phenomenon of the stress-strain, a-e, behaviour. However, the complexity of the evolving microstructure, which has been modeled by so-called internal state variable models, relies on experimental parameters for "curve fitting" purposes, and has been criticized for not representing any physicalfeatures of the microstructure[1]. By extending a kinematic microscopic model of slip [2], Diak et al. [3] have shown that the correlation between the a-s behaviour and the microplastic processes can be described by the activation volume, V, mean slip distance, X, and mean slip velocity, X: V is inversely proportional to the flow stress, af, which is the difference between the total stress, a, and the stress for inception of dislocation activity, ao (at 0.02 % offset strain); X describes the operation of a slip packet from initiation to termination and is inversely proportional to the product of af and the work hardening coefficient, da/dE; and i is a power law function of af, which incorporates the mobile dislocation density. Thus the functional relationship, a = f(,, i, T), can be evolved and the parameters X and i determined from the experimental conditions as extensively discussed [3]. Through this development the parameters calculated from the mechanical data correlate very well to the model except that the magnitude of X is about one order too large. Thus the model, which inherently assumes that all generated dislocations are stored, has to be relaxed by invoking the dynamic annihilation of dislocations. The primary evidence for such an effect is the well-known phenomenon that the stored work is only about 5 % of the mechanical work expended [2]. Therefore, the real mean slip distance is theoretically tractable from dynamic measurement of the a-e behaviour as [31 ,=( n 2Aa
dy =(nM 'b
b 2
rd-
2Aa
2
2blnde3nM afda =(2Aa(1
where n = number of successive loops from the same source, A = annihilation factor, a = strength parameter, b = Burgers vector, g = shear modulus, and r and y are the shear stress and strain, respectively, calculated from af and , using the Taylor factor, M. Tabata et al.'s [41 in situ observation of dislocation glide within cell interiors suggests that X could correlate to the cell size, d,. Field and Weilland [1] have used TEM to quantitatively assess d. in commercial purity aluminum with 100 gm grain size, D, compressed to small strains between room temperature and 523 K. They observe a more rapid decrease in mean free path for 45 Mat. Res. Soc. Symp. Proc. Vol. 578 © 2000 Materials Research Society
oriented grains versus ones, with values for the me
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