Precipitate-induced plastic anisotropy: Explicit solutions of the plastic anisotropy due to plate-shaped precipitates
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I. INTRODUCTION
TWO models have previously been proposed to address the effect of precipitates on plastic anisotropy; the plastic inclusion model and the elastic inclusion model. In this study, explicit solutions are derived for the anisotropic terms in the inclusion models for the case of plate-shaped precipitates. From these solutions, additional insight into the physical origin of yield strength anisotropy is achieved, and the differences between the two models may be readily analyzed. The plastic inclusion model proposed by Hosford and Zeisloft[1] considers that precipitates experience plastic deformation. The main assumption of this model is that only a portion of the strain in the matrix is transferred into plastic deformation of precipitates; the remainder of the strain is accommodated by rotation of precipitates and by localized plastic flow in the surrounding matrix. The orientationdependent microstructural parameter, N, that describes the yield anisotropy contribution due to precipitates is the average ratio of the effective strain in the precipitates to the applied strain in the matrix. For the plastic inclusion model, the uniaxial yield stress in a precipitate-strengthened material is given as sy 5 M(tM (1 2 f ) 1 ftPPTN )
[1]
where M is the average Taylor factor calculated from the texture of the matrix and the imposed strain increment, tM is the critical resolved shear stress for the matrix, f is the volume fraction of the precipitates, tPPT is the effective shear strength of the precipitates, and N is the average precipitate strengthening parameter calculated from the distribution of M.T. LYTTLE, formerly Graduate Student, Department of Materials Science and Engineering, University of Virginia, is Post-Doctoral Associate, Materials Research Department, Risø National Laboratory, DK-4000 Roskilde, Denmark, J.A. WERT, Professor, is with the Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22903. Manuscript submitted May 26, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS A
precipitate orientations deduced from habit plane orientation and grain orientation measurements. The elastic inclusion model proposed by Brown and Stobbs[2] and Brown and Clarke,[3] based on the accommodation problems of Eshelby,[4] considers that precipitates experience elastic deformation. The main assumption of this model is that there are elastically strained precipitates in a plastically-deformed matrix.[4,5] The orientation-dependent microstructural parameter, g, that defines the yield anisotropy contribution due to precipitates is the magnitude of the accommodation tensor, which describes the amount of elastic deformation required to deform the inclusion to fit into the hole generated by the externally imposed strain. The form of the accommodation tensor has been solved by Eshelby and considers the shape and orientation of the precipitates. For the elastic inclusion model, the uniaxial yield stress in a precipitate-strengthened material is given as
sy 5 Mtm (1 2 f ) 1 2 mf ,g , «
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