The calculation of habit planes for elastic transformations by minimization of their elastic strain energy
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DAVID R. CLARKE is Lecturer, University of California at Berkeley, Berkeley, CA 94720. Manuscript submitted August 20, 1975. METALLURGICAL TRANSACTIONS A
created by the transformation, and the stored energy in the elastic field, were shown by Eshelby to be also dependent on the shape of the inclusion and its orientation with respect to the coordinate axes of the transformation strain. More recently, Brown and Clarke10 have calculated these effects on the shear stress components of the elastic field, together with tabulating them for many of the more common inclusion geometries. The approach adopted here is firstly to decide on a plausible transformation strain that will describe the change in crystal structure that occurs on transformation; secondly to calculate the stored elastic energy when an elipsoid has this strain, and to find the shape and orientation of the ellipsoid that will minimize this energy. The same approach has been employed independently by Shibata and Ono'! to calculate successfully the habit planes for the bcc-hcp transformations in titanium. In his 1957 paper, Eshelby used the idea that a transformation would occur in such a way as to minimize its total elastic energy, in order to deduce from Zener's12 value of the heat of formation for martensite in iron that the martensite formed as discs with an aspect ratio of about 0.08. Later, Christtan'" in an attempt to relate the accommodation effects of a transformation to the Bowles and Mackenz.ie'" dilatation parameter, applied Eshelby's method to the case of a thin ellipsoidal plate subjected to a general transformation strain. He showed that the accommodation, and hence the elastic energy, was minimized as the aspect ratio of the plate decreases. Since it is based on Eshelby's method, the procedure used here is appropriate to any coherent shear transformation. For this reason Section 2 is devoted to illustrating its applicability to a variety of transformation processes: a simple twinning reaction, the simple transformation in an indium -thallium alloy, the more complex martensitic transformation in an iron-nickel alloy, and the martensitic transformation in polyethylene crystals, where a habit plane has yet to be reported.
As the purpose of this paper is to demonstrate the VOLUME 7A, MAY 1976-723
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calculation of habit planes and the shapes of the transformed products, rather than to present highly accurate calculations, the simplifications of linear isotropic elasticity and of an isotropic surface energy of the transformed inclusion are assumed. Indeed it is shown in Section 3, that unless the transformed regions are very small (;S1O-21 cm") the surface energy contribution to the total free energy is negligible compared with the elastic strain energy.
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1. BACKGROUND THEORY
The problem that Eshelby tackled in his papers was how to calculate the elastic state of a body when an internal region underwent a change in shape or size, which in the absence of the constraining material could be described by a stress-free trans
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