Nonlinear and nonlocal continuum model of transformation precursors in martensites
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I.
INTRODUCTION
A. Modulated Structures and Precursors O n e of the most interesting developments over the past two decades in the study of structural phase transitions has been the direct observation of pretransitional modulated structures. 1-5 Made possible by the great advances in experimental methods over that period, it is now recognized that these phenomena are ubiquitous to an extent hardly expected, particularly in weakly first order phase transitions where mode softening with temperature is not found to a significant degree. Precursors have been observed in alloys, 2,3 minerals, 4'5 ceramics, 6 ferroelectrics,7 Jahn-Teller materials, s'9 and perhaps even in solid helium. 10 Many types of pretransitional structures have been observed, among them the so-called "tweed" patterns. 11'12'13 Several possible causes of this pretransitional behavior have been suggested. Although phonon anomalies (consisting of broad shoulders or weak minima in the phonon dispersion curves at some nonzero wave vector) usually are found in materials which exhibit these effects,'21415 ' genuine phonon mode softening with temperature "to (near) zero" has not been found. Thus it is hard to believe that the explanation lies in soft mode transition theories. However, as far as is known, materials exhibiting anomalously low, anisotropic shear modulus softening show precursor structure in the prototype phase, 2,H i.e., materials for which the slope of a transverse acoustic branch in the long wavelength limit and/or the associated elastic shear modulus are strongly temperature dependent and become very small (or even tend to zero) at the transition temperature for weakly first (or second) order phase transitions. In addition to anomalous shear modes, for both Inl_xT1x alloys 16 and AI5 type intermetallic compounds 17-2° anomalies in elastic data and thermal expansion are seen for G.R. BARSCH is Professor of Physics, Department of Physics and Materials Research Laboratory, The Pennsylvania State University, University Park, PA 16802. J.A. KRUMHANSL is Horace White Professor of Physics, Department of Physics, Cornell University, Ithaca, NY 14853. This paper is based on a presentation made in the symposium "Pretransformation Behavior Related to Displacive Transformations in Alloys" presented at the 1986 annual A1ME meeting in New Orleans, March 2-6, 1986, under the auspices of the ASM-MSD Structures Committee. METALLURGICAL TRANSACTIONS A
tens of degrees above the transition temperature. It thus seems impossible to explain these as some form of critical fluctuations. 2~ One class of models attributes the precursors to defects or impurities. It is quite possible that Clapp's 22 "localized soft mode" concept is pertinent for particular situations; or that precipitate models230r other structural defect models 24'25 may apply in special cases. However, in view of the now recognized reproducibility and generality of precursor formation, it is important to examine alternative intrinsic mechanisms as well. The fundamental question, then, is whether
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