Abstract:
In the seminal work on the equilibrium of an elastic bar, Ericksen (1975) revealed that for an up-down-up stress-strain relation multiple-phase states can co-exist. It was found: (1) The stable states are piecewise homogeneous deformations with strain discontinuity. (2) The stress is exactly the Maxwell stress in the stable states. (3) There are infinitely many stable states, and the number of discontinuities of strain is arbitrary. The present work considers the stress-induced phase transitions in a slender 3-D shape memory alloy cylinder by taking into account both macroscopic and microscopic effects (the latter is through the volume fraction of martenisite phase). A main purpose is to examine whether more conclusive results (in comparison with Ericksen's 1-D results) can be achieved. More specifically, a constitutive model in literature is adopted, which leads to a 3-D mechanical system with an internal variable. By utilizing the smallness of the characteristic axial strain and aspect ratio, the complex governing system is reduced to three linear systems for three different regions (austenite, martensite and phase transition regions). Then, we concentrate on the inhomogeneous state with one transformation front, for which the cylinder contains one austenite region, one martensite region. Mathematically, determining the solutions becomes solving a nonlinear eigenvalue problem with a group of eigen parameters. When the two interfaces are planar, we manage to construct the closed-form solutions. The analytical solution reveals some important theoretical insights, comparing with Ericksen's 1-D results. For example, although the axial stress is inhomogeneous in the cross section for the current 3-D setting, for the optimal state (which has the smallest energy value), the average axial stress is still the 1-D Maxwell stress. Also, unlike the 1-D case of Ericksen's bar problem, for which there are infinite-many stable states with the same energy value, the current 3-D problem only has one optimal state although there are still infinite-many solutions. Gradient models are popular in numerical simulations but it is difficult to choose the proper gradient parameter(s). Here, we also make a comparison of our 3-D analytical results with those of a 1-D gradient model, which leads to a plausible choice of the gradient parameter.
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