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Bifurcation and metastability in a new one-dimensional model for martensitic phase transitions. (English) Zbl 0949.74049
Summary: Materials undergoing stress-induced martensitic phase transitions often form complex twinned microstructures with multiple phase boundaries. They also exhibit hysteretic mechanical behavior. We propose and analyze a one-dimensional model for twinning. We consider two elastic bars coupled by a system of continuously distributed linear springs. One of the bars has a two-well nonconvex elastic energy density that models a two-variant martensitic phase. The other bar is linearly elastic and is meant to model the parent austenite phase. Interfacial energy is modeled by a strain-gradient term. Various types of boundary conditions model parameter-dependent loading. A local bifurcation analysis shows that local energy minima (metastable states) often involve a large number of phase boundaries. This is confirmed by the global-bifurcation diagrams obtained numerically. We observe that this microstructure emerges via both sudden (finite) and gradual (infinitesimal) phase nucleation. We propose an energetic argument that predicts hysteresis in overall load-deformation behavior due to metastability of multiple equilibria. A limiting case with zero interfacial energy is treated analytically, yielding global solution diagrams.

MSC:
74N30 Problems involving hysteresis in solids
74G65 Energy minimization in equilibrium problems in solid mechanics
74G60 Bifurcation and buckling
Software:
AUTO; AUTO-86
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