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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.

74N30 Problems involving hysteresis in solids
74G65 Energy minimization in equilibrium problems in solid mechanics
74G60 Bifurcation and buckling
Full Text: DOI
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