Bifurcation and metastability in a new one-dimensional model for martensitic phase transitions.

*(English)*Zbl 0949.74049Summary: 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 |

##### Keywords:

stress-induced martensitic phase transitions; one-dimensional model for twinning; two-well nonconvex elastic energy density; local bifurcation analysis; local energy minima; global-bifurcation diagrams; phase nucleation; hysteresis; zero interfacial energy
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\textit{A. Vainchtein} et al., Comput. Methods Appl. Mech. Eng. 170, No. 3--4, 407--421 (1999; Zbl 0949.74049)

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