Mechanics of a discrete chain with bi-stable elements.

*(English)*Zbl 0973.74060Summary: It has become common to model materials supporting several crystallographic phases as elastic continua with non (quasi) convex energy. This peculiar property of the energy originates from the multi-stability of the system at the microlevel associated with the possibility of several energetically equivalent arrangements of atoms in crystal lattices. In this paper we study the simplest prototypical discrete system – a one-dimensional chain with a finite number of bi-stable elastic elements.

Our main assumption is that the energy of a single spring has two convex wells separated by a spinodal region where the energy is concave. We neglect the interaction beyond nearest neighbors, and explore in some detail a complicated energy landscape for this mechanical system. In particular, we show that under generic loading the chain possesses a large number of metastable configurations which may contain up to one (snap) spring in the unstable (spinodal) state. As the loading parameters vary, the system undergoes a number of bifurcations, and we show that the type of bifurcation may depend crucially on the details of the concave (spinodal) part of the energy function. In special cases we obtain explicit formulas for the local and global minima, and provide a quantitative description of the possible quasi-static evolution paths and of the associated hysteresis.

Our main assumption is that the energy of a single spring has two convex wells separated by a spinodal region where the energy is concave. We neglect the interaction beyond nearest neighbors, and explore in some detail a complicated energy landscape for this mechanical system. In particular, we show that under generic loading the chain possesses a large number of metastable configurations which may contain up to one (snap) spring in the unstable (spinodal) state. As the loading parameters vary, the system undergoes a number of bifurcations, and we show that the type of bifurcation may depend crucially on the details of the concave (spinodal) part of the energy function. In special cases we obtain explicit formulas for the local and global minima, and provide a quantitative description of the possible quasi-static evolution paths and of the associated hysteresis.

##### MSC:

74N30 | Problems involving hysteresis in solids |

74N05 | Crystals in solids |

74E15 | Crystalline structure |

82D25 | Statistical mechanical studies of crystals |

82B26 | Phase transitions (general) in equilibrium statistical mechanics |

##### Keywords:

phase transformations; twinning; crystal plasticity; two-well energy; discrete one-dimensional chain; bi-stable elastic elements; spinodal region; bifurcation; quasi-static evolution paths; hysteresis
PDF
BibTeX
XML
Cite

\textit{G. Puglisi} and \textit{L. Truskinovsky}, J. Mech. Phys. Solids 48, No. 1, 1--27 (2000; Zbl 0973.74060)

Full Text:
DOI

##### References:

[1] | Alinger, T.L.; Epstein, M.; Herzog, W., Stability of muscle fibers on the descending limb of the force-length relation, J. biomechanics, 5, 627-633, (1996) |

[2] | Braides, A., Dal, Maso G., Garroni, A. 1998 Variational formulation of softening phenomenon in fracture mechanics. Preprint · Zbl 0945.74006 |

[3] | Ericksen, J.L., Equilibrium of bars, J. elast., 5, 191-201, (1975) · Zbl 0324.73067 |

[4] | Ericksen, J.L., The Cauchy and Born hypothesis for crystals, () · Zbl 0567.73112 |

[5] | Ericksen, J.L., Local bifurcation theory for thermoelastic bravais lattices, () · Zbl 0797.73050 |

[6] | Fedelich, B.; Zanzotto, G., Hysteresis in discrete systems of possibly interacting elements with a double-well energy, J. nonlinear sci., 2, 319-342, (1992) · Zbl 0814.73004 |

[7] | Muller, I., Seelecke, S. 1996 Thermodynamic aspects of shape memory alloys. Preprint · Zbl 1066.74043 |

[8] | Muller, I.; Villaggio, P., A model for an elastic-plastic body, Arch. rat. mech. anal., 65, 25-46, (1977) · Zbl 0366.73038 |

[9] | Puglisi, G.; Truskinovsky, L., First and second order phase transitions in a discrete system with bistable elements, (), 140-152 |

[10] | Puglisi, G., Truskinovsky, L. 1998 Hysteresis in a chain with bi-stable elements. To be submitted · Zbl 1035.74045 |

[11] | Rogers, R.; Truskinovsky, L., Discretization and hysteresis, Physica B, 233, 370-375, (1997) |

[12] | Truskinovsky, L., Fracture as a phase transition, (), 322-332 |

[13] | Vainshtein, A.; Healey, T.; Rosakis, P.; Truskinovsky, L., The role of the spinodal in one-dimensional phase transitions microstructures, Physica D, 115, 29-48, (1998) · Zbl 0962.74530 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.