Still states of bistable lattices, compatibility, and phase transition

Cherkaev, Andrej; Kouznetsov, Andrei; Panchenko, Alexander
September 2010
Continuum Mechanics & Thermodynamics;Sep2010, Vol. 22 Issue 6-8, p421
Academic Journal
We study a two-dimensional triangular lattice made of bistable rods. Each rod has two equilibrium lengths, and thus its energy has two equal minima. A rod undergoes a phase transition when its elongation exceeds a critical value. The lattice is subject to a homogeneous strain and is periodic with a sufficiently large period. The effective strain of a periodic element is defined. After phase transitions, the lattice rods are in two different states and lattice strain is inhomogeneous, the Cauchy-Born rule is not applicable. We show that the lattice has a number of deformed still states that carry no stresses. These states densely cover a neutral region in the space of entries of effective strains. In this region, the minimal energy of the periodic lattice is asymptotically close to zero. When the period goes to infinity, the effective energy of such lattices has the 'flat bottom' which we explicitly describe. The compatibility of the partially transited lattice is studied. We derive compatibility conditions for lattices and demonstrate a family of compatible lattices (strips) that densely covers the flat bottom region. Under an additional assumption of the small difference of two equilibrium lengths, we demonstrate that the still structures continuously vary with the effective strain and prove a linear dependence of the average strain on the concentration of transited rods.


Related Articles

  • A model of threading dislocation density in strain-relaxed Ge and GaAs epitaxial films on Si (100). Wang, G.; Loo, R.; Simoen, E.; Souriau, L.; Caymax, M.; Heyns, M. M.; Blanpain, B. // Applied Physics Letters;3/9/2009, Vol. 94 Issue 10, pN.PAG 

    Strain relaxation in large lattice-mismatched epitaxial films, such as Ge and III-V materials on Si, introduces high threading dislocation densities (TDDs). A thermodynamic model of TDD dependence on film thickness is developed. According to this model, the quasiequilibrium TDD of a given...

  • Thermodynamic effects of linear dissipative small deformations. Anssari-Benam, Afshin; Viola, Giuseppe; Korakianitis, Theodosios // Journal of Thermal Analysis & Calorimetry;Jun2010, Vol. 100 Issue 3, p941 

    This paper presents a phenomenological model of dissipative losses manifested as heat transfer effects in small linear deformations of solid continua. The impetus is the need for a unified theory characterizing heat transfer effects (called “stretching calorimetry” in the literature)...

  • Development of New Control Techniques for Vibration Isolation of Structures using Smart Materials. Bhat, Shubha P.; Manjunath, T. C.; Ardil, C. // International Journal of Computer, Information & Systems Science;2008, Vol. 2 Issue 3, p163 

    In this paper, the effects of the restoring force device on the response of a space frame structure resting on sliding type of bearing with a restoring force device is studied. The NS component of the El - Centro earthquake and harmonic ground acceleration is considered for earthquake...

  • Inelastic deformation of flexible cylindrical shells with an elliptic hole. Chernyshenko, I. S.; Storozhuk, E. A.; Kadyrov, F. D. // International Applied Mechanics;May2007, Vol. 43 Issue 5, p512 

    The elastoplastic state of isotropic homogeneous cylindrical shells with elliptic holes and finite deflections under internal pressure is studied. Problems are formulated and numerically solved taking into account physical and geometrical nonlinearities. The distribution of stresses...

  • Comparison of physical and mechanical properties of tension and opposite wood from ten tropical rainforest trees from different species. Julien Ruelle; Jacques Beauchene; Anne Thibaut; Bernard Thibaut // Annals of Forest Science (EDP Sciences);Jul2007, Vol. 64 Issue 5, p503 

    On 10 trees from 10 species of French Guyana tropical rainforest in a clear active process of restoring verticality growth strains were measured in situ in order to determine the occurrence of tension wood within samples. Wood specimens were cut in the vicinity of the growth strains measurements...

  • A constitutive model for strain rates from 10-4 to 106 s-1. Steinberg, D. J.; Lund, C. M. // Journal of Applied Physics;2/15/1989, Vol. 65 Issue 4, p1528 

    Presents a study which developed an addition to the Steinberg-Guinan high strain-rate constitutive model that extends its validity to strain rates. Rate-dependent, shock-induced phenomena in tantalum that was reproduced with the use of the model; Data for yield strength versus strain rate at...

  • Two-dimensional growth of strained Ge0.85Si0.15 on Si(111) by liquid phase epitaxy. Hansson, P. O.; Ernst, F.; Bauser, E. // Journal of Applied Physics;9/1/1992, Vol. 72 Issue 5, p2083 

    Provides information on a study which investigated heteroepitaxy immediate to thermodynamically equilibrium for liquid phase epitaxy of Ge[sub0.85]Si[sub0.15] on Si(111). Status of the GeSi layer below and beyond the critical thickness for strain relaxation; Problem of Ge[subx]Si[sub1-x]/Si...

  • The Influence of Torsional Moments on the Elastoplastic Interaction Curves of a Symmetrical I Cross-Section. Michaltsos, George T.; Raftoyiannis, Ioannis G. // Open Mechanics Journal;2010, p26 

    This work deals with the influence of the pure torsional (St. Venant) moment on deriving the elastoplastic interaction curves of internal forces and moments of a symmetrical I-section. The problem is addressed employing the exact theory of torsion. The simultaneous action of normal and shear...

  • The Ranking of Efficiency of Structural Layouts using Form Factors Part 2: Design for Strength. Burgess, S C // Proceedings of the Institution of Mechanical Engineers -- Part C;1998, Vol. 212 Issue 2, p129 

    This paper presents a method for ranking the mass efficiency of simple structural layouts for strength design using form factors. Form factors are derived for three different frameworks and seven different beams for an example load case of a simply supported and centrally loaded structure. The...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics