Journal article
Still states of bistable lattices, compatibility, and phase transition
Continuum mechanics and thermodynamics, Vol.22(6), pp.421-444
09/2010
Handle:
https://hdl.handle.net/2376/104903
Abstract
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.
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Details
- Title
- Still states of bistable lattices, compatibility, and phase transition
- Creators
- Andrej Cherkaev - Department of Mathematics University of Utah Salt Lake City UT USAAndrei Kouznetsov - Department of Mathematics Washington State University Pullman WA USAAlexander Panchenko - Department of Mathematics Washington State University Pullman WA USA
- Publication Details
- Continuum mechanics and thermodynamics, Vol.22(6), pp.421-444
- Academic Unit
- Mathematics and Statistics, Department of
- Publisher
- Springer-Verlag; Berlin/Heidelberg
- Identifiers
- 99900546788801842
- Language
- English
- Resource Type
- Journal article