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Journal of Nonlinear Science - In the variational problems involving non-convex integral functionals, finding the binodal, the boundary of validity of the quasiconvexity of the energy density, is... 相似文献
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Power law fluctuations and scale-free spatial patterns are known to characterize steady state plastic flow in crystalline materials. In this Letter we study the emergence of correlations in a simple Frenkel-Kontorova-type model of 2D plasticity which is largely free of arbitrariness, amenable to analytical study, and is capable of generating critical exponents matching experiments. Our main observation concerns the possibility to reduce continuum plasticity to an integer-valued automaton revealing inherent discreteness of the plastic flow. 相似文献
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Yury Grabovsky Vladislav A. Kucher Lev Truskinovsky 《Continuum Mechanics and Thermodynamics》2011,23(2):87-123
In the case of Lipschitz extremals of vectorial variational problems, an important class of strong variations originates from
smooth deformations of the corresponding non-smooth graphs. These seemingly singular variations, which can be viewed as combinations
of weak inner and outer variations, produce directions of differentiability of the functional and lead to singularity-centered
necessary conditions on strong local minima: an equality, arising from stationarity, and an inequality, implying configurational
stability of the singularity set. To illustrate the underlying coupling between inner and outer variations, we study in detail
the case of smooth surfaces of gradient discontinuity representing, for instance, martensitic phase boundaries in non-linear
elasticity. 相似文献
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Buckling of slender structures under compressive loading is a failure of infinitesimal stability due to a confluence of two
factors: the energy density non-convexity and the smallness of Korn’s constant. The problem has been well understood only
for bodies with simple geometries when the slenderness parameter is well defined. In this paper, we present the first rigorous
analysis of buckling for bodies with complex geometry. By limiting our analysis to the “near-flip” instability, we address
the universal features of the buckling phenomenon that depend on neither the shape of the domain nor the degree of constitutive
nonlinearity of the elastic material.
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G. PuglisiL. Truskinovsky 《Journal of the mechanics and physics of solids》2002,50(2):165-187
The nontrivial behavior of an elastic chain with identical bi-stable elements may be considered prototypical for a large number of nonlinear processes in solids ranging from phase transitions to fracture. The energy landscape of such a chain is extremely wiggly which gives rise to multiple equilibrium configurations and results in a hysteretic evolution and a possibility of trapping. In the present paper, which extends our previous study of the static equilibria in this system (Puglisi and Truskinovsky, J. Mech. Phys. Solids (2000) 1), we analyze the behavior of a bi-stable chain in a soft device under quasi-static loading. We assume that the system is over-damped and explore the variety of available nonequilibrium transformation paths. In particular, we show that the “minimal barrier” strategy leads to the localization of the transformation in a single spring. Loaded periodically, our bi-stable chain exhibits finite hysteresis which depends on the height of the admissible barrier; the cold work/heat ratio in this model is a fixed constant, proportional to the Maxwell stress. Comparison of the computed inner and outer hysteresis loops with recent experiments on shape memory wires demonstrates good qualitative agreement. Finally we discuss a relation between the present model and the Preisach model which is a formal interpolation scheme for hysteresis, also founded on the idea of bi-stability. 相似文献
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Lev Truskinovsky 《Archive for Rational Mechanics and Analysis》1994,125(4):375-397
Subsonically propagating phase boundaries (kinks) can be modelled by material discontinuities which satisfy integral conservation laws plus an additional jump condition governing the phase-change kinetics. The necessity of an additional jump condition distinguishes kinks from the conventional shocks which satisfy the Lax criterion. We study stability of kinks with respect to the breakup (splitting) into a sequence of waves. We assume that all conventional shocks are admissible and that admissible kinks are selected by a prescribed kinetic relation. As we show, regardless of a particular choice of the kinetic relation, sufficiently fast-phase boundaries are unstable. The mode of instability includes an emission of a centered Riemann wave followed by a sonic shock (Chapman-Jouguet type phase boundary). 相似文献