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21.
Marginal stability plays an important role in nonlinear elasticity because the associated minimally stable states usually delineate failure thresholds. In this paper we study the local (material) aspect of marginal stability. The weak notion of marginal stability at a point, associated with the loss of strong ellipticity, is classical. States that are marginally stable in the strong sense are located at the boundary of the quasi-convexity domain and their characterization is the main goal of this paper. We formulate a set of bounds for such states in terms of solvability conditions for an auxiliary nucleation problem formulated in the whole space and present nontrivial examples where the obtained bounds are tight. 相似文献
22.
We propose a new method of constructing a series of nested quasicontinuum models, which describe linear elastic behavior of
crystal lattices at successively smaller scales. The relevant scales are dictated by the interatomic interactions and are
not arbitrary. The novelty of the model is in the use of a decomposition of the displacement field into the coarse part and
the micro-level corrections. The coarse contribution is the conventional homogenized displacement field used in classical
continuum elasticity. The micro-level corrections are sub-continuum fields representing the fine structure of the boundary
layers exhibited by the discrete equilibrium configuration. The model is based on a multi-point Padé approximation in the
Fourier space of the discrete Green’s function. We systematically compare the new model with the conventional strain gradient
model.
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23.
We examine some elementary interpretations of the classical theorem of Clapeyron in linear elasticity theory. As we show, a straightforward application of this theorem in the purely mechanical setting leads to an apparent paradox which can be resolved by referring either to dynamics or to thermodynamics. These richer theories play an essential part in understanding the physical significance of this theorem. 相似文献
24.
This paper addresses the non-uniqueness pointed out by Ericksen in his classical analysis of the equilibrium of a one-dimensional elastic bar with non-convex energy [1]. Following some previous work in this area, we suitably regularize the problem in order to investigate this degenerancy. Our approach gives an explicit framework for the the study of the rich variety offinite-scale equilibrium microstructures for the bar in a hard loading device, and their stability properties. In this way we clarify the role of interfacial energy in creating finitescale microstructures, by considering the combined effect of the oscillation-inducing and oscillation-inhibiting terms in the energy functional.
Sommario Il lavoro riguarda la non unicità messa in luce da J.L. Ericksen nella sua analisi dell'equilibrio di barre elastiche con energia non convessa. Seguendo le linee di precedenti lavori, per investigare questa degenerazione si ricorre ad una regolarizzazione del problema e si dà un esplicito quadro di riferimento per lo studio della ricca varietà delle microstrutture di scala finita e della loro stabilità. Si chiarisce in particolare il ruolo dell'energia di interfaccia nella creazione di microstrutture di scala finita considerando l'effetto combinato di termini inibitori e favorevoli all'insorgere di oscillazioni nel funzionale energia.相似文献
25.
In the setting of the simplest debonding problem we give a systematic comparison of the fracture models due to Griffith and Barenblatt. We prove that the Griffith model represents an asymptotic
-limit of the Barenblatt model, when the ratio of the external and internal lengths increases indefinitely. We then illustrate the character of convergence by solving explicitly two sample problems with initially rigid and initially elastic cohesive energies. The geometrical simplicity of the setting allows us to study the small parameter dependence of both global and local minimizers of the total energy.Received: 22 April 2003, Accepted: 12 September 2003, Published online: 9 January 2004PACS:
62.20.Mk, 68.35.Md, 81.40.Jj, 83.50.LhCorrespondence to: J.-J. Marigo 相似文献
26.
In order to understand the phenomenon of reversible plasticity exhibited by shape memory alloys and other smart materials,
we study an elementary prototypical model. Building on an original idea of Müller and Villaggio [17], we consider an inhomogeneous
ensemble of bi-stable elements connected in series and loaded in a soft device. To interpret the fine structure of the hysteresis
loops observed experimentally, we assume that the dynamics is maximally dissipative and investigate different evolutiona ry
strategies for a “driven” system with external force changing quasi-statically. Our main result is that the inhomogeneity
of the elastic properties leads to a distinctive hardening with serrations of a Portevin-Le Chatelier type and produces a
realistic memory structure characterized by the “congruency” and “return point memory” properties.
Received December 28, 2001 / Published online June 4, 2002
Dedicated to Ingo Müller on the occasion of his 65th birthday
Communicated by Kolumban Hutter, Darmstadt 相似文献
27.
Alexander?MielkeEmail author Lev?Truskinovsky 《Archive for Rational Mechanics and Analysis》2012,203(2):577-619
We show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete
microscopic model describing a visco-elastic crystal lattice with quenched disorder. The constitutive structure changes as
a result of two concurrent limiting procedures: the vanishing-viscosity limit and the discrete-to-continuum limit. In the
course of these limits a non-convex elastic problem transforms into a convex elastic problem while the quadratic rate-dependent
dissipation of visco-elastic lattice transforms into a singular rate-independent dissipation of an ideally plastic solid.
In order to emphasize our ideas we employ in our proofs the simplest prototypical system mimicking the phenomenology of transformational
plasticity in shape-memory alloys. The approach, however, is sufficiently general that it can be used for similar reductions
in the cases of more general plasticity and damage models. 相似文献
28.
29.
Our starting point is a parameterized family of functionals (a ‘theory’) for which we are interested in approximating the
global minima of the energy when one of these parameters goes to zero. The goal is to develop a set of increasingly accurate
asymptotic variational models allowing one to deal with the cases when this parameter is ‘small’ but finite. Since Γ-convergence
may be non-uniform within the ‘theory’, we pose a problem of finding a uniform approximation. To achieve this goal we propose
a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically
matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas
with physically meaningful examples covering a broad set of subjects from homogenization and dimension reduction to fracture
and phase transitions. In particular, we give considerable attention to the problem of transition from discrete to continuum
when the internal and external scales are not well separated, and one has to deal with the so-called ‘size’ or ‘scale’ effects.
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30.