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1.
The remarkable ability of nematic elastomers to exhibit large deformations under small applied forces is known as soft elasticity.
The recently proposed neo-classical free-energy density for nematic elastomers, derived by molecular-statistical arguments,
has been used to model soft elasticity. In particular, the neo-classical free-energy density allows for a continuous spectrum
of equilibria, which implies that deformations may occur in the complete absence of force and energy cost. Here we study the
notion of force-free states in the context of a continuum theory of nematic elastomers that allows for isotropy, uniaxiality,
and biaxiality of the polymer microstructure. Within that theory, the neo-classical free-energy density is an example of a
free-energy density function that depends on the deformation gradient only through a nonlinear strain measure associated with
the deformation of the polymer microstructure relative to the macroscopic continuum. Among the force-free states for a nematic
elastomer described by the neo-classical free energy density, there is, in particular, a continuous spectrum of states parameterized
by a pair of tensors that allows for soft deformations. In these force-free states the polymer microstructure is material
in the sense that it stretches and rotates with the macroscopic continuum. Limitations of and possible improvements upon the
neo-classical model are also discussed.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
2.
Antonin Chambolle 《Archive for Rational Mechanics and Analysis》2003,167(3):211-233
We show that in a two-dimensional bounded open set whose complement has a finite number of connected components, the vector
fields uH
1
(Ωℝ2) are dense in the space of fields whose symmetrized gradient e(u) is in L
2
(Ωℝ4). This allows us to show the continuity of some linearized elasticity problems with respect to variations of the set, with
applications to shape optimization or the study of crack evolution.
(Accepted September 18, 2002)
Published online February 4, 2003
Commmunicated by V. Šverák 相似文献
3.
Material frame indifference implies that the solution in non-linear elasticity theory for a connected body rigidly rotated
at its border is a rigid, stress-free, deformation. If the same problem is considered within linear elasticity theory, considered
as an approximation to the true elastic situation, one should expect that if the angle of rotation is small, the body still
undergoes a rigid deformation while the corresponding stress, though not zero, remains consistently small. Here, we show that
this is true, in general, only for homogeneous bodies. Counterexamples of inhomogeneous bodies are presented for which, whatever
small the angle of rotation is, the linear elastic solution is by no means a rigid rotation (in a particular case it is an
“explosion”) while the stress may even become infinite. If the same examples are re-interpreted as problems in an elasticity
theory based upon genuinely linear constitutive relations which retain their validity also for finite deformations, it is
shown that they would deliver constraint reaction forces that are not in equilibrium in the actual, deformed, state. This
furnishes another characterization of the impossibility of an exact linear constitutive theory for elastic solids with zero
residual stress.
相似文献
4.
Sergio Gutiérrez 《Journal of Elasticity》1998,53(3):215-256
The explicit computation of the effective elasticity tensor of the material produced by laminating two homogeneous elastic
media is used to show that, in 2-dimensional and 3-dimensional linear elasticity, for any isotropic material a whose elasticity tensor is strongly elliptic, but not semipositive definite, we can select very strongly elliptic materials,
so that through laminations between these with material a, we can create a nonstrongly elliptic media, whose existence contradicts properties concerning the propagation of elastic
waves.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
5.
Roberto Alicandro Marco Cicalese Antoine Gloria 《Archive for Rational Mechanics and Analysis》2011,200(3):881-943
This article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices. Under
polynomial growth assumptions, we prove that the energy functionals Fe{F_\varepsilon} stored in the deformation of an e{{\varepsilon}}-scaling of a stochastic lattice Γ-converge to a continuous energy functional when e{{\varepsilon}} goes to zero. In particular, the limiting energy functional is of integral type, and deterministic if the lattice is ergodic.
We also generalize, to systems and nonlinear settings, well-known results on stochastic homogenization of discrete elliptic
equations. As an application of the main result, we prove the convergence of a discrete model for rubber towards the nonlinear
theory of continuum mechanics. We finally address some mechanical properties of the limiting models, such as frame-invariance,
isotropy and natural states. 相似文献
6.
Isaac V. Chenchiah Kaushik Bhattacharya 《Archive for Rational Mechanics and Analysis》2008,187(3):409-479
The elastic energy of a multiphase solid is a function of its microstructure. Determining the infimum of the energy of such
a solid and characterizing the associated optimal microstructures is an important problem that arises in the modeling of the
shape memory effect, microstructure evolution, and optimal design. Mathematically, the problem is to determine the relaxation
under fixed phase fraction of a multiwell energy. This paper addresses two such problems in the geometrically linear setting.
First, in two dimensions, we compute the relaxation under fixed phase fraction for a two-well elastic energy with arbitrary
elastic moduli and transformation strains, and provide a characterization of the optimal microstructures and the associated
strain. Second, in three dimensions, we compute the relaxation under fixed phase fraction for a two-well elastic energy when
either (1) both elastic moduli are isotropic, or (2) the elastic moduli are well ordered and the smaller elastic modulus is
isotropic. In both cases we impose no restrictions on the transformation strains. We provide a characterization of the optimal
microstructures and the associated strain. We also compute a lower bound that is optimal except possibly in one regime when
either (1) both elastic moduli are cubic, or (2) the elastic moduli are well ordered and the smaller elastic modulus is cubic;
for moduli with arbitrary symmetry we obtain a lower bound that is sometimes optimal. In all these cases we impose no restrictions
on the transformation strains and whenever the bound is optimal we provide a characterization of the optimal microstructures
and the associated strain. In both two and three dimensions the quasiconvex envelope of the energy can be obtained by minimizing
over the phase fraction. We also characterize optimal microstructures under applied stress. 相似文献
7.
Journal of Elasticity - 相似文献
8.
9.
We study a notion of capacity related to elasticity which proves convenient for analyzing the concentrations of strain energy
caused by rigid displacements of some infinitesimal parts of an elastic body in two or three dimensions. By way of application,
we investigate the behavior of solutions to initial boundary value problems describing vibrations of periodic elastic composites
with rapidly varying elastic properties. More specifically, we analyze a two-phase medium whereby a set of heavy stiff tiny
particles is embedded in a softer matrix. This task is set in the context of linearized elasticity. 相似文献
10.
In this paper we derive and mathematically justify models of micropolar rods and plates from the equations of linearized micropolar elasticity. Derivation is based on the asymptotic techniques with respect to the small parameter being the thickness of the elastic body we consider. Justification of the models is obtained through the convergence result for the displacement and microrotation fields when the thickness tends to zero. The limiting microrotation is then related to the macrorotation of the cross–section (transversal segment) and the model is rewritten in terms of macroscopic unknowns. The obtained models are recognized as being either the Reissner–Mindlin plate or the Timoshenko beam type. 相似文献
11.
12.
13.
David S. McCormick James C. Robinson Jose L. Rodrigo 《Archive for Rational Mechanics and Analysis》2014,214(2):503-523
We prove the existence, uniqueness and regularity of weak solutions of a coupled parabolic-elliptic model in 2D, and the existence of weak solutions in 3D; we consider the standard equations of magnetohydrodynamics with the advective terms removed from the velocity equation. Despite the apparent simplicity of the model, the proof in 2D requires results that are at the limit of what is available, including elliptic regularity in L 1 and a strengthened form of the Ladyzhenskaya inequality $$\| f \|_{L^{4}} \leqq c \| f \|_{L^{2,\infty}}^{1/2} \|\nabla f\|_{L^{2}}^{1/2},$$ which we derive using the theory of interpolation. The model potentially has applications to the method of magnetic relaxation introduced by Moffatt (J Fluid Mech 159:359–378, 1985) to construct stationary Euler flows with non-trivial topology. 相似文献
14.
An important necessary condition for an exact relation for effective moduli of polycrystals to hold is stability of that relation
under lamination. This requirement is so restrictive that it is possible (if not always feasible) to find all such relations
explicitly. In order to do this one needs to combine the results developed in Part I of this paper and the representation
theory of the rotation groups SO(2) and SO(3). More precisely, one needs to know all rotationally invariant subspaces of the space of material moduli. This paper presents
an algorithm for finding all such subspaces. We illustrate the workings of the algorithm on the examples of 3‐dimensional
elasticity, where we get all the exact relations, and the examples of 2‐dimensional and 3‐dimensional piezoelectricity, where
we get some (possibly all) of them.
(Accepted September 24, 1997) 相似文献
15.
20世纪80年代中期准晶的发现,曾引起晶体学、物理学、化学和材料科学界的轰动.报导其发现的论文,后来被美国<物理评论快报>(Phys.Rev .Lett.)列为该杂志发表的论文中被引用最多的前10名中的第8名(到2002年,已被引用3000多次).从那时起,准晶因具有重量轻、强度高和性能稳定的特性,所以可将其制成功能... 相似文献
16.
Hans-Otto Walther 《Journal of Dynamics and Differential Equations》2010,22(3):439-462
We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form x′(t) = g(? x t , x t ). The state space is a closed subset in a manifold of C 2-functions. Applications include equations with state-dependent delay, as for example x′(t) = a x′(t + d(x(t))) + f (x(t + r(x(t)))) with \({a\in\mathbb{R}, d:\mathbb{R}\to(-h,0), f:\mathbb{R}\to\mathbb{R}, r:\mathbb{R}\to[-h,0]}\). 相似文献
17.
We propose and evaluate a linearized method to measure dynamic friction between micromachined surfaces. This linearized method
reduces the number of data points needed to obtain dynamic friction data, minimizing the effect of wear on sliding surfaces
during the measurement. We find that the coefficient of dynamic friction is lower than the coefficient of static friction,
while the adhesive pressure is indistinguishable for the two measurements. Furthermore, after an initial detailed measurement
is made on a device type, the number of trial runs required to take the data on subsequent devices can be reduced from 200
to approximately 20. 相似文献
18.
19.
Convergence of Peridynamics to Classical Elasticity Theory 总被引:1,自引:0,他引:1
The peridynamic model of solid mechanics is a nonlocal theory containing a length scale. It is based on direct interactions
between points in a continuum separated from each other by a finite distance. The maximum interaction distance provides a
length scale for the material model. This paper addresses the question of whether the peridynamic model for an elastic material
reproduces the classical local model as this length scale goes to zero. We show that if the motion, constitutive model, and
any nonhomogeneities are sufficiently smooth, then the peridynamic stress tensor converges in this limit to a Piola-Kirchhoff
stress tensor that is a function only of the local deformation gradient tensor, as in the classical theory. This limiting
Piola-Kirchhoff stress tensor field is differentiable, and its divergence represents the force density due to internal forces.
The limiting, or collapsed, stress-strain model satisfies the conditions in the classical theory for angular momentum balance, isotropy, objectivity,
and hyperelasticity, provided the original peridynamic constitutive model satisfies the appropriate conditions.
相似文献
20.
The stability of travelling waves which occur when a nematic liquid crystal is subjected to crossed electric and magnetic
fields has been studied previously by (Stewart and Faulkner, Cont. Mech. Thermodyn. (1997)) where conditions on a control
parameter q for stability to occur have been given. This article is concerned with the behaviour of the stable perturbations as time
increases. For each of the three travelling wave solutions we calculate both the essential spectrum and the eigenvalues and
use these to determine the long-term monotonic or oscillatory behaviour of the perturbations. The results are also relevant
to liquid crystals subjected to a single field.
Received January 14, 1998 相似文献