From non-linear elasticity to linear elasticity with initial stress via ��-convergence

We consider a initially stressed hyperelastic body in equilibrium in its undeformed configuration under a system of dead loads. We give sufficient conditions on the stored energy which guarantee that when the loads undergo a small perturbation, the energy functional Γ converges, after some re-scaling, to the energy functional of linear elasticity with initial stress. We also show, under stronger conditions, that quasi-minimizers of the non-linear problem converge to a minimizer of the incremental problem.     

Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond

We study global minimizers of a continuum Landau–De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W ^1,2, to a global minimizer predicted by the Oseen–Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen–Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau–De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau–De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.     

Variational formulation of the equivalent eigenstrain method with an application to a problem with radial eigenstrains

In this paper a variational formulation of the equivalent eigenstrain method is established. A functional of the Hashin–Shtrikman type is proposed such that the solution of the equivalent eigenstrain equation is a unique minimizer of the functional. Moreover, it is also shown that the equivalent eigenstrain equation is the Euler–Lagrange equation of the potential energy of the inclusions. An approximate solution of the equivalent eigenstrain equation is then found as a minimizer of the functional on a finite dimensional span of basic eigenstrains. Special attention is paid to possible symmetries of the problem. The variational formulation is illustrated by determination of effective linear elastic properties. In particular, material with a simple cubic microstructure is considered in detail. A solution for the polynomial radial basic eigenstrains approximation is found. In particular, for the homogeneous eigenstrain approximation, the effective moduli are derived in an exact closed form.     

Existence and Stability of a Screw Dislocation under Anti-Plane Deformation

We formulate a variational model for a geometrically necessary screw dislocation in an anti-plane lattice model at zero temperature. Invariance of the energy functional under lattice symmetries renders the problem non-coercive. Nevertheless, by establishing coercivity with respect to the elastic strain and a concentration compactness principle, we prove the existence of a global energy minimizer and thus demonstrate that dislocations are globally stable equilibria within our model.     

A Variational Justification of Linear Elasticity with Residual Stress

We consider a residually-stressed, uniform hyperelastic body whose stored energy is quadratic with respect to the Green–St. Venant strain. We show that, in the limit of vanishing loads, suitable minimizing sequences converge to the unique minimizer of the energy functional of linear elasticity. We also deduce the standard stress-strain relations for linear elasticity with residual stress.     

Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics

The Bogoliubov–Dirac–Fock (BDF) model is the mean-field approximation of no-photon quantum electrodynamics. The present paper is devoted to the study of the minimization of the BDF energy functional under a charge constraint. An associated minimizer, if it exists, will usually represent the ground state of a system of N electrons interacting with the Dirac sea, in an external electrostatic field generated by one or several fixed nuclei. We prove that such a minimizer exists when a binding (HVZ-type) condition holds. We also derive, study and interpret the equation satisfied by such a minimizer. Finally, we provide two regimes in which the binding condition is fulfilled, obtaining the existence of a minimizer in these cases. The first is the weak coupling regime for which the coupling constant α is small whereas αZ and the particle number N are fixed. The second is the non-relativistic regime in which the speed of light tends to infinity (or equivalently α tends to zero) and Z, N are fixed. We also prove that the electronic solution converges in the non-relativistic limit towards a Hartree–Fock ground state.     

Standing Pulse Solutions to FitzHugh–Nagumo Equations

Reaction–diffusion systems serve as relevant models for studying complex patterns in several fields of nonlinear sciences. A localized pattern is a stable non-constant stationary solution usually located far away from neighborhoods of bifurcation induced by Turing’s instability. In the study of FitzHugh–Nagumo equations, we look for a standing pulse with a profile staying close to a trivial background state except in one localized spatial region where the change is substantial. This amounts to seeking a homoclinic orbit for a corresponding Hamiltonian system, and we utilize a variational formulation which involves a nonlocal term. Such a functional is referred to as Helmholtz free energy in modeling microphase separation in diblock copolymers, while its global minimizer does not exist in our setting of dealing with standing pulse. The homoclinic orbit obtained here is a local minimizer extracted from a suitable topological class of admissible functions. In contrast with the known results for positive standing pulses found in the literature, a new technique is attempted by seeking a standing pulse solution with a sign change.     

Some remarks on the stability of the homogeneous deformation for an elastic bar

The stability of the homogeneous deformation for an elastic bar subject to uniaxial tension is studied. It is shown that the critical extension at which the homogeneous deformation becomes unstable is a decreasing function of the aspect ratio. Furthermore, it is shown that for small aspect ratios, the homogeneous deformation need not be the global minimizer of the total energy, even though it may be a local minimizer.     

The Singular Set of Lipschitzian Minima of Multiple Integrals

The singular set of any Lipschitzian minimizer of a general quasiconvex functional is uniformly porous and hence its Hausdorff dimension is strictly smaller than the space dimension     

Steady States in Galactic Dynamics

By minimizing the energy for the Vlasov-Poisson system under a constraint, Guo and Rein have constructed a large class of isotropic, spherically symmetric steady states. They have shown that an isolated minimizer is automatically dynamically stable under general (i.e., not necessarily symmetric) perturbations. The main result of this work is to remove the assumption that the minimizer must be isolated, so minimizers are stable even if they are not isolated. It is also shown that the Lagrange multipliers associated with all minimizers have the same value. Finally, an example where two distinct minimizers exist is studied numerically.     

Finite Scale Microstructures in Nonlocal Elasticity

In this paper we develop a simple one-dimensional model accounting for the formation and growth of globally stable finite scale microstructures. We extend Ericksen's model [9] of an elastic “bar” with nonconvex energy by including both oscillation-inhibiting and oscillation-forcing terms in the energy functional. The surface energy is modeled by a conventional strain gradient term. The main new ingredient in the model is a nonlocal term which is quadratic in strains and has a negative definite kernel. This term can be interpreted as an energy associated with the long-range elastic interaction of the system with the constraining loading device. We propose a scaling of the problem allowing one to represent the global minimizer as a collection of localized interfaces with explicitly known long-range interaction. In this limit the augmented Ericksen's problem can be analyzed completely and the equilibrium spacing of the periodic microstructure can be expressed as a function of the prescribed average displacement. We then study the inertial dynamics of the system and demonstrate how the nucleation and growth of the microstructures result in the predicted stable pattern. Our results are particularly relevant for the modeling of twined martensite inside the austenitic matrix.     

On the rotating rod with variable cross section

Summary Stability of a heavy rotating rod with a variable cross section is studied by energy method. Bifurcation points for the system of equilibrium equations are analyzed. It is shown that for the case when the rotation speed exceeds the critical one, the trivial solution ceases to be the minimizer of the potential energy, so that rod loses stability, according to the energy criteria. Also, a new estimate of the maximal rod deflection in the post-critical state is obtained. Accepted for publication 11 November 1996     

Stability Estimates for a Twisted Rod Under Terminal Loads: A Three-dimensional Study

The stability of an inextensible unshearable elastic rod with quadratic strain energy density subject to end loads is considered. We study the second variation of the corresponding rod-energy, making a distinction between in-plane and out-of-plane perturbations and isotropic and anisotropic cross-sections, respectively. In all cases, we demonstrate that the naturally straight state is a local energy minimizer in parameter regimes specified by material constants. These stability results are also accompanied by instability results in parameter regimes defined in terms of material constants.     

拟弹性SMA对复合材料板抗低速冲击响应性能的影响

以形状记忆合金(SMA)纤维增强复合材料板为研究对象,根据SMA拟弹性曲线的特性,建立了一种SMA拟弹性应力-应变关系的分段线性化模型;在此基础上,采用分步能量平衡法,求解了SMA增强复合材料板受低速冲击时的横向位移和应力,分析了SMA的拟弹性特性对复合材料板低速冲击性能的影响.研究结果表明,SMA的能量吸收特性能有效地增强复合材料板抗低速冲击能力,板的最大位移和最大应力都明显减少.冲击速度为10m/s的情况下,板的最大挠度和应力降低了18%左右;冲击速度为25 m/s的情况下,板的最大挠度和应力降低了42%左右.     

some general theorems and generalized and piecewise generalized variational principles for linear elastodynamics

From the concept of four-dimensional space and under the four kinds of time limit conditions, some general theorems for elastodynamics are developed, such as the principle of possible work action, the virtual displacement principle, the virtual stress-momentum principle, the reciprocal theorems and the related theorems of time terminal conditions derived from it. The variational principles of potential energy action and complementary energy action, the H-W principles, the H-R principles and the constitutive variational principles for elastodynamics are obtained. Hamilton's principle, Toupin's work and the formulations of Ref. [5], [17]-[24] may be regarded as some special cases of the general principles given in the paper. By considering three cases: piecewise space-time domain, piecewise space domain, piecewise time domain, the piecewise variational principles including the potential, the complementary and the mixed energy action fashions are given. Finally, the general formulation of piecewise variati     

On the Convexity of Nonlinear Elastic Energies in the Right Cauchy-Green Tensor

We present a sufficient condition under which a weak solution of the Euler-Lagrange equations in nonlinear elasticity is already a global minimizer of the corresponding elastic energy functional. This criterion is applicable to energies \(W(F)=\widehat{W}(F^{T}F)=\widehat{W}(C)\) which are convex with respect to the right Cauchy-Green tensor \(C=F^{T}F\), where \(F\) denotes the gradient of deformation. Examples of such energies exhibiting a blow up for \(\det F\to0\) are given.     

A Double Bubble in a Ternary System with Inhibitory Long Range Interaction

The free energy of a ternary system with a self-organization property includes an interface energy and a longer ranging, inhibitory interaction energy. In a planar domain, if the two energies are properly balanced and two of the three constituents make up an equal but small fraction, the free energy admits a local minimizer that is shaped like a perturbed double bubble. Most difficulties in the proof of this result are related to the triple junction phenomenon that the three constituents of the ternary system meet at a point. Two techniques are developed to deal with the triple junction. First, one defines restricted classes of perturbed double bubbles. Each perturbed double bubble in a restricted class is obtained from a standard double bubble by a special perturbation. The two triple junction points of the standard double bubble can only move along the line connecting them, in opposite directions, and by the same distance. The second technique is the use of the so called internal variables. These variables derive from the more geometric quantities that describe perturbed double bubbles in restricted classes. The advantage of the internal variables is that they are only subject to linear constraints, and perturbed double bubbles in a restricted class represented by internal variables are elements of a Hilbert space. A local minimizer of the free energy in each restricted class is found as a fixed point of a nonlinear equation by a contraction mapping argument. The second variation at the fixed point within its restricted class is positive. This perturbed double bubble satisfies three of the four equations for critical points of the free energy. The unsolved equation is the 120 degree angle condition at triple junction points. Then perform another minimization among the local minimizers from all restricted classes. A minimum of minimizers emerges and solves all the equations for critical points.     

Method of integro-differential relations in linear elasticity

Boundary-value problems in linear elasticity can be solved by a method based on introducing integral relations between the components of the stress and strain tensors. The original problem is reduced to the minimization problem for a nonnegative functional of the unknown displacement and stress functions under some differential constraints. We state and justify a variational principle that implies the minimum principles for the potential and additional energy under certain boundary conditions and obtain two-sided energy estimates for the exact solutions. We use the proposed approach to develop a numerical analytic algorithm for determining piecewise polynomial approximations to the functions under study. For the problems on the extension of a free plate made of two different materials and bending of a clamped rectangular plate on an elastic support, we carry out numerical simulation and analyze the results obtained by the method of integro-differential relations.     

Variational formulations and a consistent finite-element procedure for a class of nonlocal elastic continua

The structural boundary-value problem in the context of nonlocal (integral) elasticity and quasi-static loads is considered in a geometrically linear range. The nonlocal elastic behaviour is described by the so-called Eringen model in which the nonlocality lies in the constitutive relation. The diffusion processes of the nonlocality are governed by an integral relation containing a recently proposed symmetric spatial weight function expressed in terms of an attenuation function. A firm variational basis to the nonlocal model is given by providing the complete set of variational formulations, composed by ten functionals with different combinations of the state variables. In particular the nonlocal counterpart of the classical principles of the total potential energy, complementary energy and mixed Hu–Washizu principle and Hellinger–Reissner functional are recovered. Some examples concerning a piecewise bar in tension are provided by using the Fredholm integral equation and the proposed nonlocal FEM.     

Existence and Non-linear Stability of Rotating Star Solutions of the Compressible Euler–Poisson Equations

We prove the existence of rotating star solutions which are steady-state solutions of the compressible isentropic Euler–Poisson (Euler–Poisson) equations in three spatial dimensions with prescribed angular momentum and total mass. This problem can be formulated as a variational problem of finding a minimizer of an energy functional in a broader class of functions having less symmetry than those functions considered in the classical Auchmuty–Beals paper. We prove the non-linear dynamical stability of these solutions with perturbations having the same total mass and symmetry as the rotating star solution. We also prove finite time stability of solutions where the perturbations are entropy-weak solutions of the Euler–Poisson equations. Finally, we give a uniform (in time) a priori estimate for entropy-weak solutions of the Euler–Poisson equations.