A Note on Uniqueness in Linear Elastostatics

We obtain a uniqueness theorem for linear elastostatics, for a homogeneous isotropic body, under the condition of strong ellipticity, in the case of “mixed–mixed” boundary conditions. The theorem applies to bodies of restricted local concavity and convexity. Some example domains are illustrated.      

Complementary energy release rates and dual conservation integrals in micropolar elasticity

The complementary energy momentum tensor, expressed in terms of the spatial gradients of stress and couple-stress, is used to construct the and conservation integrals of infinitesimal micropolar elasticity. The derived integrals are related to the release rates of the complementary potential energy associated with a defect translation or rotation. A nonconserved integral is also derived and related to the energy release rate that is associated with a self-similar cavity expansion. The results are compared to those obtained on the basis of the classical energy momentum tensor, expressed in terms of the spatial gradients of displacement and rotation, and the release rates of the potential energy. It is shown that the evaluation of the complementary conservation integrals is of similar complexity to that of the classical conservation integrals, so that either can be effectively used in the energetic analysis of the mechanics of defects. The two-dimensional versions of the dual conservation integrals are then derived and applied to an out-of-plane shearing of a long cracked slab.     

Some Remarks Concerning a Boundary-Value Problem in Nonlinear Elastostatics

The paper is concerned with solvability of a mixed boundary value problem of place and traction that describes the straightening of annular cylindrical sectors composed of unconstrained isotropic hyperelastic solids. The existence and uniqueness of solutions is discussed and several examples are exhibited. This revised version was published online in July 2006 with corrections to the Cover Date.     

Uniqueness in boundary/initial value problems for a thermoviscoelastic solid of integral type

A general boundary/initial value problem for a linear thermoviscoelastic solid of integral typs is posed, and a uniqueness theorem is proved which holds for both bounded and unbounded domains. In the latter case we do not impose conditions on the behaviour of the unknown fields at infinity. Mild definiteness hypotheses on the initial values of the relaxation functions are assumed throughout.

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Sommario Dopo aver formulato un problema ai limiti generale per solidi lineari termoviscoelastici di tipo integrale, si stabilisce un teorema di unicità della soluzione valido per domini limitati e per domini illimitati. In questo secondo caso non viene imposta alcuna condizione sul comportamento all'infinito delle variabili di campo; si suppongono verificate solamente certe deboli condizioni di segno sui valori iniziali di alcune funzioni di rilassamento.

     

Global Continuation in Displacement Problems of Nonlinear Elastostatics via the Leray-Schauder Degree

We obtain global solution continua of forced displacement problems of nonlinear elastostatics via a Leray-Schauder scheme. We adopt strong ellipticity as a bsic constitutive hypothesis. The usual Leray-Schauder approach, based in part upon the reduction of the boundary value problem to an operator equation for the zeros of a compact vector field, apparently fails. On the one hand, strong ellipticity alone is not enough to insure “invertibility/rdquo; of the principal, quasilinear part of the differential operator. More importantly, the physical requirement of local injectivity of the deformation and the associated growth of the stored-energy function dictate that ellipticity is not uniform. We demonstrate how to overcome these difficulties. Finally, with additional, physically reasonable restrictions on the stored-energy function, we obtain unbounded branches of classical, globally injective solutions. Accepted: October 22, 1999     

Unbounded Branches of Classical Injective Solutions to the Forced Displacement Problem in Nonlinear Elastostatics

We consider classical solutions of the equilibrium equations of nonlinear elastostatics under prescribed boundary displacements and body forces, both of which depend on a loading parameter. We employ recent results on global continuation [HS] that establish general existence of global continua of solutions in nonlinear elastostatics. Under reasonable hypotheses on the stored energy function we prove the existence of an unbounded branch of globally injective solutions to the boundary value problem. This revised version was published online in August 2006 with corrections to the Cover Date.     

Uniqueness of Equilibrium Solutions in Second-Order Gradient Nonlinear Elasticity

We consider a three-dimensional elastic body whose material response function depends not only on the gradient of the deformation, but also on its second gradient. Using the elastic energy-momentum tensor as derived by Eshelby [2] we generalize a well-known uniqueness result of Knops and Stuart [8] for a Dirichlet boundary value problem associated with this response function.     

非线性非自治系统拓扑结构分析的能量法

利用能量原理,提出一种以能量为状态参数的分析非线性非自治系统拓扑结构的定性方法。分析了能量参数确定系统状态的唯一性,揭示了能量方法的正确性和有效性以及反映系统运动状态信息的丰富性。以非自治Duffing方程为对象,采用能量状态参数,应用提出的能量方法在相空间和运动空间展开了一般的定性分析,给出了能量相图和能量时间历程、能量Poincaré截面图、能量参数分岔图,分析了其中呈现出的运动规律和捕捉到的有价值信息,并与现有的方法进行了比较。     

Energy Pumping with Various Nonlinear Structures: Numerical Evidences

Numerical investigations are carried out on a linear structure, weakly coupled to a small nonlinear attachment. The essential nonlinearity of the attachment enables it to resonate with any of the linearized modes of the structure leading to energy pumping, i.e. passive, one-way, irreversible transfer of energy from the structure to the attachment. Different nonlinear structures (piecewise linear system, chaotic system) and efficiency of energy pumping are studied in each case in order to be able to apply it to civil engineering. As a specific application, attenuation of vibrations of a building is studied with two building models. In particular, the case of stochastic excitations is analyzed to examine if it is possible to process energy pumping when a seism occurs and an indicator of efficiency has been introduced.     

Designing a Linear Structure with a Local Nonlinear Attachment for Enhanced Energy Pumping

We present a design procedure for enhancing nonlinear energy pumping from a mode of a linear-damped substructure to a weakly coupled, essentially nonlinear oscillator. By this we denote the one way, irreversible passive transfer of vibrational energy from the mode to the nonlinear attachment. The design relies in the asymptotic expansion for large energies of a nonlinear normal mode of the underlying conservative system that provides an analytic estimate of the level of the amplitude reached by the nonlinear attachment in the energy pumping regime. The analytical findings are validated by direct numerical simulations.     

Energy Method for Approximate Periodic Solution of Strongly Nonlinear Nonautonomous Systems

In this paper, based upon the concept of the conservation of average energy, a theorem for the periodic solution of strongly nonlinear nonautonomous systems is proved. By using this theorem not only the necessary and sufficient conditions for the existence and stability of the periodic solutions can be obtained but, at the same time, the approximate expressions for those periodic solutions can also be obtained.     

On the Uniqueness of Energy Minimizers in Finite Elasticity

The uniqueness of absolute minimizers of the energy of a compressible, hyperelastic body subject to a variety of dead-load boundary conditions in two and three dimensions is herein considered. Hypotheses under which a given solution of the corresponding equilibrium equations is the unique absolute minimizer of the energy are obtained. The hypotheses involve uniform polyconvexity and pointwise bounds on derivatives of the stored-energy density when evaluated on the given equilibrium solution. In particular, an elementary proof of the uniqueness result of Fritz John (Commun. Pure Appl. Math. 25:617–634, 1972) is obtained for uniformly polyconvex stored-energy densities.     

Nonlinear Energy Transfer over the Wave Spectrum in Water Covered by Solid Ice

On the basis of the hydrodynamic equations for nonlinear elastic-gravity waves beneath a solid ice cover and their Hamiltonian representation, a three-wave kinetic equation for the time evolution of the wave spectrum is formulated. The properties of the kernel of the kinetic integral describing the nonlinear interactions between wave triplets are investigated. An algorithm for numerically calculating the kinetic integral is developed. The rate of nonlinear energy transfer over the wave spectrum is estimated quantitatively and its most important characteristics are found.     

Bui's path-independent integral in finite elasticity

A path-independent integral has been stated by Bui in the presence of a straight crack in a two-dimensional deformation field. Such an integral isdual to the Rice integral in the sense that it is based on the complementary stress energy density. Here we establish a boundary-independent integral in finite elasticity from which Bui's result follows as a particular case.

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Sommario Un integrale indipendente dal cammino intorno al vertice di una frattura in un campo di deformazione bi-dimensionale è stato stabilito da Bui. Tale integrale èduale all'integrale di Rice, nel senso che si basa sulla densità di energia complementare o degli sforzi. Qui si propone un integrale invariante in un continuo tridimensionale soggetto a deformazioni finite. Si mostra che il risultato di Bui segue come caseo particolare.

     

Energy Method for Computing Periodic Solutions of Strongly Nonlinear Autonomous Systems with Multi-Degree-of-Freedom

In this paper with the use of conservation of average energy, a newmethod for computing the periodic solutions of strongly nonlinearautonomous systems with multi-degree-of-freedom is suggested. Thismethod cannot only decide the existence, but also give the approximateexpressions of the periodic solutions.     

Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators

Redistribution of energy in a highly asymmetric system consisting ofcoupled linear and highly nonlinear damped oscillators isinvestigated. Special attention is paid to the excitation of a nonlinearnormal mode while the energy is initially stored in other modes of thesystem. The transition proceeds via the mechanism of subharmonicresonance which is possible because of the strong nonlinearity of thesystem. The conditions of the energy transition to NNM being effectiveare revealed and guidelines to design such a systems are formulatedin detail.     

A Remark on the Invariance of Solenoidal Vectors in Elastostatics

The linear elastostatic displacement boundary-value problem is considered for a bounded simply connected region Ω in R ⁿ in the case of a homogeneous isotropic medium. For n = 2 it is shown that if all solenoidal forcing terms result in solenoidal displacements, then Ω is a disk. It is likely that the result is true for n ≥ 3 but that problem is not resolved. This revised version was published online in August 2006 with corrections to the Cover Date.     

Energy Approach to the Formulation of Boundary Problems of Nonlinear Thermomechanics for Elastic Systems

An energy approach is proposed to derive the physical constitutive equations of nonlinear thermomechanics for inertial elastic systems. A potential of local inertial thermodynamic state and a potential of thermoelastic energy dissipation are introduced. The variational formulation of nonlinear boundary problems of thermoelasticity is implemented on the basis of the Hamiltonian energy functional. Sufficient conditions for the convexity of the functional are formulated __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 9, pp. 52–59, September 2005.     

New Principles of Work and Energy as Well as Power and Energy Rate for Continuum Field Theories

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