Variational principles in elasticity with nonlinear stress-strain relation

Since 1979, a series of papers have been published concerning the variational principles and generalized variational principles in elasticity such as [1] (1979), [6] (1980), [2,3] (1983) and [4,5] (1984). All these papers deal with the elastic body with linear stress-strain relations. In 1985, a book was published on generalized variational principles dealing with some nonlinear elastic body, but never going into detailed discussion. This paper discusses particularly variational principles and generalized variational principles for elastic body with nonlinear stress-strain relations. In these discussions, we find many interesting problems worth while to pay some attention. At the same time, these discussions are also instructive for linear elastic problems. When the strain is small, the high order terms may be neglected, the results of this paper may be simplified to the well-known principles in ordinary elasticity problems.     

Variational principles and generalized variational principles for nonlinear elasticity with finite displacement

In a previous paper (1979)^[1], the minimum potential energy principle and stationary complementary energy principle for nonlinear elasticity with finite displacement, together with various complete and incomplete generalized principles were studied. However, the statements and proofs of these principles were not so clearly stated about their constraint conditions and their Euler equations. In somecases, the Euler equations have been mistaken as constraint conditions. For example, the stress displacement relation should be considered as Euler equation in complementary energy principle but have been mistaken as constraint conditions in variation. That is to say, in the above mentioned paper, the number of constraint conditions exceeds the necessary requirement. Furthermore, in all these variational principles, the stress-strain relation never participate in the variation process as constraints, i.e., they may act as a constraint in the sense that, after the set of Euler equations is solved, the stress-strain relation may be used to derive the stresses from known strains, or to derive the strains from known stresses. This point was not clearly mentioned in the previous paper (1979)^[1]. In this paper, the high order Lagrange multiplier method (1983)^[2] is used to construct the corresponding generalized variational principle in more general form. Throughout this paper, V/.V. Novozhilov's results (1958)^[3] for nonlinear elasticity are used.     

Variational principles of asymmetric elasticity theory of finite deformation

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The mechanically-based approach to 3D non-local linear elasticity theory: Long-range central interactions

This paper presents the generalization to a three-dimensional (3D) case of a mechanically-based approach to non-local elasticity theory, recently proposed by the authors in a one-dimensional (1D) case. The proposed model assumes that the equilibrium of a volume element is attained by contact forces between adjacent elements and by long-range forces exerted by non-adjacent elements. Specifically, the long-range forces are modelled as central body forces depending on the relative displacement between the centroids of the volume elements, measured along the line connecting the centroids. Further, the long-range forces are assumed to be proportional to a proper, material-dependent, distance-decaying function and to the products of the interacting volumes. Consistently with the modelling of the long-range forces as central body forces, the static boundary conditions enforced on the free surface of the solid involve only local stress due to contact forces.The proposed 3D formulation is developed both in a mechanical and in a variational context. For this the elastic energy functionals of the solid with long-range interactions are introduced, based on the principle of virtual work to set the proper correspondence between the mechanical and the kinematic variables of the model. Numerical applications are reported for 2D solids under plane stress conditions.     

Crack-tip problem in non-local elasticity

Solutions are presented for the one- and two-dimensional Griffith crack problems in non-local elasticity. The displacements and stresses are determined in an elastic plate, weakened by a sharpedged line crack. The plate is loaded by a uniform tension perpendicular to the line of the crack at infinity. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis.     

The Peierls stress in non-local elasticity

The effect of non-locality on the Peierls stress of a dislocation, predicted within the framework of the Peierls-Nabarro model, is investigated. Both the integral formulation of non-local elasticity and the gradient elasticity model are considered. A modification of the non-local kernel of the integral formulation is proposed and its effect on the dislocation core shape and size, and on the Peierls stress are discussed. The new kernel is longer ranged and physically meaningful, improving therefore upon the existing Gaussian-like non-locality kernels. As in the original Peierls-Nabarro model, lattice trapping cannot be captured in the purely continuum non-local formulation and therefore, a semi-discrete framework is used. The constitutive law of the elastic continuum and that of the glide plane are considered both local and non-local in separate models. The major effect is obtained upon rendering non-local the constitutive law of the continuum, while non-locality in the rebound force law of the glide plane has a marginal effect. The Peierls stress is seen to increase with increasing the intrinsic length scale of the non-local formulation, while the core size decreases accordingly. The solution becomes unstable at intrinsic length scales larger than a critical value. Modifications of the rebound force law entail significant changes in the core configuration and critical stress. The discussion provides insight into the issue of internal length scale selection in non-local elasticity models.     

Variational principles in elastostatics

Summary In a previous paper three new variational principles of elastostatics in the theory of small displacements were added to the four principles previously known: that of the minimum potential energy, of Menabrea-Finzi, of Reissner and of Hu-Washizu. This paper presents another new principle plus a variant of Reissner's principle and of its dual and reviews all the principles of elastostatics known to date.

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Sommario In una precedente nota sono stati presentati tre nuovi principi variazionali dell'elastostatica nella teoria dei piccoli spostamenti che vanno ad aggiungersi ai quattro principi precedentemente noti: quello della minima energia potenziale, di Menabrea-Finzi, di Reissner e di Hu-Washizu. In questa nota è presentato un nuovo principio, viene segnalata una variante al principio di Reissner e al suo duale e infine viene data una visione globale su tutti i principi attualmente noti dell'elastostatica.

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This work has been done under the auspices of the Research Group no. 34 of C.N.R.     

Wave propagation in anisotropic media with non-local elasticity

In an effort to understand and quantify the effect of non-local elasticity on the wave propagation response of laminated composite layered media, a frequency-wavenumber domain based finite element method is employed. The developed elements are based on the exact solution in the transformed domain and thus exactly represent the dynamics of a layer. This feature enables to model a layer of any thickness by a single element and drastically reduces the cost of computation. The effect of non-locality on the dispersion relation and in turn on the wave response is compared with local (classical) elasticity solutions. A procedure and sample example is outlined to estimate the magnitude of the non-locality parameter by comparing the dispersion relation with lattice dynamics. The effect of non-locality, in terms of the mode-shift and appearance of dispersion on the modes of Lamb waves is further demonstrated.     

THE MIXED BOUNDARY-VALUE PROBLEM FOR NON-LOCAL ASYMMETRIC ELASTICITY

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On the functional form of non-local elasticity kernels

The functional form of non-local elasticity kernels is studied within the context of the integral formalism. The study is limited to linear isotropic elasticity. The kernels are derived analytically based on the discrete structure of the material at the atomic scale. Atomistic simulations are used to validate the results. Materials in which the interatomic interactions are represented by pair, as well as embedded atom-type potentials are considered. The derived kernels have a range which extends up to the cut-off radius of the interatomic potential, are positive at the origin, and become negative approximately one atomic distance away, thus departing from the commonly assumed Gaussian functional form. The functional form of the potential and the radial distribution function of interacting neighbors about a representative atom fully define their shape. This new continuum model involves two material length scales that are both derived from atomistics for a Morse solid and for Al. Two applications are considered in closure. It is shown that in strained superlattices, the non-local model predicts maximum stresses that are much larger than those obtained within the local theory. This observation has implications for defect nucleation in these structures. Furthermore, the new non-local model improves upon the Gaussian one by predicting a more realistic wave dispersion relationship, with essentially zero group velocity at the boundary of the Brillouin zone.     

The analysis of crack problems with non-local elasticity

IntroductionTheclassicalconhnuummechanicshasbeenusedtosolvemanyproblemsinmacrofracturemechanics,butencountersdifficulheswhentheeffectofITilcrocharacteristicdimensionshouldbetakenintoaccount.Thestressfieldverynearthecracktipisstillnotclear.Somephenomenaofshortcrackscannotbeexplained["']andsomemechanismoffracturehasnotbeensolvedyet.Thenon-localelashcitytheoryseemsattractivetotheseproblems.Thetheoryofnon-localelasticity,establishedanddevelopedbyEringenetal[3),connectstheclassicalcontinuummechan…     

Thermal buckling of nanorod based on non-local elasticity theory

The buckling of nanostructures including as a nanobeam, nanorod, and nanotube in a temperature field is investigated based on the non-local elasticity field theory with non-linear strain gradients first proposed by Eringen. New higher-order governing differential equations both in transverse and axial direction for buckling of such nanostructures are derived based on the exact variational principle approach with corresponding higher-order non-local boundary conditions. Based on these new governing equations and boundary conditions, new analytical solutions for some practical examples on buckling of nanostructures are presented and analyzed in detail. Subsequently, the effects of non-local nanoscale and temperature change on critical buckling load are analyzed and discussed. It is observed that those factors have great influence on the critical buckling load of the nanostructures. In particular, the non-local stress very much affects the stiffness of nanostructures and the critical buckling load is significantly increased in the presence of non-local stress. The paper concludes that at low and room temperature the critical buckling load of nanostructures increases with increasing temperature change, while at high temperature the critical buckling load decreases with increasing temperature change. A critical temperature change which causes buckling without external load is also derived and discussed.     

Variational principles in micromorphic thermoelasticity

The paper is concerned with the linear dynamic theory of micromorphic thermoelasticity We establish variational principles which fully characterize the solution of the boundary-initial value problem of thermoelastodynamics.     

Variational principles in potential theory

Summary Although the theoretical and practical importance of variational techniques in potential theory has long been secure, fresh instances of their utility are not without interest. Two cases in point are detailed, one relating to the capacitance of a condenser formed by a centrally placed strip within a circular shell and the other relating to the torsional rigidity of a circular cylindrical bar with a radial slit. The high degree of geometrical symmetry reflected in these configurations affords a corresponding diversity in approaches to the concomitant boundary value problems; and it is noteworthy that the exercise of different procedural options furnishes variational characterizations with a complementary nature. Thus, in the foregoing examples, a pair of newly devised variational principles are shown to provide opposite bounds for the capacitance and torsional rigidity, respectively, from those associated with alternative (though more readily established) principles. It appears, furthermore, that the new variational formulations are especially suited to the circumstances wherein the larger values of the ratio of the strip, or slit, width to the circular diameter obtain, whereas the others enjoy a corresponding fitness if the ratio is small compared with unity.This work was supported in part by the Office of Naval Research Contract Nonr-225(74). Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Variational principles for Cosserat body

Two variational principles are derived for the mixed boundary value problem of Cosserat solid. These principles are a generalization of the stationary principle of potential energy and the stationary principle of complementary energy from non-linear theory of elasticity.     

Multi-scale non-local approach for geomaterials

A thermodynamically consistent multi-scale, rate dependent, non local approach is developed in this work for geo-materials in conjunction with the anisotropic modified Cam Clay model. The gradient for the micro-structure is incorporated through the micro level gradient of the back-stress and volumetric strain while the gradient for macro-structure is incorporated through the macro level gradient of back-stress and volumetric plastic strain. Gradient results in the regularization of the local behavior. Visco-plasticity is also incorporated for an additional regularization of the local behavior. Therefore, the effects of two separate regularizations are naturally separated. The plastic spin is incorporated to separate the effect of micro-structural rotation from the gradient effect. The flow characteristics of the soil is also incorporated in order to separate the viscosity effect from the flow effect.Through this multi scale non local approach, a more realistic simulation of large strain problems such as shear band formation can be achieved.     

Variational approach to foam drainage equation

A variational principle is established using the semi-inverse method for the foam drainage equation.