Variational Principles In Elasticity Research Articles

Hill–Mandel condition and bounds on lower symmetry elastic crystals

Despite advances in contemporary micromechanics, there is a void in the literature on a versatile method for estimating the effective properties of polycrystals comprising of highly anisotropic single crystals belonging to lower symmetry class. Basing on variational principles in elasticity and the Hill–Mandel homogenization condition, we propose a versatile methodology to fill this void. It is demonstrated that the bounds obtained using the Hill–Mandel condition are tighter than the Voigt and Reuss [1,2] bounds, the Hashin–Shtrikman [3] bounds as well as a recently proposed self-consistent estimate by Kube and Arguelles [4] even for polycrystals with highly anisotropic single crystals.

Study on Parametric Variational Principles in Elasticity

A parametric variational principle is deduced according to the equivalent integral form of all the controlling equations and boundary conditions in elasticity, and by adjusting the parameters, all kinds of variational principles put forward past and some new variational principles can be gained, which means that the method above is more clear in concept, and more concise.

Study of the equivalent theorem of generalized variational principles in elasticity

The relations of all generalized variational principles in elasticity are studied by employing the invariance theorem of field theory. The infinitesimal scale transformation in field theory was employed to investigate the equivalent theorem. Among the results found particularly interesting are those related to that all generalized variational principles in elasticity are equal to each other. Also studied result is that only two variables are independent in the functional and the stress-strain relation is the variational constraint condition for all generalized variational principles in elasticity. This work has proven again the conclusion of Prof. Chien Wei-zang.

Estimation of ground movements due to tunnelling based on the variational principles of elasticity; application to the determination of surface settlement (In French) : Dormieux, L; De Buhan, P; Leca, E Rev Fr GeotechN59, 1992, P15–32

Estimation of ground movements due to tunnelling based on the variational principles of elasticity; application to the determination of surface settlement (In French) : Dormieux, L; De Buhan, P; Leca, E Rev Fr GeotechN59, 1992, P15–32

Bounding solutions for the pressuremeter modulus using variational principles in elasticity : Dormieux, L Int J Num Anal Meth GeomechV16, N5, May 1992, P311–321

Bounding solutions for the pressuremeter modulus using variational principles in elasticity : Dormieux, L Int J Num Anal Meth GeomechV16, N5, May 1992, P311–321

The Optimal Design for an Internal Stiffener Plate in a Penstock Bifurcation

This paper presents the optimum design of an internal stiffener plate in a penstock bifurcation. It is assumed that the monetary cost of the structure is dependent on the amount of material used. Hence, the weight (volume) of the plate is minimized subject to an allowable stress constraint. This is accomplished by determining the shape of the moving, free boundary of the plate, while the loaded boundary remains fixed. The mathematics of the problem is based on the variational principles of elasticity, the material derivative concept of continuum mechanics, and a gradient projection method of analysis. A sensitivity analysis is performed to examine the influence of the moving boundary of the plate on the stress constraint. At the optimum design the moving boundary reaches the predefined allowable stress. The finite element analysis package ANSYS is used for the stress calculations.

Bounding solutions for the pressuremeter modulus using variational principles in elasticity

AbstractIt is classically assumed that the pressuremeter modulus (determined from the initial slope of the pressuremeter expansion curve) can be used as an estimate for the soil's shear modulus, provided disturbance effects can be neglected. This approximation is only rigorous in the plane‐strain case and requires that the height of the cell be large in comparison with its diameter. The present paper allows to quantify the error that results from this approximation as a function of the height over diameter ratio. The analysis is based on an application of the variational principles of elasticity. It is shown that the static and the kinematic approaches lead, respectively, to a lower‐ and an upper‐bound estimate of the pressuremeter modulus expressed in terms of the soil's elastic coefficients.The calculations are completely carried out in the case of an infinite cavity subjected to a pressure on a finite length. It is shown that the plane‐strain approximation overestimates the shear modulus. Based on the kinematic approach, a method is developed to derive the shear modulus from the pressuremeter modulus which accounts for the finite slenderness of the cell. This method can be extended to more complex geometries and material properties, including soil heterogeneity and anisotropy.

Parametrized multifield variational principles in elasticity: II. Hybrid functionals and the free formulation

AbstractA one‐parameter family of d‐generalized hybrid/mixed variational principles for linear elasticity is constructed following a domain subdivision. The family includes the d‐generalized Hellinger‐Reissner and potential energy principles as special cases. The parametrized principle is discretized by independently varied internal displacements, stresses and boundary displacements. The resulting finite element equations are studied following a physically motivated decomposition of the stress and internal displacement fields. The free formulation of Bergan and Nygård is shown to be a special case of this element type, and is obtained by assuming a constant internal stress field. The parameter appears as a scale factor of the higher‐order stiffness.

Parametrized multifield variational principles in elasticity: I. Mixed functionals

AbstractA one‐parameter family of mixed variational principles for linear elasticity is constructed. This family includes the generalized Hellinger‐Reissner and total potential energy principles as special cases. The presence of the free parameter offers an opportunity for the systematic derivation of energy‐balanced finite elements that combine displacement and stress assumptions. It is shown that Fraeijs de Veubeke's stress‐assumption limitation principle takes a particulary elegant expression in terms of the parametrized discrete form. Other possible parametrizations are briefly discussed.

General form of the generalized variational principle in elasticity

AbstractA new method for establishing generalized variational principles, called the undetermined coefficient method, is presented. A more general form of functional for the generalized variational principles in elasticity and a concept of energy functional space are given. Based on the present functional, a general formulation for a hybrid/mixed element is established.

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.

Generalized variational principles with several arbitrary parameters and the variable substitution and multiplier method

The functional transformations of variational principles in elasticity are classified as three patterns: I relaxation pattern, II augmented pattern and III equivalent pattern.

Several patterns of functional transformation and generalized variational principles with several arbitrary parameters

In this paper, the functional transformations of variational principles in elasticity are classified as three patterns: pattern I (relaxation pattern) is a generalized equivalent pattern in which constraint conditions are transformed into natural conditions; pattern II (augmented pattern) is a generalized equivalent pattern in which augmented conditions are transformed into natural conditions; pattern III (equivalent pattern) is a pattern in which a nonconditional functional is transformed into an equivalent functional with several arbitrary parameters. Pattern I is the well-known pattern of Lagrange multipliers method: patterns II and III are new patterns proposed in this paper. On the basis of pattern III, generalized variational principles with several arbitrary parameters are formulated, and the general and simple forms of the functional are defined. Many existing functionals of variational principles in elasticity are special cases of this functional.

Classification of variational principles in elasticity

In this paper, variational principels in elasticity are classified according to the differences in the constraints used in these principles. It is shown in a previous paper [4] that the stress-strain relations are the constraint conditions in all these variational principles, and cannot be removed by the method of linear Lagrange multiplier. The other possible constraints are four of them: (1) equations of equilibrium, (2) strain-displacement relations, (3) boundary conditions of given external forces and (4) boundary conditions of given boundary displacements. In variational principles of elasticity, some of them have only one kind of such constraints, some have two kinds or three kinds of constraints and at the most four kinds of constraints. Thus, we have altogether 15 kinds of possible variational principles. However, for every possible variational principle, either the strain energy density or the complementary energy density may be used. Hence, there are altogether 30 classes of functional of variational principles in elasticity. In this paper, all these functionals are tabulated in detail.

Variational Principles Derived from the Matrix Structural Analysis Equations

ABSTRACT A method for generating the conventional variational principles of elasticity is presented. The technique consists in writing scalar potentials for different forms of the analysis equations of a trussed structure. A straightforward transformation translates the potentials and their properties into analogue functional in the continuum domain. Various formulations of the Hu-Washizu, Hellinger-Reissner, Total Potential, and Complementary Energy variational principles are thus obtained.

On patched variational principles in elasticity

In this paper we show how to use a combination of variational principles, each for a given subregion of the underlying region of space, so as to best utilize the chief benefits of the individual principles.

Variational principles of elasticity in terms of the polarization tensor

New variational principles for anisotropic and nonhomogeneous elasticity theory, in terms of the elastic polarization tensor, are derived by a straightforward method. Previous variational principles given by Hashin and Shtrikman [3] are rederived by a similar method and are generalized to take account of body forces and mixed boundary conditions.

Note on Some Variational Principles in Elasticity

Note on Some Variational Principles in Elasticity

A note on variational principles in elasticity

Zusammenfassung Bekannte Variationsprinzipe der Elastizitatstheorie fur (1) Spannungen und Verschiebungen, (2) Spannungen, Formanderungen und Verschiebungen werden verallgemeinert in solcher Weise. dass die Momentengleichgewichtsbedingungen τ xy = τ yx , etc., als Euler'sche Gleichungen der Variationsprinzipe erscheinen, und nicht als Definitionsgleichungen. Anwendungen dieser Ergebnisse beruhen auf der Idee dass es manchmal von Nutzen sein kann, bei der Aufstellung von Naherungslosungen von Randwertproblemen. Moment und Kraftgleichgewichtsbedingungen beide nur naherungsweise zu befriedigen.