Unified Variational Principle Research Articles

Generalized standard media and variational principles in classical and finite strain elastoplasticity

This paper uses the concept of generalized standard media in order to develop a rational theory of small and finite strain elastoplasticity. Invariance requirements for yield function and flow rule are derived especially for the finite strain case. It is shown that isotropic and kinematic hardening can be incorporated into the general concept in a natural way. Finally the thermodynamic background is discussed and used to derive a novel unified variational principle.

A framework for nonlinear shells based on generalized stress and strain measures

A very general class of shell theories is established, based on a unified variational principle and on expansions with respect to biorthogonal function systems. The shell models derived are able to accommodate the flow theory of large strain thermoelastoplasticity including the effects due to unloading. The specific case of a compressible neo-Hookean material with a von Mises yield condition is discussed in detail.

Electroelastic modeling of the motions of ceramic bars under large driving voltages

In relation to electroceramic media subject to large driving voltages, this paper presents the one-dimensional nonlinear equations of a cylindrical ceramic bar. The field variables of the ceramic bar are all expanded in the power series expansions of its cross-sectional coordinates. By means of a unified variational principle, which is formulated through Hamilton’s principle and Legendre’s transformation [e.g., M. C. Dökmeci, IEEE Trans. UFFC 35, 775–787 (1988)] together with the series expansions of field variables, the one-dimensional electroelastic equations are systematically derived in both differential and variational forms. They are capable of predicting the extensional, flexural, and torsional as well as coupled motions of a ceramic bar at low and high frequencies. By a proper truncation of the series expansions, the electroelastic equations incorporate as many higher-order effects as deemed desirable in any case. Special cases are investigated [cf. M. C. Dökmeci, Int. J. Solids Struct. 10, 401–409 (1974) and Proceedings of the 40th Annual Symposium on Frequency Control (IEEE, New York, 1986), pp. 168–178]. The uniqueness is examined in solutions of the linearized electroelastic equations. [Work supported in part by The Scientific and Technical Research Council of Turkey.]

The random variational principle in finite deformation of elasticity and finite element method

In the present paper, we have introduced the random materials, toads, geometrical shapes, force and displacement boundary condition directly into the functional variational formula, by use of a small parameter perturbation method, a unified random variational principle in finite deformation of elasticity and nonlinear random finite element method are established, and used for reliability analysis of structures. Numberical examples showed that the methods have the advantages of simple and conveninet program implementation and are effective for the probabilistic problems in mechanics.

The variational piezoelectromagnetic equations for elastic dielectrics

The aim of this paper is to extend the variational principles of piezoelectricity [M. C. Dökmeci, IEEE Trans. Ultrason. Ferroelec. Freq. Control UFFC-35(6), 775–787 (1988)] so as to incorporate the magnetic effects. The principle of virtual work is applied to the motions of piezoelectromagnetic medium, and an associated variational principle is derived that yields some of the fundamental equations of the medium and includes its remaining equations as constraints. By use of Lagrange multipliers, the constraints are incorporated into the variational principle and a unified variational principle is derived. Likewise, a variational principle is formulated for an electromagnetic medium with an internal surface of discontinuity. These integral and differential types of variational principles provide a standard basis for generating approximate direct solutions as well as deducing systematically lower order equations of piezoelectromagnetic medium. Also, special cases and the reciprocals of the variational principles are indicated. The results are shown to agree with the earlier variational principles [e.g., M. C. Dökmeci, J. Math. Phys. 19, 109–126 (1978) and P. C. Y. Lee, J. Appl. Phys. 69, 7470–7473 (1991)]. [Work supported in part by The Scientific and Technical Research Council of Turkey.]

The random variational principle and finite element method

In this paper, we introduced the random materials, geometrical shapes, force and displacement boundary condition directly into the functional variational formulations and developed a unified random variational principle and finite element method with the small parameter perturbation method. Numerical examples showed that the methods have the advantages of the simple and convenient program implementation, and are effective for the random mechanics problems.

Shell theory for vibrations of piezoceramics under a bias

A consistent derivation of the shell theory in invariant form for the dynamic fields superimposed on a static bias of piezoceramics is discussed. The fundamental equations of piezoelectric media under a static bias are expressed by the Euler-Lagrange equations of a unified variational principle. The variational principle is deduced from the principle of virtual work by augmenting it through Friedrich's transformation. A set of two-dimensional (2-D), approximate equations of thin elastic piezoceramics is systematically derived by means of the variational principle together with a linear representation of field variables in the thickness coordinate. The 2-D electroelastic equations accounting for the influence of mechanical biasing stress accommodate all the types of incremental motions of a polarized ceramic shell coated with very thin electrodes. Emphasis is placed on the special motions, geometry, and material of the piezoceramic shell. The uniqueness of the solutions to the linearized electroelastic equations of the piezoceramic shell is established by the sufficient boundary and initial conditions.

Dynamics of elastic ship laminae

This paper is addressed to the macromechanical analysis of dynamics of an elastic ship laminae within the effective stiffness concept of composites. The laminae that may possess arbitrary number of layers is immersed within an ideal fluid of infinite extent. In the first part, a variational principle [N. Sarigül and M. C. Dökmeci, AIAA J. 20, 1173–1175 (1984)] is extended by Friedrichs's transformation [M. C. Dökmeci, IEEE Trans. UFFC‐35, 775–787 (1988)] so as to formulate a unified variational principle of fluid‐solid interaction. In the second part, a set of two‐dimensional governing equations of laminae is systematically derived by means of a variational principle together with a series representation in the thickness coordinate for the fields of displacement and hydrodynamic pressure that are chosen as a basis of derivation. The significant effects of laminated composites are considered, including those of the dynamic interactions between, and the hydrodynamic pressure on, the layers. The governing equations accommodate all the types of motions of laminae. In the third part, emphasis is placed upon a theorem of uniqueness in solutions of the governing equations and special cases.

A nonlinear theory for high-frequency vibrations of piezoelectric crystal rods

This paper is addressed to a systematic derivation of a hierarchy of one-dimensional theories for the motions of crystal rods in view of the author's review article [Shock Vib. Dig. 20(2), 3–20 (1988)]. The nonlinear theories are derived using a unified variational principle [IEEE UFFC, 35(6) (November 1988)] together with the trigonometric series expansions of the field quantities [cf. Int. J. Solids Structures 10(4), 401–409 (1974)]. All the mechanical and electrical effects and the material nonlinearity are taken into account. The resulting equations govern all the extensional, thickness, flexural, torsional, and coupled motions of crystal rods. By a proper truncation of the series expansions, special motions and material of crystal rods are considered. Also, the fully linearized equations of crystal rods are examined. The sufficient boundary and initial conditions are enumerated so as to ensure the uniqueness in solutions of the linearized rod equations. [Work supported by U.S. Army through its European Research Office.]

Certain integral and differential types of variational principles in nonlinear piezoelectricity

Various forms of variational principles are developed and used to generate, as Euler-Lagrange equations, the fundamental differential equations of nonlinear piezoelectricity. First, Hamilton's principle is rigorously applied to the motion of an electroelastic solid with small piezoelectric coupling, and an associated variational principle is readily derived. Then, by use of the dislocation potentials and Lagrange undetermined multipliers (Friedrich's transformation), the variational principle is augmented for the motion of a piezoelectric solid region with an internal surface of discontinuity. To incorporate the constraints into the two-field variational principle, Friedrich's transformation is again applied, and a unified variational principle is systematically established. This unified variational principle is shown to produce the fundamental equations of an electroelastic solid with small piezoelectric coupling.

A new variational method for finding Einstein metrics on compact Kähler manifolds, I

Let X be a compact Kähler manifold of complex dimension m . One knows that the determination of Einstein-Kähler metrics on X can be reduced to the solution of complex Monge-Ampère equations, depending on the sign of the first Chern class of X . Here, one studies a unified variational principle in the Sobolev space W 2, m ( X ), whose Euler-Lagrange equations are the p.d.e. in question. The resulting variational problem results in a limit case for the associated Sobolev inequalities.

Extended variational principle in nonlinear theory of elasticity

From the extended energy functional, a unified variational principle and an extended variational principle are given and proved in this paper. It is about the non-conservative, abrupt and divided domains static (or kinetic) system and by using the undetermined energy functional, we can immediately deduce various variational principles.

A unified variational principle for the information theoretic analysis of data and its scatter

A single variational principle can be used to determine the maximal entropy distribution, as well as to provide a bound on the uncertainty of the Lagrange multipliers due to scatter in the data. Both the distribution and the Lagrange multipliers are considered as variational parameters.

Variational Methods for Nonstandard Eigenvalue Problems in Waveguide and Resonator Analysis

The nonstandard (general) eigenvalue problem is defined in operator form by L (Lambda) f = 0 and B (Lambda) f = 0, where L and B are linear operators, and for a standard problem L is a linear function of the parameter Lambda and B does not depend on Lambda. It is shown by examples, that nonstandard problems arise in electromagnetic problems, and a unified variational principle is formulated from which stationary functional for the nonstandard eigenvalues can be constructed. The examples include cutoff problem of a waveguide with surface reactance, propagation problem of an azimuthally magnetized ferrite-filled waveguide, the cutoff problem of a corrugated waveguide and the problem of a material insert in a resonator. It is demonstrated with these simple but nontrivial examples that the present method leads to a good engineering accuracy with very elementary test functions.

A Unified Variational Formulation of Classical and Quantum Dynamics. I

We have critically re-examined the role and status of Lagrangian concepts in both classical and quantal contexts. We find a new, unified variational principle which, subject to the superadded postulates of determinism or indeterminism, respectively, leads uniquely to the classical concept of state and Lagrange equations of motion or to the quantal concept of state and Schrödinger equations of motion.