Variational principles for the equations of porous piezoelectric ceramics

The governing equations of a porous piezoelectric continuum are presented in variational form, though they were well established in differential form. Hamilton's principle is applied to the motions of a regular region of the continuum, and a three-field variational principle is obtained with some constraint conditions. By removing the constraint conditions that are usually undesirable in computation through an involutory transformation, a unified variational principle is presented for the region with a fixed internal surface of discontinuity. The unified principle leads, as its Euler-Lagrange equations, to all the governing equations of the region, including the jump conditions but excluding the initial conditions. Certain special cases and reciprocal variational principles are recorded, and they are shown to recover some of the earlier ones.     

Conservation laws for the Navier-Stokes equations

A variety of conservation laws for the Navier-Stokes equations are derived. The conserved currents are described in terms of Lie symmetries and of adjoint variables, which give rise to a complementary variational principle for the Navier-Stokes equations.     

Variational formulation of the equations for small fields superposed on finite biasing fields in an electroelastic body

The equations for small fields superposed on finite biasing fields in an electroelastic body are obtained by an expansion procedure in the variational formulation of a nonlinear electroelastic body. The resulting equations, written with respect to the reference configuration, are the same as those reported earlier except for some algebraic errors that have been corrected in the paper. A variational principle for the incremental fields naturally results from the procedure.     

Variational principles for linear coupled dynamic theory of thermoviscoelasticity

Variational principles for linear coupled dynamic theory of thermoviscoelasticity are constructed using variational theory of potential operators. The functional derived herein gives, when varied, all the governing equations, including the boundary and initial conditions, as the Euler equations. The procedure shown herein does not require, in contrast to Gurtin's method, the transformation of field equations into an equivalent set of integro-differential equations, and includes the initial conditions of the problem explicitly. Gurtin's variational principle for dynamic theory of thermoviscoelasticity is also derived and compared with the present one. Variational principles for elastodynamics, visco-elasticity, etc. are derived as special cases of the variational principle derived herein.     

High-frequency equations for non-linear vibrations of thermopiezoelectric shells

This paper develops a system of 2D shear deformable equations so as to analyze the non-linear vibrations of shells on the basis of the 3D fundamental equations of thermopiezoelectricity with a second sound effect. First, a differential type of variational principles is presented for the 3D fundamental equations. Next, the system of 2D approximate equations of successively higher orders is deduced from the 3D fundamental equations with the aid of the variational principle and the series expansions of the field variables of thermopiezoelectric shells. The system of 2D equations which is established in invariant differential and variational forms governs all the types of vibrations of thermopiezoelectric shells at both low and high frequencies. All the mechanical, electrical and thermal effects of higher orders are taken into account for the case of large electric fields, infinitesimal temperature variations and large deflections. Lastly, attention is confined to some of special cases involving types of vibrations, geometry and material properties. Besides, the uniqueness is investigated in solutions of the system of fully linearized 2D equations of thermopiezoelectric shells and the conditions sufficient for the uniqueness are enumerated.     

A variational criterion for micropolar fluids

A variational criterion for the study of micropolar continuous media in which dissipative phenomena occur is proposed. This criterion is an extension of Lebon-Lambermont's, which is recovered when the spin becomes zero (structureless media). The Euler-Lagrange equations obtained from the variational principle are the balance equations for a micropolar medium with microisotropy. As application of the criterion, one treats numerically the Poiseuille flow of a micropolar medium.     

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.     

Formulation of a numerical process for acoustic impedance sensitivity analysis based on the indirect boundary element method

The objective of the work presented in this paper is the formulation, implementation and validation of an algorithm for computing the acoustic sensitivity with respect to the unequal impedance boundary conditions in an indirect boundary element method (IBEM). The IBEM integral equations are considered for all possible acoustic boundary conditions including velocity, pressure, unequal impedance, and simultaneous velocity and unequal impedance boundary conditions. The numerical system of equations is developed using a variational approach. The sensitivity formulation is based on analytically differentiating the system of equations formed by the variational approach with respect to the unequal acoustic impedance boundary conditions. Numerical sensitivity results obtained using the formulation developed in this paper are compared to analytical solutions in order to validate the new formulation.     

Generalization of the equations for frame-type structures; a variational approach

Summary The differential equations for frame-type structures with elastically deformable joints, derived recently by A. D. Kerr and A. M. Zarembski [1], are genealized first by including the translational inertia terms. The corresponding variational principle is then derived formally, and the mechanical meaning of each term is established. The variational principle is then generalized by including a geometrical non-linearity, the effect of thermal and variable axial forces, and the variation of sectional properties. The corresponding differential equations are derived and the admissible boundary and matching conditions are discussed. As examples, formulations for two problems are presented.With 8 FiguresResearch supported by the National Science Foundation, Washington, DC under grants CME 8001928 and CEE 8308919.     

On the unified equations of functionally graded piezoceramic beams

In this work (part I), we establish the 1D unified equations of a functionally graded piezoceramic beam from the 3D equations of piezoelectricity in both differential and variational forms. The equations of the beam, including a theorem of uniqueness, are obtained using a unified variational principle together with a kinematic-based product method of reduction. In part II, the free vibrations of the beam are considered and the basic properties of eigenvalues are examined. In part III, the equations are derived for the beam under mechanical and electrical bias. Furthermore, a solution for a piezoceramic torsion problem is given.     

Variational methods in irreversible thermoelasticity: theoretical developments and minimum principles for the discrete form

A variational approach for fully coupled dynamic irreversible thermoelasticity is developed for continua, which considers both the conservative and dissipative character in terms of mixed variables. By introducing a consistent variational scheme for the spatial and temporal discretization of the governing equations, a mixed continuum element is established under the Hamiltonian-Lagrangian formalism. The proposed method leads to the development of minimum principles in discrete form with the proper selection of state variables and temporal action sum operators. Consequently, this novel mixed variational formulation can provide the basis for a class of optimization-based methods for irreversible thermomechanics. Several applications are considered to demonstrate the robustness of the proposed variational approach, including transient dynamic response of thermoelastic media due to surface heating caused by ramp- and step-type heat fluxes, and a sequence of laser pulses.     

Lagrangian mechanics and variational integrators on two‐spheres

Euler–Lagrange equations and variational integrators are developed for Lagrangian mechanical systems evolving on a product of two‐spheres. The geometric structure of a product of two‐spheres is carefully considered in order to obtain global equations of motion. Both continuous equations of motion and variational integrators completely avoid the singularities and complexities introduced by local parameterizations or explicit constraints. We derive global expressions for the Euler–Lagrange equations on two‐spheres, which are more compact than existing equations written in terms of angles. Since the variational integrators are derived from Hamilton's principle, they preserve the geometric features of the dynamics such as symplecticity, momentum maps, or total energy, as well as the structure of the configuration manifold. Computational properties of the variational integrators are illustrated for several mechanical systems. In addition, Lie group variational integrators can be used to integrate Lagrangian flows on more general homogeneous spaces. This is achieved by lifting the discrete Hamilton's principle on homogeneous spaces to a discrete variational principle on the Lie group that is constrained by a discrete connection. Copyright © 2009 John Wiley & Sons, Ltd.     

Complementary variational principles for a class of biharmonic problems

Summary Maximum and minimum principles for certain plate bending problems are derived in a unified manner from the canonical theory of complementary variational principles for multiple operator equations. The minimum principle is known in the literature, but the maximum principle appears to be new. A new error bound for approximate variational solutions is also presented.     

Variational principles for generalized thermodiffusion theory in pyroelectricity

The universal thermodynamic variational principle proposed in the previous papers for nonlinear dielectrics and thermopiezoelectricity is extended to the thermodiffusion theory in pyroelectricity, and it is used as a fundamental physical principle to derive the simple complete governing equations of the generalized thermo-electro-diffuso-elastic theory in this paper. In the generalized thermo-electro-diffuso-elastic theory it is assumed that the variation of temperature needs the extra heat which introduces the inertial entropy, and the variation of chemical potential also needs the extra heat which introduces the inertial concentration, etc. The electro-chemical Gibbs function variational principle, the electric Gibbs function variational principle and the internal energy variational principle are derived in this paper.     

The meshless Galerkin boundary node method for Stokes problems in three dimensions

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines an equivalent variational formulation of boundary integral equations for governing equations and the moving least‐squares (MLS) approximations for generating the trial and test functions. In this approach, boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Besides, the resulting formulation inherits the symmetry and positive definiteness of the variational problems. The GBNM is developed in this paper for solving three‐dimensional stationary incompressible Stokes flows in primitive variables. The numerical scheme is based on variational formulations for the first‐kind integral equations, which are valid for both interior and exterior problems simultaneously. A rigorous error analysis and convergence study of the method for both the velocity and the pressure is presented in Sobolev spaces. The capability of the method is also illustrated and assessed through some selected numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Plate equations for piezoelectrically actuated flexural mode ultrasound transducers

This paper considers variational methods to derive two-dimensional plate equations for piezoelectrically actuated flexural mode ultrasound transducers. In the absence of analytical expressions for the equivalent circuit parameters of a flexural mode transducer, it is difficult to calculate its optimal parameters and dimensions, and to choose suitable materials. The influence of coupling between flexural and extensional deformation, and coupling between the structure and the acoustic volume on the dynamic response of piezoelectrically actuated flexural mode transducer is analyzed using variational methods. Variational methods are applied to derive two-dimensional plate equations for the transducer, and to calculate the coupled electromechanical field variables. In these methods, the variations across the thickness direction vanish by using the stress resultants. Thus, two-dimensional plate equations for a stepwise laminated circular plate are obtained.     

Coupled boundary-element scheme for eddy-current computation

The mathematical foundation of a symmetric boundary-element method for the computation of eddy currents in a linear homogeneous conductor which is exposed to an alternating magnetic field is presented. Starting from the A-based variational formulation of the eddy-current equations and a related transmission problem, the problem inside and outside the conductors is reformulated in terms of integral equations on the boundary of the conductors. Surface currents occur as new unknowns of this direct formulation. The integral equations can be coupled in a symmetric fashion using the transmission conditions for the vector potential A and the magnetic field H. The resulting variational problem is elliptic in suitable trace spaces. A conforming Galerkin boundary-element discretization is employed, which relies on surface edge elements and provides quasi-optimal discrete approximations for the tangential traces of A and H. Surface stream functions supplemented with co-homology vector fields ensure the vital zero divergence of the discrete equivalent surface currents. Simple expressions allow the computation of approximate total Ohmic losses and surface forces from the discrete boundary data.     

Electric Field Gradient Theory with Surface Effect for Nano-Dielectrics

The electric field gradient effect is very strong for nanoscale dielectrics. In addition, neither the surface effect nor electrostatic force can be ignored. In this paper, the electric Gibbs free energy variational principle for nanosized dielectrics is established with the strain/electric field gradient effects, as well as the effects of surface and electrostatic force. As regards the surface effects both the surface stress and surface polarization are considered. From this variational principle, the governing equations and the generalized electromechanical Young-Laplace equations, which take into account the effects of strain/electric field gradient, surface and electrostatic force, are derived. The generalized bulk and surface electrostatic stress are obtained from the variational principle naturally. The form are different from those derived from the flexoelectric theory. Based on the present theory, the size-dependent electromechanical phenomenon in nano-dielectrics can be predicted.