Unified Variational Principle Research Articles

Efficient variational constitutive updates for Allen–Cahn-type phase field theory coupled to continuum mechanics

This paper deals with efficient variational constitutive updates for Allen–Cahn-type phase field theory coupled to a geometrically exact description of continuum mechanics. The starting point of the implementation is a unified variational principle: A time-continuous potential is introduced, the minimizers of which describe naturally every aspect of the aforementioned coupled model—including the homogenization assumptions defining the mechanical response of the bulk material in the diffuse interface region. With regard to these assumptions, classic models such as the one by Voigt/Taylor or the one by Reuss/Sachs are included. Additionally, more sound homogenization approaches falling into the range of rank-1 convexification are also included in the unified framework. Based on a direct discretization of this time-continuous potential in time and space, an efficient numerical finite element implementation is proposed. In order to guarantee admissible order parameters of the phase field, the unconstrained optimization problem is supplemented by respective constraints. They are implemented by means of Lagrange parameters combined with the Fischer–Burmeister NCP functions. This results in an exact fulfillment of the aforementioned constraints without considering any inequality. Several numerical examples show the predictive capabilities as well as the robustness and efficiency of the final algorithmic formulation. Furthermore, the influence of the homogenization assumption is analyzed in detail. It is shown that the choice of the homogenization assumption does influence the predicted microstructure in general. However, all models converge to the same solution in the limiting case.

On the unified equations of functionally graded piezoceramic beams

ABSTRACTIn 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.

Correlation functions from a unified variational principle: Trial Lie groups

Time-dependent expectation values and correlation functions for many-body quantum systems are evaluated by means of a unified variational principle. It optimizes a generating functional depending on sources associated with the observables of interest. It is built by imposing through Lagrange multipliers constraints that account for the initial state (at equilibrium or off equilibrium) and for the backward Heisenberg evolution of the observables. The trial objects are respectively akin to a density operator and to an operator involving the observables of interest and the sources. We work out here the case where trial spaces constitute Lie groups. This choice reduces the original degrees of freedom to those of the underlying Lie algebra, consisting of simple observables; the resulting objects are labeled by the indices of a basis of this algebra. Explicit results are obtained by expanding in powers of the sources. Zeroth and first orders provide thermodynamic quantities and expectation values in the form of mean-field approximations, with dynamical equations having a classical Lie–Poisson structure. At second order, the variational expression for two-time correlation functions separates–as does its exact counterpart–the approximate dynamics of the observables from the approximate correlations in the initial state. Two building blocks are involved: (i) a commutation matrix which stems from the structure constants of the Lie algebra; and (ii) the second-derivative matrix of a free-energy function. The diagonalization of both matrices, required for practical calculations, is worked out, in a way analogous to the standard RPA. The ensuing structure of the variational formulae is the same as for a system of non-interacting bosons (or of harmonic oscillators) plus, at non-zero temperature, classical Gaussian variables. This property is explained by mapping the original Lie algebra onto a simpler Lie algebra. The results, valid for any trial Lie group, fulfill consistency properties and encompass several special cases: linear responses, static and time-dependent fluctuations, zero- and high-temperature limits, static and dynamic stability of small deviations.

On the fundamental equations of piezoelasticity of quasicrystal media

The three-dimensional fundamental equations of elasticity of quasicrystals with extension to quasi-static electric effect are expresses in both differential and variational invariant forms for a regular region of quasicrystal material. The principle of conservation of energy is stated for the regular region and the constitutive relations are obtained for the piezoelasticity of material. A theorem is proved for the uniqueness in solutions of the fundamental equations by means of the energy argument. The sufficient boundary and initial conditions are enumerated for the uniqueness. Hamilton’s principle is stated for the regular region and a three-field variational principle is obtained under some constraint conditions. The constraint conditions, which are generally undesirable in computation, are removed by applying an involutory transformation. Then, a unified variational principle is obtained for the regular region, with one or more fixed internal surface of discontinuity. The variational principle operating on all the field variables generates all the fundamental equations of piezoelasticity of quasicrystals under the symmetry conditions of the phonon stress tensor and the initial conditions. The resulting equations, which are expressible in any system of coordinates and may be used through simultaneous approximation upon all the field variables in a direct method of solutions, pave the way to the study of important dislocation, fracture and interface problems of both elasticity and piezoelasticity of quasicrystal materials.

Hamilton’s principle and associated variational principles in polar thermopiezoelectricity

We express Hamilton’s principle for a regular region of thermopiezoelectric polar materials. First, we obtain a four-field variational principle which leads, as its Euler–Lagrange equations, to the divergence equations and the associated natural boundary conditions only. Next, we adjoin the rest of the fundamental equations into the variational principle through an involutory transformation. Thus, we formulate a differential type of unified variational principles operating on all the field variables. The unified variational principle is extended for the region with a fixed internal surface of discontinuity and for a curvilinear laminated region as well. The principles derived in invariant form are expressible in a system of particular coordinate system most appropriate to the geometry of the regions. They are indicated to recover some of earlier principles as special cases.

On the fundamental equations of electromagnetoelastic media in variational form with an application to shell/laminae equations

The fundamental equations of elasticity with extensions to electromagnetic effects are expressed in differential form for a regular region of materials, and the uniqueness of solutions is examined. Alternatively, the fundamental equations are stated as the Euler–Lagrange equations of a unified variational principle, which operates on all the field variables. The variational principle is deduced from a general principle of physics by modifying it through an involutory transformation. Then, a system of two-dimensional shear deformation equations is derived in differential and fully variational forms for the high frequency waves and vibrations of a functionally graded shell. Also, a theorem is given, which states the conditions sufficient for the uniqueness in solutions of the shell equations. On the basis of a discrete layer modeling, the governing equations are obtained for the motions of a curved laminae made of any numbers of functionally graded distinct layers, whenever the displacements and the electric and magnetic potentials of a layer are taken to vary linearly across its thickness. The resulting equations in differential and fully variational, invariant forms account for various types of waves and coupled vibrations of one and two dimensional structural elements as well. The invariant form makes it possible to express the equations in a particular coordinate system most suitable to the geometry of shell (plate) or laminae. The results are shown to be compatible with and to recover some of earlier equations of plane and curved elements for special material, geometry and/or effects.

Variational principles for polar piezoelectric media in elastic range

The fundamental equations of polar piezoelectric media in differential form are alternatively established in variational forms with their well-known features. First, a 3-field variational principle with some constraint conditions is deduced for a regular region of media from a general principle of physics. The principle is modified by using an involutory transformation and a 9-field variational principle operating on all the field variables is derived. Next, this principle is extended and a unified variational principle is obtained for the region with a fixed internal surface of discontinuity. The unified variational principle is further generalized for the equations of a laminated polar region. The generalized variational principle with the only constraint of initial conditions yields all the equations of the laminae region, including the interface conditions, as its Euler-Lagrange equations. The variational principles are shown to recover some of earlier variational principles, as special cases.

Simulation of irreversible processes in nonisothermal systems

The possibility is demonstrated of simulating irreversible processes in nonisothermal systems using a unified variational principle. In the first stage, a thermodynamic description is made of quasi-equilibrium systems for which the parameters of inverse temperatures βj and of the rates of their time variation \(\dot \beta ^j \) are introduced. The second stage corresponds to kinetic simulation using the generalization of the principle of minimal production of entropy to the unsteady-state case.

Free vibrations of a polar body at elastic range

The purpose of this paper is to study certain features of the equations governing the time-harmonic free vibrations of a polar body at elastic range. The governing equations of micropolar elasticity are expressed in differential form, and then, the uniqueness of their solutions is investigated. The conditions sufficient for uniqueness are enumerated using the logarithmic convexity argument without any positive-definiteness assumptions of material elasticity. Applying a general principle of physics and modifying it through an involutory transformation, a unified variational principle is obtained that leads to all the governing equations of the free vibrations as its Euler-Lagrange equations. The governing equations are alternatively expressed in terms of the operators related to the kinetic and potential energies of the body. The basic properties of vibrations are studied and a variational principle in Rayleigh’s quotient is given. As an application, the high-frequency vibrations of an elastic plate are treated.

Free vibrations of a polar elastic body

This paper concerns certain features of the equations governing the time-harmonic free vibrations of a polar elastic body. The governing equations of micropolar elasticity are expressed in differential form; then, the uniqueness in their solutions is investigated. The conditions sufficient for the uniqueness are enumerated using the logarithmic convexity argument without any positive-definiteness assumptions of material elasticity. Applying a general principle of physics and modifying it through an involutory transformation, a unified variational principle is obtained for the free vibrations. The principle leads to all the governing equations of the free vibrations, as its Euler-Lagrange equations. The governing equations are alternatively expressed in terms of the operators related to the kinetic and potential energies of the body. The basic properties of vibrations are studied and a variational principle in Rayleigh’s quotient is given, and the high-frequency vibrations of a micropolar plate [cf., Altay and Dkmeci, Int. J. Solids Struct. (2006)] are likewise treated. [Work supported by TUBA.]

High-frequency thermoviscoelastic vibration of functionally graded thin beams

The paper presents the system of 1D equations so as to analyze the thermoviscoelastic behavior of an axially functionally graded beam of rectangular cross section at high-frequency vibration. First, the 3D fundamental equations of a hyperbolic thermoviscoelasticity are expressed by a unified variational principle in terms of the Laplace transformed field variables [cf., Dkmeci and Altay, Acta Mechanica 143, 91–11 (2000)]. Next, the temperature increment and the displacement field as well as the material density and elasticities [e.g., Sankar and Tzeng, AIAA J. 40, 1228–1232 (2002)] are expressed by the power series expansions in the thickness coordinate of the beam. A variational averaging procedure [e.g., Dokmeci and Altay, Int. J. Non-linear Mech. 37, 225–243 (2002)], together with the series expansions, is used in deriving the system of 1D beam equations in both differential and fully variational forms. The system of 1D equations governs the extensional, flexural, torsional, and thickness shear and also the coupled vibrations of the beam at high frequency. Special cases are indicated involving the material properties and/or the vibration of the beam. Also, the uniqueness is investigated in solutions of the system of beam equations. [Work supported by TUBA.]

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.

A polar theory for vibrations of thin elastic shells

In relation to a polar continuum, this paper presents a 2-D shear deformable theory for the high frequency vibrations of a thin elastic shell. To begin with, the 3-D fundamental equations of the micropolar elastic continuum are expressed as the Euler–Lagrange equations of a unified variational principle. Next, the kinematic variables of the shell are represented by the power series expansions in its thickness coordinate, and then, they are used to establish the 2-D theory by means of the variational principle. The 2-D theory is derived in invariant variational and differential forms and governs all the types of vibrations of the functionally graded micropolar shell. Lastly, the uniqueness is investigated in solutions of the initial mixed boundary value problems defined by the 2-D theory, and some of special cases are indicated in the theory.

Variational principles and vibrations of a functionally graded plate

In piezoelectromagnetism, the three-dimensional fundamental equations well known in differential form are alternatively expressed in variational form through a unified variational principle. The variational principle is deduced from Hamilton’s principle by removing its constraints, the gradient equations and the constitutive relations, by use of Legendre’s transformation. Then, a hierarchical system of the two-dimensional approximate equations is systematically derived for the vibrations of a functionally graded piezoelectromagnetic plate. In the derivation, the unified variational principle and the power series expansions in the thickness coordinate for the field variables are used for the plate. The system of the plate equations in invariant, differential and fully variational forms is capable of studying the direct problems of all the types of vibrations of the plate at both low and high frequency. The uniqueness is investigated and the conditions sufficient for the uniqueness are enumerated in solutions of the system of the plate equations. Further, the variational principle and the system of the plate equations are shown to recover some of the earlier ones, as special cases.

A non-linear rod theory for high-frequency vibrations of thermopiezoelectric materials

In relation to electroelastic media with thermopiezoelectric coupling, the system of one-dimensional equations is consistently derived so as to accommodate the high-frequency vibrations of a rod with temperature-dependent material. In the first part of the paper, a unified variational principle of differential type is presented which describes the fundamental equations of thermopiezoelectricity with second sound, including the physical and geometrical non-linearities. In the second part, the hierarchic system of rod equations is systematically deduced from the three-dimensional fundamental equations by use of Mindlin's method of reduction. The hierarchic system of equations which is derived in both differential and variational forms is capable of predicting the extensional, thickness-shear, flexural and torsional as well as coupled vibrations of the rod of uniform cross-section. All the higher-order effects are taken into account as deemed pertinent in any particular case. In the third part, attention is confined to certain cases involving special motions, materials and geometry. Besides, the uniqueness is investigated in solutions of the linearized system of rod equations and the sufficient conditions are enumerated for the uniqueness of solutions.

Coupled thermoelastic shell equations with second sound for high-frequency vibrations of temperature-dependent materials

Guided by the three-dimensional theory of coupled thermoelasticity with second sound, a system of shear-deformable shell equations is consistently derived in invariant, differential and variational forms for the high-frequency vibrations of temperature-dependent materials. The first part of the paper is concerned with a unified variational principle describing the fundamental equations of thermoelasticity. The differential type of variational principle is presented by expressing Hamilton's principle for the thermal part of a thermoelastic region and then combining it with its mechanical part. In the second part, the hierarchic system of non-isothermal shell equations is systematically established by use of the variational principle together with Mindlin's kinematic hypothesis for shells. The system of two-dimensional approximate equations which may take account of all the significant mechanical and thermal effects, including the temperature dependency of material, governs the extensional, thickness-shear, flexural and torsional as well as coupled vibrations of shells of uniform thickness. Lastly, in the third part, emphasis is placed on certain cases involving special material, geometry and kinematics. Besides, a theorem is devised so as to enumerate the initial and boundary conditions sufficient for the uniqueness in solutions of the system of non-isothermal shell equations.

Electroelastic plate equations for high frequency vibrations of thermopiezoelectric materials

In relation to high frequency vibrations of ceramic materials subject to strong electric fields and large deflections, a system of plate equations is deduced from the three-dimensional equations of thermopiezoelectricity with second sound. The system of two-dimensional approximate equations is derived in differential and variational forms by means of a unified variational principle together with the series expansions of field variables, and it governs the extensional, thickness-shear and flexural as well as coupled vibrations of electroelastic plates of uniform thickness. Certain cases involving special motions, geometry and material properties are indicated and, in particular, the linearised system of plate equations is recorded and the uniqueness of its solutions is also pointed out.

A mathematical study on a unified variational principle with two constrained parameters

In this paper, a parameterized variational principle based on a mixed functional obtained by a linear combination of the total potential energy functional, the modified Hellinger-Reissner functional, and the Hu-Washizu functional with two constrained parameters is proposed, and the mathematical characteristics of the variational equation of the principle are investigated for the analysis of boundary value problems in linear elasticity. It is first proved that the Euler-Lagrange equations of the variational equation is identical to the governing equations for the given problem. Then existence of the unique solution of the variational equation is systematically proved by showing that the energy bilinear form is weakly-coercive. As an application, the stress/strain smoothing can be obtained as a form of mixed FEM based on the variational equation.

Special issue on `Set-valued optimization': Editors' preface

In [1], we gave a unified variational principle for vector valued functions. In this paper, we give four equivalents of a corollary of this principle, generalizing the equivalence results of [3] to the vector case and improving the equivalence results of [5]. We also applied one of the equivalents to derive the vector form of the “Drop Theorem” ([3]). Applying the unified variational principle, we obtained a fixed point theorem for directional contractions as an application of our vector variational principle, generalizing a fixed point theorem for directional contractions in [9].

Finite element formulation of coupled grain-boundary and surface diffusion with grain-boundary migration

Finite element formulations are developed to model microstructure evolution by a combination of grain–boundary diffusion, grain–boundary migration and free surface diffusion. The formulations are based on a unified variational principle which allows fully coupled processes to be analysed. For example, a process can be analysed which involves grain–boundary diffusion along a curved and migrating grain–boundary network, coupled with surface diffusion along internal and/or external free surfaces which intersect with the grain–boundary network. The numerical solution provides the velocities of each individual grain and the velocities of grain–boundaries and migrating surfaces. The finite element formulations, when combined with a time integration algorithm, form a numerical technique which can be used to simulate microstructural evolution in polycrystalline materials. The technique can be applied to a wide range of physical problems including: sintering of powder compacts; grain–growth; diffusive void growth and crack propagation; superplastic deformation and the morphological evolution of electronic thin films. Various numerical examples are presented to demonstrate the effectiveness of the numerical technique.