Dynamical Variational Principles Research Articles

Incorporation of thermal effects into the energy‐momentum consistent elastic model for fiber‐bending stiffness in composites

AbstractOwing to the wide acceptance of fiber reinforced composites in the lightweight structure industry, the need to study its thermo‐elastic behavior in dynamical multi‐body systems still remains an open question. Standard continuum models, by virtue of first‐order gradient of deformation, cannot capture curvature effects and hence appropriate extensions of the standard modeling techniques are indispensable to model the accurate physical stiffness. The fiber‐bending curvature are captured by introducing a higher‐order gradient of the deformation as an independent field in the standard continuum. This work explores in detail, a further extension of this continuum to understand the thermal effects on the bending stiffness of fiber bundle in fiber‐reinforced composites. Our proposed model incorporates the temperature of the composite and its bidirectional dependency with the higher‐order gradients of the deformation into the extended mechanical continuum model. The implementation of the model is pursued in the context of multi‐field mixed finite element method and dynamic variational principle. The principle of virtual power enables our extended non‐isothermal continuum to derive an energy‐momentum scheme of higher order. The resulting time‐integrator exhibits physically consistent locking‐ free numerical behavior in dynamical simulations. Our novel approach is illustrated by simple transient dynamic simulations with a hyperelastic, transversely isotropic, polyconvex material behavior. We investigate the conservation of total energy and total momentum in the framework of thermo‐elasticity and study the spatial and temporal convergence for long‐term dynamic simulations with coarser grids and increased accurate numerical solutions.

Variational principles for spectral radius of weighted endomorphisms of 𝐶(𝑋,𝐷)

We give formulas for the spectral radius of weighted endomorphisms a α : C ( X , D ) → C ( X , D ) a\alpha : C(X,D)\to C(X,D) , a ∈ C ( X , D ) a\in C(X,D) , where X X is a compact Hausdorff space and D D is a unital Banach algebra. Under the assumption that α \alpha generates a partial dynamical system ( X , φ ) (X,\varphi ) , we establish two kinds of variational principles for r ( a α ) r(a\alpha ) : using linear extensions of ( X , φ ) (X,\varphi ) and using Lyapunov exponents associated with ergodic measures for ( X , φ ) (X,\varphi ) . This requires considering (twisted) cocycles over ( X , φ ) (X,\varphi ) with values in an arbitrary Banach algebra D D , and thus our analysis cannot be reduced to any of the multiplicative ergodic theorems known so far. The established variational principles apply not only to weighted endomorphisms but also to a vast class of operators acting on Banach spaces that we call abstract weighted shifts associated with α : C ( X , D ) → C ( X , D ) \alpha : C(X,D)\to C(X,D) . In particular, they are far-reaching generalizations of formulas obtained by Kitover, Lebedev, Latushkin, Stepin, and others. They are most efficient when D = B ( F ) D=\mathcal {B}(F) , for a Banach space F F , and endomorphisms of B ( F ) \mathcal {B}(F) induced by α \alpha are inner isometric. As a by-product we obtain a dynamical variational principle for an arbitrary operator b ∈ B ( F ) b\in \mathcal {B}(F) and that its spectral radius is always a Lyapunov exponent in some direction v ∈ F v\in F when F F is reflexive.

Electron-impact ionization of atomic hydrogen: dynamical variational treatment

A simple and straightforward calculating scheme is proposed for electron-impact single and multiple ionization of atoms. The method is based on the application of the Hulthen-Kohn dynamical variational principle. An effective charge seen by the scattered electron is determined for a certain type of trial wave functions mathematically in a rigorous way excluding any empirical assumptions. Validity of the elaborated approach is assessed by calculating triply differential cross section (TDCS) for electron-impact ionization of hydrogen. It is shown that, inclusion of the effective charge into the calculation reduces height of a “binary peak” in comparison with the first Born approximation result. The height of a “recoil peak” depends on the sign of the effective charge. The calculated TDCS are compared with the available experimental data and with the results of sophisticated theories and agreement is found.

Dynamical theoretical model and variational principles for coupled temperature–diffusion–mechanics

In this paper, firstly, the acceleration of the temperature and concentration are assumed to need the extra increment of the heat and energy. The inertial entropy and the inertial chemical potential were proposed by Kuang (Acta Mech 203:1–11, 2009, Acta Mech 214:275–289, 2010). Secondly, the expressions of dissipative energy produced by the variation of the temperature and concentration are derived by using the second law of thermodynamics. Finally, several variational principles for coupled temperature–diffusion–mechanics are established, and their corresponding governing equations and boundary conditions are naturally presented. Meanwhile, some numerical simulations are carried out to describe the coupled reciprocity, which shows concentration diffusion with a finite velocity.

Variational analysis in dynamical problems of linear elasticity

AbstractThe initial–boundary value problems in the linear theory of elasticity is considered. Based on the method of integrodifferential relations (MIDR) two dynamical variational principles is proposed and discussed. It is shown that the Hamilton principle as well as the corresponding complementary principle stated for dynamic boundary value problems follow out the variational formulations proposed. To minimize the nonnegative functional under algebraic and differential constraints a regular finite element algorithm is worked out. The algorithm allows us to estimate explicitly the local and integral quality of numerical solutions obtained. A 3D problem of lateral motions of a rectilinear elastic prism with a rectangular cross section are considered. The numerical results are presented and discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A variational approach to optimal control problems for elastic body motions

AbstractThe initial‐boundary problem for the linear theory of elasticity is considered. Based on the method of integrodifferential relations a new dynamical variational principle in which displacement, stress, and momentum functions are varied is proposed and discussed. To minimize the nonnegative functional under initial, boundary, and partial differential constraints arising in this approach a regular algorithm for approximation of the unknown functions is worked out. The algorithm gives us the possibility to estimate explicitly the local and integral quality of obtained numerical solutions. An effective numerical method for the optimization problems of controlled motions of elastic bodies with quadratic objective functionals is developed. As example, the 3D problems of optimal longitudinal motions of a rectilinear elastic prism with a quadratic cross section are considered for the terminal total mechanical energy to be minimized. The numerical results and their error estimates are presented and discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Role of secondary motifs in fast folding polymers: a dynamical variational principle.

A fascinating and open question challenging biochemistry, physics, and even geometry is the presence of highly regular motifs such as alpha helices in the folded state of biopolymers and proteins. Stimulating explanations ranging from chemical propensity to simple geometrical reasoning have been invoked to rationalize the existence of such secondary structures. We formulate a dynamical variational principle for selection in conformation space based on the requirement that the backbone of the native state of biologically viable polymers be rapidly accessible from the denatured state. The variational principle is shown to result in the emergence of helical order in compact structures.

The Adjoint Structure of Linearised Hamiltonian Theory

Hamiltonian dynamics, generalised in the usual way so that ideal (non-dissipative) models from fluid and plasma theory are included, has an interesting and useful adjoint structure. For linear theory this structure has several implications. For example, linearised Hamiltonian dynamics may be written as a Hermitian equation on Hilbert space provided the equilibrium state satisfy a certain stability criterion. Hamiltonian structure, Hermitian structure, exact dynamical invariants and variational principles are considered in this paper.

Structural theory of elastomeric composites based on fiber systems — Invariant description

A three-dimensional theory of elastomeric composites with elastomeric matrices reinforced by systems of fibers is presented. The theory is based on a structural approach in which the matrix and the reinforcement of the composite are considered separately without reduction to a medium having continuously changing characteristics. The approach is based on the idea of a vector field of macroscopic displacements given by the positions of the axial lines of the fibers in the curret (deformed) configuration of the composite. The vector field determines the current macroscopic configuration, the tensor fields of the measures of macroscopic strain, and the field of the macroscopic stress tensor in the composite. The displacement, strain, and stress fields in the elastomeric matrix and the fibers of the reinforcing systems are regarded as derivatives of the field of macroscopic displacements of the medium. Relations are presented to describe the kinematics of the fibers in the current configuration of the composite, including the evolution of their orientation and the frequency of their planar and spatial distribution. Equations are obtained for the macroscopic motion of the fiber-reinforced matrix, and the dynamic variational principle that governs this motion is established. The elastic macroscopic potential of the matrix is found and related to the components of the macroscopic stress tensor. The procedure to be followed in constructing the constitutive equations of the composite is described. The proposed system of equations, relations, and algorithms is closed and can be used to solve problems involving the deformation of products made of fiber-reinforced elastomers and the creation of elastomeric composite products, based on fiber systems, that possess the requisite properties.

Gust factors for hyperbolic cooling towers on soils

The dynamic response of hyperbolic cooling towers on soils are calculated in this paper based on a dynamic region-wise variational principle and the modes of free vibrations reported elsewhere by the authors. The spectrum method of random vibration theory and wind load data from full-scale measurements are used. Gust factors for hyperbolic cooling towers and footings on different soils are presented.

Dissipation and large thermodynamic fluctuations

The results of recent work of Kipnis, Olla, and Varadhan on the dynamic large deviations from a hydrodynamic limit for some interacting particle models are formally extended to a general hydrodynamic situation, including non-equilibrium steady states, as a fluctuation-dissipation hypothesis. The basic conjecture is that the exponent of decay in the probability of a large thermodynamic fluctuation is given by the dissipation of the force required to produce the fluctuation. It is shown that this hypothesis leads to a nonlinear version of Onsager-Machlup fluctuation theory that had previously been proposed by Graham. A direct consequence of the theory is a dynamic variational principle for the most probable thermodynamic history subject to imposed constraints (Onsager's principle of least dissipation). Following Graham, the theory leads also to a generalized potential, analogous to an equilibrium free energy, for the nonequilibrium steady state and an associated static variational principle. Finally, a formulation of nonlinear fluctuating hydrodynamics is proposed in which the noise enters multiplicatively so as to reproduce the conjectured large-deviations theory on a formal analogy with the results of Freidlin and Wentzell for finite-dimensional systems.

POROUS MEDIUM MAGNETOHYDRODYNAMICS

here we established the mathematical model of dynamics of porous solid medium saturated with compressible magneto-fluid, directly coupled the stress, flow and electromagnetic fields, studied their interactions and propagation regularity of waves and proved that there exist various P and S waves and also their mixture, which, generally, Possess damping and freguency scattering effects.In magnetic (geomagnetic and biological etc.) field, conductive fluid (seawater, oil and blood etc. ) flows to induce electric current thus changes the magnetic field; whearas elactric current flows in magnetic field to produce mechanical force, which influences the deformation of solid skeleton and the flow of the fluid; the deformation of solid skeleton and Darcy osmosis in the interstitial passage in turn change the flow of the fluidAfter the linearization of the model, the general decompositions of the displacement vectors on solid and flow fields as well as magnetic field are given. In this diphase medium, we have theoretically shown the existence of six types of various forms of waves and their behaviors.The difficulties and the abundant anticipated contents of the new theory come right from the complicated interactions. In this paper, with the exception of correcting several oversights and mistakes in some works on magnetohydrodynamics, we also deal with the (Micro-)Bermuda Effect, the possible mechanism of the luminescence in earthquake and the generalization of the solid skeleton from elasticity to viscoelasticity. The mathematical research on conservation laws, equivalent principles, similarities of Transmision Line Theory, dynamical stress concentrations, nonlinear waves and variational principles related to this theory will be presented later.Further study of this theory is undoubtedly important to geophysics, oceanography, life science lind the science of energy resources.Porous medium magnetohydrodynamics is in its initial stage. The field is rather new and only begin to be explored, a large number of unsolved problems and experimental work offer a challenge to future workers.

Dynamic and incremental variational inequality principles, differential inclusions and their applications to co-existent phases problems

The aim of the present paper is the derivation of some classes of variational principles in form of variational inequalities (unilateral problems), with respect to material laws and boundary conditions including strong physical nonlinearities and discontinuities describing the co-existence, of many phases, for instance Masson-Savart effects, sliding and adhesive friction phenomena etc. To treat the appearing nondifferentiable “potentials” and inequality subsidiary conditions, the “subdifferentiability”, a concept of convex analysis, and the related notion of “superpotential” is used. Some classes of unilateral dynamic problems are further discussed and the respective variational inequalities and differential inclusions are derived for several combinations of material laws and boundary conditions. For the respective static problems the propositions of minimum potential and complementary energy are proved and some uniqueness and existence questions are discussed. Finally some applications illustrate the theory.

Vibration analysis of laminated composites

This study presents a macromechanical analysis of vibration of laminated composites within the concept of the effective stiffnesses of composites. The laminated composite may possess arbitrary number of matrix and reinforcing layers, each with a distinct but uniform thickness, curvature, and mechanical properties. Starting with a linearized displacement field for each constituent layer and using an unconstrained, dynamic variational principle due to the author [J. Ship Res. 23, 115–122 (1979)], the set of two-dimensional governing equations of the prestressed laminated composite is systematically and consistently established. The continuity of tractions and displacements at the interfaces is maintained, and all the effects of transverse shear and normal strains and the rotary inertia of layers are included. The governing equations accommodate all the types of small incremental motions as well as those of initial finite motions of the laminated composite. Further, special cases of interest are pointed out, and in particular, the linearized version of the governing equations is considered. Then a theorem which enumerates the sufficient conditions for its solutions is constructed by means of the logarithmic convexity arguments. [Supported by AEK Tek: 2940-3-77.]

Dynamic Variational Principles for Discontinuous Elastic Fields

Various forms of variational principles are derived for the three-dimensional theory of elastodynamics. The continuity requirements on the fields of stresses or strains and/or displacements are relaxed through Friedrichs's transformation. Thus, the generalized forms of certain types of earlier variational principles' are systematically constructed using a basic principle of physics. The variational principles derived herein are shown to generate, as the appropriate Euler equations, the complete set of the governing equations of linear elastodynamics, that is, the stress equations of motion, the strain displacement relations, the mixed natural boundary conditions, the constitutive equations, the natural initial conditions, and the jump conditions. Similarly, generalized variational principles are established for the nonlinear theory of elastodynamics, for the incremental motions in linear elasticity, and for an elastic Cosserat continuum, as well.

Dynamic response of rapidly heated plate elements

through the constitutive law. Indirect application of the variational principle yields the classical equations of motion, the force-displacement boundary conditions, and the constitutive relationships between the distortions and bending (and twisting) stresses. A semidirect application of the variational principle that eliminates spatial-coordinate dependence yields generalized time-dependent ordinary differential equations of dynamic equilibrium and constitutive relations between the generalized force and displacement parameters of assumed spatial distributions. Results obtained using the dynamic thermoelastic variational principle demonstrate displacement and bending-moment-convergence characteristics far superior to conventional solutions. A study of the influence of temperature-dependent material properties shows that neglect of temperature dependence is an unconservative assumption; further it is demonstrated that incomplete consideration of temperature dependence can lead to dangerously unconservative results.