Fully Localised Solitary-Wave Solutions of the Three-Dimensional Gravity–Capillary Water-Wave Problem

A model equation derived by Kadomtsev & Petviashvili (Sov Phys Dokl 15:539–541, 1970) suggests that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal spatial direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. The theory is variational in nature. A simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties. The reduced functional is related to the functional associated with the Kadomtsev–Petviashvili equation, and a nontrivial critical point is detected using the direct methods of the calculus of variations.     

On the general form of the law of dynamics

The dynamics of Lagrangian systems is formulated with a differential geometric approach and according to a new paradigm of the calculus of variations. Discontinuities in the trajectory, non-potential force systems and linear constraints are taken into account with a coordinate-free treatment. The law of dynamics, characterizing the trajectory in a general non-linear configuration manifold, is expressed in terms of a variational principle and of differential and jump conditions. By endowing the configuration manifold with a connection, the general law is shown to be tensorial in the velocity of virtual flows and to depend on the torsion of the connection. This result provides a general expression of the EULER-LAGRANGE operator. POINCARÉ and LAGRANGE forms of the law are recovered as special cases corresponding, respectively, to the connection induced by natural and mobile reference frames. For free motions, the geodesic property of the trajectory is directly inferred by adopting the LEVI-CIVITA connection induced by the kinetic energy.     

Higher Integrability for Minimizers of the Mumford–Shah Functional

We prove higher integrability for the gradient of local minimizers of the Mumford–Shah energy functional, providing a positive answer to a conjecture of De Giorgi (Free discontinuity problems in calculus of variations. Frontiers in pure and applied mathematics, North-Holland, Amsterdam, pp 55–62, 1991).     

A non-iterative transformation method for Newton's free boundary problem

In book II of Newton's Principia Mathematica of 1687 several applicative problems are introduced and solved. There, we can find the formulation of the first calculus of variations problem that leads to the first free boundary problem of history. The general calculus of variations problem is concerned with the optimal shape design for the motion of projectiles subject to air resistance. Here, for Newton's optimal nose cone free boundary problem, we define a non-iterative initial value method which is referred in the literature as a transformation method. To define this method we apply invariance properties of Newton's free boundary problem under a scaling group of point transformations. Finally, we compare our non-iterative numerical results with those available in the literature and obtained via an iterative shooting method. We emphasize that our non-iterative method is faster than shooting or collocation methods and does not need any preliminary computation to test the target function as the iterative method or even provide any initial iterate. Moreover, applying Buckingham Pi-Theorem we get the functional relation between the unknown free boundary and the nose cone radius and height.     

Applicability of the free volume concept on relaxation phenomena in the glass transition range

Conclusion Except the discrepancies mentioned with respect to pressure investigations, which need clarification in the future, we can conclude in a general way, as follows.As far as only average parameters of macroscopic samples are considered (complex moduli, or dielectric constants, volume or heat content etc....), the free volume concept can relate the variations of molecular mobility to the changes of an average free volume in a semiquantitative way. This average free volume can no longer fully caracterize the wide variety of molecular motions involved in the kinetics of redistribution of holes in the liquid during the recovery experiments. These kinetic processes involve a wide distribution of retardation times, which may be associated with the local distribution of holes, or with that of cooperating groups of molecules, or molecular segments.On the other hand, free volume is not necessarily the fundamental molecular parameter which controls the rate of configurational changes, characterized by the variation of entropy of the liquid (14). Even if this is the case, most of the above discussion may be applied to any other average excess parameter, as far as theDoolittle equation is formally adopted, in which (f/b) is expressed in terms of the new parameter, rather than that of free volume. However, since the relaxational free volume, as determined from the W.L.F. equation, and the independently measured volume changes are often in close agreement, this means that the variations of the excess entropy (12), or those of the configurational free energy (13), and the changes in volume are closely related. Therefore, the free volume concept remains still a valuable tool for unifying different kinds of rate processes from both a theoretical and an experimental point of view, especially in the glass transition range.     

An upwind finite‐volume element scheme and its maximum‐principle‐preserving property for nonlinear convection–diffusion problem

For a class of nonlinear convection–diffusion equation in multiple space dimensions, a kind of upwind finite‐volume element (UFVE) scheme is put forward. Some techniques, such as calculus of variations, commutating operators and prior estimates, are adopted. It is proved that the UFVE scheme is unconditionally stable and satisfies maximum principle. Optimal‐order estimates in H¹‐norm are derived to determine the error in the approximate solution. Numerical results are presented to observe the performance of the scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

Asynchronous Variational Integrators

We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; (ii) the algorithms derive from a spacetime form of a discrete version of Hamilton's variational principle. As a consequence of this variational structure, the algorithms conserve local momenta and a local discrete multisymplectic structure exactly.To guide the development of the discretizations, a spacetime multisymplectic formulation of elastodynamics is presented. The variational principle used incorporates both configuration and spacetime reference variations. This allows a unified treatment of all the conservation properties of the system. A discrete version of reference configuration is also considered, providing a natural definition of a discrete energy. The possibilities for discrete energy conservation are evaluated.Numerical tests reveal that, even when local energy balance is notenforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorithm.     

A Refined Shear Deformation Theory of Bending of Isotropic Plates*

ABSTRACT

A nonlinear, in-plane displacement assumption is proposed, based on an undetermined variation df/dz of transverse shear strains through the plate thickness. A second-order ordinary differential equation for f(z) and two surface conditions, as well as a set of eighth-order partial differential equations and four associated boundary conditions, are derived from the principle of minimum potential energy. Coupling exists between the partial and ordinary differential equations. In the homogeneous solutions for the former, in addition to an interior solution contribution, there exist two edge-zone solution contributions, one of which induces self-equilibrated (in the thickness direction) boundary stresses. Three examples are calculated using the present theory. The last gives the stress couple and maximum-stress concentration factors at the free edge of a circular hole in a large bent plate. Numerical results for the examples are compared with those given by three-dimensional elasticity theory and several two-dimensional theories. It is found that the present theory can accurately predict nonlinear variations of in-plane stresses through the thickness of a plate.     

An extended formulation of calculus of variations for incommensurate fractional derivatives with fractional performance index

In this paper, we consider the main problem of variational calculus when the derivatives are Riemann?CLiouville-type fractional with incommensurate orders in general. As the most general form of the performance index, we consider a fractional integral form for the functional that is to be extremized. In the light of fractional calculus and fractional integration by parts, we express a generalized problem of the calculus of variations, in which the classical problem is a special case. Considering five cases of the problem (fixed, free, and dependent final time and states), we derive a necessary condition which is an extended version of the classical Euler?CLagrange equation. As another important result, we derive the necessary conditions for an optimization problem with piecewise smooth extremals where the fractional derivatives are not necessarily continuous. The latter result is valid only for the integer order for performance index. Finally, we provide some examples to clarify the effectiveness of the proposed theorems.     

Analysis of the Volume-Constrained Peridynamic Navier Equation of Linear Elasticity

Well-posedness results for the state-based peridynamic nonlocal continuum model of solid mechanics are established with the help of a nonlocal vector calculus. The peridynamic strain energy density for an elastic constitutively linear anisotropic heterogeneous solid is expressed in terms of the field operators of that calculus, after which a variational principle for the equilibrium state is defined. The peridynamic Navier equilibrium equation is then derived as the first-order necessary conditions and are shown to reduce, for the case of homogeneous materials, to the classical Navier equation as the extent of nonlocal interactions vanishes. Then, for certain peridynamic constitutive relations, the peridynamic energy space is shown to be equivalent to the space of square-integrable functions; this result leads to well-posedness results for volume-constrained problems of both the Dirichlet and Neumann types. Using standard results, well-posedness is also established for the time-dependent peridynamic equation of motion.     

A Variational Model of Disjoining Pressure: Liquid Film on a Nonplanar Surface

Variational methods have been successfully used in modelling thin liquid films in numerous theoretical studies of wettability. In this article, the variational model of the disjoining pressure is extended to the general case of a two-dimensional solid surface. The Helmholtz free energy functional depends both on the disjoining pressure isotherm and on the shape of the solid surface. The augmented Young–Laplace equation (AYLE) is a nonlinear second-order partial differential equation. A number of solutions describing wetting films on spherical grains have been obtained. In the case of cylindrical films, the phase portrait technique describes the entire variety of mathematically feasible solutions. It turns out that a periodic solution, which would describe wave-like wetting films, does not satisfy Jacobi’s condition of the classical calculus of variations. Therefore, such a solution is nonphysical. The roughness of the solid surface significantly affects liquid film stability. AYLE solutions suggest that film rupture is more likely at a location where the pore-wall surface is most exposed into the pore space, and the curvature is positive.     

箱形梁剪力滞效应分析中的翘曲位移函数及广义内力研究

以薄壁箱梁的弯曲计算理论为基础,从分析翼缘板的面内剪切变形和弯曲剪力流的分布规律入手,从理论上证明二次抛物线是箱形梁剪力滞效应分析中的合理翘曲位移函数。选取剪力滞效应引起的附加挠度作为广义位移,用基于最小势能原理的能量变分法建立箱形梁剪力滞效应分析的控制微分方程和边界条件。对箱梁横截面上新出现的广义内力给出严密定义,并建立了剪力滞翘曲应力的简便计算公式,它与初等梁弯曲应力公式具有相同的形式。对一个简支箱梁模型的计算表明,计算值与实测值吻合良好,从而证实了本文的分析方法和建立的公式是正确的。不同于弯矩的分布,剪力滞广义力矩具有快速衰减的分布特征。对集中荷载作用下的简支箱梁算例,剪力滞效应使其跨中挠度增大达12%,工程实践中必须认真对待。     

Non-local continuum mechanics and fractional calculus

The present work introduces fractional calculus into the continuum mechanics area describing non-local constitutive relations. Considering a one-dimensional body and assuming total stored energy depending not only upon the local strain but also upon a fractional derivative of the stain, an elastic model with non-local stress–strain behavior is introduced. Fractional calculus provides a natural framework for describing non-local constitutive relations and requires no assumptions for the interval of non-local influence. Furthermore, the proposed method works in finite intervals contrary to the existing theories requiring infinite domains.     

Energy-minimizing deformations of elastic sheets with bending stiffness

Necessary conditions for energy-minimizing deformations are derived for a theory of sheets in which the strain energy function depends on the second derivatives of the deformation as well as its first derivatives. All of these conditions are extensions of well-known necessary conditions in classical calculus of variations. The interpretation of some of these conditions as material stability conditions is explained.     

Variational Convergence for Nonlinear Shell Models with Directors and Related Semicontinuity and Relaxation Results

We use a variational convergence method to study the consistency of various Cosserat hypotheses in shell theory with the limit nonlinear membrane model derived from three-dimensional elasticity. In the course of the analysis, we introduce a generalization of quasiconvexity that is suitable for problems of the calculus of variations with two vectorial unknowns, one of which appears through its gradient, the other one through its value, in a weak W ^{1, p} ×L ^p framework. (Accepted: October 1, 1999)?Published online August 21, 2000     

An inverse for the Jaumann derivative and some applications to the rheology of viscoelastic fluids

Summary By using a generalization of the matrizant of matrix calculus, it is shown how one can construct formally an inverse, or integral, for the well-knownJaumann derivative of continuum mechanics. Some applications to fluid rheology are then considered. First, it is shown that this integral provides, via theBoltzmann super-position principle, a generalization of Oldroyd's quasi-linear fluid model, which is related to the molecular model ofBueche. Explicit expressions for the stresses arising in a general laminar shear flow are then derived for this model. Secondly, it is indicated how the operation can be used with rheological equations which are nonlinear in the deformation-rate, but quasi-linear in stress, to solve explicitly for the stress in terms of kinematic quantities. As an example, a rheological equation for suspensions of viscoelastic spheres in aNewtonian fluid is treated.     

Physical variational principle and thin plate theory in electro-magneto-elastic analysis

Under the assumption of the quasi-static electric and magnetic fields the electro-magneto-elastic analysis including medium and its environment is studied in this paper. The complete governing equations under the finite deformation can be derived from the physical variational principle. In the physical variational principle the variations of the electric potential and magnetic potential are divided into local variations and migratory variations. From the virtual change of the sum of the electromagnetic energy and the couple energy produced by the migratory variation we can get the electromagnetic force and in this case the virtual variation of the volume should be considered. It is also found that the Maxwell stress is directly related to the strain in a material with piezoelectric or piezomagnetic behavior for the finite deformation case. The thin plate theory in first order is derived from the general theory in this paper and the Maxwell stress is naturally included in the governing equations.     

Mechanically-based approach to non-local elasticity: Variational principles

The mechanically-based approach to non-local elastic continuum, will be captured through variational calculus, based on the assumptions that non-adjacent elements of the solid may exchange central body forces, monotonically decreasing with their interdistance, depending on the relative displacement, and on the volume products. Such a mechanical model is investigated introducing primarily the dual state variables by means of the virtual work principle. The constitutive relations between dual variables are introduced defining a proper, convex, potential energy. It is proved that the solution of the elastic problem corresponds to a global minimum of the potential energy functional. Moreover, the Euler–Lagrange equations together with the natural boundary conditions associated to the total potential energy functional are established with variational calculus and they coincide with analogous relations already obtained by means of mechanical considerations. Numerical analysis of a tensile specimen has been introduced to show the capabilities of the proposed approach.     

Numerical investigation of fully developed turbulent fluid flow and heat transfer in a square duct

Three-dimensional fully developed turbulent fluid flow and heat transfer in a square duct are numerically investigated with the author's anisotropic low-Reynolds-number k-ε turbulence model. Special attenton has been given to the regions close to the wall and the corner, which are known to influence the characteristics of secondary flow a great deal. Hence, instead of the common wall function approach, the no-slip boundary condition at the wall is directly used. Velocity and temperature profiles are predicted for fully developed turbulent flows with constant wall temperature. The predicted variations of both local wall shear stress and local wall heat flux are shown to be in close agreement with available experimental data. The present paper also presents the budget of turbulent kinetic energy equation and the systematic evaluation for existing wall function forms. The commonly adopted wall function forms that are valid for two-dimensional flows are found to be inadequate for three-dimensional turbulent flows in a square duct.     

The distribution of the specific pressure in rolling strips

The principle of the distribution of the specific pressure in rolling strips is used not only for calculating the total rolling pressure but also for providing the basis in calculating the widening and in designing the rational roll profile. Hitherto, the results of analysis on this subject in the references are expressed as a function of one dimension, and they cannot reflect the variation of the specific pressure along the width of the contact surface. This paper deals with the two-dimensional expression of the principle of the distribution of the specific pressure with the help of the calculus of variations.