On the bifurcations of the Lamé solutions in plane-strain elasticity

We consider the in-plane bifurcations experienced by the Lamé solutions corresponding to an elastic annulus subjected to radial tension on the curved boundaries. Numerical investigations of the relevant incremental problem reveal two main bifurcation modes: a long-wave local deformation around the central hole of the domain, or a material wrinkling-type instability along the same boundary. Strictly speaking, the latter scenario is related to the violation of the Shapiro–Lopatinskij condition in an appropriate traction boundary-value problem. It is further shown that the main features of this material instability mode can be found by using a singular-perturbation strategy.     

On the Impossibility, in Some Cases, of the Leray�CHopf Condition for Energy Estimates

Current proofs of time independent energy bounds for solutions of the time dependent Navier–Stokes equations, and of bounds for the Dirichlet norms of steady solutions, are dependent upon the construction of an extension of the prescribed boundary values into the domain that satisfies the inequality (1.1) below, for a value of κ less than the kinematic viscosity. It is known from the papers of Leray (J Math Pure Appl 12:1–82, 1993), Hopf (Math Ann 117:764–775, 1941) and Finn (Acta Math 105:197–244, 1961) that such a construction is always possible if the net flux of the boundary values across each individual component of the boundary is zero. On the other hand, the nonexistence of such an extension, for small values of κ, has been shown by Takeshita (Pac J Math 157:151–158, 1993) for any two or three-dimensional annular domain, when the boundary values have a net inflow toward the origin across each component of the boundary. Here, we prove a similar result for boundary values that have a net outflow away from the origin across each component of the boundary. The proof utilizes a class of test functions that can detect and measure deformation. It appears likely that much of our reasoning can be applied to other multiply connected domains.     

On the use of the model proposed by Leonov for the explanation of a secondary plateau of the loss modulus in heterogeneous polymer–filler systems with agglomerates

The purpose of the presented work was to test the capability of the model proposed by Leonov (J Rheol 34:1039–1068, 1990) for the prediction of secondary plateaus on the storage and loss moduli during small-amplitude oscillatory shear flow experiments on filled or heterogeneous polymer melts. Though the occurrence of a plateau on the storage modulus can be well explained in the frame of a filler network, a plateau on the loss modulus can hardly be described with the classical models. In the Leonov model, the continuum of dissipative processes is attributed to the rupture of flocs of particles. Experiments with polyolefins filled with magnesium hydroxide show that there is a clear connection between the amount of agglomerates and the occurrence of a plateau on the loss modulus. However, the value of the critical strain for floc rupture that can be calculated from the experiment shows that the processes responsible for the low-frequency dissipation are rather changes of configuration within the agglomerates than floc rupture. These processes are not described by the Leonov model, and the predicted strain dependence of the plateau is not observed experimentally.     

Coexisting solutions and their chaotic neighborhood in the dynamical systems of nonlinear optics

Coexisting periodic solutions of a dynamical system describing nonlinear optical processes of the second-order are studied. The analytical results concern both the simplified autonomous model and the extended nonautonomous model, including the pump and damping mechanism. The neighborhood of periodic solutions is studied numerically, mainly in phase portraits. As a result of disturbance, for example detuning, the periodic solutions are shown to escape to other states, periodic, quasiperiodic, or chaotic. The chaotic behavior is indicated by the Lyapunov exponents. We also investigate selected aspects of synchronization (unidirectional or mutual) of two identical systems being in two different coexisting states. The effects of quenching the oscillations are shown. The quenching seems very promising for design of some advanced signal processing.     

Chapman–Enskog solutions to arbitrary order in Sonine polynomials V: Summational expressions for the viscosity-related bracket integrals

The Chapman–Enskog solutions of the Boltzmann equations provide a basis for the computation of important transport coefficients for both simple gases and gas mixtures. These coefficients include the viscosity, the thermal conductivity, and the diffusion coefficient. In a preceding paper on simple gases (I), we have shown that the use of higher-order Sonine polynomial expansions enables one to obtain results of arbitrary precision that are free of numerical error. In two subsequent papers (II–III), we extended our original simple gas work to encompass binary gas mixture computations of the viscosity, thermal conductivity, diffusion, and thermal diffusion coefficients to high-order. In a fourth paper (IV) we derived general summational representations for the diffusion- and thermal conductivity-related bracket integrals and provided compact, explicit expressions for all of these bracket integrals needed to compute the diffusion- and thermal conductivity-related transport coefficients up to order 5 in the Sonine polynomial expansions used. In all of this previous work we retained the full dependence of our solutions on the molecular masses, the molecular sizes, the mole fractions, and the intermolecular potential model via the omega integrals up to the final point of solution via matrix inversion. The elements of the matrices to be inverted are, in each case, determined by appropriate combinations of bracket integrals which contain, in general form, all of the various dependencies. Since accurate expressions for the needed bracket integrals have not previously been available in the literature beyond orders 2 or 3, and since such expressions are necessary for any extensive program of computations of the transport coefficients involving Sonine polynomial expansions to higher orders, we have investigated alternative methods of constructing appropriately general bracket integral expressions that do not rely on the term-by-term, expansion and pattern matching techniques that we developed for our previous work. It is our purpose in this paper to report the results of our efforts to obtain useful, alternative, general expressions for the bracket integrals associated with the viscosity-related Chapman–Enskog solutions for gas mixtures. Specifically, we have obtained such expressions in summational form that are conducive to use in high-order viscosity coefficient computations for arbitrary gas mixtures and have computed and reported explicit expressions for all of the orders up to 5.     

Explicit solutions for the coupled stretching–bending problems of holes in composite laminates

Although the classical lamination theory was developed long time ago, it is still not easy to apply this theory to find the analytical solutions for the curvilinear boundary value problems especially when the stretching and bending are coupled each other. To overcome the difficulties, recently we developed a Stroh-like formalism for the general composite laminates. By using this formalism, most of the relations for the coupled stretching–bending problems can be organized into the forms of Stroh formalism for two-dimensional anisotropic elasticity problems. With this newly developed Stroh-like formalism, it becomes easier to obtain an analytical solution for the coupled stretching–bending problems of holes in composite laminates. Because the Stroh-like formalism is a complex variable formalism, the analytical solutions for the whole field are expressed in complex form. Through the use of some identities derived in this paper, the resultant forces and moments around the hole boundary are obtained explicitly in real form. Due to the lack of analytical solutions for the general cases, the comparison is made with the existing analytical solutions for some special cases. In addition, to show the generality of our analytical solutions, several numerical examples are presented to discuss the coupling effect of the laminates and the shape effect of the holes.     

On the Spectrum of the Poincaré Variational Problem for Two Close-to-Touching Inclusions in 2D

We study the spectrum of the Poincaré variational problem for two close to touching inclusions in R ². We derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero.     

Floquet–Bloch decomposition for the computation of dispersion of two-dimensional periodic,damped mechanical systems

Floquet–Bloch theorem is widely applied for computing the dispersion properties of periodic structures, and for estimating their wave modes and group velocities. The theorem allows reducing computational costs through modeling of a representative cell, while providing a rigorous and well-posed spectral problem representing wave dispersion in undamped media. Most studies employ the Floquet–Bloch approach for the analysis of undamped systems, or for systems with simple damping models such as viscous or proportional damping. In this paper, an alternative formulation is proposed whereby wave heading and frequency are used to scan the k-space and estimate the dispersion properties. The considered approach lends itself to the analysis of periodic structures with complex damping configurations, resulting for example from active control schemes, the presence of damping materials, or the use of shunted piezoelectric patches. Examples on waveguides with various levels of damping illustrate the performance and the characteristics of the proposed approach, and provide insights into the properties of the obtained eigensolutions.     

On the use of the stratified momentum balance for the deduction of shear stress in horizontal gas–liquid pipe flow

The use of the stratified flow momentum balance for the deduction of interfacial and liquid wall shear stresses from experimental measurements is examined. A systematic error analysis is applied to the governing equations using the principle of maximum uncertainty. A series of air–water experiments were conducted in 50 and 80 mm diameter pipes, in which gas pressure drop, liquid height and gas wall shear stress were measured. A framework for the correlation of the deduced shear stresses is proposed from the experimental measurements. The uncertainty analysis is used to show that the definition of mean liquid height does not significantly influence the overall results. The development of empirical equations based on such methods may lead to total uncertainties of up to 40%, irrespective of accuracy of the experimental data or the appropriateness of the correlating technique. Comparisons with state-of-the-art correlations for the liquid wall and interfacial friction factor data showed even larger discrepancies between measurement and prediction.     

Chapman–Enskog solutions to arbitrary order in Sonine polynomials IV: Summational expressions for the diffusion- and thermal conductivity-related bracket integrals

The Chapman–Enskog solutions of the Boltzmann equations provide a basis for the computation of important transport coefficients for both simple gases and gas mixtures. These coefficients include the viscosity, the thermal conductivity, and the diffusion coefficient. In a preceding paper on simple gases, we have shown that the use of higher-order Sonine polynomial expansions enables one to obtain results of arbitrary precision that are free of numerical error. In two subsequent papers, we have extended our original simple gas work to encompass binary gas mixture computations of the viscosity, thermal conductivity, diffusion, and thermal diffusion coefficients to high-order. In all of this previous work we retained the full dependence of our solutions on the molecular masses, the molecular sizes, the mole fractions, and the intermolecular potential model via the omega integrals up to the final point of solution via matrix inversion. The elements of the matrices to be inverted are, in each case, determined by appropriate combinations of bracket integrals which contain, in general form, all of the various dependencies. Since accurate, explicit, general expressions for bracket integrals are not available in the literature beyond order 3, and since such general expressions are necessary for any extensive program of computations of the transport coefficients involving Sonine polynomial expansions to higher orders, we have investigated alternative methods of constructing appropriately general bracket integral expressions that do not rely on the term-by-term, expansion and pattern matching techniques that we developed for our previous work. It is our purpose in this paper to report the results of our efforts to obtain useful, alternative, general expressions for the bracket integrals associated with the diffusion- and thermal conductivity-related Chapman–Enskog solutions for gas mixtures. Specifically, we have obtained such expressions in summational form that are conducive to use in high-order transport coefficient computations for arbitrary gas mixtures and have computed and reported explicit expressions for all of the orders up to 5.     

On the Nonhomogeneous Boundary Value Problem for the Navier–Stokes System in a Class of Unbounded Domains

The stationary Navier–Stokes system with nonhomogeneous boundary conditions is studied in a class of domains Ω having “paraboloidal” outlets to infinity. The boundary ${\partial\Omega}$ is multiply connected and consists of M infinite connected components S _m , which form the outer boundary, and I compact connected components Γ _i forming the inner boundary Γ. The boundary value a is assumed to have a compact support and it is supposed that the fluxes of a over the components Γ _i of the inner boundary are sufficiently small. We do not pose any restrictions on fluxes of a over the infinite components S _m . The existence of at least one weak solution to the Navier–Stokes problem is proved. The solution may have finite or infinite Dirichlet integral depending on geometrical properties of outlets to infinity.     

On the energy conservation and critical velocities for the propagation of a ＂steady-shock＂ wave in a bar made of cellular material

The propagation of shock waves in a cellular bar is systematically studied in the framework of continuum solids by adopting two idealized material models, viz. the dynamic rigid, perfectly plastic, locking （D-R-PP-L） model and the dynamic rigid, linear hardening plastic, locking （D-R-LHP-L） model, both considering the effects of strain-rate on the material properties. The shock wave speed relevant to these two models is derived. Consider the case of a bar made of one of such material with initial length L 0 and initial velocity v i impinging onto a rigid target. The variations of the stress, strain, particle velocity, specific internal energy across the shock wave and the cease distance of shock wave are all determined analytically. In particular the ＂energy conservation condition＂ and the ＂kinematic existence condition＂ as proposed by Tan et al. （2005） is re-examined, showing that the ＂energy conservation condition＂ and the consequent ＂critical velocity＂, i.e. the shock can only be generated and sustained in R-PP-L bars when the impact velocity is above this critical velocity, is incorrect. Instead, with elastic deformation, strain-hardening and strain-rate sensitivity of the cellular materials being considered, it is appropriate to redefine a first and a second critical impact velocity for the existence and propagation of shock waves in cellular solids. Starting from the basic relations for shock wave propagating in D-R-LHP-L cellular materials, a new method for inversely determining the dynamic stress-strain curve for cellular materials is proposed. By using e.g. a combination of Taylor bar and Hopkinson pressure bar impact experimental technique, the dynamic stress-strain curve of aluminum foam could bedetermined. Finally, it is demonstrated that this new formulation of shock theory in this one-dimensional stress state can be generalized to shocks in a one-dimensional strain state, i.e. for the case of plate impact on cellular materials, by simply making proper replacements of the elastic and plastic constants.     

Computer-Assisted Methods for the Study of Stationary Solutions in Dissipative Systems, Applied to the Kuramoto–Sivashinski Equation

We develop some computer-assisted techniques for the analysis of stationary solutions of dissipative partial differential equations, of their stability, and of their bifurcation diagrams. As a case study, these methods are applied to the Kuramoto–Sivashinski equation. This equation has been investigated extensively, and its bifurcation diagram is well known from a numerical point of view. Here, we rigorously describe the full graph of solutions branching off the trivial branch, complete with all secondary bifurcations, for parameter values between 0 and 80. We also determine the dimension of the unstable manifold for the flow at some stationary solution in each branch.     

Solitonic interactions,Darboux transformation and double Wronskian solutions for a variable-coefficient derivative nonlinear Schrödinger equation in the inhomogeneous plasmas

By symbolic computation we study a variable-coefficient derivative nonlinear Schrödinger (vc-DNLS) equation describing nonlinear Alfvén waves in inhomogeneous plasmas. Based on the Lax pair of the vc-DNLS equation, the N-fold Darboux transformation is constructed via a gauge transformation and the reduction technique. Multi-solitonic solutions in terms of the double Wronskian for the vc-DNLS equation are obtained. Two- and three-solitonic interactions are analyzed graphically, i.e., overtaking, head-on and parallel interactions. Plasma streaming and inhomogeneous magnetic field control the amplitudes and velocities of the solitonic waves, respectively. The nonuniform density affects the amplitudes of the solitonic waves. The effects of the spectral parameters on the dynamics of the two-solitonic waves are discussed. Our results might facilitate the analytic investigation on certain inhomogeneous systems in the Earth’s magnetosphere, solar winds, planetary bow shocks, dusty cometary tails and interplanetary shocks.     

On an extension of the d’Alembert solution to initial–boundary value problems in multi-layered,multi-material domains

This work introduces a method for the exact solution of initial–boundary value problems for linear, one-dimensional conservation laws in multi-layered, multi-material domains. The method is based on the geometry of the solutions of such conservation laws and represents an extension of the d’Alembert solution to initial–boundary value problems in multi-layered, multi-material domains.     

On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory： equilibrium, governing equation and static deflection

This paper has successfully addressed three critical but overlooked issues in nonlocal elastic stress field theory for nanobeams： （i） why does the presence of increasing nonlocal effects induce reduced nanostructural stiffness in many, but not consistently for all, cases of study, i.e., increasing static deflection, decreasing natural frequency and decreasing buckling load, although physical intuition according to the nonlocal elasticity field theory first established by Eringen tells otherwise？ （ii） the intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study, e.g., bending deflection of a cantilever nanobeam with a point load at its tip; and （iii） the non-existence of additional higher-order boundary conditions for a higher-order governing differential equation. Applying the nonlocal elasticity field theory in nanomechanics and an exact variational principal approach, we derive the new equilibrium conditions, do- main governing differential equation and boundary conditions for bending of nanobeams. These equations and conditions involve essential higher-order differential terms which are opposite in sign with respect to the previously studies in the statics and dynamics of nonlocal nano-structures. The difference in higher-order terms results in reverse trends of nanoscale effects with respect to the conclusion of this paper. Effectively, this paper reports new equilibrium conditions, governing differential equation and boundary condi- tions and the true basic static responses for bending of nanobeams. It is also concluded that the widely accepted equilibrium conditions of nonlocal nanostructures are in fact not in equilibrium, but they can be made perfect should the nonlocal bending moment be replaced by an effective nonlocal bending moment. These conclusions are substantiated, in a general sense, by other approaches in nanostructural models such as strain gradient theory, modified couple stress models and experiments.     

Some issues related to the modeling of interfacial areas in gas–liquid flows,II. Modeling the source terms for dispersed flows

The first part of this paper dealt with the conceptual issues encountered in the definition of the interfacial area and in the derivation of a transport equation. The second part has the following objectives: 1. to address the closure issues for the source terms appearing in the transport equation when dealing with a bubbly flow in a vertical pipe; 2. to provide a list of open questions to be answered before introducing a transport equation for the interfacial area concentration in a thermal–hydraulic computer code.     

The mixed interaction of localized,breather, exploding and solitary wave for the (3+1)-dimensional Kadomtsev–Petviashvili equation in fluid dynamics

Nonlinear Dynamics - The (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation with weak nonlinearity, dispersion and perturbation can denote the development of the long waves and the...