Constructing an analytical solution for lamb waves using the cosserat continuum approach

The problem of propagation of a Lamb elastic wave in a thin plate is considered using the Cosserat continuum model. The deformed state is characterized by independent displacement and rotation vectors. Solutions of the equations of motion are sought in the form of wave packets specified by a Fourier spectrum of an arbitrary shape for three components of the displacement vector and three components of the rotation vector which depend on time, depth, and the longitudinal coordinate. The initial system of equations is split into two systems, one of which describes a Lamb wave and the second corresponds to a transverse wave whose amplitude depends on depth. Analytical solutions in displacements are obtained for the waves of both types. Unlike the solution for Lamb waves, the solution obtained for the transverse wave has no analogs in classical elasticity theory. The solution for the transverse wave is compared with the solution for the Lamb wave. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 143–150, January–February, 2007. An erratum to this article is available at .     

Gas-dynamic approach in the nonlinear theory of ion acoustic waves in a plasma: An exact solution

An exact solution of the problem of the acoustic wave structure in a plasma is obtained. Both plasma component are treated as gases with specified initial temperatures and adiabatic exponents. The system of equations describing the wave profile is solved using an original method consisting of reducing the system to the Bernoulli equation. A numerical example of the obtained general solution of the problem of the wave profile for arbitrary parameters is given. Curves are constructed that bound the region of existence of a stationary solitary ion acoustic wave in the parameter space. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 3–11, September–October, 2007.     

Optimal control of MHD-flow deceleration

A problem of pulsed control for a three-dimensional magnetohydrodynamic (MHD) model is considered. It is demonstrated that singularities of the solution of MHD equations do not develop with time because they are suppressed by a magnetic field. The existence of an optimal control is proved. An optimality system with the solution regular in time as a whole is constructed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 3–10, September–October, 2008.     

Analytic Description of Layer Undulations in Smectic A Liquid Crystals

We investigate the layer undulations that appear in smectic A liquid crystals when a magnetic field is applied in the direction parallel to the smectic layers. In an earlier work (García-Cervera and Joo in J Comput Theor Nanosci 7:795–801, 2010) the authors characterized the critical field using the Landau–de Gennes model for smectic A liquid crystals. In this paper, we obtain an asymptotic expression of the unstable modes using Γ-convergence theory, and a sharp estimate of the critical field. Under the assumption that the layers are fixed at the boundaries, the maximum layer undulation occurs in the middle of the cell and the displacement amplitude decreases near the boundaries. Our estimate of the critical field is consistent with the Helfrich–Hurault theory. When natural boundary conditions are considered, the displacement amplitude does not diminish near the boundary, in sharp contrast with the Dirichlet case, and the critical field is reduced compared to the one calculated in the classical theory. This is consistent with the experiments carried out by Ishikawa and Lavrentovich (Phys Rev E 63:030501(R), 2001). Furthermore, we prove the existence and stability of the solution to the nonlinear system of the Landau–de Gennes model using bifurcation theory. Numerical simulations are used to illustrate the predictions of the analysis.     

The Influence of Moduli Slope of a Linearly Graded Matrix on the Bulk Moduli of Some Particle- and Fiber-Reinforced Composites

The stored energy functional of a homogeneous isotropic elastic body is invariant with respect to translation and rotation of a reference configuration. One can use Noether's Theorem to derive the conservation laws corresponding to these invariant transformations. These conservation laws provide an alternative way of formulating the system of equations governing equilibrium of a homogeneous isotropic body. The resulting system is mathematically identical to the system of equilibrium equations and constitutive relations, generally, of another material. This implies that each solution of the system of equilibrium equations gives rise to another solution, which describes the reciprocal deformation and solves the system of equilibrium equations of another material. In this paper we derive conservation laws and prove the theorem on conjugate solutions for two models of elastic homogeneous isotropic bodies – the model of a simple material and the model of a material with couple stress (Cosserat continuum). This revised version was published online in August 2006 with corrections to the Cover Date.     

Equations of foam hydrodynamics

A closed system of differential equations describing a foam as a viscoelastic compressible continuum is obtained on the basis of the general theory of the mechanics of deformable continua [3–5]. Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 91–95, September–October, 1988.     

Existence of Global Weak Solution for Compressible Fluid Models of Korteweg Type

This work is devoted to proving existence of global weak solutions for a general isothermal model of capillary fluids derived by Dunn and Serrin (Arch Rational Mech Anal 88(2):95–133, 1985) which can be used as a phase transition model. We improve the results of Danchin and Desjardins (Annales de l’IHP, Analyse non linéaire 18:97–133, 2001) by showing the existence of global weak solution in dimension two for initial data in the energy space, close to a stable equilibrium and with specific choices on the capillary coefficients. In particular we are interested in capillary coefficients approximating a constant capillarity coefficient κ. To finish we show the existence of global weak solution in dimension one for a specific type of capillary coefficients with large initial data in the energy space.     

Development of a Generalized Version of the Poisson– Nernst–Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples

A numerical scheme for the transient solution of a generalized version of the Poisson–Nernst–Planck (PNP) equations is presented. The finite element method is used to establish the coupled non-linear matrix system of equations capable of solving the present problem iteratively. The PNP equations represent a set of diffusion equations for charged species, i.e. dissolved ions, present in the pore solution of a rigid porous material in which the surface charge can be assumed neglectable. These equations are coupled to the ‘internally’ induced electrical field and to the velocity field of the fluid. The Nernst–Planck equations describing the diffusion of the ionic species and Gauss’ law in use are, however, coupled in both directions. The governing set of equations is derived from a simplified version of the so-called hybrid mixture theory (HMT). The simplifications used here mainly concerns ignoring the deformation and stresses in the porous material in which the ionic diffusion occurs. The HMT is a special version of the more ‘classical’ continuum mixture theories in the sense that it works with averaged equations at macroscale and that it includes the volume fractions of phases in its structure. The background to the PNP equations can by the HMT approach be described by using the postulates of mass conservation of constituents together with Gauss’ law used together with consistent constitutive laws. The HMT theory includes the constituent forms of the quasistatic version of Maxwell’s equations making it suitable for analyses of the kind addressed in this work. Within the framework of HTM, constitutive equations have been derived using the postulate of entropy inequality together with the technique of identifying properties by Lagrange multipliers. These results will be used in obtaining a closed set of equations for the present problem.     

Some Properties of a New Model for Slow Flow of Granular Materials

Some properties of a new continuum model for the bulk flow of a dense granular material in which neighbouring grains are in contact for a finite duration of time and in which the contact force is non-impulsive – the so called slow flow regime – are presented. The model generalises both the plastic potential and double-shearing models and contains an additional kinematic quantity – the intrinsic spin. The stress tensor is, in general, non-symmetric and separate yield conditions govern translational and rotational yield. We consider homogeneous, quasi-static loadings for the symmetric part of the stress and dynamic loading for the anti-symmetric part of the stress. A solution for the stress state in terms of a single parameter, namely the major principal direction of the symmetric part of the stress, is presented. This direction itself is determined by a consideration of the flow equations in the context both dilatant and isochoric simple shear flows. These simple flows are used to complete the characterisation of the relationship between the anti-symmetric part of the stress and the intrinsic spin.     

Discrete structures equivalent to nonlinear Dirichlet and wave equations

The Amann–Conley–Zehnder (ACZ) reduction is a global Lyapunov–Schmidt reduction for PDEs based on spectral decomposition. ACZ has been applied in conjunction to diverse topological methods, to derive existence and multiplicity results for Hamiltonian systems, for elliptic boundary value problems, and for nonlinear wave equations. Recently, the ACZ reduction has been translated numerically for semilinear Dirichlet problems and for modeling molecular dynamics, showing competitive performances with standard techniques. In this paper, we apply ACZ to a class of nonlinear wave equations in , attaining to the definition of a finite lattice of harmonic oscillators weakly nonlinearly coupled exactly equivalent to the continuum model. This result can be thought as a thermodynamic limit arrested at a small but finite scale without residuals. Reduced dimensional models reveal the macroscopic scaled features of the continuum, which could be interpreted as collective variables.      

EXTENSION OF POINCARE’S NONLINEAR OSCILLATION THEORY TO CONTINUUM MECHANICS (I)-BASIC THEORY AND METHOD

In this paper we extend Poincare’s nonlinear oscillation theory of discrete system to continuum mechanics. First we investigate the existence conditions of periodic solution for linear continuum system in the states of resonance and non-resonance. By applying the results of linear theory, we prove that the main conclusion of Poincare’s nonlinear oscillation theory can be extended to continuum mechanics. Besides, in this paper a new method is suggested to calculate the periodic solution in the states of both resonance and nonresonance by means of the direct perturbation of partial differential equation and weighted integration.     

On a Fluid-Structure Model in Which the Dynamics of the Structure Involves the Shear Stress Due to the Fluid

In this paper we consider a model for fluid-structure interaction. The hybrid system describes the interaction between an incompressible fluid in a three-dimensional container with interior a fixed domain and a thin elastic plate, the interface, which coincides with a flexible flat part of the surface of the vessel containing the fluid. The motion of the fluid is described by the linearized Navier–Stokes equations and the deformation of the plate by the classical plate equations for in-plane motions, modified to include the viscous shear stress which the fluid exerts on the plate as well as damping of Kelvin–Voigt type. We establish the existence of a unique weak solution of the interactive system of partial differential equations by considering an appropriate variational formulation. Uniform stability of the energy associated with the model is shown under the assumption that the potential plate energy is dominated by the dissipation induced by the viscosity of the fluid. The retention of the physical parameters in the problem is an a priori requirement in this physical condition.      

Periodic solutions of a system of nonlinear integro-differential equations with pulse action

We establish a condition for the existence and uniqueness of a periodic solution of a system of nonlinear integro-differential equations with pulse action. The solution is represented as the limit of periodic iterations. We give estimates for the rate of convergence and for the exact solution of the system. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 553–573, October–December, 2005.     

Traveling Wave Fronts of Reaction-Diffusion Systems with Delay

This paper deals with the existence of traveling wave front solutions of reaction-diffusion systems with delay. A monotone iteration scheme is established for the corresponding wave system. If the reaction term satisfies the so-called quasimonotonicity condition, it is shown that the iteration converges to a solution of the wave system, provided that the initial function for the iteration is chosen to be an upper solution and is from the profile set. For systems with certain nonquasimonotone reaction terms, a convergence result is also obtained by further restricting the initial functions of the iteration and using a non-standard ordering of the profile set. Applications are made to the delayed Fishery–KPP equation with a nonmonotone delayed reaction term and to the delayed system of the Belousov–Zhabotinskii reaction model. An erratum to this article is available at .     

An unsymmetric constitutive equation for anisotropic viscoelastic fluid

A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid–liquid crystalline (LC) polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie–Ericksen theory is described by the first Rivlin–Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion–extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extrudate of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a series of new anisotropic non-Newtonian fluid problems can be addressed. The project supported by the National Natural Science Foundation of China (10372100, 19832050) (Key project). The English text was polished by Yunming Chen.     

New global exponential stability result to a general Cohen–Grossberg neural networks with multiple delays

In this paper, we first discuss the existence of an equilibrium point to a general Cohen–Grossberg neural networks with multiple delays by means of using degree theory and linear matrix inequality (LMI) technique. Then by applying the existence result of an equilibrium point, linear matrix inequality technique and constructing a Lyapunov functional, we study the global exponential stability of equilibrium solution to the Cohen–Grossberg neural networks. Compared with known results, our results of global exponential stability of equilibrium point are new. In our results, the hypothesis for differentiability in existing papers on the behaved functions is removed and the hypotheses for boundedness and monotonicity in existing papers on the activation functions are also removed.     

On dislocation models of grain boundaries

Dislocation models of grain boundaries was suggested by Bragg (Proc Phys Soc 52:54–55, 1940) and Burgers (Proc Phys Soc 52:23–33, 1940). The first quantitative study of these models was given by Read and Shockley (Phys Rev 78(3):275–289, 1950). They obtained a formula for the dependence of the grain boundary energy on the misorientation of the neighboring grains, which became a cornerstone of the grain boundary theory. The Read–Shockley formula was based on a proposition that the grain boundary energy is the sum of energies of the two sets of dislocations that come from the two neighboring grains. This proposition was proved under an assumption on a quite special geometry of the slip planes. This paper aims to show that the assumption is not necessary and the proposition holds for arbitrary geometry of slip planes. Another goal of this paper is to provide all basic formulas of the theory: though the dislocation model of grain boundaries is considered in all treatises on dislocation theory, a complete analysis, including the relations for lattice rotations and displacements, has not been given. This analysis shows, in particular, that continuum theory does not yield the proper relations for the lattice misorientations, and these relations must be introduced by an independent ansatz.     

Analytical solution of plane constrained shear problem for single crystals within continuum dislocation theory

Berdichevsky and Le have recently found the analytical solution of the anti-plane constrained shear problem within the continuum dislocation theory (CMT, Contin. Mech. Thermodyn. 18:455–467, 2007). Interesting features of this solution are the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effect. In this paper an analytical solution of the plane constrained shear problem for single crystals exhibiting similar features is obtained and the comparison with the discrete dislocation simulation is provided. Dedicated to the memory of George Herrmann     

Thermal stresses in two- and three-component anisotropic materials

This paper deals with an analytical model of thermal stresses which originate during a cooling process of an anisotropic solid continuum with uniaxial or triaxial anisotropy. The anisotropic solid continuum consists of anisotropic spherical particles periodically distributed in an anisotropic infinite matrix. The particles are or are not embedded in an anisotropic spherical envelope, and the infinite matrix is imaginarily divided into identical cubic cells with central particles. The thermal stresses are thus investigated within the cubic cell. This mulfi-particle-（envelope）-matrix system based on the cell model is applicable to two- and three-component materials of precipitate-matrix and precipitate-envelope-matrix types, respectively. Finally, an analysis of the determination of the thermal stresses in the multi-par- ticle-（envelope）-matrix system which consists of isotropic as well as uniaxial- and/or triaxial-anisotropic components is presented. Additionally, the thermal-stress induced elastic energy density for the anisotropic components is also derived. These analytical models which are valid for isotropic, anisotropic and isotropic-anisotropic multi-particle- （envelope）-matrix systems represent the determination of important material characteristics. This analytical determination includes： （1） the determination of a critical particle radius which defines a limit state regarding the crack initiation in an elastic, elastic-plastic and plastic components; （2） the determination of dimensions and a shape of a crack propagated in a ceramic components; （3） the determination of an energy barrier and micro-/macro-strengthening in a component; and （4） analytical-（experimental）-computational methods of the lifetime prediction. The determination of the thermal stresses in the anisotropic components presented in this paper can be used to determine these material characteristics of real two- and three-component materials with anisotropic components or with anisotropic and isotropic components.