Theories for laminated and sandwich plates

The growing use of sandwich and laminated plates requires a theoretically based prediction of the mechanical behavior of structural elements of such type. Starting with the pioneering studies of Reissner, a great number of theories for the engineering calculations have been developed. The review deals with the classification of the theories and discusses some of them in detail.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Martin Luther Universität, Halle-Wittenberg, Fachbereich Werkstoffwissenschaften, D-06099 Halle, Germany. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 333–348, 1998.     

Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments

In this paper we deal with the numerical solutions of Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. The numerical solution is given by the numerical Green’s function. It is shown that Runge–Kutta methods preserve their original order for first-order periodic boundary value differential equations with piecewise constant arguments. We give the conditions under which the numerical solutions preserve some properties of the analytic solutions, e.g., uniqueness and comparison theorems. Finally, some experiments are given to illustrate our results.     

Wood-filled thermoplastic composites

Different physical properties of wood-filled thermoplastic materials produced by a special mixing and extrusion process are examined. The results show that the wood content and the kind of plastics are the main parameters that control the physical properties of composites. In general, wood-filled thermoplastic materials exhibit mechanical properties comparable to those of customary wood fiber products, i.e., medium density fiberboard (MDF); however, they show distinctly better behavior than the MDF and natural wood after exposure to moisture.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Universität Kassel, Institut für Werkstofftechnik, Kunstoff-und Recyclingtechnik, Mönchebergstraße 3, 34109 Kassel, Germany. Published in Mekhanika Kompozitnykh Materialov, Vol. 34, No. 6, pp. 795–802, November–December, 1998.     

Effective elastic properties of particulate-reinforced compossite materials

A numerical approach for determination of the effective properties of particulate composite materials has been developed. A representative volume element (RVE) of the composite material is analyzed with help of the finite-element method. Uniform boundary displacements or tractions are applied on the boundaries of the RVE for introducing the known average strain in the RVE. Local stress and strain distributions in the RVE are calculated using the finite-element method. Different effective elastic constants can be calculated by averaging the local fields corresponding to different sets of boundary conditions. The present approach allows us to determine the effective properties of particle-reinforced composites with acceptable accuracy. The calculated effective properties of the composite are between the upper and lower Hashin—Shtrikman bounds. The results based on the present approach lead to higher stiffness of composites in comparison with analytical approaches.Institute fur Werkstoffwissenschaften, Fachberech Werkseoffwissenschaften, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle, Germany. Published in Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 450–459, July–August, 1997.     

Convergence of the method of lines for the first periodic boundary-value problem for a second-order nonlinear parabolic equation

For solving the first generalized periodic boundary-value problem in the case of a second-order quasilinear parabolic equation of form with periodic condition and boundary conditions there is examined a longitudinal variant of the method of lines, reducing the solving of problem (1)–(3) to the solving of a two-point problem for a system ofN ^-1 first-order ordinary differential equations of form with the two-point conditions An error estimate is established. The convergence of the solutions of problem (4)–(5) to the generalized solution of problem (1)–(3) is established for two methods of choosing the functions. Convergence with orderh ² is guaranteed under the assumption of square-integrability of the third derivative of the solution of problem (1)–(3).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 90, pp. 268–276, 1979.     

On the derivation of an asymptotically correct shear correction factor for the Reissner-Mindlin plate model

In the framework of thin linear elastic plates it is known that the solutions of both the three-dimensional problem and the Reissner-Mindlin plate model can be developed into asymptotic expansions. By comparing the particular asymptotic expansions with respect to the half-thickness ɛ of the plate in the case of periodic boundary conditions on the lateral side, the shear correction factor in the Reissner-Mindlin plate model can be determined in such a way that this model approximates the three-dimensional solution with one order of the plate thickness better than the classical Kirchhoff model. This fails for hard clamped lateral boundary conditions so that the Reissner-Mindlin model is in this case asymptotically as good as the Kirchhoff model.     

A solution of the discrete minimum-time control problem

An iterative procedure for the synthesis of discrete minimum-amplitude and minimum-time controls is presented. The algorithm is based on some new relations obtained by extending well-known results on the minimum-energy control problem as given in Refs. 1–3. This approach yields a set of implicit algebraic equations from which the desired optimal control sequence is determined by the iteration procedure referred to above. The algorithm has the advantage that convergence to the optimal solution can be guaranteed. Simplicity of the recursion formulas and insensitivity to numerical errors make the procedure well suited for on-line or off-line computations.This work was done at the Institut für Regelungstechnik, Technische Universität Berlin, West Berlin, Germany. The author is indebted to Professor G. Schneider for many stimulating discussions and criticisms during the course of this research.     

Stability of transversely isotropic annular plates with low shear resistance

The stability differential equations of a cylindrically orthotropic circular plate are obtained on the assumption of an axisymmetric buckling mode with allowance for transverse shears. These equations are solved for the case of a transversely isotropic material when the inner and outer edges of the plate are identically loaded by uniformly distributed radial forces. The transcendental equations for the critical load parameter are constructed for various edge conditions. The dependence of this parameter on the boundary conditions and the relative thickness of the plate, Poisson's ratio, and the ratio of the Young's and shear moduli of the material are investigated. Certain conclusions are reached concerning the design of reinforced-plastic plates.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 872–880, September–October, 1969.     

Initial boundary value problems for linear viscoelastic flows generated by integro-differential equations

An estimate of the velocity field is obtained for the equation of motion of incompressible media. With the help of this estimate, the integro-differential equations that describe the motion of linear viscoelastic fluids in the two-dimensional case are studied. The existence is proved for a weak, global in time, solution of the Cauchy problem and of the initial boundary value problem with periodic boundary conditions. Bibliography: 18 titles.Dedicated to Olga Aleksandrovna Ladyzhenskaya on the occasion of her jubilee__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 90–98.     

Imposing boundary conditions in Sinc method using highest derivative approximation

Sinc approximate methods are often used to solve complex boundary value problems such as problems on unbounded domains or problems with endpoint singularities. A recent implementation of the Sinc method [Li, C. and Wu, X., Numerical solution of differential equations using Sinc method based on the interpolation of the highest derivatives, Applied Mathematical Modeling 31 (1) 2007 1–9] in which Sinc basis functions are used to approximate the highest derivative in the governing equation of the boundary value problem is evaluated for structural mechanics applications in which interlaminar stresses are desired. We suggest an alternative approach for specifying the boundary conditions, and we compare the numerical results for analysis of a laminated composite Timoshenko beam, implementing both Li and Wu’s approach and our alternative approach for applying the boundary conditions. For the Timoshenko beam problem, we obtain accurate results using both approaches, including transverse shear stress by integration of the 3D equilibrium equations of elasticity. The beam results indicate our approach is less dependent on the selection of the Sinc mesh size than Li and Wu’s SIHD. We also apply SIHD to analyze a classical laminated composite plate. For the plate example, we experience difficulty in obtaining a complete system of equations using Li and Wu’s approach. For our approach, we suggest that additional necessary information may be obtained by applying the derivatives of the boundary conditions on each edge. Using this technique, we obtain accurate results for deflection and stresses, including interlaminar stresses by integration of the 3D equilibrium equations of elasticity. Our results for both the beam and the plate problems indicate that this approach is easily implemented, has a high level of accuracy, and good convergence properties.     

The global attractor of the viscous Fornberg–Whitham equation

This paper aims to present a proof of the existence of the attractor for the one-dimensional viscous Fornberg–Whitham equation. In this paper, the global existence of solution to the viscous Fornberg–Whitham equation in L² under the periodic boundary conditions is studied. By using the time estimate of the Fornberg–Whitham equation, we get the compact and bounded absorbing set and the existence of the global attractor for the viscous Fornberg–Whitham equation.     

Free convection boundary layers near corners and sharp trailing edges

The problems of how a free convection boundary layer adjusts to a small but sudden change in plate angle, and how it leaves a body with a wedge-shaped trailing edge are discussed. The angle is taken to beO(G _r ^–3/28) so that the flow near the discontinuity can be described by a double deck structure. For plates with very small angle changes, the lower deck equations can be linearised and the resulting equations have been solved using Fourier transforms. In the general case, the lower deck equations have to be solved numerically and results are given for both concave and convex corners and for a range of wedge angles. It is found that for sufficiently large angles there is a region of reversed flow in each case, in the first and last cases in a region centred on the discontinuity, while for convex corners it is downstream of the corner.

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Zusammenfassung Es wird das Problem gelöst wie sich bei freier Konvektion die Grenzschicht an eine kleine aber plötzliche Änderung des Plattenwinkels anpaßt, und wie sie eine keilförmige Hinterkante verläßt. Der Winkel wird von der GrößenordnungO(G _r ^–3/28) gewählt, damit die Strömung nahe zur Diskontinuität durch eine Zweischichtstruktur beschrieben werden kann. Für Platten mit sehr kleinen Winkeländerungen kann man die Gleichungen der unteren Schicht linearisieren und durch eine Fourier-Transformation lösen. Für den allgemeinen Fall müssen die Gleichungen der unteren Schicht numerisch gelöst werden, und wir geben Resultate für konkave und konvexe Ecken sowie für einen Reihe von Keilwinkeln an. Für hinreichend große Winkel konnten wir in allen Fällen eine Umkehr-Strömung feststellen, im ersten und letzten Fall in einem Gebiet mit der Diskontinuität als Zentrum, und für konvexe Ecken stromabwärts von der Ecke.

     

Componentwise inclusion and exclusion sets for solutions of quadratic equations in finite dimensional spaces

Summary In this paper we use interval arithmetic tools for the computation of componentwise inclusion and exclusion sets for solutions of quadratic equations in finite dimensional spaces. We define a mapping for which under certain assumptions we can construct an interval vector which is mapped into itself. Using Brouwer's fixed point theorem we conclude the existence of a solution of the original equation in this interval vector. Under different assumptions we can construct an interval vector such that the range of the mapping has no point in common with this interval vector. This implies that there is no solution in this interval vector. Furthermore we consider an iteration method which improves componentwise errorbounds for a solution of a quadratic. The theoretical results of this paper are demonstrated by some numerical examples using the algebraic eigenvalue problem which is probably the best known example of a quadratic equation.This paper contains the main results of a talk given by the author on the occasion of the 25th anniversary of the founding of Numerische Mathematik, March 19–21, 1984 at the Technische Universität of Munich, Germany     

Scalar discrete nonlinear multipoint boundary value problems

In this paper we provide sufficient conditions for the existence of solutions to scalar discrete nonlinear multipoint boundary value problems. By allowing more general boundary conditions and by imposing less restrictions on the nonlinearities, we obtain results that extend previous work in the area of discrete boundary value problems [Debra L. Etheridge, Jesús Rodriguez, Periodic solutions of nonlinear discrete-time systems, Appl. Anal. 62 (1996) 119–137; Debra L. Etheridge, Jesús Rodriguez, Scalar discrete nonlinear two-point boundary value problems, J. Difference Equ. Appl. 4 (1998) 127–144].     

Global classical solution of quasi-stationary Stefan free boundary problem

We consider an one-phase quasi-stationary Stefan problem (Hele–Shaw problem) in multidimensional case. Under some reasonable conditions we prove that the problem has a classical solution globally in time. The method can be used in two-phase problem as well. We also discuss asymptotic behavior of solution as t→+∞. The method developed here can be extended to a general class of free boundary problems.     

Global existence and long-time asymptotics for rotating fluids in a 3D layer

The Navier–Stokes–Coriolis system is a simple model for rotating fluids, which allows to study the influence of the Coriolis force on the dynamics of three-dimensional flows. In this paper, we consider the NSC system in an infinite three-dimensional layer delimited by two horizontal planes, with periodic boundary conditions in the vertical direction. If the angular velocity parameter is sufficiently large, depending on the initial data, we prove the existence of global, infinite-energy solutions with nonzero circulation number. We also show that these solutions converge toward two-dimensional Lamb–Oseen vortices as t→∞.     

Some quasi-static problems for a viscoelastic half-space

Some problems for a viscoelastic half-space are solved in the case of noncommutative operators. A solution of the equilibrium equation analogous to the Boussinesq-Papkovich solution is constructed. The problem of a normal pressure acting on the boundary of a viscoelastic half-space is solved. Two forms of this solution are obtained and both are used in the following problems, the problem of a concentrated load moving over the boundary of a half-space and the problem of a circular rigid stamp. The case of periodic motion of a periodic load is investigated with reference to the example of motion in a circle. At constant Poisson's ratio the solution of the problem of a stamp can be used for determining the creep or relaxation function.Mekhanika Polimerov, Vol. 2, No. 3, pp. 392–402, 1966Presented 12 November 1965 at the Riga Conference on Polymer Mechanics.     

Solution of nonstationary boundary layer problems by fastest descent method

We consider the formation of a boundary layer above a semi-infinite plate that moves with velocity V_t8=At ⁿ in a viscous incompressible fluid. The problem is solved by Meksyn's asymptotic method. First, second, and third approximation equations are obtained. A numerical calculation is performed for various n, and especially interesting results are obtained for –0.5 < n < 0. The numerical solution of the third approximation equation is consistent with other known results.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 51–55, 1988.     

Method of solving quasistatic boundary value problems of viscoelasticity for media with time-varying properties

It is shown that, if certain conditions are satisfied, the solution of a viscoelastic mixed boundary value problem can be obtained from the solution of the corresponding elastic problem by substituting time operators for the body and surface forces or given displacements. A similar assertion is also proved for media with time-varying properties. The applicability of the method to the theory of shallow flexible viscoelastic shells is demonstrated.Moscow Lomonosov State University. Translated from Mekhanika Polimerov, No. 6, pp. 987–993, November–December, 1969.     

Construction of the solution of a mixed boundary-value problem for mechanothermodiffusion for layered bodies of canonical shape

We propose a method of constructing the solution of the coupled problem of mechanothermodiffusion for layered bodies of canonical shape (plate, sphere, cylinder). By using the known functional transformation and the Papkovich-Neiber representation for displacements, and introducing unknown functions of time into the boundary conditions, we carry out a separation of the coupled system of equations as well as the boundary conditions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 70–75.