Mathematical construction of a Reissner–Mindlin plate theory for composite laminates

A Reissner–Mindlin theory for composite laminates without invoking ad hoc kinematic assumptions is constructed using the variational-asymptotic method. Instead of assuming a priori the distribution of three-dimensional displacements in terms of two-dimensional plate displacements as what is usually done in typical plate theories, an exact intrinsic formulation has been achieved by introducing unknown three-dimensional warping functions. Then the variational-asymptotic method is applied to systematically decouple the original three-dimensional problem into a one-dimensional through-the-thickness analysis and a two-dimensional plate analysis. The resulting theory is an equivalent single-layer Reissner–Mindlin theory with an excellent accuracy comparable to that of higher-order, layer-wise theories. The present work is extended from the previous theory developed by the writer and his co-workers with several sizable contributions: (a) six more constants (33 in total) are introduced to allow maximum freedom to transform the asymptotically correct energy into a Reissner–Mindlin model; (b) the semi-definite programming technique is used to seek the optimum Reissner–Mindlin model. Furthermore, it is proved the first time that the recovered three-dimensional quantities exactly satisfy the continuity conditions on the interface between different layers and traction boundary conditions on the bottom and top surfaces. It is also shown that two of the equilibrium equations of three-dimensional elasticity can be satisfied asymptotically, and the third one can be satisfied approximately so that the difference between the Reissner–Mindlin model and the second-order asymptotical model can be minimized. Numerical examples are presented to compare with the exact solution as well as the classical lamination theory and the first-order shear-deformation theory, demonstrating that the present theory has an excellent agreement with the exact solution.     

Three-dimensional stability theory of deformed bodies. Internal instability

Conclusions In studying internal instability effects for elastic (which is fully obvious) and elastoplastic models of deformable bodies the approximate approach [12, 15] in the three-dimensional stability theory leads to results which disagree quantitatively and qualitatively with the corresponding results of the three-dimensional linearzed stability theory of deformable bodies (the second variant of the theory of small subcritical deformations). In this connection, in studying internal instability effects for various models of deformable bodies, in which elastic or elastoplastic deformations are substantial, the use of this approximate approach is recommended.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 21, No. 11, pp. 3–17, November, 1985.     

Stability of a Laminate Between Two Homogeneous Half-Spaces under Inelastic Deformation

The surface instability of inelastic laminated coatings is analyzed using a piecewise-homogeneous model and the three-dimensional linearized theory of stability. The general formulation of the problem is given, and the characteristic equations are derived. Specific problems for some viscoelastic and elastoplastic models are solved__________Translated from Prikladnaya Mekhanika,Vol. 41, No. 5, pp. 22–31, May 2005.     

Similarity of three-dimensional and axisymmetric chemically-nonequilibrium flows in the neighborhood of a plane of symmetry

Three-dimensional chemically-nonequilibrium flow past blunt bodies in the neighborhood of the plane of symmetry is investigated within the framework of viscous shock layer theory. The similarity of three-dimensional and axisymmetric flows, previously established in [1] for a uniform gas, is extended to chemically-nonequilibrium gas flows. It is shown that the problem of determining the heat fluxes and friction stress in the neighborhood of the line of flow divergence can be reduced to the problem of determining these quantities for the axisymmetric body. The validity of the axisymmetric analogy is verified by carrying out numerical calculations for bodies of different shapes re-entering the earth's atmosphere along a gliding trajectory. Various models of surface catalytic activity are considered. The use of similarity relations makes it possible to apply existing programs for calculating axisymmetric flows to the solution of three-dimensional problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 115–120, March–April, 1990.     

Stability of Two Different Half-Planes in Compression Along Interfacial Cracks: Analytical Solutions

The exact solutions of the stability problem for two different half-planes compressed along the cracked interface are considered within the framework of the three-dimensional linearized theory of stability of deformable bodies. The exact analytical solutions are constructed in a form common for finite (large) and small strains as applied to compressible and incompressible, isotropic and orthotropic, and elastic and plastic models. The solutions are derived using complex potentials of the above-mentioned theory and the Riemann–Hilbert problem methods. Mechanical effects are analyzed. This article is a complete report read at the ICTAM 2000 (Chicago, USA). An abstract was included in the ICTAM-2000 Abstract Book     

Instability due to three-dimensional disturbances

In the linear theory of the stability of parallel flows of a viscous fluid, most attention is usually given to plane-wave disturbances. The reason is the validity in many cases of the Squire theorem, which states that the critical Reynolds number R is determined by two-dimensional disturbances [1]. It is shown in the present paper that for large R the region generating the turbulence in the initial stage of its development is formed by three-dimensional disturbances. This feature applies both to the generating range of wave numbers and the dimension of the wall layer, where the fluctuating energy is produced. The consequences of the Squire transformations for parallel flows are analyzed. The contribution of resonant nonlinear triad coupling to the rapid growth of fluctuating energy is studied for the case of an explosive instability in an extended laminar mode. It is shown that the rate of turbulent energy production is not governed by the small derivatives of linear theory, but by nonlinear triad coupling of neutral and growing disturbances, with their three-dimensional nature playing an important role.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 29–34, September–October, 1976.The author thanks M. A. Gol'dshtik for his interest in the work and for discussion of the results.     

Free discontinuity finite element models in two-dimensions for in-plane crack problems

Two different free discontinuity finite element models for studying crack initiation and propagation in 2D elastic problems are presented. Minimization of an energy functional, composed of bulk and surface terms, is adopted to search for the displacement field and the crack pattern. Adaptive triangulations and embedded or r-adaptive discontinuities are employed. Cracks are allowed to nucleate, propagate, and branch. In order to eliminate rank-deficiency and perform local minimization, a vanishing viscosity regularization of the discrete Euler–Lagrange equations is enforced. Converge properties of the proposed models are examined using arguments of the Γ-convergence theory. Numerical results for an in-plane crack kinking problem illustrate the main operational features of the free discontinuity approach.     

Three-dimensional steady flows of stratified fluid and internal waves

A study is made of three-dimensional steady flows of an ideal heavy incompressible fluid stratified in each layer over a flat or asymptotically flat base. Mixed Euler-Lagrange variables are chosen in which surfaces of constant density, including the layer division boundaries, become flat and parallel to the plane of the base. The original problem is reduced to a nonlinear boundary-value problem for a system of three quasilinear equations in a plane layer. This system of equations is used to construct an asymptotic theory of long waves in the three-dimensional case, which has particular solutions in the first approximation in the form of solitons and soliton systems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 127–132, May–June, 1985.     

Surface wave generation in the process of finite deformation of the bottom of a basin

The process of generation of three-dimensional irrotational fluid motions induced by small local finite-duration displacements of part of the bottom of a basin is considered within the framework of wave linear theory for a basin of constant depth. The solution of the problem and an expression for the total wave field energy are obtained using integral transforms. The general properties of the process of unsteady wave generation induced by short-term and slow deformations of the bottom are analyzed. Within the framework of the piston generation model the energy characteristics of axisymmetric waves are compared for two time laws of bottom deformation of identical duration. In general, it is shown that under certain conditions the nature and intensity of the wave process depend on both the time law and the duration of the deformation process.Sevastopol. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 151–156, March–April, 1996.     

Stability of Hartmann flows

The stability of Hartmann flows for arbitrary magnetic Reynolds numbers is investigated in the framework of linear theory. The initial three-dimensional problem reduces to the equivalent two-dimensional problem. Perturbation theory is used to find asymptotic expressions for the eigenvalues. Distinguishing two types of disturbances — magnetic and hydrodynamic — is shown to be advantageous in a number of cases. Simple features of the stability are considered for particular cases. The well-know Lundquist result is generalized. An energy approach is applied to the problem of stability. The results of simulations involving the solution of the linear stability problem are described. A distinctive picture of stability is developed. There are several types of instability and they can develop simultaneously. The hydrodynamic and magnetic phenomena interact with each other in a very complex fashion. The magnetic field can either enhance flow stability or reduce it.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–31, November–December, 1972.     

Boussinesq–Flamant problem in gradient elasticity with surface energy

The strain gradient elasticity theory with surface energy is applied to Boussinesq–Flamant problem. The solution for the vertical displacements at the surface of half space due to the surface normal line load is presented. The theory includes both volumetric and surface energy terms. Boussinesq–Flamant problem in the strain gradient elasticity is solved by means of Fourier transform. The results obtained show that the vertical displacements of half space in the gradient elasticity are some different from that in the classical elasticity and the effects of the strain gradient parameters (material characteristic lengths) on the vertical displacements do exist.     

Cross-Properties Relations in 3D Percolation Networks: II. Network Permeability

An efficient method to estimate the absolute permeability of three-dimensional percolation networks was proposed. It uses a Kozeny–Carman relationship in the form of a scaling law to relate the network permeability to its hydraulic characteristic length. This characteristic length was determined at the network percolation threshold using a three-dimensional extension of the Hoshen–Kopelman algorithm. For developing the scaling laws, the network permeability was calculated by solving the Kirchoff’s law for all sample spanning clusters that had been identified by the three-dimensional version of the Hoshen–Kopelman algorithm. The method was tested with simple cubic site-bond network models with and without spatial correlations. The universality of the exponents in the scaling laws were also investigated. It was shown that, once the scaling law has been derived, the permeability value can be estimated 3–9 times faster using the present method.     

Plane Problem of Three-Dimensional Stability for a Hinged Plate with Two Symmetric End Cracks

The plane stability problem for a rectangular plate with two symmetric end cracks is solved in three-dimensional formulation. The three-dimensional linearized theory of stability and the finite-difference method are used. The effect of the crack parameter on the critical load is examined__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 47–52, April 2005.     

Complete model of the process of propagation of long waves and their interaction with a vertical wall

Asymptotic analysis of the nonlinear three-dimensional boundary-value problem of potential theory is carried out and a complete system of equations describing the process of propagation of long surface waves is obtained. Approximate solutions of the problems for both traveling and standing waves are constructed to the third approximation.Translated by Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 173–176, May–June, 1985.     

Quasi-three-dimensional calculation of flow in blade passages of turbines

The flow in turbomachines is currently calculated either on the basis of a single successive solution of an axisymmetric problem (see, for example, [1-A]) and the problem of flow past cascades of blades in a layer of variable thickness [1, 5], or by solution of a quasi-three-dimensional problem [6–8], or on the basis of three-dimensional models of the motion [9–11]. In this paper, we derive equations of a three-dimensional model of the flow of an ideal incompressible fluid for an arbitrary curvilinear system of coordinates based on averaging the equations of motion in the Gromek–Lamb form in the azimuthal direction; the pulsation terms are taken into account in the equations of the quasi-three-dimensional motion. An algorithm for numerical solution of the problem is described. The results of calculations are given and compared with experimental data for flows in the blade passages of an axial pump and a rotating-blade turbine. The obtained results are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 69–76, March–April, 1991.I thank A. I. Kuzin and A. V. Gol'din for supplying the results of the experimental investigations.     

Conical wing of maximum lift-drag ratio in a supersonic gas flow

The calculation of supersonic flow past three-dimensional bodies and wings presents an extremely complicated problem, whose solution is made still more difficult in the case of a search for optimum aerodynamic shapes. These difficulties made it necessary to simplify the variational problems and to use the simplest dependences, such as, for example, the Newton formula [1–3]. But even in such a formulation it is only possible to obtain an analytic solution if there are stringent constraints on the thickness of the body, and this reduces the three-dimensional problem for the shape of a wing to a two-dimensional problem for the shape of a longitudinal profile. The use of more complicated flow models requires the restriction of the class of considered configurations. In particular, paper [4] shows that at hypersonic flight velocities a wing whose windward surface is concave can have the maximum lift-drag ratio. The problem of a V-shaped wing of maximum lift-drag ratio is also of interest in the supersonic velocity range, where the results of the linear theory of [5] or the approximate dependences of the type of [6] can be used.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 128–133, May–June, 1986.We note in conclusion that this analysis is valid for those flow regimes for which there are no internal shock waves in the shock layer near the windward side of the wing.     

Plane nonisothermal poiseuille flow for four molecular models

The method of three-dimensional moments in the third approximation is used to solve the problem of plane nonisothermal Poiseuille flow. Flows due to temperature and pressure gradients, as well as the rate of thermal slip, are obtained for four molecular models. A comparison is made with the results of other authors.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 182–183, July–August, 1973.     

Analysis of the three-dimensional turbulent boundary layer on the suface of an arbitrary body at large angles of attack

Three-dimensional compressible gas flow past an arbitrary model body at large angles of attack is analyzed in the framework of the boundary layer theory with allowance for heat transfer. The equations of a three-dimensional turbulent boundary layer are solved using computer codes, the data on the external inviscid flow, and the body geometry.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 55–66, May–June, 1995.     

The nonlinear evolution of two-dimensional and three-dimensional waves in mixing layers

The authors consider problems connected with stability [1–3] and the nonlinear development of perturbations in a plane mixing layer [4–7]. Attention is principally given to the problem of the nonlinear interaction of two-dimensional and three-dimensional perturbations [6, 7], and also to developing the corresponding method of numerical analysis based on the application to problems in the theory of hydrodynamic stability of the Bubnov—Galerkin method [8–14].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhldkosti i Gaza, No. 1, pp. 10–18, January–February, 1985.     

Development of the Boundary-Element Method for Three-Dimensional Problems of Static and Nonstationary Elasticity

The boundary-element method (BEM) applied to three-dimensional problems in the linear theory of elasticity is analyzed. The solutions of test problems for spherical and cubic cavities are used as examples to consider the basic aspects and difficulties of applying the traditional BEM to static and nonstationary three-dimensional problems. It is established that using Chebyshev polynomials in the Gaussian quadrature formula to evaluate the singular segments of surface integrals reduces the computation time by a factor of 2 to 3 without loss of accuracy compared with the traditional Gauss–Legendre formula. BEM-based approaches are proposed to solve three-dimensional problems in the linear theory of elasticity