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.     

Non-Darcian Effects on the Onset of Convection in a Porous Layer with Throughflow

The Brinkman extended Darcy model including Lapwood and Forchheimer inertia terms with fluid viscosity being different from effective viscosity is employed to investigate the effect of vertical throughflow on thermal convective instabilities in a porous layer. Three different types of boundary conditions (free–free, rigid–rigid and rigid–free) are considered which are either conducting or insulating to temperature perturbations. The Galerkin method is used to calculate the critical Rayleigh numbers for conducting boundaries, while closed form solutions are achieved for insulating boundaries. The relative importance of inertial resistance on convective instabilities is investigated in detail. In the case of rigid–free boundaries, it is found that throughflow is destabilizing depending on the choice of physical parameters and the model used. Further, it is noted that an increase in viscosity ratio delays the onset of convection. Standard results are also obtained as particular cases from the general model presented here.     

Elastoplastic Deformation of a Compound Disk under Impulsive Loading

The small elastoplastic deformation theory and the finite-element method are used to analyze the behavior of a compound disk under axisymmetric impulsive loading. Numerical results are presented, which describe the development of plastic deformation, the effect of hardening and the duration of the loading impulse on the oscillatory elastoplastic deformation of the disk__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 94–99, April 2005.     

Micro–macro approach for the rock salt behaviour

Rock salt is considered as a pure aggregate of halite (mineral NaCl) crystals and its behaviour is investigated by a micro–macro approach. The behaviour of the polycrystalline aggregate is deduced from the properties of the constituent halite crystals. A model for the elastoplastic behaviour of halite crystal has been deduced from experimental data available in the literature. The basic equations of the micro–macro model for the polycrystalline medium and the calculation method are then presented and the elastoplastic behaviour of rock salt is investigated by this method. The hardening effects obtained for the polycrystal are found to be very different from those obtained for FCC metal polycrystals. The differences are explained as a consequence of differences of families of glide systems in these crystals. Finally, the internal stresses in the polycrystal are studied in order to elucidate the origin of cracking and damage of the rock salt.     

On Variational Approaches to Plate Modes

The three basic functionals of potential energy, complementary energy and Hellinger–Prange–Reissner are usedto obtain a rational derivation of Reissner–Mindlinplate models, starting from the three-dimensional theory.We show that the models so obtained are instances of the same plate theory; nevertheless, due to the different constitutive relations governing their response, they mimic the three-dimensional behaviour in three different manners.     

Dynamics of unidirectional glass-fiber-reinforced plastic

The motion problem of unidirectional glass-fiber-reinforced plastic is formulated under the assumption that the fibers are under stress-strain only, while the binder is under shear stress only. The binder and fiber inertia is calculated along a direction parallel to the fibers. The system of equations in partial derivatives obtained is reduced by Laplace transformation with respect to time to a system of ordinary differential equations in which only the fiber displacements occur. As illustration, the effect of a normal stress wave on a half space is solved. The solution is obtained in the form of an infinite series provided with an explicit law by which the terms are obtained. Curves are presented for the distribution of the normal and shearing stresses at different moments of time. The binder inertia reduces to the appearance of tangential stresses at the fiber-binder boundary, which can explain the tendency towards stratification in constructions made of glass fiber-reinforced plastic.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 139–145, July–August, 1974.     

Magma flow through elastic-walled dikes

A convection–diffusion model for the averaged flow of a viscous, incompressible magma through an elastic medium is considered. The magma flows through a dike from a magma reservoir to the Earth’s surface; only changes in dike width and velocity over large vertical length scales relative to the characteristic dike width are considered. The model emerges when nonlinear inertia terms in the momentum equation are neglected in a viscous, low-speed approximation of a magma flow model coupled to the elastic response of the rock.Stationary- and traveling-wave solutions are presented in which a Dirichlet condition is used at the magma chamber; and either a (i) free-boundary condition, (ii) Dirichlet condition, or (iii) choked-flow condition is used at the moving free or fixed-top boundary. A choked-flow boundary condition, generally used in the coupled elastic wave and magma flow model, is also used in the convection–diffusion model. The validity of this choked-flow condition is illustrated by comparing stationary flow solutions of the convection–diffusion and coupled elastic wave and magma flow model for parameter values estimated for the Tolbachik volcano region in Kamchatka, Russia. These free- and fixed-boundary solutions are subsequently explored in a conservative, local discontinuous Galerkin finite-element discretization. This method is advantageous for the accurate implementation of the choked flow and free-boundary conditions. It uses a mixed Eulerian–Lagrangian finite element with special infinite curvature basis function near the free boundary and ensures positivity of the mean aperture subject to a time-step restriction. We illustrate the model further by simulating magma flow through host rock of variable density, and magma flow that is quasi-periodic due to the growth and collapse of a lava dome.     

Nonaxisymmetric Thermoelastoplastic Analysis of Branched Shells of Revolution by the Semianalytic Finite-Element Method

A technique is proposed to solve elastoplastic deformation problems for branched shells of revolution under the action of asymmetric forces and a temperature field. The kinematic equations are derived within the framework of the linear Kirchhoff–Love theory of shells and the thermoelastic relations within the framework of the theory of small elastoplastic strains. The problem is given a variational formulation based on the virtual-displacement principle and the Fourier-series expansion of the unknown functions and loads with respect to the circumferential coordinate. The additional-load method is used to solve a nonlinear problem and the finite-elements method is used to carry out a numerical analysis. As an example, an asymmetric stress–strain analysis is performed for a cylindrical shell reinforced by a ring plate.     

Study of the Elastoplastic Axisymmetric Stress–Strain State of Irradiated Laminated Shells with a Loading History

A technique is proposed to determine the axisymmetric state of laminated shells under thermoradiation loading. The deformation is assumed simple. The axisymmetric elastoplastic stress–strain state of a two-layered continuous plate under a distributed load and a neutron flux incident on one of the plate surfaces is studied within the framework of the Kirchhoff–Love hypothesis for a layer stack. The results obtained with and without regard for the loading history are presented. An important role of the loading history in a stress analysis is confirmed     

Response of flexure–torsion coupled composite thin-walled beams with closed cross-sections to random loads

A study of the flexure–torsion coupled random response of the composite beams with solid or thin-walled closed-sections subjected to various types of concentrated and distributed random excitations is dealt with in this paper. The effects of flexure–torsion coupling, shear deformation and rotary inertia are included in the present formulations. The random excitations are assumed to be stationary, ergodic and Gaussian. Analytical expressions for the displacement response of the composite beams are obtained by using normal mode superposition method combined with frequency response function method. The present method can produce the effective solutions for the composite Timoshenko beams with circumferentially antisymmetric (CAS) configuration and more general beam assemblages of connected beams. The influences of flexure–torsion coupling, shear deformation and rotary inertia on the random response of an appropriately chosen composite beam from the literature are demonstrated and discussed.     

Simulation of three-point bending at large deflections by an elastoplastic contact analysis

Three-point bending is simulated by an elaborate numerical procedure based on an elastoplastic, large deflection, contact analysis. A minimization formulation is used, which is equivalent to the incremental form posed as partial differential equations with inequalities. A sequential quadratic programming approach based on the finite-element technique is adopted as a method of solution. To examine the validity of the simulation method, experiments are carried out for specimens that have various widths.Paper was presented at the 1988 SEM Spring Conference on Experimental Mechanics held in Portland, OR on June 5–10.     

The axisymmetric contact problem of the theory of elasticity for a half-space in the presence of body forces

We consider the axisymmetric problem of determination of the stress-strain state in an elastic half-space in the case of a circular line of separation of the boundary conditions on the boundary plane z = 0. We assume that on the entire boundary z = 0 the tangential stress _rz = 0, while inside the circle r a (z = 0) the normal displacement u_z is known and in its exterior the normal stress _z is given. In addition, we assume that body forces are acting in the half-space. The investigation of problems of similar kind presents interest in connection with the application of A. A. Il'yushin's method of elastic solutions to the problem of the indentation of punches into a nonlinear-elastic, in particular, into an elastoplastic half-space.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 110–115, July–August, 1971.The author wishes to thank M. Ya. Leonov for his valuable suggestions during the preparation of this paper.     

Static flexural analysis of elliptic Reissner–Mindlin plates on a Pasternak foundation by the triangular differential quadrature method

In this paper, static flexural analysis of elliptic Reissner–Mindlin plates resting on a Pasternak foundation is conducted using the triangular differential quadrature method (TDQM). A triangular serendipity transformation is proposed to map a curvilinear domain onto a unit isosceles right triangle. The applicability of the TDQM to problems on domains with curvilinear boundaries is improved significantly since no inner points are needed in the transformation. In the case of thin circular plates, the results of the present method are in excellent agreement with those of the available exact solution, demonstrating the effectiveness of the present method. Elliptic plates with various aspect ratios and thickness-to-width ratios under uniformly distributed load are studied.     

The Elastoplastic Axisymmetric Stress–Strain State of Flexible Laminated Shells Exposed to Radiation

A procedure is proposed to numerically study the thermoelastoplastic axisymmetric stress–strain state of laminated flexible shells exposed to radiation. The equations of thermoradiation plasticity describing simple processes are used. Results of an analysis of the elastoplastic state of a three-layer shell with regard for radiation effects are presented     

Subsonic gas-liquid cavitation flow past a disk

The problem of axisymmetric subsonic gas-liquid cavitation flow past a disk in accordance with the Riabouchinsky scheme is solved using the method of [1]. Formulas relating the main flow parameters with the cavitation number, the Mach number on the free boundary and the gas/liquid volume ratio under stagnation conditions are presented.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 202–206, March–April, 1996.     

Boundary layer of a body of revolution in a drag-reducing polymer solution

A possible method is presented for calculating an axisymmetric boundary layer when a body moves in weak polymer solutions with constant concentration. The method is based on the use of a velocity profile and system of integral equations which take into account most fully the effects of the transverse curvature of the body's surface. The computational scheme makes it possible to take into account the change in the flow regimes in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 40–48, May–June, 1981.     

Method of solution of the hypersonic boundary layer equations in the case of strong viscous-inviscid interaction

A method of solving the boundary layer equations is developed taking into account the strong interaction between the boundary layer and the outer hypersonic inviscid flow. The method is aimed at solving problems whose salient feature is the possible upstream propagation of disturbances over distances comparable with the body length. The procedure for fitting a self-consistent contour of the effective body using an artificially formulated boundary value problem for an ordinary second-order differential equation, which lies at the basis of the method, is considered in detail. The method is applied to the problem of flow around a flat plate with roughness in the form of an embankment or a trench; the calculated results are presented.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 81–89, July–August, 1995.     

Identification of the Ten Inertia Parameters of a Rigid Body

Standard experiments for identifying inertia parameters of a rigid body only provide a subset of the inertia parameters of the body [1–10]. In addition, they do not use in the estimation process the complete information included in the equations of motion of the rigid test body. The objective of the work described in this paper is the simultaneous, automatic experimental identification of the ten inertia parameters of a rigid body using the complete information hidden in the nonlinear model equations of the test body. This task has been solved in several steps:– mathematical modelling of the special motions of a rigid body in space. These model equations have been mapped into a form suitable for identification purposes (identification hypothesis)– design of a special measurement robot for performing the identification experiments– laboratory experiments providing test data used for the identification experiments– identification of the inertia parameters and accuracy tests.The accuracy of the identified parameters is satisfactory.     

Characteristic-value analysis for plastic dynamic buckling of columns under elastoplastic compression waves

The characteristic-value analysis of plastic dynamic buckling is presented for columns under the action of elastoplastic compression wave caused by an axial-step load. Two critical conditions constituting a dynamic instability criterion are derived on the basis of transformation and conservation of energy. The governing equations, the boundary conditions and the continuity conditions derived by the use of the first critical condition are the same as those given by the adjacent-equilibrium criterion and are insufficient for determining two characteristic parameters involved in the governing equations. A supplementary restraint equation for buckling deformations at the plastic-wave front and the elastic-wave front is derived by the use of the second critical condition. Then, a couple of characteristic equations for two characteristic parameters are derived on the condition that the governing equations have non-trivial solutions satisfying the boundary conditions, the continuity conditions and the supplementary restraint equation. The critical-load parameters, dynamic characteristic parameter (exponent parameter of inertia term) and dynamic buckling modes are calculated from the solutions of the characteristic equations.     

Elastic–plastic FE analysis of a notched shaft under multiaxial nonproportional synchronous cyclic loading

An elastic–plastic finite element analysis is presented for a notched shaft subjected to multiaxial nonproportional synchronous cyclic tension/torsion loading. The elastic–plastic material property is described by the von Mises yield criterion and the kinematic hardening rule of Prager/Ziegler. The finite element program system ABAQUS is used to solve the boundary value problem. Special emphasis is given to explore the effects of the stress amplitude, the mean-stress, and the mutual interactions on the local stress–strain responses at the notch root.