A geometrically exact viscoplastic membrane-shell with viscoelastic transverse shear resistance avoiding degeneracy in the thin-shell limit. Part I: The viscoelastic membrane-plate

We reduce a viscoelastic finite-strain continuum model to a two-dimensional membrane-plate. The reduction is based on assumed kinematics, analytical integration through the thickness and physically motivated simplifications. The resulting formulation is observer-invariant and accounts for thickness stretch and finite rotations.The membrane energy is a quadratic, uniformly Legendre-Hadamard elliptic, first order energy in contrast to classical membrane models and the corresponding system of balance equations remains of second order. An evolution equation for some independent rotation is appended (already in the bulk-model) introducing viscoelastic transverse shear resistance. It can be shown that this reduced membrane formulation is locally well-posed. Use is made of a dimensionally reduced version of an extended Korns first inequality.     

Variant of the geometrically nonlinear theory of anisotropic shells with allowance for transverse shear

A very simple variant of the geometrically nonlinear theory of anisotropic shells with allowance for the high compliance of the material in transverse shear is proposed. From this theory there follow, as a special case, the equations for an isotropic shell; these differ from the relations of [2] with respect to terms of the order of the ratio of the thickness of the shell to the radii of curvature small as compared with unity. The equations obtained are used to solve the problem of the stability of orthotropic shells of revolution relative to the starting axisymmetric state of stress.Translated from Mekhanika Polimerov, No. 5, pp. 863–871, September–October, 1969.     

A variational principle in a geometrically nonlinear theory of heterogeneous viscoelastic shells

The use of the hereditary theory for shells heterogeneous across their thickness is considered. A variational method is formulated for calculating thin anisotropic shells made of a material whose deformation behavior can be described by relations of the linear theory of viscoelasticity. In order to transform the corresponding functional into a form suitable for shells, some assumptions related to concepts of the theory of thin shells are introduced. In the capacity of Euler equations, physical relations, nonlinear equilibrium equations, and nonlinear boundary conditions are derived. The state equations are deduced for a multilayered shell. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 231–240, March–April, 2009.     

On one approach to studying the stress-strain state of nonlinear viscoelastic shells and plates with regard for transverse shear

Based on the theory of nonlinear viscoelasticity, we have constructed a computational scheme for determining the stress-strain state of thin taking into account transverse shells and plates with regard for transverse shear and normal stress.     

A note on: “Relaxation oscillators with exact limit cycles”

In this note we give a family of planar polynomial differential systems with a prescribed hyperbolic limit cycle. This family constitutes a corrected and wider version of an example given in the work [M.A. Abdelkader, Relaxation oscillators with exact limit cycles, J. Math. Anal. Appl. 218 (1998) 308-312]. The result given in this note may be used to construct models of Liénard differential equations exhibiting a desired limit cycle.     

Experimental investigation of the transverse shear effect in the bending of cantilevered plates with low shear stiffness

The state of stress of cantilevered plates made of different materials (steel and glass-reinforced plastic) has been experimentally investigated. The results obtained are compared with the known theoretical solutions. It is shown that in certain cases it is necessary to take into account the effects associated with the low shear stiffness of the plate.Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov; Kamenets-Podol'skii General Engineering Department of Khmel'nitskii Technological Institute. Translated from Mekhanika Polimerov, No. 4, pp. 736–738, July–August, 1971.     

Analysis of the secondary anisotropic creep of layered metal-composite plates with account of their weakened resistance to the transverse shear 1. Structural models

Iterative models describing the mechanical behavior of layered metal-composite media of regular structure whose constituent materials operate under the conditions of secondary anisotropic creep are offered. The modeling is carried out both with and without account of inelastic dilatation of the constituent materials.     

On the rate of convergence of sequential quadratic programming with nondifferentiable exact penalty function in the presence of constraint degeneracy

We analyze the convergence of a sequential quadratic programming (SQP) method for nonlinear programming for the case in which the Jacobian of the active constraints is rank deficient at the solution and/or strict complementarity does not hold for some or any feasible Lagrange multipliers. We use a nondifferentiable exact penalty function, and we prove that the sequence generated by an SQP using a line search is locally R-linearly convergent if the matrix of the quadratic program is positive definite and constant over iterations, provided that the Mangasarian-Fromovitz constraint qualification and some second-order sufficiency conditions hold. Received: April 28, 1998 / Accepted: June 28, 2001?Published online April 12, 2002     

The transmission problem in bending of plates with transverse shear deformation

The existence of weak solutions is proved for contact problemsin bending of elastic plates with transverse shear deformation,and for various boundary integral equations arising from potentialrepresentations of such solutions.     

On subcritical development of a shear crack in a composite with viscoelastic components

We present results of an investigation of the development of a transverse shear crack in a composite material with linearly viscoelastic components under external shear load. The solution is divided into the following two main stages: determination of the time dependence of the crack tip opening displacement and determination of the crack-growth kinetics as a result of the solution of integral equations. In the first stage, we use the solution of the corresponding elastic problem of determination of the crack opening displacement and the problem of determination of the effective moduli of the composite reinforced with unidirectional discrete fibers. Using the theoretically proved principle of elasto-viscoelastic analogy and the method of Laplace inverse transformation, we obtain a solution in a time domain. In the second stage, using the criterion of critical crack opening displacement for a transverse shear crack and an equation for the viscoelastic crack opening displacement of this crack, we construct an equation of crack growth. We present results of the numerical solution, which illustrate the influence of relations between the relaxation parameters of the materials of the components on the durability of the body with a crack.     

On the theoretical strength limit of the layered elastic and viscoelastic composites in compression

The theoretical strength limit in compression (TSLC) of composite material with two alternating isotropic homogeneous elastic and viscoelastic layers is studied. The investigation is made within the piecewise-homogeneous body model with the use of the Three-Dimensional Linearized Theory of Stability for elastic and viscoelastic materials. It is assumed that there is an initial imperfection in the local curving form of the reinforcing layers along the direction which is inclined to the layers. As the criterion for determination of the TSLC, the case is taken for which the aforementioned imperfections start to increase indefinitely. The numerical results for the TSLC are presented and discussed. According to these numerical results, in particular, the direction (or the surface) along which the fracture in compression takes place, is determined.     

On the bending of plates with transverse shear deformation and mixed periodic boundary conditions

The method of matched asymptotic expansions is used to find a homogenized problem whose solution is an approximation to the solution of a mixed periodic boundary value problem in the theory of bending of thin elastic plates. A critical size for the fixed parts of the boundary is found such that the boundary condition of the homogenized problem is an intermediate case between that for the clamped edge plate and that for the free boundary plate.     

Stiffness matrix of the finite element of a plate compliant in transverse shear

A four-node rectangular finite element is elaborated for a plate calculated with account of transverse shear strains. On variational realization of the finite-element method, the stiffness matrix and the vector of equivalent nodal forces are obtained for the case of a surface load. The basic nodal kinematic parameters adopted also include the angles of transverse shear.     

A note on the Stokes resistance tensors of multiparticle systems in shear fields

The Stokes resistance tensors (material tensors) of a multiparticle system subjected to a general shear flow depends on the spatial configuration of the particles, their shape, size and location of some arbitrarily fixed set of points locked to the particles by which their linear velocities are traced. The objective of this work is the determination of some symmetry properties of the material tensors and the transformation laws which govern them under translation of the set of points fixed to the particles.     

Experimental determination of the transverse shear modulus of a wound glass-reinforced plastic in creep

It is proposed to determine the interlayer shear modulus in creep starting from the results of bending tests on simply supported beams with a concentrated load. It is assumed that the increase in deflection with time is caused by interlayer creep. The data of creep tests on wound glass-reinforced plastic beams are presented and used to determine the long-time interlayer shear modulus.Leningrad. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 370–372, March–April, 1969.