The small free oscillations of a strip

The refined equations of the free oscillations of a rod-strip, constructed previously in a first approximation by reducing the two-dimensional equations to one-dimensional equations by using trigonometric basis functions and satisfying the static boundary conditions on the boundary surfaces are analysed. These equations, the solutions of which are obtained for the case of hinge-supported end sections of the rod, are split into two independent systems of equations. The first of these describe non-classical fixed longitudinal-transverse forms of free oscillations, which are accompanied by a distortion of the plane form of the cross section. It is shown that the oscillation frequencies corresponding to them depend considerably on Poisson's ratio and the modulus of elasticity in the transverse direction, while for a rod of average thickness for the same value of the frequency parameter (the tone) they may be considerably lower than the frequencies corresponding to the classical longitudinal forms of free oscillations, which are performed while preserving the plane form of the cross sections. The second system of equations describes transverse flexural-shear forms of free oscillations, whose frequencies decrease as the transverse shear modulus decreases. They are practically equivalent in quality and content to the similar equations of well-known versions of the refined theories, but, unlike them, when the number of the tone increases and the relative thickness parameter decreases they lead to the solutions of the classical theory of rods.     

预测纳米纤维复合材料有效弹性性能的界面模型和界面相模型北大核心CSCD

基于广义自洽法,同时采用Gurtin-Murdoch界面模型和界面相模型研究了纳米纤维复合材料的有效弹性性能,获得了两种模型下有效体积模量的封闭解析解和计算有效面内剪切模量数值解的全部公式.基于界面模型的解答,讨论了有效体积模量和有效面内剪切模量的界面效应.证明了界面模型的解答可由界面相模型的解答退化得到,其中有效体积模量可以实现解析退化,有效面内剪切模量则可以数值退化.以含纳米孔洞的金属铝为例,比较了两种模型计算结果的差异.结果表明,当纳米孔洞半径较小时,两个模型的结果存在很大差异,而当半径较大时两个模型的结果差别不大.     

Non-uniformly scaled buckling modes of reinforcing elements in fiber reinforced plastic

We consider a problem about non-uniformly scaled buckling modes of isolated fiber (without accounting of interaction with the surrounding epoxy) or bundle of fibers, which are structural elements of fiber reinforced plastics under the transverse tension (compression) and shear stresses in prebuckling state. Such initial state is formed in fibers and bundles of fibers at tension-compression tests of flat specimens from cross ply composites with unidirectional fibers. For problem statement we use equations recently constructed by reduction of consistent version of geometrically nonlinear equations of theory of elasticity to one dimensional equations of rectilinear beams. Equations are based on refined shear S. P. Timoshenko model with accounting of tension-compression stresses in transverse directions. We give theoretical explanation of developed phenomenon as reducing shear modulus of elasticity of fiber reinforced plastic during the increasing of shear strains. We show that under the loading process of specimens under review uninterruptedly structure reconstruction of composite trough implementation and uninterruptedly changing of internal buckling modes at changing wave parameter is feasible.     

Shear Buckling Form of a Circular Sandwich Ring under Uniform External Pressure

The problem on the stability of a circular sandwich ring under uniform external pressure is considered. It is shown that, along with a mixed flexural-shear buckling form (BF), a pure shear BF can be realized in the core. This form is accompanied by rotation of the load-carrying layers at the cost of the transverse shear strain (constant in the circumferential direction) in the core. It is found that the simplified equations of the theory of shallow shells cannot describe this nonclassical BF. The critical loads corresponding to the shear BF may prove to be smaller than the critical load of the classical mixed flexural BF for a circular sandwich ring of medium thickness at a low shear modulus of the core. The results obtained contribute greatly to the understanding of buckling mechanisms of sandwich structures and supplement the existing classification of BFs.     

Coupling effect of thickness and shear deformation on size-dependent bending of micro/nano-scale porous beams

A unified nonlocal strain gradient beam model with the thickness effect is developed to investigate the static bending behavior of micro/nano-scale porous beams. Size-dependent governing equations and corresponding analytical solutions for the bending of hinged-hinged beams are obtained by employing minimum total potential energy principle, the Navier solution method as well as the variational-consistent boundary conditions. For nonlocal strain gradient theory (NSGT) with thickness effect, virtual strain energy function of shear beams can contain additional nonlocal shear stress and high-order nonlocal shear stress related to the thickness direction in comparison with that of Euler–Bernoulli beam, so the coupling of the shear and thickness effects should be drawn huge attention. By means of detailed numerical analysis, it is found that, the stiffness-hardening effect is underestimated in NSGT without the thickness effect, and the stiffness-hardening and stiffness-softening effects of NSGT with the thickness effect can be not only length-dependent but also thickness-dependent. Interestingly, the generalized Young’s modulus depends on half-wave number, which means that the generalized Young’s modulus may be different due to applied load types. In the context of NSGT with the thickness effect, the deflection of Euler–Bernoulli beam predicted is smaller than that of shear beam, especially for thick beams. Furthermore, porosities distributed in the top or bottom of beams can possess a greater influence on the decrease of overall stiffness of beam than those distributed in the vicinity of the middle plane of beams.     

Geometrically nonlinear bending analysis of laminated composite plate

In this work, a transverse bending of shear deformable laminated composite plates in Green–Lagrange sense accounting for the transverse shear and large rotations are presented. Governing equations are developed in the framework of higher order shear deformation theory. All higher order terms arising from nonlinear strain–displacement relations are included in the formulation. The present plate theory satisfies zero transverse shear strains conditions at the top and bottom surfaces of the plate in von-Karman sense. A C⁰ isoparametric finite element is developed for the present nonlinear model. Numerical results for the laminated composite plates of orthotropic materials with different system parameters and boundary conditions are found out. The results are also compared with those available in the literature. Some new results with different parameters are also presented.     

The effective shear modulus for an n-layered composite sphere

This work presents the derivation of the effective shear modulus for a heterogeneous material composed of multi-layered composite spheres embedded in a linear elastic matrix. It is based on the composite spheres model known from the literature. In contrast to previous publications the effective shear modulus is obtained by equating the results of two models: In the first model, a heterogeneous sphere is embedded in an equivalent homogeneous material, whereas in the second model, the heterogeneous sphere is replaced by an equivalent homogeneous sphere. In the context of both, a shear stress approach and a shear deformation approach, this results into an overdetermined system of equations which is solved with the least squares method. In a numerical study our results are compared to effective moduli and bounds from the literature. Furthermore, a convincing agreement with experimental data for glass microspheres embedded in a polyester matrix is demonstrated. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The indentation of a flat punch into an ideal rigid-plastic half-space under the action of shear contact stresses

A general plane problem of the impression of a flat punch into a rigid-plastic half-space under the action of transverse and longitudinal shear contact stresses is considered. The condition of complete plasticity and the hyperbolic equations of the general plane problem of the theory of ideal plasticity [1] are used. The reduction of the limit pressure on the punch is determined as a function of the shear contact stresses.     

The consistent equations of the theory of plane curvilinear rods for finite displacements and linearized problems of stability

Based on a previously constructed, consistent version of the geometrically non-linear equations of elasticity theory, for small deformations and arbitrary displacements, and a Timoshenko-type model taking into account transverse shear and compressive deformations, one-dimensional equations of an improved theory are derived for plane curvilinear rods of arbitrary type for arbitrary displacements and revolutions and with loading of the rods by follower and non-follower external forces. These equations are used to construct linearized equations of neutral equilibrium that enable all possible classical and non-classical forms of loss of stability (FLS) of rods of orthotropic material to be investigated, ignoring parametric deformation terms in the equations. These linearized equations are used to find accurate analytical solutions of the problem of plane classical flexural-shear and non-classical flexural-torsional FLS of a circular ring under the combined and separate action of a uniform external pressure and a compression in the radial direction by forces applied to both faces.     

Hyperbolic submodels of an incompressible viscoelastic Maxwell medium

A two-dimensional motion of an incompressible viscoelastic Maxwell continuum is considered. The system of quasilinear equations describing this motion has both real and complex characteristics. A class of effectively one-dimensionalmotions is analyzed for which the original system of equations is decomposed into a hyperbolic subsystem and a quadrature. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. When one chooses the Jaumann corotational derivative as the invariant derivative, the equations of the submodel remain quasilinear. They can be represented in the form of conservation laws, which allows one to analyze discontinuous solutions to these equations. When one chooses the upper or lower convected derivative, the equations of one-dimensional hyperbolic submodels turn out to be linear. The problem of shear motion between parallel plates and the problem of interaction between the stress field that does not depend on one of the coordinates and a transverse shear flow with initially constant vorticity are studied in detail. It is established that a plane Couette flow in the model with the corotational derivative is unstable in the linear approximation in the class of shear flows if the Weissenberg number is greater than one. The development of small perturbations gives rise to discontinuities in tangential velocities and stresses. The hysteresis phenomenon is observed when the Weissenberg number successively increases and decreases while passing through a critical value. The Couette flow in models with the upper and lower convected derivative remains stable with respect to one-dimensional perturbations.     

Necessary conditions and sufficient conditions of irregular shearlet frames

In this paper, necessary conditions and sufficient conditions for the irregular shearlet systems to be frames are studied. We show that the irregular shearlet systems to possess upper frame bounds, the space‐scale‐shear parameters must be relatively separated. We prove that if the irregular shearlet systems possess the lower frame bound and the space‐scale‐shear parameters satisfy certain condition, then the lower shearlet density is strictly positive. We apply these results to systems consisting only of dilations, obtaining some new results relating density to the frame properties of these systems. We prove that for a feasible class of shearlet generators introduced by P. Kittipoom et al., each relatively separated sequence with sufficiently hight density will generate a frame. Explicit frame bounds are given. We also study the stability of shearlet frames and show that a frame generated by certain shearlet function remains a frame when the space‐scale‐shear parameters and the generating function undergo small perturbations. Explicit stability bounds are given. Using pseudo‐spline functions of type I and II, we construct a family of irregular shearlet frames consisting of compactly supported shearlets to illustrate our results.     

Error estimates for Gaussian quadrature formulae

Summary We derive both strict and asymtotic error bounds for the Gauss-Jacobi quadrature formula with respect to a general measure. The estimates involve the maximum modulus of the integrand on a contour in the complex plane. The methods are elementary complex analysis.     

A general method for constructing Timoshenko-type theories

The equations of the plane theory of for the elasticity bending of a long strip are reduced by the method of simple iterations to the solution of a system of two equations for the displacement of the axis of the strip and the shear stress. If the transverse load varies slowly along the strip, the resolvent equations reduce to a single equation that is identical to the classical equation for the bend of a beam. When a local load is applied, the resolvent equation acquires an additional singular term that is the solution of the equation for the shear stresses under the assumption that the displacement (deflection) is a function of small variability. The convergence of the solution in an asymptotic sense is demonstrated. The application of the method of simple iterations to the dynamic equations for the bending of a strip also leads to a system of two resolvent equations in the displacement of the axis of the strip and the shear stress. These equations reduce to a single equation that is identical with the well-known Timoshenko equation. Hence, the procedure for using the method of simple iterations that has been developed can be classified as a general method for obtaining Timoshenko-type theories. An equation is derived for the bending of a strip on an elastic base with an isolated functional singular part with two bed coefficients, corresponding to the transverse and longitudinal springiness of the base.     

Generalized shear deformation theory for bending analysis of functionally graded plates

In this study, the static response is presented for a simply supported functionally graded rectangular plate subjected to a transverse uniform load. The generalized shear deformation theory obtained by the author in other recent papers is used. This theory is simplified by enforcing traction-free boundary conditions at the plate faces. No transversal shear correction factors are needed because a correct representation of the transversal shearing strain is given. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The equilibrium equations of a functionally graded plate are given based on a generalized shear deformation plate theory. The numerical illustrations concern bending response of functionally graded rectangular plates with two constituent materials. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, and volume fraction distributions are studied. The results are verified with the known results in the literature.     

A Gaussian upper bound for the fundamental solutions of a class of ultraparabolic equations

We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolic equations. These estimates are independent of the modulus of continuity of the coefficients and generalize the classical upper bounds by Aronson for uniformly parabolic equations.     

The Evolution of Linearized Perturbations of Parallel Flows

The equations of an incompressible fluid are linearized for small perturbations of a basic parallel flow. The initial-value problem is then posed by use of Fourier transforms in space. Previous results are systematized in a general framework and used to solve a series of problems for prototypical examples of basic shear flow and of initial disturbance. The prototypes of shear flow are (a) plane Couette flow bounded by rigid parallel walls, (b) plane Couette flow bounded by rigid walls at constant pressure, (c) unbounded two-layer flow with linear velocity profile in each layer, (d) a piecewise linear profile of a boundary layer on a rigid wall. The prototypes of initial perturbation are the fundamental ones: (i) a point source of the field of the transverse velocity (represented by delta functions), (ii) an unbounded sinusoidal field of the transverse velocity, (iii) a point source of the lateral component of vorticity, (iv) a sinusoidal field of the lateral vorticity. Detailed solutions for an inviscid fluid are presented, but the problem for a viscous fluid is only broached.     

Deformative characteristics of plastics reinforced with high-modulus anisotropic fibers

The independent elastic constants of plastics unidirectionally reinforced with transversely isotropic fibers have been determined. It has been assumed that the distribution of reinforcement in a transverse section of the plastic is regularly rectangular or hexagonal. To determine the transverse elastic modulus and the shear modulus in the plane of reinforcement, a constancy-of-plane-sections hypothesis was used. Values of deformative characteristics determined by the assumed calculational dependences have been compared with the experimental ones for plastics reinforced with graphite fibers.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 631–639, July–August, 1972.     

自然弯扭梁广义翘曲坐标的求解

提出了自然弯扭梁受复杂载荷作用时静力分析的一种理论方法,重点在于对控制方程的求解,其中考虑了与扭转有关的翘曲变形和横向剪切变形的影响．在特殊的情况下,可以比较容易地得到这些方程的解答,包括各种内力、应力、应变和位移的计算．算例给出了平面曲梁受水平和垂直分布载荷作用时广义翘曲坐标的求解方法．计算结果表明,求得的应力和位移的理论值和三维有限元结果非常接近．此外,该理论不限于具有双对称横截面的自然弯扭梁,同样可推广至具有一般横截面形状的情况．     

On the Moments of the Modulus of Continuity of Itô Processes

The modulus of continuity of a stochastic process is a random element for any fixed mesh size. We provide upper bounds for the moments of the modulus of continuity of Itô processes with possibly unbounded coefficients, starting from the special case of Brownian motion. References to known results for the case of Brownian motion and Itô processes with uniformly bounded coefficients are included. As an application, we obtain the rate of strong convergence of Euler–Maruyama schemes for the approximation of stochastic delay differential equations satisfying a Lipschitz condition in supremum norm.