An experimental and analytical investigation of the rail shear-test method as applied to composite materials

This report presents the results from an experimental and analytical investigation of the stress distributions occurring in a rail shear test. The effects of nonuniform stresses induced by differential thermal expansion, rail flexibility and specimen aspect ratio on measured shear modulus and ultimate strength of composite laminates are shown. A two-dimensional linearly elastic finite-element model was used to analytically determine how various geometric parameters influenced the magnitude and distribution of inplane normal and shear stresses in a tensile-rail-shear specimen. Rail shear tests were conducted at room temperature and 589 K (600°F) on selected graphite-polyimide composite laminates using two titanium rail configurations. The analysis and test methods are discussed, and the results of the effects of the various parameters on shear modulus and ultimate strength are presented.     

Axisymmetric static and dynamics snap-through phenomena in a thermally postbuckled temperature-dependent FGM circular plate

Thermal postbuckling analysis and the axisymmetric static and dynamic snap-through phenomena due to static/sudden uniform lateral pressure in a thermally postbuckled functionally graded material circular plate are performed in this research. Plate is formulated using the first order shear deformation plate theory. Thermo-mechanical properties of the plate are assumed to be temperature dependent where dependency is described according to the higher order Touloukian representation. Two types of temperature loading are considered. Uniform temperature rise and heat conduction across the thickness direction. The one dimensional heat conduction equation in the thickness direction is obtained and discreted via the central finite difference method. The obtained system of equations is nonlinear since the thermal conductivity itself is a function of the unknown nodal temperatures. Using the von-Kármán assumptions, the governing equations of the plate are obtained in a matrix representation with the aid of the conventional Ritz method whose shape functions are developed using the Gram-Schmidt process. At first thermal postbuckling analysis is performed which is a nonlinear problem with respect to both temperature and displacements. Afterwards, response of the bulged thermally postbuckled plate is obtained under the static and dynamic uniform pressure. Snap-through phenomenon may be observed in both static and dynamic loading cases, due to the immovability of the edge of the plate and the initial deflection caused by postbuckling deflection. To capture the snapping phenomenon and trace the path beyond the limit loads, cylindrical arch-length technique is used. In dynamic snap-through analysis, the effect of structural damping is also included. Numerical results of this study reveal that the structure is sensitive to the initial deflection caused by thermal postbuckling load. Increasing the temperature prior to mechanical loads enhances the snap-through intensity and also increases both the upper and lower limit loads. As shown, dynamic snap-through loads are lower than the static ones, however dynamic snap-through intensity is more than the static snap-though intensity. Furthermore, structural damping enhances the dynamic buckling loads of the plate and decreases the dynamic postbuckling deflection of the plate.     

Kármán-type equations for a higher-order shear deformation plate theory and its use in the thermal postbuckling analysis

Kármán-type nonlinear large deflection equations are derived occnrding to the Reddy’s higher-order shear deformation plate theory and used in the thermal postbuckling analysis The effects of initial geometric imperfections of the plate areincluded in the present study which also includes th thermal effects.Simply supported,symmetric cross-ply laminated plates subjected to uniform or nomuniform parabolictemperature distribution are considered. The analysis uses a mixed GalerkinGolerkinperlurbation technique to determine thermal buckling louds and postbucklingequilibrium paths.The effects played by transverse shear deformation plate aspeclraio, total number of plies thermal load ratio and initial geometric imperfections arealso studied.     

Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment

Free vibration of statically thermal postbuckled functionally graded material （FGM） beams with surface-bonded piezoelectric layers subject to both temperature rise and voltage is studied. By accurately considering the axial extension and based on the Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surface-bonded piezoelectric layers subject to thermo-electro- mechanical loadings are formulated. It is assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate, and the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of the beam vibration is small and its response is harmonic, the above mentioned non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations. One is for the postbuckling, and the other is for the linear vibration of the beam superimposed upon the postbuckled configuration. Using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, the response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subject to transversely nonuniform heating and uniform electric field is obtained. Thermo-electric postbuckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity, and the material gradient parameters are plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with the increase of the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.     

Non-linear analysis of functionally graded plates in cylindrical bending under thermomechanical loadings based on a layerwise theory

A layerwise theory is used to analyze analytically displacements and stresses in functionally graded composite plates in cylindrical bending subjected to thermomechanical loadings. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The non-linear strain–displacement relations in the von Kármán sense are used to study the effect of geometric non-linearity. The equilibrium equations are solved exactly and also by using a perturbation technique. Numerical results are presented to show the effect of the material distribution on the deflections and stresses.     

Nonlinear stability and vibration of pre/post-buckled microstructure-dependent FGPM actuators

The present study deals with the study of the nonlinear stability and small free vibration of microstructure-dependent functionally graded piezoelectric material (FGPM) beams in pre/post-buckling regimes. The Timoshenko beam theory with various inplane and out-of-plane boundary conditions are considered under different types of mechanical and thermal loads. The beam is assumed to be under inplane mechanical, thermal, and electrical excitations. Each thermo-electro-mechanical property of the beam is graded across the thickness (i.e., height) of the beam, based on a power law model. The von Kármán type geometric nonlinearity is included to account for the large deflection behavior of the beam under inplane loads. The modified couple stress theory is included to account for the size effects. A weak-form, displacement-based, finite element formulation is developed to discretize the equations of motion. The resulting system of nonlinear algebraic equations is solved using Newton’s iterative method. The numerical results of frequencies and lateral deflections as a function of load parameters reveal the existence of bifurcation or critical states in some cases. The effects of load type, microstructural dependency, boundary conditions, beam geometry, composition rule of the constituents, and actuator voltage are discussed through the various parametric studies.     

Growth of an arc-crack in bonded dissimilar materials

The problem of growth of a crack lying along the interface of a circular inclusion embedded in an infinite plate is studied within the framework of linear elasticity. The plate is subjected to a uniform uniaxial stress at infinity at any angle of inclination relatively to the crack. The critical load for unstable crack growth, the angle of initial crack extension and the subsequent crack path are investigated using the strain energy density fracture criterion. The combined effect of crack length and orientation on the fracture stress is considered for the case of an aluminum-epoxy composite.     

切应力对牛骨蠕变变形的影响研究

骨尤其是湿骨,在恒定载荷作用下会发生蠕变变形。为了确定切应力是否影响骨的蠕变变形,采用对骨薄板试样分别施加集中载荷和均布载荷的方式,测量试样挠度实时的变化曲线。结果显示,在载荷恒定时,骨试样的挠度随时间不断增加,体现了典型的蠕变特性。集中荷载下骨的蠕变变形远大于均布荷载下骨的蠕变变形,湿骨的蠕变位移比干骨高近7倍。分析认为,对试样粘弹性性质的影响不仅有正应力的作用,也有切应力的作用;切应力产生的蠕变变形约为正应力所产生蠕变变形的0.85倍。     

Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam

In this paper, an exact analytical solution is presented for a transversely isotropic functionally graded magneto-electro-elastic (FGMEE) cantilever beam, which is subjected to a uniform load on its upper surface, as well as the concentrated force and moment at the free end. This solution can be applied for any form of gradient distribution. For the basic equations of plane problem, all the partial differential equations governing the stress field, electric, and magnetic potentials are derived. Then, the expressions of Airy stress, electric, and magnetic potential functions are assumed as quadratic polynomials of the longitudinal coordinate. Based on all the boundary conditions, the exact expressions of the three functions can be determined. As numerical examples, the material parameters are set as exponential and linear distributions in the thickness direction. The effects of the material parameters on the mechanical, electric, and magnetic fields of the cantilever beam are analyzed in detail.     

On the Role of In-Plane Compliance in Edge Wrinkling

Bifurcations of a thin circular elastic plate subjected to uniform normal pressure are investigated by taking into account the in-plane compliance of the edge restraint. This effect amounts to introducing a Hookean spring relating the radial components of the membrane stress tensor and the corresponding in-plane displacement fields. The addition of this new feature gives rise to an adaptive radial stretching of our configuration, which is intimately linked to the strength of the applied pressure. The Föppl-von Kármán nonlinear plate theory, in conjunction with singular perturbation arguments, help us to establish the nature of the localised wrinkling observed in numerical simulations. Asymptotic analysis of the problem provides some simple qualitative predictions for the dependence of the critical load on a number of key dimensionless parameters.     

Shear deformable bars of doubly symmetrical cross section under nonlinear nonuniform torsional vibrations—application to torsional postbuckling configurations and primary resonance excitations

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross section, taking into account the effects of geometrical nonlinearity (finite displacement—small strain theory) and secondary twisting moment deformation. The bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are subjected to the most general axial and torsional (twisting and warping) boundary conditions. The resulting coupling effect between twisting and axial displacement components is also considered and a constant along the bar compressive axial load is induced so as to investigate the dynamic response at the (torsional) postbuckled state. The bar is assumed to be adequately laterally supported so that it does not exhibit any flexural or flexural–torsional behavior. A coupled nonlinear initial boundary value problem with respect to the variable along the bar angle of twist and to an independent warping parameter is formulated. The resulting equations are further combined to yield a single partial differential equation with respect to the angle of twist. The problem is numerically solved employing the Analog Equation Method (AEM), a BEM based method, leading to a system of nonlinear Differential–Algebraic Equations (DAE). The main purpose of the present contribution is twofold: (i) comparison of both the governing differential equations and the numerical results of linear or nonlinear free or forced vibrations of bars ignoring or taking into account the secondary twisting moment deformation effect (STMDE) and (ii) numerical investigation of linear or nonlinear free vibrations of bars at torsional postbuckling configurations. Numerical results are worked out to illustrate the method, demonstrate its efficiency and wherever possible its accuracy.

     

Two nonmixed symmetric end-loadings of an elastic waveguide

This investigation treats the response of the semi-infinite plate with free faces subjected to suddenly applied normal loads on its end. The plate is elastic and in plane strain. The normal loads are symmetric and act in the absence of shear stress, i.e. the plate has nonmixed end conditions. A double Laplace transform technique is used to obtain long-time information for two particular normal loads; the uniform load and the line-load. Near-field and far-field approximations are found. Results in the long-time near-field for the uniform load reduce to elementary forms; for the line-load however, the corresponding forms are quite complex entailing singular terms and some numerically evaluated contributions. The far-field approximations give rise to integral of the Airy function forms for both loads and, if the forces applied under the two loadings are equal, these far-field responses are shown to be identical.     

Nonlinear bending behavior of 3D braided rectangular plates subjected to transverse loads in thermal environments

Nonlinear bending behavior of 3D braided rectangular plates subjected to transverse loads is investigated. A new micro-macro-mechanical model of unit cells is suggested. In this model, a 3D braided composite may be considered as a cell system and the geometry of each cell is deeply dependent on its position in the cross-section of the plate. The material properties of the epoxy are expressed as a linear function of temperature. Based on Reddy’s higher-order shear deformation plate theory and general von Kármán-type equations, analytical solutions for nonlinear bending behavior of simply supported 3D braided rectangular plates are obtained using mixed Galerkin-perturbation method. The numerical examples concern effects of geometric parameters, of fiber volume fraction, braiding angle and load boundary condition.     

An investigation into non-linear vibrations of thin plates

The variational and modified forms of the von Kármán-type non-linear plate equations are considered in the context of the Rayleigh-Ritz and Galerkin methods. An approximate analysis of the non-linear vibrations of thin elastic plates including inplane inertia is presented. The quantitative study confirms that the inplane inertia effects are negligible for thin plates provided the non-linearity is not too large. It is observed that the non-linear inertia terms in the transverse equation of motion should be retained in any such study. The analysis is simplified by neglecting the inplane inertia and applied to constrained and unconstrained plates. A different type of inplane boundary condition termed ‘the partially constrained’ is studied, and the inadequacy of replacing the unconstrained condition by means of an average-zero stress condition is clearly demonstrated. It is observed that in most of the cases considered the Galerkin method yields lower bounds for the non-linear coefficient of the modal equation. In all cases the Galerkin results yield less stiff models than the Rayleigh-Ritz method. The general significance of the convergence of the two methods beyond the scope of the title problem is highlighted.     

Stochastic nonlinear bending response of laminated composite plates with system randomness under lateral pressure and thermal loading

In this paper, the effect of sensitivity of randomness in system parameters on the nonlinear transverse central deflection response of laminated composite plates subjected to transverse uniform lateral pressure and thermal loading is examined. System parameters such as the lamina material properties, expansion of thermal coefficients, lamina plate thickness and lateral load are modelled as basic random variables. A higher order shear deformation theory in the von-Karman sense is used to model the system behavior of the laminated plate. A direct iterative-based C ⁰ nonlinear finite element method in conjunction with the first-order perturbation technique developed by the authors is extended for thermal problem to obtain the second-order response statistics, i.e., mean and variance of the nonlinear transverse deflection of the plate. Typical numerical results of composite plates with temperature independent and dependent material properties subjected to uniform temperature and combination of uniform and transverse temperature are obtained for various combinations of geometric parameters, uniform lateral pressures, staking sequences and boundary conditions. The results have been compared with those available in the literature and an independent Monte Carlo simulation.     

A new technique for large deflection analysis of non-prismatic cantilever beams

This paper studies the very large deflection behavior of prismatic and non-prismatic cantilever beams subjected to various types of loadings. The formulation is based on representing the angle of rotation of the beam by a polynomial on the position variable along the deflected beam axis. The coefficients of the polynomial are obtained by minimizing the integral of the residual error of the governing differential equation and by applying the beam’s boundary conditions. Several numerical examples are presented covering prismatic and non-prismatic cantilever beams subjected to uniform, non-uniform distributed loads and tip concentrated loadings in vertical and horizontal directions. The loads considered in this study are restricted to the non-follower type loads. Cases with different loadings and geometries are compared with MSC/NASTRAN computer package. However, for some very large deflection case, the MSC/NASTRAN failed to predict the deflected shape due to divergence problems.     

NONLINEAR DYNAMIC ANALYSIS OF A LAMINATED HYBRID COMPOSITE PLATE SUBJECTED TO TIME-DEPENDENT EXTERNAL PULSES

Nonlinear dynamic responses of a laminated hybrid composite plate subjected to time-dependent pulses are investigated. Dynamic equations of the plate are derived by the use of the virtual work principle. The geometric nonlinearity effects are taken into account with the von Kármán large deflection theory of thin plates. Approximate solutions for a clamped plate are assumed for the space domain. The single term approximation functions are selected by considering the nonlinear static deformation of plate obtained using the finite element method. The Galerkin Method is used to obtain the nonlinear differential equations in the time domain and a MATLAB software code is written to solve nonlinear coupled equations by using the Newmark Method. The results of approximate-numerical analysis are obtained and compared with the finite element results. Transient loading conditions considered include blast, sine, rectangular, and triangular pulses. A parametric study is conducted considering the effects of peak pressure, aspect ratio, fiber orientation and thicknesses.     

Buckling of a flush-mounted plate in simple shear flow

The buckling of an elastic plate with arbitrary shape flush-mounted on a rigid wall and deforming under the action of a uniform tangential load due to an overpassing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated for the buckled state resulting in a fourth-order partial differential equation with position-dependent coefficients parameterized by the Poisson ratio. The governing equation also describes the deformation of a plate clamped around the edges on a vertical wall and buckling under the action of its own weight. Solutions are computed analytically for a circular plate by applying a Fourier series expansion to derive an infinite system of coupled ordinary differential equations and then implementing orthogonal collocation, and numerically for elliptical and rectangular plates by using a finite-element method. The eigenvalues of the resulting generalized algebraic eigenvalue problem are bifurcation points in the solution space, physically representing critical thresholds of the uniform tangential load above which the plate buckles and wrinkles due to the partially compressive developing stresses. The associated eigenfunctions representing possible modes of deformation are illustrated, and the effect of the Poisson ratio and plate shape is discussed.     

Analytical solution for a strained reinforcement layer bonded to lip-shaped crack under remote mode III uniform load and concentrated load

In this paper, the analytical solution of stress field for a strained reinforcement layer bonded to a lip-shaped crack under a remote mode III uniform load and a concentrated load is obtained explicitly in the series form by using the technical of conformal mapping and the method of analytic continuation. The effects of material combinations, bond of interface and geometric configurations on interfaciai stresses generated by eigenstrain, remote load and concentrated load are studied. The results show that the stress concentration and interfaciai stresses can be reduced by rational material combinations and geometric configurations designs for different load forms.     

Large deflections of prestressed simply supported rectangular plates under uniform pressure

The von Kármán large deflection equations for laterally loaded rectangular plates are extended to include uniform prestresses parallel to the edges and are solved for uniform load and for edges which are simply supported against movement normal to the plane of the plate and which are either held or free to move as a rigid body in the plane of the plate. Calculated values of center deflection and maximum stress parameters are given as functions of the load parameter for plates of various aspect ratios.