Green’s functions for bending problem of a thin anisotropic plate with an elliptic hole

The Green’s functions have not been studied in open literatures for the bending problem of an anisotropic plate with an elliptic hole subjected to a normal concentrated force and a concentrated moment. In this paper, the problem is investigated and the Green’s functions are first obtained by using the complex potential approach. The techniques of conformal mapping transformation and analytic continuation are used to derive the closed-form complex stress functions. The Green’s functions obtained have some potential applications in the analysis of composite structures such as the modification of the displacement compatibility model for notched stiffened composite panels and the formulation of a new method for interlaminar stress analysis around holes of laminates.     

A new trigonometric shear deformation theory for isotropic,laminated composite and sandwich plates

A new trigonometric shear deformation theory for isotropic and composite laminated and sandwich plates, is developed. The new displacement field depends on a parameter “m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature. The results show that the present model performs as good as the Reddy’s and Touratier’s shear deformation theories for analyzing the static behavior of isotropic and composite laminated and sandwich plates.     

Analysis of an interfacial crack in a piezoelectric bi-material via the extended Green’s functions and displacement discontinuity method

Based on the extended Stroh formalism, we first derive the extended Green’s functions for an extended dislocation and displacement discontinuity located at the interface of a piezoelectric bi-material. These include Green’s functions of the extended dislocation, displacement discontinuities within a finite interval and the concentrated displacement discontinuities, all on the interface. The Green’s functions are then applied to obtain the integro-differential equation governing the interfacial crack. To eliminate the oscillating singularities associated with the delta function in the Green’s functions, we represent the delta function in terms of the Gaussian distribution function. In so doing, the integro-differential equation is reduced to a standard integral equation for the interfacial crack problem in piezoelectric bi-material with the extended displacement discontinuities being the unknowns. A simple numerical approach is also proposed to solve the integral equation for the displacement discontinuities, along with the asymptotic expressions of the extended intensity factors and J-integral in terms of the discontinuities near the crack tip. In numerical examples, the effect of the Gaussian parameter on the numerical results is discussed, and the influence of different extended loadings on the interfacial crack behaviors is further investigated.     

Theoretical and experimental studies on nonlinear oscillations of symmetric cross-ply composite laminated plates

The nonlinear oscillations and resonant responses of the symmetric cross-ply composite laminated plates are investigated theoretically and experimentally. The governing equations of motion for the composite laminated plate are derived by using the von Karman type equation, Reddy’s third-order shear deformation plate theory, and Galerkin method with the geometric nonlinearity. The four-dimensional averaged equation is obtained by using the method of multiple scales. The frequency-response functions are analyzed under the consideration of strongly coupled of two modes. The influences of the resonance case on the softening and hardening type of nonlinearity are analyzed with different parameters for the composite laminated plates. The numerical results indicate that there exist the hardening and softening types of the composite laminated plate in the specific resonant case. The variation of the response amplitudes is studied for the composite laminated plate under combined the transverse and in-plane excitations. A sweep frequency experiment is performed to obtain the hardening and softening nonlinearities of a composite laminated plate. The experimental results coincide with the numerical results qualitatively. The influences of the excitation amplitudes on the softening and hardening types of nonlinearity are also analyzed for the composite laminated plate. The amplitude spectrums of the test plate also demonstrate that the change of the nonlinear dynamic responses may be caused by the subharmonic resonance.     

Large amplitude free vibration of shear deformable laminated composite annular sector plates by a sector p-element

A sector p-element is presented for the large amplitude free vibration analysis of laminated composite annular sector plates. The effects of out-of-plane shear deformations, rotary inertia, and geometric non-linearity are taken into account. The shape functions are derived from the shifted Legendre orthogonal polynomials. The element stiffness and mass matrices are integrated analytically with the aid of symbolic computing. The method consists of modeling the annular sector plate as one element. The accuracy of the solution is improved simply by increasing the polynomial order. The time-dependent coefficients are described by a truncated Fourier series. The equations of free motion are obtained using the harmonic balance method and solved by the linearized updated mode method. Results for the linear and non-linear frequencies of clamped laminated composite annular sector plates are obtained. The case of a clamped isotropic annular sector plate is also shown. The linear frequencies are found to converge rapidly downwards as the polynomial order is increased. Comparisons of the linear frequencies with published results show excellent agreement. The effects of sector angle, inner-to-outer radius ratio, thickness-to-outer radius ratio, moduli ratio, number of plies, and layup sequence on the backbone curves are also investigated. It is shown that the hardening behavior increases or decreases depending on geometric and lamination parameters.     

复合材料层合板在动集中力作用下的结构声强特性

本文研究了复合材料正交异性层合板在动集中力作用下的结构声强特性。应用MSC/-NASTRAN商业软件计算了复合材料正交异性层合板在动集中力作用下各单元的内力和速度,再应用MATLAB软件得出复合材料层合板的结构声强。算例表明,复合材料正交异性层合板的结构声强流线图与各向同性板存在明显不同的特性。复合材料正交异性层合板的结构声强流线图受边界条件、层合板叠层顺序和层数的影响。从结构声强向量图和流线图可获得关于能量传递路径、源位置和能量汇合点的许多信息。进一步,结构振动产生的噪声可根据上述信息加以控制。     

Elastodynamic behavior of laminated orthotropic plates under initial stress

A finite element method of analysis of the vibrational and wave propagational characteristics is presented for a laminated orthotropic plate under initial stress. The plate may have an arbitrary number of bonded elastic orthotropic layers, each with distinct thickness, density and mechanical properties, and the analysis is capable of treating a completely arbitrary three-dimensional state of initial stress. Biot's theory for incremental elastic deformations of a stressed solid forms the basis for this study. A homogeneous, isotropic plate under two different states of initial stress was analyzed and their numerical results showed excellent correlation with those from an exact solution. Further examples of a three layer composite plate and a sandwich plate are offered to add some general insight to the physical behavior of such plates.     

Large deflection analysis of unsymmetrically laminated composite plates: analytical-numerical type approach

Large deflection analysis of laminated composite plates is considered. The Galerkin method along with Newton-Raphson method is applied to large deflection analysis of laminated composite plates with various edge conditions. The von Kármán plate theory is utilized and the governing differential equations are solved by choosing suitable polynomials as trial functions to approximate the plate displacement functions. The solutions are compared to that of Dynamic Relaxation and finite elements. A very close agreement has been observed with these approximating methods. In the solution process, analytical computation has been done wherever it is possible, and analytical-numerical type approach has been made for all problems.     

Thin plate spline radial basis functions for vibration analysis of clamped laminated composite plates

A meshless method based on thin plate spline radial basis functions and higher-order shear deformation theory are presented to analyze the free vibration of clamped laminated composite plates. The singularity of thin plate spline radial basis functions is eliminated by adding infinitesimal to the zero distance. Convergence characteristics of the present thin plate spline radial basis functions for the vibration analysis of the clamped laminated plates are investigated. The frequencies computed by the present method agree well with the available published results.     

复合材料层合板自由振动分析的无网格自然邻接点Petrov-Galerkin法

基于一阶剪切变形理论,提出了复合材料层合板自由振动分析的无网格自然邻接点Petrov-Galerkin法。计算时在复合材料层合板中面上仅需要布置一系列的离散节点,并利用这些节点构建插值函数。在板中面上的局部多边形子域上,采用加权余量法建立复合材料层合板自由振动分析的离散化控制方程,并且这些子域可由Delaunay三角形方便创建。自然邻接点插值形函数具有Kronecker delta函数性质,因而无需经过特别处理就能准确地施加本质边界条件。对不同边界条件、不同跨厚比、不同材料参数和不同铺设角度的复合材料层合板,由本文提出的无网格自然邻接点Petrov-Galerkin法进行自由振动分析时均可得到满意的结果。数值算例结果表明,本文方法求解复合材料层合板的自由振动问题是行之有效的。     

ANALYSIS OF INTER-LAMINAR STRESSES FOR COMPOSITE LAMINATED PLATE WITH INTERFACIAL DAMAGE

A constitutive model for composite laminated plates with the damage effect of the intra-layers and inter-laminar interface is presented. The model is based on the general six-degrees-of-freedom plate theory, the discontinuity of displacement on the interfaces are depicted by three shape functions, which are formulated according to solutions satisfying three equilibrium equations, By using the variation principle, the three-dimensional non-linear equilibrium differential equations of the laminated plates with two different damage models are derived. Then, considering a simply supported laminated plate with damage, an analytical solution is presented using finite difference method to obtain the inter-laminar stresses.     

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.     

Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory

In this paper, we present an effectively numerical approach based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) for geometrically nonlinear analysis of laminated composite plates. The HSDT allows us to approximate displacement field that ensures by itself the realistic shear strain energy part without shear correction factors (SCFs). IGA utilizing basis functions namely B-splines or non-uniform rational B-splines (NURBS) enables to satisfy easily the stringent continuity requirement of the HSDT model without any additional variables. The nonlinearity of the plates is formed in the total Lagrange approach based on the small strain assumptions. Numerous numerical validations for the isotropic, orthotropic, cross-ply and angle-ply laminated plates are provided to demonstrate the effectiveness of the proposed method.     

ON THE REFINED FIRST-ORDER SHEAR DEFORMATION PLATE THEORY OF KARMAN TYPE

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Delamination of laminated composite plates by p-convergent partial discrete-layer elements with VCCT

The simulation of the delamination process in laminated composite plates is quite complex and requires advanced finite element modeling techniques. Failure analysis tools must be able to predict initiation, size and propagation of delamination process. This paper presents the p-convergent partial discrete-layer elements with the virtual crack closure technique (VCCT) for the delamination analysis of laminated composite plates. The proposed element can be formulated by the suitable dimensional reduction from three-dimensional solid to two-dimensional plate. It is assumed that the piecewise linear variation of in-plane displacements and the constant value of out-of-plane displacements across the thickness. The higher-order approximation based on integrals of Legendre polynomials is used to define displacement fields. The three-dimensional VCCT is also slightly modified to incorporate with the proposed elements to estimate the energy release rate. The initiation of delamination occurs when the energy release rate for a displacement increment is same as the critical energy release rate corresponding to fracture toughness. The approach is to use a fracture mechanics criterion, but to avoid the complex moving mesh technique. At first, the validation and characteristic of the proposed elements are investigated on isotropic plates and orthotropic laminated plates, compared with referenced values. Then for fracture analysis, the efficiency of proposed approach is demonstrated with the help of additionally two problems such as the double-cantilever-beam test and the orthotropic laminated square plate with interior delamination.     

Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk

This paper considers the bending of transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate, subject to a transverse load in the form of qr^k (k is zero or a finite even number). The differential equations satisfied by stress functions for the particular problem are derived. An elaborate analysis procedure is then presented to derive these stress functions, from which the analytical expressions for the axial force, bending moment and displacements are obtained through integration. The method is then applied to the problem of transversely isotropic functionally graded circular plate subject to a uniform load, illustrating the procedure to determine the integral constants from the boundary conditions. Analytical elasticity solutions are presented for simply-supported and clamped plates, and, when degenerated, they coincide with the available solutions for an isotropic homogenous plate. Two numerical examples are finally presented to show the effect of material inhomogeneity on the elastic field in FGM plates.     

Integral transforms of differential equations for deflections of isotropic sandwich plates

Integral transforms of derivatives of functions appearing in the equations for deflections of sandwich plates are presented in terms of transforms of those functions and of appropriate boundary values. The use of the generalized Green's formula enables us to present these transforms in an invariant form. A rectangular plate is analyzed as an example.     

Green’s function for T-stress of semi-infinite plane crack

Green’s function for the T-stress near a crack tip is addressed with an analytic function method for a semi-infinite crack lying in an elastical, isotropic, and infinite plate. The cracked plate is loaded by a single inclined concentrated force at an interior point. The complex potentials are obtained based on a superposition principle, which provide the solutions to the plane problems of elasticity. The regular parts of the potentials are extracted in an asymptotic analysis. Based on the regular parts, Gre...     

Multi-pulse chaotic motions of high-dimension nonlinear system for a laminated composite piezoelectric rectangular plate

This paper investigates the multi-pulse global bifurcations and chaotic dynamics of the high-dimension nonlinear system for a laminated composite piezoelectric rectangular plate by using an extended Melnikov method in the resonant case. Using the von Karman type equations, Reddy’s third-order shear deformation plate theory and Hamilton’s principle, the equations of motion are derived for the laminated composite piezoelectric rectangular plate with combined parametric excitations and transverse excitation. Applying the method of multiple scales and Galerkin’s approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:2 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used for the first time to investigate the Shilnikov type multi-pulse homoclinic bifurcations and chaotic dynamics of the laminated composite piezoelectric rectangular plate. The necessary conditions of the existence for the Shilnikov type multi-pulse chaotic dynamics of the laminated composite piezoelectric rectangular plate are analytically obtained. Numerical simulations also illustrate that the Shilnikov type multi-pulse chaotic motions can also occur in the laminated composite piezoelectric rectangular plate. Overall, both theoretical and numerical studies demonstrate that the chaos in the Smale horseshoe sense exists for the laminated composite piezoelectric rectangular plate.     

A comprehensive analysis of functionally graded sandwich plates: Part 1—Deflection and stresses

A two-dimensional solution is presented for bending analysis of simply supported functionally graded ceramic–metal sandwich plates. The sandwich plate faces are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity and Poisson’s ratio of the faces are assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Several kinds of sandwich plates are used taking into account the symmetry of the plate and the thickness of each layer. We derive field equations for functionally graded sandwich plates whose deformations are governed by either the shear deformation theories or the classical theory. Displacement functions that identically satisfy boundary conditions are used to reduce the governing equations to a set of coupled ordinary differential equations with variable coefficients. Numerical results of the sinusoidal, third-order, first-order and classical theories are presented to show the effect of material distribution on the deflections and stresses.