A comparison of finite-difference and finite-element methods for calculating free edge stresses in composites

This study considers the accuracy of the finite difference method in the solution of linear elasticity problems that involve either a stress discontinuity or a stress singularity. Solutions to three elasticity problems are discussed in detail: a semi-infinite plane subjected to a uniform load over a portion of its boundary; a bimetallic plate under uniform tensile stress; and a long, midplane symmetric, fiber-reinforced laminate subjected to uniform axial strain. Finite difference solutions to the three problems are compared with finite element solutions to corresponding problems. For the first problem a comparison with the exact solution is also made. The finite difference formulations for the three problems are based on second order finite difference formulas that provide for variable spacings in two perpendicular directions. Forward and backward difference formulas are used near boundaries where their use eliminates the need for fictitious grid points. Moreover, forward and backward finite difference formulas are used to enforced continuity of interlaminar stress components for the third problem. The study shows that the finite difference method employed in this investigation provides solutions to the three elasticity problems considered that are as accurate as the corresponding finite element solutions. Furthermore, the finite difference method appears to give a solution for the laminate problem that characterizes the stress distributions near an interface corner in a more realistic manner than the finite element method.     

Nonlinear analysis of edge effects in angle-ply laminates

An initial stress iteration method using a quasi-three-dimensional finite element has been developed for the nonlinear analysis of Carbon Fibre Reinforced Plastic (CFRP) laminates subjected to a uniform inplane load. Linear and nonlinear analyses of the free edge problem were carried out for the [0/90]_s, and [ ± 45]_s laminates with and without resin layers included between plies of laminae. The nonlinear results show that the peak shear stress in the [0/90]_s laminate and all the stress components in the [±45], laminates are reduced. A maximum strain failure criterion was used and failure modes were predicted.     

A new finite element for free edge effect analysis in laminated composites

This paper presents a finite element model based on the Multiparticle Model of Multilayered Materials (M4) developed in Institut Navier-LAMI during the last years. The laminated plate is considered as a superposition of Reissner plates coupled by interfacial stresses. An eight-node multiparticle element is developed here. This element has 5n degrees of freedom per node (n is the layer’s number of the laminate). A finite element program called MPFEAP has been developed for the implementation of the multiparticle element. The proposed finite element model is capable of computing interlaminar stresses and other localized effect which is impossible with classical 2D finite element model. Some examples of free edge problem are provided to illustrate the accuracy of the present element in predicting interlaminar stresses.     

A 3-D finite difference solution for orthotropic laminated composites using curvilinear coordinates

The static equilibrium equations for orthotropic composite laminates are written in body fitted curvilinear coordinate system. A 3-D Finite Difference code has been developed to solve these equations. This combination extends the capability of the Finite Difference method to analyse effectively stress concentrations near arbitrary 2-D defects in composites. The stress field around a circular hole in an angleply composite laminate (?45°, 45°|_s) is analysed as a test case.     

Application of a three-dimensional mixed finite element model to the flexure of sandwich plate

The bending analysis of sandwich plates consisting of very stiff face sheets and a comparatively flexible core material offers challenge due to large variation in the magnitude of stress and strain components in the face and in the core regions of the plate. Similarly, the displacement fields do vary in zigzag manner at the layer interface of stiff face sheet and the soft core, thereby making the transverse strains highly discontinuous at such layer interfaces. All these behavioural aspects indicate that only an individual layerwise model can appropriately analyze sandwich plates. A layerwise (three-dimensional), mixed, 18-node finite element (FE) model developed by Ramtekkar et al. [Mech. Adv. Mater. Struct. 9 (2002) 133] has been employed for the accurate evaluation of transverse stresses in sandwich laminates. The FE model consists of six degrees-of-freedom (three displacement components and three transverse stress components τ_xz, τ_yz, σ_z, where z is the thickness direction) per node which ensures the through thickness continuity of transverse stress and displacement fields. Results obtained by using the FE model have shown excellent agreement with the available elasticity solutions for sandwich plates. Additional results on the variation of transverse strains have also been presented to highlight the magnitude of discontinuity in these quantities due to difference in properties of the face and the core materials of sandwich plates.     

The variation of the stress singularity at the notch tip as notch angle and radius of curvature

This paper concerns itself with the analysis of the deviations of the stress singularity at the blunted notch tip from the square root singularity which is associated with an ideal crack. The stress distributions and the stress intensity factors for V and U notches were analyzed for several blunting angles and the radii of curvature, using finite element method. The results showed that the differences of the stress intensity factors between V notch tip and an ideal crack were quite little in the region of that the notch angle is from 0 to 80°.     

Buckling analysis of laminated composite plates with an elliptical/circular cutout using FEM

In this study, a buckling analysis was carried out of a woven–glass–polyester laminated composite plate with an circular/elliptical hole, numerically. In the analysis, finite element method (FEM) was applied to perform parametric studies on various plates based on the shape and position of the elliptical hole. This study addressed the effects of an elliptical/circular cutout on the buckling load of square composite plates. The laminated composite plates were arranged as symmetric cross-ply [(0°/90°)₂]_s and angle-ply [(15°/−75°)₂]_s, [(30°/−60°)₂]_s, [(45°/−45°)₂]_s. The results show that buckling loads are decreased by increasing both c/a and b/a ratios. The increasing of hole positioned angle cause to decrease of buckling loads. Additionally, the cross-ply composite plate is stronger than all other analyzed angle-ply laminated plates.     

Three-dimensional finite element analysis of interlaminar stresses in thick composite laminates

The three-dimensional finite element computer program has been developed to investigate interlaminar stresses in thick composite laminates. The finite element analysis is based on displacement formulation employing curved isoparametric 16-node elements. By using substructure technique, the program developed is capable of handling any composite laminates which consist of any number of orthotropic laminae and any orientations. In this paper, solid laminates and laminates with a circular hole were taken to study interlaminar stresses at the straight edge and the curved edge, respectively. Various solid laminates such as [45_n/0_n − 45_n/90_n]_s, [45/0/ − 45/90]_ns, and [45/0/ − 45/90]_sn (n = 1˜4) were analyzed. Also, [45/0/ − 45/90]_sn laminates with a circular hole were studied for n = 1 ˜ 20. The effect of laminate thickness and stacking sequence on the interlaminar stresses near the free edge was investigated. Interlaminar stresses were governed by stacking sequence rather than laminate thickness. The boundary layer width did not increase with laminate thickness but with the number of plies in the repeating unit.     

Stress distributions in an elastic body due to molecular interactions considering one-dimensional periodic material distribution based on Mindlin’s solution

A method to calculate the stress distributions in a semi-infinite elastic body caused by molecular interactions has been established based on Mindlin’s solution. A molecular interaction force derived from the Lennard-Jones potential considering a one-dimensional periodic material distribution in the in-plane direction, i.e., x-direction, was used as a concentrated force in Mindlin’s solution. The stresses acting on the (100) and (001) surfaces (σ_x, σ_z, and τ_zx) for the distribution of two materials were calculated as a typical example of a periodic material distribution. The stress distributions were shown by color maps and the basic characteristics of the stress distribution, especially the effects of the surface distance, were quantitatively clarified. The asymptotic values of σ_x, σ_z, and τ_zx at the position far from the surface were analytically derived. Those of σ_x and σ_z had the same value, depending on the surface distance, while that of τ_zx was always zero. This difference comes from the existence of a non-fluctuation contribution.

     

The effects of additional deformations in box-beam bridges on the longitudinal stresses and transverse moments

The study of transverse moments m_s, and longitudinal stress σ₁₁ that are developed due to the deformations of a concrete box-beam from eccentric loading is discussed.The basic equations for calculating the transverse moments m_s, and the longitudinal stresses σ₁₁, due to deformations of the box beam are developed. The differential equation for calculating angular deformations is also developed. Further, the analogy of the differential equation of beam-on-elastic foundation is performed. In addition, the analogy of stress resultants of a box-beam to those of a beam-on-elastic foundation is also being carried out. A general method of calculating moments m_s, and stresses σ₁₁ of the beam useful for practical applications is also given, followed by plotting the diagrames of the variation of moments m_s, and stresses σ₁₁ for various span lengths and for different heights of the box beam. Some useful conclusions have been drawn from studying these diagrames.     

A traction-free-edge hybrid-stress element for the analysis of edge effects in cross-ply laminates

A hybrid-stress formulation is presented for the analysis of multilayer cross-ply laminates under uniform inplane strain and a special-purpose element is developed for which the traction-free edge conditions are exactly satisfied. Results obtained for interlaminar stress distributions are shown to converge to large, but finite, values near the free edge. The importance of exact representation of the traction-free edge conditions is examined by comparison of results with those obtained by other investigators and by using an analogous multilayer hybrid-stress element in which the free-edge conditions are approximately satisfied. It is concluded that exact representation of the free-edge conditions is required if accurate point-by-point stress distribution in the vicinity of the free-edge are essential, as in the study of laminate strength and failure.     

Laminated composite structures by continuum-based shell elements with transverse deformation

In this paper a first-order shear/fourth-order transverse deformation theory of laminated composite shells is presented. A nonlinear continuum-based (degenerated 3D) finite element model with a strain/stress enhancement technique is developed in such a way that the nonzero surface traction boundary conditions and the interlaminar shear stress continuity conditions are all satisfied identically. Analytical integration through the shell thickness is performed. The resultants of the stress integrations are expressed in terms of the laminate stacking sequence. Consequently, the shell laminate characteristics in the normal direction can be evaluated precisely and the computational cost of the overall analysis is reduced. The numerical results are compared with analytical solutions and other finite element solutions to demonstrate the effectiveness of the theory and the computational procedure developed herein.     

A class of C finite elements for the static and dynamic analysis of laminated plates

A general higher-order deformation theory is developed to analyse the behaviour of an arbitrary laminated fibre-reinforced composite plate. Three-dimensional effects such as the warping of sections and the presence of interlaminar stress field components are taken into account assuming a power series expansion of displacements along the thickness. A class of C⁰ finite element models based on this theory is then developed for mono- and bi-dimensional elements. Applications of the models to bending and vibration of laminated plates are then discussed. The present solutions are compared with those obtained using the three-dimensional elasticity theory, classical laminate theory and other higher-order theories.     

A consistent control of spurious singular modes in the 9-node Lagrange element for the laplace and mindlin plate equations

When 2 × 2 quadrature is used with the 9-node Lagrange element, which is essential in C⁰ plate elements to avoid locking, spurious singular modes appear on the element level which can lead to singularity or near-singularity of the global equations. Here these modes are controlled by a procedure that introduces additional generalized stresses and strains so that the spurious modes are eliminated and the consistency of the resulting finite difference equations is not impaired; hence the procedure passes the patch test. Applications to the diffusion and Mindlin plate equations are presented. Results show that h³ convergence in the L₂-norm is almost retained.     

On the convergence of Uzawa's method for the solution of biharmonic problem

The convergence of Uzawa's method for the solution of biharmonic problem in the mixed finite element method had been proved in [2] and [3]. In this paper, we obtained an asymptotic rate of convergence. Furthermore, in order to keep the same order of error between the solutions of Uzawa's iteration and biharmonic problem, as the order of error between the solutions of the mixed finite element method and biharmonic problem, we estimated the timesn of Uzawa's iteration as follows:n≥≥?(75/vσ ²)h ^?1 lnh for 2-degree Lagrange elements, andn≥?(18/vσ ²)h ^?1 lnh?(12/vσ ²)h ^?1 ln(?lnh) for linear elements.     

A high precision coupled bending–extension triangular finite element for laminated plates

It is well recognized that the estimation of interlaminar stresses and strain energy release rates is important in designing laminated composite panels. Generally coupled bending–extension finite elements are necessary to study laminates to include the effects of coupling and/or combined transverse and extensional loads. Such elements are normally formulated adapting the classical theory of bending and extension. While the classical laminated plate theory of bending has provision to obtain interlaminar stresses due to transverse loading, it is necessary to include certain higher order terms in the extensional theory in order to obtain the interlaminar stresses due to inplane loads. A high precision triangular element based on a theory which includes both the bending and extension with necessary higher order terms is presented in this paper. The performance of this element is validated with the aid of examples. Numerical results for displacements in symmetric and unsymmetric laminates under bending loads have been given. Numerical results for interlaminar stresses in symmetric and unsymmetric laminates have been given for the well-known benchmark problem of a coupon with free edges. Strain energy release rate components at the delamination tip in coupons with unsymmetric sublaminates have been given. The effects of delamination length and location on the components of the strain energy release rate have been studied. Results indicated that with the use of this element, the interlaminar stresses can be estimated reasonably accurately, over a major part of the laminate except in a small local region close to the free edge. Global–local analysis with three-dimensional elements in the local region, is suggested to obtain local stresses more accurately. Interlaminar stresses at the boundary of a hole in a perforated plate under extension have been obtained to illustrate the use of the present element in a global–local analysis strategy.     

Error analysis in the series evaluation of the exponential type integral ezE1(z)

The method of convolution algebra is used to compute values of the exponential type integral e^zE₁(z),

by expansion of the integrand in a string of Taylor series' along the real s-axis for any complex parameter z, accurate within ±1 of the last digit of seven-digit computation. Accuracy is verified by comparison with existing tables of E₁ and related integrals. This method is used to assess the accuracy of the error estimates of all subsequent computations.Three errors of a Taylor series are identified. These consist of a Taylor series truncation error, a digital truncation error, and a stability error. Methods are developed to estimate the error. By iteration a numerical radius of convergence for a given accuracy is determined.The z-plane is divided into three regions in which three different types of series are used to expand the function f(z) directly in z. Around the center the well-known Frobenius series is used. In the outer region the well-known asymptotic approximation is used. Their accuracy boundaries are determined. In the near-annular region in between, a set of Taylor series is introduced.As the result, the function f(z) can be computed fast with the appropriate series for any complex argument z, to an accuracy within less than relative error of 5 × 10⁻⁷.     

A new empirical model to retrieve soil moisture and roughness from C-band radar data

A new empirical model for the retrieval, at a field scale, of the bare soil moisture content and the surface roughness characteristics from radar measurements is proposed. The derivation of the algorithm is based on the results of three experimental radar campaigns conducted under natural conditions over agricultural areas. Radar data were acquired by means of several C-band space borne (SIR-C, RADARSAT) or helicopter borne (ERASME) sensors, operating in different configurations of polarization (HH or VV) and incidence angle. Simultaneously to radar acquisitions, a complete ground truth data base was built up with different surface condition measurements of the mean standard deviation (rms) height s, the correlation length l, and the volumetric surface moisture M_v. This algorithm is more specifically developed using the radar cross-section σ⁰ (HH polarization and 39° incidence angle off nadir), namely, σ_0HH,39, and the differential (HH polarization) radar cross-section Δσ₀=σ_0,23°−σ_0,39° in terms of an original roughness parameter, Z_s, namely Z_s=s²/l, and M_v. A good agreement is observed between model outputs and backscattering measurements over different test fields. Eventually, an inversion technique is proposed to retrieve Z_s and M_v from radar measurements.     

Listing closed sets of strongly accessible set systems with applications to data mining

We study the problem of listing all closed sets of a closure operator σ that is a partial function on the power set of some finite ground set E, i.e., σ:F→F with F⊆P(E). A very simple divide-and-conquer algorithm is analyzed that correctly solves this problem if and only if the domain of the closure operator is a strongly accessible set system. Strong accessibility is a strict relaxation of greedoids as well as of independence systems. This algorithm turns out to have delay O(|E|(T_F+T_σ+|E|)) and space O(|E|+S_F+S_σ), where T_F, S_F, T_σ, and S_σ are the time and space complexities of checking membership in F and computing σ, respectively. In contrast, we show that the problem becomes intractable for accessible set systems. We relate our results to the data mining problem of listing all support-closed patterns of a dataset and show that there is a corresponding closure operator for all datasets if and only if the set system satisfies a certain confluence property.