Free-edge Effects Research Articles

Free-Edge Effects in Composite Laminates

There are many technical applications in the field of lightweight construction as, for example, in aerospace engineering, where stress concentration phenomena play an important role in the design of layered structural elements (so-called laminates) consisting of plies of fiber reinforced plastics or other materials. A well known stress concentration problem rich in research tradition is the so-called free-edge effect. Mainly explained by the mismatch of the elastic material properties between two adjacent dissimilar laminate layers, the free-edge effect is characterized by the concentrated occurrence of three-dimensional and singular stress fields at the free edges in the interfaces between two layers of composite laminates. In the present contribution, a survey on relevant literature from more than three decades of scientific research on free-edge effects is given. The cited references date back to 1967 and deal with approximate closed-form analyses, as well as numerical investigations by the finite element method, the finite difference method, and several other numerical techniques. The progress in research on the stress singularities which arise is also reviewed, and references on experimental investigations are cited. Related problems are also briefly addressed. The paper closes with concluding remarks and an outlook on future investigations. In all, 292 references are included.

Plane Strain Test for Metal Sheet Characterization

This article shows the influence of a plane strain test specimen geometry on the measurable strain field and the influence of free edge effects over the stress computation. The experimental strain field distribution is measured over the whole deformable zone of a plane strain test specimen by an optical strain gauge. The chosen material is the DC06 IF steel of 0.8 mm thickness. The stress field is computed for several geometries at different strain levels by a Finite Element (FE) commercial code (Samcef ®). The results show that the stress field is sensitive to the specimen’s geometry and also to the tested material (strain field behavior is independent of material) and, based on results, an optimal specimen geometry is proposed in order to minimized the stress computation error.

407 Cross-Ply CFRP積層板の高サイクル疲労におけるトランスバースクラック及び層間はく離進展挙動の調査(高分子/高分子基複合材料1)

Damage growth behavior of transverse crack and delamination in cyclic loading was studied with cross-ply laminates of carbon fiber reinforced plastics (CFRPs) composite. Damage of the specimens during the fatigue test was observed by soft X-ray photography. It was found that delamination growth behavior depended on the inter-laminer stress and the out-plane stress in thickness direction caused by free-edge effect. Quasi-isotropic, [45/0/-45/90]_s, laminates, are applied large peering stress at free edges of the specimen when the specimen is subjected to tensile stress. Therefore they showed edge delamination. On the other hands, cross-ply, [0/90_6]_s laminates, are applied large inter-laminer shear stress at transverse crack tips, and show local delamination originated by a transverse crack. Under the high-cycle loading, it was observed that depending on the applied stress level delamination growth behavior differed.

An enhanced global–local higher-order theory for the free edge effect in laminates

According to the double-superposition hypothesis proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Methods Eng 1997;40:1197–212], an enhanced global–local higher-order theory (EGLHT- mn ) is developed to analyze the edge-effect problems in laminates. The in-plane displacement field consists of m th-order (9 ⩾ m ⩾ 3) polynomial in global coordinate z along the thickness direction and order 3 power series in local coordinate ζ within each layer whereas the transverse deflection is represented by a n th-order (9 ⩾ n ⩾ 3) polynomial of global coordinate z . By imposing the free surface conditions and the geometric and the stress continuity conditions at interfaces, the number of variables of the higher-order theory is independent of the number of layers of the laminates. As an improvement to the global–local theory proposed by Li and Liu (1997), the present higher-order theory takes into account all the effects of shear and normal stresses. A three-node triangular element satisfying the requirement of C 1 weak-continuity conditions between elements is also presented. Comparing to previous published results, it is found that the present higher-order theory is capable of treating free-edge problems in symmetric and unsymmetric laminates under extension, bending and thermal loading. Other characteristic of the present theory is that transverse shear stresses can be accurately computed directly from the constitutive equations without smoothing. However, to determine transverse normal stresses, the local equilibrium equation approach has to been adopted.

The Effect of Epoxy Molding Compound on Thermal/Residual Deformations and Stresses in IC Packages During Manufacturing Process

Thermal/residual deformations and stresses in plastic integrated circuit (IC) packages caused by epoxy molding compound (EMC) during the manufacturing process are investigated experimentally (only for deformations), theoretically, and numerically. A real-time Twyman-Green interferometry is used for measuring the out-of-plane thermal and residual deformations of die/EMC bi-material specimens. Dynamic mechanical analysis (DMA) and thermomechanical analysis (TMA) are for characterizing thermomechanical properties of the EMC materials. A finite element model (FEM) and theory associated with experimental observations are employed for understanding the thermal/residual deformations and stresses of IC packages due to EMC encapsulation. It is shown that EMC materials must be fully cured so that the material properties are stable enough for applications. Experimental results show that the EMC material experiences stress relaxation due to its viscoelastic behavior during the post mold curing (PMC) process. As a result, the strains (stresses) resulted from the chemical shrinkage of the EMC curing could be relaxed during the PMC process, so that the chemical shrinkage has no effect on the residual strains (stresses) for the plastic packages being post cured. Compared with numerical and theoretical analyses, the experimental results have demonstrated that die/EMC bi-material structure at high temperature (above Tg) warps less than expected, as a result of viscoelastic stress relaxation of EMC at high temperature (during solder reflow process). Meanwhile, this stress relaxation can also cause shifting this zero-stress temperature to the higher one, so that the residual deformations (stresses) of die/EMC bi-material specimens were found to increase by about 40% after the solder reflow process. The residual and thermal stresses have been resolved by FEM and theoretical analyses. The results suggest that the pure bending stresses (without shear and peel stresses) of the bi-material specimens are only limited in the region from x= 0 (the center) to x= 0.75 L due to the free edge effects, but this region is shrunk down to x= 0.4L at 200degC. And the maximum warpage and bending stress per unit temperature change is occurred around 165degC (Tg of the EMC). This study has demonstrated that the Twyman-Green experiment with associated bi-material plate theory and FEM can provide a useful tool for studying the EMC-induce residual/thermal deformations and stresses during the IC packaging fabrication

On kinematic coupling equations within the scaled boundary finite-element method

The scaled boundary finite element method is a semi-analytical analysis technique, which combines the advantages of the finite element method and the boundary-element method. Assuming that the geometry of the governing structure can be represented by mapping its boundary with respect to the so-called scaling coordinate, the problem can be handled in a closed-form analytical manner in the scaling direction and by a finite-element approximation in the other directions. Thus, a discretization of the boundary is sufficient and the nodal degrees of freedom are functions of the scaling coordinate. In some situations, such as the analysis of the free-edge effect in laminated plates, it is useful to introduce kinematic coupling equations, which are valid not only on the boundary, but also within the domain. The implementation of linear kinematic coupling equations within the method is presented for the case of a three-dimensional structure with scaling in a fixed Cartesian direction. Rigid-body modes are handled by using the concept of generalized inverse matrices. In some benchmark examples the efficiency of the approach is demonstrated and comparison with the results of the finite-element method shows good accordance.

The Pipes–Pagano-problem revisited: Elastic fields in boundary layers of plane laminated specimens under combined thermomechanical load

This paper addresses the development of a layerwise variational approach to free-edge effects in layered plates with arbitrary layups under thermomechanical load. In the course of this method, the physical layers of the laminate are discretized into an arbitrary number of mathematical layers through the plate thickness in each of which a C0-continuous displacement field is formulated. Beside terms corresponding to Classical Laminate Plate Theory, the layerwise displacement fields consist of unknown inplane functions in the layer interfaces and a linear interpolation through the layer thickness. The inplane functions are governed by systems of coupled Euler–Lagrange second order differential equations which are derived from the principle of minimum elastic potential and which in the presently considered cases allow for a closed-form analytical solution. The boundary conditions of traction free laminate edges are satisfied in an average sense by using an integral formulation. The present approach is straightforwardly applied, requires astonishingly low computational effort and is in excellent agreement with the results of other authors. Even though a discretization scheme is employed, the method allows for closed-form solutions for all state variables in the considered laminate which renders this approach a semi-analytical methodology.

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 5 n 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.

Strain-rate Dependence of Interlaminar Stress Fields in the Vicinity of Free Edges of AS4-PEEK Composite Laminates Under Uniaxial Tension

Using finite element program generator (FEPG) of Beijing Feijian Software Corporation, a relative finite element procedure is developed for the three-dimensional viscoplasticity model which can characterize well the nonlinear behavior and strain-rate dependence behavior of thermoplastic composite, AS4-PEEK. Interlaminar stresses of angle-ply laminates []sfor different strain rates under uniaxial tension are calculated. The effects of the variation of strain rates on interlaminar stresses in the vicinity of free edges of AS4-PEEK composite laminates are investigated. From the three-dimensional plots of the interlaminar stresses, it can be seen that distant free-edge effects and end-edge effects exist in the angle-ply laminates []s. The interlaminar stresses in free-edges and two ends increase with increasing strain rates. The extents to which the strain rates affect interlaminar stresses are depicted.

Influence of free edge intralaminar stresses on damage process in CFRP laminates under thermal cycling conditions

The purpose of this study is to understand the damage mechanisms of CFRP laminate subjected to repeated thermo-mechanical loads. Thermal cycling tests in oxygen show that the oxidation at high temperature associated with the “fatigue” phenomenon due to the cyclic thermal stresses lead to many matrix cracks. By means of optical microscopy and X-ray technique, some observations permit to characterize and quantify the initiation, build-up and propagation kinetics of these transverse cracks.A thermo-elastic calculation of the initial intralaminar transverse thermal stresses has been carried out in 3D finite-element models of the undamaged specimens. The comparison of experimental data and FEM results highlights a significant “free edge effect” on transverse matrix cracking onset, depending on both the direction of the edge referred to the fibres and the position of the layer in the lay-up. From this observation, a scenario of damage development dealing with a coupling between the local transverse stresses and the oxidation process is proposed.

Analysis of moisture-induced stresses in an FRP composites reinforced masonry structure

It has been shown in many studies that fiber reinforced polymer (FRP) composite laminates can be used to effectively strengthen structures constructed by unreinforced masonry units. For practical application of this reinforcement method, the long-term interface bonding degradation due to moisture and temperature environmental effects needs to be addressed further. In this study, a finite-element modeling procedure for analyzing moisture-induced stresses in a multi-layered structure constructed with distinct permeable materials was developed. The modeling procedure was used to analyze moisture-induced stresses in a concrete block reinforced with a unidirectional glass–epoxy FRP composite laminate partially covering one lateral surface. The nonlinear humidity transport properties of both the concrete and FRP materials were taken into account in the analysis, and the convergence issue of the interfacial shear stress components associated with the free edge effect was addressed by the use of the submodeling technique. It was demonstrated that the moisture-induced stresses at the FRP–concrete interface critical to structural integrity could be determined for any time instant. The results showed that the interfacial stresses increased with the increase of the humidity diffusion time and monotonically approached the stress level at the steady-state condition of the humidity diffusion. It was also shown that the analysis by assuming constant humidity transport properties resulted in a significant underestimation on the maximum interfacial stresses. The current finite-element analysis procedure provides a general method for determining moisture-induced stresses in a multi-layered permeable structure. It can be used to aid with the design of FRP–masonry structures or other similar structures for minimizing interfacial stresses induced due to the mismatch of moisture swelling properties of the constituent materials.

A Variational Model for Boundary Layer Effects in Cross-ply Laminates based on a C0-Continuous Layerwise Displacement Formulation

A variational approach to free-edge and free-corner effects in cross-ply laminates under thermal load is presented. Based on the discretization of the laminate into mathematical layers with respect to the thickness coordinate and the postulation of a C0-continuous layerwise displacement formulation with unknown in-plane functions and a linear interpolation through the layer thickness, the application of the principle of minimum potential yields a set of coupled ordinary differential equations, which govern the unknown in-plane functions. Due to some simplifying assumptions in the assumed displacement fields, the resultant Euler-Lagrange equations can be decomposed with respect to the two in-plane coordinates which eventually enables a closed-form solution. After fulfilling the given boundary conditions of traction-free laminate edges in an integral sense, the complete stress field in the cross-ply laminate is obtained in a closed-form manner. The approach allows for an easy application, requires very little computational effort, and is in excellent agreement with comparative finite element calculations.

Analyses of mode I edge delamination by thermal stresses in multilayer systems

Multilayer systems are generally subjected to thermal stresses because of the thermal mismatch between layers, and localized stresses normal to interfaces; i.e., interfacial peeling stresses, exist at edges because of the free-edge effect. These peeling stresses can result in mode I edge delamination. However, because of the complexity of the problem, exact closed-form solutions for peeling stresses are unattainable, even for bilayer systems. As a result, the peeling moment resulting from these localized peeling stresses was proposed recently as an alternative to study edge delamination. Exact closed-form solutions for the peeling moment at each interface in multilayer systems are derived in the present study, and approximate solutions for the special case of multilayer of thin films on a thick substrate are also presented. Specific results are calculated for GaAs/Si bilayers and GaP/GaAs 0.5P 0.5/GaAs trilayers. However, the present solutions are readily to be applied to any elastic multilayer system for characterizing mode I edge delamination by thermal stresses.

3D quantitative image analysis of open-cell nickel foams under tension and compression loading using X-ray microtomography

The deformation behaviour and fracture of an open-cell nickel foam were analysed using X-ray microtomography at the ESRF, Grenoble, France. In situ tensile and compression tests were performed at a resolution of 2 and 10 μm. The initial morphology of the foam was studied using 3D image analysis. Parameters such as the cell volume and strut length distributions, number of faces per cell, number of nodes per face and the shape of the most representative cells were determined. The cells are shown to be non-spherical due to the initial geometrical anisotropy of the polyurethane foam template and to the load applied to the nickel foam during processing. This geometrical anisotropy is shown to be related to the observed anisotropy of the elastic properties of the material using a simple beam model. In tension, bending, stretching and alignment of struts are observed. A tensile test in the longitudinal direction is shown to reinforce the privileged orientations of the cells. In contrast, a tensile test in the transverse direction leads to a more isotropic distribution of the cells. These features are illustrated by pole figures of the three axes of equivalent ellipsoids for all cells at different strain levels. Compression tests are associated with strain localization phenomena due to the buckling of struts in a weaker region of the foam. Finally, study of open-cell nickel foam fracture shows that cracks initiate at nodes during tensile tests and that the damaged zone is about five cells wide. Free edge effects on crack initiation are also evidenced.

Thickness effect on the compressive strength of T800/924C carbon fibre–epoxy laminates

In this study, the effect of laminate thickness on the compressive behaviour of composite laminates was investigated through systematic experimental work using the stacking sequences, [04]ns, [45/0/−45/90]ns and [45n/0n/−45n/90n]s (n=2–8) with a 3 mm diameter open hole at the centre. Parameters such as fibre volume fraction, void content, fibre waviness and interlaminar stresses, influencing compressive strength with increasing laminate thickness were also studied experimentally and theoretically. Furthermore the stacking sequence effects on failure strength of multidirectional laminates were examined. For this, two different scaling techniques were used: (1) ply-level technique [45n/0n/−45n/90n]s and (2) sublaminate level technique [45/0/−45/90]ns. An apparent thickness effect existed in the lay-up with blocked plies, i.e. unidirectional specimens ([04]ns) and ply-level scaled multidirectional specimens ([45n/0n/−45n/90n]s). Fibre waviness and void content were found to be main parameters contributing to the thickness effect on the compressive failure strength. However, the compressive strength of the sublaminate level scaled specimens ([45/0/−45/90]ns) was almost unaffected regardless of the specimen thickness (since ply thickness remains constant). From the investigation of the stacking sequence effect, the strength values obtained from the sublaminate level scaled specimens were slightly higher than those obtained from the ply level scaled specimens. This applied to all specimen thicknesses regardless of the presence of an open hole. The reason for this effect was explained by the fibre waviness, void content, free edge effect and stress redistribution in blocked 0° plies and unblocked 0° plies. Finally the average strengths of open hole specimens obtained from both stacking sequences increased with increasing specimen thickness. The measured failure strengths were compared with the predicted values.

Thermoelastic Fields in Boundary Layers of Isotropic Laminates

An approximate approach to the calculation of displacements, strains, and stresses near edges and corners in symmetric rectangular layered plates of dissimilar isotropic materials under thermal load is presented. In the thickness direction the plate is discretized into an arbitrary number of sublayers/mathematical layers. A layerwise linear displacement field is formulated such that the terms according to classical laminate plate theory are upgraded with unknown in-plane functions and a linear interpolation scheme through the layer thickness in order to describe edge and corner perturbations. By virtue of the principle of minimum potential energy the governing coupled Euler–Lagrange differential equations are derived, which in the case of free-edge effects allow a closed-form solution for the unknown inplane functions. Free-corner effects are investigated by combining the displacement formulations of the two interacting free-edge effects. Hence, all state variables in the plate are obtained in a closed-form manner. Boundary conditions of traction free plate edges are satisfied in an integral sense. The present methodology is easily applied and requires only reasonable computational expenses.

Numerical Investigations of Perforated Laminates in the Presence of Residual Ply Stresses

Finite element-based first-ply failure strength investigations of perforated laminates in the presence of residual ply stresses are presented. Unitcell models are applied, where inter-and intra-laminar failures as well as free-edge effects are taken into account on the ply level. The critical stress states contain constant and variable stress parts, which mean that nonradial loading paths are considered. Failure envelopes are computed for different layups and residual stress levels. This way the strength evaluation of structures containing perforated laminates can be performed on the macroscopic, i.e. structural, level.

Free-edge effect in cross-ply laminated plates under a uniform extension

Based on the basic equations of elasticity, free-edge effects on stresses in cross-ply laminated plates are found by using the state space method. The laminates are subjected to uniaxial-uniform extension plate, which is a typical example of general plane strain problem. The study takes into account material constants of all individual material layers and the state equation of a laminate is solved analytically in the through thickness direction. By this approach, a composite plate may be composed of an arbitrary number of orthotropic layers, each of which may have different material properties and thickness. The solution provides a continuous displacement and inter-laminar stress fields across all material interfaces and an approxiamte prediction to the singularity of stresses occurring in the boundary layer region of a free-edge. Numerical solutions are obtained and compared with the results obtained from an alternative numerical method.

Coupled and uncoupled analyses of piezoelectric free-edge effect in laminated plates

The present paper considers the effect of electromechanical coupling on the interlaminar stresses and the electric field strengths at free edges of laminated plates with piezoelectric material properties. A comparison of coupled and uncoupled piezoelectric analyses is performed using the finite element method. Two symmetric cross-ply plates and one symmetric angle-ply laminate are investigated under uniaxial tension. It is shown that the interlaminar stresses at the free edge are significantly higher in the coupled case for cross-ply laminates, whereas the coupling effect for the symmetric angle-ply laminate is of minor significance. In addition, the occurrence of electric field strengths with singular character is revealed.

Straight free-edge effects in laminated composites

A new three-dimensional hybrid stress element with a traction-free planar surface has been developed for efficient analysis of the interlaminar stresses near traction-free straight boundaries. All 3D stress components are included and are interpolated independently within each layer. The expressions of six stress components are derived, such that the homogeneous equilibrium equations in each layer, the continuity of transverse stresses on the interlayer surface, the traction-free boundary conditions over the planar surface and lower surface of the laminated composite are satisfied a priori, while the interelement reciprocity condition is satisfied a posteriori through a Lagrange multiplier method. The use of such special element in the finite element solution has been shown to be highly accurate when only a very coarse element mesh is used near the free-edge.