Abstract

The design and analysis of composite structures in the form of layered plates or shells is often driven by stress concentration phenomena that occur due to geometric or material discontinuities. One prominent example is the so-called free-edge effect that manifests itself in the form of significant localized interlaminar stress fields in the vicinity of free laminate edges and that is given rise to due to the mismatch of the elastic properties of the individual laminate layers. The free-edge effect has been under scientific investigation for more than five decades, and this paper aims at providing an overview of recent developments and scientific advances in this specific field wherein an emphasis is placed on investigations that were published in the time range between the years 2005 and 2020. This paper reviews closed-form analytical methods as well as semi-analytical and numerical analysis approaches and summarizes the recent state of the art concerning the investigation of stress singularities and experimental characterization of free-edge effects. This paper also reviews advanced problems such as free-edge effects in curved laminated structures and in piezoelectric laminates as well as in the vicinity of holes and other geometric discontinuities, and two new aspects in the field of free-edge effects, namely, the development and application of a new semi-analytical method (the so-called scaled boundary finite element method (SBFEM)) and the fracture mechanical strength assessment, also by novel approaches such as finite fracture mechanics, are also discussed. This paper closes with a summary and an outlook on future investigations.

1 Introduction

1.1 The Free-Edge Effect.

The last five decades have seen a significant shift in the lightweight engineering industry from classical metallic structures to the use of fiber-reinforced composite materials in the form of composite laminated plates and shells (Fig. 1). Composite laminates are thin-walled layered structures wherein the fibers (in many technical applications carbon or glass fibers) are embedded in a matrix (, e.g., a polymer matrix) oriented in certain directions where the laminate architecture—i.e., the layup and the specific orientation angles of the individual layers—is designed according to the specific purpose of the considered structural element. The main advantages of composite laminated structures are high stiffness and strength properties at a relatively low density and advantageous fatigue properties, thus making such material especially suitable for lightweight engineering applications such as aeronautics and astronautics, but also for use in automotive or civil engineering. Quite naturally, the layered architecture of composite laminates, along with the inherent orthotropy of fiber-reinforced materials, make the structural response and thus also the analysis of such structures much more sophisticated than is the case when classic isotropic materials such as steel or aluminum are treated. For this purpose, a number of laminated plate theories have been developed over the last decades (the interested reader is referred to the textbooks [17]), and the most common one is the so-called classical laminated plate theory (short: CLPT). CLPT is based on the kinematics of Kirchhoff's plate theory, i.e., straight lines normal to the laminate middle plane remain normals and straight in the deformed state and do not undergo extensions while a state of plane stress is assumed with respect to the thickness direction of the laminate. The constitutive behavior of a laminate in the framework of CLPT can be represented in the following form:
(NxxNyyNxyMxxMyyMxy)=[A11A12A16B11B12B16A12A22A26B12B22B26A16A26A66B16B26B66B11B12B16D11D12D16B12B22B26D12D22D26B16B26B66D16D26D66](εxx0εyy0γxy0κxxκyyκxy)
(1)
Fig. 1
An exemplary composite laminate, consisting of a number of fiber-reinforced layers
Fig. 1
An exemplary composite laminate, consisting of a number of fiber-reinforced layers
Close modal

Therein, Aij=hQ¯ijdz are the membrane stiffnesses of the laminate, Bij=hQ¯ijzdz are the so-called coupling stiffnesses, and Dij=hQ¯ijz2dz denote the bending stiffnesses (with i,j=1,2,6), wherein Q¯ij are the transformed reduced stiffnesses of the individual laminate layers under the assumption of a plane state of stress. The quantities Nαβ and Mαβ (α,β=x,y) are the resultant force and moment fluxes, respectively. The laminate midplane strains are expressed by the displacements u0, v0, and w0 and can be represented as εxx0=u0/x,εyy0=v0/y, and γxy0=u0/y+v0/x. An index 0 signifies quantities that are defined with respect to the laminate middle plane at z =0. The curvatures and the twist of the laminate middle plane are defined as κxx=2w0/x2,κyy=2w0/y2,κxy=22w0/xy. For a homogeneous isotropic or orthotropic plate, A16=A26=Bij=D16=D26=0 holds so that CLPT includes classical thin-walled structures such as isotropic disks and plates as a special case. A laminate is called balanced if for each layer with the fiber angle θ another layer with the angle θ exists from which A16=A26=0 follows. Symmetric laminates always exhibit vanishing coupling stiffnesses so that Bij = 0. Cross-ply laminates are structures where the fiber angles θ are either θ=0deg or θ=90deg for which A16=A26=D16=D26=0. Laminates where the fiber orientation angles always appear in pairs [±ϑ] are commonly addressed as angle-ply laminates for which A16=A26=0 holds.

Classical laminated plate theory, being essentially a two-dimensional theory, is based on the assumption of a plane stress state with respect to the thickness direction z of the laminate. A direct result of these assumptions is that the shear strains γxz=γyz=0 vanish within this laminate theory. Consequently, the so-called interlaminar shear stresses τxz=τyz=0, i.e., the shear stresses in the thickness direction, cannot be determined by use of a constitutive law. As a result, the interlaminar normal stress σzz is not accessible either when CLPT is employed. Nonetheless, CLPT delivers results of sufficient accuracy for many engineering tasks where a “global” structural response is the aim of the computations such as the determination of deflections of composite laminated plates and shells under static load, or the analysis of the buckling loads and eigenfrequencies of thin-walled composite laminated structures. However, there are certain problems that are inherent to laminated composite structures which require analysis frameworks beyond the capabilities of CLPT.

A prominent example for such advanced problems is the so-called free-edge effect that manifests itself by significant and potentially singular localized three-dimensional stress fields in the vicinity of interfaces between dissimilar laminate layers. Quite naturally, it is outside the scope of a laminate theory such as CLPT to capture such interlaminar stress fields so that more advanced means of analysis need to be employed. Free-edge stress fields are known to significantly perturb the plane state of stress that can be expected to occur in the innermost regions of a laminated plate and that can be adequately described by CLPT or other two-dimensional laminate theories. It is well-known that interlaminar free-edge stress fields decay rapidly with increasing distance from the laminate edges, and in many cases it can be anticipated that significant interlaminar stresses arise in a free-edge region with inplane dimensions of about one laminate thickness.

The free-edge effect can be motivated by considering a plane tensile specimen (thickness h) with a four-layered cross-ply layup [0deg/90deg]S (layer thickness h/4) under plane strain εxx as shown in Fig. 2. The coordinate axes x, y, and z are oriented as indicated and have their origin in the laminate middle plane that is generated by the x- and y-axis wherein x is the longitudinal axis. Unless mentioned otherwise, the transverse axis y is measured from the free edge pointing inward. The specimen given in Fig. 2 is assumed to be infinitely long in the longitudinal direction x so that under this elementary load case, all stresses and strains as well as the displacements v and w are uncoupled from x and are only functions of y and z. An analysis of this elementary situation by CLPT will predict a stress state consisting of the two inplane normal stresses σxx and σyy whereas in a pure cross-ply layup under inplane strain εxx no inplane shear stress τxy will occur. As outlined earlier, the interlaminar stresses τxz, τyz and the interlaminar normal stress σzz (which is oriented perpendicularly to the fiber direction and thus especially critical) cannot be determined by CLPT though their occurrence can be explained rather straightforwardly by the situation given in Fig. 2.

Fig. 2
The free-edge effect in a symmetric cross-ply laminate under uniaxial tension in the x-direction: If the individual layers are considered without a fixed bond between them (top left), strongly heterogenic transverse strains due to the different fiber orientations occur. In a laminate with firmly bonded layers (center left), intralaminar transverse stresses σyy and thus also interlaminar shear stresses τyz (detail A) must therefore arise. For reasons of equilibrium (see the free-body diagram of the upper laminate layer shown on the lower right), this causes peel stresses/interlaminar normal stresses σzz (detail B), which in this specific case occur as tensile stresses in the vicinity of the free edge and can be the trigger for edge delamination. Characteristic distributions of the interlaminar stresses τyz and σzz are shown in the lower part of this figure. The occurrence of such local interlaminar stress fields is commonly referred to as free-edge effect.
Fig. 2
The free-edge effect in a symmetric cross-ply laminate under uniaxial tension in the x-direction: If the individual layers are considered without a fixed bond between them (top left), strongly heterogenic transverse strains due to the different fiber orientations occur. In a laminate with firmly bonded layers (center left), intralaminar transverse stresses σyy and thus also interlaminar shear stresses τyz (detail A) must therefore arise. For reasons of equilibrium (see the free-body diagram of the upper laminate layer shown on the lower right), this causes peel stresses/interlaminar normal stresses σzz (detail B), which in this specific case occur as tensile stresses in the vicinity of the free edge and can be the trigger for edge delamination. Characteristic distributions of the interlaminar stresses τyz and σzz are shown in the lower part of this figure. The occurrence of such local interlaminar stress fields is commonly referred to as free-edge effect.
Close modal
We now consider the case that the layers of the cross-ply laminate under consideration are not bonded at all and are free to deform independently under the imposed longitudinal extension (Fig. 2, top left). Since the 0deg and the 90deg-layers exhibit largely different contraction properties, it is clear that in this case the outer 0deg-plies will suffer much greater displacements v than the inner 90deg-layers. As a result, the inplane displacement v will become discontinuous at the interfaces between the 0deg and the 90deg-layers. In an actually bonded laminate, this discontinuity of v is of course not possible. Consequently, inplane normal stresses σyy would have to be applied in order to equalize the displacement v in the interfaces (Fig. 2, middle left, where σyy is depicted as an edge stress) wherein at the 0deg-layers a tensile stress would need to be applied while at the 90deg-plies compressive stresses would arise so that v becomes continuous at the interface. Quite apparently, the values of these stresses σyy are exactly the stresses that would result from a computation of this specific situation using CLPT. Since this group of inplane stresses is required to be self-equilibrating the absolute value of σyy in each layer is identical. Equilibrium of forces in the y-direction thus yields
h2h2σyydz=0
(2)
The given situation dictates that the inplane stresses σyy that are depicted as edge stresses in Fig. 2, middle, of course have to vanish at the free laminate edge y =0. As a result, the interlaminar shear stress τyz will arise in the vicinity of the free edge in order to maintain equilibrium of forces in the ydirection (see the free-body diagram of the 0deg-layer in Fig. 2 where the interlaminar shear stress τyz is shown at the interface at z=h4, detail A). A typical distribution of τyz along the interface between the 0deg and the 90 deg-layer for the given situation is shown in Fig. 2, bottom left. Equilibrium of forces in the y-direction in this interface requires that the condition
0yτyzdŷ=h4h2σyydz
(3)

holds for sufficiently large values of y.

The resultants of the stresses σyy and τyz as shown in the free-body diagram given in Fig. 2 do not share a common line of action so that equilibrium of moments about the xaxis dictates that the interlaminar normal stress σzz will have to arise as well in the interface between the 0deg and the 90 deg-layers at z=h4 (cf. Fig. 2, detail B). The following equilibrium condition can be deduced:
0yσzzŷdŷ=h4h2σyy(zh4)dz
(4)
As there is no other stress component acting in the z-direction at z=h4 at the given example, the interlaminar normal stress σzz needs to be self-equilibrating, i.e.,
0yσzzdŷ=0
(5)

given that sufficiently large values of y are considered. In order for Eq. (5) to be fulfilled, σzz will have to change its sign at least once along the y-direction, and a typical distribution of σzz in the interface between the 0deg and 90 deg-layer is given in Fig. 2, bottom right.

For the given situation, it is important to point out that σzz occurs as a tensile stress at the free edge, thus acting as a tensile stress perpendicularly to the fiber directions of both adjacent layers and making this laminate especially prone to delamination failure in the vicinity of the free edge. It is a direct result of the theory of linear elasticity that due to the assumed discontinuous change of the elastic properties in the interface between the 0deg and 90 deg-plies σzz is governed by a mathematical singularity at the point y =0, z=h4, thus rendering this particular location of the laminate especially critical concerning the onset of delaminations.

Figure 3 shows a three-dimensional representation of the stress field in the vicinity of the free edge of a cross-ply laminate [0deg/90deg]S (taken from Mittelstedt and Becker [8]). The given stress results clearly show that while the interlaminar stress fields become significant in the interfaces between dissimilar layers directly at the free edge, they decay rapidly so that in regions at a certain distance from the free edge, the stress field according to CLPT is restored. It can generally be assumed that free-edge stress fields decay after a distance from the free edge that amounts to about one laminate thickness h (see also Fig. 2).

Fig. 3
Free-edge stress fields in a symmetric cross-ply laminate under uniaxial extension [8]: Besides the intralaminar stresses σxx and σyy, the interlaminar stresses τyz and σzz also occur, wherein especially the latter stress component shows a highly localized edge concentration which is governed by a stress singularity and which may trigger edge delaminations
Fig. 3
Free-edge stress fields in a symmetric cross-ply laminate under uniaxial extension [8]: Besides the intralaminar stresses σxx and σyy, the interlaminar stresses τyz and σzz also occur, wherein especially the latter stress component shows a highly localized edge concentration which is governed by a stress singularity and which may trigger edge delaminations
Close modal
Interlaminar stress concentrations can also be motivated by considering a symmetric angle-ply laminate under uniaxial extension εxx. Figure 4 shows an angle-ply laminate with the layup [±45deg]S. First, consider the case of unbonded layers where the laminate plies can deform independently from each other (Fig. 4, top left). Due to the imposed axial extension εxx the unbonded layers will exhibit inplane shear strains γxy where the shear strains in the +45deglayers will exhibit a different sign than those in the 45degplies wherein the absolute values will be identical. As a result, this will lead to a discontinuity of the longitudinal displacement u in the interfaces between the +45deg and the 45deglayers. In order to maintain continuity of the interface displacements, intralaminar shear stresses τxy will be called forth (Fig. 4, middle, depicted as edge stresses), with magnitudes as they would result from a calculation using CLPT. Given the present load case and structural situation, the inplane shear stress τxy is self-equilibrating, thus
h2h2τxydz=0
(6)
Fig. 4
On the occurrence of the free-edge effect in an angle-ply layup under uniaxial extension: due to the mismatch in the inplane shear behavior of the individual laminate layers (top left and bottom left), the intralaminar shear stress τxy is called forth (center left). Equilibrium of forces in the longitudinal direction (center right) gives rise to the interlaminar shear stress τxz (detail A and bottom right) that is governed by a singularity at the free edge of the laminate.
Fig. 4
On the occurrence of the free-edge effect in an angle-ply layup under uniaxial extension: due to the mismatch in the inplane shear behavior of the individual laminate layers (top left and bottom left), the intralaminar shear stress τxy is called forth (center left). Equilibrium of forces in the longitudinal direction (center right) gives rise to the interlaminar shear stress τxz (detail A and bottom right) that is governed by a singularity at the free edge of the laminate.
Close modal

Accordingly, a calculation using CLPT will lead to layerwise shear stresses τxy with identical absolute values yet opposite signs wherein in the +45deglayers τxy will exhibit a positive sign while in the 45deglayers τxy will be negative. Of course, edge stresses τxy cannot occur due to the boundary conditions of traction free laminate edges so that as a result the interlaminar shear stress τxz will occur (Fig. 4, middle) in the interfaces between the +45deg and the 45deglayers at z=±h4. This can also be explained by the free-body diagram given in Fig. 4 where the appearance of τxz can be understood by the requirement of equilibrium of forces in the xdirection (see also detail A).

Given that the laminate has a sufficient width concerning the y-direction so that in the innermost regions, CLPT can be assumed to hold, from the free-body diagram given in Fig. 4 the following equilibrium condition can be deduced:
0yτxzdŷ=h4h2τxydz
(7)

A characteristic distribution of τxz in the interface between the +45deg and the 45deglayers at z=±h4 with respect to the ydirection is given in Fig. 4, bottom right. The interlaminar shear stress τxz exhibits its maximum value at the free edge where it is dominated by a mathematical singularity and decays rapidly with increasing distance from the free edge. The typical warping deformations as they occur in angle-ply laminates under uniaxial extension are given qualitatively in Fig. 4, bottom left.

Summing up, the free-edge effect in angle-ply layups is called forth by the mismatch of the elastic shear properties between dissimilar adjacent layers and is mainly characterized by the appearance of the interlaminar shear stress τxz which is known to decay rapidly remote from the free edge. Figure 5 shows three-dimensional representations of the interlaminar shear stress τxz as well as the interlaminar normal stress σzz that is also known to play a role when free-edge effects in angle-ply laminates are considered. The results given in Fig. 5, taken from Mittelstedt and Becker [8], demonstrate that these two interlaminar stresses are dominating the free-edge stress field while decaying rapidly with increasing edge distance.

Fig. 5
Free-edge stress fields in a symmetric angle-ply laminate under uniaxial extension [8]: this example of a free-edge effect is characterized by the two interlaminar stress components τxz and σzz which are confined to a small edge region and which are governed by a stress singularity
Fig. 5
Free-edge stress fields in a symmetric angle-ply laminate under uniaxial extension [8]: this example of a free-edge effect is characterized by the two interlaminar stress components τxz and σzz which are confined to a small edge region and which are governed by a stress singularity
Close modal

1.2 Scope of This Paper.

An exact solution of the governing differential equation system that describes the free-edge effect, in conjunction with the underlying boundary and continuity conditions, does not exist. This explains the considerable scientific efforts in establishing closed-form analytical, semi-analytical, and numerical analysis approaches for the analysis of free-edge effects in composite laminates over a time span of more than 50 years since the pioneering work of Pipes and Pagano [9] in 1970. A number of review papers exist (see Refs. [1016]), and the current paper understands itself as a follow-up contribution to the review paper by Mittelstedt and Becker [16] that summarized references from 1967 to 2005. The current paper places its emphasis on the survey and summary of relevant references concerning the free-edge effect in composite laminates wherein closed-form analytical and semi-analytical methods are discussed in Sec. 2, followed by Sec. 3 that is devoted to numerical methods. Sections 47 deal with advanced problems concerning the free-edge effect in composite laminates, namely, stress concentrations in laminated shells (Sec. 4), free-edge effects at curved edges of holes in laminates (Sec. 5), free-corner effects (Sec. 6), and free-edge effects in piezoelectric laminates (Sec. 7). A novel semi-analytical method especially suited for the analysis of stress concentration phenomena in composite laminates is the so-called scaled boundary finite element method (SBFEM), which is the topic of Sec. 8. Section 9 summarizes references that report experimental data, and Sec. 10 deals with the analysis of the onset and propagation of free-edge delaminations. This paper closes with a summary and an outlook on future investigations in Sec. 11.

2 Closed-Form and Semi-Analytical Methods

An analytical method was developed [8,1721] employing an inner solution using CLPT and predicting the free-edge effects using mathematical layers obtained by discretization of the physical plies with respect to the thickness direction, wherein the displacement field for each mathematical layer consists of unknown interlaminar in-plane functions and those interpolated along the numerical layer thickness. Besides CLPT, the first-order shear deformation theory (FSDT) and a layerwise laminate theory have been employed as well for predicting the unknown parameters that exist in the reduced displacement field of elasticity and for the calculation of the local interlaminar stresses within the boundary layer areas of the laminate [22,23].

A closed-form analytical solution to determine the in-plane functions can be obtained using the principle of minimum elastic potential, which generates a set of coupled Euler–Lagrange second-order differential equations. The following layerwise displacement functions (see Fig. 6) for layer (k) were employed in Refs. [8] and [17]:
u(k)=U10(k)(x,y)+U11(k)(y,z)v(k)=U20(k)(y)+U21(k)(y,z)w(k)=U30(k)(z)+U31(k)(y,z)
(8)

where the displacement functions v and w are uncoupled from the longitudinal axis x. Moreover, for the laminate under the applied load, U10(k) and U20(k) are linear along the depth and are obtained by integrating the CLPT in-plane strains. The displacement component U30(k) is a layerwise linear function through z. Furthermore, the perturbation terms are given as U11(k),U21(k), and U31(k), which are supposed to include unknown in-plane interlaminar components, multiplied by linear interpolation functions through the numerical layer thickness. Through the plies, inside the laminate, CLPT is supposed to hold, whereas the above-mentioned functions, which are assumed to demonstrate the free-edge perturbations, are required to decay completely with increasing distance from the free edges.

Fig. 6
Analysis model for the free-edge effect in a laminate under thermomechanical load, Mittelstedt and Becker [8,17]: the individual physical layers of the laminate are subdivided into an arbitrary number of mathematical layers
Fig. 6
Analysis model for the free-edge effect in a laminate under thermomechanical load, Mittelstedt and Becker [8,17]: the individual physical layers of the laminate are subdivided into an arbitrary number of mathematical layers
Close modal
The layerwise perturbation terms are given as
U11(k)(y,z)=U1(k)ψ1(k)+U1(k+1)ψ2(k)U21(k)(y,z)=U2(k)ψ1(k)+U2(k+1)ψ2(k)U31(k)(y,z)=U3(k)ψ1(k)+U3(k+1)ψ2(k)
(9)

wherein the quantities ψ1(k) and ψ2(k) are layerwise linear interpolation functions, interpolating between the perturbation functions U1, U2, and U3 in the layer interfaces. Selected results taken from Ref. [8] are shown in Fig. 7, which highlights the accuracy of the analysis method, at a fraction of the computational cost of full-scale finite element method (FEM) computations.

Fig. 7
Interlaminar normal stress σzz in the interface of a cross-ply laminate [0 deg/90 deg]S, comparison with the results of Pipes [18], Pagano [19], Kassapoglou and Lagace [20] (top); interlaminar stresses through the thickness of a quasi-isotropic laminate, comparison with results by Reddy [1] (bottom)
Fig. 7
Interlaminar normal stress σzz in the interface of a cross-ply laminate [0 deg/90 deg]S, comparison with the results of Pipes [18], Pagano [19], Kassapoglou and Lagace [20] (top); interlaminar stresses through the thickness of a quasi-isotropic laminate, comparison with results by Reddy [1] (bottom)
Close modal

The closed-form shapes of the stresses through the interfaces at the free edges of finite length composite laminates under tension or shear conditions can be determined utilizing the three-dimensional multiterm extended Kantorovich method [24,25]. The principle of minimum elastic potential determines the system of governing coupled ordinary differential equations. Bending-torsion conditions and thermal loading effects can be studied by this method [26]. Kapuria and Dhanesh [27] improved this method to the mixed-field multiterm extended Kantorovich method form (MMEKM). This method causes a quicker convergence with high validity in predicting stresses at the free edges of composite structures in different loading conditions using an analytical three-dimensional elasticity approach [28]. Kapuria and Kumari [29] improved the single-term extended Kantorovich method basically presented in 1968 [30] for two-dimensional elasticity problems to the three-dimensional approach for composite laminates. However, the stresses close to the edges could not be estimated correctly using the single-term method. Therefore, they generalized it to the multiterm solution [31]. For this purpose and for ensuring an identical validity order in finding stress and displacement fields, they utilized the Reissner-type mixed variational principle. It should be mentioned that numerous important elasticity problems, such as the free-edge effects in composite laminates in various loading conditions can be solved precisely by the above-mentioned method. Kapuria and Dhanesh [32] analyzed free-edge effects in flat composite plates with imperfections through the interfaces using a three-dimensional elasticity approach. They generalized the MMEKM for the three-dimensional (3D) investigation of completely bonded composite panels to analyze the displacement discontinuities and consequently involve the interfacial compliance. Furthermore, Kumari et al. [33] proposed another 3D elasticity solution for composite panels with Levy-type boundary conditions by improving the above-mentioned MMEKM in conjunction with the Fourier series.

A stress-based equivalent single-layer has been applied as a useful method in the preliminary design of structural components in the presence of noticeable free-edge effects. Moreover, it has been found that in comparison with a displacement-based solution, the stress-based approach is appropriate to solve the boundary value problem with prescribed stresses efficiently and accurately [34]. Lee et al. [34] determined stresses through the interfaces of a laminated patch under bending load. Moreover, stress-based functions in the framework of a single layer theory were used by Huang et al. [35], for the determination of free-edge stress fields under consideration of viscoelastic effects, with proposing a closed-form solution.

The state space equation method can be employed for the determination of continuous shape functions of stresses and displacements through the interfaces of the plies in the vicinity of free edges and cracks [36]. Furthermore, a novel layerwise model called SCLS1 (Statically Compatible Layerwise Stresses) for multilayered plates was presented by Baroud et al. [37].

For the analysis of laminates containing delaminations under uniaxial extension, a method has been proposed that in its first part [38] extended the application of LS1 (Layerwise Stress Model). In the second part [39], a refined approach was presented for the determination of free-edge effects and stress concentrations at the tips of cracks in composite plates, specifically those with angle-ply stacking sequence.

Hamidreza Yazdani Sarvestani and Mohammadreza Yazdani Sarvestani [40] obtained the stresses throughout the interfaces close to the free edges of laminates under different loading conditions using a closed-form solution. Elasticity displacement functions were developed for a long laminate. FSDT has been applied to obtain coefficients that exist in the displacement shape function, thus enabling the prediction of the general deformation of the plate. For examination of the stresses of the boundary layer through a laminate, a layerwise theory was utilized analytically and numerically. In another research, Yazdani Sarvestani et al. [41] utilized the above-mentioned method for cross-ply plates under bending. Furthermore, they proposed an analytical solution using layerwise theory according to Reddy [42]. Moreover, Yazdani Sarvestani and Naghashpour [43] obtained the free-edge stresses throughout the interfaces in composite plates under various loading conditions using higher-order equivalent single-layer theory. After obtaining the three-dimensional stress field through the whole laminate, it was concluded that the computational effort has been reduced by using a higher-order equivalent single-layer theory in comparison with a layerwise theory.

The readers are suggested to refer to Refs. [4455] for further information about closed-form analytical and semi-analytical methods for the analysis of free-edge effects in composite laminates.

3 Numerical Investigations

It remains a very difficult problem to determine the singular stresses at free edges, tips of cracks, or notches in composite laminates, or through the interfaces of multimaterial junctions using a closed-form solution. FEM is a suitable alternative to overcome this problem. This method has been employed for the analysis of free-edge effects in numerous academic and industrial researches. In the current study, a selection of works has been listed that include utilizing FEM for obtaining interlaminar stresses and other localized effects such as the free-edge problem.

A meso-scale FEM model was used for the investigation of coupon specimens and a large plate made of triaxially braided composites subjected to extension [56]. The mentioned method was utilized to improve a representative unit cell model of a triaxial braided composite on the basis of the composite fiber volume ratio, specimen thickness, and microscopic image analysis in Ref. [57]. Researchers examined single-layer triaxially braided composites and evaluated the influence of the free edge on the mechanical response of using the present method and tested and analyzed straight-sided specimens under tensile load transversely in Ref. [58].

Vidal et al. [59] considered multilayered composite plates and employed an enriched model using higher order terms for the in-plane and the transverse displacements, and a layer refinement was assessed using a refined sinusoidal model. Numerous tests were performed to demonstrate the applicability and limitations of the solution. Furthermore, for the modeling of the free-edge effect in composite laminates, the separation of variables was investigated in Ref. [60].

Nguyen and Caron [61] proposed an FEM model resulting from the multiparticle model of multilayered materials (M4) using the coupling of Reissner plates and a developed eight-node multiparticle element. In another study, they employed a multiparticle finite element for laminates and showed that global and local responses can be predicted at the same time [62]. They performed a classical bending validation primarily and then analyzed the free-edge effects of composite laminates under various loading conditions using the above-mentioned eight-node layerwise finite element.

Lo et al. [63] improved enhanced global-local higher-order theory based on the double superposition hypothesis presented by Li and Liu [64] to study the free-edge effect in composite laminates. C1 weak continuity was satisfied in developing a 3-node triangular element. Zhen and Wanji [65], two researchers from the above-mentioned group [63], proposed a higher order displacement model in order to study the stress concentration effects in laminated plates. Both the transverse shear stress continuity and free surface conditions were satisfied by the model. The investigation of stress concentrations in composite laminates with arbitrary layup was performed using the presented model in Ref. [63]. Moreover, for the analysis of the stresses at curved free boundaries under in-plane loading conditions, they developed a higher order model [66] in the form of a single-layer approach. The capability of the solution was illustrated by numerical examples of curved free-edge problems to show that the developed model correctly depicts the stress field in the vicinity of a circular hole.

Ramtekkar and Desai [67] investigated the free-edge effect and the initiation of delamination at free edges of fiber-reinforced plastic composites using a layerwise 3D mixed FEM. Dhadwal and Jung [68] evaluated free-edge stresses through the interfaces of laminates utilizing a multifield FE beam sectional formulation based on the Hellinger-Reissner principle regarding a 3D material constitutive model. Islam and Prabhakar [69] developed a Quasi-two-dimensional plane strain formulation in the framework of FEM for estimating interlaminar stress fields in composite laminates.

Jain and Mittal [70] investigated the stress and displacement fields in composite laminates with a circular hole at the center of a rectangular plate under transverse static loading using FEM. Ahn et al. [71] investigated the distributions of stresses close to a circular hole of the composite plate in tensile conditions and proposed a p-convergent layerwise global–local model. The special capability of this model was to mix two-dimensional and 3D elements in a designed mesh. Zhao et al. [72] investigated the interlaminar stress fields through centrally notched angle-ply AS4/PEEK laminates using FEM. Furthermore, they studied centrally notched quasi-isotropic APC-2/AS-4 laminates under tensile load by FEM [73]. The analysis demonstrated that the interlaminar shear stress is more concentrated and its maximum value can be observed close to the hole edge. The methodology of using FEM for determining interlaminar stresses in composite plates containing circular holes also has been traced by Babu and Pradhan [74], Hosseini-Toudeshky et al. [75], and Suemasu et al. [76].

Espadas-Escalante et al. [77] studied plain woven composite laminates in tension conditions to determine the free-edge effects and relative layer shifting in the interlaminar stresses using FEM. Ballard and Whitcomb [78] studied the free-edge effect in a thin cross-ply laminated composite under uniaxial tension using FEM. It was concluded that the stress fields through the interfaces are affected by the Poisson ratios of the fibers and matrix remarkably. Also, the interlaminar normal stresses are less affected by the microstructure. In order to optimize the composite strain energy Hosoi and Kawada [79] presented a solution to estimate the 3D stresses in cross-ply laminates with transverse cracks in the 90 plies using the principle of minimum complementary energy.

Further numerical studies are available with e.g., Refs. [8096] and [97].

4 Free-Edge Effects in Laminated Shells

Interlaminar stresses in composite laminates are a phenomenon that is not only inherent to the free-edge effect but that also arises quite prominently in curved structures. Lekhnitskii [4,98] e.g., derived closed-form analytical solutions for anisotropic circular cylindrically curved beams and pointed out that due to the curvature of the structural elements, through-the-thickness stresses are induced which, consequently, pose a potential threat to delamination in laminated shells, even at a sufficient distance from the boundary-layer region.

Although two-dimensional approaches such as those provided by Lekhnitskii are not capable to predict the interlaminar stress concentrations in close proximity to the traction-free edges of laminated shells, they are frequently being employed in order to obtain an initial assessment of the underlying stress state in a highly efficient way. Recent works were published by González-Cantero et al. [99,100], who developed novel efficient semi-analytical models to compute the interlaminar stresses in composite laminates with highly curved parts under different mechanical loadings that are prone to delamination failure. Herein, the authors assumed two-dimensional conditions with respect to the axial direction and verified the quality of the approaches by means of finite element computations, which confirmed the good accordance between both methods. Thurnherr et al. [101,102], on the other hand, utilized the principle of minimum complementary energy in order to derive higher order beam models that are able to accurately predict the three-dimensional stresses in curved laminates, even in close proximity to localized features. On the basis of numerical computations and experimental results from the literature, Thurnherr et al. draw the conclusion that interlaminar stresses especially have to be considered during the assessment of thick curved composite beams since their studies showed that those structures will most likely fail due to delamination of adjacent plies. For the failure analysis of thin laminated shells, however, out-of-plane stress components play a rather secondary role and particular focus has to be played to in-plane loadings.

Concerning three-dimensional analysis methods, Miri and Nosier [103105] developed different displacement-based layerwise approaches to analytically investigate the free-edge effect in composite shells undergoing mechanical as well as hygrothermal loadings. Herein, Miri and Nosier were able to derive an appropriate reduced elasticity displacement field in which the unknown local displacement functions were specified in terms of Reddy's layerwise theory and verified by comparison with results obtained by first-order shear deformation shell theory. The analysis of curved sandwich structures, which are subjected to transverse loadings under consideration of the free-edge effect, was examined by Afshin et al. [106]. Herein, the face sheets were considered as cross-ply laminated shells and the core was modeled as isotropic, linear elastic medium. Numerical results revealed that the maximum magnitude of the interlaminar stress components occurs in close proximity to the traction-free edges of the considered sandwich structure and that the transverse stresses at the bottom face-core interface are larger than those at the top face-core interface. Another study concerning the hygrothermal stresses in the boundary-layer region of sandwich cylinders with arbitrary laminated faces was conducted by Ahmadi [107]. Herein, Ahmadi developed a displacement-based layerwise approach in which the through-the-thickness temperature as well as the moisture content has been modeled as uniform as well as nonuniform. Ahmadi [108,109] also developed three-dimensional Galerkin-based layerwise formulations for circular cylindrical composite shells with arbitrary laminations and thickness which were subjected to either pure bending or distributed transverse loads. The numerical results were verified via available results in the literature and numerous parametric studies revealed the influence of different geometric quantities on the free-edge stress components. Tahani et al. [110] presented an analysis method based on the 3DMTEKM in order to examine the free-edge stresses of thick laminated shell panels with general layer stacking subjected to pressure load under consideration of various boundary conditions. The interlaminar stresses in the boundary-layer region of cross-ply laminated shells were analyzed by Schnabel et al. [111] (Fig. 8). Similar to the approach of Mittelstedt and Becker [8], they combined a closed-form analytical solution initially introduced by Ko and Jackson [112] with a linear layerwise displacement-based approach (Fig. 8) and obtained the governing equations by making use of the principle of minimum total potential energy. The accuracy was confirmed by comparing the numerical results of the efficient semi-analytical method with full-scale three-dimensional finite element computations which, further on, revealed that although the underlying displacement, strain, and stress fields showed good agreement for several analyzed cases, the approach exhibited certain discrepancies concerning the through-the-thickness shear stresses. The employed FEM mesh is shown in Fig. 9, selected results for the interlaminar normal stress σrr in unsymmetric cross-ply laminated shells consisting of two layers are given in Fig. 10. Therein, the z-axis has its origin at the free edge of the curved laminate and is pointing inward. Kappel and Mittelstedt [113] modified the analysis method by Schnabel et al. [111] by introducing a higher order displacement-based approach and a new, adjustable discretization scheme with respect to the thickness direction of the composite shells. Comparisons with numerical results obtained by Schnabel et al. [111] as well as three-dimensional finite element calculations clearly demonstrated that those modifications lead to an improvement of both, quality of results concerning the out-of-plane stress components and computational efficiency. A third-order shear and normal deformable plate/shell theory wherein the transverse shear and the normal stress components are computed with a one-step stress recovery scheme was presented by Shah and Batra [114] in order to assess the stress concentrations arising in close proximity to the edges of doubly curved composite laminated shells undergoing tangential and normal loadings.

Fig. 8
Analysis model [111] for a curved laminate under bending (top and center): the individual physical layers of the curved laminate are subdivided into an arbitrary number of mathematical layers (bottom)
Fig. 8
Analysis model [111] for a curved laminate under bending (top and center): the individual physical layers of the curved laminate are subdivided into an arbitrary number of mathematical layers (bottom)
Close modal
Fig. 9
Finite element mesh for free-edge effects in curved composite laminates [111] with significant edge refinements
Fig. 9
Finite element mesh for free-edge effects in curved composite laminates [111] with significant edge refinements
Close modal
Fig. 10
Results for the interlaminar normal stress σrr in curved cross-ply composite laminates [111]
Fig. 10
Results for the interlaminar normal stress σrr in curved cross-ply composite laminates [111]
Close modal

Hélénon et al. [115] presented the so-called high stress concentration method, which enables an objective prediction concerning the criticality of the highly localized through-the-thickness stress concentrations, occurring during the linear elastic analysis of composite structures. Numerical as well as experimental investigations with regard to the failure mode of T-shaped laminated composite structures under bending and tensile load were also conducted by Hélénon et al. [116,117]. The studies revealed that linear elastic FE computations can be employed with confidence in order to obtain the state variables in the curved segments of T-piece laminated composite structures. Herein, the authors also stated that during failure analysis, the localized free-edge stresses must be considered since they can have a significant impact on the underlying structural strength. The three-dimensional finite element modeling of the free-edge effect in L-shaped composite laminates has been addressed by Nagle et al. [118]. Herein, the authors focused on the out-of-plane normal stress component arising due to the curvature, the effect of various lay-ups on the localized free-edge effect as well as on determining how induced torsion due to constraints affects the interlaminar stress distribution. Reinarz et al. [119] presented a new high-performance finite element analysis tool for complex and large-scale composite structures by making use of novel iterative solvers and a preconditioner (GenEO). The considered structural problem was the analysis of the interlaminar stresses caused by unfolding of a curved laminate. The accuracy of the presented FEA tool was exemplified by comparing the underlying numerical results with those obtained by the commercial finite element software abaqus.

Based on the computational predictions of delamination emergence and delamination growth in L-shaped laminates, Wimmer et al. [120] developed a test design in which test specimens with and without initial delaminations are examined in order to specify load cases for which delamination is the dominant failure mechanism. In addition, Wimmer et al. [121] presented a combined strength and energy approach for the numerical treatment of laminated composite components, which allows for a conservative assessment concerning the critical size and position of the initial delamination. Herein, the delamination initiation due to the free-edge effect is evaluated by means of Puck's first-ply failure while the assessment of delamination propagation is realized through the virtual crack-closure technique. Zimmermann et al. [122], on the other hand, performed various numerical and experimental investigations for ultrathick T-shaped composites under multiple load cases for the development of a main landing gear with composite side stay fitting. Further experimental investigations concerning the deformation and strength of carbon/epoxy laminated curved beams with variable curvature and thickness by means of the digital speckle correlation method and a four-point-bending test configuration were carried out by Hao et al. [123]. Apart from that, Charrier et al. [124] proposed a novel protocol in order to identify the tensile strength of L-shaped specimens composed of unidirectional plies. By examining the results of a test campaign with L-shaped unidirectional carbon-fiber-reinforced laminates, González-Cantero et al. [125] draw the conclusion that the delamination at the traction-free edges of curved laminated shell panels may be induced through a kinking matrix crack which, under increasing load, propagates instantaneously. The influence of the stacking sequence on the failure of narrow L-shaped laminates was experimentally studied by Pan et al. [126]. Herein, two different laminate stacking sequences were loaded quasi-statically under four-point-bending conditions, and it was found that the failure load for one of the test specimens was reduced by over 30% due to the free-edge effect while the carrying capacity of the other stacking sequence was similar to L-shaped laminates under plane-strain conditions. Thus, they concluded that the failure mode is controlled by the length-to-thickness ratio and that the edge effect in narrow specimens can be reduced significantly by optimizing the stacking sequence. Journoud et al. [127] employed the discrete ply model in order to compute four-point bending tests on L-shaped specimens and compared them to experimental studies. Furthermore, the authors conducted numerical sensitivity analyses on frictional coefficient, intralaminar matrix cracking, transverse tensile and shearing strength, and critical energy release rate in modes I and II. The results revealed that the matrix transverse tensile strength drives the failure of L-shaped test specimens. The interaction of both, interlaminar delamination and intralaminar matrix cracking, during failure analysis of curved cross-ply laminates with traction-free edges were examined by Cao et al. [128]. Herein, the authors compared their specimens, which were tested under four-point-bending conditions with the results of a three-dimensional cohesive zone model wherein zero-thickness cohesive elements were utilized in circumferential and radial direction in order to depict matrix crack induced delamination or delamination kinking.

A new free-edge treatment of curved laminates was presented by Fletcher et al. [129]. Herein, they applied resin to the polished and plasma-treated free edges and performed four-point-bending tests. On the basis of finite element computations, it was demonstrated that the new resin edge treatment, on the one hand, increases the strength of narrow circular cylindrical laminated shells by 16% and on the other hand, removes the potential singular, interlaminar stress concentrations at the traction-free edges and thus allowing the through-thickness stresses of FE models to converge. Ranz et al. [130] experimentally studied the interlaminar tensile strength of carbon/epoxy laminated curved beams, which were reinforced through-the-thickness by means of tufting technology. Given the background that the interlaminar tensile strength of the specimens was improved by 40% for 4-layered coupons and 12% for 12-layered coupons, they concluded that tufting technology is most effective for thinner specimens. They also mentioned that although the out-of-plane properties are becoming better through higher stitching densities, this, however, will have a disadvantageous influence on the in-plane properties since fibers are damaged through the reinforcement. In contrast, Ju et al. [131] investigated the through-the-thickness reinforcement effect of grooved Z-pins on the delamination failure strength of composite laminated curved beams wherein the pin diameter as well as the pin areal densities was varied. Several four-point bending tests revealed that the interlaminar tensile strength could be improved significantly by realizing a pin density of 2% while for larger pin diameters, the interlaminar strength was decreasing due to the fact that the reinforcement caused substantially more damage and distortion of the carbon fibers which was shown with the help of microscopic analyses.

Under consideration of several assumptions as well as previous research results, Louhghalam et al. [132] introduced new closed-form expressions for the dynamic edge effect in cylindrical composite pipes. Ahmadi [133], on the other hand, employed the layerwise theory of Reddy in order to conduct interlaminar stress analyses at the traction-free edges of thick composite cylinders with arbitrary laminations undergoing uniform and nonuniform distributed radial pressure. Apart from that, analyses concerning the free-edge effects in corrugated laminates were performed by Filipovic and Kress [134].

5 Free-Edge Effects in Laminates With Holes

Implementing holes in structures such as plates (Fig. 11) is necessary in many technical situations for reasons of weight optimization or for enabling connections to other structural components such as cutouts or holes in wing spars and cover panels to access hydraulic systems and for maintainability. Furthermore, aircraft wing ribs are usually lightened by incorporating holes. However, these holes redistribute the membrane stresses in the plates (Fig. 12) and may remarkably reduce their strength and stability. Quite naturally, this also holds for holes in composite laminates, and in addition to the in-plane stress concentrations as depicted in Fig. 12, free-edge effects can also occur at curved edges at laminate holes.

Fig. 11
Laminated plate containing a circular hole
Fig. 11
Laminated plate containing a circular hole
Close modal
Fig. 12
In-plane stress distributions at the edge of a circular hole in an isotropic plate under uniaxial tensile load
Fig. 12
In-plane stress distributions at the edge of a circular hole in an isotropic plate under uniaxial tensile load
Close modal

In the available literature, some investigations have been performed analytically on this topic with the purpose of calculating resultant forces, but they do not include any prediction of the free-edge effects. Pastorino et al. [135] improved a closed-form methodology for determining the resultant forces in a finite composite plate, weakened by the presence of elliptical cutouts, under membrane loads. The methodology to be developed was based on the Lekhnitskii formalism [4]. The finite dimensions of the plate were regarded using the boundary collocation points, as described in Ref. [136]. Ukadgaonker and Kakhandki [137] utilized an analytical approach of solution for an irregularly shaped hole in an orthotropic laminate. They employed Savin's complex variable formulation [138]. Ghannadpour and Mehrparvar [139] developed a method to model laminates with circular and elliptical holes. Since the perforated plates were moderately thick, the FSDT was used to embody the shear effects along the thickness direction. Moreover, Von Kármán's assumptions were considered to incorporate the geometric nonlinearity. The formulations were founded upon the principle of minimum potential energy and the approximation of displacement fields were based on the Ritz method and calculated by Chebyshev polynomials.

Nonetheless, a number of researchers have devoted their attention to free-edge effects at holes in composite laminated plates a selection of which is cited in the following: It is noteworthy that due to the obvious complexity of the laminate hole situation, most available investigations resort to numerical analysis approaches. Nguyen and Caron [61] presented a finite element model based on the assumption of Mindlin's plate theory for each individual layer of a laminate and the coupling of the layers by transfer of interlaminar stresses. An eight-node finite element for the free-edge effect analysis was developed from this, which was successfully used to calculate edge stresses in straight plane laminates, as well as in perforated structures. A numerically driven fracture mechanics investigation of edge delaminations at circular holes in plane laminates was described by Babu and Pradhan [74]. Jain and Mittal [70] performed finite element analyses for laminates with a circular hole. Ramesh et al. [82] developed a triangular finite element for the analysis of fiber composite laminates based on Reddy's third-order shear deformation theory (see Reddy [1]) in conjunction with layerwise formulations. Chaudhuri [140] devised a three-dimensional shell formulation, from which a finite element formulation for analyzing perforated laminated shells was derived. Zhao et al. [7173] performed three-dimensional finite element analyses on laminates with circular holes. Zhen and Wanji [66] used a single-layer model to determine the stress fields in the vicinity of circular holes in laminates under plane loading, from which a finite element could be derived based on CLPT that allowed continuous interlaminar stress distributions. Hosseini-Toudeshky et al. [75] investigated the delamination behavior of perforated laminates and used a three-dimensional modeling approach using special interface elements. Fagiano et al. [81] presented an efficient improvement of standard elements for the highly accurate calculation of interlaminar stresses in fiber composite laminates, which ensured, among other things, continuous interlaminar stresses across element boundaries. Ahn et al. [71] presented a global–local numerical analysis model for calculating the three-dimensional stress fields in the vicinity of the free edges of circular holes in fiber composite laminates under mechanical load, using three-dimensional displacement formulations in the vicinity of the hole. Baroud et al. [37] developed a layerwise analysis model for the calculation of laminate structures, which was implemented in a finite element formulation for the analysis of hole edge effects. Tian et al. [86] used hybrid laminate elements for edge effect analysis at edges of holes in fiber composite laminates. The formulations were setup to satisfy both the equilibrium conditions and the requirement for continuous interlaminar stresses and traction-free edges. In conclusion, the formulation was based on an adapted complementary energy principle. Hu and Madenci [141] used a particle-based method, called the peridynamics method, to analyze the fatigue and residual strength behavior of fiber composite laminates. The method is not only highly accurate, but also requires considerable computational effort.

In addition to the numerical and analytical treatment of stress concentration problems in the vicinity of holes in laminates, experimental investigations are of course also of importance. Without going into more detail on this special topic, a selection of corresponding works is given here with [73,76,142144].

6 Free-Edge Effects at Laminate Corners

Free laminate corners, i.e., a structural situation where two straight free edges intersect to form a corner of arbitrary opening angle, are potentially also critical with respect to three-dimensional and possible even singular interlaminar stress fields. Since this situation can be interpreted to be a culmination of two free-edge effects, it is appropriate to speak about the so-called free-corner effect at this point (Fig. 13). As pointed out in, e.g., Refs. [145150] or Ref. [151], the free-corner effect is also characterized by three-dimensional stress fields with significant stress gradients and a corner stress singularity.

Fig. 13
Interlaminar stresses in the vicinity of a free laminate corner: Due to the intralaminar stresses σxx, σyy, τxy the interlaminar shear stresses τxz, τyz and the interlaminar normal stress/the peel stress σzz are called forth in the vicinity of a free laminate edge
Fig. 13
Interlaminar stresses in the vicinity of a free laminate corner: Due to the intralaminar stresses σxx, σyy, τxy the interlaminar shear stresses τxz, τyz and the interlaminar normal stress/the peel stress σzz are called forth in the vicinity of a free laminate edge
Close modal

Only a handful of works on the free-corner effect in the timeframe as relevant for this paper exists. The SBFEM (see Sec. 8) was employed in Refs. [152154] for the determination of the orders and modes of stress singularities at free laminate corners with arbitrary opening angles. A layerwise displacement approach based on a discretization of the physical laminate layers into mathematical layers concerning the thickness direction was discussed in Refs. [155] and [156] for isotropic as well as anisotropic laminates. The method was shown to work with high accuracy while requiring only a fraction of the computational effort of similar FEM computations.

7 Piezoelectric Composite Structures

Adaptive composite laminated structures often incorporate piezoelectric layers for the active control of such structures, as well as for sensoric purposes. Quite naturally, it can be assumed that the free-edge effect also occurs in piezoelectric laminates, which is the topic of this section. The analysis of such concentration problems can be treated by using closed-form analytical methods, semi-analytical approaches, or purely numerical methods such as the finite element method wherein all mentioned approaches rely on energetic considerations as follows (see, e.g., Ref. [157]): The mechanical and electrical quantities are inter-related by the following equations in a Cartesian system x1, x2, x3:
σ¯=C¯¯ε¯e¯TE¯D¯=e¯¯ε¯+ϕ¯E¯
(10)
wherein the stress and strain components are summarized in the vectors σ¯ and ε¯ as
σ¯=(σ11,σ22,σ33,σ23,σ13,σ12)Tε¯=(ε11,ε22,ε33,γ23,γ13,γ12)T
(11)
The electrical quantities, i.e., the electric flux vector D¯ and the field vector E¯ are defined as
D¯=(D1,D2,D3)TE¯=(E1,E2,E3)T
(12)
The matrices C¯¯,e¯¯, and ϕ¯¯ in (10) are the stiffness matrix, the piezoelectric coupling matrix, and the dielectrical matrix, respectively, defined as
C¯¯=[C11C12C13C14C15C16C12C22C23C24C25C26C13C23C33C34C35C36C14C24C34C44C45C46C15C25C35C45C55C56C16C26C36C46C56C66]e¯¯=[e11e12e13e14e15e16e21e22e23e24e25e26e31e32e33e34e35e36]ϕ¯¯=[ϕ11ϕ12ϕ13ϕ12ϕ22ϕ23ϕ13ϕ23ϕ33]
(13)
Equilibrium equations are given for the mechanical and electrical quantities as
σi1x1+σi2x2+σi3x3+fi=0D1x1+D2x2+D3x3q=0
(14)

with i =1, 2, 3. Herein, the quantities fi and q are volume forces and the electrical body charge, respectively.

The weak form of (14) can be given in the form of volume integrals over the domain Ω with admissible virtual displacements δui (i =1, 2, 3) and electrical potential variations δφ
Ωi=13[(σi1x1+σi2x2+σi3x3+fi)δui]dΩ+Ω(D1x1+D2x2+D3x3q)δφdΩ=0
(15)
Considering material and electrical linearity, the following kinematic equations and relations between the electric field vector components Ei and the electric potential φ hold (i,j=1,2,3):
εii=uixiγij=uixj+ujxiEi=φxi
(16)
Using Eqs. (16) and (10) employing the divergence theorem, Eq. (15) transforms into
Ωε¯TC¯¯δε¯dΩ(a)Ω(ε¯Te¯¯δE¯+E¯Te¯¯δε¯)dΩ(b)+ΩE¯Tϕ¯¯δE¯dΩ(c)=Ωf¯δu¯dΩ+Γt¯δu¯dΓ(d)+ΩqδφdΩ+ΓQδφdΓ(e)
(17)
Therein, t¯ are the surface tractions while Q denotes the electrical surface charges on the surface Γ. Equation (17) thus constitutes the variational expression that is required for piezoelectric computations. The part denoted as (a) is the inner virtual energy of the considered piezoelectric solid, while(b) and (c) constitute the piezoelectric part and the electrical part of the virtual inner energy. The parts (d) and (e) are the mechanical and electrical virtual works.

In a fundamental work, Artel and Becker [157] treated the effect of electromechanical coupling on free-edge stress fields and the electric field strengths in the vicinity of free edges of composite laminates under uniaxial extension with piezoelectric material properties. The analyses were performed using the finite element method, and a comparison was made between coupled and uncoupled piezoelectric analyses. A similar numerical study was performed by Yang et al. [158] at the example of sandwich beams with two piezoelectric face sheets. Tahani and Mirzababaee [159] presented a semi-analytical method for analyzing free-edge stress fields in composite laminates under uniaxial extension considering electromechanical coupling. The analysis method is based on an adequate displacement formulation in the framework of a higher order shear deformation theory and an energy principle. The authors studied stress fields in cross-ply and angle-ply layups and found the effect of electromechanical coupling to be especially relevant in cross-ply laminates. A further study by the authors was published with Ref. [160]. Izadi and Tahani [161] developed an analytical displacement-based method for the interlaminar stress analysis in piezoelectric laminates under transverse mechanical loads. The analysis employs a single-layer shear deformation theory of second order. Kapuria and Kumari [162] considered the free-edge effect in piezoelectric laminates by use of a layerwise formulation for Levy-type rectangular laminates. The authors used a mixed formulation and reduced the governing equations to a system of ordinary differential equations. Results were presented for piezoelectric laminated and sandwich plates and were found to be in good agreement with FEM simulations. Han et al. [163] considered free-edge effects in piezoelectric cross-ply laminates under uniaxial extension by using the method of transfer matrices. Huang and Kim [164] presented a free-edge effect analysis approach employing stress functions under uni-axial extension. By means of the governing equations that were derived from the principle of complementary virtual work, the authors studied cross-ply and angle-ply layups as well as quasi-isotropic laminates and drew comparisons with FEM computations. Huang and Kim [165] employed an iterative analysis method wherein stress fields were determined by use of the extended Kantorovich method. Huang et al. [166] employed a single-layer theory in conjunction with stress functions for the determination of free-edge stress fields in magneto-electro-elastic laminates under uniaxial extension and magnetic load. The analysis approach uses a separation formulation wherein the transverse stress functions are formulated in the form of harmonic and hyperbolic terms. The in-plane functions are later derived by imposing an energy principle which leads to an eigenvalue problem. The resultant formulations are finally adapted to the underlying boundary conditions, and results are generated for cross-ply and angle-ply laminates. Kapuria and Dhanesh [167] presented an analytical approach based on three-dimensional piezoelasticity for the analysis of interlaminar stress fields in laminates including piezoelectric transducer layers where the loading scenario included extension, bending, torsion, and electric field actuation. The iterative analysis approach handles arbitrary laminate layups and full electromechanical coupling and is based on Reissner's mixed variational principle and enables the fulfillment of all given boundary and continuity conditions. Solutions were obtained by use of a mixed-field multiterm extended Kantorovich method. Dhanesh and Kapuria [168] developed an analytical method for piezoelectric laminates under thermal load. The thermal problem was solved by a separate heat transfer analysis, whereas the free-edge problem was treated in a semi-analytical manner by employing a Reissner-type mixed variational principle and a mixed-field multiterm extended Kantorovich method as developed in preceding publications by the authors. Results were presented for sandwich and hybrid composite laminates under thermal loads. The three-dimensional multiterm extended Kantorovich method was also used by Andakhshideh et al. [169] for the free-edge analysis of arbitrarily laminated piezoelectric laminates under uni-axial extension.

8 The Scaled Boundary Finite Element Method

For the numerical analysis of the free-edge effect, currently the finite element method is probably the most employed tool. This is due to the overall availability, the general-purpose character of this method, and the fact that almost all engineers are experienced with this tool. Due to the strong localization of free-edge stresses, however, in most cases a sophisticated and well-adapted mesh refinement is needed in the finite element discretization close to the free edge and close to the considered interfaces. In particular when the interlaminar peeling stress σzz and/or the interlaminar shear stress τxz occurs as singularities, it is a big challenge to get sufficiently precise stress predictions using finite elements with elementwise linear or quadratic stress representations. Also, there is no good access to the singularity orders of the occurring stress singularities by means of these finite elements.

For the numerical analysis of free-edge stresses in many cases, a much more efficient method is the so-called SBFEM. Although not so well-known, this method has proven its advantageous properties in a large number of application cases for linear systems. Strictly speaking, the SBFEM is a semi-analytical method that combines the advantages of the standard finite element method (FEM) and the boundary element method (BEM). Similarly as in the case of the BEM, the SBFEM needs only the discretization of the boundary (or a part of the boundary) so that the problem dimension is reduced by one and the discretization is much easier. As the FEM, the SBFEM is based on a variational principle, in contrast to the BEM, however, without the need of a fundamental solution. Song and Wolf [170,171] can be considered as pioneers of the SBFEM. Since the method is capable to resolve stress singularities efficiently, it has been advantageously employed in linear elastic fracture mechanics problems [172174]. A good review of the application of the SBFEM for linear elastic fracture mechanics problems can be found in Ref. [175]. It is worth mentioning that the SBFEM can also be applied to complex three-dimensional crack problems where cracks, interfaces, and edges are meeting and it gives deep insight into the combined occurrence of stress singularities of dissimilar orders [176]. Of course, the application of the SBFEM is also very advantageous for the stress analysis and the assessment of the free-edge effect. In a recent publication, this has, for instance, been demonstrated for the example of angle-ply laminates by Dölling et al. [177].

In order to get an idea about the SBFEM and the analysis approach, Fig. 14 shows a [±ϑ]S-laminate under tensile loading. For the analysis of the free-edge behavior, it is sufficient to focus on the indicated quarter model comprising the upper two layers. A cross section within the yz plane can be covered and analyzed by means of scaled coordinates ξ and η as they are shown in Fig. 15 where the coordinate ξ starts at the point of the stress singularity (the so-called scaling center), and the coordinate η runs along the boundary part given in red. For the unknown displacements u(ξ,η), an approximate representation of the kind
u(ξ,η)=N(ξ)û(η)
(18)
is chosen where N denotes a matrix of shape functions within the line elements along the boundary, as they are employed in standard finite element method. The quantities û(ξ) are unknown functions for each of the boundary nodes indicated in Fig. 15. Substituting representation (18) into the principle of virtual work and taking into account kinematics and Hooke's law after some calculations leads to a system of Euler–Cauchy differential equations of the kind
ξ2E0û,ξξ(ξ)+ξ(E0+E1TE1)û,ξ(ξ)E2û(ξ)=ξ(SξSη)
(19)

together with a set of corresponding boundary conditions. In Eq. (19) the quantities E0,E1, and E2 are matrices resulting from the given material stiffnesses and the quantities Sξ and Sη are arrays representing the given loading. For more details, see Ref. [177]. The general homogeneous solution of the system (19) is a linear combination of power functions ξλ where the lowest exponents λ correspond to the existent stress singularities. The particular solution is of the type of the right-hand side of Eq. (19). The coefficients of the general solution can be identified from the given boundary conditions.

Fig. 14
Laminated angle-ply coupon (top) and quarter model (bottom) employing symmetry boundary conditions
Fig. 14
Laminated angle-ply coupon (top) and quarter model (bottom) employing symmetry boundary conditions
Close modal
Fig. 15
SBFEM discretization with global axes x, y, z and local axes ξ,η: in the framework of the SBFEM, it is sufficient to only discretize the boundary Ωd of the considered domain Ω
Fig. 15
SBFEM discretization with global axes x, y, z and local axes ξ,η: in the framework of the SBFEM, it is sufficient to only discretize the boundary Ωd of the considered domain Ω
Close modal

The decisive advantage of the sketched SBFEM is that already a rather low number of scaled boundary elements gives surprisingly accurate stress results, in particular in the vicinity of the stress concentrations. That makes this method highly interesting and advantageous for the analysis of the free-edge effect. Further works that deal with the application of the SBFEM to problems in the framework of linear elastic fracture mechanics and the investigation of stress concentration phenomena are cited with Refs. [176,178180].

9 Experimental Investigations

Besides the theoretical consideration of free-edge effects using analytical or numerical analysis methods, the experimental investigation of such problem fields is of utmost importance and of interest for a deeper understanding of the underlying mechanics and the occurring failure mechanisms.

Lecomte-Grosbras et al. [181] investigated angle-ply laminates consisting of 15 deg and 15deg layers experimentally. Measurements were carried out under tensile loading on edges of the samples. In another paper, the authors coupled two-dimensional and three-dimensional measurements and employed X-ray tomography and kinematic field measurements [182]. Duan et al. [183] investigated the free-edge effect at the edges of circular holes in composite laminates. A theoretical model, which was in view of cutting area and damage factor, was originally developed to predict the hole-edge stress concentration in composites with regard to the hole size effect. Corresponding physical tests were performed under tensile loading monitored by a digital image correlation system. The experimental results showed that the influenced area of the stress concentration is within the range of four times of the hole radius. Han et al. [142] performed an experimental investigation on laminates containing holes, wherein a stitching reinforcement was applied. The strain distribution and concentration were studied analytically and experimentally for different stitching parameters, applied load, and edge location of the hole. Wisnom and Hallet [143] compiled results of several series of open hole tension tests.

For further experimental works, the interested reader is referred to, e.g., Refs. [57,58,87,89,144], and [184186]. For reasons of brevity, however, we will not go into any deeper details at this point.

10 Prediction of Delamination Initiation

Due to the singular nature of the interlaminar stresses σzz and τxz in the framework of linear elasticity, the engineer in many cases is somewhat lost with an assessment or a conclusion of the actual practical consequences of free-edge effects. As according to our common understanding, no material is able to withstand infinite (singular) stresses, and thus it is very hard to draw realistic conclusions from stress fields with singularities. A purely stress-based strength-of-materials-typical assessment would always lead to the conclusion that there will be local material failure in the form of free-edge delamination. That means, for instance, that with a sufficiently fine finite element discretization, we always can formally predict structural failure. This, of course, is misleading, since many laminate structures with a free-edge effect have sufficient load-carrying capacity without developing free-edge delaminations. On the other hand, with a sufficiently coarse finite element discretization, we could erroneously conclude that a structural situation is harmless although the free-edge effect might lead to free-edge delamination and subsequent disintegration.

This difficulty of the right conclusions and the right assessment of realistic strength are not limited to laminate mechanics and the free-edge effect. It occurs in most situations of stress singularities, for instance, in the cases of geometrically given notches or in the case of crack-like defects.

The analysis of structural situations with one or more cracks is the typical subject of consideration in fracture mechanics. Fracture mechanics as a special discipline of solid mechanics has developed mainly after the first world war. With its idea to live with stress singularities at crack tips and its concepts to assess the criticality of cracks by means of the so-called stress intensity factors or, alternatively, by the so-called energy release rates, fracture mechanics was not accepted immediately by the solid mechanics community and by application engineers. It took some time to convince people of the new concepts, but today it is commonly accepted and established and we know, that existent cracks in structures of sufficiently brittle materials can be assessed with very reasonable strength prediction capability. Due to their 1/r-behavior, the stresses around crack tips go along with a finite nonzero energy release rate G and it is well-approved that a crack will grow when the energy release rate exceeds a critical value, namely, the so-called fracture toughness Gc. That means the energetical requirement GGc has to be fulfilled for the growth of a crack. On the other hand, the local overloading of the crack tip vicinity (in terms of stresses) is not sufficient.

The energetical concept of fracture mechanics, however, does no longer work in the case of a notch with a nonzero notch opening angle and it does not either work in the case of a bimaterial interface at the free edge of a laminate. In both cases namely, the energy release rate of a new growing crack starts with a zero energy release rate so that the energetical requirement is not met for very short cracks.

There is a pragmatic way to handle this difficulty, which has already been suggested many decades ago by Neuber, namely, not to take the stresses directly at the location of the singularity, but some distance a away, and then to implement just a strength-of-materials assessment [187]. This kind of concept has also been suggested by Whitney and Nuismer for the strength assessment of laminates with holes or notches [188]. More recently, a decade ago, Taylor newly suggested this concept again, now calling it “Theory of Critical Distances” [189,190]. Although it is easy to work with this concept and although in many cases it gives reasonably good predictions, the problem is that it requires the specific length quantity a. This has to be identified in tests and plays the role of an additional material parameter. Unfortunately, it cannot always be transferred from one geometrical situation to another which is a serious weakness of the concept. A similar alternative concept is not to take the stresses at a certain point but to average them over a certain distance (so-called line method). This has also been suggested by Whitney and Nuismer [188] and has been used, e.g., by Lagunegrand et al. for the prediction of delamination in laminates [191]. An alternative pragmatic method where an internal length parameter is introduced is to work with a numerical discretization length (, e.g., the introduction of artificial laminate layers of a certain thickness). This leads to finite stresses and is the way Diaz and Caron [192] assessed the delamination tendency in laminates.

Similarly, in the case of the free-edge delamination, an initial delamination crack of length a could be postulated. This leads to a nonzero finite energy release rate and the applicability of standard fracture mechanics. This concept, however, is rather artificial when in reality there is no initial crack. Also, the length a again is an additional new parameter.

A good alternative to the use of a critical distance is the implementation of cohesive zone models where the actual material decohesion process is considered to happen in a concentrated strip-like zone. This modeling idea goes back to the Dugdale and Barenblatt models in fracture mechanics [193,194] but can be generalized for many other scenarios [195], for instance, also for delamination cracks. The problem is that a traction-separation law has to be identified or postulated along the cohesive zone(s). Another problem is that a respective finite element analysis with cohesive zones is highly nonlinear and requires a larger number of iterative analysis steps. Nevertheless, cohesive zone modeling has proven its usefulness in countless examples and it is rather common worldwide.

After many decades, where it was not really clear why structures and materials can sustain excessively high stresses as long as these stresses are sufficiently localized, Leguillon presented a paper in 2002 that can be considered as a new milestone [196]. In that paper, Leguillon postulated that in the situation of singular or nonsingular stress concentrations, an instantaneous initiation of a crack of a finite length a will happen if and only if two necessary and sufficient conditions are fulfilled, namely, the material has to be overloaded along the crack (i.e., a stress based strength criterion has to be met) and at the same time the energy released by that crack formation has to be sufficiently large to make the crack formation energetically possible. Mathematically, this can be formulated as follows:
f(σij)σcalongcracklengthG¯GcwhereG¯=ΔΠΔAGc
(20)
Herein, the quantity σc is a characteristic strength value of the employed material, the failure function f should be appropriate for the underlying material, the quantity G¯ is the so-called incremental energy release rate defined by the change ΔΠ of the total potential, ΔA is the generated crack surface area and Gc is the common fracture toughness of the employed material or the corresponding interface in the case of a generated interface crack. In the quite general case, the orientation and length of a crack both may be unknown. If the failure load for a structural situation is to be determined, we have to look for the orientation and the length of such a crack for which the conditions (20) are fulfilled for the first time. The incremental energy release rate G¯ is connected to the more common “differential energy release rate” G=dΠ/dA by the following interrelation:
G¯=1a0aG(ã)dã
(21)

When in fracture mechanics the instantaneous initiation of a crack of finite size is postulated and considered, this is also called “finite fracture mechanics,” a nomination that originally goes back to Hashin [197]. Since in Leguillon's criterion (20) stress and energy requirement are combined, it can also be called a “coupled criterion” or a “hybrid criterion.”

It is not hard to accept this hybrid criterion since we are all familiar with strength-of-materials approaches and also know that a balance of energy should be given. Nevertheless, this hybrid criterion has far-going consequences and bridges the gap between strength-of-material approaches on the one hand and fracture mechanical analysis on the other hand. Of course the hybrid criterion contains the commonly accepted strength-of-materials approach for all situations without considerable stress concentrations as a limiting case and it also contains standard fracture mechanics of cracks as another limiting case. Beyond this, it allows the assessment of structural situations with more or less pronounced stress concentrations (singular or nonsingular).

Some examples with high practical relevance are cracks at V-notches, cracks within adhesive joints, intra- and interlaminar cracks in laminates, debonding of inclusions, crack initiation at open holes in plates, and the formation of crack patterns in surface layers. For a comprehensive overview, please consider the review article of Weißgraeber et al. [198] and, more specifically, for instance, the works [199210].

The assessment and prediction of delamination by means of the coupled concept first has been done by Hebel and Becker [210]. They predicted brittle delamination crack initiation by means of the coupled criterion in a very special laminate, namely, a Solid Oxide Fuel Cell Stack where the glass-ceramic sealings between the metallic layers tend to trigger delamination crack initiation due the present thermal mismatch and the high temperature load. This prediction was in good agreement with experimental observations. Two years later, Hebel and Becker extended his consideration to angle-ply laminates of carbon fiber-reinforced plastics, again with good correspondence to experimental findings [204]. In the same year, also Martin et al. [211] predicted the effective strength of angle-ply laminate coupons by means of the coupled criterion. By means of regression analysis he identified the interface strength and the interface toughness in such a way that a pretty good agreement to experimental findings was obtained. However, this leads to some questionable values for the respective material properties [212]. Just recently, Dölling et al. [213] followed the same kind of approach, however, based on more realistic physical values for strength and toughness and performing the analysis by means of a newly formulated scaled boundary finite element (SBFEM, see Sec. 8) analysis. This kind of analysis keeps the discretization effort very low and makes the analysis numerically highly efficient.

11 Summary and Conclusions

This paper shows that the topic of free-edge effects in composite laminates is still a very active field of study in the international scientific community, even though early investigations range back as far as 1967 (cf. Refs. [15] and [16]), and an impressive body of work has been established over a time span of more than 50 years. The current paper has the aim of extending the scope of the works [15,16] to more recent investigations on free-edge effect problems and spans closed-form analytical, numerical, and experimental works on plane and curved laminated structures as well as perforated laminates and piezoelectric laminated structures. Quite naturally, in the light of the high number of available publications, this paper cannot claim completeness so that it is appropriate to speak of a selective literature survey at this point.

Besides the classical free-edge situation of a plane elastic laminated specimen under in-plane load such as uniaxial extension, more recent works also focus on the analysis of free-edge effects in curved laminated structures (Sec. 4) as well as on perforated laminates (Sec. 5) where significant three-dimensional stress states are known to occur at the curved edges. This also holds for stress states in the vicinity of corners of composite laminates (Sec. 6), and it can be expected that future investigations on closed-form analytical and semi-analytical methods as well as numerical approaches will have a focus on such geometrical situations. A further rather recent research focus is the coupled electromechanical analysis of piezoelectric laminates (Sec. 7), and it can be expected that future works will also focus on this field of study.

The SBFEM (Sec. 8) is a rather new analysis approach that has been found to exhibit the advantages of the BEM using standard finite element formulations while avoiding the specific disadvantages of the BEM such as the need for a fundamental solution. This method relies on a surface discretization of the given structure using standard finite element formulations while enabling closed-form solutions in the inner regions of the structure under consideration. The SBFEM has been found to be a very suitable method in order to analyze three-dimensional and potentially even singular stress states in composite laminates with excellent numerical efficiency, and it can be expected that future investigations will focus on more advanced problem fields also in the framework of the mechanics of composite structures. Especially in conjunction with the fracture assessment and initiation of delaminations (Sec. 10) in free-edge situations, the SBFEM is a promising analysis tool with high future potential. Furthermore, assessing fracture criticalities of composite laminated structures by means of finite fracture mechanics (Sec. 10) is a promising new field of research also when free-edge effects and related problems are to be considered.

References

1.
Reddy
,
J. N.
,
2004
,
Mechanics of Laminated Composite Plates and Shells
, 2nd ed.,
CRC Press
,
Boca Raton, FL
.
2.
Ambartsumyan
,
S. A.
,
1970
,
Theory of Anisotropic Plates
,
Technomic Publishing
,
Stamford, CT
.
3.
Mittelstedt
,
C.
, and
Becker
,
W.
,
2016
,
Strukturmechanik Ebener Laminate
,
Verlag Studienbereich Mechanik
,
Technische Universität Darmstadt, Darmstadt
(in German, ‘Structural Mechanics of Plane Composite Laminates’).
4.
Lekhnitskii
,
S. G.
,
1968
,
Anisotropic Plates
,
Gordon and Breach
,
London, UK
.
5.
Ashton
,
J. E.
, and
Whitney
,
J. M.
,
1970
,
Theory of Laminated Plates
,
Technomic Publishing
,
Stamford, CT
.
6.
Jones
,
R. M.
,
1975
,
Mechanics of Composite Materials
,
Scripta Book Co
,
Washington, DC
.
7.
Altenbach
,
H.
,
Altenbach
,
J.
, and
Kissing
,
W.
,
2018
,
Mechanics of Composite Structural Elements
,
Springer
,
Singapore
.
8.
Mittelstedt
,
C.
, and
Becker
,
W.
,
2007
, “
The Pipes-Pagano-Problem Revisited: Elastic Fields in Boundary Layers of Plane Laminated Specimens Under Combined Thermomechanical Load
,”
Compos. Struct.
,
80
(
3
), pp.
373
395
.10.1016/j.compstruct.2006.05.018
9.
Pipes
,
R. B.
, and
Pagano
,
N.
,
1970
, “
Interlaminar Stresses in Composite Laminates Under Uniform Axial Extension
,”
J. Compos. Mater.
,
4
(
4
), pp.
538
548
.10.1177/002199837000400409
10.
Salamon
,
N. J.
,
1980
, “
An Assessment of the Interlaminar Stress Problem in Laminated Composites
,”
J. Compos. Mater.
,
14
(
1
), pp.
177
194
.10.1177/002199838001400114
11.
Herakovich
,
C. T.
,
1989
, “
Free Edge Effects in Laminated Composites
,”
Handbook of Composites
, Vol.
2
,
Elsevier Science Publishers B.V
,
Amsterdam
, The Netherlands, pp.
187
230
.
12.
Pagano
,
N. J
et al
1989
, “
Composite Materials Series
,”
Interlaminar Response of Composite Materials
, Vol.
5
,
Elsevier
,
Amsterdam
et al.
13.
Reddy
,
J. N.
, and
Robbins
,
D. H.
,
1994
, “
Theories and Computational Models for Composite Laminates
,”
ASME Appl. Mech. Rev.
,
47
(
6
), pp.
147
169
.10.1115/1.3111076
14.
Kant
,
T.
, and
Swaminathan
,
K.
,
2000
, “
Estimation of Transverse/Interlaminar Stresses in Lam-Inated Composites - a Selective Review and Survey of Current Developments
,”
Compos. Struct.
,
49
(
1
), pp.
65
75
.10.1016/S0263-8223(99)00126-9
15.
Mittelstedt
,
C.
, and
Becker
,
W.
,
2004
, “
Interlaminar Stress Concentrations in Layered Structures - Part I: A Selective Literature Survey on the Free-Edge Effect Since 1967
,”
J. Compos. Mater.
,
38
(
12
), pp.
1037
1062
.10.1177/0021998304040566
16.
Mittelstedt
,
C.
, and
Becker
,
W.
,
2007
, “
Free-Edge Effects in Composite Laminates
,”
ASME Appl. Mech. Rev.
,
60
(
5
), pp.
217
245
.10.1115/1.2777169
17.
Mittelstedt
,
C.
, and
Becker
,
W.
,
2006
, “
Fast and Reliable Analysis of Free-Edge Stress Fields in a Thermally Loaded Composite Strip by a Layerwise Laminate Theory
,”
Int. J. Numer. Methods Eng.
,
67
(
6
), pp.
747
770
.10.1002/nme.1631
18.
Pipes
,
R. B.
,
1972
, “
Solution of Certain Problems in the Theory of Elasticity for Laminated Anisotropic Systems
,” Ph.D. dissertation thesis,
University of Texas
,
Arlington, TX
.
19.
Pagano
,
N. J.
,
1974
, “
On the Calculation of Interlaminar Normal Stress in Composite Laminate
,”
J. Compos. Mater.
,
8
(
1
), pp.
65
81
.10.1177/002199837400800106
20.
Kassapoglou
,
C.
, and
Lagace
,
P. A.
,
1986
, “
An Efficient Method for the Calculation of Interlaminar Stresses in Composite Materials
,”
ASME J. Appl. Mech.
,
53
(
4
), pp.
744
50
.10.1115/1.3171853
21.
Yao
,
W.
,
Nie
,
Y.
, and
Xiao
,
F.
,
2011
, “
Analytical Solutions to Edge Effect of Composite Laminates Based on Symplectic Dual System
,”
Appl. Math. Mech.
,
32
(
9
), pp.
1091
1100
.10.1007/s10483-011-1483-7
22.
Nosier
,
A.
, and
Bahrami
,
A.
,
2006
, “
Free-Edge Stresses in Antisymmetric Angle-Ply Laminates in Extension and Torsion
,”
Int. Journal Solids Structures
,
43
(
22–23
), pp.
6800
6816
.10.1016/j.ijsolstr.2006.02.006
23.
Nosier
,
A.
, and
Bahrami
,
A.
,
2007
, “
Interlaminar Stresses in Antisymmetric Angle-Ply Laminates
,”
Compos. Struct.
,
78
(
1
), pp.
18
33
.10.1016/j.compstruct.2005.08.007
24.
Andakhshideh
,
A.
, and
Tahani
,
M.
,
2013
, “
Interlaminar Stresses in General Thick Rectangular Laminated Plates Under in-Plane Loads
,”
Compos. Part B Eng.
,
47
, pp.
58
69
.10.1016/j.compositesb.2012.10.020
25.
Tahani
,
M.
, and
Andakhshideh
,
A.
,
2012
, “
Interlaminar Stresses in Thick Rectangular Laminated Plates With Arbitrary Laminations and Boundary Conditions Under Transverse Loads
,”
Compos. Struct.
,
94
(
5
), pp.
1793
1804
.10.1016/j.compstruct.2011.12.027
26.
Andakhshideh
,
A.
, and
Tahani
,
M.
,
2013
, “
Free-Edge Stress Analysis of General Rectangular Composite Laminates Under Bending, Torsion and Thermal Loads
,”
Eur. J. Mech.-A/Solids
,
42
, pp.
229
240
.10.1016/j.euromechsol.2013.06.002
27.
Kapuria
,
S.
, and
Dhanesh
,
N.
,
2016
, “
Free Edge Stresses in Composite Laminates With Imperfect Interfaces Under Extension, Bending and Twisting Loading
,”
Int. J. Mech. Sci.
,
113
(
2016
), pp.
148
161
.10.1016/j.ijmecsci.2016.04.017
28.
Dhanesh
,
N.
,
Kapuria
,
S.
, and
Achary
,
G.
,
2017
, “
Accurate Prediction of Three-Dimensional Free Edge Stress Field in Composite Laminates Using Mixed-Field Multiterm Extended Kantorovich Method
,”
Acta Mech.
,
228
(
8
), pp.
2895
2919
.10.1007/s00707-015-1522-0
29.
Kapuria
,
S.
, and
Kumari
,
P.
,
2011
, “
Extended Kantorovich Method for Three Dimensional Elasticity Solution of Laminated Composite Structures in Cylindrical Bending
,”
ASME J. Appl. Mech.
,
78
(
6
), p.
061004
.10.1115/1.4003779
30.
Kerr
,
A. D.
,
1968
, “
An Extension of the Kantorovich Method
,”
Q. Appl. Math.
,
26
(
2
), pp.
219
229
.10.1090/qam/99857
31.
Kapuria
,
S.
, and
Kumari
,
P.
,
2012
, “
Multiterm extended Kantorovich Method for Three-Dimensional Elasticity Solution of Laminated Plates
,”
ASME J. Appl. Mech.
,
79
(
6
), p.
061018
.10.1115/1.4006495
32.
Kapuria
,
S.
, and
Dhanesh
,
S.
,
2015
, “
Three-Dimensional Extended Kantorovich Solution for Accurate Prediction of Interlaminar Stresses in Composite Laminated Panels With Interfacial Imperfections
,”
J. Eng. Mech.
,
141
(
4
), p.
04014140
.10.1061/(ASCE)EM.1943-7889.0000860
33.
Kumari
,
P.
,
Kapuria
,
S.
, and
Rajapakse
,
R. K. N. D.
,
2014
, “
Three-Dimensional Extended Kantorovich Solution for Levy-Type Rectangular Laminated Plates With Edge Effects
,”
Compos. Struct.
,
107
, pp.
167
176
.10.1016/j.compstruct.2013.07.053
34.
Lee
,
J.
,
Cho
,
M.
, and
Kim
,
H. S.
,
2011
, “
Bending Analysis of a Laminated Composite Patch Considering the Free-Edge Effect Using a Stress-Based Equivalent Single-Layer Composite Model
,”
Int. J. Mech. Sci.
,
53
(
8
), pp.
606
616
.10.1016/j.ijmecsci.2011.05.007
35.
Huang
,
B.
,
Kim
,
H. S.
,
Wang
,
J.
,
Du
,
J.
, and
Guo
,
Y.
,
2016
, “
Time-Dependent Stress Variations in Symmetrically Viscoelastic Composite Laminates Under Uniaxial Tensile Load
,”
Compos. Struct.
,
142
, pp.
278
285
.10.1016/j.compstruct.2016.01.101
36.
Zhang
,
D.
,
Ye
,
J.
, and
Sheng
,
H. Y.
,
2006
, “
Free-Edge and Ply Cracking Effect in Cross-Ply Laminated Composites Under Uniform Extension and Thermal Loading
,”
Compos. Struct.
,
76
(
4
), pp.
314
325
.10.1016/j.compstruct.2005.04.021
37.
Baroud
,
R.
,
Sab
,
K.
,
Caron
,
J. F.
, and
Kaddah
,
F.
,
2016
, “
A Statically Compatible Layerwise Stress Model for the Analysis of Multilayered Plates
,”
Int. J. Solids Struct.
,
96
, pp.
11
24
.10.1016/j.ijsolstr.2016.06.030
38.
Saeedi
,
N.
,
Sab
,
K.
, and
Caron
,
J. F.
,
2012
, “
Delaminated Multilayered Plates Under Uniaxial Extension. Part I: Analytical Analysis Using a Layerwise Stress Approach
,”
Int. J. Solids Struct.
,
49
(
26
), pp.
3711
3726
.10.1016/j.ijsolstr.2012.08.005
39.
Saeedi
,
N.
,
Sab
,
K.
, and
Caron
,
J. F.
,
2012
, “
Delaminated Multilayered Plates Under Uniaxial Extension. Part II: Efficient Layerwise Mesh Strategy for the Prediction of Delamination Onset
,”
Int. J. Solids Struct.
,
49
(
26
), pp.
3727
3740
.10.1016/j.ijsolstr.2012.08.003
40.
Yazdani Sarvestani
,
H.
, and
Yazdani Sarvestani
,
M.
,
2012
, “
Free-Edge Stress Analysis of General Composite Laminates Under Extension, Torsion and Bending
,”
Appl. Math. Modell.
,
36
(
4
), pp.
1570
1588
.10.1016/j.apm.2011.09.028
41.
Yazdani Sarvestani
,
H.
,
Naghashpour
,
A.
, and
Heidari-Rarani
,
M.
,
2013
, “
Prediction of Interlaminar Stresses of an Unsymmetric Cross-Ply Laminate Using Layerwise and Higher-Order Equivalent Single-Layer Theories
,”
Int. J. Aerosp. Lightweight Struct.
,
03
(
04
), pp.
419
444
.10.3850/S2010428614000014
42.
Yazdani Sarvestani
,
H.
,
Naghashpour
,
A.
, and
Heidari-Rarani
,
M.
,
2015
, “
Bending Analysis of a General Cross-Ply Laminate Using 3D Elasticity Solution and Layerwise Theory
,”
Int. J. Adv. Struct. Eng.
,
7
(
4
), pp.
329
340
.10.1007/s40091-014-0073-2
43.
Yazdani Sarvestani
,
H.
, and
Naghashpour
,
A.
,
2014
, “
Analysis of Free Edge Stresses in Composite Laminates Using Higher Order Theories
,”
Indian J. Mater. Sci.
,
2014
, pp.
1
15
.10.1155/2014/253018
44.
Zhang
,
D.
,
Ye
,
J.
, and
Lam
,
D.
,
2007
, “
Free-Edge and Ply Cracking Effect in Angle-Ply Laminated Composites Subjected to In-Plane Loads
,”
J. Eng. Mech.
,
133
(
12
), pp.
1268
1277
.10.1061/(ASCE)0733-9399(2007)133:12(1268)
45.
Kim
,
H. S.
,
Lee
,
J.
, and
Cho
,
M.
,
2012
, “
Free-Edge Interlaminar Stress Analysis of Composite Laminates Using Interface Modeling
,”
J. Eng. Mech.
,
138
(
8
), pp.
973
983
.10.1061/(ASCE)EM.1943-7889.0000399
46.
Nosier
,
A.
, and
Maleki
,
M.
,
2008
, “
Free-Edge Stresses in General Composite Laminates
,”
Int. J. Mech. Sci.
,
50
(
10–11
), pp.
1435
1447
.10.1016/j.ijmecsci.2008.09.002
47.
Afshin
,
M.
, and
Taheri-Behrooz
,
F.
,
2015
, “
Interlaminar Stresses of Laminated Composite Beams Resting on Elastic Foundation Subjected to Transverse Loading
,”
Comput. Mater. Sci.
,
96
, pp.
439
447
.10.1016/j.commatsci.2014.06.027
48.
Liang
,
W. Y.
,
Tseng
,
W. D.
, and
Tarn
,
J. Q.
,
2014
, “
Exact Analysis of Stress Fields in Composite Laminates Under Extension
,”
J. Mech.
,
30
(
5
), pp.
477
489
.10.1017/jmech.2014.44
49.
Huang
,
B.
,
Wang
,
J.
,
Du
,
J.
,
Guo
,
Y.
,
Ma
,
T.
, and
Yi
,
L.
,
2016
, “
Extended Kantorovich Method for Local Stresses in Composite Laminates Upon Polynomial Stress Functions
,”
Acta Mech. Sin.
,
32
(
5
), pp.
854
865
.10.1007/s10409-016-0570-6
50.
Hajikazemi
,
M.
, and
van Paepegem
,
W.
,
2018
, “
A Variational Model for Free-Edge Interlaminar Stress Analysis in General Symmetric and Thin-Ply Composite Laminates
,”
Compos. Struct.
,
184
, pp.
443
451
.10.1016/j.compstruct.2017.10.012
51.
Ahmadi
,
I.
,
2018
, “
Three-Dimensional Stress Analysis in Torsion of Laminated Composite Bar With General Layer Stacking
,”
Eur. J. Mech./A Solids
,
72
, pp.
252
267
.10.1016/j.euromechsol.2018.05.003
52.
Romera
,
J. M.
,
Carbajal
,
N.
, and
Mujika
,
F.
,
2020
, “
A Simple Analytical Method for Determining Interlaminar Shear Stresses in Symmetric Laminates
,”
Structures
,
25
, pp.
683
695
.10.1016/j.istruc.2020.03.053
53.
Ahmadi
,
I.
,
2020
, “
Stress Analysis in Transverse Loading of Soft Core Sandwich Plates With Various Boundary Conditions
,”
J. Sandwich Struct. Mater.
,
22
(
8
), pp.
2692
2734
.10.1177/1099636218816107
54.
Ju
,
S. H.
,
Liang
,
W. Y.
,
Hsu
,
H. H.
, and
Tarn
,
J. Q.
,
2020
, “
Analytic Solution of Angle-Ply Laminated Plates Under Extension, Bending, and Torsion
,”
J. Compos. Mater.
,
54
(
8
), pp.
1093
1106
.10.1177/0021998319873025
55.
Pipes
,
R. B.
,
Goodsell
,
J. G.
,
Ritchey
,
A.
,
Dustin
,
J.
, and
Gosse
,
J.
,
2010
, “
Interlaminar Stresses in Composite Laminates: Thermoelastic Deformation
,”
Compos. Sci. Technol.
,
70
(
11
), pp.
1605
1611
.10.1016/j.compscitech.2010.05.026
56.
Zhang
,
C.
, and
Binienda
,
W. K.
,
2014
, “
Numerical Analysis of Free-Edge Effect on Size-Influenced Mechanical Properties of Single-Layer Triaxially Braided Composites
,”
Appl. Compos. Mater.
,
21
(
6
), pp.
841
859
.10.1007/s10443-014-9386-3
57.
Zhang
,
C.
, and
Binienda
,
W. K.
,
2014
, “
A Meso-Scale Finite Element Model for Simulating Free-Edge Effect in Carbon/Epoxy Textile Composite
,”
Mech. Mater.
,
76
, pp.
1
19
.10.1016/j.mechmat.2014.05.002
58.
Zhang
,
C.
,
Binienda
,
W. K.
, and
Goldberg
,
R. K.
,
2015
, “
Free-Edge Effect on the Effective Stiffness of Single-Layer Triaxially Braided Composite
,”
Compos. Sci. Technol.
,
107
, pp.
145
153
.10.1016/j.compscitech.2014.12.016
59.
Vidal
,
P.
,
Polit
,
O.
,
D'Ottavio
,
M.
, and
Valot
,
E.
,
2014
, “
Assessment of the Refined Sinus Plate Finite Element: Free Edge Effect and Meyer-Piening Sandwich Test
,”
Finite Elem. Anal. Des.
,
92
, pp.
60
71
.10.1016/j.finel.2014.08.004
60.
Vidal
,
P.
,
Gallimard
,
L.
, and
Polit
,
O.
,
2015
, “
Assessment of Variable Separation for Finite Element Modeling of Free Edge Effect for Composite Plates
,”
Compos. Struct.
,
123
, pp.
19
29
.10.1016/j.compstruct.2014.11.068
61.
Nguyen
,
V. T.
, and
Caron
,
J. F.
,
2006
, “
A New Finite Element for Free Edge Effect Analysis in Laminated Composites
,”
Comput. Structures
,
84
(
22–23
), pp.
1538
1546
.10.1016/j.compstruc.2006.01.038
62.
Nguyen
,
V. T.
, and
Caron
,
J. F.
,
2009
, “
Finite Element Analysis of Free-Edge Stresses in Composite Laminates Under Mechanical an Thermal Loading
,”
Compos. Sci. Technol.
,
69
(
1
), pp.
40
49
.10.1016/j.compscitech.2007.10.055
63.
Lo
,
S.
,
Zhen
,
W.
,
Cheung
,
Y.
, and
Wanji
,
C.
,
2007
, “
An Enhanced Global–Local Higher-Order Theory for the Free Edge Effect in Laminates
,”
Compos. Struct.
,
81
(
4
), pp.
499
510
.10.1016/j.compstruct.2006.09.013
64.
Li
,
X.
, and
Liu
,
D.
,
1997
, “
Generalized Laminate Theories Based on Double Superposition Hypothesis
,”
Int. J. Numer. Methods Eng.
,
40
(
7
), pp.
1197
1212
.10.1002/(SICI)1097-0207(19970415)40:7<1197::AID-NME109>3.0.CO;2-B
65.
Zhen
,
W.
, and
Wanji
,
C.
,
2009
, “
A Higher-Order Displacement Model for Stress Concentration Problems in General Lamination Configurations
,”
Mater. Des.
,
30
(
5
), pp.
1458
1467
.10.1016/j.matdes.2008.08.013
66.
Zhen
,
W.
, and
Wanji
,
C.
,
2009
, “
Stress Analysis of Laminated Composite Plates With a Circular Hole According to a Single-Layer Higher-Order Model
,”
Compos. Struct.
,
90
(
2
), pp.
122
129
.10.1016/j.compstruct.2009.02.010
67.
Ramtekkar
,
G.
, and
Desai
,
Y.
,
2009
, “
On Free-Edge Effect and Onset of Delamination in FRPC Laminates Using Mixed Finite Element Model
,”
J. Reinforced Plast. Compos.
,
28
(
3
), pp.
317
341
.10.1177/0731684407084243
68.
Dhadwal
,
M. K.
, and
Jung
,
S. N.
,
2020
, “
Free-Edge Stress Evaluation of General Laminated Composites Using a Novel Multifield Variational Beam Formulation
,”
Compos. Struct.
,
233
, p.
111705
.10.1016/j.compstruct.2019.111705
69.
Islam
,
M.
, and
Prabhakar
,
P.
,
2017
, “
Modeling Framework for Free Edge Effects in Laminates Under Thermo-Mechanical Loading
,”
Compos. Part B Eng.
,
116
, pp.
89
98
.10.1016/j.compositesb.2017.01.072
70.
Jain
,
N.
, and
Mittal
,
N.
,
2008
, “
Finite Element Analysis for Stress Concentration and Deflection in Isotropic, Orthotropic and Laminated Composite Plates With Central Circular Hole Under Transverse Static Loading
,”
Mater. Sci. Eng. A
,
498
(
1–2
), pp.
115
124
.10.1016/j.msea.2008.04.078
71.
Ahn
,
J.-S.
,
Kim
,
Y.-W.
, and
Woo
,
K.-S.
,
2013
, “
Analysis of Circular Free Edge Effect in Composite Laminates by p-Convergent Global–Local Model
,”
Int. J. Mech. Sci.
,
66
, pp.
149
155
.10.1016/j.ijmecsci.2012.11.003
72.
Zhao
,
G. H.
,
Tong
,
J. W.
,
Shen
,
M.
,
Aymerich
,
F.
, and
Priolo
,
P.
,
2009
, “
Numerical Analysis of Interlaminar Stresses of Angle-Ply AS4/PEEK Laminate With a Central Hole
,”
J. Thermoplast. Compos. Mater.
,
22
(
4
), pp.
383
406
.10.1177/0892705709098154
73.
Zhao
,
G. H.
,
Tong
,
J. W.
, and
Shen
,
M.
,
2010
, “
Numerical Analysis and Experimental Validation of Interlaminar Stresses of Quasi-Isotropic APC-2/as-4 Laminate With a Central Hole Loaded in Tension
,”
J. Thermoplast. Compos. Mater.
,
23
(
4
), pp.
413
433
.10.1177/0892705709344563
74.
Babu
,
P. R.
, and
Pradhan
,
B.
,
2007
, “
Effect of Damage Levels and Curing Stresses on Delamination Growth Behaviour Emanating From Circular Holes in Laminated FRP Composites
,”
Compos. Part A
,
38
(
12
), pp.
2412
2421
.10.1016/j.compositesa.2007.08.010
75.
Hosseini-Toudeshky
,
H.
,
Jalalvand
,
M.
, and
Mohammadi
,
B.
,
2009
, “
Delamination Analysis of Holed Composite Laminates Using Interface Elements
,”
Procedia Eng.
,
1
(
1
), pp.
39
42
.10.1016/j.proeng.2009.06.011
76.
Suemasu
,
H.
,
Takahashi
,
H.
, and
Ishikawa
,
T.
,
2006
, “
On Failure Mechanisms of Composite Laminates With an Open Hole Subjected to Compressive Load
,”
Compos. Sci. Technol.
,
66
(
5
), pp.
634
641
.10.1016/j.compscitech.2005.07.042
77.
Espadas-Escalante
,
J. J.
,
van Dijk
,
N. P.
, and
Isaksson
,
P.
,
2018
, “
The Effect of Free-Edges and Layer Shifting on Intralaminar and Interlaminar Stresses in Woven Composites
,”
Compos. Struct.
,
185
, pp.
212
220
.10.1016/j.compstruct.2017.11.014
78.
Ballard
,
M. K.
, and
Whitcomb
,
J. D.
,
2019
, “
Effect of Heterogeneity at the Fiber–Matrix Scale on Predicted Free-Edge Stresses for a [0/90]s Laminated Composite Subjected to Uniaxial Tension
,”
J. Compos. Mater.
,
53
(
5
), pp.
625
639
.10.1177/0021998318788915
79.
Hosoi
,
A.
, and
Kawada
,
H.
,
2008
, “
Stress Analysis of Laminates of Carbon Fiber Reinforced Plastics, Containing Transverse Cracks, Considering Free-Edge Effect and Residual Thermal Stress
,”
Mater. Sci. Eng. A
,
498
(
1–2
), pp.
69
75
.10.1016/j.msea.2007.11.153
80.
Wowk
,
D.
,
Marsden
,
C.
, and
Thibaudeau
,
D.
,
2020
, “
Predicting the Relative Magnitude of Interlaminar Stresses Due to Edge Effects in Thin Angle-Ply Laminates Using Macroscopic Finite Element Modeling
,”
Compos. Struct.
,
242
, p.
112164
.10.1016/j.compstruct.2020.112164
81.
Fagiano
,
C.
,
Abdalla
,
M. M.
,
Kassapoglou
,
C.
, and
Gürdal
,
Z.
,
2010
, “
Interlaminar Stress Recovery for Three-Dimensional Finite Elements
,”
Compos. Sci. Technol.
,
70
(
3
), pp.
530
538
.10.1016/j.compscitech.2009.12.013
82.
Ramesh
,
S. S.
,
Wang
,
C.
,
Reddy
,
J.
, and
Ang
,
K.
,
2009
, “
A Higher-Order Plate Element for Accurate Prediction of Interlaminar Stresses in Laminated Composite Plates
,”
Compos. Struct.
,
91
(
3
), pp.
337
357
.10.1016/j.compstruct.2009.06.001
83.
Yan
,
X.
,
Ding
,
S.
,
Tong
,
J.
,
Shen
,
M.
, and
Huo
,
Z.
,
2009
, “
Numerical Elastic-Plastic Simulation of Interlaminar Stresses in a Notched Angle-Ply Thermoplastic Composite Laminate
,”
Mech. Compos. Mater.
,
45
(
3
), pp.
293
302
.10.1007/s11029-009-9081-x
84.
Guo
,
Z.
,
Han
,
X.
, and
Zhu
,
X.
,
2012
, “
Finite Element Analysis of Interlaminar Stresses for Composite Laminates Stitched Around a Circular Hole
,”
Appl. Compos. Mater.
,
19
(
3–4
), pp.
561
571
.10.1007/s10443-011-9234-7
85.
Ding
,
S. R.
,
Tong
,
J.
, and
Shen
,
M.
,
2005
, “
The Three-Dimensional Elastic-Plastic Analysis of Interlaminar Stresses in Notched Thermoplastic Composites
,”
J. Reinforced Plast. Compos.
,
24
(
11
), pp.
1151
1158
.10.1177/0731684405048838
86.
Tian
,
Z. S.
,
Yang
,
Q. P.
, and
Wang
,
A. P.
,
2016
, “
Three-Dimensional Stress Analyses Around Cutouts in Laminated Composites by Special Hybrid Finite Elements
,”
J. Compos. Mater.
,
50
(
1
), pp.
75
98
.10.1177/0021998315570509
87.
Esquej
,
R.
,
Castejon
,
L.
,
Lizaranzu
,
M.
,
Carrera
,
M.
,
Miravete
,
A.
, and
Miralbes
,
R.
,
2013
, “
A New Finite Element Approach Applied to the Free Edge Effect on Composite Materials
,”
Compos. Struct.
,
98
, pp.
121
129
.10.1016/j.compstruct.2012.09.043
88.
D'Ottavio
,
M.
,
Vidal
,
P.
,
Valot
,
E.
, and
Polit
,
O.
,
2013
, “
Assessment of Plate Theories for Free-Edge Effects
,”
Compos. Part B
,
48
, pp.
111
121
.10.1016/j.compositesb.2012.12.007
89.
Guillamet
,
G.
,
Turon
,
A.
,
Costa
,
J.
, and
Linde
,
P.
,
2016
, “
A Quick Procedure to Predict Free-Edge Delamination in Thin-Ply Laminates Under Tension
,”
Eng. Fract. Mech.
,
168
, pp.
28
39
.10.1016/j.engfracmech.2016.01.019
90.
Das
,
S.
,
Choudhury
,
P.
,
Halder
,
S.
, and
Sriram
,
P.
,
2013
, “
Stress and Free Edge Delamination Analyses of Delaminated Composite Structure Using ANSYS
,”
Procedia Eng.
,
64
, pp.
1364
1373
.10.1016/j.proeng.2013.09.218
91.
Miguel
,
A. G.
,
Carrera
,
E.
,
Pagani
,
A.
, and
Zappino
,
E.
,
2018
, “
Accurate Evaluation of Interlaminar Stresses in Composite Laminates Via Mixed One-Dimensional Formulation
,”
AIAA J.
,
56
(
11
), pp.
4582
4594
.10.2514/1.J057189
92.
Peng
,
B.
,
Goodsell
,
J.
,
Pipes
,
R. B.
, and
Yu
,
W.
,
2016
, “
Generalized Free-Edge Stress Analysis Using Mechanics of Structure Genome
,”
ASME J. Appl. Mech.
,
83
(
10
), p.
101013
.10.1115/1.4034389
93.
Cater
,
C.
,
Xiao
,
X.
,
Goldberg
,
R.
, and
Gong
,
X.
,
2018
, “
Gong, X: Multiscale Investigation of Micro-Scale Stresses at Composite Laminate Free Edge
,”
Compos. Struct.
,
189
, pp.
545
552
.10.1016/j.compstruct.2018.01.098
94.
Solis
,
A.
,
Sánchez-Sáez
,
S.
, and
Barbero
,
E.
,
2018
, “
Influence of Ply Orientation on Free-Edge Effects in Laminates Subjected to in-Plane Loads
,”
Compos. Part B Eng.
,
153
, pp.
149
158
.10.1016/j.compositesb.2018.07.030
95.
Ullah
,
Z.
,
Kaczmarczyk
,
L.
,
Zhou
,
X. Y.
,
Falzon
,
B. G.
, and
Pearce
,
C. J.
,
2020
, “
Hierarchical Finite Element-Based Multi-Scale Modelling of Composite Laminates
,”
Compos. Part B
,
201
, p.
108321
.10.1016/j.compositesb.2020.108321
96.
Guo
,
Y.
, and
Ruess
,
M.
,
2015
, “
A Layerwise Isogeometric Approach for NURBS-Derived Laminate Composite Shells
,”
Compos. Struct.
,
124
, pp.
300
309
.10.1016/j.compstruct.2015.01.012
97.
Meng
,
M.
,
Le
,
H. R.
,
Rizvi
,
M. J.
, and
Grove
,
S. M.
,
2015
, “
3D FEA Modelling of Laminated Composites in Bending and Their Failure Mechanisms
,”
Compos. Struct.
,
119
, pp.
693
708
.10.1016/j.compstruct.2014.09.048
98.
Lekhnitskii
,
S. G.
,
1963
, “
Theory of Elasticity of an Anisotropic Elastic Body
,”
Holden-Day Series in Mathematical Physics
,
Holden-Day
,
San Francisco, CA
.
99.
González-Cantero
,
J. M.
,
Graciani
,
E.
,
Blázquez
,
A.
, and
París
,
F.
,
2016
, “
A New Analytical Model for Evaluating Interlaminar Stresses in the Unfolding Failure of Composite Laminates
,”
Compos. Struct.
,
147
, pp.
260
273
.10.1016/j.compstruct.2016.03.025
100.
González-Cantero
,
J. M.
,
Graciani
,
E.
,
París
,
F.
, and
López-Romano
,
B.
,
2017
, “
Semi-Analytic Model to Evaluate Non-Regularized Stresses Causing Unfolding Failure in Composites
,”
Compos. Struct.
,
171
, pp.
77
91
.10.1016/j.compstruct.2017.02.016
101.
Thurnherr
,
C.
,
Groh
,
R. M. J.
,
Ermanni
,
P.
, and
Weaver
,
P. M.
,
2016
, “
Higher-Order Beam Model for Stress Predictions in Curved Beams Made From Anisotropic Materials
,”
Int. J. Solids Struct.
,
97–98
, pp.
16
28
.10.1016/j.ijsolstr.2016.08.004
102.
Thurnherr
,
C.
,
Groh
,
R. M. J.
,
Ermanni
,
P.
, and
Weaver
,
P. M.
,
2017
, “
Investigation of Failure Initiation in Curved Composite Laminates Using a Higher-Order Beam Model
,”
Compos. Struct.
,
168
, pp.
143
152
.10.1016/j.compstruct.2017.02.010
103.
Nosier
,
A.
, and
Miri
,
A. K.
,
2010
, “
Boundary-Layer Hygrothermal Stresses in Laminated, Composite, Circular, Cylindrical Shell Panels
,”
Arch. Appl. Mech.
,
80
(
4
), pp.
413
440
.10.1007/s00419-009-0323-0
104.
Miri
,
A. K.
, and
Nosier
,
A.
,
2011
, “
Out-of-Plane Stresses in Composite Shell Panels: Layerwise and Elasticity Solutions
,”
Acta Mech.
,
220
(
1–4
), pp.
15
32
.10.1007/s00707-011-0471-5
105.
Miri
,
A. K.
, and
Nosier
,
A.
,
2011
, “
Interlaminar Stresses in Antisymmetric Angle-Ply Cylindrical Shell Panels
,”
Compos. Struct.
,
93
(
2
), pp.
419
429
.10.1016/j.compstruct.2010.08.038
106.
Afshin
,
M.
,
Sadighi
,
M.
, and
Shakeri
,
M.
,
2010
, “
Free-Edge Effects in a Cylindrical Sandwich Panel With a Flexible Core and Laminated Composite Face Sheets
,”
Mech. Compos. Mater.
,
46
(
5
), pp.
539
554
.10.1007/s11029-010-9170-x
107.
Ahmadi
,
I.
,
2018
, “
Edge Stresses Analysis in Laminated Thick Sandwich Cylinder Subjected to Distributed Hygrothermal Loading
,”
J. Sandwich Struct. Mater.
,
20
(
4
), pp.
425
461
.10.1177/1099636216657681
108.
Ahmadi
,
I.
,
2019
, “
Free Edge Stress Prediction in Thick Laminated Cylindrical Shell Panel Subjected to Bending Moment
,”
Appl. Math. Modell.
,
65
, pp.
507
525
.10.1016/j.apm.2018.08.029
109.
Ahmadi
,
I.
,
2020
, “
A Three-Dimensional Formulation for Levy-Type Transversely Loaded Cross-Ply Shell Panels
,”
Int. J. Mech. Sci.
,
167
, p.
105224
.10.1016/j.ijmecsci.2019.105224
110.
Tahani
,
M.
,
Andakhshideh
,
A.
, and
Maleki
,
S.
,
2016
, “
Interlaminar Stresses in Thick Cylindrical Shell With Arbitrary Laminations and Boundary Conditions Under Transverse Loads
,”
Compos. Part B Eng.
,
98
, pp.
151
165
.10.1016/j.compositesb.2016.05.013
111.
Schnabel
,
J. E.
,
Yousfi
,
M.
, and
Mittelstedt
,
C.
,
2017
, “
Free-Edge Stress Fields in Cylindrically Curved Symmetric and Unsymmetric Cross-Ply Laminates Under Bending Load
,”
Compos. Struct.
,
180
, pp.
862
875
.10.1016/j.compstruct.2017.08.002
112.
Ko
,
W. L.
, and
Jackson
,
R. H.
,
1989
, “
Multilayer Theory for Delamination Analysis of a Composite Curved Bar Subjected to End Forces and End Moments
,” NASA Research Center, Edwards, CA, Report No. 4139.
113.
Kappel
,
A.
, and
Mittelstedt
,
C.
,
2020
, “
Free-Edge Stress Fields in Cylindrically Curved Cross-Ply Laminated Shells
,”
Compos. Part B: Eng.
,
183
, p.
107693
.10.1016/j.compositesb.2019.107693
114.
Shah
,
P. H.
, and
Batra
,
R. C.
,
2017
, “
Stress Singularities and Transverse Stresses Near Edges of Doubly Curved Laminated Shells Using TSNDT and Stress Recovery Scheme
,”
Eur. J. Mech. A/Solids
,
63
, pp.
68
83
.10.1016/j.euromechsol.2016.11.007
115.
Hélénon
,
F.
,
Wisnom
,
M. R.
,
Hallett
,
S. R.
, and
Allegri
,
G.
,
2010
, “
An Approach for Dealing With High Local Stresses in Finite Element Analyses
,”
Compos. Part A Appl. Sci. Manuf.
,
41
(
9
), pp.
1156
1163
.10.1016/j.compositesa.2010.04.014
116.
Hélénon
,
F.
,
Wisnom
,
M. R.
,
Hallett
,
S. R.
, and
Trask
,
R. S.
,
2012
, “
Numerical Investigation Into Failure of Laminated Composite T-Piece Specimens Under Tensile Loading
,”
Compos. Part A Appl. Sci. Manuf.
,
43
(
7
), pp.
1017
1027
.10.1016/j.compositesa.2012.02.010
117.
Hélénon
,
F.
,
Wisnom
,
M. R.
,
Hallett
,
S. R.
, and
Trask
,
R. S.
,
2013
, “
Investigation Into Failure of Laminated Composite T-Piece Specimens Under Bending Loading
,”
Compos. Part A Appl. Sci. Manuf.
,
54
, pp.
182
189
.10.1016/j.compositesa.2013.07.015
118.
Nagle
,
A.
,
Wowk
,
D.
, and
Marsden
,
C.
,
2020
, “
Three-Dimensional Modelling of Interlaminar Normal Stresses in Curved Laminate Components
,”
Compos. Struct.
,
242
, p.
112165
.10.1016/j.compstruct.2020.112165
119.
Reinarz
,
A.
,
Dodwell
,
T.
,
Fletcher
,
T.
,
Seelinger
,
L.
,
Butler
,
R.
, and
Scheichl
,
R.
,
2018
, “
Dune-Composites – a New Framework for High-Performance Finite Element Modelling of Laminates
,”
Compos. Struct.
,
184
, pp.
269
278
.10.1016/j.compstruct.2017.09.104
120.
Wimmer
,
G.
,
Kitzmüller
,
W.
,
Pinter
,
G.
,
Wettemann
,
T.
, and
Pettermann
,
H. E.
,
2009
, “
Computational and Experimental Investigation of Delamination in L-Shaped Laminated Composite Components
,”
Eng. Fract. Mech.
,
76
(
18
), pp.
2810
2820
.10.1016/j.engfracmech.2009.06.007
121.
Wimmer
,
G.
,
Schuecker
,
C.
, and
Pettermann
,
H. E.
,
2009
, “
Numerical Simulation of Delamination in Laminated Composite Components – a Combination of a Strength Criterion and Fracture Mechanics
,”
Compos. Part B: Eng.
,
40
(
2
), pp.
158
165
.10.1016/j.compositesb.2008.10.006
122.
Zimmermann
,
K.
,
Zenkert
,
D.
, and
Siemetzki
,
M.
,
2010
, “
Testing and Analysis of Ultra Thick Composites
,”
Compos. Part B Eng.
,
41
(
4
), pp.
326
336
.10.1016/j.compositesb.2009.12.004
123.
Hao
,
W.
,
Ge
,
D.
,
Ma
,
Y.
,
Yao
,
X.
, and
Shi
,
Y.
,
2012
, “
Experimental Investigation on Deformation and Strength of Carbon/Epoxy Laminated Curved Beams
,”
Polym. Test.
,
31
(
4
), pp.
520
526
.10.1016/j.polymertesting.2012.02.003
124.
Charrier
,
J. S.
,
Laurin
,
F.
,
Carrere
,
N.
, and
Mahdi
,
S.
,
2016
, “
Determination of the Out-of-Plane Tensile Strength Using Four-Point Bending Tests on Laminated L-Angle Specimens With Different Stacking Sequences and Total Thicknesses
,”
Compos. Part A Appl. Sci. Manuf.
,
81
, pp.
243
253
.10.1016/j.compositesa.2015.11.018
125.
González-Cantero
,
J. M.
,
Graciani
,
E.
,
López-Romano
,
B.
, and
París
,
F.
,
2018
, “
Competing Mechanisms in the Unfolding Failure in Composite Laminates
,”
Compos. Sci. Technol.
,
156
, pp.
223
230
.10.1016/j.compscitech.2017.12.022
126.
Pan
,
Z. Y.
,
Duan
,
Q. F.
,
Zhong
,
Y. C.
,
Li
,
S. X.
, and
Cao
,
D. F.
,
2018
, “
Stacking Sequence Effect on the Fracture Behavior of Narrow L-Shaped Cross-Ply Laminates: Experimental Study
,”
Strength Mater.
,
50
(
1
), pp.
203
210
.10.1007/s11223-018-9960-2
127.
Journoud
,
P.
,
Bouvet
,
C.
,
Castanié
,
B.
,
Laurin
,
F.
, and
Ratsifandrihana
,
L.
,
2020
, “
Experimental and Numerical Analysis of Unfolding Failure of L-Shaped CFRP Specimens
,”
Compos. Struct.
,
232
, p.
111563
.10.1016/j.compstruct.2019.111563
128.
Cao
,
D.
,
Hu
,
H.
,
Duan
,
Q.
,
Song
,
P.
, and
Li
,
S.
,
2019
, “
Experimental and Three-Dimensional Numerical Investigation of Matrix Cracking and Delamination Interaction With Edge Effect of Curved Composite Laminates
,”
Compos. Struct.
,
225
, p.
111154
.10.1016/j.compstruct.2019.111154
129.
Fletcher
,
T. A.
,
Kim
,
T.
,
Dodwell
,
T. J.
,
Butler
,
R.
,
Scheichl
,
R.
, and
Newley
,
R.
,
2016
, “
Resin Treatment of Free Edges to Aid Certification of Through Thickness Laminate Strength
,”
Compos. Struct.
,
146
, pp.
26
33
.10.1016/j.compstruct.2016.02.074
130.
Ranz
,
D.
,
Cuartero
,
J.
,
Miravete
,
A.
, and
Miralbes
,
R.
,
2017
, “
Experimental Research Into Interlaminar Tensile Strength of Carbon/Epoxy Laminated Curved Beams
,”
Compos. Struct.
,
164
, pp.
189
197
.10.1016/j.compstruct.2016.12.010
131.
Ju
,
H.
,
Nguyen
,
K. H.
,
Chae
,
S. S.
, and
Kweon
,
J. H.
,
2017
, “
Delamination Strength of Composite Curved Beams Reinforced by Grooved Stainless-Steel Z-Pins
,”
Compos. Struct.
,
180
, pp.
497
506
.10.1016/j.compstruct.2017.08.018
132.
Louhghalam
,
A.
,
Igusa
,
T.
, and
Tootkaboni
,
M.
,
2014
, “
Dynamic Characteristics of Laminated Thin Cylindrical Shells: Asymptotic Analysis Accounting for Edge Effect
,”
Compos. Struct.
,
112
, pp.
22
37
.10.1016/j.compstruct.2014.01.031
133.
Ahmadi
,
I.
,
2017
, “
Interlaminar Stress Analysis in General Thick Composite Cylinder Subjected to Nonuniform Distributed Radial Pressure
,”
Mech. Adv. Mater. Struct.
,
24
(
9
), pp.
773
788
.10.1080/15376494.2016.1196782
134.
Filipovic
,
D. T.
, and
Kress
,
G. R.
,
2020
, “
Free-Edge Effects of Corrugated Laminates
,”
Curved Layered Struct.
,
7
(
1
), pp.
101
124
.10.1515/cls-2020-0009
135.
Pastorino
,
D.
,
Blazquez
,
A.
,
López-Romano
,
B.
, and
París
,
F.
,
2019
, “
Closed-Form Methodology for Stress Analysis of Composite Plates With Cutouts and Non-Uniform Lay-Up
,”
Compos. Struct.
,
212
, pp.
389
397
.10.1016/j.compstruct.2019.01.013
136.
Lin
,
C. C.
, and
Ko
,
C. C.
,
1988
, “
Stress and Strength Analysis of Finite Composite Laminates With Elliptical Holes
,”
J. Compos. Mater.
,
22
(
4
), pp.
373
385
.10.1177/002199838802200405
137.
Ukadgaonker
,
V.
, and
Kakhandki
,
V.
,
2005
, “
Stress Analysis for an Orthotropic Plate With an Irregular Shaped Hole for Different in-Plane Loading Conditions–Part 1
,”
Compos. Struct.
,
70
(
3
), pp.
255
274
.10.1016/j.compstruct.2004.08.032
138.
Savin
,
G. N.
,
1961
,
Stress Concentration Around Holes
,
Pergamon
,
London, UK
.
139.
Ghannadpour
,
S.
, and
Mehrparvar
,
M.
,
2018
, “
Energy Effect Removal Technique to Model Circular/Elliptical Holes in Relatively Thick Composite Plates Under in-Plane Compressive Load
,”
Compos. Struct.
,
202
, pp.
1032
1041
.10.1016/j.compstruct.2018.05.026
140.
Chaudhuri
,
R. A.
,
2009
, “
A New Three-Dimensional Shell Theory in General (Non-Lines-of-Curvature) Coordinates for Analysis of Curved Panels Weakened by Through/Part-Through Holes
,”
Compos. Struct.
,
89
(
2
), pp.
321
332
.10.1016/j.compstruct.2008.07.005
141.
Hu
,
Y. L.
, and
Madenci
,
E.
,
2017
, “
Peridynamics for Fatigue Life and Residual Strength Prediction of Composite Laminates
,”
Compos. Struct.
,
160
, pp.
169
184
.10.1016/j.compstruct.2016.10.010
142.
Han
,
X. P.
,
Li
,
L. X.
,
Zhu
,
X. P.
, and
Yue
,
Z. F.
,
2008
, “
Experimental Study on the Stitching Reinforcement of Composite Laminates With a Circular Hole
,”
Compos. Sci. Technol.
,
68
(
7–8
), pp.
1649
1653
.10.1016/j.compscitech.2008.02.017
143.
Wisnom
,
M. R.
, and
Hallett
,
S. R.
,
2009
, “
The Role of Delamination in Strength, Failure Mechanism and Hole Size Effect in Open Hole Tensile Tests on Quasi-Isotropic Laminates
,”
Compos. Part A
,
40
(
4
), pp.
335
342
.10.1016/j.compositesa.2008.12.013
144.
Solis
,
A.
,
Barbero
,
E.
, and
Sánchez-Sáez
,
S.
,
2020
, “
Analysis of Damage and Interlaminar Stresses in Laminate Plates With Interacting Holes
,”
Int. J. Mech. Sci.
,
165
, p.
105189
.10.1016/j.ijmecsci.2019.105189
145.
Herrmann
,
K. P.
, and
Linnenbrock
,
K.
,
2002
, “
Three-Dimensional Thermal Crack Growth in Self-Stressed Bimaterial Joints: Analysis and Experiment
,”
Int. J. Fract.
,
114
(
2
), pp.
133
151
.10.1023/A:1015034803792
146.
Becker
,
W.
,
Jin
,
P. P.
, and
Neuser
,
P.
,
1999
, “
Interlaminar Stresses at the Free Corners of a Laminate
,”
Compos. Struct.
,
45
(
2
), pp.
155
162
.10.1016/S0263-8223(99)00019-7
147.
Griffin
,
O. H.
,
1988
, “
Three-Dimensional Thermal Stresses in Angle-Ply Composite Laminates
,”
J. Compos. Mater.
,
22
(
1
), pp.
53
70
.10.1177/002199838802200104
148.
Koguchi
,
H.
,
1997
, “
Stress Singularity Analysis in Three-Dimensional Bonded Structure
,”
Int. J. Solids Struct.
,
34
(
4
), pp.
461
480
.10.1016/S0020-7683(96)00028-5
149.
Labossiere
,
P. E. W.
, and
Dunn
,
M. L.
,
2001
, “
Fracture Initiation at Three-Dimensional Bimaterial Interface Corners
,”
J. Mech. Phys. Solids
,
49
, pp.
609
634
.10.1016/S0022-5096(00)00043-0
150.
Dimitrov
,
A.
,
Andrä
,
H.
, and
Schnack
,
E.
,
2001
, “
Efficient Computation of Order and Mode of Corner Singularities in 3D-Elasticity
,”
Int. J. Numer. Methods Eng.
,
52
(
8
), pp.
805
–8
24
.10.1002/nme.230
151.
Dimitrov
,
A.
,
Andrä
,
H.
, and
Schnack
,
E.
,
2002
, “
Singularities Near Three-Dimensional Corners in Composite Laminates
,”
Int. J. Fract.
,
115
(
4
), pp.
361
375
.10.1023/A:1016320103641
152.
Mittelstedt
,
C.
, and
Becker
,
W.
,
2005
, “
Asymptotic Analysis of Stress Singularities in Composite Laminates by the Boundary Finite Element Method
,”
Compos. Struct.
,
71
(
2
), pp.
210
219
.10.1016/j.compstruct.2004.10.003
153.
Mittelstedt
,
C.
, and
Becker
,
W.
,
2005
, “
Semi-Analytical Computation of 3D Stress Singularities in Linear Elasticity
,”
Commun. Numer. Methods Eng.
,
21
(
5
), pp.
247
257
.10.1002/cnm.742
154.
Mittelstedt
,
C.
, and
Becker
,
W.
,
2006
, “
Efficient Computation of Order and Mode of Three-Dimensional Stress Singularities in Linear Elasticity by the Boundary Finite Element Method
,”
Int. J. Solids Struct.
,
43
(
10
), pp.
2868
2903
.10.1016/j.ijsolstr.2005.05.059
155.
Mittelstedt
,
C.
, and
Becker
,
W.
,
2005
, “
Thermoelastic Fields in Boundary Layers of Isotropic Laminates
,”
ASME J. Appl. Mech.
,
72
(
1
), pp.
86
101
.10.1115/1.1827247
156.
Mittelstedt
,
C.
, and
Becker
,
W.
,
2005
, “
A Variational Model for Boundary Layer Effects in Cross-Ply Laminates Based on a C0-Continuous Layerwise Displacement Formulation
,”
J. Compos. Mater.
,
39
(
20
), pp.
1789
1818
.10.1177/0021998305051121
157.
Artel
,
J.
, and
Becker
,
W.
,
2005
, “
Coupled and Uncoupled Analyses of Piezoelectric Free-Edge Effect in Laminated Plates
,”
Compos. Struct.
,
69
(
3
), pp.
329
335
.10.1016/j.compstruct.2004.07.015
158.
Yang
,
Q. S.
,
Qin
,
Q. H.
, and
Liu
,
T.
,
2006
, “
Interlayer Stress in Laminate Beam of Piezoelectric and Elastic Materials
,”
Compos. Struct.
,
75
(
1–4
), pp.
587
592
.10.1016/j.compstruct.2006.04.024
159.
Tahani
,
M.
, and
Mirzababaee
,
M.
,
2009
, “
Higher-Order Coupled and Uncoupled Analyses of Free Edge Effect in Piezoelectric Laminates Under Mechanical Loadings
,”
Mater. Des.
,
30
(
7
), pp.
2473
2482
.10.1016/j.matdes.2008.10.004
160.
Mirzababaee
,
M.
, and
Tahani
,
M.
,
2009
, “
Accurate Determination of Coupling Effects on Free Edge Interlaminar Stresses in Piezoelectric Laminated Plates
,”
Mater. Des.
,
30
(
8
), pp.
2963
2974
.10.1016/j.matdes.2009.01.005
161.
Izadi
,
M.
, and
Tahani
,
M.
,
2010
, “
Analysis of Interlaminar Stresses in General Cross-Ply Laminates With Distributed Piezoelectric Actuators
,”
Compos. Struct.
,
92
(
3
), pp.
757
768
.10.1016/j.compstruct.2009.09.003
162.
Kapuria
,
S.
, and
Kumari
,
P.
,
2012
, “
Boundary Layer Effects in Levy-Type Rectangular Piezoelectric Composite Plates Using a Coupled Efficient Layerwise Theory
,”
Eur. J. Mech. A/Solids
,
36
, pp.
122
140
.10.1016/j.euromechsol.2012.02.015
163.
Han
,
C.
,
Wu
,
Z.
, and
Niu
,
Z.
,
2014
, “
Accurate Prediction of Free-Edge and Electromechanical Coupling Effects in Cross-Ply Piezoelectric Laminates
,”
Compos. Struct.
,
113
, pp.
308
315
.10.1016/j.compstruct.2014.03.027
164.
Huang
,
B.
, and
Kim
,
H. S.
,
2014
, “
Free-Edge Interlaminar Stress Analysis of Piezo-Bonded Composite Laminates Under Symmetric Electric Excitation
,”
Int. J. Solids Struct.
,
51
(
6
), pp.
1246
1252
.10.1016/j.ijsolstr.2013.12.016
165.
Huang
,
B.
, and
Kim
,
H. S.
,
2015
, “
Interlaminar Stress Analysis of Piezo-Bonded Composite Laminates Using the Extended Kantorovich Method
,”
Int. J. Mech. Sci.
,
90
, pp.
16
24
.10.1016/j.ijmecsci.2014.11.003
166.
Huang
,
B.
,
Kim
,
H. S.
,
Wang
,
J.
, and
Du
,
J.
,
2016
, “
Free Edge Stress Prediction for Magneto-Electro-Elastic Laminates Using a Stress Function Based Equivalent Single Layer Theory
,”
Compos. Sci. Technol.
,
123
, pp.
205
211
.10.1016/j.compscitech.2015.12.019
167.
Kapuria
,
S.
, and
Dhanesh
,
N.
,
2017
, “
Free Edge Stress Field in Smart Piezoelectric Composite Structures and Its Control: An Accurate Multiphysics Solution
,”
Int. J. Solids Struct.
,
126-127
, pp.
196
207
.10.1016/j.ijsolstr.2017.08.007
168.
Dhanesh
,
N.
, and
Kapuria
,
S.
,
2018
, “
Edge Effects in Elastic and Piezoelectric Laminated Panels Under Thermal Loading
,”
J. Therm. Stresses
,
41
(
10–12
), pp.
1577
1596
.10.1080/01495739.2018.1524732
169.
Andakhshideh
,
A.
,
Rafiee
,
R.
, and
Maleki
,
S.
,
2019
, “
3D Stress Analysis of Generally Laminated Piezoelectric Plates With Electromechanical Coupling Effects
,”
Appl. Math. Modell.
,
74
, pp.
258
279
.10.1016/j.apm.2019.04.060
170.
Song
,
C.
, and
Wolf
,
J. P.
,
1995
, “
Consistent Infinitesimal Finite-Element-Cell Method: Out-of-Plane Motion
,”
J. Eng. Mech.
,
121
(
5
), pp.
613
619
.10.1061/(ASCE)0733-9399(1995)121:5(613)
171.
Song
,
C.
, and
Wolf
,
J. P.
,
1997
, “
The Scaled Boundary Finite-Element Method - Alias Consistent Infinitesimal Finite-Element Cell Method - for Elastodynamics
,”
Comput. Methods Appl. Mech. Eng.
,
147
(
3–4
), pp.
329
355
.10.1016/S0045-7825(97)00021-2
172.
Song
,
C.
,
2005
, “
Evaluation of Power-Logarithmic Singularities, T-Stresses and Higher Order Terms of in-Plane Singular Stress Fields at Cracks and Multi-Material Corners
,”
Eng. Fract. Mech.
,
72
(
10
), pp.
1498
1530
.10.1016/j.engfracmech.2004.11.002
173.
Song
,
C.
,
Tin-Loi
,
F.
, and
Gao
,
W.
,
2010
, “
A Definition and Evaluation Procedure of Generalized Stress Intensity Factors at Cracks and Multi-Material Wedges
,”
Eng. Fract. Mech.
,
77
(
12
), pp.
2316
2336
.10.1016/j.engfracmech.2010.04.032
174.
Saputra
,
A. A.
,
Birk
,
C.
, and
Song
,
C.
,
2015
, “
Computation of Three-Dimensional Fracture Parameters at Interface Cracks and Notches by the Scaled Boundary Finite Element Method
,”
Eng. Fract. Mech.
,
148
, pp.
213
242
.10.1016/j.engfracmech.2015.09.006
175.
Song
,
C.
,
Ooi
,
E. T.
, and
Natarajan
,
S.
,
2018
, “
A Review of the Scaled Boundary Finite Element Method for Two-Dimensional Linear Elastic Fracture Mechanics
,”
Eng. Fract. Mech.
,
187
, pp.
45
73
.10.1016/j.engfracmech.2017.10.016
176.
Hell
,
S.
, and
Becker
,
W.
,
2015
, “
The Scaled Boundary Finite Element Method for the Analysis of 3D Crack Interaction
,”
J. Comput. Sci.
,
9
, pp.
76
81
.10.1016/j.jocs.2015.04.007
177.
Dölling
,
S.
,
Hahn
,
J.
,
Felger
,
J.
,
Bremm
,
S.
, and
Becker
,
W.
,
2020
, “
A Scaled Boundary Finite Element Method Model for Interlaminar Failure in Composite Laminates
,”
Compos. Struct.
,
241
, p.
111865
.10.1016/j.compstruct.2020.111865
178.
Artel
,
J.
, and
Becker
,
W.
,
2006
, “
On Kinematic Coupling Equations Within the Scaled Boundary Finite-Element Method
,”
Arch. Appl. Mech.
,
76
(
11–12
), pp.
617
633
.10.1007/s00419-006-0052-6
179.
Goswami
,
S.
, and
Becker
,
W.
,
2012
, “
Computation of 3-D Stress Singularities for Multiple Cracks and Crack Intersections by the Scaled Boundary Finite Element Method
,”
Int. J. Fract.
,
175
(
1
), pp.
13
25
.10.1007/s10704-012-9694-2
180.
Dieringer
,
R.
, and
Becker
,
W.
,
2015
, “
A New Scaled Boundary Finite Element Formulation for the Computation of Singularity Orders at Cracks and Notches in Arbitrarily Laminated Composites
,”
Compos. Struct.
,
123
, pp.
263
270
.10.1016/j.compstruct.2014.12.036
181.
Lecomte-Grosbras
,
P.
,
Paluch
,
B.
, and
Brieu
,
M.
,
2013
, “
Characterization of Free Edge Effects: Influence of Mechanical Properties, Microstructure and Structure Effects
,”
J. Compos. Mater.
,
47
(
22
), pp.
2823
2834
.10.1177/0021998312458817
182.
Lecomte-Grosbras
,
P.
,
Réthoré
,
J.
,
Limodin
,
N.
,
Witz
,
J. F.
, and
Brieu
,
M.
,
2015
, “
Three-Dimensional Investigation of Free-Edge Effects in Laminate Composites Using X-Ray Tomography and Digital Volume Correlation
,”
Exp. Mech.
,
55
(
1
), pp.
301
311
.10.1007/s11340-014-9891-1
183.
Duan
,
S.
,
Zhang
,
Z.
,
Wei
,
K.
,
Wang
,
F.
, and
Han
,
X.
,
2020
, “
Theoretical Study and Physical Tests of Circular Hole-Edge Stress Concentration in Long Glass Fiber Reinforced Polypropylene Composite
,”
Compos. Struct.
,
236
, p.
111884
.10.1016/j.compstruct.2020.111884
184.
Charkviani
,
R. V.
,
Pavlov
,
A. A.
, and
Pavlova
,
S. A.
,
2017
, “
Interlaminar Strength and Stiffness of Layered Composite Materials
,”
Procedia Eng.
,
185
, pp.
168
172
.10.1016/j.proeng.2017.03.335
185.
Kappel
,
E.
,
2021
, “
Experimental Study on How Free-Edge Effects Impede CTE Measurements
,”
Compos. Part C Open Access
5
, p.
100129
.10.1016/j.jcomc.2021.100129
186.
Lagunegrand
,
L.
,
Lorriot
,
T.
,
Harry
,
R.
, and
Wargnier
,
H.
,
2005
, “
Design of an Improved Four Point Bending Test on a Sandwich Beam for Free Edge Delamination Studies
,”
Compos. Part B
,
37
(
2–3
), pp.
127
136
.10.1016/j.compositesb.2005.07.002
187.
Neuber
,
H.
,
1937
,
Kerbspannungslehre
, Springer,
Berlin
.
188.
Whitney
,
J. M.
, and
Nuismer
,
R. J.
,
1974
, “
Stress Fracture Criteria for Laminated Composites Containing Stress Concentrations
,”
J. Compos. Mater.
,
8
(
3
), pp.
253
265
.10.1177/002199837400800303
189.
Taylor
,
D.
,
2007
,
The Theory of Critical Distances
,
Elsevier
,
Oxford, UK
.
190.
Taylor
,
D.
,
2008
, “
The Theory of Critical Distances
,”
Eng. Fract. Mech.
,
75
(
7
), pp.
1696
1705
.10.1016/j.engfracmech.2007.04.007
191.
Lagunegrand
,
L.
,
Lorriot
,
T.
,
Harry
,
R.
,
Wargnier
,
H.
, and
Quenisset
,
J. M.
,
2006
, “
Initiation of Free-Edge Delamination in Composite Laminates
,”
Compos. Sci. Technol.
,
66
(
10
), pp.
1315
1327
.10.1016/j.compscitech.2005.10.010
192.
Diaz
,
A. D.
, and
Caron
,
J.-F.
,
2006
, “
Prediction of the Onset of Mode III Delamination in Carbon-Epoxy Laminates
,”
Compos. Struct.
,
72
(
4
), pp.
438
445
.10.1016/j.compstruct.2005.01.014
193.
Dugdale
,
D. S.
,
1960
, “
Yielding of Steel Sheets Containing Slits
,”
J. Mech. Phys. Solids
,
8
(
2
), pp.
100
104
.10.1016/0022-5096(60)90013-2
194.
Barenblatt
,
G. I.
,
1962
, “
The Mathematical Theory of Equilibrium Cracks in Brittle Fracture
,”
Adv. Appl. Mech.
,
7
, pp.
55
129
.10.1016/S0065-2156(08)70121-2
195.
Sorensen
,
B. F.
,
2010
,
Cohesive Laws for Assessment of Materials Failure: Theory, Experimental Methods and Application
,
Technical University of Denmark (DTU)
,
Roskilde, Denmark
.
196.
Leguillon
,
D.
,
2002
, “
Strength or Toughness? A Criterion for Crack Onset at a Notch
,”
Eur. J. Mech. A/Solids
,
21
(
1
), pp.
61
72
.10.1016/S0997-7538(01)01184-6
197.
Hashin
,
Z.
,
1996
, “
Finite Thermoelastic Fracture Criterion With Application to Laminate Cracking Analysis
,”
J. Mech. Phys. Solids
,
44
(
7
), pp.
1129
1145
.10.1016/0022-5096(95)00080-1
198.
Weißgraeber
,
P.
,
Leguillon
,
D.
, and
Becker
,
W.
,
2016
, “
A Review of Finite Fracture Mechanics: Crack Initiation at Singular and Non-Singular Stress Raisers
,”
Arch. Appl. Mech.
,
86
(
1–2
), pp.
375
401
.10.1007/s00419-015-1091-7
199.
Leguillon
,
D.
, and
Yosibash
,
Z.
,
2003
, “
Crack Onset at a v-Notch. Influence of the Notch Tip Radius
,”
Int. J. Fract.
,
122
(
1/2
), pp.
1
21
.10.1023/B:FRAC.0000005372.68959.1d
200.
Carpinteri
,
A.
,
Cornetti
,
P.
,
Pugno
,
N.
,
Sapora
,
A.
, and
Taylor
,
D.
,
2008
, “
A Finite Fracture Mechanics Approach to Structures With Sharp V-Notches
,”
Eng. Fract. Mech.
,
75
(
7
), pp.
1736
1752
.10.1016/j.engfracmech.2007.04.010
201.
Weißgraeber
,
P.
, and
Becker
,
W.
,
2011
, “
A New Finite Fracture Mechanics Approach for Assessing the Strength of Bonded Lap Joints
,”
Key Eng. Mater.
,
471–472
, pp.
1075
1080
.10.4028/www.scientific.net/KEM.471-472.1075
202.
Stein
,
N.
,
Weißgraeber
,
P.
, and
Becker
,
W.
,
2015
, “
A Model for Brittle Failure in Adhesive Lap Joints of Arbitrary Joint Configuration
,”
Compos. Struct.
,
133
, pp.
707
718
.10.1016/j.compstruct.2015.07.100
203.
Garcia
,
I.
,
Mantic
,
V.
,
Blazquez
,
A.
, and
Paris
,
F.
,
2014
, “
Transverse Crack Onset and Growth in Cross-Ply [0/90], Laminates Under Tension. Application of a Coupled Stress and Energy Criterion
,”
Int. J. Solids Struct.
,
51
, pp.
3844
3856
.10.1016/j.ijsolstr.2014.06.015
204.
Hebel
,
J.
,
Dieringer
,
R.
, and
Becker
,
W.
,
2010
, “
Modeling Brittle Crack Formation at Geometrical and Material Discontinuities Using a Finite Fracture Mechanics Approach
,”
Eng. Fract. Mech.
,
77
(
18
), pp.
3558
3572
.10.1016/j.engfracmech.2010.07.005
205.
Mantic
,
V.
,
2009
, “
Interface Crack Onset at a Circular Cylindrical Inclusion Under a Remote Transverse Tension. Application of a Coupled Stress and Energy Criterion
,”
Int. J. Solids Struct.
,
46
, pp.
1287
1304
.10.1016/j.ijsolstr.2008.10.036
206.
Weißgraeber
,
P.
,
Felger
,
J.
,
Geipel
,
D.
, and
Becker
,
W.
,
2016
, “
Cracks at Elliptical Holes: Stress Intensity Factor and Finite Fracture Mechanics Solution
,”
Eur. J. Mech. A/Solids
,
55
, pp.
192
198
.10.1016/j.euromechsol.2015.09.002
207.
Felger
,
J.
,
Stein
,
N.
, and
Becker
,
W.
,
2017
, “
Asymptotic Finite Fracture Mechanics Solution for Crack Onset at Elliptical Holes Iin Composite Plates of Finite-Width
,”
Eng. Fract. Mech.
,
182
, pp.
621
634
.10.1016/j.engfracmech.2017.05.048
208.
Rosendahl
,
P.
,
Weißgraeber
,
P.
,
Stein
,
N.
, and
Becker
,
W.
,
2017
, “
Asymmetric Crack Onset at Open-Holes Under Tensile and in-Plane Bending Loading
,”
Int. J. Solids Struct.
,
113–114
, pp.
10
23
.10.1016/j.ijsolstr.2016.09.011
209.
Leguillon
,
D.
,
Haddad
,
O.
,
Adamowska
,
M.
, and
da Costa
,
P.
,
2014
, “
Crack Pattern Formation and Spalling in Functionalized Thin Films
,”
Procedia Mater. Sci.
,
3
, pp.
104
109
.10.1016/j.mspro.2014.06.020
210.
Hebel
,
J.
, and
Becker
,
W.
,
2008
, “
Numerical Analysis of Brittle Crack Initiation at Stress Concentrations in Composites
,”
Mech. Adv. Mater. Struct.
,
15
(
6–7
), pp.
410
420
.10.1080/15376490802135266
211.
Martin
,
E.
,
Leguillon
,
D.
, and
Carrere
,
N.
,
2010
, “
A Twofold Strength and Toughness Criterion for the Onset of Free-Edge Shear Delamination in Angle-Ply Laminates
,”
Int. J. Solids Struct.
,
47
(
9
), pp.
1297
1305
.10.1016/j.ijsolstr.2010.01.018
212.
Dölling
,
S.
,
Hell
,
S.
, and
Becker
,
W.
,
2018
, “
Investigation of the Laminate free-edge Effect by Means of the Scaled Boundary Finite Element Method
,”
PAMM—Proc. Appl. Math. Mech.
,
18
(
1
), p.
e201800129
.10.1002/pamm.201800129
213.
Dölling
,
S.
,
Felger
,
J.
,
Hahn
,
J.
,
Bremm
,
S.
, and
Becker
,
W.
, 9th–13th July
2019
, “
An Application of the Scaled Boundary Finite Element Method to Laminates: Prediction of Interlaminar Crack Onset Caused by the Free-Edge Effect
,”
Presentations at the 10th ICCM2019
,
G. R.
Liu
,
F.
Cui
,
G. X.
Xiangguo
, eds.,
ScienTech Publisher
,
Singapore
, Paper No. 3747.