## 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
Fig. 1
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 $κxx=−∂2w0/∂x2, κyy=−∂2w0/∂y2, κxy=−2∂2w0/∂x∂y$. 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 $θ=0 deg$ or $θ=90 deg$ 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 $[0 deg/90 deg]S$ (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
Fig. 2
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 $0 deg$ and the $90 deg$-layers exhibit largely different contraction properties, it is clear that in this case the outer $0 deg$-plies will suffer much greater displacements v than the inner $90 deg$-layers. As a result, the inplane displacement v will become discontinuous at the interfaces between the $0 deg$ and the $90 deg$-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 $0 deg$-layers a tensile stress would need to be applied while at the $90 deg$-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 $y−$direction (see the free-body diagram of the $0 deg$-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 $0 deg$ 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 $x−$axis dictates that the interlaminar normal stress σzz will have to arise as well in the interface between the $0 deg$ and the 90 deg-layers at $z=h4$ (cf. Fig. 2, detail B). The following equilibrium condition can be deduced:
$∫0yσzzŷdŷ=−∫h4h2σyy(z−h4)dz$
(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 $0 deg$ 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 $0 deg$ 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 $[0 deg/90 deg]S$ (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
Fig. 3
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 $[±45 deg]S$. 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 $+45 deg−$layers will exhibit a different sign than those in the $−45 deg−$plies 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 $+45 deg$ and the $−45 deg−$layers. 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
Fig. 4
Close modal

Accordingly, a calculation using CLPT will lead to layerwise shear stresses τxy with identical absolute values yet opposite signs wherein in the $+45 deg−$layers τxy will exhibit a positive sign while in the $−45 deg−$layers 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 $+45 deg$ and the $−45 deg−$layers 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 $x−$direction (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 $+45 deg$ and the $−45 deg−$layers at $z=±h4$ with respect to the $y−$direction 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
Fig. 5
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
Fig. 6
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
Fig. 7
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.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
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
Fig. 11
Close modal
Fig. 12
Fig. 12
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
Fig. 13
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
$∂σi1∂x1+∂σi2∂x2+∂σi3∂x3+fi=0∂D1∂x1+∂D2∂x2+∂D3∂x3−q=0$
(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 $δφ$
$∫Ω∑i=13[(∂σi1∂x1+∂σi2∂x2+∂σi3∂x3+fi)δui]dΩ+∫Ω(∂D1∂x1+∂D2∂x2+∂D3∂x3−q)δφdΩ=0$
(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 ($i,j=1,2,3$):
$εii=∂ui∂xiγij=∂ui∂xj+∂uj∂xiEi=−∂φ∂xi$
(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.

## 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
$ξ2E0û,ξξ(ξ)+ξ(E0+E1T−E1)û,ξ(ξ)−E2û(ξ)=−ξ(Sξ−Sη)$
(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
Fig. 14
Close modal
Fig. 15
Fig. 15
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 $−15 deg$ 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 $G≥Gc$ 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:
$f(σij)≥σc along crack length∧G¯≥Gcwhere G¯=−ΔΠΔA≥Gc$
(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:
$G¯=1a∫0aG(ã)dã$
(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.

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