Cohesive zone models and impact damage predictions for composite structures

Abstract

This article surveys existing cohesive zone models (CZM) and their use in numerical simulations for analysis impacts on composite structures and predicting the damage induced. These models are used for matrix cracks and delamination. The first part of article gives the required background on failure criteria for predicting the onset of delaminations, on fracture mechanics and the various types of CZM. The second part discusses applications of CZM to several types of impact problems with composite structures. Applications for these models to other problems are briefly mentioned at the end. CZM are now part of state of the art numerical simulations with progressive damage analysis.

1 Introduction

Predicting impact induced damage in composite structures requires the ability to predict the onset of the various damage modes and the growth of that damage during the impact. Cohesive zone models (CZM) used to predict the growth of various types of cracks are reviewed in this article with a special focus on impact problems. These models are generally used in a numerical progressive damage analysis that accounts for damage development and the degradation of the properties of the plies and the interfaces. This article provides an overview of cohesive zone models including their connection to fracture mechanics and their use in prediction impact damage in composite structures. Such damage prediction requires failure criteria to determine when damage initiates and the study of the subsequent damage propagation. The article presents a brief overview of the failure criteria used to predict damage initiation including delamination. With CZMs, the propagation of delaminations is based on a fracture criterion so some basic concepts from fracture mechanics are added to make the presentation self-contained and easier to follow. CZMs are included in several commercial finite element codes and are widely used for various applications. In the analysis of impact damage on composite structures, this article will show that CMZs have resulted in accurate predictions that were not previously possible and greater insights.

Several types of damage can be induced: (1) interply debonding or delamination; (2) intraply damage in the form of shear cracks, bending cracks, and fiber failure. These two types of damage interact to create complex patterns. At a microscopic level, fiber-matrix debonding and microscopic matrix cracks develop and eventually coalesce and form failure surfaces at the ply level. Foreign object impacts can cause delaminations and matrix cracks that can severely reduce the load carrying capacity of the structure [1]. Delaminations also occur when discontinuities induce interlaminar stress concentrations. Examples include: free edges, holes, matrix cracks, ply drop offs, bonded joints, and bolted joints. Extensive research has been conducted on delaminations since this type of failure is induced rather easily and can have a dramatic effect on the residual properties. Several reviews of the literature on delaminations in composite structures are available [2–6]. Failure criteria for predicting the onset of delamination are discussed in Sect. 2.

Once an interfacial crack is created, its propagation under load should be determined using a fracture mechanics approach. Basic fracture mechanics concepts are recalled and their application to the propagation of delaminations is discussed in Sect. 3 that includes a discussion of widely used numerical technique called the Virtual Crack Closure Technique (VCCT). In recent years, the propagation of delaminations and matrix cracks has been analyzed using CZM that are presented in Sect. 4. Applications of the CZM to study damage in composite structures is discussed in Sect. 5 with emphasis on foreign object impacts on composite and sandwich structures and energy absorbing structures made out of composite materials.

2 Failure criteria for predicting the onset of delamination

Debonding between adjacent layers depends on the stresses acting on that interface: the normal componentand the two shear stresses and $\sigma_{23}$. References [7–9] predicted delamination using a maximum stress criterion for the normal stress and a quadratic criterion for the two shear components

$$ \sigma_{33} /Z_{T} \ge 1\;\quad {\text{or}}\quad \;\left( {\frac{{\sigma_{23} }}{{S_{3} }}} \right)^{2} + \left( {\frac{{\sigma_{13}^{{}} }}{{S_{3}^{{}} }}} \right)^{2} \ge 1 $$

(2.1)

$\sigma_{33}$ where S₃ is the shear strength and Z_T $\sigma_{13}$ the tensile strength in the thickness direction. It is assumed that no failure occurs when $\sigma_{33} < 0$. The criterion proposed by Christensen and DeTeresa [10]

$$ \left( {\frac{{\sigma_{13} }}{{S_{13} }}} \right)^{2} + \left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2} \ge 1 $$

(2.2)

allows for different strengths for $\sigma_{13}$ and $\sigma_{23}$ but does not account for interactions between the normal and shear stresses acting at the interface.

Several criteria accounting for the interaction between the three stress components acting at the interface have been introduced. References [11–19] postulated that the onset of delamination is governed by

$$ \left( {\frac{{\sigma_{33} }}{{Z_{T} }}} \right)^{2} + \left( {\frac{{\sigma_{13} }}{{S_{13} }}} \right)^{2} + \left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2} \ge 1\quad {\text{when}}\quad \sigma_{33} > 0 $$

(2.3)

where S₁₃ and S₂₃ are the shear strength in the through-thickness and fiber plane. Again, References [12, 16, 17, 20] assumed that S₁₂ = S₂₃. Cesari et al. [11] used the same criterion when $\sigma_{33} < \,\,0$ but with Z_c (compressive strength in the through-thickness direction) instead of Z_T in that case. The ellipsoid defined by Eq. 2.3 accounts for the interaction between the three stress components acting at the interface.

References [8, 21–26] use the delamination criterion proposed by Choi and Chang [27]. In the latter, originality lies in the fact that the stresses are averaged over the thickness of the layer above the interface (layer n + 1) and the strengths can be those of either the layer above or the layer below (n).

The quadratic delamination criterion of Brewer and Lagace [28] is similar to Eq. (2.3) and can be written as

$$ \left( {\frac{{\sigma_{33} }}{Z}} \right)^{2} + \left( {\frac{{\sigma_{13} }}{{S_{13} }}} \right)^{2} + \left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2} \ge 1 $$

(2.4)

where Z = Z_t when $\sigma_{33} > \,\,0$ and Z = Z_c when $\sigma_{33} < \,\,0$. Thus, the value of Z makes the difference between tensile normal stress (that are opening) and compressive normal stress (that are closing). Naik et al. [29–31] used Eq. 2.4 and averaged the values of the stresses through the thickness of the ply. Li et al. [32] used the Brewer-Lagace criterion as defined when $\sigma_{33} > \,\,0$ and omitted the effect of the transverse normal stress when $\sigma_{33} < \,\,0$. In that case, we recover the criterion proposed by Yeh and Kim [33] which predicts that tensile delaminations occurs when $\sigma_{33} > 0$ and

$$ \left( {\frac{{\sigma_{33} }}{{Z_{T} }}} \right)^{2} + \left( {\frac{{\sigma_{13} }}{{S_{13} }}} \right)^{2} + \left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2} \ge 1 $$

(2.5)

and shear delaminations occurs when σ₃₃ < 0 and

$$ \left( {\frac{{\sigma_{13} }}{{S_{13} }}} \right)^{2} + \left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2} \ge 1 $$

(2.6)

Equation 2.6 is identical to Eq. 2.2, the criterion proposed by Christensen and DeTeresa [10]. Huang and Lee [25], Liu and Wang [34] used Yeh’s criterion (Eqs. 2.5, 2.6).

Zhao and Cho [18] used only the tensile part of the criterion (Eq. 2.5). Chen [35] included the effect of the in-plane transverse stress σ₂₂ and predict a damage propagation between the layer (n) and (n + 1) by the relation

$$ \left( {\frac{{{}^{n}\sigma_{33} }}{{Z_{C} }}} \right)^{2} \,+\, D_{1} \left( {\frac{{{}^{n}\sigma_{23}^{2} + {}^{n + 1}\sigma_{13}^{2} }}{{S^{2} }}} \right) + D_{2} \left( {\frac{{{}^{n + 1}\sigma_{22} }}{Y}} \right)^{2} = e_{D} $$

(2.7)

where Y represents the in-plane transverse tensile (σ₂₂ ≥ 0) or compressive (σ₂₂ < 0) strength of laminates. D₁, D₂ are experimental constants and they are only related to the material properties of laminates. Damage occurs when $e_{D} \ge 1$.

Hou et al. [13, 14] assumed that delamination occurs when

$$\left( {\frac{{\sigma_{33} }}{{Z_{T} }}} \right)^{2} + \frac{{\sigma_{23}^{2} + \sigma_{13}^{2} }}{{S_{13}^{2} \left( {d_{ms} d_{fs} + \delta } \right)}} \ge 1\quad {\text{when}}\quad \sigma_{33} \ge 0 $$

(2.8)

or

$$ \frac{{\sigma_{23}^{2} + \sigma_{13}^{2} - 8\sigma_{33}^{2} }}{{S_{13}^{2} \left( {d_{ms} d_{fs} + \delta } \right)}} \ge 1\quad {\text{when}}\quad - \sqrt {\left( {\sigma_{23}^{2} + \sigma_{13}^{2} } \right)/8} \le \sigma_{33} < 0 $$

(2.9)

They also assumed that no delamination occurs when

$$ \sigma_{33} < - \sqrt {\left( {\sigma_{23}^{2} + \sigma_{13}^{2} } \right)/8} $$

(2.10)

In Eqs. (2.8, 2.9), d_ms is a damage coefficient of matrix cracking and d_fs is a damage coefficient of fibre failure and δ is the ratio between interlaminar stresses before and after matrix or fiber failure.

Zou et al. [13, 14] proposed a single criterion that accounts for different strength for tension and compression in the transverse direction and for the effect of the two transverse shear stresses

$$ \frac{{\sigma_{33}^{2} }}{{Z_{t} Z_{c} }} + \frac{{\sigma_{13}^{2} + \sigma_{23}^{2} }}{{S^{2} }} + \left( {\frac{1}{{Z_{t} }} - \frac{1}{{Z_{c} }}} \right)\sigma_{33} \ge 1 $$

(2.11)

Fenske and Vizzini [36] extended the Brewer-Lagace criterion by including the effects of inplane stresses.

These various criteria attempt to predict the onset of delamination at the interface between two adjacent plies in terms of the stresses acting at that interface. It is important to remember that the behavior is very different depending on the sign of the transverse normal stress.

3 Fracture mechanics approach

Once interface failure is predicted, the stress distribution in the surrounding material becomes singular and the presence of the crack should be accounted for in the analysis. This section presents a brief historical overview of the development of fracture mechanics followed by a discussion of its application to the propagation of delaminations at bimaterial interfaces, and an introduction to the Virtual Crack Closure Technique (VCCT) used for numerical simulations.

3.1 Historical background

Initially, the first studies concerning fracture mechanics, which are recalled in this part, were focused on crack propagation in isotropic metallic materials. The later developpements of fracture mechanics theories for orthotropic materials are based on these theories.

Leonardo da Vinci (1452–1519) conducted tests to determine the strength of iron wires and found an inverse relationship between the strength and the length of the wire [37]. This early evidence of a size effect was confirmed by several other experiments on iron bars and wires and glass fibers. A possible explanation for this size effect is the presence of flaws in the material that would significantly reduce the strength.

In 1898, Kirsch (in [38]) studied the stress concentrations created by a circular hole in a plate subjected to uniaxial tension and, in 1913, Inglis [39] extended that work to determine the stress field around a plate containing an elliptical rather than circular hole (Fig. 1). It was shown that, at the end of the major axis of the ellipse, the normal stress in the tangential direction is given by

$$ \sigma_{y} \, = \sigma \,\,\left( {1 + 2\sqrt {a/\rho } } \right) $$

(3.1)

where $\sigma \,$ is the remote stress, a is the length of the major semi-axis which is perpendicular to the loading direction, and $\rho$ is the radius of curvature at the end of the major axis of the ellipse. As becomes much smaller than a, $\sigma_{y}$ becomes infinite. This result would imply that for a body with a sharp crack the strength would be near zero. For a circular hole, $a = \rho$ and Eq. (3.1) gives the well-known result $\sigma_{y} \, = 3\sigma \,$ in that case.

Fig. 1

Elliptic flaw in an infinite plate subjected to a uniaxial stress

The work of Griffith [40] was motivated by the fact that the bulk strength of glass (172 MPa) was much smaller than the strength of a thin glass tube (2372 MPa) or that of glass fibers (1500–6200 MPa) [37]. Griffith introduced an artificial flaw in specimens (Fig. 2) and experimental results indicated that, at fracture, the product $\sigma \,\sqrt a$ remained nearly constant.

Fig. 2

An edge crack flaw in a material

Westergaard [41] showed that the stress near a crack tip for infinite plate and through thickness cracks is approximately equal to

$$ \begin{aligned} \sigma_{x} &= \frac{{K_{I} }}{{\sqrt {2\pi r} }} \cdot \cos \frac{\theta }{2}\left( {1 - \sin \frac{\theta }{2}\sin \frac{3\theta }{2}} \right) \hfill \\ \sigma_{y}& = \frac{{K_{I} }}{{\sqrt {2\pi r} }} \cdot \cos \frac{\theta }{2}\left( {1 + \sin \frac{\theta }{2}\sin \frac{3\theta }{2}} \right) \hfill \\ \sigma_{xy} &= \frac{{K_{I} }}{{\sqrt {2\pi r} }} \cdot \cos \frac{\theta }{2}\sin \frac{\theta }{2}\cos \frac{3\theta }{2} \hfill \\ \end{aligned} $$

(3.2)

where $K_{I} = \sigma \sqrt {\pi a}$ is the stress intensity factor for mode I fracture, r is the distance from the crack tip and $\theta$ is the orientation angle (Fig. 3). Near the crack tip, the stress is maximum for $\theta = 0$ and

$$ \upsigma_{{\rm y}} |_{\theta = 0} = \upsigma {\sqrt{\frac{{\rm a}}{2{\rm r}}}} = \frac{\upsigma {\sqrt {\pi {\rm a}}}}{{\sqrt {2\pi {\rm r}}}} = \frac{{\rm K}_{{\rm I}}}{{\sqrt {2\pi {\rm r}}}} $$

(3.3)

The stress is singular at the crack tip and the singularity is of order r^−1/2.

Fig. 3

Cylindrical coordinates at crack tip

Irwin [42] studied this problem for an infinite plate and plane stress conditions. He showed that for isotropic ductile materials a plastic zone is created near the crack tip and that the total energy release rate G is the sum of the surface energy release rate $2\gamma$ and a plastic energy release rate G_p. For brittle materials such as glass the surface energy release rate dominates and $G \approx 2\gamma = 2\,{\text{J}}/{\text{m}}^{2}$. For ductile materials such as steel G_p dominates and $G \approx 1000 {\text{J}}/{\text{m}}^{2}$.

$$ \sigma_{f} \sqrt {\pi a_{c} } = \sqrt {GE} $$

(3.4)

G is the strain energy release rate and E is the Young modulus.

So far, we discussed the effect of a Mode I crack—Opening mode with a tensile stress normal to the plane of the crack. Figure 4 shows two other fracture modes:

Fig. 4

The three fracture modes: a Mode I; b Mode II; c Mode III

• Mode II crack—Sliding mode (a shear stress acting parallel to the plane of the crack and perpendicular to the crack front)

• Mode III crack—Tearing mode (a shear stress acting parallel to the plane of the crack and parallel to the crack front)

Three stress intensity factors can be defined as

$$ \left\{ {\begin{array}{*{20}c} {K_{I} } \\ {K_{II} } \\ {K_{III} } \\ \end{array} } \right\} = \mathop {\lim }\limits_{r \to 0} \sqrt {\pi r} \left\{ {\begin{array}{*{20}c} {\sigma_{22} } \\ {\sigma_{12} } \\ {\sigma_{23} } \\ \end{array} } \right\} $$

(3.5)

Failure criteria described in Sect. 2 can be used to predict the onset of delamination. Once initiated, the propagation of a delamination is a dynamic fracture event. A Griffith type criteria can be used to determine whether an existing delamination extends or not based on the value of the total strain energy release rate: the delamination does not extend if $G \le G_{c}$ and it extends if $G > G_{c}$. This approach requires a detailed stress analysis near the crack tip and the ability to calculate the change in strain energy as the delamination increases by a small amount. However, it does not require knowledge of the individual strain energy release rates G_I, G_II, G_IlI.

In general it is assumed that a mixed-mode failure criterion can be developed in the following form

$$ f\left( {\frac{{G_{I} }}{{G_{IC} }},\,\,\frac{{G_{II} }}{{G_{IIC} }},\,\,\frac{{G_{III} }}{{G_{IIIC} }}} \right) \ge 1 $$

(3.6)

In developing such a criterion, the two challenges are: determining the form of the function f and conducting tests to identify the critical energy release rates and other parameters that may be present in specific failure criteria. The developments of fracture mechanics for composite materials are based on the principles defined for isotropic materials. Examples of mixed-mode failure criteria are the power law criterion

$$ \left( {\frac{{G_{I} }}{{G_{IC} }}} \right)^{\alpha } + \left( {\frac{{G_{II} }}{{G_{IIC} }}} \right)^{\alpha } = 1 $$

(3.7)

first introduced by Wu and Reuter [43] and the Benzeggagh and Kenane criterion [44]

$$ G_{TC} = G_{IC} + \left( {G_{IIC} - G_{IC} } \right)\left( {\frac{{G_{II} }}{{G_{T} }}} \right)^{\alpha } $$

(3.8)

where $G_{T} = G_{I} + G_{II}$ is the total energy release rate, $G_{TC}$ is the total critical energy and $\alpha$ is a parameter. Mixed-mode fracture criteria are needed in the formulation of cohesive elements and the two mentioned here (Eqs. 3.7, 3.8) are the most commonly used.

3.2 Fracture mechanics approach to delamination propagation

Fracture mechanics can be used to study the propagation of impact induced delaminations at ply interfaces. The objective is to predict the extent of delaminations at each interface. Individual plies may have cracks because of matrix failure due to tensile bending stresses, transverse shear stresses, or residual thermal stresses [45]. The second objective is to determine whether these matrix cracks will lead to interface debonding.

Interfacial cracks between two isotropic materials with different elastic properties have been studied extensively, For cracks at the interface of two isotropic bodies with different elastic properties, single mode loading produces both opening and shearing modes (Williams [46]). For a bimaterial interfacial crack, the near-tip normal and shear stresses are given by

$$ \sigma_{yy} + i\sigma_{xy} = \frac{{\left( {K_{I} + iK_{II} } \right)r^{i\varepsilon } }}{{\sqrt {2\pi r} }} $$

(3.9)

where $i = \sqrt { - 1} , \, K_{I} \quad {\text{and}}\quad K_{II}$ are components of the complex stress intensity factor K = K_I + iK_II and $\varepsilon$ is the oscillation index

$$ \varepsilon = \frac{1}{2\pi }\ln \left[ {\frac{1 - \beta }{1 + \beta }} \right] $$

(3.10)

where $\beta$ is one of two Dundurs constants. For plane strain, the Dundurs constants are

$$ \alpha = \frac{{\bar{E}_{1} - \bar{E}_{2} }}{{\bar{E}_{1} + \bar{E}_{2} }}\quad \beta = \frac{{G_{1} \left( {1 - 2\nu_{2} } \right) - G_{2} \left( {1 - 2\nu_{1} } \right)}}{{2\left[ {G_{1} \left( {1 - 2\nu_{2} } \right) + G_{2} \left( {1 - 2\nu_{1} } \right)} \right]}} $$

(3.11)

where $\bar{E} = E/\left( {1 - \nu^{2} } \right)$. E is the elastic modulus, G is the shear modulus, $\nu$ is Poisson’s ratio and the subscripts 1 and 2 refer to the materials above and below the interface. Stresses oscillate as the distance from the crack tip increase, as indicated by the $r^{i\varepsilon }$ term in Eq. 3.9, but this oscillation is limited to a very small region near the crack tip [47]. The individual rates G_I and G_II oscillate very close to the crack tip but Mulville and Mast [48] showed that the total strain energy release rate remains almost constant as the crack propagates. When the materials above and below the interface are identical, $\varepsilon = 0,\;\alpha = 0,\;\beta = 0$. Then, stresses do not oscillate as expected for a crack in a homogeneous material.

A crack impinging on an interface joining two dissimilar materials may arrest or may advance by either penetrating the interface or deflecting into the interface (Fig. 5). He and Hutchinson [49] showed that, in order for a crack impinging the interface at any angle to be deflected, the toughness of the interface must be less than one quarter of the toughness of the material on the other side of the interface when $- 0.5 < \alpha < 0.25$. It was shown that residual thermal stresses have a significant effect on this problem [50].

Fig. 5

Several scenarios for a crack impinging on a bi-material interface

He and Hutchinson [51] studied the kinking of a crack out of an interface (Fig. 6). Assuming that the condition for propagation in the interface is G_o = G_oc and that for propagation in material 2, G_II = G_IIc. If G_IIc is sufficiently large compared to G_oc the crack does not kink into material 2. If these two critical energy release rates are comparable there is a loading range such that the crack stays in the interface and above that the crack will kink into material 2. Ryoji [52] gave a stress-based criterion for predicting the kink angle. Rudas [53] gave a simple formula for predicting the kink angle based on the mode mixity ratio K_I/K_II. Carlsson and Prasad [54–56] used the results obtained in [51] to study interfacial cracks in sandwich structures and the kinking of these cracks into the core.

Fig. 6

Geometry for crack kinking of a bi-material interface [51]

Experiments using a modified DCB (Double Cantilever Beam) specimen to produce high speed Mode I crack propagation showed that the Mode I dynamic fracture toughness of unidirectional AS4/3501-6 is equal to the static fracture toughness for crack speeds up to 200 m/s [57]. Dynamic delamination experiments on modified ENF specimens of unidirectional AS4/3501-6 graphite/epoxy composite under three point bending showed that the dynamic fracture toughness is not affected significantly by crack speed up to 1100 m/s [58, 59]. Dynamic initiation fracture toughness of S2/8552 and IM7/977-3 composites were obtained using a wedge insertion fracture (WIF) test method indicated that both the dynamic initiation fracture toughness and the fracture toughness during dynamic delamination cracking at speeds up to 1000 m/s remain approximately equal the static fracture toughness [60]. Studies of the effects of loading rates on fracture toughness for composites include Refs [61, 62] for mode I, [48, 63–66] for mode II, and [67] for mode III.

Analytical studies of dynamic interfacial crack propagation were presented by Gol’dshtein [68], Brock and Achenbach [69], Atkinson [70], Willis [71], Deng [72]. Willis [71] presented a two-dimensional analysis of the stress field around a crack on the plane interface between two bonded dissimilar anisotropic elastic half-spaces. Yang et al. [73] examined the singular fields around a crack running dynamically along the interface between two anisotropic substrates.

Experimental studies of dynamic interface crack propagation started with the work of Tippur and Rosakis [74] in 1991. In a homogeneous solid, the crack tip velocity cannot exceed the Rayleigh wave speed of the material [75]. A series of experiments [76–83] showed that, at the interface of bimaterial plates, crack tip velocities between steel and PMMA (polymethylmethacrylate) can reach the intersonic regime. That is, the range between the shear wave velocity and the longitudinal wave velocity of PMMA. Large scale contact behind the crack tip is observed when $c_{s} < v < \sqrt 2 c_{s}$. An analysis of these experiments confirms those experimental findings [84].

A study of dynamic crack growth along the interface of a fiber-reinforced polymer composite–homalite bimaterial subjected to impact shear loading [85] led to the first conclusive experimental evidence of interfacial crack speeds faster than any characteristic elastic wave speed of the more compliant material and the first experimental observation of a mother–daughter crack mechanism allowing a subsonic crack to evolve into an intersonic crack.

Lambros and Rosakis [86] applied the same experimental techniques to laminated composite plates. In [87] laminated composite plates where subjected to transverse impacts and delamination speeds measured ranged from 500–1800 m/s. Accurate optical measurements of the transverse displacements were used to estimate the size of delaminations and the speed at which they grew.

Elder et al. [88] provide a quick overview of currently available methods for predicting delaminations induced by low velocity impacts. Davies et al. [89–94] presented a very simple approach to predict the delamination threshold load (DTL) using a simply supported circular plate and a clamped circular plate with an existing delamination of radius a located on the midplane (Fig. 7). For plates without a delamination, bending stresses are zero and transverse shear stresses reach a maximum on the midplane. One can then expect that delaminations might occur on the midplane under mode II fracture.

Fig. 7

Simplified model for predicting the delamination damage threshold load [375]

The deflection of the central portion an undamaged clamped plate of thickness 2 h under the force P is

$$ \delta_{u} = \frac{{3Pa^{2} \left( {1 - \nu^{2} } \right)}}{{4\pi E\left( {2h} \right)^{3} }} $$

(3.12)

The central portion of the damaged plate can be considered as two plates each having a thickness h and subjected to a force P/2. The deflection of the central portion of the damage plate is then

$$ \delta_{d} = 3\frac{P}{2}\frac{{a^{2} \left( {1 - \nu^{2} } \right)}}{{4\pi Eh^{3} }} $$

(3.13)

The energy available to drive the crack is

$$ U = \frac{1}{2}P\left( {\delta_{d} - \delta_{u} } \right) = \frac{{9P^{2} a^{2} \left( {1 - \nu^{2} } \right)}}{{64\pi Eh^{3} }} $$

(3.14)

As the crack advances from a to $a + \delta a,\;G.2\pi a\delta a = \frac{\partial U}{\partial a}\delta a$. The DTL is then given by

$$ P_{c}^{2} = \frac{{64\pi^{2} Eh^{3} }}{{9\left( {1 - \nu^{2} } \right)}}G_{IIc} $$

(3.15)

which indicates that this load is independent of the initial radius of the delamination, that it depends of the Mode II critical energy release rate, and that it varies with h^3/2. Schoeppner and Abrate [95] showed that the DTLs predicted by this equation are in good agreement with experimental results for a large number of tests. Equation 3.15 is also used by Olsson [96–99] and others (e.g. [100]) to predict the DTL.

3.3 Virtual crack closure technique

The finite element method is used extensively in linear elastic fracture mechanics and a first review of the literature is presented by Banks-Sills [101, 102]. The virtual crack closure technique (VCCT) proposed by Rybicki and Kanninen [103] is widely used and several reviews have been presented by Krueger [104]. The principle of the two-step VCCT [104] can be described as follows:

• 1. A finite element model is used to model a solid with pairs of coincident nodes coupled together ahead of the crack (Fig. 8a).

  Fig. 8

  

  Two step VCCT. a First step—crack closed. b Second step—crack extended

• 2. When the load reaches a critical value, the coupled nodes at the crack tip are released and the crack extends one element length $\Delta a$(Fig. 8b).

• 3. The energy released during the crack extension is assumed to be equal to the energy required to close the crack extension back.

$$ \Delta E = \frac{1}{2}\left[ {X_{1L} \cdot \Delta u_{2L} + Z_{1L} \cdot \Delta w_{2L} } \right] $$

(3.16)

where $\Delta u_{2L}$ and $\Delta w_{2L}$ are the differences in shear and opening displacements at node L in the second finite element model and X_1L and Z_1L are the forces applied at node L in the first model to close the crack extension.

The modified VCCT or one-step VCCT is illustrated in Fig. 9. Considering the extension of the crack from node i to node k, it is assumed that the displacements at node i after the extension will be the same as those at node L in the current configuration. The forces shown at node i are those needed to prevent the extension from i to k. The energy released is

$$ \Delta E = \frac{1}{2}\left[ {X_{i} .\Delta u_{L} + Z_{i} .\Delta w_{L} } \right] $$

(3.17)

Using four-noded elements as in Fig. 9, the mode I and mode II energy release rates are given by

$$ G_{I} = \frac{1}{2\Delta a}Z_{i} .\Delta w_{L},\quad G_{II} = \frac{1}{2\Delta a}X_{i} .\Delta u_{L} $$

(3.18)

Bonhomme et al. in [105], determine the energy release rate under mode I in a AS4/8552 carbon/epoxy laminates by the one step and the two step VCCT method. Results were compared with empirical data obtained from double cantilever beam (DCB) tests. Both one and and Two-step VCCT methods converge as element length decreases.

Fig. 9

Modified VCCT or one-step VCCT

Extensions of this basic approach to 2D problems modeled usind 8-noded elements, to 3D problems using solid elements and problems modeled using plate or shell elements are discussed in Krueger [104]. The additional dimension allowed to calculate the distribution of the energy release rates along the delamination front and makes it possible to obtain G_III.

The use of elements with quadratic shape functions gives a kinematically incompatible interpenetration which caused initial problems. Element-wise opening (where edge and midside nodes are released) corrects this and yields reliable results.

Special attention should be given to the calculation of the strain energy release rate components when the element lengths (Δa) for the element in front of the crack tip and behind are not identical. The following relationships, which only depend on the length of the elements in front and behind the crack tip, account for this effect.

$$ G_{I} = \frac{1}{{2\Delta a_{1} }}Z_{i} \cdot \Delta w_{L} \cdot \left( {\frac{{\Delta a_{2} }}{{\Delta a_{1} }}} \right)\quad G_{II} = \frac{1}{{2\Delta a_{1} }}X_{i} \cdot \Delta u_{L} \cdot \left( {\frac{{\Delta a_{2} }}{{\Delta a_{1} }}} \right) $$

(3.19)

where Δa₁ and Δa₂ are the element length in front and behind the crack tip. A similar correction can be used when the elements along the delamination front have different widths.

The modified VCCT is used in [101, 102] to characterize respectively the energy release rate in mode I, mode II, mode III, mixed mode I + II of glass/epoxy laminates and the energy release rate in mode I of carbon fiber reinforced composites. Improvements for tracking delamination fronts made by Pietropaoli and Riccio [105–108] consist in adapting the load step size to the size of the mesh by an iterative calculation until the convergence is reached, which corresponds to the superposition of the crack front with mesh nodes. To allow more complex shapes of the delamination front to be taken into account, this improvements is associated with the use of weight factors expressed in terms of fraction of debonded area in order to take into account the node position in the delaminated lobe.

The VCCT method is used by Yoshimura [109] to calculate the energy release rate of the delamination crack in a stitched carbon fiber reinforced composite laminate. In order to consider the bridging effect, a linear spring element is used to connect the neighboring layer. Simulation results revealed that the improvement in impact damage resistance due to stitching became greater as the delamination area grew larger.

4 Cohesive element approach

The virtual crack closure technique used to estimate the energy release rates has several drawbacks: (1) it requires a very fine mesh near the interfacial crack tip; (2) an existing crack with given shape and size is needed; (3) it requires complex moving mesh techniques to advance the crack front. A different approach has been developed recently and is used extensively for studying delaminations in composite materials and failure of bonded joints. This approach is based on the cohesive zone concept proposed by Barenblatt [110, 111] and Dugdale [112]. It is assumed that ahead of the crack tip, there is a very thin layer separating two solids in which the damage mechanisms leading to fracture are localized. The behavior of this fracture process zone is characterized by a traction–separation law called the cohesive law. Shet and Chandra [113] listed eleven popular CZM with different cohesive laws. Cohesive laws are classified into two categories: (a) intrinsic cohesive laws that have an initial elastic slope; and (b) extrinsic cohesive laws that are initially rigid [114].

For finite element analyses, several types of elements based on this CZM were developed to model the behavior of the interface. These elements called “cohesive elements” or “decohesion elements” can be: (1) point decohesion elements or discrete cohesive elements [115–118] which are essentially three dimensional nonlinear springs; (2) continuous decohesion elements connecting line elements, 2D or 3D solid elements, or plate or shell elements. This strategy allows the modelling of complex shapes by the use of refined meshes.

Considering the interface between two elastic layers, Fig. 10 shows the lower layer, the interface, the delamination front, and the local coordinate system to be used in the following. The x₁-x₂ plane is the plane of the interface, x₃ is normal to the interface, and the delamination extends in the x₁ direction. In that coordinate system, stresses acting on the interface have two shear components $\tau_{1}$ and $\tau_{2}$, and one normal component $\tau_{3}$. At a given point on the interface, two nodes are initially in contact: one is on the lower layer and the other is on the upper layer. The displacements of the upper and lower layers are denoted by $u_{i}^{U}$ and $u_{i}^{L}$ where i = 1, 2, 3. The relative displacements defined by $\delta_{i} = u_{i}^{U} - u_{i}^{L}$ consist of two sliding displacements (δ₁ and δ₂) and one opening displacement $\delta_{3}$.

Fig. 10

Coordinate system at a delaminating interface

The behavior of the interface is defined by a cohesive law that relates the relationship between the interfacial stresses and the relative displacements. Many cohesive laws have been formulated [119]. Indeed, CZMs have been used to simulate the fracture process in a number of material systems and under various loading conditions. Here, the cohesive laws most used in the frameworks of impacts on composite structures are presented. The use of cohesive elements for the modelling of interlaminar and intralaminar cracks in composite laminates under impact is treated in part 5.

4.1 Bilinear cohesive law

The most commonly used cohesive law used in the development of cohesive elements is the bilinear law illustrated in Fig. 11 for pure mode I fracture. The normal stress $\tau_{3}$ increases linearly with the relative displacement $\delta_{3}$ until the onset of the decohesion process when these two quantities reach the critical values N and softening is observed as the relative displacement increases beyond $\delta_{3}^{o}$ as shown in the figure. Complete failure of the interface occurs when $\tau_{3} = 0$ and the displacement reaches its final value. The area under the curve is the energy dissipated in the process and must be equal to the critical energy release rate G_Ic in this case.

Fig. 11

Bilinear cohesive law

The behavior of the cohesive zone is assumed to be similar for Mode II and pure Mode III loading. Therefore, the behavior for the initial phase can be written as

$$ \tau_{i} = k_{i} \delta_{i} $$

(4.1)

where k_i is the initial stiffness of the interface. If the limits for the tractions in the tangential directions and the normal direction are T, S, and N, then

$$ \delta_{1}^{o} = T/k_{1} ,\quad\delta_{2}^{o} = S/k_{2},\quad\delta_{3}^{o} = N/k_{3} $$

(4.2)

The area under the curves must be equal to the corresponding critical strain energy release rate. That is,

$$ G_{Ic} = \int_{o}^{{\delta_{3}^{f} }} {\tau_{3} d\delta_{3} } ,\quad G_{IIc} = \int_{o}^{{\delta_{1}^{f} }} {\tau_{1} d\delta_{1} } ,\quad G_{IIIc} = \int_{o}^{{\delta_{2}^{f} }} {\tau_{2} d\delta_{2} } $$

(4.3)

Therefore, for this bilinear cohesive law, the relative displacements at failure are

$$ \delta_{1}^{f} = 2G_{IIc} /T,\quad\delta_{2}^{f} = 2G_{IIIc} /S,\quad\delta_{3}^{f} = 2G_{Ic} /N $$

(4.4)

In the case of mixed mode loading, the relative displacements are $\delta_{1m} ,\,\delta_{2m} ,\delta_{3m}$ where the subscript m is added to distinguish quantities for mixed mode loading from the same quantities in the case of pure mode loading. Introducing the normalized relative displacements $\lambda_{i} = \delta_{im} \,/\,\delta_{i}^{f}$, the total effective relative displacement is defined as

$$ \lambda = \sqrt {\lambda_{1}^{2} + \lambda_{2}^{2} + \lambda_{3}^{2} } $$

(4.5)

Under mixed mode loading the decohesion process starts when $\lambda$ reaches a value

$$ \lambda^{o} = \sqrt {\left( {\lambda_{1}^{o} } \right)^{2} + \left( {\lambda_{2}^{o} } \right)^{2} + \left( {\lambda_{3}^{o} } \right)^{2} } = \sqrt {\left( {\frac{{\delta_{1m}^{o} }}{{\delta_{1}^{f} }}} \right)^{2} + \left( {\frac{{\delta_{2m}^{o} }}{{\delta_{2}^{f} }}} \right)^{2} + \left( {\frac{{\delta_{3m}^{o} }}{{\delta_{3}^{f} }}} \right)^{2} } $$

(4.6)

Complete failure occurs when

$$ \lambda^{f} = \sqrt {\left( {\lambda_{1}^{f} } \right)^{2} + \left( {\lambda_{2}^{f} } \right)^{2} + \left( {\lambda_{3}^{f} } \right)^{2} } = \sqrt {\left( {\frac{{\delta_{1m}^{f} }}{{\delta_{1}^{f} }}} \right)^{2} + \left( {\frac{{\delta_{2m}^{f} }}{{\delta_{2}^{f} }}} \right)^{2} + \left( {\frac{{\delta_{3m}^{f} }}{{\delta_{3}^{f} }}} \right)^{2} } $$

(4.7)

A stress based failure criterion is used to determine $\lambda^{o}$ and a mixed mode fracture criterion is used to determine $\lambda^{f}$.

During the unloading phase $\left( {\lambda^{o} \le \lambda \le \lambda^{f} } \right)$ the tractions reduce linearly according to

$$ \tau_{i} = k_{i} \frac{{\lambda^{o} }}{\lambda }\frac{{\lambda^{f} - \lambda }}{{\lambda^{f} - \lambda^{o} }}\delta_{im} $$

(4.8)

The present description follows that of Bui et al. [120] and is very similar to that used by many authors. Because for pure mode loading the maximum tractions are the strengths of the interface and the area under the cohesive law is equal to the fracture toughness, this approach predicts the onset of delamination using a stress-based criterion and the propagation of delaminations using an energy criterion based on fracture mechanics.

To be more specific, following Pinho et al. [121], we define the equivalent relative displacement in shear and the shear traction as

$$ \lambda_{\text{shear}} = \sqrt {\lambda_{1}^{2} + \lambda_{2}^{2} } ,\quad t_{\text{shear}} = \sqrt {t_{1}^{2} + t_{2}^{2} } $$

(4.9)

The effective relative displacement can be written as

$$ \lambda = \sqrt {\left( {\lambda_{shear} } \right)^{2} + \left\langle {\lambda_{3} } \right\rangle^{2} } $$

(4.10)

Since compressive normal stresses do not induce delaminations, the second term under the radical is written using the McCauley bracket so that $\left\langle {\lambda_{3} } \right\rangle = \hbox{max} \left\{ {0,\;\lambda_{3} } \right\}$. An equivalent Yeh quadratic delamination criterion is used for the prediction of delamination onset. It’s expressed in terms of relative displacements. A mixed-mode propagation criterion (power law type) establishes the state of complete decohesion for different ratios of applied mode I and shear mode energy release rates.

May and Hallet [122], like many others, used the quadratic stress criterion (2.5) for predicting the onset of delamination and the linear multi-mode fracture criterion (Eq. 3.7 with $\alpha = 1$) for delamination propagation. Camanho et al. [123, 124] same penalty stiffness in Modes I, II and III. Many other choices can be made. For example, references [123–128] used the Benzeggagh and Kenane criterion (Eq. 3.8) for delamination propagation and Jiang et al. [129] used the power criterion (Eq. 3.7) for predicting the onset of delaminations.

For very high initial stiffnesses the bilinear model in Fig. 11 becomes the linear cohesive law used by many authors (e.g. [130]) that consist only of the decreasing portion of the curve.

The major advantage of bilinear cohesive laws is that it is a very simple traction/separation law that provides results good enough to model with accuracy matrix cracking and delamination. Computational implementation is straightforward and the parameters can be easily identified from simple tests (DCB, ENF…). Nevertheless, this cohesive law represents brittle materials and can not account for dissipative phenomena like pseudo-plasticity or friction. More, the linear softening of the traction load can lead to computational instability issues when the decreasing slope is too high [131].

4.2 Other cohesive laws

Even though bilinear cohesive laws are widely used for the modelling of matrix cracks and delamination during impact loading on composite laminates, several other cohesive laws are frequently used. These laws, initially developed for metallic or ceramic materials, can also be used for composite materials. This subsection describes several commonly used cohesive laws.

4.2.1 Exponential cohesive laws

Several other cohesive laws are also used by various researchers. One of them is the exponential law

$$ \frac{{\tau_{i} }}{{\tau_{ic} }} = \frac{{\delta_{i} }}{{\delta_{ic} }}\exp \left[ {\frac{{1 - \left( {\delta_{i} /\delta_{ic} } \right)^{\beta } }}{\beta }} \right] $$

(4.11)

used by Goyal et al. [132] and illustrated in Fig. 12. When $\beta$ = 1, this expression recovers the Smith-Ferrante universal binding law used by several investigators [133–138]. Ortiz and Pandolfi [139] defined the surface tractions $\bar{\tau }$ in terms of a single potential function $\phi$

$$ \bar{\tau } = \frac{\partial \phi }{{\partial \bar{\delta }}} $$

(4.12)

The Smith-Ferrante cohesive law is obtained using the function

$$ \phi = e\tau_{c} \delta_{c} \left[ {1 - \left( {1 + \frac{\delta }{{\delta_{c} }}} \right)e^{{ - \delta /\delta_{c} }} } \right] $$

(4.13)

Other types of exponential cohesive laws are used. Alfano [140] used different exponential functions for the increasing and decreasing portions of the curve. Corigliano [141, 142] used the cohesive law proposed by Rose et al. [143, 144] and used by Xu and Needleman [145, 146] and Camacho and Ortiz [147] (Fig. 13)

$$ \tau_{3} = K\delta_{3} \exp \left[ { - \beta \frac{{\delta_{3} }}{{\delta_{3}^{o} }}} \right] $$

(4.14)

and introduced rate effects by assuming that the parameter $\beta$ depends on $\dot{\delta }_{3}$, the velocity of the opening displacement. Other exponential functions were used by Song and Waas [148, 149].

Fig. 12

Exponential cohesive law (Eq. 4.11)

Fig. 13

Generalized cohesive law (Eq. 4.14)

Alfano [135] showed that exponential cohesive laws are optimal in terms of finite element approximation. However, the computational cost is higher than bilinear laws and the parameters are identified less easily.

4.2.2 Polynomial cohesive laws

Zerbst et al. [150] discuss the use of different cohesive laws including polynomial laws like Needleman’s law

$$ \frac{{\tau_{i} }}{{\tau_{ic} }} = \frac{27}{4}\frac{{\delta_{i} }}{{\delta_{ic} }}\left( {1 - \frac{{\delta_{i} }}{{\delta_{ic} }}} \right)^{2} $$

(4.15)

shown schematically in Fig. 14. Cubic polynomial laws such as Eq. 4.15 are also used in [125, 151], and Blackman et al. [152] who obtained analytical and finite element results for the DCB (Double Cantilever Beam) test, the tapered DCB test and the 90^o peel test. These laws present a similar shape for the traction/separation behavior than for the exponential cohesive laws.

Fig. 14

Cubic polynomial cohesive law (Eq. 4.15)

4.2.3 Linear exponential cohesive laws

The linear-exponential cohesive law (Fig. 15) used by Bouvet [153] and others [154] consists of an initial linear part given by $\tau_{i} = k_{i} \,\delta_{i}$ followed by a softening curve given by

$$ \frac{{\tau_{i} }}{{\tau_{ic} }} = \frac{{\delta_{i} }}{{\delta_{ic} }}\exp \left[ { - \beta \left( {\delta_{i} - \delta_{ic} } \right)} \right] $$

(4.16)

The two curves meet when the relative displacements reaches the critical value $\delta_{ic}$. Other linear-exponential cohesive laws are used by Liu et al. [155]. This law is a combination of the bilinear cohesive law and the exponential cohesive law. The main advantage of this law is that instability issues due to the quick decreasing of the traction load in the bilinear law [131] are avoided. More, as for the bilinear law, the initial stiffness can easilybe identified.

Fig. 15

Linear-exponential [153, 154]

4.2.4 Trapezoidal cohesive law

The trapezoidal cohesive law (Fig. 16) proposed by Tvergaard and Hutchinson [156, 157] for elastic–plastic problems is used by many investigators (i.e. [140, 158–165]. Multilinear cohesive laws (Fig. 17) are also employed [159, 162, 166, 167]. The effect of the shape of the cohesive load was found to be problem-dependent [140]. In some cases it makes very little difference on the accuracy of the solution while on some examples differences up to 15 % are reported [140]. Another concern is that the shape of the cohesive law affects the stability of the solution. In that respect, the trapezoidal cohesive model performs worse than the exponential or bilinear laws. On the other hand there are examples such as those treated by Pinto [166] in which, because of the highly nonlinear behavior of the adhesive, a multilinear model performs best when modeling a bonded joint.

Fig. 16

Trapezoidal cohesive law

Fig. 17

Multilinear cohesive laws [162]

4.2.5 Loading rate, moisture and other complicating effects

In addition to the Corigliano’s method discussed above [141, 142], rate dependent cohesive laws have been proposed in [168–175] and the effect of friction between delaminated interfaces was introduced in cohesive laws Alfano and Sacco [176], Yang and Cox [177]. The mechanical behavior of polymer matrix materials and adhesives is usually affected by moisture absorption. Moisture dependent CZM properties were determined by several investigators and Refs. [178–182] showed that the cohesive strength and the cohesive fracture energy decrease with moisture absorption.

When delamination cracks grow in laminated composites, fibers may bridge the two laminas behind the crack tip. Similarly, with z-fiber pinning the crack is intentionally bridged by the transverse fiber reinforcement. Then, there is both a bridging zone and a cohesive zone. Delaminations have been studied using CZM in several publications (i.e. [183–187]). Dantuluri [188] modeled z-pinned laminates with bilinear cohesive law for interface and also a nonlinear spring with a bilinear force–displacement law for the z-pins. With this model the z-pin are effective only in the normal direction. Cui et al. [189] use interface elements for laminates with z-pins that provide Mode I–Mode II bridging. For predicting failure of the z-pins themselves, cohesive laws for mode I and mode II are used separately.

4.3 Interface elements formulation

The development of cohesive finite elements can follow classical lines like for an eight node solid finite element with the top surface connected to elements representing the layer above the interface and the bottom surface connected to elements modeling the layer below the interface. Instead of finite thickness elements, zero thickness elements can also be developed. In both cases, interpolation functions are used to determine various quantities from their values at the nodes. In other words, in continuous interface elements nodal forces depend on all nodal displacements.

The following outlines some of the early developments of finite elements with CZM. Hillerborg [190] is considered to be the first to use cohesive crack tip model in finite element analysis as stated by Geißler and Kaliske [171]. Some of the first applications of cohesive elements for the analysis of composite material include Cui and Wisnom [117], Allix, Ladeveze and Corigliano [191], and Schellekens and de Borst [192].

Mi et al. [193] developed zero-thickness interface elements to be used in a 2D finite element analysis between 8-noded elements. A bilinear cohesive law and a linear fracture criterion were used. Alfano and Crisfield [194] developed cohesive elements and discussed problems related to mesh size leading to spurious oscillations in the solution. The number of integration points in the isoparametric formulation of the element is also shown to have a significant effect on the stability of the solution. Cohesive models have also been developed for use between beam, plate or shell elements [195–197].

A different approach consists of introducing spring elements at the nodes. With these discrete cohesive elements, zero-length springs connect nodes initially at the same location and depend only on the displacements at those nodes. Several authors have used this approach [116, 155, 158, 159, 198, 199]. This approach is sometimes called discrete cohesive zone model (DCZM).

The force–separation relationship for the discrete spring element is based on the continuum damage evolution law governing the material behavior. Generally, the form is bilinear or trapezoidal. Each fracture mode (I, II, and III) requires three or four parameters according the form of the law. The required parameters are the critical energy release rates, the critical strengths or critical separation for damage initiation, the shape factors that define the plasticity and the initial stiffnesses. The critical forces in the spring depend on the material cohesive strengths.

Borg et al. [118] consider this method as a penalty method tying coincident nodes with three orthogonal springs. The penalty forces P_i are related to $\delta_{i}$, the relative displacements for the coincident nodes by $P_{i} = \left( {1 - \omega_{i} } \right)\,k_{i} \,\delta_{i}$ where k_i is the penalty stiffness in direction i and $\omega_{i}$ are damage variables with values between 0 and 1. This penalty approach can also be used when layers above and below the interface are modeled as beams, plates or shells. Forces and moments are applied to force the transverse displacements to be the same and to eliminate shear deformations at the interface [200].

To sum up, CZMs are widely used to model crack propagation in composite laminates. A large number of cohesive laws have been developed and studied. The most used is the bilinear cohesive law. Alfano [135] shows that the bilinear laws represent the best compromise between computational cost and approximation.

These laws have been implemented in specific finite elements (zero thickness 8-noded elements and zero length spring elements) that have to cope with specific numerical issues. Many authors [201–203] showed that the element size was very important for the accuracy of the calculation and its convergence. At the crack tip, the variation of stress can be very high in a small area. More details on the calculation of cohesive elements length are given in part 5.1.1. More, independently of the element size, some cohesive laws leads to difficult convergence, like the trapezoidal law [135].

In the following, a description is given of how these cohesive elements are used for the modelling of impact damages in composite structures.

5 Applications of cohesive elements

The study of foreign object impacts on laminated composite structures goes back several decades and led to the publication of thousands of articles. The most challenging aspect is the prediction of impact damage and in particular the size and location of delaminations. Until recently, it was only possible to predict the onset of damage but predicting the sequence of events leading to the final damage pattern and the details through the thickness was not possible. The development of cohesive elements and their availability in well-known finite element programs has made detailed simulations of the impact possible. Realistic results are obtained and compare well with experimental results. Similarly the introduction of cohesive elements enabled significant progress in the simulation of composite structures under crash impacts. In the following, we briefly discuss the use of this type of elements in the analysis of foreign object impacts on composite and sandwich structures and the crushing of energy absorbing structures.

The use of cohesive elements for simulating impacts on composites in recent years can be attributed in part to the availability of elements such as element COH3D8, a three dimensional, 8 node, zero thickness element in ABAQUS. Table 1 lists 45 references published between 2008 and 2015 in which this element is used. ABAQUS also offers the 6 node, three dimensional cohesive element COH3D6 to be used with tetrahedral solid elements [204–208] and the 4 node, two- dimensional cohesive element COH2D4 [209–211]. Cohesive elements are also available in many commercial finite element programs including LS-DYNA, MSC.Marc and ANSYS [212]. Table 2 provides a quick reference to other publications cited in this section in which cohesive elements are used.

Table 1 Articles in which the cohesive element COH3D8 in ABAQUS is used

Table 2 Other references using cohesive zone models to study impact problems

5.1 Impact on composite laminates

Failure initiation in laminated composite materials under foreign object impacts is predicted using intralaminar failure criteria for matrix cracks and fiber failures and interlaminar failure criteria such as those discussed in Sect. 2 for delamination. Once the damage is initiated it might grow under further loading and may cause other type of damage to develop. For example, matrix cracks may cause delaminations as they reach an interface between plies. Similarly, a delamination crack may induce a matrix crack in one of the adjacent plies which in turn will induce a delamination at the next interface. The simulation should treat the initial failures as cracks and, using cohesive interface elements, it is possible to track their evolution throughout the impact and obtain accurate predictions of the damage state and get an understanding of the damage development process. This section focuses on transverse impacts by hard projectiles.

5.1.1 Low velocity impacts

Cohesive elements are used to predict delaminations during low velocity impacts (LVI) on laminated composites. In most cases, interface elements with a bilinear cohesive law are placed between adjacent plies [128, 209, 211, 213–228]. A linear elastic phase is followed by a linear softening. Unloading after damage onset is expected to follow a linear path to the origin. In some studies a different cohesive law is used [229, 230]. Some simulations use the tie-break interface in LS-DYNA which also provides for a bilinear traction–separation law [231]. Airoldi [228] showed examples of multiple delaminations that propagate unstably and lead to immediate loss of load-carrying capability and also of slowly-propagating interlaminar damage that does not affect the overall response.

Cohesive elements can also be used to model intra-ply failures in composites. This includes matrix cracks induced by transverse tensile stresses, transverse shear stresses, and inplane shear stresses. Reference [212] reviews application of cohesive zone interface elements to modelling discrete matrix dominated failures in polymer composites. Fiber failures can also be modeled by cohesive elements. In a series of articles, Bouvet et al. [199, 232–234] use cohesive elements to predict both intraply and interlaminar damage during low velocity impacts. This approach was also used to study the compression after impact behavior of laminates [235]. In [236] damage in a beam under three-point bending is predicted using cohesive elements for both matrix cracks and delaminations. Other studies also include interface elements to predict both intralaminar and interlaminar impact damage [237, 238].

While most studies assume a linear stress–strain behavior for each ply, several studies account for nonlinear stress–strain behavior under inplane shear. For example, in [214, 238] the shear stresses ${\uptau}_{\text{ij}}$ are given in terms of the ultimate shear strength ${\text{S}}_{\text{ij}}$, the shear modulus ${\text{G}}_{\text{ij}}^{\text{o}}$ as

$$ \tau_{ij} = S_{ij} \left[ {1 - \exp \left( { - G_{ij}^{o} \gamma / S_{ij} } \right)} \right] $$

(5.1)

for i, j = 1–3. The cubic polynomial shear stress–stress relationship

$$ \tau_{ij} = c_{1} \gamma_{ij} + c_{2} \gamma_{ij}^{2} + c_{3} \gamma_{ij}^{3} $$

(5.2)

is adopted in several publications [239–241]. Other approaches for modeling nonlinear shear behavior have been proposed [242, 243].

A significant issue in using cohesive elements is the small element size required to achieve convergence. Typically, elements must be smaller than 0.5 mm [244]. The main reason is that the stress at the interface varies greatly over a distance of about 4 or 5 mm. Several elements are needed to capture this phenomenon since they usually provide linear stress variations. In Hillerborg et al. [190], the characteristic length for isotropic materials is a material property given by $l_{c} = E G_{c} / \sigma_{max}^{2}$. The length of the cohesive zone can be estimated with

$$ \hbox{l}_{{\rm cz,z}} = \left[ {\hbox{E}_{\rm I}^{'} \hbox{G}_{{\rm Ic}} / \left( {\upsigma_{{\rm z}}^{{\rm o}} } \right)^{2} } \right]^{1/4} \hbox{h}^{3/4} $$

(5.3)

$$ l_{cz,zx} = \left[ {\hbox{E}_{II}^{'} \hbox{G}_{IIc} \hbox{h} / \left( {\tau_{zx}^{o} } \right)^{2} } \right]^{1/2} $$

(5.4)

for Mode I and Mode II loadings respectively [245]. In these formulas, h is the half thickness of the laminate, ${\upsigma}_{\text{z}}^{\text{o}}$ and $\tau_{zx}^{o}$ are the maximum interfacial strengths, and ${\text{E}}_{\text{I}}^{{\prime }}$ and ${\text{E}}_{\text{II}}^{\prime }$ are the equivalent moduli under Mode I and Mode II loadings. Reference [246] provides expressions for calculating the equivalent moduli in terms of the elastic moduli E_i, the Poisson ratios ${\upnu}_{\text{ij}}$, and the shear moduli G_ij where i, j = 1, 2, 3. Several other approaches for estimating the length of the cohesive zone are given in [247, 248]. Usually, ${\text{l}}_{{{\text{cz}},{\text{z}}}} \ll {\text{l}}_{{{\text{cz}},{\text{zx}}}}$ and it is recommended to have at least two or three elements in the cohesive zone defined by ${\text{l}}_{{{\text{cz}},{\text{z}}}}$ in order to capture the stress distribution in that zone.

Several references pointed out that, for successful modeling using interface elements, two conditions must be met: (1) the compliance introduced by the cohesive element before crack propagation should be negligible; (2) the element size should be less than the length of the cohesive zone [126, 248].

Several strategies were developed to reduce the number of elements needed for obtaining stable solutions. In [244], interface elements are enriched with the analytical solution of an idealized beam on elastic foundation. Accurate predictions are obtained with element sizes of 5 mm. Further developments of this approach can be found in [249–251]. Other approaches have been presented including one in which, ahead of the delamination, the initial stiffness and strength are lowered while keeping the same onset displacement and the same fracture toughness which will increase the final displacement in the cohesive law [169, 252].

The effects of preloads on impact damage have received considerable attention since most structures in normal operating conditions are subjected to stress while impacted. One study indicated that preloads can have a significant effect [253]. Tensile preloads result in smaller delaminations since deflections are reduced. Compressive preloads resulted in larger size delaminations because larger deflections. Realistic delamination sizes are obtained when cohesive elements are used at many interfaces through the thickness.

Several aircraft components are exposed to the risk of bird strike [254]: windshield, window frame, radome, wings and empennage leading edges, engine inlets and fan blades. Models with stacked shell elements and bilinear cohesive elements (element COH3D8R in Abaqus) were developed [254–256]. Large deflections and the effect of preloads were found to be important. Similar approaches to predict damage to composite parts induced by bird strikes can be found in [242, 257–259].

Some studies focus on impacts on particular components instead of generic plates or shells: composite aircraft wing flap support impacted by a wheel rim fragment [201], impacts on scarf joints [260], on helmets [261, 262] or on composite pipes [263]. Most numerical investigations are conducted using the finite element method and often some well-known commercial codes. For example, Abaqus is used by many authors and the bilinear cohesive element COH3D8R [201, 253–256, 264]. Other numerical approaches are also used. For example: the element free method for both matrix cracks and delaminations in [265–267], modeling of delaminations by the XFEM [268], and peridynamics [269, 270].

Jalalvand [271] use interface elements to model fiber failure in plies with carbon fibers and interface elements to predict delaminations between glass fiber reinforced plies and plies reinforced by carbon fibers.

The results presented by Bouvet and others show that the use of cohesive models results in very accurate and detailed prediction of low velocity impact damage that was not possible before. The procedure used is a progressive damage analysis: damage initiates, propagates, and leads to the initiation of other damage modes. For example, a matrix crack initiates, reaches a ply interface and induces a delamination. Delaminations can propagate in a self-similar fashion for a while and then induce a matrix crack in the next ply which will create another delamination in the next interface. This process is sometimes called “ply-jumping” or delamination migration. Those transitions can be tracked as shown in some of the references cited. Often, damage is modeled discretely for matrix cracks and fiber failures for example. Some investigators account for this type of damage using continuum damage models that represent the material as a homogeneous material with reduced properties. The literature on continuum modeling is very extensive and to give a coherent presentation of those models is beyond the scope of the present article. However, that approach is used by many authors (e.g. [241]).

5.1.2 High velocity impacts

Ballistic impacts of composite structures are modeled using bilinear cohesive elements in several publications. For example, Loikkanen [272] analyze the complete penetration by spherical and cylindrical steel projectiles with velocities from 100 to 900 ft/s. Varas et al. [273] study the ballistic impact of a woven fabric laminate by a cylindrical flat-ended steel projectile and showed that delamination absorbs relatively little energy in that process. The delaminated area increases as projectile velocity increases until the ballistic limit is reached. As the initial velocity increases further, both experiments and simulations showed that the delaminated area decreases. For high velocity impacts on plate with either tensile or compressive preloads, matrix cracking and delaminations are the major impact damage modes under medium velocities with rebounding projectile [253]. For higher velocities, fibre rupture and fibre blow-out also appeared before full penetration. Tensile preloading leads to a reduced extent of delaminations resulting from a reduced bending deflection of the plate under impact. Compressive preload leads to increased delamination size due to increased bending deflections of the plate (Fig. 18).

Fig. 18

Damage induced by ballistic impact on laminated composites [376]

Accurate predictions in case of complete perforations typically require a very fine mesh. In a typical example, a 110 mm × 110 mm plate with 12 layers and a total thickness of 2.4 mm was modeled by 3D solid elements with one element per layer and cohesive elements at each of the 11 interfaces [274]. A total of 270,000 elements were used in this model. Similar examples are presented in [275].

This subsection shows that the availability of cohesive interface elements has led to significantly improved damage predictions with remarkable agreement with experimental results for impacts by both rigid and soft projectiles. Simulations require a detailed model of the laminate and a very small time step at the same time.

5.2 Impact on composite sandwich structures

In sandwich structures, the behaviour with respect to the crack propagation is strongly depending on the type of core: structural foams, honeycomb, other type. Indeed, delaminations either grow near the facesheet-core interface or kink into the core material. Typically, cracks are induced near the core-top facing interface and may not propagate in a self-similar manner but grow in an inclined direction instead (Fig. 19). The kink angle ${\varOmega}$ can be estimated using the Erdogan-Shi formula [276]

$$ \varOmega = 2 \tan^{ - 1} \left\{ {\left.\left[ {\sqrt {1 + 8\left( {K_{II} /K_{I} } \right)^{2} } - 1} \right] \right/\left(4 K_{II} /K_{I} \right)} \right\} $$

(5.5)

where K_I and K_II are stress intensity factors. These mode I and mode II stress intensity factors should be determined numerically using the VCCT for example [277].

Fig. 19

Crack kinking in the foam core of a sandwich beam

To use cohesive elements, one should anticipate the direction of propagation of the delamination which is not possible when the crack kinks into the core [278, 279]. The issue of crack kinking from the core-facing interface and its relation to previous work dealing with crack kinking at bimaterial interfaces is discussed in [54–56]. Further studies of crack kinking foam core sandwich structures include [277, 280].

Cohesive elements are used to predict delaminations induced by low velocity impacts on sandwich structures. Often a bilinear cohesive law is used [281–285]. Some studies include the nonlinear behavior of the facesheet material under inplane shear loading [281]. Other cohesive laws are used: triangular [286], cubic polynomial [283, 284]. Reference [278] assume that the delamination is a mode I interfacial crack and uses a cubic polynomial cohesive law of type proposed by Needleman. Reference [287] shows that this model can predict stable and unstable delamination growth when large deflections are considered.

Sun and Chen [288] proposed a bilinear rate-dependent CZM of the viscoelastic interface for sandwich structures by analogy with the standard linear solid model in viscoelasticity. That model was used to studying loading rate effects on delamination of sandwich structures.

Measuring critical strain energy release rates and cohesive strengths for interfaces between the facesheet and core of sandwich panel is difficult. Several studies have addressed this problem and developed tests that produce reliable data [285, 289–294].

Heimbs [295] used a stacked shell model with cohesive interface to predict composite facesheet delaminations in sandwich structures with foldcores subjected to impact. Stacked shell elements with cohesive elements at the interfaces were also used to model facesheets in [296].

Gopalakrishnan et al. [297] study the coupled buckling-debonding of sandwich beam with aluminium honeycomb core, use a surface potential $\phi$ of the form similar to that proposed by Volokh and Needleman [298]

$$ \phi = \exp \left( 1 \right) \sigma_{\hbox{max} } \bar{\delta }_{n} \left[ {1 - \left( {1 + \Delta_{n} } \right)\exp \left( { - \Delta_{n} - \Delta_{t}^{2} } \right) } \right] $$

(5.6)

where $\Delta_{n} = \delta_{n} /\bar{\delta }_{n}$ and ${\Delta }_{\text{t}}={\updelta}_{\text{t}} /\bar{\delta }_{\text{t}}$ ${\updelta}_{\text{n}}$ and ${\updelta}_{\text{t}}$ are the normal and tangential separations and ${\updelta}_{\text{n}}$ $\bar{\delta }_{\text{n}}$ and $\bar{\delta }_{\text{t}}$ are their maximum values. ${\upsigma}_{ \hbox{max} }$ is the normal strength of the interface. The normal and tangential surface tractions T_n and T_t are obtained by differentiating $\phi$ with respect to ${\updelta}_{\text{n}}$ and ${\updelta}_{\text{t}}$. A similar approach is taken in [299] to study the same problem.

5.3 Composite energy absorbing structures

Crashworthiness is defined as the ability of a structure to protect its occupants during an impact. This requires dissipating the kinetic energy of the vehicle by plastic deformation of metallic structural elements or by failure of members made out of composite materials. The literature dealing composite structures subjected to crash impacts is reviewed in several articles [300–304] and books [301].

A typical crash absorbing structure is a composite tube under axial crushing. It is designed not to fail in a global mode such as column buckling but in a local progressive mode. The tube is splayed into several fronts and energy is dissipated by fiber breakage and delamination of the fronds and by friction between the fronds and the surface they come in contact with. Simulating this continuous process is a challenging task and the following will show that significant progress was made using cohesive elements.

5.3.1 Automotive applications

In automotive applications, several components made out of composite materials are designed to absorb energy during crashes: bumpers [305–307], parts of the frame to absorb energy during frontal or rear impacts, side-door impact beams [308, 309]. The front end of race cars usually have a shell type crash absorber made of composite materials [310–315]. They also have a similar type of crash absorber mounted on the back of the car for rear impacts [316]. On buses for public transportation, composite materials are used to reinforce roofs to prevent rollovers and to fabricate rollbars to protect passengers in case of rollover [317–321]. Composite materials are also found in roolbars for farm tractors [322]. Composite materials are also used in the construction of guardrails [323–325].

Research in this area typically focuses on composite tubes with different cross-sections. Such structures are usually loaded axially and the basic design requirements are that: (1) the maximal force transmitted be limited in order to limit the deceleration of the vehicle during the crash; (2) a significant amount of energy be dissipated. This precludes failure by global or local buckling, or by fracture. Instead, progressive failure is sought. Typical failure during axial crushing of composite tubes (Fig. 20) involve: (1) splaying of the tube wall into several fronds (Mode I fracture), the bending of the fronds, fiber fracture, and delamination of the fronds (Mode II fracture).

Fig. 20

Failure mechanisms for axial crushing of composite tubes [377]

The crushing of laminated composite tubes has been studied numerically using finite elements. With the “stacked shell” approach several plies or sublaminates are each modeled by shell elements connected by cohesive elements to allow for delaminations. Palanivelu et al. [203, 326] study the crushing of circular and square tubes using the Abaqus finite element software with a bilinear cohesive law and claim to be the firsts to use cohesive elements for this problem [326]. With the stacked-shell approach, ply interfaces can also be modeled using one-dimensional cohesive elements [327] or “tie breaks” in LS-DYNA [328–331].

Another approach consists of using three dimensional finite elements with interface elements for both interlaminar and intralaminar failure. This approach revealed that, for the case of a braided composite tube under axial crushing, delamination played a very minor role in energy absorption [332]. The dominant failure modes were splaying, fragmentation, progressive folding and catastrophic failure. Pinho et al. [333] used solid elements and bilinear cohesive elements to model interface delamination during the crushing of square composite tubes. Most composite tubes used as crash absorbers have unidirectional or woven fiber reinforcement. However, Ref. [334] considers circular tubes made out of a glass mat thermoplastic material with a polypropylene matrix and random fiber mat. The finite element model of these tubes used 3D solid elements and tie-break contact interfaces to model delaminations.

A study of a crash impact attenuator with a shape described as a square frusta was modeled using two approaches: one using shell and the other using 3D solid elements and cohesive elements to account for delaminations [312].

Cohesive elements are used to model the failure of composite beams used in structures designed to resist side impacts or to be used as car bumpers. Greve [335] used a bilinear cohesive law to model delaminations between shell elements for the numerical analysis of a composite thin-walled beam with a complex cross section under three point bending with an axial loading. This is an idealization of a pole impact on an automotive side sill structure.

To model impacts on motorcycle helmets, Ref. [261] used six 3D brick elements through the thickness of the composite shell of the helmet with bilinear cohesive elements at the interfaces.

5.3.2 Aerospace applications

Composite materials are used in the design of energy absorbing structures in case of crash landing of aircraft structures. Typically, studies focus on the vertical drop of a fuselage section impact either a rigid surface [336], soil, or water. The sub-cargo area of the fuselage first impacts the ground (Fig. 21). Beams in the cargo floor are designed to absorb energy. Plastic hinges in the frames are a second energy absorbing mechanism [336, 337]. Struts connecting the lower frames to the passenger floor are designed to also absorb energy. A significant portion of the kinetic energy is absorbed by the crushing of composite tubes used in the design of these struts connecting the two floors. In one design, end fittings cut the tube into strips that are then crushed by bending [338, 339]. Similarly, composite tubes are used as energy absorbers integrated in the design of helicopter landing gears [340]. Bolted single lap joints have been suggested as energy absorbing devices [341]. Energy dissipation occurs as metallic bolts are pulled through composite members. Another type of composite structures designed to absorb energy during crash is the subfloor structure of helicopters which generally consist of a network of I-beams running in two orthogonal directions [342, 343]. The crashworthiness of final designs is based on two main criteria: (1) the loads applied to passengers; (2) loads applied to the frame of the structure [337]. For hard-landings and survival crash scenarios, the maximum deceleration at passenger seats should be limited to 16 g for 90 ms in cases of forward landing and 14 g for 80 ms for downward loadings [344].

Fig. 21

Typical fuselage section [378]

Cohesive elements are used to model delaminations during the crushing of composite structures aerospace applications. Composite tubes in energy absorbing struts were modeled by two layers of shell elements connected by interface elements [338, 339]. The analysis predicted the crushing load but predicting the peak loads in the crushing process would require a much more refined model.

The crushing of laminated plates is thought to be simpler because it involves fewer failure modes than that of laminated tubes [345–348]. Experiments showed significant differences between the static and dynamic crushing of composite materials. In some cases, changes in crushing modes led to dramatic reductions in crushing loads for some material systems. High speed video analyses show that the trigger plays an important role during the initial of the crushing process that induces delamination and buckling of individual plies and evolves in a process of splaying and fracture. Israr et al. [349–351] model the crushing of laminated composite plates by meshing each ply of the laminate, using bilinear cohesive elements between each ply. The analysis predicts the major failure modes (splaying and fragmentation) and provides an estimate of the energy absorbed by each damage mode.

While there is a strong emphasis of the crushing of composite tubes in the literature, Joosten et al. [352] analyzed the crushing thin-walled open section composite profiles using stacked shell models with bilinear cohesive interfaces. David and Johnson considered thin-walled open sections consisting of a half circle and two flanges experimentally [353]. The numerical simulation of these test involved the finite element modeling of the composite profile using the stacked-shell approach with cohesive interface elements [354]. The crushing of sandwich plates under edge compression is studied by Pickett [355].

6 Conclusion

CZM are becoming the method of choice for modeling delaminations in composite materials particularly since they are available in some widely available finite element codes. This article reviews the failure criteria used to predict the onset of delaminations, basic fracture mechanics concepts including approaches for predicting crack kinking, and the basic formulations of the various CZM. Afterwards a review of the applications of CZM to impact problems is presented including low velocity and ballistic impacts by foreign objects on composite and sandwich structures, and energy absorbing components of automotive or aerospace structures during crash.

A significant advantage of this method is that it can be implemented in a general purpose finite element code in a way that is not problem-dependent. Results presented by various authors show that impact damage can be predicted with remarkable accuracy and a high level of detail. However, a very refined mesh is required to ensure the stability of the solution process. The size of the elements is determined from the length of the cohesive zone which can be estimated from simple approximate formulas. The computational effort increases rather dramatically with the number of delaminations considered. The choice of technique to model impact on composite laminates depends on the level of precision needed and on the size of the modeled structure. A refined finite element model is required to determine interfacial stresses accurately and small size elements are also needed to avoid stability problems with cohesive elements. This results in the need for very small time steps in the analysis and the combination of a large number of elements and a small time step make such analyses computationally expensive. At this time such analyses are only performed on small test size components. For instance, modeling intra-laminar failure with cohesive elements provides a good representation of damage scenario but it is hardly feasible for a large structure.

CZMs are also used to analyze many other problems for composite structures with delaminations. For example, they are used in numerical models to determine the residual strength of structures with impact-induced damage, to determine the strength of bonded or bolted joints and to determine the effectiveness of various repairs.