Impact-induced delamination of composites: a 2D simulation
Introduction
Composite laminates are used in many engineering applications which expose them to low-velocity impact by foreign objects. A typical example is that of an aircraft structure subjected to the impact of a dropped tool or the collision with runway debris. The relatively weak behavior of composite materials under localized impact has always been one of the major weaknesses limiting their use: unlike metals, polymeric matrix composites do not have the ability to deform plastically to absorb the kinetic energy of the impactor. The energy absorption mechanism of these materials consists in the creation of large fracture area, especially at the weaker interfaces between the composite layers, in a process referred to as impact-induced delamination.
Vast amounts of experimental data and observations have been collected over the past two or three decades on this issue, providing important insight on the various phenomena involved in the delamination process. However, a reliable analytical tool is still missing, which would allow designers to accurately predict the location and extent of impact-induced damage. The accurate prediction of the impact-induced damage is the mandatory preliminary step in the determination of the post-impact properties of these structures. The current lack of a precise modeling tool has led, for example, the marine industry to impose safety factors of the order of ten when dynamic loading of laminated composite structures is involved, about five times the safety factors typically used for the static loading of these structures[1].
The extensive amount of experimental work on impact-induced delamination of composites, recently reviewed by Abrate2, 3and Cantwell and Morton[4], has allowed the scientific community to achieve a relatively sound understanding of the phenomena leading to the impact-induced failure of composite laminates. The basic mechanisms are depicted schematically in Fig. 1, and involve various steps. The initial failure occurs at relatively low energy levels and involves a series of small transverse matrix cracks oriented towards the impact point. As the energy increases, these cracks extend until they reach the interface of a neighboring ply with a different fiber orientation. At this point, the matrix cracks can no longer propagate in their original direction and are deflected onto the ply interface, thereby starting the delamination process. The matrix crack which actually initiates the delamination is usually referred to as the critical matrix crack [Fig. 1(a)]. The delamination continues toward the point of impact on the top interface and away from the point of impact on the bottom one [Fig. 1(b)]. As the delamination grows further, additional transverse matrix cracks tend to appear [Fig. 1(c)]. The extent of delamination damage depends on a wide range of factors, such as the spatial distribution, amplitude and duration of the loading stress waves, the delamination strength of the interlayer, the number and orientation of the plies, etc. In many situations, the internal fracture surfaces nucleate along more than one interlayer, creating a complex three-dimensional fracture problem[5].
But, while the basic mechanisms of the delamination process seem to be well understood, much work is still needed in the development of analytical tools to quantify and predict the impact-induced damage nucleation and growth. Most of the existing analytical investigations have involved various levels of simplification regarding the laminate geometry, the delamination process and/or the dynamic effects. For example, in the case of low-velocity impact in which the time of contact is assumed to be much longer than the time of travel of elastic waves through the laminate thickness, most analyses have been based on the so-called quasi-static assumption, in which all dynamic effects are essentially neglected. The delamination process is then investigated statically, based on the maximum observed impact load6, 7. This approach has, however, been qualified as “unwise” by Collombet et al.[1], who observed differences of 50–100% in the extent of damage between purely static and slow impact experiments. They even noted a difference in damage events between the two types of tests, as the static tests presented some fiber ruptures absent in the dynamic situations. The need to incorporate dynamic effects in the analysis has also been emphasized by Choi et al.[8]who compared the static and dynamic interlaminar stresses obtained numerically for the same overall deflection of the composite plate and showed a strong influence of the dynamic effects, especially on the distribution of intralaminar shear stresses, which drives the initiation of internal matrix cracking.
Many experimental observations have emphasized the geometric complexity of the delamination process. However, a large number of numerical simulations are based on the simplified laminate beam and plate theories5, 7, 9, 10. This geometrical simplification does not, however, allow for a precise description of the fundamental role of the transverse shear stress components in the damage process (unless higher-order theories are used), and rules out the simulation of the major effects associated with the stress concentration in the vicinity of the propagating delamination front. A simplified investigation which would not include a description of individual layers would also preclude the simulation of multiple delamination zones, which are often detected experimentally.
Finally, and perhaps more importantly, while many investigations have emphasized the importance of precise modeling of the damage process5, 11, few analyses have incorporated a specific delamination model which accounts for the creation of new surfaces between the various layers. A detailed modeling of the nucleation and propagation of the interfacial fracture surfaces and of the associated stress and strain concentrations is expected to yield results very different from an “in-ply” failure model such as the Tsai–Wu model[12], a continuum damage mechanics approach[13], or a volumetric stress-based degradation model[11]. The evolution of the delamination front has recently been the focus of a numerical analysis by Zheng and Sun[7]who used an elaborate crack closure algorithm to advance the crack front between two laminate plates. Although the approach showed promising results, the numerical scheme showed important mesh sensitivity and did not take dynamic effects into account. Other approaches to simulate the delamination process and the creation of new surfaces used nonlinear “springs” between interlaminar nodes[14].
One of the most complete investigations of impact-induced delamination of composites has been performed by Chang and his co-workers8, 15, 16, 17, who have presented detailed experimental and numerical 2D and 3D analyses of the damage process. The damage process used in their dynamic finite element simulations was primarily based on matrix cracking through an empirical expression of the transverse tensile strength of the various plies. The main objective of their simulations was to predict the location of the critical matrix cracking. Matrix degradation was modeled by modifying the material properties of the plies in the failing elements. They showed that, associated with the matrix failure process, a stress singularity results along the interfaces situated just above and below the locations of the critical matrix cracks. They then suggested that these singularities could be the driving force behind the onset of the delamination process. However, in addition to the empirical aspect of the matrix cracking criterion used there, the time- and space-resolution of their model was relatively limited, and their approach did not allow for the direct simulation of the delamination process itself.
The main objective of the research project summarized hereafter is to develop and implement a numerical scheme aimed at modeling both the matrix cracking and delamination processes within a unified framework. Special emphasis is put on capturing the stress concentration associated with the spontaneously propagating delamination front(s). The method also incorporates the dynamic (inertial) effects and the geometrically nonlinear (stiffening) effects. The approach is based on the cohesive/volumetric finite element (CVFE) scheme which has recently shown promise in a variety of dynamic fracture problems, such as dynamic crack branching in brittle materials[18], dynamic interfacial crack growth19, 20, impact of ceramic materials[21]and crack growth in elastic–viscoplastic solids[22]. The CVFE scheme is based on a dual representation of the mechanical response of the material: a volumetric constitutive model to represent its “bulk behavior”, and a cohesive model to simulate the spontaneous creation of internal failure surfaces.
In this preliminary “feasibility” study, the numerical analysis is limited to two dimensions and is performed within the framework of the theory of plane strain. The results are compared to those of the 2D impact experiments conducted by Choi et al.[16], in which a composite plate was clamped along two ends and impacted by a cylindrically nosed impactor.
The paper is organized as follows: the numerical scheme and its implementation are discussed in Section 2. Then, we present a couple of test simulations aimed at validating the numerical method. Finally, in Section 4, we summarize the results of the delamination simulations and compare them with the aforementioned experiments of Choi et al.
Section snippets
Concept and implementation of the cohesive/volumetric finite element scheme
The basic idea behind the cohesive/volumetric finite element (CVFE) method can be found in the pioneering work of Dugdale[23]and Barrenblatt[24]who introduced the concept of cohesive failure to provide some structure to the failure process taking place in the vicinity of the crack tip. To allow for the spontaneous initiation and propagation of fracture surfaces, a series of cohesive failure surfaces are introduced within a conventional finite element mesh (Fig. 2). These cohesive elements act
Convergence of the CVFE scheme: dynamic crack propagation in an elastic strip
In this section, we present the results of a series of simulations aimed at investigating the convergence of the proposed CVFE scheme. This preliminary analysis is performed for a simpler dynamic fracture problem involving a mode I crack propagating spontaneously in a pre-strained elastic strip. Of particular interest is the ability of the numerical scheme to capture the cohesive failure model and to describe the energetics of the fracture process.
The use of the cohesive failure law introduces
Problem description and FE discretization
As indicated earlier, we chose to simulate the 2D composite impact experiments performed by Choi et al.[16]in which a laminated composite plate with two opposite sides clamped and the other two free was impacted with a cylindrical nose impactor along a line located between the two clamped ends. The specific plate to be modeled hereafter is a 10 cm long [06/904/06] laminate of T300/976 graphite/epoxy with a thickness of 2.3 mm. Since the impact is centered, the problem is symmetric about the
Conclusions
A numerical method based on the cohesive/volumetric finite element (CVFE) scheme has been developed specifically to investigate the delamination occurring in polymeric matrix fiber reinforced composite laminates subjected to low-velocity impact. The numerical results obtained from preliminary simulations agree relatively well with the 2D line-impact experiments performed on a graphite/epoxy doubly clamped composite plate by Choi et al.8, 16. The method is able to capture the locations of the
Acknowledgements
This paper has been written as part of J. S. Baylor's M.Sc. thesis work, supported partially by a grant from the University of Illinois Campus Research Board. The authors wish to acknowledge various discussions with Professor Alan Needleman from Brown University. Most of the large scale simulations presented in this paper have been performed on the Convex Exemplar available at the National Center for Supercomputing Applications located on the University of Illinois Urbana–Champaign campus.
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