Mode separation of energy release rate for delamination in composite laminates using sublaminates

Abstract

Individual energy release rates for delamination in composite laminates do not exist according to two- or three-dimensional elastic theory due to the oscillatory characteristics of the stress and displacement fields near the delamination tip (Sun, C.T., Jih, C.J., 1987. Engng. Fracture Mech. 28, 13–20; Raju, I.S., Creus Jr., J.H., Aminpour, M.A., 1988. Engng. Fracture Mech. 30, 383–396.) In this paper, sublaminates governed by transverse shear deformable laminate theory are adopted to model such delamination. Oscillatory singular stresses around the delamination tip are avoided as a result. Instead, stress resultant jumps are found in the sublaminates across the delamination tip. It transpires that mode I, II and III energy release rates can then be obtained using the virtual crack closure technique. The results produced by this approach for a homogeneous double cantilever beam and an edge-delamination in a non-homogeneous laminate show good agreement with those available in the literature. The approach produces both total and individual components of energy release rate for delamination, which converge as the sublaminate division is refined and the sizes of the delamination tip elements decrease.

Introduction

One of the most frequently encountered forms of damage in composite laminates is delamination, essentially interface cracks between the plies. Theoretical studies of the propagation of existing delaminations have to date been carried out mainly by adopting fracture mechanics to deal with the singularity at a delamination leading edge or tip. The propagation of an existing delamination is governed by the magnitudes of stress intensity factors or energy release rates. However, unlike cracks in homogeneous bodies, linear elastic fracture mechanics based upon a two- or three-dimensional theory has encountered considerable difficulties when dealing with these interfacial cracks which have not yet been overcome satisfactorily. The mismatch of material properties across the interface always results in coupled fracture modes. The stress fields around the crack tip show an oscillatory singularity (Williams, 1959), as do the relative displacements between the surfaces of the crack (England, 1965). This leads to physically inadmissible interpenetration of crack surfaces near crack tips.

Several types of stress intensity factors have been introduced to characterise crack tip stress fields Rice and Sih, 1965, Wang, 1983, Rice, 1988, Suo, 1990, Wu, 1990. However defined, for interfacial cracks in dissimilar media, these stress intensity factors do not carry the classical physical interpretations which identify three independent singular fields, referred to as the three modes of singularity, as in a homogeneous body. This leaves a gap which needs to be bridged before the theory can be applied to practical problems.

It has also been found that individual components of energy release rate for an interface crack expressed in terms of classical crack closure integrals or the virtual crack closure technique (VCCT) for solid finite element analysis are not well defined because they do not converge. Rather, they show oscillatory behaviour as do the stresses and displacements Sun and Jih, 1987, Raju et al., 1988. Although the total energy release rate does converge to a definite value, using it as a criterion for delamination growth is limited when a mixed mode is involved (Hutchinson and Suo, 1992).

Some researchers Hwu and Hu, 1992, Toya et al., 1997 modified the definition of energy release rates by using a finite crack extension, larger than the size of the oscillation region, instead of the infinitesimal one in the conventional definition. However, choosing the magnitude of the finite crack extension lacks sound theoretical or experimental grounds. Raju et al. (1988) modelled resin-rich layers of about 0.01 mm thickness between neighbouring plies as physical entities. A crack was assumed to exist centrally within this resin layer. As a result, the oscillation vanished. The numerical results showed that the individual components as well as the total energy release rates remain unchanged when the sizes of the crack tip elements decrease. Unfortunately, the resin-rich layer is too thin to be modelled in practical problems.

For delaminations in composite laminates, it is preferable to use laminate theory rather than three-dimensional elasticity theory. It is computationally expensive to use solid finite elements because a large number of elements through the laminate thickness are required, especially in the case of multiple delamination problems. Attempts to obtain individual fracture modes from laminate theory can be found in a number of publications. Williams (1988) suggested that mode I delamination be represented by a pair of moments and transverse shear forces acting in opposite directions applied to the opposite sides of the delamination. Mode II is obtained when the curvatures in the two parts of a delaminated laminate are the same. The underlying justification for this approach is associated with the relative displacements between the surfaces of the delamination around its tip. The opening displacement produces mode I while mode II corresponds to an in-plane sliding displacement. However, a pair of moments acting in opposite directions applied to a split beam with two arms of different thickness, for example, results in non-zero relative sliding displacement. This means that it is a mixed mode rather than a pure mode I problem (Suo and Hutchinson, 1990). With the help of a two-dimensional asymptotic numerical solution for a semi-infinite interface crack between two elastic layers given by Suo and Hutchinson, 1990, Toya et al., 1997 calculated the energy release rates for mode I and mode II by a straightforward application of the crack closure method with finite crack extension. Strictly speaking, this approach is valid only for laminates consisting of only two different layers. This is far too restrictive to be applied to practical laminates. Sheinman and Kardomateas (1997) used the elastic property smearing technique to convert the problem of a delaminating beam into an equivalent homogeneous problem with orthotropic behaviour through the beam thickness and separated the individual modes. In such smearing techniques, the effects of ply stacking sequence are ignored completely. This may cause errors depending on the nature of the beam. When laminate theory is used and the laminate is considered to be comprised of two sublaminates in the delaminated region and a single intact laminate in the undelaminated region (the model adopted widely in the literature), the moments contributing to both mode I and mode II will inevitably be involved in the expression for total energy release rate. Individual components cannot be separated directly. Various assumptions have to be made to reach this goal which require justification.

Unlike the above laminate models, in this paper, the laminate is divided into sublaminates in both delaminated and undelaminated regions with transverse shear-deformable laminate theory being adopted for each of the sublaminates. The use of laminate theory eliminates the stress singularity and the oscillatory behaviour involved in two- or three-dimensional linear elastic fracture mechanics theory for this problem. Instead, stress resultants may show discontinuities across the delamination tip which reflects the interfacial stress singularity there. Since there are no interfacial moments between the sublaminates (as will be discussed later), individual components of energy release rate can be obtained using the VCCT. They are all well defined according to their classical definitions and converge to definite values as the magnitude of the virtual delamination extension reduces. This approach will be applied to standard delamination problems concerning a homogeneous double cantilever beam and a non-homogenous edge delaminated composite laminate. The results show good agreement with those in the literature. As will be shown, usually more than two sublaminates through the laminate thickness are required even for single delaminations in order to reflect the three-dimensional nature of the problem with reasonable accuracy.