A solution for the initial postbuckling and growth behavior of internal delaminations

This paper presents a closed form solution for the initial postbuckling and growth behavior of delaminations in plates. The solution is derived with no restrictive assumptions on the delamination thickness and plate length, i.e. the usual thin film assumptions are relaxed. A perturbation procedure is used, based on an asymptotic expansion of the load and deformation quantities in terms of the distortion parameter of the delaminated layer, the latter being considered a compressive elastica. In addition to determining the load and mid-point delamination deflection as a function of the applied compressive displacement the analysis produces closed form expressions for the energy-release rate and the mixity ratio (i.e. Mode I1 vs Mode I) at the delamination tip. A higher Mode I component is found to be present during the initial postbuckling phase for delaminations of increasing ratio of delamination thickness over plate thickness, h/T (i.e. delaminations further away from the surface). Moreover, the energy release rate corresponding to the same applied strain is larger for a higher h/T ratio. The reduced growth resistance of these configurations is verifed by experimental results on unidirectional composite specimens with internal delaminations. Introduction Delaminations or interlayer cracks are developed as a result of imperfections in production technology or due to service loads which may include impact by foreign objects. As a consequence, structural elements with delaminations under compression suffer a degradation of their stiffness and buckling strength and potential loss of integrity from possible growth of the interlayer crack. Besides strength, delaminations can influence other performance characteristics, such as the energy absorption capacity of composite beam systems’. Delamination buckling in plates under compression has received considerable at tention and numerous contributions have addressed related issues in both onedimensional and two-dimensional However, although the critical point can be fairly well determined and has been extensively studied, limited work has focused on the postbuckling behavior, which ultimately governs the growth characteristics of the delamination. The one configuration most thoroughly studied is the one-dimensional delamination, consisting of a delamination in an infinitely thick plate. In this model2, which has also been called “thin film” model, the unbuckled (base) plate is assumed to be subject t o a uniform compressive strain. Closed form expressions for the energy release rate from the thin film model were derived by Chai et a12 by using the strain energy expressions before and after delamination buckling. The other configuration most extensively studied is the axisymmetric counterpart to the one-dimensional delamination, i.e. a circular delamination in a perfectly rigid supporting plate. The latter relates also to the socalled blister test used to determine adhesive and cohesive material properties. For this configuration, Evans and Hutchinson5 derived a formula for the energy release rate by using an asymptotically valid solution to the system of governing equations for small buckling deflections. Results for the energy release rate of a circular delamination were also given by Chai7 and calculated through a path-independent integral approach by Yin’. In the same context, Stordkers and Anderssong derived general potential energy theorems and associated bounds for composite plates within the kinematical assumptions usually attributed to von Karman, and studied in detail the efficiency of different analytical and numerical means for this circular delamination case. For delaminations in plates that cannot fullfil the “thin film” model assumptions, both the critical load and the post-critical behavior are expected to deviate from the predictions of Chai et al’. To this extent, Simitses et a13 studied the citical load for a delamination of arbitrary thickness and size in a finite plate. Their results showed indeed a range of critical load vs thin film load ratios, depending on delamination and base plate dimensions, as well as base plate end fixity (simplysupported vs clamped). Concerning the post-critical behavior of delaminations of arbitrary size, Kardomateas” provided a formulation for studying the postbuckling behavior by using elastica theory for representing the deflections of the buckled layer; this work resulted in a system of nonlinear equations rather than closed form expressions. In this paper, the initial postbuckling behavior of delaminated composites (with no restrictive assumptions on the delamination dimensions) is studied by using a perturbation procedure based on an asymptotic. Copyright 01993 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 2679 expansion of the load and deformation quantities in terms of the distortion parameter of the delaminated layer, the latter being considered a compressive elastica. The analysis will lead to closed form solutions for the load versus applied compressive displacement and the near tip resultant moments and forces. Subsequently, the bimaterial interface crack solutions for the energy release rate and the mode mixity in terms of the resultant moments and forces, as derived by Suo and Hutchinson” will be employed to study the growth characteristics of the delamination. Numerical results for a range of relative delamination thicknesses and lengths are presented and discussed. Moreover, the predicted growth characteristics are compared with experimental observations from compression tests of unidirectional composite specimens with internal debonds. Initial Postbucklinv Solution In a delaminated system, which can be thought of as an aggregate of constitutive parts such as the delaminated layer and the base plate, the conditions of geometrical continuity at the common sections (Le. where the delamination starts or ends) play a particularly important part in the realization of equilibrium states which follow non-linear paths. The exact laws that govern the behavior of single compressive elements elastically restrained at the ends by means of concentrated forces and moments constitutes the elastica theory’’. Generalized coordinates of deformation are the distortion parameter, a, which represents the tangent rotation at an inflection point from the straight position, and the amplitude variable a(.). The initial postbuckling deformations are relatively small, so the exact expressions may be expanded in Taylor series in terms of the distortion parameter. Exact dependence of the end moments, end rotations and the flexural contraction is through elliptic functions; however the asymptotic expressions that will be given in this work are in terms of trigonometric functions. Consider a plate of half-length L (and unit width) with a through-the-width delamination of half-length e, symmetrically located (Fig. 1). The delamination is at an arbitrary position through the thickness T . Over the delaminated region, the laminate consists of the part above the delamination, of thickness h referred to as the “delaminated” part, and the part below the delamination, of thickness H = T h, referred to as the “substrate” part. The remaining, intact laminate, of thickness T and length b = L e, is referred to as the “base” plate. Without loss of generality, it may be assumed that h < T/2. Accordingly, the subscript i = d, s, b refers to the delaminated part, the substrate or the base plate, respectively. In the following, we shall also denote by Di the bending stiffness, Di = Et:/ [12 (1 v ’ ) ] , ti being the thickness of the corresponding part, E the modulus of elasticity and v the Poisson’s ratio. For simplicity reasons, the properties of the material are assumed homogeneous, linearly elastic and isotropic (orthotropic properties can be accounted by using ~ 1 2 ~ 2 1 instead of v’ and E the modulus of elasticity along the 5 1 axis). Notice also that in Fig. 1 the end moments and tangent rotations are assumed positive clockwise. The buckled configuration of the delaminated layer is part of an inflectional elastica with end amplitude @ d and distortion parameter E . At the critical state, the end amplitude is (a:. Suppose that in the slightly buckled configuration @ d can be expanded in the form: @ d = @: + d ! ) E + d&)€’ + 0 ( E 3 ) . (1) Then the end rotation at the common section B is given by expanding the relevant expression’’ in Taylor series in terms of t: (notice that at the critical state 6’ = 0): ’ 1 6 = (sin -(sin 24 @ d cos’ + , . . =