Development of a micro-indentation model simulating ... - CiteSeerX

Composites Science and Technology 61 (2001) 369±375

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Development of a micro-indentation model simulating di€erent mechanical responses of the ®bre/matrix interface Mondher Zidi a,c, Luc Carpentier a, Antoine Chateauminois b,*, Ph. Kapsa a, FrancËois Sidoro€ a a Laboratoire de Tribologie et Dynamique des SysteÁmes, UMR CNRS 5513, Ecole Centrale de Lyon, BP 163, 69131 Ecully Cedex, France Laboratoire d'IngeÂnierie et Fonctionnalisation des SysteÁmes, UMR CNRS 5621, Ecole Centrale de Lyon, BP 163, 69131 Ecully Cedex, France c Laboratoire de MeÂcanique des Solides, Ecole Nationale d'IngeÂnieurs de Monastir, Avenue Ibn Eljazzar, 5019 Monastir, Tunisia

b

Received 16 July 1999; received in revised form 22 November 1999; accepted 16 May 2000

Abstract An analytical shear-lag model has been developed for quantifying the interfacial shear strength of glass-reinforced composites from micro-indentation experiments. The model takes into account the local ®bre environment, together with the occurrence of debonding and ®bre sliding. In order to simulate the experimental indentation curves, various interfacial laws have been implemented. In a ®rst approach, it was assumed that the shear stress in the debonded part of the interface was constant and proportional to the debonding stress. A more re®ned generalised interface law relating the shear stress to the ®bre displacement was subsequently introduced to describe a progressive transition from an adhesive to a sliding state. The model has been successfully applied to the analysis of experimental reduced indentation curves giving the displacement of the ®bre surface as a function of the applied load. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Micro-indentation; Shear lag; Glass/epoxy composites; Interface law; Fibre/matrix debonding

1. Introduction The durability of polymer-matrix composites under mechanical and environmental loading is known to be strongly dependent upon the strength of the ®bre/matrix interface. As a result, many methods based on both macro- or micro-mechanical tests have been developed for measuring the level of adhesion at the interface [1]. Among the many micro-techniques for interfacial property characterisation, the micro-indentation test is very attractive because it is an in-situ testing method conducted on real composites, thus allowing for evaluation of the processing or environmental conditions encountered either during manufacturing or service. Initially introduced by Mandell et al. [2,3], this test consists in indenting a single ®bre in the polished cross-section of a composite specimen until the occurrence of interfacial debonding. * Corresponding author. Tel.: +33-4-72-18-6453; fax: +33-4-7833-1140. E-mail address: antoine.chateauminois@ec- lyon.fr (A. Chateauminois).

The corresponding debonding load is generally identi®ed either from the force/displacement curves or from speci®c procedures involving microscopic observations after testing at increasing indentation loads [4±7]. The derivation of the interfacial shear strength from the measured debonding load is, however, complicated by the highly heterogeneous stress ®eld induced by the indenter. From parametric ®nite-element studies, Ho et al. [8] and Mandell et al. [3] have also demonstrated that the local ®bre arrangement can strongly a€ect the perturbed stress ®eld around the indented ®bres. As a result, the scatter in the local ®bre packing can result in very di€erent debonding loads. In order to overcome these diculties, Mandell and co-workers proposed a data reduction scheme based on a linear axi-symmetric ®nite-element analysis. In a ®rst step, the experimental debonding loads need to be shifted to an adjusted value for a given ratio tm =df , where tm is the average matrix thickness between the tested ®bre and its nearest neighbours and df is the ®bre diameter. The interfacial shear strength is subsequently calculated from this adjusted debonding load using a maximum shear stress criterion

0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(00)00123-8

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M. Zidi et al. / Composites Science and Technology 61 (2001) 369±375

and the results of the ®nite-element analysis. The use of this procedure is, however, complicated by the ambiguities surrounding the experimental determination of tm in the real non axi-symetric con®guration. Alternative methods based on mono-dimensional shear lag models have also been used to analyse microindentation data. Although these approaches imply very crude assumptions [9,10], the resulting analytical relationships are very appropriate for the systematic identi®cation of parameters from experimental data. This latter point is especially relevant in the ®eld of micro-indentation tests, where the large data scatter requires the statistical analysis of many experiments. In a previous study [11,12], the present authors have derived and validated by ®nite-element modelling (FEM) simulations such a model for polymer matrix composites. This model incorporates to main features: 1. the local environment around the tested ®bre is taken into account through a single parameter, which is calculated directly from the experimental elastic sti€ness of the indented system. 2. the identi®cation of the debonding load is based on a ®tting of the entire loading curve, instead of the identi®cation of an always dubious threshold for debonding. In order to ful®l this last requirement, it was necessary to incorporate into the model the possibility of ®bre sliding in the debonded part of the interface. In a ®rst approach, this was done by considering that the interface obeyed a very simple Tresca criterion. The interfacial shear stress in the debonded area was thus assumed to be constant and equal to the debonding stress. This model provided a satisfactory description of the micro-indentation curves obtained using unidirectional glass/epoxy composites [12]. Some limitations in the application of the model were, however, encountered with other systems such as glass/polyester composites [11]. From scanning electron microscopy (SEM) observations of the indented ®bres, the latter were related to changes in the debonding mechanisms. This indicated the need of further re®nements of the interface law, in order to take into account the di€erent debonding behaviour observed experimentally. The objective of this paper is therefore to present some theoretical developments of the micro-indentation model, which include more re®ned interface laws. The ability of various interface laws to describe di€erent kinds of micro-indentation curves is especially considered. Experimental results are provided in order to demonstrate the potential of the theoretical model for the simulation of di€erent experimental indentation responses. The systematic analysis of micro-indentation data using these models will, however, be considered in a further companion paper.

2. Background 2.1. Extraction of reduced indentation curves The micro-indentation experiments provide raw indentation curves giving the applied load as a function of the overall displacement of the indenter. As a consequence of the compliance of the polymer matrix, it is, however, very dicult to accurately detect the debonding load from such a curve. To overcome this problem, a data reduction scheme has been proposed, which is based on the removal of the displacement component uep caused by the elasto-plastic indentation of the ®bre surface by the indenter. This later contribution was assessed from micro-indentation tests carried out using bulk-glass specimens whose chemical composition is the same as for the ®bres. Full details regarding the development of this procedure and its validation by means of FEM simulations can be found in Refs. [5,11]. The subtraction of uep from the overall displacement results in a so-called reduced indentation curve giving the applied load as a function of the displacement of the ®bre surface only. These reduced curves contain all the useful information regarding the analysis of the interfacial behaviour. Depending on the magnitude of the applied load, the reduced indentation curves can exhibit either a linear elastic behaviour (at low loads) or a non-linear response (at high loads) during loading (Fig. 1). By systematic SEM and optical microscope observation of the ®bres after indentation, it was established that the non-linear behaviour is associated to ®bre/matrix debonding (Fig. 2). Some limitations were encountered in the ®ltering of the experimental indentation behaviour of bulk glass at low loads (P< 50 mN) during the unloading step. As a result, the corresponding part of the reduced indentation curve was not reliable and is not represented in Fig. 1 and in all the subsequent ®gures showing reduced curves. 2.2. Basic equations used to model the reduced indentation curves In order to analyse the reduced indentation curves, a mono-dimensional shear-lag model has been developed. This model is able to simulate the loading and unloading steps of the indentation experiments, while taking into account interfacial debonding. The occurrence of debonding is predicted by means of a maximum shearstress criterion. The model in its basic con®guration has been fully detailed elsewhere [11,12] and only the basic assumptions and equations will be recalled here. An axi-symmetric con®guration with a single ®bre (radius a) embedded in a polymer matrix is considered (Fig. 3). The displacement, u…x†, of the ®bre is supposed to be uniform in a given cross-section. This assumption allows treating the problem as mono-dimensional. Radial and axial stresses as a result of residual curing

M. Zidi et al. / Composites Science and Technology 61 (2001) 369±375

371

Fig. 2. SEM picture of an indented glass ®bre showing the occurrence of debonding (glass/epoxy unidirectional composite, peak load 0.2 N).

Fig. 3. Geometry of the shear-lag model. Fig. 1. Typical reduced indentation curves showing either (a) a linear behaviour (below the debonding load) or (b) a non-linear behaviour (above the debonding load). Solid line in (b): theoretical curve using Eqs. (10)±(12). (glass/epoxy system, indentation speed 0.2 mm sÿ1).

stresses and Poisson's e€ects are not taken into account. Boundary conditions are established by considering that: i. the axial stress,  0, on the ®bre surface is uniform and related to the applied load P through: P 0 ˆ 2 ; a

…1†

ii. the displacement and the axial stress in the ®bre decrease when the depth is increased (the specimen thickness is much greater than the ®bre diameter), i.e.: u ! 0;  ! 0;

for x ! 1

…2†

Elastic equilibrium conditions in the ®bre are written as follows: d 2 ˆÿ dx a

…3†

where  is the interfacial shear stress. From Hooke's law for the ®bre, the following equation can be written:

du  ˆÿ dx E

…4†

The interfacial shear stress,  …x†, is assumed to be linearly related to the displacement, w…x†, of the matrix for r ˆ a, i.e.:  ˆ kw

…5†

where k is a global sti€ness constant including the elastic properties of the ®bre and the matrix, together with the local ®bre environment. By combining Eqs. (3) and (4), the following equilibrium relationship can be written: d2 u 2…x† ˆ0 ÿ dx2 aE

…6†

when u ˆ w, i.e. before local sliding, Eq. (6) reduces to: d2 u ÿ n2 u ˆ 0 dx2 with r 2k nˆ aE

…7†

…8†

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M. Zidi et al. / Composites Science and Technology 61 (2001) 369±375

After the onset of debonding, it becomes necessary to take into account a debonded length, h, where  …x† is de®ned by the interface law. In the basic version of the model, an expression relating the stress  to the relative displacement v…x† ˆ u…x† ÿ w…x† between the ®bre and the matrix was considered. It was at ®rst assumed that the shear stress was constant and equal to the shear strength, d , in the debonded area. The corresponding interface law can thus be expressed as follows:  ˆ d

dv jdvj

…9†

From knowledge of the interface law, Eqs. (6) and (7) can be solved using the appropriate set of boundary conditions. Accordingly, the displacement, uo , of the ®bre surface can be expressed as a function of the applied stress, 0 : a. during loading uo ˆ

0 nE

for

0 < d

…10†

where d is the applied axial stress at the onset of debonding.   1 02 ‡ d …11† for 0 > d u0 ˆ 2nE d b. during unloading During unloading, the model can simulate the occurrence of sliding in the debonded part of the ®bre, when the applied axial stress is less than a limiting value, g . For the studied glass/thermoset systems and the considered interface law, the calculation predicted that such a sliding would require negative loads, i.e. a pull-out con®guration[12]. As a result, only the purely elastic response will be considered herein:  2  1 max ‡ d ‡ 2…0 ÿ max † …12† u0 ˆ 2nE d where max is the maximum load applied during loading. The model requires the identi®cation of two parameters, namely n and the debonding load Pd ˆ d a2 . The former coecient is measured from the slope of the initial elastic part of the loading curve [cf. Eq. (10)]. It provides some information regarding the local environment of the tested ®bres: n is increased, i.e. the global sti€ness of the system is increased, when the local ®bre packing is enhanced. The value of the debonding load is identi®ed by a least-square ®tting of the experimental loading curve using expressions (10) and (11). The interfacial shear strength is subsequently calculated using the following expression:

d ˆ

nPd 2a

…13†

Fig. 1(b) shows an example of the application of the model to an experimental reduced indentation curve obtained using an E-glass/epoxy system. It can be seen that the linear response during unloading is consistent with the assumption of a purely elastic response of the system during this stage. 3. Improved interfacial laws The model presented above is based on a very crude assumption regarding the interfacial behaviour. It proved, however, to give realistic values of the interfacial shear strength (in the order of 70 MPa) for a wide range of glass/epoxy systems [11]. With some other systems, such as glass/polyester composites, it appeared, however, that the theoretical expressions were unable to ®t the experimental indentation curves. This was observed especially when a poor interfacial strength led to a premature and extensive debonding. These processes are often associated with a `kink' in the reduced indentation curve at the end of the linear stage, probably caused by the debonding instability (Fig. 4). Above this critical point, the displacement u0 of the ®bre surface occurred generally to a greater extent than for the glass/epoxy systems. In addition, a non-linear behaviour, probably associated with the occurrence of sliding, was also observed during unloading. The theoretical curve obtained using the initial version of the model with the experimental data reported in Fig. 4 is also shown. This result clearly demonstrates the inability of Eqs. (10)± (12) to describe the experimental behaviour. Improved interfacial laws were thus introduced to reproduce the experimental response.

Fig. 4. Reduced indentation curve obtained using a glass/polyester system (0.2 mm sÿ1)., (*) experimental data; solid line, theoretical curve using g ˆ d [Eqs. (16)±(18),  ˆ 0:27, d ˆ 123 MPa, n ˆ 0:04 mmÿ1); dashed line, theoretical curve using g ˆ d [Eqs. (10)±(12), d ˆ 65 MPa, n ˆ 0:04 mmÿ1).

M. Zidi et al. / Composites Science and Technology 61 (2001) 369±375

3.1. Interface law including a sliding stress di€erent from the debonding stress In the basic con®guration of the model, it was assumed that the shear stress in the debonded area is constant and equal to the debonding strength. It is physically more realistic to consider that the transition from an adhesive to a sliding state is associated with a change in the interfacial stress. In a ®rst approach, it was assumed that sliding occured at a constant stress, g , proportional to d : g ˆ d