A generic approach to constitutive modelling of composite delamination under mixed-mode loading conditions

https://doi.org/10.1016/j.compscitech.2011.11.012Get rights and content

Abstract

A generic approach to constitutive modelling of composite delamination under mixed mode loading conditions is developed. The proposed approach is thermodynamically consistent and takes into account two major dissipative mechanisms in composite delamination: debonding (creation of new surfaces) and plastic/frictional deformation (plastic deformation of resin and/or friction between crack surfaces). The coupling between these two mechanisms, experimentally observed at the macro scales through the stiffness reduction and permanent crack openings, is usually not considered in depth in many cohesive models in the literature. All model parameters are shown to be identifiable and measurable from experiments. The model prediction of mixed-mode delamination is in good agreement with benchmarked mixed-mode bending experiments. It is further shown that accounting for all major dissipative mechanisms in the modelling of delamination is the key to the accurate prediction of both resistance and damage of the interface.

Introduction

In fibre-reinforced polymer composites, the interfacial region between plies of different orientations is particularly vulnerable to damage, due to the increased likelihood of voids and defects. Because of the geometrical constraint of stiff fibres in adjacent plies, delaminating cracks are constrained to evolve within the resin-rich interface which typically has very little resistance to fracture growth. Consequently, this mode of failure can easily lead to catastrophic loss of the structural integrity of a laminated composite component. Therefore, it is crucial, in the modelling of delamination, to accurately predict both the resistance and damage induced delamination of the interfacial region.

Experimental evidence [1], [2] shows that the nature of delamination, including the associated energy loss, is strongly dependent on the fracture properties of the polymeric resin. Thermoset epoxies, preferred in the early days due to their superior manufacturing and in-service properties, are gradually being replaced by thermoplastic or particulate-enhanced resins, which offer higher fracture toughness thanks to substantial plastic deformation. Large permanent crack openings, a testimony of frictional/plastic dissipation at the crack faces, have been witnessed in experiments presented by Fan [3] on mixed-mode bending (MMB) tests and Rikards et al. [4] on double-cantilever beam (DCB) tests on composite laminates with fibre surface treatment. Not only is the plastic/frictional dissipation concomitant to the actual damage process, as noted by Carlsson et al. [5] for the End-Notched Flexure (ENF) test, but it may also exist a posteriori, should the newly created crack surfaces happen to come into contact. The need to consider the irreversible permanent deformation provides a rationale for the development of an improved interface constitutive model that is capable of handling both (i) mixed-mode loading conditions, and (ii) the coupling between debonding and plastic dissipation or friction that is witnessed experimentally.

In the literature, cohesive zone models have been widely used for the prediction of interfacial behaviour of laminated composites [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. The combination of both strength and fracture criteria in these models give them advantages over models based on linear elastic fracture mechanics or virtual crack closure technique (e.g. [6], [7]). The issue of mode interaction under mixed-mode loading has been treated with such strategies as a fully coupled interface stiffness matrix [12] or equivalent mixed-mode interface separation [13]. Furthermore, to handle the variable mode ratio that occurs in realistic loading situations, Turon et al. [14] proposed a strategy akin to switching between constitutive laws that are computed, a priori, under conditions of fixed-mode mixity. Although having gained in popularity to the extent of being considered as “standard models” for composite delamination, most existing cohesive models in the literature are based on damage theory and do not take into account the plastic/frictional dissipation. This treatment is appropriate only if this dissipative mechanism is negligible compared to the release of energy due to the creation of new surface areas, which is not always the case. It has been recently shown [8] that overlooking the plastic/frictional dissipation, even in pure mode I, leads to inaccurate prediction of the damage state, even when the structure’s resistance can be well predicted. It is therefore rational to question the capability of damage-based cohesive models in predicting interfacial damage.

In the context of delamination modelling, there are not many cohesive models possessing coupling behaviour between damage and plasticity. In fact, Schipperen and De Borst [30] suggested, by comparing a pure damage model to two coupled damage/plasticity models (isotropic and orthotropic hardening), that pure damage models are computationally more efficient and just as accurate in terms of predicting the extent of delamination. It will be shown that, because friction and fracture are both dissipative mechanisms contributing to the interface damage process, a numerical model for delamination can only have true predictive capabilities if those physical mechanisms are adequately accounted for. For instance, Tvergaard and Hutchinson [32] proposed a traction separation law where the post-peak regime consisted of a region of pure plastic yielding followed by softening. More recently, Kolluri [22] followed the same principle by introducing a parameter, called the plastic limit, defining the boundary between pure plastic dissipation and pure damage dissipation. Su et al. [31] presented a formulation with separate yield functions for the normal and shear failure modes and a mixed-mode coupling parameter linking only the two plastic dissipations but resulting in a non-smooth yield surface that could cause numerical difficulties. The reader is also referred to Scheider [21] for a more extensive review of the literature on cohesive models. Existing coupled damage/plasticity models primarily aim to reproduce a particular form of the traction versus separation response that may be observed experimentally. Hence, the most common strategies are either to make use of phenomenological functions to fit the interface constitutive response, or enhancing classical plasticity models with stiffness reduction to capture the unloading response. Although these approaches (e.g. in [22]) are able to successfully produce the observed permanent deformation and stiffness reduction, the coupling parameter is usually not easy to identify, let alone calibrate. This is because the connection between the dissipative mechanisms represented by damage and plasticity remains unclear in those coupled models.

In this study, the development of cohesive zone models is approached from a different angle underpinned by physically based concepts and fundamentals. The formulation of a new class of cohesive models is put in a thermodynamic framework featuring strong coupling between different dissipative mechanisms [9]. The resulting constitutive models are thermodynamically consistent and possess contributions from damage and plastic/frictional mechanisms. Emphasis is put on establishing links between the coupled dissipative processes and a measurable quantity: the ratio between the damage dissipation and the total fracture energy. The mode interaction is adequately dealt with, thanks to the use of an explicitly defined dissipation potential. The proposed approach allows different experimentally observed strength and fracture criteria to be incorporated to guide the model predictive capability in mixed mode loading conditions.

The paper is organised as follows. First a general framework for the development of cohesive models featuring coupling between normal and shear modes, as well as between damage and plasticity, is presented. A cohesive model is derived from the general formulation using typical strength and fracture criteria in the literature. Its behaviour is then assessed in pure and variable mode loading conditions. The parameter identification shows that it is able to calibrate all parameters from experiments. This model was implemented in the finite element package ABAQUS/Explicit [23] and used for the prediction of failure in mixed-mode delamination benchmarks available in the literature. The importance of accounting for all major dissipative mechanisms in the constitutive modelling and its consequence in the correct prediction of both resistance and impact-induced delamination is addressed in the simulation of an impact test.

Section snippets

A thermo-mechanical formulation

The thermodynamic formulation is based on earlier developments [8], [9], where further details can be found. The following expression for the Helmholtz energy potential is proposed:Ψ=12(1-D)Tnune2+Tsuse2+12DTn-une2where D is a scalar variable representing the interface damage; u is the interfacial separation, connected to the elastic une and permanent unp parts through the incremental relationship: δu = δue + δup; Tn and Ts are the initial normal and shear stiffnesses of the interface,

Dissipative properties and parameter identification

The dissipative properties of the proposed formulation in both pure and mixed mode conditions are directly obtained from the above formulation, without having to integrate the constitutive equations:Φ=ΦD+Φnp+Φsp=χδD+tnδunp+tsδuspwhere ΦD, Φnp and Φsp are the (rates of) dissipations due to damage, and plastic/frictional deformations in the normal and shear directions, respectively. Using the flow rules (13), (14), (15) and yield condition (16), the total dissipation can be expressed in terms of

Model behaviour

As mentioned, the model has its own strength criterion given by the yield function. This is obtained by substituting Eqs. (5), (6) into the yield function (16):12F(k,D)(1-D)2tn2Tn+ts2Ts-1=0,tn>012F(k,D)(1-D)2ts2Ts-1=0,tn0

This loading surface in stress space, obtained from the above calibration procedure with the Benzeggagh–Kenane (BK) fracture criterion [26](cf. Table 2) and shown in Fig. 2, “shrinks” towards the origin as the value of the damage variable increases. This is consistent with the

Numerical implementation

This model was implemented as a user-defined interface element in the explicit finite element code ABAQUS/explicit [23]. The topology of such elements, which can be of zero or finite thickness, is presented in several papers [14], [15], [19]. Only the stress-return algorithm is described here.

  • Step 1:

    Stress predictor For a typical displacement-controlled loading, the predictor (upper prefix p) for elastic separations and tractions are obtained as:tnt+Δtp=ttn+(1-D)TnΔun,ttn>0ttn+TnΔun,ttn0tst+Δtp=ts12t+

Mixed-mode delamination benchmarks

The model was utilised in pure damage mode (cos2α = 1) for the simulation of mixed-mode delamination experiments performed by Reeders and Crews [27] on AS4/PEEK unidirectional laminates, and presented in Camanho et al. [19], to which reference is made here for the details of material properties and specimen geometry. There are three mixed-mode bending (MMB) tests with mode ratios k = 0.2; 0.5; 0.8, as well as a double cantilever beam (DCB) and End-Notched Flexure (ENF) tests corresponding,

Conclusion

A thermodynamically-consistent approach with a strong coupling between damage and plasticity was proposed for the analysis of mixed-mode delamination in composite laminates. The formulation is capable of accurately predicting the initiation and propagation of mixed-mode delamination. Two significant contributions are:

  • 1.

    The proposed coupling between the energy dissipation resulting from the creation of new surfaces (damage) and the plastic/frictional dissipation is directly calibrated from

Acknowledgements

The authors wish to thank A/Prof Itai Einav for insightful discussions. This research was supported by the Australian Research Council (project number DP1093485). Support from the University of Sydney Postdoctoral Fellowship scheme to the second author is also gratefully acknowledged.

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