Damage modelling strategies for unidirectional laminates subjected to impact using CZM and orthotropic plasticity law

Abstract

In the present paper, the crashworthiness of a thick unidirectional carbon fibre reinforced polymer is investigated. This material is manufactured via compression moulding process. A Low-Velocity Impact (LVI) test is implemented on a quasi-isotropic laminate for the experimental evaluation of the energy absorption due to impact, while the internal failure mechanism is detected using Computerized Tomography (CT). Two different Finite Element (FE) models are applied to model the damage onset and propagation: firstly, a shell-based model and, secondly, a solid-based model using Cohesive Zone Method (CZM). In the CZM, the analytical modelling of the cohesive element properties is adopted, and the critical force evaluated experimentally is related to the energy release rate of mode II, and to the equivalent elastic properties of the laminate. The strength and weakness of the proposed approach in mimicking the impact behaviour and the actual failure mechanism, are discussed, and validated versus numerical simulation. The models are in good agreement with the experimental results; in fact, the relative error of the maximum force is about 4 per cent, and it occurs in the shell-based model.

Introduction

The design of composite structures typically follows the Building Block Approach (BBA) [1], suitable for investigating the components at various levels of complexity. Using FE models, the above method aims to mimic the mechanical behaviour of complex structures starting from coupons under simple loading scenarios, and moving through tests of sub-components and components, up to the full-scale mechanical system [2], [3]. Since, the BBA is costly and time-consuming process the minimization of the experimental tests is pursued as well as the development of FE models has to comply with the accuracy of the failure loads and with a moderate computational effort.

In the framework of BBA, the experimental evaluation of Carbon-Fibre Reinforced Polymer (CFRP) crashworthiness and its prediction using FE analysis are applied. In fact, the impact tests are investigated at low- [4], high- [5], and ballistic velocity [6]. The LVI induces a minimal surface damage and internal extended failure mechanisms [7]; this consists of intralaminar damage (e.g. fibre kinking due to compression in the top part of impact sample and matrix cracking due to bending on the bottom ply), and interlaminar delamination among adjacent layers [8].

Analytical studies apply a fracture mechanics approach to predict the delamination threshold forces and damaged area due to impact on laminates [9], [10], [11], [12], [13], [14], [15]. In fact, Davies et al. have demonstrated that the delamination mechanism is driven by the interlaminar sliding shear failure mode (mode II) and by the associated energy release rate (G_IIC) [9], [10], [12]; Christoforou have derived the same formula in a normalized form [15], whereas other authors have included the number of delaminated interfaces for the evaluation of the threshold force [11], [13], [14]. The limitation of these methods lies in the hypothesis that the delamination area is equal at each interface among plies, which does not find the experimental counterpart. Nevertheless, in the present work, the analytical model developed in [9] has been adopted for the evaluation of the energy release rate of mode II (G_IIC) replacing peculiar experimental test.

Alternatively, FE models are powerful in simulating the impact response of composite laminates; however, their accuracy is affected by the material law as well as by the damage modelling technique [16]. The constitutive material law needs to mimic the plastic field, the failure onset (i.e. the intralaminar and interlaminar failures) and the crack propagation.

Including the plastic field of the material, some studies consider the in-plane-shear stress–strain by fitting the experimental data using a) an exponential law [17], b) a third-grade polynomial [18] or c) a quadratic yield surface [19]. However, the above models assume that the material is linear elastic in compression before damage. Since the laminate plastic response is driven by the matrix properties in compression, this assumption might lead to underestimate the loss energy due to impact, see for instance [20]. Therefore, concerning this aspect, an orthotropic plasticity law has been developed here, to accurately predict the permanent indentation and the energy loss induced by the non-linear response of matrix in compression.

The intralaminar damage onset has been analysed by the implementation of failure criteria. Li et al. compare the force–time and the force–displacement curves due to LVI via FE for the following [20] criteria: maximum stress, Hashin [21], Hou [22], Tsai-Wu [23], and Puck [24]. In [20], it has been demonstrated that the laminate mechanical response is well-predicted by those failure criteria; however, the Tsai-Wu criterion exhibits a higher permanent indentation since it does not consider separated damage modes.

Furthermore, the Continuum Damage Mechanics approach (CDM) is used for the intralaminar damage propagation modelling; in fact, the crack propagation is investigated at the macroscopic level defining peculiar damage variables which reduce the material mechanical properties [25], [26].

Regarding the FE models, the selection of the element spatial scale is crucial since it affects the capability of damage modelling and the computational effort. The FE models are tuned to investigate the material at various scale-levels, ranging from the macro (i.e. layered shell model), to the micro-scale, see Fig. 1 of [26]. A shell-based macro-scale model combines an accurate prediction of the intralaminar failure mechanism [27]. However, the models lack in the prediction of delamination.

Recently, the literature has been endowed with meso-scale models, this approach has been applied for the simulation of the LVI [28], [29], [30], [31], of the compression after impact test [32], [33], and of the repeated impact response [34], and also for the experimental–numerical characterization of hybrid aluminium CFRP laminate [35]. In such models, each ply is modelled by at least one solid element through its thickness, and the interface between the plies is mimicked by CZM [36], [37].

At the micro-scale level, Miao et al. [38] have investigated the impact strength of a 3D fabric structure by FE. These models are used to collect the average behaviour of the material which are adopted on macro-scale models.

On the other hand, accurate models based on global-to-local approach have been developed by [31], [39], [40], [41]. In the above works, the laminate is modelled with a fine, meso-scale solid mesh, located at the impact zone where the damage occurs, while a coarser macro-scale mesh is adopted in those areas of the laminate where damage does not occur.

For instance, the global-to-local approach is applied to simulate automotive and high-value aerospace components during crash events when the impact damage occurs only locally [26]. Unfortunately, an impact involves many components and extended areas, for instance the peculiarities of a car crash event are discussed in [42], [43], [44]. Thus, the adoption of solid elements is minimized to contain the computational effort of these FE analyses. Therefore, for a general approach, the global-to-local strategy is abandoned in the present work.

This paper reports a methodology to simulate damage evolution in composite structures subjected to impact in the early phase of the BBA. An experimental–numerical correlation of a thick UniDirectional (UD) CFRP laminate subjected by a LVI test is pursued.

Firstly, the crashworthiness and the elastic–plastic CFRP properties are detected during a LVI test according to the ASTM D7136 standard [45]. The internal damage is evaluated and discussed by means of non-destructive micro-Computerized Tomography (CT) analysis.

Secondly, two numerical strategies have been developed for the impact prediction of the UD lamina using a) a shell-based model and b) a solid-based model. The composite damage onset is modelled using a modified Tsai-Wu criterion which includes separated failure modes, a CDM approach are employed to simulate the damage propagation mechanism, whereas in the solid model, the delamination is mimicked using CZM.

A non-conventional approach has been proposed for the evaluation of the energy release rate associated with mode II (G_IIC) by means of analytical formulation, this model rationalize the delamination propagation of a quasi-isotropic laminate subjected to transverse load. The proposed analytical approach allows to reduce the number of mandatory tests for the BBA approach with respect to a more traditional one.

The FE forecasts are in good agreement with the experimental data in terms of internal damage, force, energy, and displacement versus time. In fact, the error on maximum impact force is 4.0 and −2.3 per cent for shell-based and solid-based model, respectively.