A consistent finite displacement and rotation formulation of the Linear Elastic Brittle Interface Model for triggering interlaminar damage in fiber-reinforced composites
Introduction
The investigation on interface modelling techniques has experienced a tremendous increase in the recent years. This interest arises from the fact that interfaces are present in many practical applications, and they can be identified as critical regions that are especially prone to the onset and progression of failure events in engineering structures [1], [2]. Thus, damage events at interfaces can be found in many engineering materials under different loading conditions and also at different scales of observation, comprising micromechanics of fiber-reinforced composites [3], layered ceramics [4], adhesive joints in structural components [5] or bio-mimicking elements [6], [7].
Inspired by such applications, a proper understanding of the damage process [8], crack onset [9], fracture energy dissipation and other failure features along interfaces can be understood as a key aspect in order to produce substantial improvements on innovative and firmly established designs of engineering products. Particularly, within the context of fibrous-reinforced composites, for a complete comprehension of the mechanical behaviour of interfaces, several scenarios involving different failure mechanisms have been thoroughly investigated so far implying fiber–matrix debonding [10], delamination [11] between layers, among many other cases. Although these problems have been examined from different standpoints [12], [13], one of the principal trends for their numerical treatments concerns the employment of cohesive zone models (CZMs), whose underlying hypotheses were originally introduced in the 60s of the previous century by Barenblatt [14] and Dugdale [15]. In brief, CZMs comply with the assumption through which the singular stress field at the crack tip according to Linear Elastic Fracture Mechanics (LEFM) can be removed and substituted by a specific relation stating that: the tractions around the crack tip, , can be expressed as a nonlinear function of the displacement jumps, , and the softening evolution process upon complete failure at this particular location can be characterized by a damage variable (irreversible state variable), d, ahead of the crack tip, according to
This general assumption implies the existence of the well-known fracture process zone (FPZ), whereby the stress field is generally governed by a nonlinear traction-separation law (TSL). Accordingly, the energy release rate G can be computed as the work developed by the tractions, , and displacements, , at the FPZ, and, it can be compared with the fracture toughness for triggering crack progression. This material property, , is specifically inserted into the corresponding CZMs as the area under the particular choice of the TSL.
Owing to its versatility in terms of numerical implementation and applicability, CZMs have been employed for investigations involving a wide variety of material types (covering ductile, quasi-brittle failure, hydrogen embrittlement or cycle-dependent behaviour [16]), and whose responses can be described via different nonlinear profiles of TSLs: bilinear law [17], exponential profile [18], trapezoidal law [19], among others. Moreover, in the last two decades, CZMs have been extensively employed for failure analysis of weak or imperfect interfaces and thin adhesives between solids whose stiffness is much higher than the stiffness associated with the interface. In this setting, the present investigation is focused on a limit case of a cohesive law devoted to the study of brittle interfaces, that corresponds to the so-called Linear Elastic-Brittle Interface Model (LEBIM). This model was introduced in [20] and significantly extended by the authors in [21], [22], [23], [11]. The perfectly brittle approach provides some advantages over other TSL profiles endowing: (i) the simple linear elastic behaviour prior complete and abrupt failure, (ii) the preclusion of the FPZ, since no progressive stiffness deterioration is accounted for in the LEBIM (which relaxes the discretization requirements in comparison with alternative CZMs with nonlinear behaviour), and, as a consequence, (iii) a notable simplification of the numerical implementation tasks. Furthermore, relying on the predictions of the investigations aforementioned, the LEBIM has evidenced high-level characteristics in terms of numerical robustness, simplicity and computational efficiency.
With focus on the numerical implementation of interface failure modelling recalling a non zero-thickness approach, in general terms, despite the existence of different TSLs in the related literature, the procedure to compute the stress and displacement field follows the same scheme regardless the particular profile of the TSL. Basically, the displacements at the interface are calculated according to a local coordinate system whose origin is located on the midplane of the interface, see Fig. 1 for 2D applications, and the particular orientation of the director vectors characterizes the displacement field contributions associated with the fracture modes II and I, respectively. Then, the displacements at the crack flanks can be decomposed in their corresponding normal, shear and in-plane counterparts, and , respectively, and they are used to determine the stress field via the TSL. Nonetheless, usually some assumptions are made to simplify the model and the constitutive equations in the CZM, as the preclusion of potential material distortions. For instance, in the commercial Finite Element (FE) code ABAQUS® [24], for 2D cases, the normal and shear relative displacements are the specific components that are considered within the cohesive element formulation (COH2D4 for 2D analysis) for the evaluation of the TSL, the in-plane deformation effects being neglected. That is, normal and shear tractions, and , and their conjugated displacements counterparts, and , are computed within this kind of element topology. Based on these aspects, considerations regarding the in-plane deformation at interfaces cannot be taken into account employing regular cohesive elements that are present in most of the commercial FE libraries. Differing from this, recent studies have confirmed the influence of the in-plane effects in the failure response of adhesives [25], [26], which might have a remarkable influence.
In addition to the previous considerations, there are situations in which rigid body translations and rotations become significant, and therefore the nonlinear Continuum Mechanics theory [27] should be taken as underlying modelling framework. In this regard, many of conventional cohesive elements in general purpose FE-codes include an appropriate formulation for geometrically nonlinear analysis, but many TSLs implemented by means of user material subroutine, e.g. UMAT in ABAQUS®, have been mostly developed considering infinitesimal strain theory, as occurred in [22], [11] for the LEBIM. Therefore, the application of the baseline LEBIM requires some modifications in order to consistently account for nonlinear effects via its integration as user-defined material capability (UMAT). Note that alternatively to this modelling option, the authors in [28], [8] proposed the development of interface elements for large deformation analysis for microstructures made of fibrils, whose implementation tasks required the formulation, derivations, and coding of the element kinematics.
Apart from the discrepancies in terms of formulations between small and large displacement theories, it is worth mentioning the different reference systems between general-purpose cohesive and continuum/solid element topologies in their respective local configurations. Thus, on the one hand, the cohesive basis is generally referred to the midplane of the interface, whereas, on the other hand, the solid coordinate system is related to the principal directions of strain (this aspect not being very comprehensively treated in the related literature). Hence, tractions and displacements in the continuum elements need to be expressed in the cohesive basis (located on the midplane of the interface) for a correct evaluation of the corresponding TSL.
In order to address the previous aspects herewith outlined, the main objective of this investigation is the development of a computational procedure that enables the robust determination of the relative displacement field (with potential inclusion of in-plane deformation effects) that is required for the computation of a particular TSL for interface failure modelling using a continuum-like approach (i.e. non-zero thickness interface model) under large displacement hypotheses. In other words, the principal aim is to overcome the previously listed issues associated with the existence of finite rigid body motion through the development of a formulation suitable for its implementation into general purpose FE packages as user material subroutine (avoiding the coding of the complete element kinematics), allowing normal, shear and in-plane relative displacements referred to a coordinate system located at the interface midplane to be computed. In particular, this innovative method for determining cohesive-like displacements at interfaces is applied, without loss of generality, to the LEBIM following a nonlinear FE numerical scheme and examined through the prediction of a Horizontal Drum Peel (HDP) test response. Note, as will be recalled in the forthcoming developments, that the current procedure can be employed following two basic approaches, namely, either its incorporation in standard cohesive elements or in continuum/solid elements, the latter endowing in-plane deformation effects of the interface.
The manuscript is structured as follows. Section 2 describes the two basic procedures proposed herein for the reliable computation of cohesive relative displacements (relative to the midplane of the interface) in both cohesive and solid elements of ABAQUS®. Section 3 verifies the current methodology through its application of representative benchmark problems. The employment of the developed technique using LEBIM at the interface in the FE simulation of a Horizontal Drum Peel test is shown in Section 4. Finally, Section 5 highlights the convenience of the proposed procedure and summarizes the fundamental contributions of this research.
Section snippets
Finite displacement formulation for LEBIM with small finite thickness
This Section presents the basic aspects of the two approaches developed herein for the integration of the LEBIM into built-in elements of ABAQUS® for large displacements applications. Specifically, we follow a continuum-like strategy for the interface conception, i.e. an initial small thickness of the interface is assumed, and therefore the LEBIM can be integrated into this general purpose package via the user-defined material routine UMAT. This notably simplifies the required implementation
Validation of the interface model under large displacement scenarios: benchmark problems in one-element configurations
This Section presents a set of benchmark cases in order to verify the accuracy of the current methodology in the displacement calculation in a single interface element. Some deformed configurations within a single element with prescribed displacements are shown. These examples include the basic strain state that an interface finite element could experiment, that is, normal, shear, in-plane deformations and rigid body motions.
For the sake of clarity, the deformations selected develop a uniform
Application of the novel interface model: Horizontal Drum-Peel test
The suitability of the previous formulation is discussed along this Section. In particular, the novel relative displacement calculation is tested in an engineering application regarding the quality of bonded joints: the Horizontal Drum Peel (HDP). In practice, the fracture toughness or critical energy release rate is a fracture property needed in the design process of many structural elements and the development of numerical tools are valuable for the knowledge of the testing procedure. In
Conclusions
In this paper, a novel consistent procedure to compute the relative displacements needed in a traction-separation law (TSL) suitable for Finite Element codes has been developed, in the framework of large displacements hypotheses and applied into the software ABAQUS®. Two different methods have been carried out in this 2D analysis: (i) an approach based upon cohesive element technology and (ii) an approach founded on nonlinear continuum mechanics. In the former case normal and shear relative
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This study was supported by the Spanish Ministry of Science, Innovation and Universities and European Regional Development Fund (Project PGC2018-099197-B-I00), the Consejería de Economía y Conocimiento of the Junta de Andalucía (Spain) for financial support under the contract US-1265577 and US-1266016-Programa Operativo FEDER Andalucía 2014–2020 and AT17-5908-USE (Acciones de transferencia del conocimiento).
References (51)
- et al.
Microscopical observations of interface cracks from inter-fibre failure under compression in composite laminates
Compos. Part A: Appl. Sci. Manuf.
(2018) - et al.
Crack deflection at an interface between dissimilar elastic materials
Int. J. Solids Struct.
(1989) - et al.
On the mechanics of mother-of-pearl: a key feature in the material hierarchical structure
J. Mech. Phys. Solids
(2007) - et al.
A consistent anisotropic damage model for laminated fiber-reinforced composites using the 3d-version of the puck failure criterion
Int. J. Solids Struct.
(2017) - et al.
Crack paths formed by multiple debonds in lfrp composites
Mech. Res. Commun.
(2017) - et al.
Effects of the presence of compression in transverse cyclic loading on fibre–matrix debonding in unidirectional composite plies
Compos. Part A: Appl. Sci. Manuf.
(2007) - et al.
On the 3D extension of failure models for adhesive joints under mixed-mode fracture conditions: LEBIM and CZM
Theoret. Appl. Fract. Mech.
(2019) The mathematical theory of equilibrium cracks in brittle fracture
Adv. Appl. Mech.
(1962)Yielding of steel sheets containing slits
J. Mech. Phys. Solids
(1960)- et al.
A cohesive zone framework for environmentally assisted fatigue
Eng. Fract. Mech.
(2017)