Numerical validation of a crack equivalent method for mixed-mode I + II fracture characterization of bonded joints

https://doi.org/10.1016/j.engfracmech.2013.05.008Get rights and content

Highlights

  • New data reduction scheme proposed for the analysis of Spelt apparatus testing.

  • Compliance based beam method (CBBM) applied to the Spelt Jig.

  • Mixed-mode I + II fracture envelope assessment.

  • Fracture process zone characterization for the Spelt test.

  • Fracture energy release rate obtained from the load–displacement data with an equivalent crack length.

Abstract

The present work is dedicated to development of a crack equivalent data reduction scheme applied to the load jig previously developed by Fernlund and Spelt [1] in order to characterize fracture of bonded joints under mixed-mode I + II loading. The jig allows for easy alteration of the mode-mixity and permits covering the full range of mixed-mode I + II combinations. A data reduction scheme based on specimen compliance, beam theory and crack equivalent concept is proposed to overcome several difficulties inherent to the test analysis. The method assumes that the performed test can be viewed as a combination of the double cantilever beam and asymmetrically loaded end-notched flexure tests, which provide modes I and II fracture characterization, respectively. A numerical analysis including a cohesive mixed-mode I + II damage model was performed considering different mixed-mode loading conditions to validate the proposed data reduction scheme. Issues regarding self-similar crack growth and fracture process zone development are discussed. It was verified that the considered in-plane mix mode fracture criterion is well captured using the proposed data reduction scheme.

Introduction

Bonded joints are being increasingly applied in structures involving risk, as is the case of the aeronautical, automotive, and civil infrastructure industries. The classical strength prediction based on stress or strain analysis may not be adequate in the presence of singularities which occur frequently in bonded joints. As a result, the development of sophisticated design criteria including progressive damage analysis is of fundamental importance. In this context cohesive zone modeling that combine stress-based criteria to simulate damage initiation and fracture mechanics criteria to deal with damage growth acquires special relevancy [1], [2], [3], [4], [5]. Fracture mechanics-based criteria require prior characterization of the joint under mixed-mode loading, since bonded joints in real applications often experience such situations. In fact, a crack or debond within an adhesive bond is usually obliged to propagate in a pre-defined plane (thin adhesive layer), independent of the general loading, which induces mixed-mode loading conditions. Consequently, the development of expedited procedures to perform mixed-mode I + II fracture characterization of bonded joints becomes a fundamental issue.

Several tests proposed in the literature can be applied to fracture characterization of bonded joints under mixed-mode I + II loading. Some of these are limited in the range of possible variation of mode mixity, which means that a complete description of the fracture envelope under mixed-mode I + II loading is not possible in such a configuration. This is the case of the asymmetric double cantilever beam (ADCB), the single leg bending (SLB) and the cracked lap shear (CLS) [6]. Nevertheless, there are alternatives that overcome this drawback. This is the case of the mixed-mode bending (MMB) test [7], which can be viewed as a combination of the double cantilever beam (DCB) and end-notched flexure (ENF) tests frequently used for fracture characterization under pure mode I and II loading, respectively. This test allows a large range of mode mixities and an easy alteration of the mode mixity by changing the lever length of the loading arm. However, a special apparatus with considerable dimensions is required, especially for fracture characterization of bonded joints with stiff adherends [8], [9]. Sørensen et al. [10] proposed the DCB specimen loaded by bending moments at the two free beams by means of a special device specially conceived. The mode mixity of the applied loading can be varied altering the ratio between the two applied moments. Högberg and Stigh [11] proposed the mixed mode double cantilever beam specimen based on the geometry of a semi-infinite symmetric DCB specimen. The specimen is loaded by a pair of self-balancing forces whose orientation can vary to alter the mode mixity. Singh et al. [12], have proposed the dual actuator load (DAL) method, which can be viewed as a DCB test subjected to non-symmetric loading. Two independent hydraulic actuators load the arms of a standard DCB specimen clamped at the other extremity. This test allows easy variation of the mode mixity by applying different displacement rates or loads to the specimen arms by means of the two independent hydraulic actuators. Fernlund and Spelt [13] proposed a special jig which allows mixed-mode fracture testing of adhesive joints and composite laminates over the entire range of mode mix using a standard DCB specimen. The authors used elementary beam theory as a data reduction scheme, which requires monitoring the crack extension during propagation. This task is not easy to be accomplished with the required accuracy, especially in cases when mode II loading predominates, which does not occur in mode I predominant tests, where crack tip is opened, thus facilitating the identification of its tip.

The objective of this work is to propose a simple and expedited data reduction scheme for the test developed by Fernlund and Spelt [13]. The method is based on specimen compliance, beam theory and crack equivalent concept and is proposed to overcome some difficulties inherent to the test analysis, namely crack length monitoring during its growth. The model is validated numerically by means of a detailed numerical analysis using cohesive mixed-mode I + II zone modeling. The numerical model is used to simulate damage initiation and growth for several different combinations of mode-mixity and the results treated through the proposed data reduction scheme. Some aspects related to self-similar crack growth and fracture process zone development are discussed. The resulting fracture envelope is compared with the input mixed-mode I + II fracture criterion.

Section snippets

Loading jig

The loading jig developed by Fernlund and Spelt [13] consists primarily of two rigid beams linked to each other, to the specimen, and to a base plate (Fig. 1). Different jig geometries can be achieved by altering the four distances, s1s4, thereby varying the mode-mixity of the induced loading. Changing the above referred distances leads to different loads, F1 and F2, applied to the upper and lower adherends, respectively, of the tested specimens (Fig. 2). The jig also permits the realization

Compliance based beam method

The classical data reduction schemes based on compliance calibration and beam theories are based on crack length monitoring during its propagation. However, there are two limitations related to this aspect. In fact, this task is not easy to be accomplished with the required accuracy namely in cases where mode II loading predominate, since the crack faces remain in contact during its propagation. The second limitation is related to the energy dissipation at the fracture process zone (FPZ) ahead

Numerical analysis

In order to verify the performance of the proposed method as well as the ability of the jig to provide a rigorous characterization of the fracture envelope under mixed-mode I + II of bonded joints, a numerical analysis including cohesive zone modeling was performed. The specimen geometry and mechanical properties used in the simulations are presented in Fig. 2 and Table 1, respectively. Fig. 4 presents the considered mesh and corresponding boundary conditions. The loading device was simulated by

Results and discussion

In order to validate the proposed data reduction scheme when applied to the Spelt device seven scenarios were considered in the GIGII space, including the pure mode cases (Table 2). The mode mixity between different scenarios was changed by altering the distances s1s4 (Fig. 1).

Fig. 6 represents the deformed shape and the corresponding load–displacement and R-curves for three different cases representing predominantly mode I (GI/GII = 3.6) and mode II (GI/GII = 0.24) and a balanced mixed-mode

Conclusions

The objective of this work is to propose a suitable data reduction scheme applied to the test developed by Fernlund and Spelt [13] in order to characterize fracture of bonded joints under different mixed-mode I + II loading conditions. The method is based on specimen compliance, beam theory and crack equivalent concept. It provides a simple mode partitioning method and does not require crack length monitoring during the test, which is advantageous since measurement errors can originate incorrect

Acknowledgements

The authors would like to thank the “Fundação Luso-Americana para o Desenvolvimento” (FLAD) for the support through project 314/06, 2007 and Instituto de Engenharia Mecânica (IDMEC).

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http://paginas.fe.up.pt/idmec.

2

http://www.fe.up.pt

3

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