On the determination of crack initiation directions under mixed mode loading
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On the determination of crack initiation directions under mixed mode loading Goswami, S. 1 ,M¨ uller, A. *1 , Hohe, J. 2 , and Becker, W. 3 1 Universit¨ at Siegen, IMR, Campus Paul-Bonatz-Straße, 57068 Siegen, Germany 2 Fraunhofer-Institut f¨ ur Werkstoffmechanik (IWM), 79108 Freiburg, Germany 3 Technische Universit¨ at Darmstadt, Fachbereich Mechanik, 64289 Darmstadt, Germany An important issue in the mechanics of multilayered structures is the knowledge of local stress fields. Light-weight hetero- geneous structures with crack- or notch-like discontinuities possess great danger of stress concentration and possible crack initiation leading to failure. In the present study, computational procedure, namely the finite element method, has been used to analyze bimaterial medium with various notch opening angles under arbitrary loading conditions. The prediction of the direction of crack initiation is important for wedge type constructions to gather a first hand knowledge of a potential damage zones which may undermine the integrity of the structure as a whole. The application of computational procedure to predict potential directions of crack initiation gives the flexibility required for different structural configurations, lamina orientations and incorporation of various boundary conditions. Superposition of various external loads to simulate pure mode I, mode II and mixed mode cases can also be carried out with excellent results in computational procedure. All the analysis results are based on the hypothesis of Erdogan and Sih. 1 Introduction The advent of light-weight and high-strength materials opened the door for many recent innovative applications. The highly improved manufacturing process has greatly contributed to the extensive application of multi-layered materials in primary components of lightweight structures. Large multi-component structures, i. e. aircrafts, automobiles, satellite bodies etc., require a number of joints and connections of different shape and material. Sharp corners and joints are the weakest points of the structures because of high stress concentrations, see [1] or [2], and the possibility of crack nucleation, if the strength of the joint or the interface is not sufficient. A bimaterial joint is a great challenge because of its frequency in practice. The difference of material properties for two-component bodies and a possible material discontinuity may initiate cracks which may lead to debonding inside the structure. From the structural integrity point of view, computational tools are essential for reliable prediction of damage initiation in the form of cracks which is based on the determination of circumferential and shear stress in the vicinity of the wedge or crack tip. The finite element method has been employed to analyze all the different structural configurations. The results have been compared to other available solutions. The computation of crack initiation direction is based on the criterion of Erdogan and Sih [3], knowing that e. g. Grenestedt et al. [4] describe further methods. 2 Determination of directions for crack initiation Fig. 1 Model and presumed arbitrary mixed mode loading. At a bimaterial wedge under arbitrary loading, a crack may be initiated not only at the interface but also at a certain angle ϕ 0 to the interface. The identification of these angles ϕ 0 is based on the hypothesis of Erdogan and Sih [3]. As shown in fig. 1, * Corresponding author: e-mail: [email protected], Phone: +49.271.740.2225, Fax: +49.271.740.2461 PAMM · Proc. Appl. Math. Mech. 4, 280–281 (2004) / DOI 10.1002/pamm.200410121 © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Transcript of On the determination of crack initiation directions under mixed mode loading
On the determination of crack initiation directions under mixed modeloading
Goswami, S.1, Muller, A.∗1, Hohe, J.2, andBecker, W.31 Universitat Siegen, IMR, Campus Paul-Bonatz-Straße, 57068 Siegen, Germany2 Fraunhofer-Institut fur Werkstoffmechanik (IWM), 79108 Freiburg, Germany3 Technische Universitat Darmstadt, Fachbereich Mechanik, 64289 Darmstadt, Germany
An important issue in the mechanics of multilayered structures is the knowledge of local stress fields. Light-weight hetero-geneous structures with crack- or notch-like discontinuities possess great danger of stress concentration and possible crackinitiation leading to failure. In the present study, computational procedure, namely the finite element method, has been usedto analyze bimaterial medium with various notch opening angles under arbitrary loading conditions. The prediction of thedirection of crack initiation is important for wedge type constructions to gather a first hand knowledge of a potential damagezones which may undermine the integrity of the structure as a whole. The application of computational procedure to predictpotential directions of crack initiation gives the flexibility required for different structural configurations, lamina orientationsand incorporation of various boundary conditions. Superposition of various external loads to simulate pure mode I, mode IIand mixed mode cases can also be carried out with excellent results in computational procedure. All the analysis results arebased on the hypothesis of Erdogan and Sih.
1 Introduction
The advent of light-weight and high-strength materials opened the door for many recent innovative applications. The highlyimproved manufacturing process has greatly contributed to the extensive application of multi-layered materials in primarycomponents of lightweight structures. Large multi-component structures, i. e. aircrafts, automobiles, satellite bodies etc.,require a number of joints and connections of different shape and material. Sharp corners and joints are the weakest pointsof the structures because of high stress concentrations, see [1] or [2], and the possibility of crack nucleation, if the strengthof the joint or the interface is not sufficient. A bimaterial joint is a great challenge because of its frequency in practice. Thedifference of material properties for two-component bodies and a possible material discontinuity may initiate cracks whichmay lead to debonding inside the structure. From the structural integrity point of view, computational tools are essential forreliable prediction of damage initiation in the form of cracks which is based on the determination of circumferential and shearstress in the vicinity of the wedge or crack tip. The finite element method has been employed to analyze all the differentstructural configurations. The results have been compared to other available solutions. The computation of crack initiationdirection is based on the criterion of Erdogan and Sih [3], knowing that e. g. Grenestedt et al. [4] describe further methods.
2 Determination of directions for crack initiation
Fig. 1 Model and presumed arbitrary mixed mode loading.
At a bimaterial wedge under arbitrary loading, a crack may be initiated not only at the interface but also at a certain angleϕ0
to the interface. The identification of these anglesϕ0 is based on the hypothesis of Erdogan and Sih [3]. As shown in fig. 1,
∗ Corresponding author: e-mail:[email protected], Phone: +49.271.740.2225, Fax: +49.271.740.2461
PAMM · Proc. Appl. Math. Mech. 4, 280–281 (2004) / DOI 10.1002/pamm.200410121
© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
a rectangular plane model with a notch-like opening, and the crack tip at the the centre of singularityS, has been analyzedby the finite element method and the boundary finite element method, see [5]. The lower and the upper edge of the modelhave been kept straight and of equal length. No translations of the bottom boundary of the structure are allowed while the topedge nodes are tied to each other with a single master pointP ? by multi-point constraints. The load components applied atthat point are noted asFx andFy. A load factorβ characterizing situations of arbitrary mixed mode loading is introduced as:β = Fx/(Fx + Fy). In that sense, pure opening mode I corresponds toβ = 0, and pure sliding mode II toβ = 1. Thus, anyvalue between−1 and+1 represents a mixed mode case of the applied loading. The range of the predicted directions of crackpropagation under various mixed mode situations is depicted in fig. 2. For the homogeneous wedge of linear-elastic material,one conclusion can be drawn directly, namely that under pure mode I loading, crack initiation starts at the interface and underpure mode II loading the well-known characteristic angleϕ0 = −70.5 is calculated. The graph on the right side illustrates thedirections of crack initiation for unsymmetric bimaterial configurations.
Fig. 2 Directions of crack nucleation for different notches, homogeneous material (left) and unsymmetric notch opening [ϑI , ϑII ] in abimaterial withEI/EII = 0.4 (right).
3 Conclusions
The work concentrates on the application of computational tools for the determination of crack direction. Results from thefinite element analyses have been compared with other available solutions and found to be in good agreement. Using reliablenumerical means, it is possible to identify the crack initiation direction and the probable damage zone in a multi-materialstructural component. The analyses have been carried out for various material combinations and structural configurations(variation of the notch angles). By the superposition of various external loading conditions pure mode I, mode II and mixedmode situations have been simulated. Computational procedures have found to be flexible enough to apply for virtually anyloading, boundary, material and structural configurations and suit quite well for many real life situations where analytical toolsare intractable.
References
[1] M ULLER, A., HOHE, J.; BECKER, W.: Material- und Geometrieabhangigkeit der Spannungssingularitaten an Bimaterialkerben. In:Deutscher Verband fur Materialforschung und -prufung, Bericht 234, pp. 109-118, 2002.
[2] YANG, Y.: Spannungssingularitaten in Zweistoffverbunden bei mechanischer und thermischer Belastung. VDI-Verlag GmbH,Dusseldorf, 1992.
[3] ERDOGAN, F., SIH , G. C.: On the crack extension in plates under plane loading and transverse shear. In: Journal of Basic Engineering,December, pp. 519-525, 1963.
[4] GRENESTEDT, J. L., HALLSTROM, S.: Crack initiation from homogeneous and bimaterial corners. In: Journal of Applied Mechanics,Vol. 64, pp. 811-818, 1999.
[5] M ULLER, A., HOHE, J., GOSWAMI, S.AND BECKER, W.: Thermal effects on fracture initiation at bimaterial notches. In: L. Librescuand P. Marzocca, editors. Proceedings of the5th International Congress on Thermal Stresses and Related Topics· TS2003, VirginiaPolytechnic Institute and State University, Blacksburg, Virginia, USA, Vol. 1, pp. MA-7-4-1 – MA-7-4-4, 2003.
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