Availableonlineatwww.sciencedirect.com SCIENCE DIRECT JOURNAL OF THE IECHANICS AND Journal of the Mechanics and Physics of Solids HYSICS OF SOLIDS 54(2006)266-287 www.elsevier.comlocate/jmps The roles of toughness and cohesive strength crack deflection at interfaces J.P. Parmigiana, I. M.D. Thouless, b, Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA Department of Materials Science, Engineering, Unirersity of Michigan, Ann Arbor, MI 48109, US.A Received 23 February 2005: received in revised form I September 2005: accepted 6 September 2005 Abstract In order to design composites and laminated materials, it is necessary to understand the issues that govern crack deflection and crack penetration at interfaces. Historically, models of crack deflection have been developed using either a strength-based or an energy-based fracture criterion. However, in general, crack propagation depends on both strength and toughness. Therefore in this paper, crack deflection has been studied using a cohesive-zone model which incorporates both strength and toughness parameters simultaneously. Under appropriate limiting conditions, this model reproduces earlier results that were based on either strength or energy considerations alone. However, the general model reveals a number of interesting results. Of particular note is the apparent absence of any lower bound for the ratio of the substrate to interface toughness to guarantee crack penetration. It appears that, no matter how tough an interface is, crack deflection can always be induced if the strength of the interface is low enough compared to the strength of the substrate. This may be of significance for biological applications where brittle organic matrices can be bonded by relatively tough organic layers. Conversely, it appears that there is a lower bound for the ratio of the substrate strength to interfacial strength, below which penetration is guaranteed no matter how brittle the interface. Finally, it is noted that the effect of modulus mismatch on crack deflection is very sensitive to the mixed-mode failure criterion for the interface, particularly if the cracked layer is much stiffer than the substrate C)2005 Elsevier Ltd. All rights reserved Keywords: Crack deflection; Crack penetration; Interfacial fracture; Toughness; Cohesive strength Corresponding author. Tel. +1734 7635289: fax: +17346473170 E-mail address: thouless(@ umich.edu(M. D. Thouless). Current address: Department of Mechanical Engineering, Oregon State University, Corvallis, OR 97331 USA 022-5096/S.see front matter o 2005 Elsevier Ltd. All rights reserved doi:10.1016/ . mps.2005.09.002Journal of the Mechanics and Physics of Solids 54 (2006) 266–287 The roles of toughness and cohesive strength on crack deflection at interfaces J.P. Parmigiania,1, M.D. Thoulessa,b,
a Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA b Department of Materials Science, & Engineering, University of Michigan, Ann Arbor, MI 48109, USA Received 23 February 2005; received in revised form 1 September 2005; accepted 6 September 2005 Abstract In order to design composites and laminated materials, it is necessary to understand the issues that govern crack deflection and crack penetration at interfaces. Historically, models of crack deflection have been developed using either a strength-based or an energy-based fracture criterion. However, in general, crack propagation depends on both strength and toughness. Therefore, in this paper, crack deflection has been studied using a cohesive-zone model which incorporates both strength and toughness parameters simultaneously. Under appropriate limiting conditions, this model reproduces earlier results that were based on either strength or energy considerations alone. However, the general model reveals a number of interesting results. Of particular note is the apparent absence of any lower bound for the ratio of the substrate to interface toughness to guarantee crack penetration. It appears that, no matter how tough an interface is, crack deflection can always be induced if the strength of the interface is low enough compared to the strength of the substrate. This may be of significance for biological applications where brittle organic matrices can be bonded by relatively tough organic layers. Conversely, it appears that there is a lower bound for the ratio of the substrate strength to interfacial strength, below which penetration is guaranteed no matter how brittle the interface. Finally, it is noted that the effect of modulus mismatch on crack deflection is very sensitive to the mixed-mode failure criterion for the interface, particularly if the cracked layer is much stiffer than the substrate. r 2005 Elsevier Ltd. All rights reserved. Keywords: Crack deflection; Crack penetration; Interfacial fracture; Toughness; Cohesive strength ARTICLE IN PRESS www.elsevier.com/locate/jmps 0022-5096/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2005.09.002
Corresponding author. Tel.: +1 734 7635289; fax: +1 734 6473170. E-mail address: thouless@umich.edu (M.D. Thouless). 1 Current address: Department of Mechanical Engineering, Oregon State University, Corvallis, OR 97331, USA