A local-global matching method for the single fiber pullout problem with perfectly bonded interface

Abstract

This paper presents a local-global matching method to effectively determine the detail spatial structure and magnitude of the locally singular stress field as well as the complete global stress distribution in the single fiber pullout model. The motivation to solve for the local stress field is the belief that these accentuated stresses, strains and energy are likely to induce damage. The local-global matching method consists of three components: a local analysis, a global analysis, and proper matching of the local asymptotic field to the global complete stress field. The method as developed is applicable for various fiber-matrix interfacial conditions: namely perfectly bonded interface, partially debonded interface with interfacial crack, or debonded interface with frictional interfacial sliding. In this paper, results for perfectly bonded fiber-matrix interface are presented to illustrate key features of the local-global matching method. For this problem, the local stress field is asymptotically singular at the location where the fiber protrudes from the matrix. A local analysis at the fiber protrusion point reveals that, when the fiber is stiffer than the matrix, the most dominant singular stresses are of the from: $$\sigma _{ij} (\rho ,\theta ) = K\rho ^\lambda \tilde \sigma _{_{ij} } (\theta )$$ where the exponent λ is real-valued and λ<0. The local analysis can solve for the spatial structure of the local field: its radial dependence ρλ and angular variations $$\tilde \sigma _{ij} (\theta )$$ . The actual magnitude of the local stress field is scaled by the amplitude factor K which depends upon externally applied load and global boundary conditions. The global analysis, performed using a finite element model, can be subjected to arbitrary fiber-pulling load and/or thermal load. With solutions from both the local analysis and global analysis, a local-global matching method based on angular variation of stresses is developed to accurately determine K. In local-global matching, a proper region is selected in which the angular variation of stresses of the local field is scaled to match the angular variation of the finite-element computed full-field stresses. Several monitoring parameters are developed to measure the quality of the matching and to determine the region of dominance of the local asymptotic field. The local analysis shows that in many composite material systems, there are two singular terms in the local field: $$\sigma _{ij} K = \rho ^{\lambda _1 } \tilde \sigma _{ij}^1 (\lambda ;{\text{ }}\theta ) + K_2 \rho ^{\lambda 2} \tilde \sigma _{ij}^2 (\lambda ;{\text{ }}\theta ), - 1 < \lambda _1 < \lambda _2 < 0$$ . Hence, local-global matching procedures have been developed for both one-term field and two-term field. The matching method is further generalized to determine complex-valued K for composites having complex-valued λ. The local-global matching method may also be applied to problems with material nonlinearity.