Abstract
In this paper, a semi-analytic solution of the problem associated with an elliptic inclusion embedded within an infinite matrix is developed for plane strain deformations. The bonding at the inclusion-matrix interface is assumed to be homogeneously imperfect. The interface is modeled as a spring (interphase) layer with vanishing thickness. The behavior of this interphase layer is based on the assumption that tractions are continuous but displacements are discontinuous across the interface.
Complex variable techniques are used to obtain infinite series representations of the stresses which, when evaluated numerically, demonstrate how the peak stress along the inclusion-matrix interface and the average stress inside the inclusion vary with the aspect ratio of the inclusion and a representative parameter h (related to the two interface parameters describing the imperfect interface in two-dimensional elasticity) characterizing the imperfect interface. In addition, and perhaps most significantly, for different aspect ratios of the elliptic inclusion, we identify a specific value (h *) of the (representative) interface parameter h which corresponds to maximum peak stress along the inclusion-matrix interface. Similarly, for each aspect ratio, we identify a specific value of h (also referred to as h * in the paper) which corresponds to maximum peak strain energy density along the interface, as defined by Achenbach and Zhu (1990). In each case, we plot the relationship between the new parameter h *and the aspect ratio of the ellipse. This gives significant and valuable information regarding the failure of the interface using two established failure criteria.
Similar content being viewed by others
REFERENCES
Aboudi, J., Damage in composites-modeling of imperfect bonding. Compos. Sci. Techn. 28 (1987) 103–128.
Achenbach, J.D. and H. Zhu, Effect of interfacial zone on mechanical behaviour and failure of hexagonal-array fibre composites. J. Mech. Phys. Solids 37 (1989) 381–393.
Achenbach, J.D. and H. Zhu, Effect of interphase on micro and macromechanical behaviour and failure of fibre-reinforced composites. J. Appl. Mech. 57 (1990) 956–963.
Benveniste, Y., On the efffect of bonding on the overall behaviour of composite materials. Mech. Mater. 3 (1984) 349–358.
England, A.H., Complex Variable Methods in Elasticity. Wiley-Interscience, London (1971).
Eshelby, J.D., The determination of the elastic field of an ellipsiodal inclusion and related problems. Proc. Roy. Soc. London A 241 (1957) 376–396.
Eshelby, J.D., The elastic field outside an ellipsiodal inclusion. Proc. Roy. Soc. London A 252 (1959) 561–569.
Gao, J., A circular inclusion with imperfect interface: Eshelby's tensor and related problems. J. Appl. Mech. 62 (1995) 860–866.
Gong, S.X. and S.A. Meguid, On the elastic fields of an elliptical inhomogeneity under plane deformation. Proc. Roy. Soc. London A 443 (1993) 457–471.
Hashin, Z., Thermoelastic properties of fiber composites with imperfect interface. Mech. Mater.. 8 (1990) 333–348.
Hashin, Z., Thermoelastic properties of particulate composites with imperfect interface. J. Mech. Phys. Solids 39 (1991a) 745–762.
Hashin, Z., The spherical inclusion with imperfect interface. J. Appl. Mech. 58 (1991b) 444–449.
Hashin, Z., Extremum principles for elastic heterogeneous media with imperfect interfaces and their application to bounding of effective moduli. J. Mech. Phys. Solids 40 (1992) 767–781.
Huang, Y. and K.X. Hu, A generalized self-consistent mechanics method for solids containing elliptical inclusion. J. Appl. Mech. 62 (1995) 566–572.
Jasiuk, I. and Y. Tong, The effect of interface on the elastic stiffness of composites. In: J.N. Reddy et al. (eds), Mechanics of Composite Materials and Structures, ASME AMD, Vol. 100 (1989) pp. 49–54.
Jasiuk, I. and M.W. Kouider, The effect of an inhomogeneous interphase on the elastic constants of transversely isotropic composites. Mech. Mater. 15 (1993) 53–63.
Muskhelishvili, N.I., Some Basic Problems of the Mathematical Theory of Elasticity. Noodhoff, Groningen (1963).
Pagano, N.J. and G.P. Tandon, Modeling of imperfect bonding in fibre reinforced brittle matrix. Mech. Mater. 9 (1990) 49–64.
Qu, J., Eshelby tensor for an elastic inclusion with slightly weakened interface. J. Appl. Mech. 60 (1993a) 1048–1050.
Qu, J., The effect of slightly weakened interface on the overall elastic properties of composite materials. Mech. Mater. 14 (1993b) 269–281.
Ru, C.Q. and P. Schiavone, A circular inclusion with circumferentially inhomogeneous interface in antiplane shear. Proc. Roy. Soc. London A 453 (1997) 2551–2572.
Ru, C.Q., A circular inclusion with circumferentially inhomogeneous sliding interface in plane elastostatics. J. Appl. Mech. 65 (1998) 30–38.
Shen, H., P. Schiavone, C.Q. Ru and A. Mioduchowski, An elliptic inclusion with imperfect interface in anti-plane shear. Internat. J. Solids Struct. 37 (2000) 4557–4575.
Stagni, L., Elastic field perturbation by an elliptic inhomogeneity with a sliding interface. J. Appl. Math. Phys. (ZAMP) 42 (1991) 811–819.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Shen, H., Schiavone, P., Ru, C. et al. Stress Analysis of an Elliptic Inclusion with Imperfect Interface in Plane Elasticity. Journal of Elasticity 62, 25–46 (2001). https://doi.org/10.1023/A:1010911813697
Issue Date:
DOI: https://doi.org/10.1023/A:1010911813697