Overall elastic properties of a plate containing inhomogeneities of irregular shape

Luca Lanzoni1, Enrico Radi2, Igor Sevostianov3
1DIEF, Università di Modena e Reggio Emilia, 41125 Modena, Italy
E-mail: luca.lanzoni@unimore.it
2DISMI, Università di Modena e Reggio Emilia, 42122 Reggio Emilia, Italy
E-mail: enrico.radi@unimore.it

3Mechanical & Aerospace Engineering, New Mexico State University, Las Cruces,
NM 88001, USA. 
E-mail: igor@nmsu.edu

Keywords:  stiffness contribution tensor; stress functions; bipolar coordinates.

The present works deals with the stiffness properties of an infinite 2D isotropic elastic system containing inhomogeneities having a contour formed by union of two overlapping circles of generally different radii. Starting from this general layout, the cases of a matrix with lenticular, perfectly circular, semi-circular, “C-shaped” and thin straight inclusions can be obtained as limit cases. Owing to the geometry of the system, reference is made to bipolar cylindrical coordinates (a, b), which are linked to the Cartesian ones (x1, x2) through the conformal map [2]:
			             (1)
The effective elastic properties of the system is analytically investigated by introducing a fourth-order compliance contribution tensor H, which represents the effect induced by the inhomogeneity on the compliance of the system according to [1]
eij = Sijklskl + Hijklskl,								            (2)
being S the compliance tensor for the homogeneous elastic matrix and s the stress field. It is remarked that the last term in eq (1) denotes the correction De acting on the strain field owing to the presence of the inclusions. 
 The system without inhomogeneities and subjected to a remote stress field is considered first. The corresponding fundamental stress field s(0) within the matrix does not accomplish the BCs at the contour of the inhomogeneities. Thus, following the Jeffery approach, an auxiliary stress field s(1) deduced by a biharmonic stress function in bipolar coordinates is introduced and tensor H is then evaluated by performing proper contour integrals involving the total stress distribution s(0) + s(1) along the contour of the inclusions. 
The study allows evaluating the effective elastic properties of a wide class of inhomogeneous materials, with particular reference to composites reinforced with natural or synthetic fibres having optimized cross sections.

References
[1]	Sevostianov, I., and Kachanov, M., “Explicit cross-property correlations for anisotropic
two-phase composite materials” Journal of the Mechanics and Physics of Solids, 50, 253-282 (2002).
[2]	Korn, G.A. and Korn, T.M., Mathematical handbook for scientists and engineers. Definitions, Theorems and Formulas for Reference and Review, Dover, New York (1968).