The Effective Elastic Properties in Anisotropic Composites with Ellipsoidal Inclusions

Abstract

Parallel series of ellipsoidal inclusions, aligned along their major axes, whose centers are arranged in arrays inside a matrix, result generally in an orthotropic composite. When these arrays of inclusions constitute stacks placed at equal spacings through the thickness of the composite, the material is transformed into a transversely isotropic material. Considering this simplified problem, the effective elastic properties, i.e., the longitudinal and transverse elastic and shear moduli, as well as the respective Poisson's ratios, are evaluated in this paper. The solution of the problem was made in terms of the aspect ratio of the inclusions, their volume fraction and, finally, the succession of the constituent phases inside the composite. Employing Eshelby's theory for the stress field around a single ellipsoidal inhomogeneity, which is surrounded by the effective anisotropic composite medium, and Mori-Tanaka's concept for considering the mutual interaction of the neighboring inclusions inside the matrix, we evaluated the effective properties of composites containing ellipsoidal inclusions parallel dispersed inside the matrix. The whole analysis provides, in a closed-form solution, interesting results concerning the anisotropy, as well as the stiffness of anisotropic composites, reinforced with ellipsoidal inclusions. These findings reveal that the anisotropy is merely increased in composites consisting of soft matrices and reinforced by hard needle-shaped inclusions, whereas the stiffness is affected by both the shape of inclusions, as well as by their mutual concentrations. Furthermore, introduction of different constituent phases inside the composites indicated that intense variations of stiffness occur, especially for composites consisting of hard matrices containing soft inclusions. The results of the paper concerning effective elastic properties of anisotropic composites with ellipsoidal inclusions can be applied in case of unidirectional discontinuous fiber composites, where the fibers are simulated by ellipsoidal inclusions. Also, the evaluation of the elastic properties in hard composites with enough soft inclusions provide information for the mechanical as well as the fracture behavior of hard composites weakened by internal through cracks.

Simplified Model for the Influence of Inclusion Aspect Ratio on the Stiffness of Aligned Reinforced Composites

“…VT [2], MT [5], MC [6], GMC [7], SCMC [8] Simple rule-of-mixture for (1/E), Chamis [13], de Morais [14] Particulate reinforced composites VT [2], SC [3,15], GSC [4], MT [5], GMC [7], SCMC [8], MC [16] Paul [9], Counto [10], Ishai [11], Ishai and Cohen [12] Short UD fiber composites (longitudinal) SC [15], MC [16], EMT [17,18] HT [20] Short UD fiber composites (transverse) SC [15], MC [16], EMT [17,18] HT [20]. Fiber length: diameter ratio not considered.…”

“…Fiber length: diameter ratio not considered. Aligned flake composites (in-plane) SC [15], MC [16], EMT [17,18] HT [20] Aligned flake composites (out-of-plane) SC [15], MC [16], EMT [17,18] HT [20]. Flake thickness: side ratio not considered.…”

“…Flake thickness: side ratio not considered. Aligned composites (any inclusion aspect ratio) SC [15], MC [16], EMT [17,18] Nil Note: VT: variational theorem of elasticity; SC: self-consistent approach; GSC: generalized selfconsistent model; MT: Mori-Tanaka method; MC: method-of-cells; generalized method-of-cells; SCMC: strain-compatible method-of-cells; EMT: combined Eshelby and Mori-Tanaka method; HT: Halpin-Tsai's equations. distances between an inclusion's centroid to its neighboring ones, measured along the X, Y and Z axes, are 2Áx, 2Áy and 2Áz respectively.…”

“…Moreover, upon substituting R ¼ RðÞ into Equations ( 17) and (18), the normalized composite stiffness should be ensured to be bounded within…”

“…For composites with aligned inclusions, formal solutions have been developed by Chu et al [15] based on the self-consistent method for spheroids and by Aboudi [16] based on the method-of-cells of triply periodic array of parallelpiped fibers. Tandon and Weng [17] and Demakos and Theocaris [18] combined Eshelby's stress field [19] around an ellipsoid and Mori-Tanaka's [5] averaged stress description for investigating the effects of inclusion aspect ratio of aligned composites. From the theoretical viewpoint, the analysis of ellipsoid, or more specifically spheroid, inclusion is of considerable interest due to its ability to be reduced to special cases of continuous UD fiber composites (extremely elongated spheroids), short UD fiber composites (elongated spheroids), particulate composites (spheres), flake composites (flattened spheroids) and periodically bilaminated composites (extremely flattened spheroids).…”