Archive of Applied Mechanics

, Volume 88, Issue 11, pp 2081–2099 | Cite as

A least squares approach for effective shear properties in an \({{\varvec{n}}}\)-layered sphere model

  • Rolf MahnkenEmail author
  • Peter Lenz
  • Christian Dammann


This work presents the derivation of the effective shear modulus for a heterogeneous material composed of multilayered composite spheres embedded in a linear elastic matrix. It is based on the composite spheres model known from the literature. In contrast to Herve and Zaoui (Int J Eng Sci 31:1–10, 1993), the effective shear modulus is obtained by equating the results of two models: In the first model, a heterogeneous sphere is embedded in an equivalent homogeneous material, whereas in the second model, the heterogeneous sphere is replaced by an equivalent homogeneous sphere. In the context of both, a shear stress approach and a shear deformation approach, this results in an overdetermined system of equations which is solved with the least squares method. In a numerical study, our results are compared to effective moduli and bounds from the literature. Furthermore, a convincing agreement with experimental data for glass microspheres embedded in a polyester matrix is demonstrated.


Heterogeneous materials Effective properties Composite sphere model RVE Homogenization 



This work is based on investigations of the “SPP 1712 - Intrinsische Hybridverbunde für Leichtbautragstrukturen,” which is kindly supported by the Deutsche Forschungsgemeinschaft (DFG).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Chair of Engineering MechanicsUniversity of PaderbornPaderbornGermany

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