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Residual stresses in random elastic composites: nonlocal micromechanics-based models and first estimates of the representative volume element size

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Abstract

Random elastic composites with residual stresses are examined in this paper with the aim of understanding how the prestress may influence the overall mechanical properties of the composite. A fully non-local effective response is found in perfect analogy with the un-prestressed case examined in (Drugan and Willis, J. Mech. Phys. Solids 44(4):497–524, 1996). The second gradient approximation is considered and the impact of the residual stresses on the estimate of the RVE size is studied whenever the local response is used to describe the mechanical properties of the heterogeneous medium. To this aim, total and incremental formulations are worked out in this paper and the influence of both uniform and spatially varying prestresses are studied. Among other results, it is shown how rapid oscillations of relatively “small” residual stresses in most cases may result in the impossibility of describing the overall behavior of the composite with a local constitutive equation. On the other hand, prestresses with relatively high amplitudes and slow spatial oscillations may even reduce the RVE size required for approximating the mechanical properties of un-prestressed heterogeneous media with a local constitutive equation.

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Notes

  1. Here (A⊠B)U:=AUB T, for any triple of second order tensors A, B, U.

  2. We note that ℂ∗(x)e(x)=ℂ(x)e(x), since e(x)=sym[H(x)].

  3. The further hypothesis of constant eigenstress t r within each phase r is fundamental in solving the integral equation for the polarization stress ρ through Fourier transforming.

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Acknowledgements

L. Deseri gratefully acknowledges financial support from grant MIUR-PRIN2008 (2010–2012) “Multiscale modelling, numerical analysis and testing of complex materials and structures for innovative applications”, the University of Trento and the Department of Mathematical Sciences and the Center for Nonlinear Analysis through the NSF Grant No. DMS-0635983. L. Deseri and F. Dal Corso gratefully acknowledge financial support from the grant PIAP-GA-2011-286110-INTERCER2, “Modelling and optimal design of ceramic structures with defects and imperfect interfaces”. The authors warmly acknowledge Walter J. Drugan for his helpful advice.

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Authors and Affiliations

  1. Department of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123, Trento, Italy

    Francesco Dal Corso

  2. Center for Nonlinear Analysis and Department of Mathematical Sciences, Carnegie Mellon University, 4811 Frew St., Pittsburgh, PA, 15213-3890, USA

    Luca Deseri

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  1. Francesco Dal Corso
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  2. Luca Deseri
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Correspondence to Luca Deseri.

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Dal Corso, F., Deseri, L. Residual stresses in random elastic composites: nonlocal micromechanics-based models and first estimates of the representative volume element size. Meccanica 48, 1901–1923 (2013). https://doi.org/10.1007/s11012-013-9713-z

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  • DOI: https://doi.org/10.1007/s11012-013-9713-z

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