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The problem of sharp notch in microstructured solids governed by dipolar gradient elasticity

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Abstract

In this paper, we deal with the asymptotic problem of a body of infinite extent with a notch (re-entrant corner) under remotely applied plane-strain or anti-plane shear loadings. The problem is formulated within the framework of the Toupin-Mindlin theory of dipolar gradient elasticity. This generalized continuum theory is appropriate to model the response of materials with microstructure. A linear version of the theory results by considering a linear isotropic expression for the strain-energy density that depends on strain- gradient terms, in addition to the standard strain terms appearing in classical elasticity. Through this formulation, a microstructural material constant c is introduced, in addition to the standard Lamé constants (λ, μ). The faces of the notch are considered to be traction-free and a boundary-layer approach is followed. The boundary value problem is attacked with the asymptotic Knein-Williams technique. Our analysis leads to an eigenvalue problem, which, along with the restriction of a bounded strain energy, provides the asymptotic fields. The cases of a crack and a half-space are analyzed in detail as limit cases of the general notch (infinite wedge) problem. The results show significant departure from the predictions of the standard fracture mechanics.

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  1. Mechanics Division, National Technical University of Athens, Zographou Campus, Zographou, 15773, Greece

    P. A. Gourgiotis, M. D. Sifnaiou & H. G. Georgiadis

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  1. P. A. Gourgiotis
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  2. M. D. Sifnaiou
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  3. H. G. Georgiadis
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Correspondence to H. G. Georgiadis.

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Gourgiotis, P.A., Sifnaiou, M.D. & Georgiadis, H.G. The problem of sharp notch in microstructured solids governed by dipolar gradient elasticity. Int J Fract 166, 179–201 (2010). https://doi.org/10.1007/s10704-010-9523-4

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