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Major Accomplishments in Composite Materials and Sandwich Structures pp 799–816Cite as

Fracture of Brittle Lattice Materials: A Review

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Abstract

The mechanics of failure for elastic-brittle lattice materials is reviewed. Closed-form expressions are summarized for fracture toughness as a function of relative density for a wide range of periodic lattices. A variety of theoretical and numerical approaches has been developed in the literature and in the main the predictions coincide for any given topology. However, there are discrepancies and the underlying reasons for these are highlighted. The role of imperfections at the cell wall level can be accounted for by Weibull analysis. Nevertheless, defects can also arise on the meso-scale in the form of misplaced joints, wavy cell walls and randomly distributed missing cell walls. These degrade the macroscopic fracture toughness of the lattice.

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Notes

  1. 1.

    It is worth emphasising the physical origin of each side of the fracture criterion K =K C . The stress intensity factor on the left hand side is a mechanics parameter, its value determined by the geometry, stress level and crack length in the component, and is a measure of the cracking effort being applied to the component;whilst the right hand side is a material constant, a measure of the particular material's ability to resist rapid crack advance.

  2. 2.

    Tensile fracture strength σ TS and modulus of rupture σ f are used indistinctively in this review. The modulus of rupture σ f is the maximum surface stress in a bent beam at the instant at which it fractures. If the beam is made of a brittle solid, like the cell wall discussed here, the fracture initiates at a microcrack (usually a surface microcrack) in the wall and propagates catastrophically. On average, the modulus of rupture is a little larger than the tensile strength σ TS because, in bending, only one surface of the beam sees the maximum tensile stress;in simple tension the entire beam is stressed uniformly, so a given microcrack is less likely to be stressed in bending than in simple tension. The statistics of the problem (see, for example, Davidge [12]) show that the modulus of rupture is typically about 1.1 times larger than the tensile strength.

  3. 3.

    Classical continuum theory is well-suited for situations where the variations in stresses and strains are smooth. However, in many circumstances this is not necessarily the case (e.g. near crack tips or boundary layers) and one resorts to enhanced continuum theories (also called enriched or generalised). Such theories include information on the microstructure and take into account the non-uniformity of stresses and strains at the micro-scale. One of the simplest generalised theories is Cosserat (or micro-polar) theory, in which the interaction between neighbouring material points is governed by a moment vector in addition to the force vector from classical continuum theory.

  4. 4.

    The second term in the series expansion of the mode I crack tip field is the so-called T -stress parallel to the crack plane. The T -stress scales linearly with the remote applied stress, but its magnitude depends upon the specimen configuration. The T -stress vanishes for mode II loading, by a symmetry argument. For conventional, fully dense, elastic-brittle solids the T -stress plays a relatively minor role in influencing the fracture process at the crack tip. In ductile fracture and fatigue, however, the T -stress becomes important at short crack lengths and reveals itself in a dependence of inferred fracture toughness upon specimen geometry, see for example [33, 34].

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Authors and Affiliations

  1. Cambridge University Engineering Department, Trumpington St, Cambridge, CB2 1PZ, UK

    Ignacio Quintana-Alonso & Norman A. Fleck

Authors
  1. Ignacio Quintana-Alonso
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  2. Norman A. Fleck
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Correspondence to Ignacio Quintana-Alonso .

Editor information

Editors and Affiliations

  1. Dept. Civil &Environmental Engineering, Northwestern University, 2137 Tech Drive, Catalysis Bldg., Evanston, IL, 60208, USA

    I. M. Daniel

  2. Dept. of Civil Engineering Lab. for Applied Mechanics, Democritus University of Thrace, Xanthi, 67100, Greece

    E. E. Gdoutos

  3. Office of Naval Research, Solid Mechanics Program, Arlington, VA, 22203-1995, USA

    Y. D. S. Rajapakse

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Quintana-Alonso, I., Fleck, N.A. (2009). Fracture of Brittle Lattice Materials: A Review. In: Daniel, I.M., Gdoutos, E.E., Rajapakse, Y.D.S. (eds) Major Accomplishments in Composite Materials and Sandwich Structures. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3141-9_30

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