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Fractal materials, beams, and fracture mechanics

Abstract.

Continuing in the vein of a recently developed generalization of continuum thermomechanics, in this paper we extend fracture mechanics and beam mechanics to materials described by fractional integrals involving D, d and R. By introducing a product measure instead of a Riesz measure, so as to ensure that the mechanical approach to continuum mechanics is consistent with the energetic approach, specific forms of continuum-type equations are derived. On this basis we study the energy aspects of fracture and, as an example, a Timoshenko beam made of a fractal material; the local form of elastodynamic equations of that beam is derived. In particular, we review the crack driving force G stemming from the Griffith fracture criterion in fractal media, considering either dead-load or fixed-grip conditions and the effects of ensemble averaging over random fractal materials.

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Correspondence to Martin Ostoja-Starzewski.

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Ostoja-Starzewski, M., Li, J. Fractal materials, beams, and fracture mechanics. Z. Angew. Math. Phys. 60, 1194 (2009). https://doi.org/10.1007/s00033-009-8120-8

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Keywords.

  • Fractal materials
  • beams
  • fracture mechanics