Keywords: Euler-Bernoulli beams, fatigue cracks, variational formulation, lumped crack flexibility, finite element model.

A crack on an elastic structural element introduces considerable local flexibility due to the strain energy concentration in the vicinity of the crack tip under load. Long ago, this effect was recognized and the idea of an equivalent spring, a local compliance, was used to quantify in a macroscopic way the relation between the applied load and the strain concentration around the tip of the crack [1]. Dimarogonas [2,3] introduced the local flexibility model for a crack for vibration analysis of cracked beams.

A variational approach has been used by Chondros et al. [4,5] to develop a continuous vibration model for the lateral vibration of cracked Euler-Bernoulli beams with open single-edge or double-edge cracks. The Hu-Washizu-Barr variational formulation was used to develop the differential equation and boundary conditions for flexural vibration of simply supported and cantilever beams considering the beam as a one-dimensional continuum. The crack was modelled with continuous flexibility using the displacement field in the vicinity of the crack found using fracture mechanics methods. Fracture mechanics methods were used to model the crack as a continuous flexibility in the vicinity of the crack region investigating the displacements field.

The damage state of a system corresponds to various aspects among which the identification of the existence of damage or a flaw, its location, type and severity and damage tolerance. This design philosophy is largely based upon fracture mechanics and envisages sufficient strength and structural integrity of the aircraft to sustain major damage and to avoid catastrophic failure [6]. A comparative study based on the Hu-Washizu-Barr variational formulation, a lumped crack flexibility approach and a finite element alternative is applied for the derivation of the cracked beam frequency change for vibration analysis and crack identification. The analytical and numerical results were correlated with experimental results obtained on aluminium beams with open fatigue cracks.

1

Irwin G.R., "Fracture", in Handbuch der Physik, 6, Springer-Verlag, Heidelberg, 551-590, 1958.

2

Dimarogonas A.D., "Vibration for Enginees", Second Edition Prentice-Hall Upper Saddle River, 1996.

3

Chondros T.G., Dimarogonas A.D., "Identification of cracks in welded joints of complex structures", Journal of Sound and Vibration, 69, 531-538, 1980. doi:10.1016/0022-460X(80)90623-9

4

Chondros T.G., Dimarogonas A.D., Yao J., "A Continuous Cracked Beam Vibration", Theory Journal of Sound and Vibration, 215(1), 17-34, 1998. doi:10.1006/jsvi.1998.1640

5

Chondros T.G., "The continuous crack flexibility method for crack identification", Fatigue & Fracture of Engineering Materials & Structures, 24, 643-650, 2001. doi:10.1046/j.1460-2695.2001.00442.x

6

Sih G.C., "Multiple hierarchical scale-dependency on physical mechanisms of material damage: macromechanical, microstructural and nanochemical", Particle and Continuum Aspects of Mesomechanics, Mesomechanics 2007, Editors G.C. Sih, M. Nait-Abdelaziz, T. Vu-Khanh, ISTE Ltd, 2007.

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