Indeterminate Continuously Inhomogeneous Beam under End Rotation: A Longitudinal Fracture Study

The present paper describes a theoretical analysis of the effect of the end rotation of a statically indeterminate beam structure on the behaviour of a longitudinal crack. The beam is made of a material that is continuously inhomogeneous along the beam thickness. Besides, the material has non-linear elastic behaviour. The left end of the beam is supported with an axial double rod, while the right end is rigidly fixed. There is no external mechanical loading on the beam. Thus, the only influence on the beam is the rotation of the beam left end. Equations for determination the curvatures and the coordinates of the neutral axes are worked out and solved together with equations for resolving the static indeterminacy. The response of the longitudinal crack to the rotation of the beam left end is studied by deriving the strain energy release rate. For this purpose the balance of the energy in the beam is analyzed. The strain energy release rate is verified by applying the integral, J. A parametric analysis is performed to evaluate the influence of the beam end rotation and other factors on the longitudinal fracture.