Bending of Timoshenko beam with effect of crack gap based on equivalent spring model

Considering the effect of crack gap, the bending deformation of the Timoshenko beam with switching cracks is studied. To represent a crack with gap as a nonlinear unidirectional rotational spring, the equivalent flexural rigidity of the cracked beam is derived with the generalized Dirac delta function. A closed-form general solution is obtained for bending of a Timoshenko beam with an arbitrary number of switching cracks. Three examples of bending of the Timoshenko beam are presented. The influence of the beam＇s slenderness ratio, the crack＇s depth, and the external load on the crack state and bending performances of the cracked beam is analyzed. It is revealed that a cusp exists on the deflection curve, and a jump on the rotation angle curve occurs at a crack location. The relation between the beam＇s deflection and load is bilinear, each part corresponding to an open or closed state of crack, respectively. When the crack is open, flexibility of the cracked beam decreases with the increase of the beam＇s slenderness ratio and the decrease of the crack depth. The results are useful in identifying non-destructive cracks on a beam.

Effect of electric boundary conditions on crack propagation in ferroelectric ceramics

In this paper, the effect of electric boundary conditions on Mode I crack propagation in ferroelectric ceramics is studied by using both linear and nonlinear piezoelectric fracture mechanics. In linear analysis, impermeable cracks under open circuit and short circuit are analyzed using the Stroh formalism and a rescaling method. It is shown that the energy release rate in short circuit is larger than that in open circuit. In nonlinear analysis, permeable crack conditions are used and the nonlinear effect of domain switching near a crack tip is considered using an energy-based switching criterion proposed by Hwang et al.（Acta Metal. Mater.,1995）. In open circuit, a large depolarization field induced by domain switching makes switching much more diffcult than that in short circuit. Analysis shows that the energy release rate in short circuit is still larger than that in open circuit, and is also larger than the linear result. Consequently,whether using linear or nonlinear fracture analysis, a crack is found easier to propagate in short circuit than in open circuit, which is consistent with the experimental observations of Kounga Njiwa et al.（Eng. Fract. Mech., 2006）.

Estimation of the internal and external damping from the forward and backward spectrum of a rotor with a fatigue crack

Both internal and external damping are concurrently present in a rotor system.However,the internal damping that originates from various sources often leads to instability in the rotor system,and hence its accurate estimation is crucial.The goal of the present paper is to distinctly estimate both external and internal damping in a Jeffcott rotor containing transverse crack.The modelling of the fatigue crack is based on the assumption of a switching crack,which additionally provides the internal damping due to rubbing on crack faces,which is hardly addressed in literature.Moreover,the fatigue crack leads to the forward and backward whirls due to asymmetry in the rotor.In the process of development of the identification procedure in the form of the regression equation for the estimation of internal damping,other important unknown rotor system parameters,like the external damping,unbalance and loss of stiffness of the shaft due to the crack,are also identified.This reduces the estimation error of the internal damping due to reduction in deviations of other model parameters.Numerically simulated responses are obtained from equations of motion of the rotor system in the form of a closed-form expression and by the Runge-Kutta integration method.The full-spectrum response data(both magnitude and phase)are utilized during the testing of the proposed identification procedure.Phase corrections have been incorporated in the full-spectrum.Measurement noise is added in numerical responses during identification of parameters to check the robustness of the algorithm and found to be working satisfactorily.