The Mechanics of an Elastically Deforming Cantilever Beam with an Embedded Sharp Crack and Subjected to an End Transverse Loading
Creators: name: Fang, Xiaomin
Publisher: name: Maryland Shared Open Access Repository
Year: 2013

description: Over the last two decades, the frequency response of a component or a structure has been used to assess structural health as one of several methods used in damage detection and structural health monitoring. In studies involving simply supported or cantilever beam components, the natural frequencies and mode shapes for linearly vibrating beams have been established using approximate Euler-Bernoulli or the shear-enhanced Timoshenko beam theories. The latter theory accounts for shear effects that become important in low aspect ratio beams as well in calculating high frequencies in otherwise slender beams. In most studies, the presence of damage is modeled as a localized reduction in the structural stiffness EI of the beam. While the Timoshenko theory allows for the independent reduction of the elastic stiffness E and beam cross-sectional properties such as the cross sectional area A or the second moment of inertia I, both the Euler and Timoshenko theories assumed a symmetric reduction of the beam properties leading to a symmetric reduction of the effective beam stiffness EI. As such, current techniques do not have the refinement needed to detect along with the location of damage, its shape and its orientation.  In this study, both analytical and numerical methods are employed in developing robust models for damage detection. The study utilizes a cantilever beam with a fully embedded sharp crack and subjected to an end applied transverse force as the binary geometry for model development. Initially, a broad parametric finite element meshing algorithm capable of generating model geometries with fully embedded cracks at prescribed location, length and orientation is developed. The above algorithm is employed in conducting broad parametric studies of beams with a fully embedded sharp crack. The simulations helped to establish the load transfer and deformation mechanics of the cracked beam. In addition, for each model considered, the near-tip fracture mechanics are established revealing critical issues related to fracture mode mixity, stress intensity and crack surface closure effects.  In parallel studies, analytical methods are employed in establishing for the first time simple but robust Mechanics of Materials models capable of predicting the mechanics of a cantilever beam containing a sharp horizontal crack. Model predictions on the dominant force and moment resultants in the crack region as well as on the extent of the transition zones in the near-tip region have been obtained and compared to their counterparts obtained numerically via the method of finite elements. A remarkable agreement was found between the beam model and finite element predictions.  The beam model outcomes also enabled the development of analytical solutions for the complex near-tip mixed mode fracture fields. The latter models utilize the force and moment resultants in the crack region to obtain analytical expressions for the linear elastic energy release rate at each of the crack tips using both the compliance and the J-integral methods. Arguments regarding the structure of the transition fields and induced force and moment discontinuities at the crack tip cross sections led to unique separation of the mode mixity thus revealing the dominant fracture mode II nature of the beams and loading considered.  Initially revealed by the numerical finite element models and then validated by the analytical beam model, surface curvatures of the cracked beam emerged as a viable predictor of crack damage. More specifically, it has been shown both numerically and analytically, that a measurable deviation from a smooth profile occurs in the free surface curvatures in beams containing a fully embedded sharp crack. Intriguing features on the curvature profiles provide critical clues on the location and extent of the crack. Such findings provide compelling evidence that non-model efficient methods can be developed in detecting damage. Crack surface closure effects are examined to improve the linear elastics structural response. Their effects on the load transfer and deformation mechanics of damage detection are investigated.