Free transverse vibrations of nanobeams with multiple cracks

A nonlocal model is formulated to study the size-dependent free transverse vibrations of nanobeams with arbitrary numbers of cracks. The effect of the crack is modeled by introducing discontinuities in the slope and transverse displacement at the cracked cross-section, proportional to the bending moment and the shear force transmitted through it. The local compliance of each crack is related to its stress intensity factors assuming that the crack tip stress field is undisturbed (non-interacting cracks).The kinematic field is defined based on the Bernoulli-Euler beam theory, and the small-scale size effect is taken into account by employing the constitutive equation of the stress-driven nonlocal theory of elasticity. In this manner, the curvature at each cross-section is defined as an integral convolution in terms of the bending moments at all the cross-sections and a kernel function which depends on a material characteristic length parameter. The integral form of the nonlocal constitutive equation is elaborated and converted into a differential equation subjected to a set of mathematically consistent boundary and continuity conditions at the nanobeam's ends and the cracked cross-sections. The equation of motion in each segment of the nanobeam between cracks is solved separately and the variationally consistent and constitutive boundary and continuity conditions are imposed to determine the natural frequencies. The model is applied to nanobeams with different boundary conditions and the natural frequencies and the mode shapes are presented at the presence of one to four cracks. The results of the model converge to the experimental results available in the literature for the local cracked beams and to the solutions of the intact nanobeams when the crack length goes to zero. The effects of the crack location, crack length, and nonlocality on the natural frequencies are investigated, also for the higher modes of vibrations. Novel findings including the amplification and shielding effects of the cracks on the natural frequencies are presented and discussed.