Local–nonlocal integral theories of elasticity with discontinuity for longitudinal vibration analysis of cracked rods

We present a size-dependent formulation for the longitudinal vibration study of cracked thick rods based on both the strain and stress-driven local/nonlocal mixture theories of elasticity with discontinuity. Due to the presence of the crack, the rod is divided into two segments connected by a linear spring, and compatibility conditions are given to describe the geometric discontinuity caused by the crack. The equations of motion of the discrete rods are formulated based on Rayleigh rod theory, and the two classes of local–nonlocal constitutive equations are integrated into an equivalent differential form, equipped with a set of constitutive boundary conditions at two ends of the whole structure and a set of constitutive continuity conditions at the junction of the sub-structures. The differential quadrature method (GDQM), together with the interpolation quadrature formula, is introduced to solve all the equations of motion of the sub-rods, the above constraint condition and the definite integrals occurring therein, simultaneously, through which we extract the dimensionless frequencies of the cracked rods with different boundary edges. After conducting comparison studies with the existing literature, numerical studies reveal that the present local–nonlocal model with discontinuity can effectively address the softening (or hardening) phenomenon as the structure’s size reduces. Moreover, the influence of crack location, crack severity, inertia of lateral motions and nonlocal parameters on the rods’ vibration frequencies is examined in detail.