Local-nonlocal stress-driven model for multi-cracked nanobeams

Exact closed-form solutions for multi-cracked Euler-Bernoulli nanobeams are provided by proposing two equivalent approaches. The considered multi-cracked nanobeams are modeled with a well-posed local-nonlocal stress-driven model and contain an arbitrary number n of isolated damaged cross-sections. In order to simulate the damaged sections, it is assumed that the nanobeam bending flexibility contains n Dirac delta functions, with a Dirac delta located at each damaged section: this representation of cracked sections, already used in local problem, is adopted for the first time here in the local-nonlocal stress-driven model. The first approach providing closed-form solutions is based on the integral definition of the local-nonlocal stress-driven model, where the bending curvature is given by the superposition of the local and nonlocal phases, and the nonlocal phase is an integral convolution of the bending moment. This approach provides an integro-differential equation, which is solved by taking firstly the Laplace transform and then the anti-transform. In the second approach, it is shown that the integral definition of the stress-driven multi-cracked nanobeam model is equivalent to a differential equation together with suitable constitutive boundary conditions. This equation is solved by making use of Laplace transform again. The two approaches provide the same solution, with the second one resulting computationally faster. Interesting results show that a jump of the rotation at the damaged cross-sections occurs in local beam, but does not occur in pure nonlocal nanobeams (i.e. local phase is absent): contrary to the local elasticity theory, the bending curvature of pure nonlocal nanobeams has no singularity at damaged sections. The typical stiffness increase, due to the increase in nonlocal fraction of the mixture or to the increase in the length-scale parameter, also appears in multi-cracked nanobeams and this stiffening nonlocal effect is more pronounced in more constrained nanobeams.