J. Fernández-Sáez

Torsion of cracked nanorods using a nonlocal elasticity model

This paper presents a nonlocal cracked-rod model from which we have analysed the torsional vibrations of a carbon nanotube with a circumferential crack. Several types of boundary conditions, including the consideration of a buckyball at the end of the nanotube, have been studied. The nonlocal Eringen elasticity theory is used to formulate the problem. The cracked rod is modelled by dividing the cracked element into two segments connected by a torsional linear spring whose stiffness is related to the crack severity. The effect of the nonlocal small-scale parameter, crack severity, cracked section position, different boundary conditions and attached mass are examined in this work.

First order solutions for the buckling loads of weakened Timoshenko columns

In this work, closed-form expressions for the buckling loads of weakened Timoshenko columns with different boundary conditions and shear force approaches (proportional to the bending rotation or to the total slope) are presented. The crack model used promotes discontinuities in both transversal displacements and rotation due to bending. To solve the buckling problem, the perturbation method is used, considering that the solutions for both the cracked and the uncracked columns are slightly different. This procedure leads to first-order closed-form expressions for the buckling loads of the Timoshenko cracked column, which were compared with those found by directly solving the corresponding eigenvalue problem, establishing validity limits for these solutions.

A Note on the Use of Approximate Solutions for the Bending Vibrations of Simply Supported Cracked Beams

The main goal of this note is to discuss the applicability of approximate closed-form solutions to evaluate the natural frequencies for bending vibrations of simply supported Euler-Bernoulli cracked beams. From the well-known model, which considers the cracked beam as two beams connected by a rotational spring, different approximate solutions are revisited and compared with those found by a direct method, which has been chosen as reference.

Axisymmetric free vibration of closed thin spherical nanoshells with bending effects

Nanoscale spheres have led to a growing interest in their potential applications in a wide range of technological fields. Hollow nanospheres can be modelled as closed spherical shells in the analysis of their mechanical behaviour, the ratio of thickness to radius being used to ascertain the pertinence of considering only membrane forces, or both membrane forces and bending moments. Nonlocal elasticity theory has been widely used to analyze the mechanical behaviour of nanostructures. This paper investigates the free axisymmetric vibration of nanoscale spherical shells accounting for both types of internal force, by extending the Kirchhoff–Love plate theory to Eringen nonlocal elasticity theory. The influence of coupled size and bending effects on the frequencies and modal shapes is studied, revealing specific features that cannot be observed in an uncoupled analysis. This study could be useful in biomedical and bioengineering applications as well as in other fields related to nanotechnology.

First-Order Solutions for the Buckling Loads of Euler-Bernoulli Weakened Columns

In this work, closed-form expressions for the buckling loads of a weakened column with different boundary conditions are presented. The cracked-column model is based on the well-known method consisting of dividing the column into two segments connected by a rotational linear spring whose flexibility is related to the crack size and the geometry of the cross section. For the formulation of closed-form expressions, the perturbation method is used and the results are compared with those found by directly solving the eigenvalue problem.

Identification of two cracks with different severity in beams and rods from minimal frequency data

It has been known for a long time that the problem of identifying two small cracks in a simply supported beam from the first three natural frequencies can be analytically formulated and solved if the two cracks have equal severity. In this paper we extend this result to the case of cracks with different severity. Each crack is simulated by a rotational elastic spring and the inverse problem is solved in terms of the damage-induced changes in the first four natural frequencies. Closed-form expressions of the damage parameters in terms of the measured frequencies are obtained. The results can be extended to the identification of two cracks in a longitudinally vibrating beam based on a suitable set of natural frequency and antiresonant frequency data. Numerical simulations support the theory, and show that if accurate input data are available and the cracks are not too close, then damage identification leads to satisfactory results.

An analysis of the static and dynamic fracture behaviour of a pipeline steel

This work deals with the dependence of fracture behaviour on the strain rate of a commercial pipeline steel. Low-blow impact tests, using a Charpy pendulum setup, and conventional static fracture tests were carried out with this material. Experimental results showed that the material fracture toughness increases slightly with strain rate. Numerical analyses of all the experiments were also performed, using a micromechanical damage model that explains the influence of the strain rate on the fracture toughness. Particular attention was paid to the blunting process at the crack tip under dynamic conditions.

Identification of an open crack in a beam with variable profile by two resonant frequencies

We consider the identification of a single open crack in a simply supported beam having nonuniform smooth profile and undergoing infinitesimal in-plane flexural vibration. The profile is assumed to be symmetric with respect to the mid-point of the beam axis. The crack is modeled by inserting a rotational linearly elastic spring at the damaged cross-section. We establish sufficient conditions for the unique identification of the crack by a suitable pair of natural frequency data, and we present a constructive algorithm for determining the damage parameters. The result is proved under a technical a priori assumption on the zeros of a suitable function determined in terms of the eigenfunctions of the problem. Extensions to beams under different sets of end conditions are also discussed. Theoretical results are confirmed by an extensive numerical investigation, both on simulated and experimental data.

A 2D discrete model with a bilinear softening constitutive law applied to dynamic crack propagation problems

In the present work, we propose the incorporation of a bilinear softening constitutive law in a purely linear elastic discrete model previously developed by the authors. The new model includes information of the fracture energy for the constitutive law. Three numerical examples of dynamic crack propagation are provided to demonstrate the effectiveness and robustness of the modification proposed. The obtained results are presented in terms of the crack path, the crack speed ,and the dissipated energy. These results are compared with those obtained with a previous discrete model developed by the authors and other numerical results published by other researchers. Final comments on the performance of the new model are carried out.

On the use of variable-separation method for the analysis of vibration problems with time-dependent boundary conditions

In this article, the axial vibrations of a rod with a clamped end and the transversal vibrations of a cantilever beam, both with a time-dependent and non-harmonic force applied on their free ends, are analysed. These are problems in which the traction and the shear, for the rod and the beam, respectively, prescribed in the boundaries of the bodies vary with time. The problems can be solved by the method proposed by Mindlin and Goodman. However, it is usual to solve this problem by the classic variable-separation method (which does not properly fulfil the time-dependent boundary conditions). The displacements and the forces along the systems are derived from both cited methods, and the results are compared. These results highlight the importance of using the proper solution method for the vibration problems with time-dependent boundary conditions.