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A dislocation solution to evaluate the mode III stress intensity factors (SIFs) of the crack tips, torsional rigidity and the hoop stress around the cavities in an infinite rectangular cross-section bar coated by an FG layer under torsion is developed based on the Saint–Venant theory. The solution of the rectangular cross section with the FG coating weakened by a screw dislocation under torsion was first obtained in terms of the dislocation density function. Then, the problem is transformed into a group of Cauchy integral equations in the rectangular cross-section bar with the FG coating layer via the distributed dislocation approach. The singular integral equations are reduced to a set of algebraic equations to evaluate them numerically. After solving the integral equations, the SIFs of the crack tips, the hoop stresses around the cavities and the torsional rigidity of the entire domain are found. The obtained solution is validated by performing comparison studies in multiple cases. It is shown that the interaction between the cracks, the cracks length, variation of the shear module of the FG coating layer and distance of the crack tips from stress-free surfaces can significantly affect the SIFs of crack tips and torsional rigidity of the rectangular cross-section bar coated by an FG layer.

A nonlinear finite element model based on the modified strain gradient theory (MSGT) in combination with the first-order shear deformation theory (FSDT) of shells for the free vibration of multilayer functionally graded graphene platelets reinforced composite (FG-GPLRC) toroidal micropanels is developed. The nine-noded shell elements with five degrees of freedom per node are used to accurately simulate thin as well as moderately thick toroidal micropanels without shear locking phenomena. In addition to initial thermal stresses, the influences of nonlinear elastic foundation and the elastic rotational springs at the micropanel edges are considered. The formulation and corresponding computer codes are validated by performing convergence study and doing comparison studies with some existing results in the literature. Then, the impacts of amplitude ratio, geometric parameters, ratio of thickness-to-material length scale parameter (MLSP), temperature rise, coefficients of elastic foundation and rotational springs, and different GPLs distribution patterns on the nonlinear vibrational behaviors of the multilayer FG-GPLRC toroidal micropanels are investigated. The results show that the size dependence of material properties increases the impact of geometric nonlinearity on the vibrational behavior of the micropanels. In addition, it affects the nonlinear vibrational behavior more than the linear one in a widespread range of thickness-to-MLSP ratio.

This paper presents the reinforcement impact of a functionally graded (FG) coating layer on the induced stress distribution of circular cross-section bars weakened by multiple cracks and cavities under torsional moment. The entire cross-section of the bar is under torsional loading governed by Saint-Venant’s principle. The shear modulus of the FG coating layer is assumed to vary exponentially in the radial direction. The solution procedure begins by finding the stress field corresponding to a Volterra-type screw dislocation in a circular cross-section bar with an FG coating layer using the Fourier transform. After that, the dislocations are distributed along the surfaces of cracks and cavities. The dislocation stress field is employed to derive singular integral equations in terms of the unknown dislocation density function on the surface of cracks and cavities for the evaluation of multiple defects with arbitrary patterns. These integral equations are of the Cauchy singular kinds and are solved numerically to determine the dislocation density functions on the surface of defects. The obtained results are employed to calculate the torsional rigidity of the domain under consideration, and the mode III stress intensity factors (SIFs) at the crack tips. Finally, several numerical examples are presented to study the effects of the FG coating layer and multiple cracks and cavities with arbitrary patterns on the stress distribution, stress intensity factors (SIFs) and torsional rigidity of weakened bar by cracks and cavities.

The vibrational behavior of multilayer functionally graded graphene platelets reinforced composite (FG-GPLRC) toroidal panels with elastically restrained against rotation edges is investigated. The first-order shear deformation theory (FSDT) of shells together with the finite element method (FEM) is employed to develop a general formulation and solution procedure. Nine-noded isoparametric shell elements with five degrees of freedom per node are used to discretize the spatial domain. After validating the approach, the influences of different distribution patterns of graphene platelets (GPLs) through the panel thickness, the geometrical parameters, and the elastic coefficients of the rotational springs are studied. It is shown that among the multilayer FG-GPLRC toroidal panels with different GPLs distribution patterns, the toroidal panels with X-type pattern have the highest frequency values whereas the minimum values belong to the shells with O-type and Λ -type for the first and second frequency parameters, respectively. Also, the numerical results reveal that by increasing the GPLs volume fraction and the elastic coefficients of rotational springs, the frequency parameters increase, and these parameters have considerable effects on the frequency parameters. Also, the mode shapes of multilayer FG-GPLRC toroidal panels for different GPLs distribution patterns are depicted.