Beam theories such as the Timoshenko beam theory are in agreement with the elasticity theory. However, due to the different nonlocal averaging processes, they are expected to yield different results in their nonlocal forms. In the present work, the free vibration behavior of nonlocal nanobeams is studied using the nonlocal integral Timoshenko beam theory (NITBT) and two-dimensional nonlocal integral elasticity theory (2D-NIET) with different kernels and their results are compared. A new kernel, termed the compensated two-phase (CTP) kernel, is introduced, which entirely compensates for the boundary effects and does not suffer from the ill-posedness of previous kernels. Using the finite element method, the free vibration analysis is performed for different boundary conditions based on the first three natural frequencies. For both the NITBT and 2D-NIET with both the two-phase (TP) and CTP kernels, the nonlocal parameter has a softening effect on the natural frequencies for all the boundary conditions, without observing the paradoxical behaviors of the nonlocal differential theory. For both theories, the softening effect of the nonlocal parameter is more pronounced for the TP kernel compared to the CTP kernel. The sensitivity of the 2D-NIET to the nonlocal parameter is found to be higher than that of the NITBT. Also, the softening effects for different vibration modes are compared to each other for both theories and both kernels. The obtained results can be extended for various important beam problems with nonlocal effects and help obtain a better understanding of applicable nonlocal theories.
Thanks to the advancement of additive manufacturing technologies, mechanical metamaterials have attracted a great deal of attention in recent years. With the employment of such technologies, materials with exceptional or tailored mechanical properties can be easily manufactured mainly by 3D printing of different microstructures rather than by changing the material composition. These lattice materials can provide remarkable material properties in spite of being significantly lighter than typical bulk materials. Due to the large number of degrees of freedom for engineering structures, single-scale numerical simulation of such cellular materials is computationally demanding. Therefore, two-scale computational homogenization approaches, such as FE 2 and FE-FFT, can perform a key role in the cost-effective numerical modeling of metamaterials. Twoscale computational homogenization methods rely on solving a boundary value problem (BVP) for each of the macroscopic and microscopic scales in a nested procedure. Although representative homogenization techniques have been widely used to study materials with heterogeneous microstructures, there still exist some challenges in their employment for lattice materials. This study addresses main challenges in two-scale-based computational homogenization methods for numerical modeling of mechanical metamaterials. High dependence of convergence rate and accuracy on phase contrast for fast Fourier transform (FFT) solvers and comparable macro and micro characteristic lengths in metamaterials (i.e. the applicability of the principle of scale separation) are some examples of such challenges.
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