Topology optimization design for buckling analysis related to the size effect

Owing to the excellent performance of microstructures or nanomaterials with well-designed topological configuration, the characteristic scale of structural design is gradually shifting from macroscopic to nanoscale or microscale structural design. However, the size effect that emerges from the small-scale structures may not be explained effectively with the hypothesis of classical mechanics owing to the lack of microscopic parameters in the classical constitutive model. In addition, slender beams within such small-scale structures are prone to buckling failure, which puts forward additional requirements for the stability design of the structure except for the overall compliance of the structure. Therefore, a topology optimization framework combining the modified couple stress theory with the solid isotropic material penalization (SIMP) model is constructed to illustrate the size effect on topology optimization. Numerical results show that the size effect affects the compliance, buckling performance, and topological configurations of the evolutionary structures.

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness – Part II: determination of the spherical part of the couple-stress

The indeterminacy of the spherical part of couple-stress is a well-known drawback of any theoretical formulation stemming from the Cosserat couple-stress theory of elasticity. The relevant theory of finite elastic deformations of solids reinforced by a family of fibres that resist bending is not an exception. The present communication extends and completes that theory in a manner that enables it to measure the spherical part of the couple-stress tensor outside the conventional equilibrium considerations. To achieve this, the present study reconsiders an extra piece of information that has surprisingly emerged already but, so far, has been left unexplained and unexploited; namely, the fact that the energy stored in a fibrous composite elastic solid with fibre-bending stiffness involves a couple-stress generated term that does not influence the relevant couple-stress constitutive equation. The thus resulting new theoretical development complements the theory previously presented without dismissing any of the theoretical results detailed or the conclusions drawn there. Its validity embraces boundary value problems concerning both finite and infinitesimal elastic deformations of polar fibrous composites. In the latter case, its applicability is also tested and verified through the successful determination of the spherical couple-stress of a polar transversely isotropic elastic plate subjected to pure bending.

Non-linear vibration analysis of specially orthotropic tapered micro-plates with arbitrary located crack: A non-classical analytical approach

The present work aims to study the non-linear vibrations in a cracked orthotropic tapered micro-plate. Linear and parabolic variation in the plate thickness is assumed in one as well as two directions. The partial crack is located in the centre, and it is continuous; this crack’s location is arbitrary and can be varied within the centre-line. Based on classical plate theory, the equilibrium principle is applied, and the governing equation of tapered orthotropic plate is derived. Additionally, the microstructure’s effect has been included in the governing equation using the non-classical modified couple stress theory. The simplified line spring model is used to consider the impact of partial crack on the plate dynamics and is incorporated using in-plane forces and bending moments. The introduction of Berger’s formulation brings the non-linearity in the model in terms of in-plane forces. Here, Galerkin’s method has been chosen for converting the derived governing equation into time-dependent modal coordinates, which uses an approximate solution technique to solve the non-linear Duffing equation. The crack is considered along the fibres and across the fibres to show the effect of orthotropy. Results are presented for an orthotropic cracked plate with non-uniform thickness. The effects of the variation of taper constants, crack location, crack length, internal material length scale parameter on the fundamental frequency are obtained for two different boundary conditions. The non-linear frequency response curves are plotted to show the effect of non-linearity on the system dynamics using the method of multiple scales, and the contribution of taper constants and crack parameters on non-linearity is shown with bending-hardening and bending-softening phenomenon. It has been found that vibration characteristics are affected by the taper parameters and fibre direction for a cracked orthotropic plate.