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For the first time, the nonlinear buckling responses of shallow complex curved caps and circular plates made from three-dimensional graphene foams reinforced composite (3DGF-RC) subjected to uniformly distributed pressure and thermal loads are presented and analyzed. The four curvature types of complexly curved caps are considered to be spherical, parabolic, ellipsoid, and sinusoid caps. The governing formulations are established using the first-order shear deformation theory (FSDT) and the von Kármán geometrical nonlinearities. The trigonometric solution forms of deflection and rotation angle are proposed. The equilibrium equations in the nonlinear algebraic forms are obtained approximately by applying the Galerkin method. The results are flexibly applied to 3DGF-RC caps with complex curved shell designs in engineering. From the investigated results, it is possible to evaluate the nonlinear thermal and mechanical buckling responses of 3DGF-RC complexly curved caps and circular plates with different geometrical and material input parameters. Some useful remarks on the nonlinear buckling responses of the considered types of caps and plates can be recognized from the numerical examples. Especially, the advantage of the sinusoid caps in terms of thermal and mechanical postbuckling load-carrying capacity can be clearly observed in most investigated cases.

The stress–strain relationship of the active system can be characterized by odd elasticity proposed in continuum mechanics, which leads to complex mechanical behavior by converting energy into nonconservative forces. However, current research on odd elasticity has predominantly concentrated on non-Hermitian dynamics. In this paper, the buckling model of odd elastic circular plates is established in the polar coordinate system, and the critical load and buckling modes are obtained by the separation variable method and the Galerkin method. The results show that the non-Hermitian eigenvalue matrix leads to the existence of the imaginary component of the critical load, and the buckling deformation is localized in the center or the periphery of the circular plate. The results also show that the existence of active forces makes odd elastic circular plates exhibit anomalous tensile buckling and active buckling, which both display chiral deformation patterns. The research in this paper can provide a theoretical reference for the design of directional flexion control and intelligent energy-absorbing devices.

Modern biomedical and tribological systems are increasingly deploying combinations of nanofluids and bioconvecting microorganisms which enable improved control of thermal management. Motivated by these developments, in this study, a new mathematical model is developed for the combined nanofluid bioconvection axisymmetric squeezing flow between rotating circular plates (an important configuration in, for example, rotating bioreactors and lubrication systems). The Buongiorno two-component nanoscale model is deployed, and swimming gyrotactic microorganisms are considered which do not interact with the nanoparticles. Thermal radiation is also included, and a Rosseland diffusion flux approximation is utilized. Appropriate similarity transformations are implemented to transform the nonlinear, coupled partial differential conservation equations for mass, momentum, energy, nanoparticle species and motile microorganism species under suitable boundary conditions from a cylindrical coordinate system into a dimensionless nonlinear ordinary differential boundary value problem. An efficient scheme known as differential transform method (DTM) combined with Padé-approximations is then applied to solve the emerging nonlinear similarity equations. The impact of different non-dimensional parameters i.e. squeezing Reynolds number, rotational Reynolds number, Prandtl number, thermophoresis parameter, Brownian dynamics parameter, thermal radiation parameter, Schmidt number, bioconvection number and Péclet number on velocity, temperature, nanoparticle concentration and motile gyrotactic microorganism density number distributions is computed and visualized graphically. The torque effects on both plates, i.e. the lower and the upper plate, are also determined. From the graphical results, it is seen that momentum in the squeezing regime is suppressed clearly as the upper disk approaches the lower disk. This inhibits the axial flow and produces axial flow retardation. Similarly, by enhancing the value of squeezing Reynolds number, the tangential velocity distribution also decreases. More rigorous squeezing clearly therefore also inhibits tangential momentum development in the regime and leads to tangential flow deceleration. Tables are also provided for multiple values of flow parameters. The numerical values obtained by DTM-Padé computation show very good agreement with shooting quadrature. DTM-Padé is shown to be a precise and stable semi-numerical methodology for studying rotating multi-physical flow problems. Radiative heat transfer has an important influence on the transport characteristics. When radiation is neglected, different results are obtained. It is important therefore to include radiative flux in models of rotating bioreactors and squeezing lubrication dual disk damper technologies since high temperatures associated with radiative flux can impact significantly on combined nanofluid bioconvection which enables more accurate prediction of actual thermofluidic characteristics. Corrosion and surface degradation effects may therefore be mitigated in designs.

In this work the exact axisymmetric vibration frequencies of circular and annular variable thickness plates are found. The solution is obtained using the exact element method developed earlier. It allows for the exact solution of problems with general polynomial variation in thickness using infinite power series. The solution is exact up to the accuracy of the computer. The natural frequencies of vibration are found as the solutions of the frequency equation. Normalized values for the natural frequencies are given for linear, parabolic and cubic variations of the plate thickness, for circular and annular plates, with four types of boundary conditions on the inner and outer boundaries.

This paper is concerned with the axisymmetric vibration problem of polar orthotropic circular plates of quadratically varying thickness and resting on an elastic foundation. The problem is solved by using the Rayleigh–Ritz method with boundary characteristic orthonormal polynomials for approximating the deflection function. Numerical results are computed for frequencies, nodal radii and mode shapes. Three-dimensional graphs are also plotted for the first four normal modes of axisymmetric vibration of plates with free, simply-supported and clamped edge conditions for various values of taper, orthotropy and foundation parameters.

Geometrically nonlinear analysis of functionally graded circular plates subjected to mechanical and thermal loads is carried out in this paper. The Green–Lagrange strain tensor in its entirety is used in the analysis. The locally effective material properties are evaluated using homogenization method which is based on the Mori–Tanaka scheme. In the case of thermally loaded plates, the temperature variation through the thickness is determined by solving a steady-state heat transfer (i.e. energy) equation. As an example, a functionally gradient material circular plate composed of zirconium and aluminum is used and results are presented in graphical form.

An analysis of the geometrically nonlinear dynamics of thin circular plates on a two parameter elastic foundation is presented in this paper. The nonlinear partial differential equations obtained from von Karman's large deflection plate theory have been solved by using the harmonic differential quadrature method in the space domain and the finite difference numerical integration method in the time domain. Winkler-Pasternak foundation model is considered and the influence of stiffness of Winkler (K) and Pasternak (G) foundation on the geometrically nonlinear analysis of the circular plates has been investigated.

Numerical examples demonstrate the satisfactory accuracy, efficiency and versatility of the presented approach. From the numerical computation, it can be concluded that the present coupled methodology is an efficient method for the nonlinear static and dynamic analysis of circular plates with or without an elastic medium.

This paper is concerned with a simplified approach of estimating the effects of the impact of a projectile on a circular dome. The procedure to be introduced involves simplifying the impactor (projectile) and the target (the dome) by a two-degree-of-freedom (2DOF) system which is made up of two lumped masses connected by elastic springs. This modeling approach has only been adapted for analyzing the impact response behavior of beams and plates. The original contributions of this paper is the development of equations and charts for estimating the value of the lumped mass and spring stiffness in the 2DOF lumped mass model to emulate the response behavior of circular domes. Linear elastic behavior of the dome is assumed but nonlinear behavior of the impactor has been taken into account. The developed calculation procedure has been validated and illustrated by case studies.