Isogeometric analysis for functionally graded microplates based on modified couple stress theory

Analysis of static bending, free vibration and buckling behaviours of functionally graded microplates is investigated in this study. The main idea is to use the isogeometric analysis in associated with novel four-variable refined plate theory and quasi-3D theory. More importantly, the modified couple stress theory with only one material length scale parameter is employed to effectively capture the size-dependent effects within the microplates. Meanwhile, the quasi-3D theory which is constructed from a novel seventh-order shear deformation refined plate theory with four unknowns is able to consider both shear deformations and thickness stretching effect without requiring shear correction factors. The NURBS-based isogeometric analysis is integrated to exactly describe the geometry and approximately calculate the unknown fields with higher-order derivative and continuity requirements. The convergence and verification show the validity and efficiency of this proposed computational approach in comparison with those existing in the literature. It is further applied to study the static bending, free vibration and buckling responses of rectangular and circular functionally graded microplates with various types of boundary conditions. A number of investigations are also conducted to illustrate the effects of the material length scale, material index, and length-to-thickness ratios on the responses of the microplates.Comment: 57 pages, 14 figures, 18 table

Nonlinear static and transient isogeometric analysis of functionally graded microplates based on the modified strain gradient theory

The objective of this study is to develop an effective numerical model within the framework of an isogeometric analysis (IGA) to investigate the geometrically nonlinear responses of functionally graded (FG) microplates subjected to static and dynamic loadings. The size effect is captured based on the modified strain gradient theory with three length scale parameters. The third-order shear deformation plate theory is adopted to represent the kinematics of plates, while the geometric nonlinearity is accounted based on the von Kármán assumption. Moreover, the variations of material phrases through the plate thickness follow the rule of mixture. By using Hamilton’s principle, the governing equation of motion is derived and then discretized based on the IGA technique, which tailors the non-uniform rational B-splines (NURBS) basis functions as interpolation functions to fulfil the C2-continuity requirement. The nonlinear equations are solved by the Newmark’s time integration scheme with Newton-Raphson iterative procedure. Various examples are also presented to study the influences of size effect, material variations, boundary conditions and shear deformation on the nonlinear behaviour of FG microplates

Post-buckling of functionally graded microplates under mechanical and thermal loads using isogeometric analysis

The present study uses the isogeometric analysis (IGA) to investigate the post-buckling behavior of functionally graded (FG) microplates subjected to mechanical and thermal loads. The modified a strain gradient theory with three length scale parameters is used to capture the size effect. The Reddy third-order shear deformation plate theory with the von Kámán nonlinearity (i.e., small strains and moderate rotations) is employed to describe the kinematics of the microplates. Material variations in the thickness direction of the plate are described using a rule of mixtures. In addition, material properties are assumed to be either temperature-dependent or temperature-independent. The governing equations are derived using the principle of virtual work, which are then discretized using the IGA approach, whereby a C2-continuity requirement is fulfilled naturally and efficiently. To trace the post-buckling paths, Newton’s iterative technique is utilized. Various parametric studies are conducted to examine the influences of material variations, size effects, thickness ratios, and boundary conditions on the post-buckling behavior of microplates

Generalized conforming Trefftz element for size-dependent analysis of thin microplates based on the modified couple stress theory

In this work, a new displacement-based Trefftz plate element is developed for size-dependent bending analysis of the thin plate structures in the context of the modified couple stress theory. This is achieved via two steps. First, the Trefftz functions, that are derived by introducing the thin plate bending assumptions into the three-dimensional governing equations of the modified couple stress elasticity, are adopted as the basis functions for designing the element's displacement interpolations. Second, the generalized conforming theory is employed to meet the interelement compatibility requirements in weak sense for ensuring the convergence property. The resulting 4-node displacement-based plate element performs like nonconforming models on coarse meshes and gradually converges into a conforming one with the mesh refinement. Numerical tests reveal that the new element can efficiently capture the size-dependent mechanical responses of thin microplates and exhibits satisfactory numerical accuracy and distortion tolerance. Moreover, as the element has only three degrees of freedom (DOF) per node, it can be easily incorporated into the commonly available finite element programs

State-space Levy solution for size-dependent static, free vibration and buckling behaviours of functionally graded sandwich plates

The size-dependent static, free vibration and buckling behaviours of functionally graded (FG) sandwich plates are analysed in this study. Utilising the modified couple stress theory and variational principle, governing equations of motion are developed with a refined shear deformation theory. The rectangular plates embedded on two opposite simply-supported edges with the arbitrary combinations of the other two. Based on the state-space Levy solution, the deflections, stresses, natural frequencies and critical buckling loads are analytically solved for the closed-form formulations. The effects of material distribution and graded schemes, geometric parameters and boundary conditions are also investigated to examine the size-dependent behaviours of FG sandwich microplates

Geometrically nonlinear polygonal finite element analysis of functionally graded porous plates

In this study, an efficient polygonal finite element method (PFEM) in combination with quadratic serendipity shape functions is proposed to study nonlinear static and dynamic responses of functionally graded (FG) plates with porosities. Two different porosity types including even and uneven distributions through the plate thickness are considered. The quadratic serendipity shape functions over arbitrary polygonal elements including triangular and quadrilateral ones, which are constructed based on a pairwise product of linear shape functions, are employed to interpolate the bending strains. Meanwhile, the shear strains are defined according to the Wachspress coordinates. By using the Timoshenko's beam to interpolate the assumption of the strain field along the edges of polygonal element, the shear locking phenomenon can be naturally eliminated. Furthermore, the C0–type higher-order shear deformation theory (C0–HSDT), in which two additional variables are included in the displacement field, significantly improves the accuracy of numerical results. The nonlinear equations of static and dynamic problems are solved by Newton–Raphson iterative procedure and by Newmark's integration scheme in association with the Picard methods, respectively. Through various numerical examples in which complex geometries and different boundary conditions are involved, the proposed approach yields more stable and accurate results than those generated using other existing approaches

FREE VIBRATION INVESTIGATION ON RVE OF PROPOSED HONEYCOMB SANDWICH BEAM AND MATERIAL SELECTION OPTIMIZATION

In this paper, free vibration, modal and stress state analyses of honeycomb sandwich structures with different boundary conditions was investigated and major factors affecting the sandwich frequencies and stiffness due to material or parameter changes were determined. The representative volume element (RVE) method used in this work were analytically and numerically validated by comparing the obtained results to those available in literature. Firstly, unit cell method was used to capture the entire effects of different parameters on the free vibration of honeycomb sandwich structure in ANSYS. This study analyzed the natural frequencies of honeycomb sandwich structures with different core materials combination. The effects of foil thickness, boundary conditions, materials selection, density and presence of crack on sandwich structure were taken into consideration and examined. The proposed core had an inbuilt shaped reinforcement with different materials for effective resonance, fatigue and deformation resistance at much higher frequency

On time-dependent nonlinear dynamic response of micro-elastic solids

A new approach to the mechanical response of micro-mechanic problems is presented using the modified couple stress theory. This model captured micro-turns due to micro-particles' rotations which could be essential for microstructural materials and/or at small scales. In a micro media based on the small rotations, sub-particles can also turn except the whole domain rotation. However, this framework is competent for a static medium. In terms of dynamic investigations of micro materials, it is required to involve micro-rotations' mass inertias. This fact persuades us to pay particular attention to the micro mechanics' samples and directed us to re-derive the modified couple stress model to propose and represent a new micro-mechanic approach which is well-deserved, especially for dynamic studies of microstructures. In carrying out this job, the classical beam has provided the basic form of formulation procedure. The continuum medium has been limited to a square flat non-porous beam deducing a homogeneous isotropic micromaterial. As long as the time-dependent results are concerned due to studying micro-mass inertia in time history, there would be two solution steps. The Galerkin decomposition technique is imposed in accord with an analytical postulate to issue the algebraic problem distributing time-dependent equations. The latter, the Homotopy perturbation method delivers time-dependent outcomes. The solution methods have been validated by building numerical models in Abaqus software. On the new achievements of this study, one can declare that both static and dynamic length scale parameters are very effective in order to study vibrations of microstructures. If the values of these characteristic lengths are considerable, the nonlinear frequency analysis will be essential. Furthermore, the stiffness of the structure will be higher if the values of both length scale parameters increase

Investigating parametric homogenization models for natural frequency of FGM nano beams

This research focuses on exploring the free vibration behavior of functionally graded (FG) nano-beams. To calculate the effective properties of the FG nano-beam, which varies solely in the thickness direction, the four homogenization schemes Mori-Tanaka, Tamura, Reuss and Voigt are employed. This study employs high-order shear deformation nano-beam theory and derives the governing equations of motion using nonlocal differential constitutive relations of Eringen. Hamilton's principle is utilized in conjunction with the refined three variables beam theory. The consideration of a length scale parameter accounts for small-scale effects. Analytical solutions are obtained for a simply supported FG nano-beam and compared with existing literature solutions. The research also investigates the influence of different homogenization schemes, the nonlocal parameter, beam aspect ratio and various material compositions on the dynamic response of the FG nano-beam

A Framework for Size-dependent Structural Analysis of Smart Micro/nanoplates

This age has witnessed a proliferation of technological advancements that affected all facets of civilisation. Driven by the joint force of the evolution of sophisticated design tools, tailored material characteristics, and robust mechanics-based analyses, smart composite materials are widely used in high-performance engineering applications. Meanwhile, there is a growing interest in micro/nanoscopic structures in academia and industry due to the overwhelming trend toward portability, miniaturisation and integration in engineering. Therefore, the theoretical, computational, and experimental research communities have developed various effective methodologies to understand the structural behaviour of smart small-scale structures comprehensively. This dissertation introduces two size-dependent continuum theories, modified strain gradient and nonlocal strain gradient theories, to build the analytical framework for exploring application-driven micro/nanoplates made of smart composite materials. As examples of promising candidates for power supply and nano/microelectromechanical systems, organic solar cells and thermo-magneto-elastic sandwich nanoplates are studied. Size-dependent continuum models combined with various shear deformation plate theories are adopted to derive the governing equations. The size-sensitive static and dynamic mechanical responses, including bending, buckling, and free vibration behaviours of these ultra-fine-size structures, are predicted by capturing the size effect with material length scale or nonlocal parameters. The numerical results underlying size-dependent theories pose a new insight into the structural analysis of functional micro/nanoscopic plate-like structures. Some typical size-involving mechanical characteristics are revealed by comparing the present estimation with those from size-independent models. Moreover, the simulation outcomes thoroughly investigate several practical factors, such as boundary conditions, geometric configuration, and elastic foundation modelling parameters. In this endeavour, taking advantage of the computational efficiency and accessible operation of nonclassical continuum-based theories, the current analytical framework is suitable for exploring the size-tendency of the smart micro-/nanosized structures. The present work may serve as a benchmark for following numerical simulations and as a guide for evolving new engineering tools for modelling relevant responses by designers and manufacturers