Higher-order Deformation Theory Research Articles

A robust five-unknowns higher-order deformation theory optimized via machine learning for functionally graded plates

This article presents a new kind of higher-order deformation theory, called Parametric Higher-order Deformation Theory (PHDT), for the static analysis of functionally graded plates (FGPs). The novelty of the PHDT is the use of strain shape functions that are calibrated by a set of tuning parameters to approximate 3D results along the plate thickness. The tuning parameters are assumed to vary with side-to-thickness ratios and power-law indexes. In contrast to higher-order shear deformation theories (HSDTs), the PHDT is not mathematically constrained to satisfy the traction-free boundary condition on the bottom plate’s surface. The proposed plate model is based on a 5-unknown HSDT previously presented by one of the authors. The governing equations are derived from the principle of virtual works, and Navier-type closed form solutions have been obtained for simply supported FGPs subjected to bisinuisoidal transverse pressure. A general methodology that uses genetic algorithms to determine the optimal tuning parameters of PHDTs for FGPs with various side-to-thickness ratios and power-law indexes is presented. The accuracy of the PHDT is assessed by comparing the results of numerical examples with a 3D elasticity solution, HSDTs reported in the literature, and the well-known Carrera Unified Formulation. The results show that quasi-3D displacement and stress distribution are obtained using a set of tuning parameters to form adaptable strain shape functions that are optimized for the given structural problem.

The effect of the viscoelastic support and GRPL-reinforced foam material on the thermomechanical vibration response of piezomagnetic sandwich nanosensor plates

This paper investigates the vibration characteristics of a sandwich nanosensor plate composed of piezoelectric materials, specifically barium and cobalt, in the upper and lower layers, and a core material consisting of either ceramic (silicon nitride) or metal (stainless steel) foams reinforced with graphene (GPRL). The study utilized the novel sinosoidal higher-order deformation theory and nonlocal strain gradient elasticity theory. The equations of motion for nanosensor sandwich graphene were derived using Hamilton's principle, considering the thermal, electroelastic, and magnetostrictive characteristics of the piezomagnetic surface plates. These equations were then solved using the Navier method. The core element of the sandwich nanosensor plate can be represented using three distinct foam variants: a uniform foam model, as well as two symmetric foam models. The investigation focused on analyzing the dimensionless fundamental natural frequencies of the sandwich nanosensor plate. This analysis considered the influence of three distinct foam types, the volumetric graphene ratio, temperature variation, nonlocal parameters, porosity ratio, electric and magnetic potential, as well as spring and shear viscoelastic support. Furthermore, an analysis was conducted on the impact of the metal and ceramic composition of the central section of the sandwich nanosensor plate on its dimensionless fundamental natural frequencies. In this context, the use of ceramic as the central material results in a mean enhancement of 33% in the fundamental natural frequencies. In contrast, the incorporation of graphene into the core material results in an average enhancement of 27%. The thermomechanical vibration behavior of the nanosensor plate reveals that the presence of graphene-supported foam and a viscoelastic support structure in the core layer leads to an increase in thermal resistance. This increase is dependent on factors such as the ratio of graphene, porosity ratio of the foam, and parameters of the viscoelastic support. Metal foam or ceramic foam has been found to enhance thermal resistance when compared to solid metal or ceramic core materials. The analysis results showed that it is important to take into account the temperature-dependent thermal properties of barium and cobalt, which are piezo-electromagnetic materials, and the core layer materials ceramics and metal, as well as the graphene used to strengthen the core. The research is anticipated to generate valuable findings regarding the advancement and utilization of nanosensors, transducers, and nano-electromechanical systems engineered for operation in high-temperature environments.

A New High-order Deformation Theory and Solution Procedure Based on Homogenized Strain Energy Density

A New High-order Deformation Theory and Solution Procedure Based on Homogenized Strain Energy Density

Review and Assessment of Fatigue Delamination Damage of Laminated Composite Structures.

Fatigue delamination damage is one of the most important fatigue failure modes for laminated composite structures. However, there are still many challenging problems in the development of the theoretical framework, mathematical/physical models, and numerical simulation of fatigue delamination. What is more, it is essential to establish a systematic classification of these methods and models. This article reviews the experimental phenomena of delamination onset and propagation under fatigue loading. The authors reviewed the commonly used phenomenological models for laminated composite structures. The research methods, general modeling formulas, and development prospects of phenomenological models were presented in detail. Based on the analysis of finite element models (FEMs) for laminated composite structures, several simulation methods for fatigue delamination damage models (FDDMs) were carefully classified. Then, the whole procedure, range of applications, capability assessment, and advantages and limitations of the models, which were based on four types of theoretical frameworks, were also discussed in detail. The theoretical frameworks include the strength theory model (SM), fracture mechanics model (FM), damage mechanics model (DM), and hybrid model (HM). To the best of the authors' knowledge, the FDDM based on the modified Paris law within the framework of hybrid fracture and damage mechanics is the most effective method so far. However, it is difficult for the traditional FDDM to solve the problem of the spatial delamination of complex structures. In addition, the balance between the cost of acquiring the model and the computational efficiency of the model is also critical. Therefore, several potential research directions, such as the extended finite element method (XFEM), isogeometric analysis (IGA), phase-field model (PFM), artificial intelligence algorithm, and higher-order deformation theory (HODT), have been presented in the conclusions. Through validation by investigators, these research directions have the ability to overcome the challenging technical issues in the fatigue delamination prediction of laminated composite structures.

Dynamic Analysis of a Thick-Walled Cylindrical Shell Subjected to an Internal Moving Pressure Load and Temperature Effect Based on Higher Order Deformation Theory

Dynamic Analysis of a Thick-Walled Cylindrical Shell Subjected to an Internal Moving Pressure Load and Temperature Effect Based on Higher Order Deformation Theory

Frequency parameter analysis of bi-directional functionally graded multi-span sandwich beam

This paper presents the frequency parameter analysis of bi-directional functionally graded (2D-FG) multi-span sandwich beams by using a high order deformation theory. The beams consist of three layers, the upper face of the sandwich beam is made of ceramic, the lower face is made of 2D-FG and the core is made of 1D-FG. The material properties of the beam are assumed to vary continuously in the thickness and longitudinal directions by a power-law distribution. The frequency parameter of the sandwich multi-span beam is computed with the finite element method. The accuracy of the derived formulation is confirmed by comparing the obtained results with the published data. The effects of material and number of spans on the frequency parameter of the beam are examined and discussed.

2D-GDQM and adaptively tuned deep neural network for frequency analysis of the sandwich disk with honeycomb resting on elastic foundation

The 2D-GDQM (Two-Dimensional Generalized Differential Quadrature Method) and adaptively tuned deep neural network are two computational methods that have been proposed for the frequency analysis of sandwich disks with honeycomb cores. Sandwich disks are commonly used in aerospace and automotive industries due to their lightweight and high strength properties. However, accurately predicting their natural frequencies is crucial for ensuring their structural integrity. The 2D-GDQM is a numerical method that discretizes the equations of motion of the sandwich disk using a grid-based approach. The method has been used to obtain accurate and efficient frequency solutions for various types of sandwich structures. In this study, the 2D-GDQM was applied to analyze the frequency response of the sandwich disk with honeycomb core with first and higher-order deformation theories to model displacement field. Additionally, an adaptively tuned deep neural network was used to predict the natural frequencies of the sandwich disk. This method involves training a deep neural network with a dataset of frequency solutions obtained from the 2D-GDQM simulations. The neural network is then optimized to provide accurate predictions for new cases. The results of this study showed that both the 2D-GDQM and the adaptively tuned deep neural network can provide accurate predictions of the natural frequencies of the sandwich disk with honeycomb core. The 2D-GDQM was found to be more computationally efficient, while the neural network approach can be more flexible and adaptable to new geometries or material properties.

Nonlinear thermal post-buckling analysis of rectangular FG plates using CUF

In this study, a numerical analysis examines the thermal post-buckling response of rectangular functionally graded (FG) plates. Since the post-buckling is analyzed in a thermal environment, temperature distribution across the thickness is computed by an accurate numerical method. Here, for the geometrically nonlinear analysis because of assuming large displacement, FEM is utilized according to Carrera Unified Formulation (CUF), a precise formulation and higher-order deformation theory. The principle of virtual work is employed to achieve the nonlinear equilibrium equations. The governing equations are solved by operating the arc-length method. The influence of parameters such as geometric aspect ratio, length-to-thickness ratio, boundary conditions, type of temperature distribution, and volume fraction index on the FG plates’ temperature-deflection path has been investigated. In addition, a comparison among the present study method with another reference about thermal post-buckling and also linear buckling analysis (LBA) is performed. Nonlinear post-buckling analysis shows that the bifurcation point occurs only at fully clamped boundary conditions and low aspect ratios. It proved the results of LBA in the mentioned state are verifiable. The outcomes confirm the usefulness of utilizing CUF on thermal post-buckling since the temperature curves in the figures are at a lower level compared to the similar reference.

A Legendre-Ritz solution for bending, buckling and free vibration behaviours of porous beams resting on the elastic foundation

By using high-order deformation theory, this paper presents a novel approach for analysing the mechanical behaviours of functionally graded porous beams resting on a Winkler-Pasternak elastic foundation. The governing equations are derived from the Lagrange principle. Legendre-Ritz polynomial functions, which possess the simplicity of algebraic polynomials and the efficiency of orthogonal polynomials, are developed to solve the problem. Three boundary conditions accompanied by two porosity types, including symmetric and asymmetric distributions of beams, are considered. The effects of porosity, boundary condition, foundation parameter, span-to-height ratio, and distribution type on the mechanical behaviours of porous beams are investigated. Some new results are presented for the first time as a benchmark for future studies.

Static analysis and boundary effect of FG-CNTRC cylindrical shells with various boundary conditions using quasi-3D shear and normal deformations theory

In this paper, static analysis and the stress concentration phenomenon of functionally graded carbon nanotubes reinforced composite cylindrical shells with various boundary conditions are explored by using higher-order shear-normal deformation theory and an analytical approach. Five types of distributions of reinforced carbon nanotubes are considered, that is, uniform and four kinds of functionally gradient distribution of carbon nanotubes along with the thickness of the shell. Effective material properties of carbon nanotubes reinforced composite are estimated to the rule of mixture. The governing equations for FG-CNTRC cylindrical shells with various boundary conditions are solved analytically using simple trigonometric series and the Laplace transform. Moreover, the transverse shear and normal stress are reconstructed using 3D linear elasticity equilibrium equations. The present model is validated by comparing its numerical results with those of published works, including the 3D exact model. Using higher-order shear-normal deformation theory, this article has focused on investigating the stress concentration phenomenon, the distribution of stress components in the near-boundary region, and also assessing the effects of various parameters on the stress state of the FG-CNTRC cylindrical shells.

Micro-cantilever bending tests for understanding size effect in gradient elasticity

Higher-order deformation theories, such as the couple stress and strain gradient theory, have been widely used to predict the mechanical behavior of micro/nano-scale structures. In this paper, the additional length scale parameter introduced in the couple stress theory is measured by performing bulk-scale tensile and micro-scale cantilever bending experiments. Bulk-scale characterization provided microstructural information of the polycrystalline copper plate along with macroscopic mechanical properties. Micro-scale cantilevers with thicknesses ranging from 1.6 µm to 8.6 µm were fabricated within the copper plate using femtosecond laser machining followed by focused ion beam milling. Line load was applied on these samples utilizing a nanoindenter. Finite element analysis was performed to exclude the effect of substrate deformation. The measured effective elastic modulus increases from 94 GPa to 215 GPa with decreasing thickness. The increase of the bending rigidity is analyzed based on the couple stress theory, and the length scale parameter is measured as 0.78 µm. The physical origin of the length scale parameter is discussed by considering mobile dislocations escaping via the free surfaces during loading.

Mechanically induced electric and magnetic fields in the bending and symmetric-shear deformations of a microstructure-dependent FG-MEE composite beam

A new FG-MEE composite microbeam model is developed using a general higher-order deformation theory (GHDT) to account for the symmetric thickness-shear and thickness-stretch deformations of a beam and a modified couple stress theory (MCST) to describe the microstructure-dependent size effect. Based on the basic assumption of field variables and Hamilton’s principle, the one-dimensional second-order coupled equations of motion and complete boundary conditions are obtained simultaneously. The present coupled equations include the microstructure dependence and higher-order extensions, which can reduce to the size-dependent Bernoulli-Euler, Timoshenko and Mindlin-Medick theories. Static problems of a simply supported beam loaded by uniform loading and a concentrated second-order moment are analytically solved by directly applying the newly developed equations. For the bending case, the numerical results show that the magnitudes of displacement, electric, magnetic, stress, electric displacement and magnetic flux fields are smaller than those predicted by classical theory. In addition, it is found that FG parameter can be used to control the electric and magnetic fields. For the concentrated second-order moment case, a new phenomenon of local electric and magnetic fields are demonstrated in the present FG-MEE composite microbeam structure. These results are useful for theoretically/numerically estimating or designing the MEMS of FG-MEE composites.

Effect of porosity on the bending analysis of various functionally graded sandwich plates

A new porosities distribution is proposed for bending analysis of new model of functionally graded material (FGM) sandwich plates. Material properties of FGM layers are assumed to vary continuously across the plate thickness according to either power-law or sigmoid function in terms of the volume fractions of the constituents. The face layers are considered to be FG across each face thickness while the core is made of a ceramic homogeneous layer. New higher-order deformation theory is proposed to derive the field equations of the FG sandwich plates with simply-supported edge conditions. Numerical results are presented to show the effect of the material distribution, the sandwich plate geometry and the porosity on the deflections and stresses of FG sandwich plates. The accuracy of this theory is ascertained by comparing it with other published results.

A layerwise finite element formulation for vibration and damping analysis of sandwich plate with moderately thick viscoelastic core

Most previous studies of viscoelastic sandwich plates were based on the assumption that damping was only resulting from shear deformation in the viscoelastic core. However, extensive and compressive deformations in the viscoelastic core should also be considered especially for sandwich plates with moderately thick viscoelastic core. This paper presents a finite element formulation for vibration and damping analysis of sandwich plates with moderately thick viscoelastic core based on a mixed layerwise theory. The face layers satisfy the Kirchhoff theory while the viscoelastic core meets a general high-order deformation theory. The viscoelastic core is modeled as a quasi-three-dimensional solid where other types of deformation such as longitudinal extension and transverse compression are also considered. To better describe the distribution of the displacement fields, auxiliary points located across the depth of the sandwich plate are introduced. And based on the auxiliary points, the longitudinal and transverse displacements of the viscoelastic core are interpolated independently by Lagrange interpolation functions. Quadrilateral finite elements are developed and dynamic equations are derived by Hamilton’s principle in the variation form. Allowing for the frequency-dependent characteristics of the viscoelastic material, an iterative procedure is introduced to solve the nonlinear eigenvalue problem. The comparison with results in the open literature validates the remarkable accuracy of the present model for sandwich plates with moderately thick viscoelastic core, and demonstrates the importance of the higher-order variation of the transverse displacement along the thickness of the viscoelastic core for the improvement of the analysis accuracy. Moreover, the influence of the thickness and stiffness ratios of the viscoelastic core to the face layers on the damping characteristics of the viscoelastic sandwich plate is discussed.

Optimization of Laminates for Maximum Buckling Load

Objective of this work is to optimize the structural configuration of laminates in terms of design for the maximization of buckling resistance of laminates under the effect of constraints. The predominant design parameter is the orientation of the fibers in the layers of the laminate. High-order deformation theory is used in the mathematical formulation in the buckling analysis of stratified composite plates. For this purpose, we will use the optimization technique genetic algorithm taking into account the various factors: number of fibers, geometric ratio, load ratio and boundary conditions.

Effect of thickness deformation on large-amplitude vibrations of functionally graded rectangular plates

Nonlinear forced vibrations of moderately thick functionally graded (FG) rectangular plates are investigated by considering higher-order shear deformation theories that take into account the thickness deformation effect. The geometrically nonlinear strain–displacement relationships are derived retaining full nonlinear terms in the in-plane and transverse displacements and the three-dimensional constitutive equations are used by removing the assumption of zero transverse normal strain. Both simply supported movable and immovable boundary conditions are considered at the plate edges. The equations of motion are obtained by using multi-modal energy approach. A code based on pseudo arc-length continuation and collocation scheme is utilized for numerical continuation and bifurcation analysis. Results show that higher-order thickness deformation theories yield a significant accuracy improvement for nonlinear vibrations of highly pressurized functionally graded plates.

THERMAL BUCKLING RESPONSE OF LAMINATED COMPOSITE PLATE WITH RANDOM SYSTEM PROPERTIES

This paper deals with second-order statistics of buckling temperature of shear deformable laminated composite plates resting on elastic foundation with random system properties. The temperature field is assumed to be a uniform distribution over the plate surface and through the plate thickness. The material properties are assumed to be independent of temperature. Based on the higher-order deformation theory, C0 finite element in conjunction with mean-centered first-order perturbation technique is proposed and applied to derive the random thermal eigenvalue problems. A detailed parametric study is carried out to highlight the sensitivity of the response. The results are compared with existing and Monte Carlo simulation.

Thermal buckling of functionally graded plates according to a 2D higher-order deformation theory

A two-dimensional (2D) global higher-order deformation theory is presented for thermal buckling of plates made of functionally graded materials (FGMs). The modulus of elasticity of functionally graded (FG) plates is assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of displacement components, a set of fundamental equations of a 2D higher-order theory for rectangular functionally graded (FG) plates is derived through the principle of virtual work. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of FG plates with simply supported edges. In order to assure the accuracy of the present theory, convergence properties of the critical temperature are examined in detail. A comparison of the present critical temperatures of isotropic and FG plates is also made with previously published results. Critical temperatures of simply supported FG plates are obtained for uniformly and linearly distributed temperatures through the thickness of plates. Modal transverse shear and normal stresses are calculated by integrating the three-dimensional (3D) equations of equilibrium in the thickness direction satisfying the stress boundary conditions at the top and bottom surfaces. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. It is noticed that the present higher-order approximate theories can predict accurately the critical temperatures of simply supported FG plates.

Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory

A two-dimensional (2D) higher-order deformation theory is presented for vibration and buckling problems of circular cylindrical shells made of functionally graded materials (FGMs). The modulus of elasticity of functionally graded (FG) shells is assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal deformations, and rotatory inertia is derived through Hamilton’s principle. Several sets of truncated M th order approximate theories are applied to solve the eigenvalue problems of simply supported FG circular cylindrical shells. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency for the fundamental mode r=s=1r=s=1 are examined in detail. A comparison of the present natural frequencies of isotropic and FG shells is also made with previously published results. Critical buckling stresses of simply supported FG circular cylindrical shells subjected to axial stress are also obtained and a relation between the buckling stress and natural frequency is presented. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses are calculated by integrating the three-dimensional (3D) equations of motion in the thickness direction satisfying the stress boundary conditions at the outer and inner surfaces. The 2D higher-order deformation theory has an advantage in the analysis of vibration and buckling problems of FG circular cylindrical shells.

Vibration and Stability of Functionally Graded Plates Based on a Higher-order Deformation Theory

In this article, the governing equations of motion for a functionally graded material plate (FGP) based on a higher-order deformation theory in a general state of non-uniform initial stress are derived. The properties of FGP are assumed varied continuously along the thickness of the plate, according to a simple power law of volume fractions of constituents. With the derived governing equations, the natural frequencies and buckling loads of ceramic—FGM—metal plates subjected to an initial stress are investigated. The initial stress is taken to be a combination of a uniaxial extensional stress and a pure bending stress. A ceramic—FGM—metal plate can become an all-FGM, all-ceramic plate, or all-metal plate by modifying the value of material parameter and volume fraction index. The effects of various parameters and initial stresses on the natural frequencies and buckling loads of FGPs are studied.