Higher-order Deformation Research Articles

Higher-order nonlinear modeling and multi-modal performance of axially restrained diverse flexible shaft-disk assemblies

Flexible shaft-disk combinations are essential in engines, turbines, and machinery. Understanding their modal characteristics is vital for industry-wide performance, reliability, and safety. This study uses higher-order modeling and multi-modal analysis to grasp complex dynamics, especially with multiple disks of varied sizes. Employing Hamilton’s method and Galerkin’s technique, equations of motion are formulated and then utilized for multi-modal analysis considering structural nonlinearity. Various shaft-disk arrangements are analyzed, including disk number, size, position, and shaft material, to understand their impact on modal behavior. The findings reveal the substantial impact of disk count, sizes, and placements on modal dynamics and intricate vibration characteristics across diverse spinning speeds. This effect is accentuated by integrating higher-order modeling into multi-modal analysis. Notably, higher-order deformation enhances overall resilience capacity and imposes more effective vibration restrictions than the linear approach. Furthermore, the investigation explores how different shaft materials influence modal characteristics, critical speeds, and vibration amplitudes in various shaft-disk combinations. Theoretical findings are validated using numerical results from 3D finite element-based models simulated with ANSYS 3D for accuracy. Ultimately, the study introduces a theoretical framework adaptable to numerous flexible shaft-disk combinations, offering a valuable resource for understanding and enhancing system behavior, benefiting engineers and researchers alike.

Static and dynamic stabilities of modified gradient elastic Kirchhoff–Love plates

The static and dynamic stabilities of modified gradient elastic Kirchhoff–Love plates (MGEKLPs), which incorporate two length-scale parameters related to strain gradient and rotation gradient effects, are comprehensively analyzed under various load forms and boundary conditions (BCs). The study of static stability employs static balance method and an improved energy method by introducing higher-order deformation gradients and corresponding energy terms. Utilizing the variational method, a sixth-order fundamental buckling differential equation for MGEKLPs under both transverse and in-plane loads is derived, serving as the foundation for the static balance method. The static stability analysis of MGEKLPs examines the combined effects of strain and rotation gradients on size-dependent critical buckling loads. Building on generalized strain energy with higher-order deformation energy, the energy method of classical elastic thin plate model is enhanced and applied to the static stability analysis of MGEKLPs. This approach enables the investigation of static stability without being constrained by the need to solve complex differential equations, making it applicable to various BCs and load scenarios. While static stability provides a description of stable state of an elastic system, dynamic stability offers a more scientific and rigorous analysis. The dynamic stability of simplified gradient elastic Kirchhoff–Love plates (SGEKLPs) with curved edges and different BCs is further investigated by combining the generalized strain energy with Lyapunov’s second stability method, presenting the dynamic stability criterion in the form of norms. A strict description of the dynamic stability of a SGEKLP over the entire time domain is provided for different supporting conditions, including case where all edges are supported and case with free edges. The analysis of size-dependent static and dynamic stabilities offers theoretical guidance for designing elastic thin plates with microstructures.

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.

Nonlinear Geometrical Effect on Deflection Characteristics of Fiber/Metal Laminates

The deflection behavior of the fiber-metal laminated structures is envisaged in this research using higher-order deformation kinematics and large (finite) deformations of geometries via Green–Lagrange strain. The panel governing equation is derived using variational principle and solved numerically considering the nonlinear finite element (FE) steps. The necessary responses are obtained using in-house MATLAB computer code and compared with an in-house simulation model (prepared in ANSYS) and experimental data (using in-house experimental set-up). The results are verified and agree with the published data available in the open domain. The proposed model results show better consequences compared to the simulation results. The detailed responses are computed for the variable geometrical conditions (i.e. aspect ratio, end support, side-to-thickness ratio), loading and stacking sequences, including the metal layer placing. The inferences indicate the importance of nonlinearity in the framework of a higher-order kinematic model for analysing the metal laminated structure.

Data-driven surrogate model for aerodynamic design using separable shape tensor method

In the context of increasing dimensionality of design variables and the complexity of constraints, the efficacy of Surrogate-Based Optimization (SBO) is limited. The traditional linear and nonlinear dimensionality reduction algorithms are mainly to decompose the mathematical matrix composed of design variables or objective functions in various forms, the smoothness of the design space cannot be guaranteed in the process, and additional constraint functions need to be added in the optimization, which increases the calculation cost. This study presents a new parameterization method to improve both problems of SBO. The new parameterization is addressed by decoupling affine transformations (dilation, rotation, shearing, and translation) within the Grassmannian submanifold, which enables a separate representation of the physical information of the airfoil in a high-dimensional space. Building upon this, Principal Geodesic Analysis (PGA) is employed to achieve geometric control, compress the design space, reduce the number of design variables, reduce the dimensions of design variables and enhance predictive performance during the surrogate optimization process. For comparison, a dimensionality reduction space is defined using 95% of the energy, and RAE 2822 for transonic conditions are used as demonstrations. This method significantly enhances the optimization efficiency of the surrogate model while effectively enabling geometric constraints. In three-dimensional problems, it enables simultaneous design of planar shapes for various components of the aircraft and high-order perturbation deformations. Optimization was applied to the ONERA M6 wing, achieving a lift-drag ratio of 18.09, representing a 27.25% improvement compared to the baseline configuration. In comparison to conventional surrogate model optimization methods, which only achieved a 17.97% improvement, this approach demonstrates its superiority.

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.

Research on the control strategy of coning amount and ASU adjustment considering profile and high-order flatness cooperative control

ABSTRACT In the production of a 20-high mill, high-order flatness defects occur frequently. To achieve collaborative control of the profile and high-order flatness of a 20-high mill, an integrated coupling model of the roll system and strip has been established based on ABAQUS software. It has been found that as the coning amount of the intermediate taper roll increased, the profile evaluation index C25 decreased, while the work roll axis exhibited high-order deflection deformation, which could lead to the generation of high-order flatness defects. Subsequently, a novel control strategy of the coning amount and backup roll segmented reduction (ASU) control adjustment has been proposed to ensure that the profile evaluation index C25 meets the requirements while eliminating the risk of high-order flatness defects caused by roll shifting (coning amount) control. The specific combination form of the control strategy has been determined by using the particle swarm optimisation (PSO) algorithm. The Industrial application results show that the maximum compressive stress in the area 75 mm-300 mm away from the edge can be reduced by more than 40% after using the novel control strategy.

Unravelling Size-Dependent and Coupled Properties in Mechanical Metamaterials: A Couple-Stress Theory Perspective.

The lack of material characteristic length scale prevents classical continuum theory (CCT) from recognizing size effect. Additionally, the even-order material property tensors associated with CCT only characterize the materials' centrosymmetric behavior and overlook their intrinsic chirality and polarity. Moreover, CCT is not reducible to 2D and 1D space without adding couples and higher-order deformation gradients. Despite several generalized continuum theories proposed over the past century to overcome the limitations of CCT, the broad application of these theories in the field of mechanical metamaterials has encountered significant challenges. These obstacles primarily arise from a limited understanding of the material coefficients associated with these theories, impeding their widespread adoption. Implementing a bottom-up approach based on augmented asymptotic homogenization, a consistent and self-sufficient effective couple-stress theory for materials with microstructures in 3D, 2D, and 1D spaces is presented. Utilizing the developed models, material properties associated with axial-twist, shear-bending, bending-twist, and double curvature bending couplings in mechanical metamaterials are characterized. The accuracy of these homogenized models is investigated by comparing them with the detailed finite element models and experiments performed on 3D-printed samples. The proposed models provide a benchmark for the rational design, classification, and manufacturing of mechanical metamaterials with programmable coupleddeformation modes.

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.

Computational aeroelastic analysis of wings based on the structural discontinuous Galerkin and aerodynamic vortex lattice methods

An original computational framework for the aeroelastic analysis of wings featuring general transverse section is developed. The framework is based on the coupling between a novel discontinuous Galerkin structural model and an aerodynamic vortex lattice method, which is implemented in both the planar and non-planar version. The structural model, which constitutes the novelty of the present work, allows generalized kinematics and is thus able to capture higher-order structural deformation modes. With respect to other more used structural representations, the discontinuous Galerkin approach is based on the use of discontinuous basis functions and suitably-defined boundary terms to enforce the inter-element continuity and boundary conditions. Such features naturally enable high-order accuracy, ease of parallelization and, specifically for this work, straightforward coupling with the vortex lattice method. The framework is validated through benchmark tests, providing favourable matching with reference literature data.

Theoretical evaluation of nonlinear thermomechanical deflection characteristics of damaged (cracked) layered composite structural panel bonded with shape memory alloy (SMA) fibre and experimental verification

Thermomechanical deflections of the cracked multi-layered composite panels were investigated numerically using full-scale geometrical nonlinearity and higher-order deformation kinematics. The panel is embedded with shape memory alloy (SMA) fiber to improve the structural strength and integrity under thermomechanical loadings. The theoretical model for the SMA-reinforced cracked laminated composite shell panel is derived considering the material and geometrical nonlinearity. The numerical and experimental cases were performed for the intact and damaged cylindrical panels by varying the environmental conditions. Additional examples of structural and geometrical parameters have been conducted, including SMA-related data.

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.

Numerical prediction of thermoacoustic responses of CNT reinforced natural (luffa) fibre/epoxy hybrid composite and experimental verification

This research reported the thermoacoustic radiation characteristics of luffa fibre/epoxy hybridised with MW-CNT (multi-walled carbon nanotube) composite flat panel structures. The acoustic responses are evaluated under harmonic excitation and uniform thermal environments. The panel model is derived from a coupled scheme by merging two different computational techniques, i.e., finite element method, FEM (structural modelling) and boundary element method, BEM (fluid part), via higher-order deformation kinematics. Additionally, the higher-order panel model utilises an equivalent single-layer theory for computational purposes. The responses are obtained through MATLAB computer code from the derived mathematical formulation by customising the parameters (geometry and material). A few frequency responses with and without environmental effects are compared with experimental data from home-grown experimentation facilities. The impact of individual parameters is thoroughly analysed, and highlighted their significance in controlling the panel’s acoustic radiation behaviour. The findings of this study contribute to the understanding of the acoustic radiation properties of luffa fibre/epoxy hybridised with MW-CNT composite flat panels. The higher-order coupled FEM-BEM scheme-based tailored MATLAB code also provides a reliable computational tool for analysing various composite structural components for futuristic design. This research would be a valuable reference for optimising composite flat panel structural design and performance in engineering applications requiring enhanced vibroacoustic control.

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.

Finite deformation analysis of the elastic circular plates under pressure loading

As one of the most commonly used engineering components, thin plates usually undergo large deflections when subjected to intense pressure loadings. In addition to the material nonlinearity, such as plastic yielding, the geometric nonlinearity caused by this finite deformation also has a great influence over its total response. In this study, a finite deformation approximation solution is given theoretically for the elastic circular plate, based on the von Kármán equations and the uniform membrane strain assumption. Through comparisons with the corresponding finite element simulations and the existing theoretical solution, the present solution is confirmed to be both accurate and efficient, for thin plate under uniform pressure loading, in both static and dynamic loading conditions. Thereafter, the competing effects and the respective dominant ranges of bending and stretching are investigated.Besides, based on the accurate stress prediction provided by the proposed high-order deformation model, the limit state of the elastic stage is analyzed, including the location that first reached the elastic limit and the corresponding instant. Thus, the present approximate solution will be valuable in future studies of failure criteria and subsequent inelastic behavior.

Time-domain viscoelastic analysis of laminated composite plates by using a unified formulation

This paper presents the time-domain viscoelastic formulation based on the Carrera unified formulation (CUF) to obtain the response of composite structures submitted to bending loads. The governing equations are obtained through a modified form of the principle of virtual displacement (PVD) for a viscoelastic analysis. The problem is solved using the Laplace transform instead of direct integration to reduce the computational cost. The results demonstrate the strong capability of the time-domain viscoelastic formulation. Overall, the proposed methodology allows the direct implementation of new optimized high order deformation theories including the thickness stretching effect that may be implemented in futures works.

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 and Dynamic Analysis of Linear Piezoelectric Structures Using Higher Order Shear Deformation Theories

This paper explores the effects of shear deformation on piezoelectric materials and structures that often serve as substrate layers of multilayer composite sensors and actuators. Based on higher-order shear elastic deformation and electric potential distribution theories, a general mathematical model is derived. Governing equations and the associated boundary conditions for a piezoelectric beam are developed using a generalized Hamilton’s principle. The static and dynamic behavior of the piezoelectric structure is investigated. A bending problem in static analysis and a free vibration problem in dynamic analysis are solved. The obtained results are in very good agreement with the results of the exact two dimensional solution available in the literature.