Functionally Graded Materials Research Articles

Hygrothermal buckling analysis of thin rectangular FGM plate with variable thickness

PurposeThis paper aims to contribute novel insights into the analysis of thin functionally graded material (FGM) plates with variable thickness, considering both temperature-dependent and independent material properties, focusing on critical linear buckling temperature rise and the effect of critical linear moisture for various moisture concentrations.Design/methodology/approachThe study derives stability and equilibrium equations for thin rectangular FGM plates under hygrothermal loading, employing classical plate theory (CPT). Buckling behavior is examined using Galerkin’s method to obtain pre-buckling force resultants.FindingsThe findings highlight significant increases in critical buckling temperature with aspect ratio, distinct temperature sensitivity between materials and increasing moisture susceptibility with larger aspect ratios. These insights inform material selection and design optimization for FGM plates under hygrothermal loading, enhancing engineering applications.Research limitations/implicationsThis research primarily focuses on hypothetical scenarios and mathematical model development and analysis.Originality/valueThis paper presents original contributions in the field by addressing the hygrothermal buckling analysis of thin FGM rectangular plates with variable thickness, utilizing CPT, thereby enriching the understanding of structural behavior in varying environmental conditions.

A numerical investigation for the development of functionally graded Ti/HA tibial implant for total ankle replacement: Influence of material gradation law and volume fraction index.

Stress shielding is one of the major concerns for total ankle replacement implants nowadays, because it is responsible for implant-induced bone resorption. The bone resorption contributes to the aseptic loosening and failure of ankle implants in later stages. To reduce the stress shielding, improvements can be made in the implant material by decreasing the elastic mismatch between the implant and the tibia bone. This study proposes a new functionally graded material (FGM) based tibial implant for minimizing the problem of stress shielding. Three-dimensional finite element (FE) models of the intact tibia and the implanted tibiae were created to study the influence of material gradation law and volume fraction index on stress shielding and implant-bone micromotion. Different implant materials were considered that is, cobalt-chromium, titanium (Ti), and FGM with Ti at the bottom and hydroxyapatite (HA) at the top. The FE models of FGM implants were generated by using different volume fractions and the rule of mixtures. The rule of mixtures was used to calculate the FGM properties based on the local volume fraction. The volume fraction was defined by using exponential, power, and sigmoid laws. For the power and sigmoid law varying volume fraction indices (0.1, 0.2, 0.5, 1, 2, and 5) were considered. The geometry resembling STAR® ankle system tibial implant was considered for the present study. The results indicate that FGMs lower stress shielding but also marginally increase implant-bone micromotion; however, the values were within the acceptable limit for bone ingrowth. It is observed that the material gradation law and volume fraction index influence the performance of FGM tibial implants. The tibial implant composed of FGM using power law with a volume fraction index of 0.1 was the preferred option because it showed the least stress shielding.

Recent developments in modeling and analysis of rotating functionally graded panels: A review

This article presents a comprehensive review on recent developments in the techniques for modeling and exploration of rotating functionally graded material (FGM) panels carried out between periods 2000 to 2023. Special attention is given on literature pertaining to analysis of rotating FGM panels subjected to thermo-mechanical loading. This study scrutinizes the advancements made in analysis of panels such as beam, plate, shell and blades that includes effects like porosity, thermal stress, and piezo-electric in the mathematical model. The present article also highlights the importance and ease in implementation of rule of mixture for calculation of different temperature dependent material properties used in the analysis of rotating FGM panels. An in-depth review of literature clearly indicates the importance of taking thermal effect in the estimation of elastic properties for accurate analysis of rotating FGM panels. Further, a detailed analysis of temperature profile viz., linear/non-linear, is carried out. Finally, the article concludes with the future areas that need to be explored for the analysis of rotating FGM panels.

EBSD-based image quality analysis of in-situ tempered martensitic steel generated by L-PBF

Microstructures resulting from additive manufacturing technologies, such as laser powder bed fusion (L-PBF), can exhibit both intended and unintended gradation. Intended in form of functionally graded materials (FGM) providing improved material performance. Un-intended in form of part specific local changes due to variations in thermal boundary conditions resulting from the geometry at hand (e.g. slender lattices or downskin areas). To characterize, avoid and optimize such gradations, vast spacial measurements are necessary. This work investigates if electron backscatter diffraction (EBSD) can provide such information, to characterize the microstructural gradation of low alloy steel. Central to this work is the analysis of the image quality (IQ), since the tempering of martensitic steel microstructures results in a reduction of dislocation density, changes in carbide precipitation as well as crystal structure and thus varying IQ. Based on the multi-peak IQ approach and information about the thermal history, the microstructure is considered as purely martensitic. The analysis of single laser track experiments reveals that the impact of cooling rates as well as in-situ tempering effects can be measured, which present the two major effects regarding microstructural formation in L-PBF. Experiments on in-situ and conventionally heat-treated samples show correlating trends regarding tempering states, hardness and IQ distribution. Vast measurements over a sample size of 10 × 10 mm2 demonstrate the possibility to analyze FGM. Furthermore, micro-gradations are observed within each laser track within the material. Consistent trends and correlations indicate that microstructural changes dominate changes in IQ values, while the impact of EBSD parameter and sample preparation appears to be minor. The results promote the suitability of IQ analysis for the spatial characterization of low alloy steel generated by L-PBF.

Effect of bi-directional material gradation on thermo-mechanical bending response of metal-ceramic FGM sandwich plates using inverse trigonometric shear deformation theory

PurposeThe purpose of this study is to investigate the bending analysis of metal (Ti-6Al-4V)-ceramic (ZrO2) functionally graded material (FGM) sandwich plate with material property gradation along length and thickness direction under thermo-mechanical loading using inverse trigonometric shear deformation theory (ITSDT). FGM sandwich plate with a ceramic core and continuous variation of material properties has been modelled using Voigt’s micro-mechanical model following the power law distribution method. The impact of bi-directional gradation of material properties over the bending response of FGM plate under thermo-mechanical loading has been investigated in this work.Design/methodology/approachIn this study, gradation of material properties for FGM plates is considered along length and thickness directions using Voigt’s micromechanical model following the power law distribution method. This type of FGM is called bi-directional FGMs (BDFGM). Mechanical and thermal properties of BDFGM sandwich plates are considered temperature-dependent in the present study. ITSDT is a non-polynomial shear deformation theory which requires a smaller number of field variables for modelling of displacement function in comparison to poly-nominal shear deformation theories which lead to a reduction in the complexity of the problem. In the present study, ITSDT has been utilized to obtain the governing equations for thermo-mechanical bending of simply supported uni-directional FGM (UDFGM) and BDFGM sandwich plates. Analytical solution for bending analysis of rectangular UDFGM and BDFGM sandwich plates has been carried out using Hamilton’s principle.FindingsThe bending response of the BDFGM sandwich plate under thermo-mechanical loading has been analysed and discussed. The present study shows that centre deflection, normal stress and shear stress are significantly influenced by temperature-dependent material properties, bi-directional gradation exponents along length and thickness directions, geometrical parameters, sandwich plate layer thickness, etc. The present investigation also reveals that bi-directional FGM sandwich plates can be designed to obtain thermo-mechanical bending response with an appropriate selection of gradation exponents along length and thickness direction. Non-dimensional centre deflection of BDFGM sandwich plates decreases with increasing gradation exponents in length and thickness directions. However, the non-dimensional centre deflection of BDFGM sandwich plates increases with increasing temperature differences.Originality/valueFor the first time, the FGM sandwich plate with the bi-directional gradation of material properties has been considered to investigate the bending response under thermo-mechanical loading. In the literature, various polynomial shear deformation theories like first-order shear deformation theory (FSDT), third-order shear deformation theory (TSDT) and higher-order shear deformation theory (HSDT) have been utilized to obtain the governing equation for bending response under thermo-mechanical loading; however, non-polynomial shear deformation theory like ITSDT has been used for the first time to obtain the governing equation to investigate the bending response of BDFGM. The impact of bi-directional gradation and temperature-dependent material properties over centre deflection, normal stress and shear stress has been analysed and discussed.

Optimizing mechanical behavior of middle ear prosthesis using finite element method with material degradation FGM in three functions

Advancements in technology have revolutionized healthcare, with notable impacts on auditory health. This study introduces a novel approach aimed at optimizing materials for middle ear prostheses to enhance auditory performance. We developed a finite element (FE) model of the ear incorporating a pure titanium TORP prosthesis, validated against experimental data. Subsequently, we applied Functionally Graded Materials (FGM) methodology, utilizing linear, exponential, and logarithmic degradation functions to modify prosthesis materials. Biocompatible materials suitable for auditory prostheses, including Stainless Steel, titanium, and Hydroxyapatite, were investigated. Our findings indicate that combinations such as Stainless Steel with titanium and Hydroxyapatite offer improved outcomes compared to pure titanium and Hydroxyapatite ceramic, in terms of both displacement and stress. Additionally, personalized prostheses tailored to individual patient needs are feasible, underscoring the potential for further advancements in auditory healthcare.

A new higher-order shear deformation theory for flutter instability of a hybrid sandwich panel resting on a viscoelastic foundation

Magnetorheological elastomers (MREs) are smart materials characterized by their ability to adjust mechanical properties through the manipulation of applied magnetic fields, facilitating rapid and reversible tunability. The incorporation of reinforcing fibers introduces distinctive materials termed Magnetorheological Elastomer Composites (MRECs), combining tunability with enhanced rigidity. This study delves into the flutter analysis of a sandwich panel with a mid-layer composed of laminated MRECs and face sheets composed of functionally graded materials (FGMs) featuring porosity. The sandwich panel is situated on a viscoelastic foundation and exposed to a supersonic air current. Employing a higher-order (hyperbolic) shear deformation theory (HSDT), the displacement field of the sandwich plate is estimated. This selection eliminates the requirement for shear correction factors. The mechanical characteristics of the three sheets are determined using the Halpin-Tsai micromechanical method and the rule of mixture. The First-order Piston theory is employed to evaluate the effect of aerodynamic pressure and airflow damping on the sandwich panel. The derivation of the motion equation is achieved through Hamilton's principle. In order to solve the motion equation of the structure, the Finite Element Method (FEM) is utilized. The investigation systematically explores the influence of diverse parameters on the flutter boundary of the sandwich panel.

Nonlinear thermoelastic wave propagation in general FGM sandwich rectangular plates

The present work is dedicated to investigating the thermoelastic wave propagation behavior of sandwich rectangular plates (SRP) made of functionally graded material (FGM). The main contribution lies in the partial modification of basic theoretical expressions and solution methods to improve the accuracy of practical system models. An analytical model with three types of general configurations is established. The porosity distribution in FGM layers depends on the degree of mixture of the constituent materials, with the FGM layers without porosity taken as a reference model. The effect of porosity within FGMs is addressed through a refined analytical formulation of material properties, and the temperature-dependent material properties of FGM sandwich structures (FGMSS) maintain continuity through the thickness. This improved framework introduces a porosity function encompassing three distinct structural and geometrical functions: the core-to-thickness ratio (CTR), porosity volume fraction (PVF), and porosity distribution function (PDF). It is worth mentioning that the theoretical expressions maintain good continuity and reliability under the influence of thermal conditions and system parameters of the proposed structures. Furthermore, considering the generation of thermal strain energy (TSE) caused by thermal expansion of the structure in the normal direction, an improved analytical approach for determining TSE in rectangular plate structures is then investigated by introducing the Green's nonlinear strain (GNS). Hamilton's principle is applied to derive the wave motion equations and analytical solutions for the wave dispersion relations are derived. Furthermore, accurate numerical simulation is performed and the solution is verified with data available in published resources. In addition, we present a systematic parametric analysis to examine the effects of porosity, configuration, power-law exponent (PLE), PVF, CTR, temperature, and wave number on the thermoelastic wave propagation behavior of FGMSRP.

Multilayer heterostructure inhomogeneous model for the functionally graded spherical shell with rotation effect for arbitrarily varying material properties

With the development of functionally graded materials (FGMs) composite technology, structural safety analysis related to FGMs has been widely concerned. An innovative multilayer heterostructure power-law inhomogeneous (MHPI) model is taken to solve the rotating FGMs hollow spherical shell with arbitrarily varying material properties (elastic modulus and density) along the radial direction. The basic idea of MHPI model is that the rotating FGMs spherical shell is divided into multiple sublayers, and two power functions approximate the elastic modulus and density of each sublayer respectively. For each single sublayer, the analytical solutions include two undetermined constants, which can be obtained from the boundary and continuity conditions. Many numerical examples of the rotating FGMs spherical shell are discussed, involving seven property profiles gradient laws assumptions, three boundary conditions, and nine volume fraction gradient models. Finally, this study further discusses the rotating FGMs spherical shell optimization design. The greatest advantage of this study is that the elastic response of the rotating FGMs spherical shell under various gradient assumptions is completely discussed. The numerical results show that the MHPI model can solve the problem of circumferential stress oscillation well and have good convergence and high accuracy. For many numerical examples, the error of the MHPI model results is about 5‰ when sublayer number reaches 15.

Effect of interdiffusion layer of functionally graded Al–Cu composites fabricated by spark plasma sintering on mechanical and thermal properties

The effect of the microstructure of multi-layer Al–Cu functionally graded materials (FGMs) fabricated through spark plasma sintering (SPS) on the mechanical and thermal properties was systemically investigated. Uniformly mixed Al–25, 50, 75 Cu (vol%) powder samples were prepared by mechanical ball milling and used for fabricating two types of FGMs: FGM 1 with three layers (Cu/Al–50Cu/Al) and FGM 2 with five layers (Cu/Al–75Cu/Al–50Cu/Al–25Cu/Al). These FGMs were then sintered at 400 °C under a compressive pressure of 250 MPa for 5 min. The microstructures of the sintered FGMs exhibited gradual variations in the Al and Cu contents. An interdiffusion layer composed of Cu9Al4 and CuAl2 at the layer/layer interfaces was observed, the thickness of which was analyzed by energy-dispersive spectroscopy line mapping. The Vickers hardness of FGM 1 and FGM 2 were higher than that calculated by rule of mixture due to the formation of Al–Cu IC such as Cu9Al4 and CuAl2. The drastic enhancement in a Al–75Cu layer was obtained by dominant formation of Cu9Al4. On the other hand, the thermal conductivity of the prepared FGMs was lower than that of Al due to the formation of Al–Cu ICs, enhanced heat dissipation compared with the Al–Cu composite materials. The mechanical and thermal properties of Al–Cu FGMs can be controlled by designing the layer composition or the number of layers.

Thermoelastic analysis of disk with FEM for FGM variable parameters

Functionally graded materials (FGMs) using the finite element method (FEM) with C 0 elements, focusing on the examination of material variable parameters. The research advances the understanding of FGMs through a thermo-structural analysis of an FGM disk, assessing its response under thermal and mechanical loads. Thermal and thermo-mechanica analysis demonstrates the precision of the proposed approach in delineating the complex behavior of FGMs under diverse conditions. The research contributes to enrich the comprehension of FGM behavior using simple FEM models.

Achieving High-Temperature Stable Structural Color through Nanostructuring in Polymer-Derived Ceramics.

Structural colors offer a myriad of advantages over conventional pigment-based colors, which often rely on toxic chemical substances that are prone to UV degradation. To take advantage of these benefits in demanding environments, there is growing interest in producing structural colors from ceramics. Polymer-derived ceramics (PDCs) emerge as a compelling choice, presenting two distinct advantages: their enhanced shape ability in their polymeric state associated with impressive temperature resistance once converted to ceramics. This study pioneers the fabrication of noniridescent structural colors from silicon oxycarbide (SiOC) PDC, enabled by the nanostructuring of an inverse photonic glass within the PDC material. This design, a functionally graded material with an inverse photonic glass (FGM-PhG) structure, leverages the innate light-absorbing properties of SiOC, yielding a vivid structural color that maintains its saturation even in white surroundings. This study elucidates the process-structure-properties relationship for the obtained structural colors by investigating each layer of the functionally graded material (FGM) in a stepwise coating deposition process. To further emphasize the exceptional processing flexibility of PDCs, the three-step process is later transferred to an additive manufacturing approach. Finally, the FGM-PhG structural colors are demonstrated to have remarkable thermal stability up to 1000 °C for 100 h, possibly making them the most thermally stable ceramic structural colors to date.

Investigation on dual wire TIG Arc additive manufacturing of IN625 and SS316L FGM for continuous gradient and sandwich structures

Fabrication of functionally graded material (FGM) has been studied using dual wire WAAM-TIG (wire arc additive manufacturing - tungsten inert gas) process. The experimental research aims to perform the feasibility study of different designs of FGMs, including continuous gradient (CG-FGM) and sandwich structures (SW-FGMs). The weight percentage of two wire materials, SS316L and IN625, was precisely controlled using individual wire feeding systems. Process parameters were identified to obtain the defect-free deposition for the desired combination of weight percentage. The governing elements, i.e., Ni and Fe, showed a gradual variation in the case of CG-FGM and alternate peaks for SW-FGMs while examined using energy-dispersive X-ray spectroscopy (EDS). The alternate material distinction was noticeable with the increase in the number of individual layers for each set of SW-FGMs. The microstructure evolution of CG-FGM showed a smooth transition from equiaxed to columnar dendrites along the build direction, while SW-FGM showed heterogeneous behaviour. The hardness was steadily increasing in CG-FGMs and formed a wavy pattern in SW-FGMs. X-ray diffraction (XRD) analysis confirmed the formation of the austenitic phase, and variation in peak heights correlates to the change in microstructure and hardness. The conducted tribological study showed the specific wear rate gradually decreased with an increase of Ni content in CG-FGM, whereas in SW-FGM, it showed a strong dependency on the adopted design. The alloying effect in SW-FGM with different design modalities and its comparison with equal weight percentages were also studied in detail and presented.

Introducing ANN-GP algorithm to estimate transient bending of the functionally graded graphene origami-enabled auxetic metamaterial structures

This article presents a new method called the artificial neural networks-genetic programming (ANNs-GP) algorithm, which effectively predicts the bending behavior of functionally graded graphene origami-enabled auxetic metamaterial (FG-GORAM) structures under transient conditions. Functionally graded materials (FGMs) display spatial heterogeneity in their composition and microstructure, resulting in distinctive mechanical characteristics that make them well-suited for a wide range of engineering applications. The objective of this study is to create a prediction model that can accurately capture the intricate transient bending behavior of FGM structures. To do this, the researchers have used the ANN-GP technique, which combines ANNs with GP. The ANN component acquires knowledge from a dataset including actual or simulated bending data, while the GP component fine-tunes the structure and parameters of the neural network to improve its ability to make accurate predictions. The proposed algorithm combines the strengths of ANNs and GP to accurately predict the bending behavior of FG-GORAM structures. This algorithm is robust and efficient, allowing designers and engineers to optimize the performance and reliability of these structures in various applications. The effectiveness of the ANN-GP method is proved by comparing it to experimental or simulated data. This shows that the algorithm has the potential to be a useful tool for designing and analyzing sophisticated materials and structures.

Nonlinear vibration control of interconnected functionally graded fluid-conveying pipeline

In this study, an exploratory vibration experiment is established and an interconnected functionally graded material (FGM) pipeline system is installed. This experiment represents the first time that NiTiNOL-steel wire ropes (NiTi-ST) are applied to FGM pipeline system. And the experimental results demonstrate that the NiTi-ST has a practical vibration damping effect. Based on the experiment, a further theoretical pipeline system model is created. The effects of fluid velocity, temperature difference, and clamp stiffness on the modal and response of the pipeline system are added. The modal analysis part is solved through the implementation of the Rayleigh-Ritz approach using a set of modified orthogonal polynomials. In order to verify the accuracy of the Raylei-Ritz calculations, the finite element method (FEM) is also used. The response analysis part is solved through the Galerkin truncation method (GTM) and harmonic balance method (HBM). To match the harmonic balance solutions, the Runge-Kutta method (RK) are adopted. The results of the response analysis reveal that NiTi-ST demonstrates significant nonlinear vibration control capabilities to the interconnected FGM pipeline system under various conditions.

Effect of Three Layered Centrifugal Casting on Tribological and Microstructural Characteristics of Mg Based Functionally Graded Material Alloy

Objectives: The aim of this study is to examine the tribological properties and microstructural of functionally graded material (FGM) composites based on magnesium (Mg) base material. Magnesium alloys are commonly employed in the development of biomaterials for implant applications owing to their favorable corrosion properties. The research objective is to produce Zn/Mo reinforced functionally graded magnesium composites using the centrifugal casting. Methods: A centrifugal process was employed to fabricate a functionally graded material (FGM) consisting of three layers with a cylindrical shape. The base material used for this FGM was Magnesium, which was alloyed with 10% of Zn and 10% of Mo. The developed FGMs have been analyzed for their mechanical and microstructural characteristics. The microstructure of the samples were analyzed via the Optical Microscope (OM). It is identified that denser particle molybdenum (Mo) have influenced the mechanical and microstructural characteristics of the FGM composites. Findings: Results recommend that, all the three layered testing’s, Mg (80%) +Zn (10%) + Mo (10%) composite exhibited favorable mechanical and microstructural properties. It is identified that denser particle of Mo which is influenced the microstructural characteristics. The alteration in micro hardness in the direction of centrifugal force is observed, and it is observed that the minimum wear loss for sliding wear samples A, B & C of Mg based FGM alloy were found to be 0.0018 g, 0.028 g and 0.031 g respectively, while the maximum wear loss for sliding wear samples A, B & C of FGM alloy were found to be 0.0021 g, 0.41 g and 0.31 g respectively. Novelty: In this study, a novel three-layered centrifugal casting technique was devised. Owing to its rapid degradability, the anticipated duration of the implants within the human body would be significantly shorter in comparison to alternative biomaterials such as Titanium and Stainless steel. Furthermore, the findings from the conducted tests strongly advocate for the utilization of this technique in biomedical implantations. Keywords: Functionally graded material (FGM), Centrifugal casting, Tribological characteristics, microstructural behavior, and bio­implants

Machine learning-based prediction of heat transfer performance in annular fins with functionally graded materials

This paper presents a study investigating the performance of functionally graded material (FGM) annular fins in heat transfer applications. An annular fin is a circular or annular structure used to improve heat transfer in various systems such as heat exchangers, electronic cooling systems, and power generation equipment. The main objective of this study is to analyze the efficiency of the ring fin in terms of heat transfer and temperature distribution. The fin surfaces are exposed to convection and radiation to dissipate heat. A supervised machine learning method was used to study the heat transfer characteristics and temperature distribution in the annular fin. In particular, a feedback architecture with the BFGS Quasi-Newton training algorithm (trainbfg) was used to analyze the solutions of the mathematical model governing the problem. This approach allows an in-depth study of the performance of fins, taking into account various physical parameters that affect its performance. To ensure the accuracy of the obtained solutions, a comparative analysis was performed using guided machine learning. The results were compared with those obtained by conventional methods such as the homotopy perturbation method, the finite difference method, and the Runge–Kutta method. In addition, a thorough statistical analysis was performed to confirm the reliability of the solutions. The results of this study provide valuable information on the behavior and performance of annular fins made from functionally graded materials. These findings contribute to the design and optimization of heat transfer systems, enabling better heat management and efficient use of available space.

Active layer damping of bi-directionally tapered functionally graded sandwich plates with 1-3 piezoelectric composites

This article investigates the effect of smart damping on bi-directionally tapered functionally graded sandwich plates. The substrate comprises FG material on both sides of the core of either soft herex or ceramic material. The viscoelastic layer of ALD is restrained, while the compelling layer consists of 1-3PZC. The finite element formulation developed incorporates layer-wise and first-order-shear-deformation theory. The plate’s damping is actively controlled using velocity feedback control incorporating piezoelectric patches. The effects of various parameters of taper ratio and patch positions on vibration control are investigated. The efficacy of the ALD in improving the structural performance of plates is investigated.

Three-dimensional nonlinear stress analysis of functionally graded shells subjected to displacement-dependent loads

The nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading, developed in the companion paper, is here extended to the stress analysis of functionally graded material (FGM) shells. The term GeX means that the middle surface is described by analytical functions, including spline functions, and, therefore, the coefficients of the first and second fundamental forms can be taken exactly at element nodes. The FGM solid-shell element formulation is based on the choice of N sampling surfaces (SaS) parallel to the middle surface and located at Chebyshev polynomial nodes within the shell as well as on the outer surfaces, to introduce the displacements of these surfaces as basic unknowns. The use of Lagrange polynomials of degree N–1 in the distributions of displacements, strains, stresses, and elastic coefficients through the thickness of the shell leads to an efficient 3N-parameter shell formulation. To develop the FGM four-node solid-shell element based on the hybrid-mixed method, in which strains and stresses are utilized as primary variables, the Hu-Washizu variational principle is applied. Due to the analytical integration used to evaluate the tangent stiffness matrix, the developed FGM solid-shell element subjected to displacement-dependent loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered benchmark problems. It has been established that the difference between the second Piola-Kirchhoff and Cauchy stress tensors in non-conservative problems for FGM shells undergoing moderately large strains can be significant.

Galerkin finite block method with Lagrange multipliers method for cracked solids in functionally graded materials

This paper presents the application of the Galerkin Finite Block Method (GFBM) to address cracked solids associated with Functionally Graded Materials (FGMs), leveraging the foundational principles of the Galerkin method. The equilibrium equations pertinent to FGMs are articulated in their weak form. Employing Chebyshev polynomials as shape functions, the GFBM integrates mapping techniques to accommodate irregular finite or semi-infinite physical domains. Boundary and continuity conditions are enforced through the Lagrange Multiplier Method. The domain integrals are calculated through either analytical or numerical integration. The proposed method can easily solve the edge crack or diagonal crack problem by using the crack opening displacement method. The accuracy and convergence of the proposed method are illustrated through a selection of numerical examples. The obtained numerical solutions are verified with analytical solutions and the results from the Finite Element Method.