Gradation Index Research Articles

Analytical and FE modeling of a bi-directional functionally graded Timoshenko beam under thermomechanical loading

Functionally graded materials (FGMs) have variable volume fractions in different directions, making structural analysis challenging due to shear deformation. This study investigates the behavior of a two-directional functionally graded material (TDFGM) Timoshenko beam under thermomechanical loads. Varied material properties in both axial and transverse directions are analyzed using a mathematical approach and a finite element model. The mathematical approach involves transforming the system of two ordinary differential equations to a single fourth-order differential equation to determine deflection and rotation. Additionally, a bilinear quadrilateral element is used to validate the mathematical solution. The results show that increasing grading parameters reduced transverse deflection, with a maximum reduction of 38.97% for CF beams and 39.26% for SS beams when the gradation index az increased from 0 to 1 under mechanical loading. Under combined mechanical and thermal loading, higher temperatures on the beam’s hot side lead to increased deflection, with notable increases at higher gradation indices. Moreover, increased maximum deflection by 6.29% for CF beams and 4.05% for SS beams when considering temperature-dependent properties. It was observed that when TDFGM Timoshenko beams were heated for thermal loading, the SS beam deflected in the opposite direction of the applied mechanical load. The results agree with previous research and indicate alignment between mathematical and finite element solutions. The impact of boundary conditions on the deflection of Timoshenko beams made of TDFGM is discussed. The introduced methodology enhances TDFGM beam bending analysis, allowing control over behavior through grading parameters and boundary conditions selection.

Nonlinear postbuckling and snap-through instability of movable simply supported BDFG porous plates rested on elastic foundations

This study presents, for the first time, a novel mathematical model that can address the nonlinear buckling, postbuckling, and snap-through of bidirectional functionally graded porous simply supported plates rested on elastic foundation and subjected to uniaxial/biaxial compressive loads. The parabolic shear deformation plate theory and the von Kármán strain nonlinearity are employed to derive governing equilibrium equations relative to neutral surface plane. The governing equations that consist of four nonlinear-coupled variable-coefficients partial differential equations are discretized by using the differential/integral quadrature method. The solution methodology depends on whether the discretized nonlinear algebraic system is homogeneous (with zero force vector) or nonhomogeneous. A homogeneous system can be formulated and solved as a nonlinear eigenvalue problem. Pseudo-arc-length continuation is implemented with a proposed iterative method to predict the load-deflection paths whether the system is homogeneous or not. Theoretical analysis and numerical results indicate that the type of porosity and position of in-plane loading have significant effects on the pitchfork-bifurcation or snap-through instability response of the bidirectional functionally graded porous plate. Parametric studies are presented to illustrate the impact of gradation indices, geometrical properties, porosity, and foundation constants on the postbuckling responses of bidirectional functionally graded porous plates. The proposed model may be used in designing nuclear, marine, aerospace, and civil structures with bi-directional functionally graded material constituents.

Analysis of hot band effects on the vibration and stability of a rotating FGM heavy vehicle disc brake

In this paper, vibration and stability analysis of a rotating Functionally Graded Material (FGM) disc brake considering the hot band phenomena are studied. In order to model the hot band phenomena, the disc brake is divided into two sections, the inner disc and the outer ring. The latter is where thermal and mechanical loads are applied. Based on Hamilton’s total potential energy principle and applying first-order shear deformation theory (FSDT), the governing equations are derived. Furthermore, the derived equations of motion are solved via the Generalized Differential Quadrature (GDQ) method. The results reveal that using a functionally graded material would improve the mechanical and thermal characteristics of a disc brake, most notably higher buckling temperature and smaller deflection in a disc brake. Moreover, using a functionally graded material for a disc brake will result in increasing the natural frequency of the disc, that is to say, the buckling will occur at higher rotational speeds. Following this further, the effects of the radius of loading annular ring, where thermal and mechanical loads are applied, as well as gradation index and mechanical and thermal loading, are also studied. The results show there is a non-linear relation between the radius of the loading annular ring and deflection as well as critical buckling temperature. Furthermore, a comparison of results with those available in the literature shows good agreement.

Natural frequency analysis of bidirectional functionally graded plates based on a refined shear theory using nonconforming finite elements

In this study, we investigate the free vibrations of functionally graded plates using the refined shear deformation theory. These plates have material properties that vary in two directions: the z-axis and the x-axis. The variation follows a bidirectional gradation described by a power law based on the Voigt rule of mixture. The displacement field is characterized by four variables, chosen in such a way that shear strain and stress vanish, and traction is free at the upper and lower surfaces of the plate. This eliminates the need for a shear correction factor, as required in the first-order shear deformation theory. We derive the equations of motion and related boundary conditions using Hamilton’s principle and a finite element approach. The proposed element ensures C1-continuity by including the first derivative of the transverse deflection in the kinematic field. For spatial discretization, we employ a rectangular element with four nodes and two interpolation schemes. Lagrangian shape functions are used for in-plane displacement, while a Hermitian scheme is employed for transverse deflection. This enables us to determine the natural frequencies of the 2D-FG plates. We demonstrate the validity, convergence, and accuracy of the proposed element through illustrative examples and compare it with previous literature. Furthermore, we conduct numerical analyses to examine the effects of transverse shear function, boundary conditions, plate aspect ratio, slenderness ratio, and gradation indices on the natural frequencies. The developed model provides insights into the significance of natural frequencies in the design of various structural elements, including 2D-FG plates.

Differential-integral quadrature numerical solution for free and forced vibration of bidirectional functionally graded Terfenol-D curved beam

The dynamic behaviour of a bidirectional functionally graded Terfenol-D curved beam under a moving load is the focus of this study. Combined differential and integral quadrature solve the curved beam boundary value problem. A bidirectional functionally graded Terfenol-D curved beam's damping, damped frequencies for different boundary conditions, and mode shapes were examined in free vibration investigations. Based on accessible literature data, the findings are reliable. Furthermore, DQ-IQ numerical technique findings examine the impact of many factors, including control gain, thickness, moving load velocity, bidirectional gradation index, and multi-moving load.

Experimental investigation on the pore structure and Water Inrush Evolution Law of weakly cemented fault fracture zone with different filling gradations

Water and mud inrush caused by fault is a geological disaster characterized by high frequency and huge destructiveness. It is important to study the evolutionary laws of water inrush in fault fracture zones with various filling types. The effect of filling gradation on the mesoscopic structure and seepage characteristics of fault fracture zones was investigated. The law of water inrush evolution and water inrush characteristics of fault-fractured zones with different filling gradations and strong zoning filling characteristics were studied. The results showed that for the larger Talbot gradation indices, the mass of water inrush and the fractal dimension of the lost particles were larger, the peak water pressure and the mass of the lost particles were smaller, and the duration of the initial impermeability stage was shorter for the same loading water pressure. For the fault fracture zones with strong zoning filling characteristics, the peak water pressure, the mass of water surges, and the mass of lost particles were larger, the fractal dimension of the lost particles was smaller, and the duration of the initial impermeability stage was shorter for the fracture zones with larger filling gradation were used as the initial impermeability zones. Furthermore, with larger filling gradation, we observed a greater proportion of large pores, a larger equivalent throat radius, higher pore connectivity, and coordination numbers. Filling gradation and confining pressure greatly affected the permeability of the fault. The permeability decreased by 98.71% when the Talbot gradation indices decreased from 1.25 to 0.6, and decreased by 58.4% when the confining pressure increased from 5 MPa to 15 MPa.

Thermoelastic Analysis of Functionally Graded Cylinder

In the present work, Thermo elastic analysis is performed on a functionally graded cylinder to investigate combined effect of pressure and thermal load. It is assumed that the material of the cylinder is functionally graded in which the properties are vary continuously along the thickness based on a power law function. Results are obtained using Finite element-based software ANSYS. To know the accuracy of the present approach, results are confirmed with available literature. Furthermore, the effect of the temperature rising on the distribution of the displacement and stress for different values of gradation index are investigated in the paper. From the results it is concluded that replacing the traditional material with functionally graded material is helpful in the design of pressure vessels.

Dynamic analysis of 2DFGM porous nanobeams under moving load with surface stress and microstructure effects using Ritz method

This paper investigates the dynamic behavior of micro/nanobeams made of two-dimensional functionally graded porous material (2DFGPM) under accelerated, decelerated, and uniform moving harmonic load, using surface elasticity and modified couple stress theories. The key feature of this formulation is that it deals with a higher order shear deformation beam theory. The non-classical equilibrium equations are developed using Lagrange's equation and the concept of physical neutral surface. The equations of motion are derived using the same approach, accounting for the porosity effect and the modified power-law distribution of material properties. The trigonometric Ritz method is used with sinusoidal trial functions for the displacement field, and the Newmark method is applied to obtain the dynamical response of 2DFGPM nanobeams. The results are compared with previous studies, and the impact of critical parameters such as gradation indices, volume fraction ratio, pattern of porosity, velocity, frequency, and motion type of the applied force are explored. This study highlights the importance of considering the porosity effect, as neglecting it can lead to significant errors in the predicted results. Additionally, the study found that the accelerated and decelerated motions of the applied load have a greater impact on the dynamical deflection of 2DFGPM nanobeams than the uniform motion. The findings of this study can provide guidance for the optimal design of micro/nanobeams subjected to a moving force with multifunctional properties.

Experimental Research on High-Temperature Stability of Asphalt Concrete Panels of Impermeable Layers

In order to study the slope stability of an impervious layer asphalt concrete panel, in this study, the maximum aggregate size used was 19 mm, and a slope flow value test was carried out after changing the gradation index, filler content and bitumen aggregate ratio. The test results showed that the relationship curve between the slope flow value and the test time was mainly divided into three stages for the slope flow value: an almost linear growth stage, a gradual stabilization stage, and a stable stage. The grading index, bitumen aggregate ratio and filler content had an effect on the slope flow value of asphalt concrete. The slope flow value decreased with the increase in the grading index. A reasonable increase in the grading index can increase the slope stability of the asphalt concrete panel. The slope flow value increased with the increase in the filler content and bitumen aggregate ratio. When the filler content exceeded 13%, the slope flow value significantly increased. At the same time, it was also verified that the asphalt concrete slope with the maximum aggregate size of 19 mm had good thermal stability. On this basis, a prediction model of asphalt concrete slope flow value and test time was established. The model considered the effect of different parameters of mix proportion on the slope flow value. The calculation results were in good agreement with the test results.

Novel incremental procedure in solving nonlinear static response of 2D-FG porous plates

The manuscript presents a nonlinear mathematical model capable to investigate the nonlinear bending response of Bi-directional functionally graded (BDFG) plate resting on elastic foundations including large deformations in the sense of von Kármán. A unified higher order shear plate theory is used to implement the shear influence with parabolic distribution through thickness direction. The gradation of materials is portrayed by power law function through axial (x-direction) and thickness (z-direction). The governing equations consist of a set of four nonlinear coupled partial differential equations. The assumption that the material properties change in the x- direction complicates the governing equations since they have variable-coefficients. Winkler–Pasternak model is exploited to present the elastic foundations with normal and shear deformations. In this work, a novel incremental–iterative method is proposed, and some tricky computational issues are performed to ensure stable and fast convergence rates even with large incremental load steps. The differential/integral quadrature method (DIQM) is developed to numerically discretize the variable-coefficient governing equations on a rectangular plate with different boundary conditions. Parametric studies are presented to illustrate the impact of nonlinear strains, gradation indices, geometrical properties, and boundary conditions on the transverse displacement, normal and shear stresses.

Analysis of stress shielding reduction in bone fracture fixation implant using functionally graded materials

Bone density reduction occurs when implants are used to arrest fracture propagation. Bones are stress shielded by the implants when enough loads are not placed on the bones. Stress-shielding occurs when stiffness mismatch between the bone and the fracture fixation implant is significant. The aim of this study was to perform an analysis of stress-shielding reduction using functionally graded material (FGM) implants when Young’s modulus vary along the length of the implant. In this study, the finite element method (FEM) was used for the simulation of functionally graded implant mounted on a bone. The effect of FGMs on the reduction of stress-shielding imposed on a bone was studied parametrically by conducting simulations of various two-dimensional (2D) and three-dimensional (3D) computational models. First, a low-fidelity 2D model was created where it was observed that the FGMs were capable of reducing the stress-shielding substantially. Then, high-fidelity models such as single-sided with and without screws as well as double-sided FGM implants with different shapes for consideration of partial cracks were simulated. Several simulations were carried out to investigate the minimum required number of layers in an FGM to avoid stress concentration at the FGM layer interfaces. Furthermore, various material gradations in an FGM were tested to present the FGM gradation index that can reduce the stress shielding. Lastly, it was established convincingly through the finite element simulations that FGMs were able to reduce the stress-shielding on a bone with a crack.

Bending of Bi-directional inhomogeneous nanoplates using microstructure-dependent higher-order shear deformation theory

In contrast of experimental analysis in the mechanical behavior of the complicated structures, the progress of theoretical research has many difficulties, especially with regard to the modelling of structures. The main objective of this research is to analyze analytically the static problem of a new model of functionally graded materials, namely the “bi-coated FGM” plate. The materials are graded continuously through 2-directional by using a power law function. Two types of coated FG plates are investigated, Hardcore and Softcore FG plates. Based on the generalized field of displacement, a Quasi-3D higher-order shear deformation plate theory is proposed in this work by reducing the number of variables from six to five variables. The equilibrium equations are performed based on the principle of virtual work and solved using the Galerkin method to cover different boundary conditions. The nonlocal strain gradient theory is employed to capture the microstructure-dependent effect. The model is validated compared to recent relevant research interests. Numerical results are performed to present the impact of hardcore and softcore distributions, gradation indices, and boundary conditions on static bending deflection and stresses of bi-coated FGM plate.

Electro-Elasto-Statics of porosity-gradient smart functionally graded plates with piezoelectric fibre-reinforced composite

Smart structures offer functional advantages over conventional composites through capabilities of material tailoring and controlled response under external actuation/sensing. The porosity induced in functionally graded (FG) materials due to manufacturing processes can be detrimental to the strength and durability characteristic of aerospace/automobile components. On the other hand, porosities act favourably for biomedical implants with blood circulation functionality in connecting tissues and stress shielding effect with compatibility between the natural and implant components. To accurately model smart porous FG structures under electrical and mechanical loads, a polynomial-based higher order shear and normal deformation theory (HOSNT) is utilized for the first time in this work. The governing equations are derived using the variational energy principle, and analytical solutions are obtained using Navier's solution method. The transverse shear stresses are obtained using a one-step stress recovery process. The influence of various porosity distribution models is studied on the bending response of smart porous FG plates. The effect of actuation voltage, porosity coefficient, material gradation index are critically assessed, and benchmark solutions for the class of problems are proposed. It is observed that stress response through thickness doesn't change monotonically with the change in porosity coefficient; instead, regions of higher and lower stresses are obtained with respect to the non-porous plate. Voltage-actuated FG plates are shown to have a steeper fall in stresses away from the piezo-layer compared to the conventional composites.

Scenario Simulation and Effects Assessment of Co-control on Pollution and Carbon Emission Reduction in Beijing

Focused on the key areas of energy, buildings, industry, and transportation, with 2020 as the base year and 2035 as the target year, we respectively designed the baseline scenario, policy scenario, and enhanced scenario, calculated the emission reduction potential of air pollutants and CO2 of Beijing, and constructed an assessment method of co-control effect gradation index to evaluate the co-control effect of air pollutants and CO2 in the policy scenario and enhanced scenario. The results showed that in the policy scenario and enhanced scenario, the reduction rates of air pollutants emissions will reach 11%-75% and 12%-94%, respectively, and reduction rates of CO2 emissions will reach 41% and 52%, respectively, compared with those from the baseline scenario. Optimizing vehicle structure had the largest contribution to the emission reduction of NOx, VOCs, and CO2, and the emission reduction rates will reach 74%, 80%, and 31% in the policy scenario and 68%, 74%, and 22% in the enhanced scenario, respectively. Replacing coal-fired with clean energy in rural areas had the largest contribution to the emission reduction of SO2; the emission reduction rates will reach 47% and 35% in the policy scenario and enhanced scenario, respectively. Improving the green level of new buildings had the largest contribution to the emission reduction of PM10; the emission reduction rates will reach 79% and 74% in the policy scenario and enhanced scenario, respectively. Optimizing travel structure and promoting green development of digital infrastructure had the best co-control effect. The co-control effect of replacing coal-fired with clean energy in rural areas, optimizing vehicle structure, and promoting green upgrading of the manufacturing industry will be improved to a better status in the enhanced scenario. More attention should be paid to improving the proportion of green trips, implementing the promotion of new energy vehicles, and the green transportation of goods to reduce emissions in the field of transportation. At the same time, with the continuous improvement in electrification level in the end energy consumption structure, the proportion of green electricity should be increased by expanding local renewable energy power production and increasing external green electricity transmission capacity, to enhance the co-control effect of pollution and carbon reduction.

Thermomechanical vibration response of nanoplates with magneto-electro-elastic face layers and functionally graded porous core using nonlocal strain gradient elasticity

The thermal vibration and buckling behavior of a functionally graded nanoplate are examined in this study. The nanoplate is made up of a silicon nitride/stainless steel core plate and two cobalt-ferrite/barium-titanate face plates. Four alternative porosity models were used to simulate the porosity of the nanoplate, and numerous variables that can affect the nanoplate’s behavior were taken into account. The study found that the thermomechanical behavior of nanoplates with magneto-electro-elastic face layers and a functionally graded porous core plate is affected by material gradation indices, porosity ratios, nonlocal variables, and different core plate material porosity models.

Nonlinear modelling of unimorph and bimorph magneto-electro-elastic energy harvesters

This paper nonlinearly models cantilever-based functionally graded magneto-electro-elastic energy harvesters (FGMEEEH) for the first time. The coupled magneto-electro-mechanical model is obtained on the basis of the Euler-Bernoulli beam theory. A hybrid procedure including Ritz's method is then utilized to generate reduced order models for both asymmetric unimorph and symmetric bimorph configurations. The resulting sets of initial value problems, whose convergence will be examined, are then analytically solved using the method of multiple time scales for both the free vibrations and primary resonance cases. The analytical time-histories of the system are compared by those obtained numerically and excellent agreements between them are observed. In addition, simplifying the frequency response function of the system in the primary resonance case, the present findings are validated by those available in the literature for linear unimorph systems. The influences of the harvester configurations, base acceleration amplitude, the value of the tip mass, the material gradation index as well as the resistances of the piezoelectric and magnetic circuits on the nonlinear response of the system are also studied in detail. It is observed that FGMEEEEHs enjoy much more efficiency in comparison to piezoelectric-based systems.

Free Vibration Characteristics of Bidirectional Graded Porous Plates with Elastic Foundations Using 2D-DQM

This manuscript develops for the first time a mathematical formulation of the dynamical behavior of bi-directional functionally graded porous plates (BDFGPP) resting on a Winkler–Pasternak foundation using unified higher-order plate theories (UHOPT). The kinematic displacement fields are exploited to fulfill the null shear strain/stress at the bottom and top surfaces of the plate without needing a shear factor correction. The bi-directional gradation of materials is proposed in the axial (x-axis) and transverse (z-axis) directions according to the power-law distribution function. The cosine function is employed to define the distribution of porosity through the transverse z-direction. Equations of motion in terms of displacements and associated boundary conditions are derived in detail using Hamilton’s principle. The two-dimensional differential integral quadrature method (2D-DIQM) is employed to transform partial differential equations of motion into a system of algebraic equations. Parametric analysis is performed to illustrate the effect of kinematic shear relations, gradation indices, porosity type, elastic foundations, geometrical dimensions, and boundary conditions (BCs) on natural frequencies and mode shapes of BDFGPP. The effect of the porosity coefficient on the natural frequency is dependent on the porosity type. The natural frequency is dependent on the coupling of gradation indices, boundary conditions, and shear distribution functions. The proposed model can be used in designing BDFGPP used in nuclear, marine, aerospace, and civil structures based on their topology and natural frequency constraints.

Static Response of 2D FG Porous Plates Resting on Elastic Foundation Using Midplane and Neutral Surfaces with Movable Constraints

The current manuscript develops a novel mathematical formulation to portray the static deflection of a bi-directional functionally graded (BDFG) porous plate resting on an elastic foundation. The correctness of the static response produced by middle surface (MS) vs. neutral surface (NS) formulations, and the position of the boundary conditions, are derived in detail. The relation between in-plane displacement field variables on NS and on MS are derived. Bi-directional gradation through the thickness and axial direction are described by the power function; however, the porosity is depicted by cosine function. The displacement field of a plate is controlled by four variables higher order shear deformation theory to satisfy the zero shear at upper and lower surfaces. Elastic foundation is described by the Winkler–Pasternak model. The equilibrium equations are derived by Hamilton’s principles and then solved numerically by being discretized by the differential quadrature method (DQM). The proposed model is confirmed with former published analyses. The numerical parametric studies discuss the effects of porosity type, porosity coefficient, elastic foundations variables, axial and transverse gradation indices, formulation with respect to MS and NS, and position of boundary conditions (BCs) on the static deflection and stresses.

Deformation failure and acoustic emission characteristics of continuous graded waste rock cemented backfill under uniaxial compression.

In order to study the effect of backfill aggregate particle size on the compressive strength and failure mode of cemented backfill, uniaxial compression tests were carried out on seven kinds of cemented backfills with different particle size gradations. By analyzing the AE characteristics during the failure process of the backfill, the damage evolution mechanism of the cemented backfill with different particle size gradations was discussed. The test results show that with the increase of the Talbot gradation index n, the compressive strength of the backfill specimens first increases and then decreases, and the failure mode gradually changes from shear failure to tensile failure. With the increase of particle size gradation, the particle size of aggregate increases, the interface between aggregate and cement matrix is more likely to be fractured, and the characteristic parameters of acoustic emission are more active. During the failure process of backfill, the AE energy rate increases rapidly in the plastic development stage, and reaches maximum value before and after the peak stress, which can be used as the precursor to judge the failure of waste rock cemented backfill. According to the test results, the damage model and constitutive equations of waste rock cemented backfill with different Talbot particle size gradations are established, which can provide engineering guidance for filling mined-out areas with waste rock to ensure safe production of mines.

Stability and dynamic behaviour of bi-directional functionally graded beam subjected to variable axial load

The current study emphasizes static stability and dynamic characteristics of bi-directional functionally graded beams subjected to variable axial loads using the Ritz method and Reddy’s beam theory. The material property is varied as a function of the gradation pattern along with the length and thickness directions. The solutions procedures are tested against the results in the literature to show the accuracy of the present method. The influence of uniform, linear, and parabolic axial loads along the length of the beam on buckling and vibration responses are investigated. There is a remarkable variation observed in both the responses, by changing the material properties from isotropic to bi-direction functionally graded. Furthermore, the study reveals that higher stiffness is achieved by the material gradation index increment along the thickness direction compared to the lengthwise gradation index increment. Even the variations in the aspect ratios and end conditions are depicting significant variations in the buckling and vibration responses. Buckling and free vibration modes are also highly sensitive to the nature of variable axial loads and gradation index.