Volume Fraction Exponent Research Articles

Thermally Induced Bistable Functionally Graded Nanocomposite Plate

This paper explores the feasibility of using functionally graded carbon nanotube-reinforced composite (FG-CNTRC) in designing bistable plates. Single-walled carbon nanotubes (SWCNTs) in this nanocomposite are assumed to have a functionally graded distribution across the thickness direction following a power law. By varying the distribution type, volume fraction, and volume fraction exponent of SWCNTs, laminates with multiple thermal properties can be generated. A higher-order Rayleigh–Ritz model is presented to investigate the bistability and buckling behaviours of the proposed bistable plates. Their out-of-plane displacements and snap-through forces are predicted using this model. In the buckling analysis, transverse concentrated forces are applied to the panel corners to induce snap-through. To trace the load-displacement path during the loading process, the arc-length method is used to solve derived nonlinear equilibrium equations. The analytical results are compared using nonlinear finite element (FE) analysis for validation. A comprehensive parametric study is conducted to analyse the effects of curing temperature, distribution type, volume fractions, and volume fraction exponents of SWCNTs on the bistable behaviours of FG-CNTRC plates. It is found that, by varying these factors, bistable FG-CNTRC plates with multiple stable shapes and a wide range of snap-through forces can be generated. FG- CNTRC thus offers a rich design space for producing bistable plates.

Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells

The nonlinear vibration responses of functionally graded shells with different cone angles under external load are studied in this paper. Firstly, the Voigt model was employed to describe the physical properties along the thickness direction of FGMs conical shells. Then, the motion equations were derived based on the first shear deformation theory, von Kármán geometric nonlinearity and Hamilton’s principle. Next, the Galerkin method was applied to discretize the motion equations and the governing equations was simplified into a single degree of freedom nonlinear vibration differential equation based on the Volmir’s assumption. Finally, the nonlinear motion equation was solved by the harmonic balance method and Runge-Kutta method, and the amplitude frequency response characteristic curves of FGMs conical shell were obtained. The effects of different material distribution function and the ceramic volume fraction exponents on the amplitude frequency response curve of conical shell were discussed. The bifurcation diagrams of conical shells with different cone angles and time process diagrams and phase diagrams with different excitation amplitudes were described. The motion characteristic was characterized by Poincaré map. The results show that FGMs conical shell presents the nonlinear characteristics of "hardening" spring. The chaotic motion of FGMs conical shell is restrained and FGMs conical shell is not prone to produce motion instability with the increase of cone angle. The motion of FGMs conical shell presents a process from periodic motion to multi-periodic motion and then to chaos with the increase of excitation amplitude.

Design and finite element analysis of femoral stem prosthesis using functional graded materials

Conventionally biometals were used for design and development of bioimplants. However, the Young’s Modulus (YM) of these bioimplants is higher than that of a natural bone. Asymmetric load transfer from a bone to the bioimplant results in aseptic loosening and stress shielding. Here-in, the use of functionally graded materials (FGM) has been introduced to design the femoral stem prosthesis as a model bioimplant using computational biomechanics. The material properties variations in these FGMs in longitudinal and radial directions are explored to minimize the aseptic loosening and stress-shielding that plays a vital role in defining the performance and longevity of the prosthesis. Three groups of FGM (Ti-HA, SS316L-HA and CoCr alloy-HA) have been explored to design the stem prosthesis and the finite element analysis (FEA) was carried out using computational biomechanics. The stress distribution profile in the designed stem prosthesis demonstrated an increase in the stress values with an increase in the volume fraction exponent. The results corroborated with the stress distribution obtained from the simulation results of a cortico-cancellous bone. The stress distribution in the Ti-HA prosthesis is observed to be more uniform than CoCr-HA and SS316L-HA prosthesis. In addition, the reduced number of stress shielding points were observed for the Ti-HA prosthesis when compared with the CoCr-HA and SS 316 L-HA stem prostheses. Hence, the results suggested that the Ti-HA prosthesis could be considered as a mechanically stable prosthesis and the same could offer safe design for further development of a femoral bioimplant.

Free Vibration of Bi-Directional Functionally Graded Sandwich Plates Using Layerwise Finite Element Approach

In this paper, layerwise finite element analysis for the free vibration behavior of two-dimensional functionally graded sandwich plates with different boundary conditions is presented. The plates consist of three layers; a functionally graded layer embedded between ceramic and metal isotropic layers. The layerwise approach is based on the third order shear deformation theory for the middle layer, while the first order shear deformation theory is used to model both the upper and lower isotropic layers. Quadrilateral 8-noded element with 13-degrees of freedom per node is used for this purpose. The present results show very good agreements with the published analytical results of plates consist of a single functionally graded layer. Furthermore, for sandwich plates good agreements were obtained when the present results are compared with similar problems solved by other methods in literature. Parametric studies were investigated for various plate parameters including applied boundary conditions, volume fraction exponents and plate side to thickness ratio.

Upheaval thermal buckling of functionally graded subsea pipelines

This study deals with the upheaval thermal buckling of functionally graded subsea pipeline subjected to thermal loadings. The temperature-dependent material properties of the functionally graded subsea pipeline are considered, which distribute through the cross-section of the pipe. The governing equations of the functionally graded subsea pipeline are derived by using the nonlinear von-Kármán assumption and Euler-Bernoulli beam theory. An accurate analytical solution is derived to evaluate the thermal post-buckling response. The proposed mathematical model is verified by comparing it with the results in the literature. Numerical results are provided to explore the influences of the temperature-dependent material properties and the volume fraction exponent. The results show that the temperature-dependent material property of the thermal expansion coefficient has a great influence on the upheaval thermal buckling behavior. In addition, both the minimum critical temperature and the maximum stress will reduce when the functionally graded subsea pipeline is employed. Thus, the functionally graded materials have potential application to the subsea pipeline transporting high-temperature oil and gas.

Vibration response analysis of PZT-4/PZT-5H based functionally graded tapered plate subjected to electro-mechanical loading

In this paper, the vibration responses of the tapered functionally graded piezoelectric (FGP) square plate are analyzed. The governing equations are obtained using the first-order shear deformation theory (FSDT) with the variational (Hamilton's) principle, and the solution is obtained using nine-noded isoperimetric elements based finite element method. The effective electrical and mechanical properties are varying power-law distribution along the thickness direction. The tapered FGP plate is composed of PZT-4 (Top-side) and PZT-5H (bottom-side). Convergence and comparison tests have been carried out to validate the exactness of the current method. The numerical results are presented to analyze the effect of side to thickness(a/h)ratio, tapered ratio(ℏ), power law exponent(N), applied electric voltage(V)and the different boundary conditions for the tapered FGP square plate. The results show that the FGP tapered plate frequencies are influenced by the tapered ratio, volume fraction exponent, side to thickness ratio, and the applied electric voltage. These findings can be used to design smart structures based on the functionally graded piezoelectric material.

Nonlinear dynamic analysis of FG carbon nanotube/epoxy nanocomposite cylinder with large strains assuming particle/matrix interphase using MLPG method

In this paper, a first known study on the large deformation dynamic behavior of functionally graded carbon nanotube-reinforced (CNTR) nanocomposite cylinder considering the particle/matrix interphase property is presented. The geometrically nonlinear dynamic analysis is carried out using the meshless local Petrov–Galerkin (MLPG) method based on the total Lagrangian approach. The obtained nonlinear time dependent equations are solved using the incremental-iterative Newmark/Newton–Raphson technique. The effective properties of CNTR cylinder is evaluated by employing the three-phase Halpin–Tsai model and rule of mixture. In these models, three constituent phases including nanoparticle, interphase and matrix is considered. It is assumed that CNTs is dispersed in the resin matrix with four different function along the radial direction to obtain the optimum grading pattern. The effects of interphase properties, CNT's weight fraction, CNT distribution pattern and volume fraction exponent on the dynamic characteristics of the cylinder are discussed in details. Numerical results demonstrated the high performance of the proposed MLPG method for geometrically nonlinear dynamic analysis of functionally graded carbon nanotube/epoxy nanocomposites due to its advantage in eliminating the mesh distortion and high capability in FG material modeling. Moreover, it is found that the interphase properties may dramatically affect the dynamic behavior of nanocomposite cylinder especially at the higher CNT volume fractions. So, it is important to consider the interphase properties to correctly model the nanocomposite structures.

Static bending and free vibration analysis of functionally graded porous plates laid on elastic foundation using the meshless method

This paper presents a numerical approach for static bending and free vibration analysis of the functionally graded porous plates (FGPP) resting on the elastic foundation using the refined quasi-3D sinusoidal shear deformation theory (RQSSDT) combined with the Moving Kriging–interpolation meshfree method. The plate theory considers both shear deformation and thickness-stretching effects by the sinusoidal distribution of the in-plane displacements, satisfies the stress-free boundary conditions on the top and bottom surfaces of the plate without shear correction coefficient. The advantage of the plate theory is that the displacement field of plate is approximated by only four variables leading to reduce computational efforts. Comparison studies are performed for the square FGPP with simply supported all edges to verify the accuracy of the present approach. The effect of the aspect ratio, volume fraction exponent, and elastic foundation parameters on the static deflections and natural frequency of FGPP are also investigated and discussed.
 Keywords:
 meshless method; Moving Kriging interpolation; refined quasi-3D theory; porous functionally\break graded plate; Pasternak foundation.

Using finite element analysis to predict the damage in FGM-3D notched plate under tensile load; Different geometric concept

FGM (functionally graded materials) have experienced rapid expansion in recent years in several industrial fields. The analysis of their mechanical behavior by the finite element method requires the use of subroutines established under a FORTRAN compiler given the difficulty in implementing the type of behavior of the materials constituting the FGM. Our work aims to establish a program in MATLAB (UMM) making possible to introduce the behavior of the FGM by element in the mesh model of the structure, hence the advantage and reliability of graded the material in three dimensions for two or even three constituents of FGM and secondly, for different geometric designs. This technique is more sensitive to the mesh, hence the adaptation strategy of a driven mesh in order to improve the precision and the convergence of results under minimal computation time. The constitutive law of our model follows the von Mises equivalent stress flow theory with a hardening variable in incremental form. Moreover, to describe the elastic–plastic behavior of FGM, the model of TTO (Tamura-Tomota-Ozawa) was used for both methods that of UMAT in the form of a numerical algorithm and for our new method in the form of calculated values by the MATLAB code. The parameters used in the two calculation methods are taken from the results of stress–strain tests of the two materials; ceramic (TiB; Titanium mono biorde) and metal (Ti; Titanium). A comparison between the two methods has taken place in the form of load–displacement curves in order to present the capacities of our technique with respect to the usual UMAT technique. However, for damage, the use of the XFEM technique in our UMM method has shown its advantage in the separation of the structure after crack initiation. The crack path as well as its propagation speed under the effect of the volume fraction exponent of the FGM has been highlighted. The two models used in this study (UMM and UMAT) were validated on the one hand with analytics through the TTO model and on the other hand by the experimentation of a tensile test.

A continuum shell element in layerwise models for free vibration analysis of FGM sandwich panels

A finite element model based on continuum shell elements available in ABAQUS software has been developed for the free vibration analysis of FGM sandwich panels. Applications to sandwich plates with different combinations of FGM core and/or FGM face sheets satisfying through-the-thickness material gradations in the form of either power (P-FGM) or sigmoid (S-FGM) or exponential (E-FGM) distributions are considered. A user-defined material subroutine UMAT was used to implement functionally graded properties of the constitutive FGM layer. Numerical studies are given for free vibrations of sandwich plates subjected to different boundary conditions and with different structural parameters such as span to thickness, face sheet–core–face sheet thickness and aspect ratios, and volume fraction exponent. The studies showed very good agreement between the present results and those existing in the literature that confirmed the accuracy of the developed model. A series of numerical solutions obtained in the research extends the results of testing examples and may serve as additional benchmark data for other researchers.

Effect of porosity on the bending of functionally graded plates integrated with PFRC layer

This paper will present the analysis of static bending of functionally graded porous plates attached to a piezoelectric fiber-reinforced composite layer. The plate is exposed to sinusoidal electrical and mechanical loads. The displacement field is presented by using a sinusoidal shear deformation theory. The virtual work principle will be used for the derivation of the equilibrium equations and Navier’s method for the solution. Displacements and stresses are confirmed by comparing them with corresponding literature. Lastly, numerical results confirm that the porosity factor, applied voltage, volume fraction exponent, Young’s modulus ratio, aspect ratio, and side-to-thickness ratio have a huge influence on the electromechanical bending of functionally graded porous plates.

Transient thermal stress analysis for a short thick hollow FGM cylinder with nonlinear temperature-dependent material properties

In this study, nonlinear transient thermo-elastic analysis for a thick hollow 1D-FGM (functionally graded material) axisymmetric cylinder with finite length is investigated using higher-order graded finite element method. The material property gradations are along the radial direction and the power-law volume fraction and simple rule of mixture scheme are utilized to perform the effective material properties. As the cylinder exposed to high temperature, the temperature dependency of material properties is considered. So it led to nonlinear thermal conduction equation and in spite of the material distribution along one-direction by changing the temperature, the material properties are varied along two-direction with time. The nonlinear heat equation leads to 2D-temperature distribution for an axisymmetric cylinder. Also the thermal stress field would be different from an infinite cylinder which has been studied before. The temperatures, displacements and stresses for different values of volume fraction exponents along radial direction are investigated. It is revealed that the stresses and temperature in FGM cylinder are lower than the ceramic rich one. In other words, functionally graded cylinder wall gives better results in both stress and temperature cases than non-FGM cylinder.

Spatially resolved measurements of soot and gaseous precursors in ethylene counterflow diffusion flames up to 32 atm

We investigate the effect of pressure on both flame structure and soot formation in nitrogen diluted counterflow diffusion flames of ethylene in the 8–32atm pressure range. Capillary-probe gas sampling is performed to resolve spatially the profiles of gaseous species up to three-ring aromatics by GC/MS analysis and multi-color pyrometry is used to quantify the soot volume fraction and dispersion exponent. Self-similarity of flames is preserved by keeping constant mixture fraction and strain rate, so that profiles of concentrations and temperature, normalized with respect to their peak values, are unaffected by changes in pressure, once the axial coordinate is nondimensionalized with respect to the pressure-dependent diffusion length scale. When conditions are chosen so that the overall soot loading is approximately constant and compatible with the diagnostics, it is found that both the soot volume fraction and the profiles of key aromatics in the high-temperature nucleation region are virtually invariant. For it to happen, a twofold increase in pressure must be compensated by a ∼100 K decrease in peak flame temperature and, therefore, in the temperature across the soot forming region. The implication is that from the perspective of the chemical kinetics of soot formation these two actions counterbalance each other. As pressure increases (and temperature decreases) the peak production rate of the high-temperature soot mechanism decreases and, further downstream, towards the particle stagnation plane, a low-temperature soot mechanism sets in, yielding an increase in soot H/C content. This mechanism is enhanced as the pressure is raised, causing a higher overall soot volume production rate in the 16atm flame and, especially, in the 32atm one. The role of C4/C2 species in the formation of C6H6 increases with increasing pressure and dominates over the recombination of propargyl radical at sufficiently high pressures. A comprehensive database is established for soot models at high pressures of relevance to applications.

Static response of sandwich plates with FG core and piezoelectric faces under thermo-electro-mechanical loads and resting on elastic foundations

This study presents an analysis of the static bending of a sandwich plate composed of a functionally graded (FG) core and face sheets manufactured of piezoelectric material. The sandwich plate is resting on Pasternak's elastic foundations and subjected to sinusoidal thermo-electro-mechanical loads. Four-unknown shear deformation theory is utilized to define the displacement components. The equilibrium equations are established using the virtual work principle and solved by following Navier's solution method. The influences of applied voltage, thermal expansion coefficients, volume fraction exponent, elastic foundations parameters, side to-thickness ratio, and aspect ratio have been discussed. The current formulation is supported by comparisons with the published results.

Thermo-rotational buckling and post-buckling analyses of rotating functionally graded microbeams

Buckling and post-buckling responses of rotating clamped–clamped functionally graded microbeams in thermal environment are examined on the basis of the Euler–Bernoulli beam assumption. To enrich the formulation with the size effect the modified couple stress theory is employed. The temperature dependency of material properties is considered. The nonlinear finite element technique alongside with the Newton–Raphson technique is utilized to extract the prestressed deformation. Moreover, the direct iteration method is employed to determine the buckling point and post-buckling equilibrium path. The impacts of material length scale parameter, volume fraction exponent, rotor radius, microbeam length to its thickness proportion and rotation speed on the presented outcomes are examined.

Thermal buckling of porous symmetric and non-symmetric sandwich plate with homogenous core and S-FGM face sheets resting on Pasternak foundation

The effect of porosity and thermal environment on buckling responses for the sandwich plates with different boundary conditions is analyzed and discussed in the present study. The author’s recently developed sandwich plate using a new modified sigmoid function based functionally graded material (S-FGM) plate of different symmetric and asymmetric configurations is considered. A new temperature distribution through the thickness based on modified sigmoid law is proposed in which temperature is assumed to be conducted in the thickness direction. Three different types of porosity are considered viz. even, uneven but symmetric and uneven but non-symmetric. A new uneven non-symmetric porosity model is used in which micro-voids are varied in accordance with material property variation in the thickness direction to capture the accurate distribution of voids on the plate. The principle of virtual work is employed to derive equilibrium equations. An exact solution is obtained using the assumed solution with shape functions satisfying the edge boundary conditions. This is implemented using Galerkin Vlasov’s method. The analysis has been performed to study the effect of porosity, geometric configurations, inhomogeneity parameter, and foundation parameter. It is observed that the elastic foundation parameters have the least effect on non-symmetric plate configuration subjected to nonlinear temperature distribution in comparison to other symmetric and non-symmetric plate configurations and temperature distribution. Besides, the non-symmetric plate configuration possesses maximum thermal buckling temperature in comparison to a symmetric configuration. The volume fraction exponent with value 2 is observed as a changeover point where the effect of porosity distribution is null and void for 2-1-1 plate configuration irrespective of boundary conditions. The calculated outcomes and interpretations can be useful as a validation study for the imminent investigation of sandwich S-FGM plates having porosities in the thermal environment.

Nonlinear vibrations of fluid-conveying FG cylindrical shells with piezoelectric actuator layer and subjected to external and piezoelectric parametric excitations

Nonlinear vibrations of a fluid-conveying functionally graded (FG) cylindrical shell with piezoelectric actuator layer and subjected to external excitation and piezoelectric parametric excitation are analyzed theoretically. Considering the electric-thermo-fluid-structure interaction effect, a nonlinear dynamic model of fluid-conveying FG cylindrical shell with piezoelectric actuator layer is developed based on Hamilton’s principle and von-Karman geometrical nonlinearity. The inviscid, incompressible, isentropic and irrotational fluid is coupled into governing equations using the linearized potential theory. The nonlinear coupled differential governing equations of system are obtained by using Galerkin’s method with two modes. The developed coupled model is validated by comparing with the prior data and good agreements are observed. Multiple scales method is used to obtain nonlinear dynamic behaviors of the coupled system in case of 1:2 internal resonance, primary parametric resonance and 1/2 subharmonic resonance. The effects of three coupling cases between two modes on the dynamic behaviors of system are explored. Considering strongly coupled effect between two modes, the effect of external excitation, damping coefficient, piezoelectric harmonic voltage, fluid flow velocity, temperature and volume fraction exponent of FG cylindrical shell on the frequency-response curves are investigated. The influence of detuning parameters on force-amplitude response and voltage-amplitude response of system are also discussed.

Thermo-elastic buckling and post-buckling analysis of functionally graded thin plate and shell structures

In this paper, we extend the Kirchhoff–Love model to thermal buckling and post-buckling analysis of functionally graded structures. The kernel idea of the proposed model consists of the consideration of large displacements and finite rotation to accurately model the thermal effects on buckling and post-buckling behavior of such structures. Both uniform and nonuniform temperature distributions are considered. Material properties of the FG structures are graded in the thickness direction and assumed to obey a power law distribution of the volume fraction of the constituents. The effectiveness and usefulness of the proposed model are highlighted through different numerical examples, and the effects of the volume fraction exponent, thermal loads, length-to-thickness ratio, boundary conditions and geometrical parameters on the buckling and post-buckling behavior of FGM structures are also examined.

Applying isogeometric approach for free vibration, mechanical, and thermal buckling analyses of functionally graded variable-thickness blades

This article presents free vibration and buckling analyses of functionally graded blades with variable thickness subjected to mechanical and thermal loading using isogeometric analysis as a powerful numerical method. The proposed method is based on deployment of Hamilton’s principle to the two-dimensional kinematics of blades. The governing equations are derived in the context of a modified form of higher order shear deformation plate theory that merely needs C0-continuity (C0-higher order shear deformation plate theory). Without the necessity of defining a shear correction factor, the theory can accurately predict the solution for different thickness-to-length ratios. The numerical predictions for the buckling loads and natural frequencies are successfully compared with the available solutions in the published articles and in the lack of relevant results, finite element analysis using ANSYS is used for verification of the model. The effects of variable thickness and aspect ratio on the natural frequencies and mode shapes known as the frequencies loci veering phenomena are assessed for the first time, which is an important design factor for the blades. The proposed method uses non-uniform rational B-spline element, which is able to approximate linear and nonlinear thickness distribution and the couple modes with an excellent numerical consistency. The influences of aspect ratio, thickness variation, taper ratio, volume fraction exponent, and boundary conditions on the free vibration and buckling of variable-thickness functionally graded blades are also examined.

EFFECT OF POROSITY ON BEHAVIOURS OF PLATE STRUCTURES

In this paper, effect of porosity on nonlinear analysis of plate structures is presented. Two porous distributions are considered. Governing equations are expressed by using isogeometric analysis (IGA) and the third-order shear deformation theory (TSDT). With these approaches, it is easy to fulfil the C1-continuity requirement of the plate model. The obtained results demonstrate the significance of porosity volume fraction, porosity distributions and volume fraction exponent on nonlinear analysis of the plate structures.