Modified Power Law Research Articles

Effect of circular and elliptical cutouts on the free vibration analysis of sandwich FGM plate under the elastic foundations

The present article investigates the effect of circular and elliptical cutouts on vibrational characteristics of porous sandwich functionally graded material (SFGM) plates with FGM core resting on Pasternak elastic foundation. The structural kinematics of the plate are formulated based on nonpolynomial-based higher-order shear deformation theory (HSDT). Geometric discontinuities have been incorporated in terms of circular and elliptical cutouts in the SFGM plates. The sandwich structure is considered to be comprised of two homogeneous face sheets and a porous FGM core with various cutout shapes and sizes. The variation of material properties from the homogeneous face sheet to the FGM core has been carefully modeled using modified power law distribution. Further, a C0 continuous finite element formulation with a four-noded, isoparametric quadrilateral element with seven degrees of freedom (DOFs) per node has been employed to accomplish the results. The accuracy of the present results has been demonstrated through convergence and validation studies. A comprehensive study has been carried out to investigate the influence of volume fraction index, even and uneven porosity distribution, circular and elliptical cutouts, as well as H-elliptical and V-elliptical type cutout shapes on the frequency parameter of SFGM plates with the elastic foundation. The numerical results demonstrate the aforementioned parameters significantly impact the vibrational frequency of the SFGM plates.

Density functional theory (DFT) study of water autoionization in solvated clusters.

We have implemented a cluster-continuum method using density functional theory to model water clusters and various charged species derived from water. The two aims of this study are to determine the minimal basis required for proper modeling of water autoionization and to determine the minimum number of explicit water molecules required to properly model the energetics of solvation. The thermodynamics of water autoionization converge following a modified power law to deliver chemically accurate values of the Gibbs energy change for autoionization with tractably small clusters. Convergence is slower and not exponential as assumed in previous work. We identify the n = 21 set of clusters as the first effectively bulk water like clusters that can capture the energetic influence of both the first and second solvation shells. In this cluster, a water molecule is encapsulated in the center of a closed shell of other water molecules that hydrogen bond to form five-membered rings. The total energy change for clusters with n ≥ 21 calculated using both the RPBE-D3 and ωB97X-D exchange-correlation functionals with the 6-311+G** basis set is shown to deliver good approximations to the free energy change at 298K. This is true even though neither functional models the individual enthalpy or entropy contributions particularly well.

Bulk and fracture process zone contribution to the rate-dependent adhesion amplification in viscoelastic broad-band materials

The contact between a rigid Hertzian indenter and an adhesive broad-band viscoelastic substrate is considered. The material behavior is described by a modified power law model, which is characterized by only four parameters, the glassy and rubbery elastic moduli, a characteristic exponent n and a timescale τ0. The maximum adherence force that can be reached while unloading the rigid indenter from a relaxed viscoelastic half-space is studied by means of a numerical implementation based on the boundary element method, as a function of the unloading velocity, preload and by varying the broadness of the viscoelastic material spectrum. Through a comprehensive numerical analysis we have determined the minimum contact radius that is needed to achieve the maximum amplification of the pull-off force at a specified unloading rate and for different material exponents n. The numerical results are then compared with the prediction of Persson and Brener viscoelastic crack propagation theory, providing excellent agreement. However, comparison against experimental tests for a glass lens indenting a PDMS substrate shows data can be fitted with the linear theory only up to an unloading rate of about 100μm/s showing the fracture process zone rate-dependent contribution to the energy enhancement is of the same order of the bulk dissipation contribution. Hence, the limitations of the current numerical and theoretical models for viscoelastic adhesion are discussed in light of the most recent literature results.

Structural properties, dielectric relaxation, and impedance spectroscopy of NASICON type Na3+xZr2−xPrxSi2PO12${\rm Na}_{3+x} {\rm Zr}_{2-x} {\rm Pr}_{x} {\rm Si}_2 {\rm PO}_{12}$ ceramics

AbstractWe investigate the dielectric and impedance spectroscopic investigation of Pr‐doped NASICON type () samples as a function of temperature and frequency. The Rietveld refinement of X‐ray diffraction patterns confirms the monoclinic phase having C2/c space groups for all the samples. The scanning electron microscopy shows the granular‐like structure and energy dispersive X‐ray analysis confirms the desired compositions. The temperature (90–400 K) and frequency (20 Hz–2 MHz) dependence of electric permittivity are explained using Maxwell–Wagner–Sillars (MWS) polarization and space charge polarization mechanisms. The dielectric relaxation shows nearly equal activation energy for all the samples with a non‐Debye type of relaxation in the measured temperature range. The complex impedance analysis shows the presence of broad grain and grain boundary relaxation peaks. The stretched exponent analysis of electric modulus using the Kohlrausch–Williams–Watts (KWW) function further confirms the non‐Debye type of relaxation. Moreover, scaling analysis of the electric modulus shows a similar type of relaxation for all the samples. The ac conductivity data are fitted using modified power law, where the temperature dependence of exponent () confirms the correlated barrier hopping (CBH) type conduction for all the samples. Our results indicate that the Pr‐doped NASICON samples are potential candidates for charge storage devices due to their large electric permittivity.

Predicting Damage in Notched Functionally Graded Materials Plates through extended Finite Element Method based on computational simulations

Presently, Functionally Graded Materials (FGMs) are extensively utilised in several industrial sectors, and the modelling of their mechanical behaviour is consistently advancing. Most studies investigate the impact of layers on the mechanical characteristics, resulting in a discontinuity in the material. In the present study, the extended Finite Element Method (XFEM) technique is used to analyse the damage in a Metal/Ceramic plate (FGM-Al/SiC) with a circular central notch. The plate is subjected to a uniaxial tensile force. The maximum stress criterion was employed for fracture initiation and the energy criterion for its propagation and evolution. The FGM (Al/SiC) structure is graded based on its thickness using a modified power law. The plastic characteristics of the structure were estimated using the Tamura-Tomota-Ozawa (TTO) model in a user-defined field variables (USDFLD) subroutine. Validation of the numerical model in the form of a stress-strain curve with the findings of the experimental tests was established following a mesh sensitivity investigation and demonstrated good convergence. The influence of the notch dimensions and gradation exponent on the structural response and damage development was also explored. Additionally, force-displacement curves were employed to display the data, highlighting the fracture propagation pattern within the FGM structure.

Optimization of the Electrochemical Nature of Ni/Co MOF Assisted by Zn Metal Centres for Better Electrode Performance in Hybrid Devices

Owing to the exceptional porous coordination chemistry and expanded redox culture of mixed metal organic framework (MMOF) materials are now presenting a hot matter for their potential usage in battery-supercapacitor electrodes. Here, we synthesized Ni/Co-MOF and its based Zn x -(Ni/Co)y-MOF nanoparticles containing different transition metals ratio; x: y = 0.25:0.75, 0.50:0.50, and 0.75:0.25 by hydrothermal process and named as ZMOF1, ZMOF2 and ZMOF3. Their electrochemical profile was carried out by CV, GCD and EIS characterization in three electrode setup. Among the MOFs nanoparticles, partially Zn enriched ZMOF1 shows prominent specific capacity of 177.23 and 181.12 C g−1 in 1.0 and 3.0 M KOH electrolyte solution at current density of 0.3 A g−1 along with good rate capability performance. Meanwhile, it retains brilliant specific capacity ∼ 86% of its original value compared to other displayed by ZMOF2 and ZMOF3 (80% and 69%) after charging-discharging for 3000 cycles at j = 4.0 A g−1. Moreover, modified power law was utilized to estimate the battery-type charge storage worth of ZMOF1 from CV cycle (at 5 mV s−1) in 1.0 and 3.0 M electrolytes medium thereby found the contribution 91.58% and 94.32%, respectively. These features of ZMOF1 attributed to particle diverse morphology, enriched redox sites and admiring electrical conductivity.

Modeling the Training Effect in Exchange‐Biased Bilayers for Large Numbers of Magnetization Reversal Cycles

The exchange bias (EB) is a unidirectional anisotropy that occurs, e.g., upon field‐cooling ferromagnet/antiferromagnet systems. In many material systems, the EB field is reduced from one hysteresis loop to the next measurement. This so‐called training effect (TE) has been investigated in experiments and by means of theoretical efforts by many research groups. The reduction of the EB field as a result of subsequent magnetization reversal processes is often fitted by a power law, usually with the exception of n = 1, or with an equation based on the discretized Landau–Khalatnikov equation, as first suggested by Binek. Few other models, usually with more fitting parameters, have been proposed yet. Herein, it is shown that for large numbers of subsequent magnetization reversal processes in Co/CoO thin film samples, a modified power law or a logarithmic fit can model the TE in most cases as well as the abovementioned, commonly used models.

Analysis of Stress Distribution in a Curved Functionally Graded Porous Beam Using the Unified Shear Deformation Theory

Using unified shear deformation theory (USDT) and a modified power law, the current study examines bending properties of two-dimensional functionally graded curved porous beam. In order to improve accuracy, this method incorporates equilibrium equations, potential energy, and the idea of a neutral surface. The analysis uses a boundary conditions, namely simply supported . A functionally graded beam composed of metal and ceramic with both even and unequal porosity is modeled. The formulation takes into account the symmetrical material gradation, which guarantees alignment between the geometrical and physical neutral surfaces. A displacement-based formulation and energy concepts are used, which leads to a more thorough and accurate beam analysis. This approach effectively regulates the constant changing of material characteristics in FGMs, takes into consideration higher-order shear deformation effects, and does away with the requirement for shear correction factors. As a result, it improves structural behavior predictions, which makes USDT very useful for advanced material applications. The equilibrium equations for the beams are derived using the Hamilton technique and solved with the Kuhn-Tucker conditions.

Impact of the Porosity and Elastic Foundation on Frequency and Buckling Response of Bidirectional Functionally Graded Piezoelectric Porous Plate

This work presents a thermoelectric vibration and buckling behavior of a bidirectional functionally graded piezoelectric porous (BD-FGPP) plate resting on the elastic foundation with arbitrary boundary conditions. The variation of the material properties in the FGPP plate is bidirectional, i.e. along longitudinal and thickness directions, using the modified power law distributions. Also, the BD-FGPP plate carries various porosity distributions, i.e. even, uneven and symmetric center types. The governing equation of the FGPP plate has been obtained through Hamilton’s principle and solved through a higher-order finite element approach. The correctness and usefulness of the current technique are confirmed and compared with existing model outcomes available in the literature. A comprehensive parametric study has investigated the effect of the elastic foundations, porous exponent with various porosity distributions, bidirectional material exponent, temperature change, electrical loading and boundary conditions. The attained results are more valuable for designing functionally graded piezoelectric-based smart structures by considering the porosity distribution in a thermoelectric environment.

A simplified algorithm for predicting creep settlement of high fills based on modified power law model

The calculation and prediction of creep settlement is an important basis for judging whether the settlement of high fills is stable and arranging subsequent construction. In this study, a modified power law model is established based on Hypothesis B and Bjerrum's classification of creep deformation. The modified model can uniformly describe the one-dimensional creep deformation of clay, sand and soil-rock mixture, and satisfies the requirement that the one-dimensional creep deformation tends to a constant value over time rather than increasing indefinitely. Then a simplified algorithm for predicting creep settlement is proposed based on the modified model. The simplified algorithm has only three parameters with explicit physical meaning, which are determined by settlement monitoring data. By the settlement monitoring data from high fills at Chengde Airport, Chengchi Expressway and Panzhihua South Railway Station, the fitting and predicted results of the simplified algorithm were compared with other three prediction algorithms commonly used in geotechnical engineering. The results show that the proposed simplified algorithm can predict the creep settlement of high fills more effectively and accurately.

Thermal and hygrothermal buckling characteristics of porous magneto-electro-elastic skewed plates using third-order shear deformation theory

The implementation of third order shear deformation theory using finite element method is presented in the current article to analyze the stability characteristics of porous magneto-electro-elastic (MEE) plates. This study aims to present the obscure behavior of FGMEE plate considering the loading to be hygrothermal in nature for buckling. Several temperature and porosity distribution are involved in this study in order to understand their influence on the buckling characteristics. The material properties of the porous MEE plate are assumed to vary continuously throughout its thickness according to rule of mixture. The porosity is considered as a locally distributed density estimated using the modified power law. The effect of temperature and moisture on buckling of MEE plates is of interest in this study. Skewness is incorporated through a transformation matrix to account for the angle. The stability characteristics of the porous FGMEE plate are examined under thermal, hygrothermal, electric, and magnetic loads to determine the critical buckling temperature. The influence of various variables, such as porosity, porosity distribution, and porosity volume, on the buckling behavior is extensively studied to realize that as porosity increases, the critical buckling temperature decreases in both load forms and noticed to be highest in Vo and Vx porosity distribution. Skew angle directly influenced the critical buckling temperature and was seen to be increasing with increase in skew angle between the plates. The study is also extended to assess the effect of geometrical parameters, such as aspect ratio, thickness ratio, and boundary conditions, on the stability characteristics that established an inverse relation between the parameters.

Deflection of FGM sandwich conoidal shell with porous core

The maximum deflection of functionally graded sandwich conoidal (FGSC) shell with porous core is studied in this paper. Even and uneven porosity distribution is considered to investigate the effect of porosity in the core of the FGSC shell. The inclusion of porosity (even or uneven) in the core makes the sandwich shell lighter. The modified power law is used to account the porosity distribution. The governing equations are developed based on improved mathematical model. An isoparametric finite element (FE) having 9 nodes and 63 degrees of freedom have been used. The improved mathematical model is executed in the present FE code. The numerical results in terms of maximum deflections are compared with available results in the literature. A broad parametric study is carried out to check the effect of porosity, volume fraction index and material properties on the maximum deflection of FGSC shell with porous core.

Assessment of porosity influence on thermo-mechanical stability characteristics of skew functionally graded plates using higher order shear deformation theory

The current article presents the modeling and investigation of thermo-mechanical buckling behavior for the porous functionally graded (FG) plates of rectangular and skew geometry. In this article, the modified rule of mixture and power law distribution is incorporated in the Third order Shear Deformation Theory (TSDT) to establish the governing equations using principle of minimum potential energy (PMPE). Finite element meshing is introduced using four node rectangular elements to discretize the FG plate for stability analysis. The constituents of the FG plate are continuous and vary over the thickness in accordance with modified power-law. Porosity is associated as locally distributed density and estimated using modified power law. To ascertain the effective boundary restraints skewness to the FG plate is imposed via transformation matrix. The computation of the critical non-dimensional buckling load for the structures subjected to uni-axial compressive and different thermal loads is executed via Gauss numerical integration. The assessment of the present formulation is performed against the buckling parameter available in the reference literature. Stability characteristics is evaluated under three different temperature distributions. The impact of material compositions, different thermal load, porosity distributions and various other parameters on the stability characteristics of porous FG structures are thoroughly investigated.

A microstructure-based multiscale approach to predict the formability of multiphase steels

A multiscale approach was proposed to predict the macroscopic forming limit of Q&P980 steel. The microstructure-based simulation was implemented in the microscopic simulation to obtain the macroscopic stress-strain curve. The representative volume element (RVE) was established using the data extracted from the characterised results of electron back-scattered diffraction (EBSD). The microscopic constitutive model of each constituent phase was determined based on a new method. The acquired macroscopic stress-strain curve was extrapolated using the modified Power-Law (MPL) strain hardening model. In the macroscopic simulation, the Nakajima and Marciniak-Kuczynski (M-K) simulations were combined to obtain the macroscopic forming limit only based on the macroscopic stress-strain curve. The microscopic and macroscopic simulations show good agreement with the experiment results, verifying the multiscale approach's accuracy. This study provides a solution for the direct connection between the microstructure and macroscopic forming limit.

Free vibration analysis of the structural integrity on the porous functionally graded plates using a novel Quasi-3D hyperbolic high order shear deformation theory

In this study, a novel quasi-three dimensional hyperbolic high-order shear deformation theory (quasi-3D HHSDT) is developed for free vibration analysis of porous functionally graded plates (FGPs). There are six unknowns in the current displacement field, and no shear correction factor is required. The mechanical properties are varied continuously through the thickness of porous FG plates using a modified power law function while considering the effect of porosities on the plate’s structural integrity. Two distinct porosity distribution models are considered, including even and uneven porosity distributions. The Navier technique is employed to obtain the closed-form solutions of motion's equations. An exhaustive parametric study is presented to show the influence of the different parameters on the fundamental frequencies.

Finite element analysis of functionally graded magneto-electro-elastic porous cylindrical shells subjected to thermal loads

Functionally graded materials with porosity have received increased application in satellites, space vehicles, aircraft, and other transportation systems. The multi-physics coupled modeling technique for structural response analysis remains a big challenge. This paper develops a finite element formulation based on the first-order shear deformation (FOSD) hypothesis for evaluation of the static and dynamic behavior of functionally graded magneto-electro-elastic porous (FG-MEEP) cylindrical shells under thermal loads. Four types of thermal loads such as uniform, linear, sinusoidal, and heat conduction temperature rise are included in the FE model. Furthermore, FG-U, FG-V, FG-O, and FG-X distributions of porosity is considered. The material parameters of FG-MEEP are obtained by modified power law. Two forms of material gradation in the framework of ‘B’-rich bottom and ‘F’-rich bottom are used. The correctness of the present model is verified by comparing with the results of literature. Finally, parametric studies are adopted to analyze the static and dynamic response of FG-MEEP cylindrical shell by varying functionally graded pattern, gradient index, porosity volume, temperature, porosity distribution, etc. These numerical results can serve as benchmarks for the future study of porous structures in thermal environment.

Elastic damage characterization of an ilmenite smelter freeze lining

A furnace freeze lining is necessary for safety and economic reasons in several smelting operations. The integrity of the freeze lining is put at risk by furnace power imbalances. Although numerous models have been derived to model the growth and depletion of the freeze lining due to power imbalances, no model exists for thermomechanical damage to the freeze lining. This study provides an initial pathway for modelling thermomechanical damage to the freeze lining in an ilmenite smelting furnace using information from the literature and experimental work. A methodology under the framework of continuum damage mechanics is proposed to model mechanical damage to the freeze lining due to phase changes, thermophysical changes, and constrained thermal expansion. Drill cores of solidified slag ingots were used to represent the freeze lining. The modified power law proved to be the best predictor of the softening response of the drill cores. Damage driving parameters were extracted from the raw data, and governing equations of the parameters with respect to temperature were derived for use in a finite element method (FEM) code.

Stability/instability of electrorheological nano pipes used in braking nanosystems by the machine learning method

ABSTRACT Electro-rheological fluids change their viscosity properties due to their application in an electrical field. This property could be acquired and maintained by accurately controlling the external electrical field. Besides brake systems, such electro-rheological (ER) fluids could be utilized in small-scale damping systems. In the present study, for the first time, the effects of using an electrical field in improving the dynamic stability of nanostructure containing ER fluids are presented. In this regard, a cylindrical sandwich structure, with an inner and outer layer composed of two-directional functionally graded (2D-FG) material and ER fluid core, is considered. Employing energy methods and modified power law the equations of motion of the structure are derived. Between FG and ER layers’ compatibility conditions are imposed in terms of displacement and strains. Moreover, to bypass computational complications, the inputs and outputs are utilized for training a deep neural network (DNN). In this way, the inputs and outputs are related to each other through the regression method without numerically solving the equations of motion all over again. Finally, the effects of different parameters on the frequency and loss factor characteristics of the current nanostructure will be presented.

Studies on modified power law inflation

In this study, we evaluate power law inflation (PLI) with a monomial potential and obtain a novel exact solution. It is well known that the conventional PLI with an exponential potential is inconsistent with the Planck data. Unlike the standard PLI, the present model does not encounter the graceful exit problem, and the results agree fairly well with recent observations. In our analysis, we calculate the spectral index and the tensor-to-scalar ratio, both of which agree very well with recent observational data and are comparable with those of other modified inflationary models. The employed technique reveals that the large cosmological constant decreases with the expansion of the universe in the case of the PLI. The coupling of the inflaton with gravitation is the primary factor in this technique. The basic assumption here is that the two metric tensors in the gravitational and inflaton parts correspond to different conformal frames, which contradicts with the conventional PLI, where the inflaton is directly coupled with the background metric tensor. This fact has direct applications to different dark energy models and the assisted quintessence theory.

Post-buckling characteristics of functionally graded fluid-conveying pipe with geometric defects on Pasternak foundation

In this paper, the static buckling and post buckling dynamic characteristics of functionally graded pipeline with geometric imperfections on Pasternak foundation are studied. The functional gradient material is modeled based on the modified power law distribution, and the pores generated in the material during manufacturing are described based on the Voigt model. Based on the Euler-Bernoulli beam theory, considering Pasternak foundation and von-Kaman nonlinearity, the control equation is derived by using Hamilton principle. The closed solution of nonlinear stability is given. The effects of maximum defect amplitude, porosity, power law index and Pasternak foundation on the static buckling and vibration characteristics near the first-order buckling configuration are discussed. The numerical results show that the maximum defect amplitude and Pasternak foundation have significant effects on the critical buckling load, post-buckling structure and natural frequency of the pipeline.