Dynamic stability analysis of porous functionally graded beams under hygro-thermal loading using nonlocal strain gradient integral model

We present a study on the dynamic stability of porous functionally graded(PFG)beams under hygro-thermal loading.The variations of the properties of the beams across the beam thicknesses are described by the power-law model.Unlike most studies on this topic,we consider both the bending deformation of the beams and the hygro-thermal load as size-dependent,simultaneously,by adopting the equivalent differential forms of the well-posed nonlocal strain gradient integral theory(NSGIT)which are strictly equipped with a set of constitutive boundary conditions(CBCs),and through which both the stiffness-hardening and stiffness-softening effects of the structures can be observed with the length-scale parameters changed.All the variables presented in the differential problem formulation are discretized.The numerical solution of the dynamic instability region(DIR)of various bounded beams is then developed via the generalized differential quadrature method(GDQM).After verifying the present formulation and results,we examine the effects of different parameters such as the nonlocal/gradient length-scale parameters,the static force factor,the functionally graded(FG)parameter,and the porosity parameter on the DIR.Furthermore,the influence of considering the size-dependent hygro-thermal load is also presented.

Development of Mathematical Model on Preparation of Functionally Graded Material by Co-sedimentation

From the process of sedimentation the mathematical relationships among deposition volume and powder properties as well as sedimentation parameters were deduced. Based on the formula a mathematical model was set up and simulated through the computer. At last the validity of mathematical model was supported by the representative experiment on Ti-Mo system FGM prepared by co-sedimentation.

Hybrid graded element model for transient heat conduction in functionally graded materials

This paper presents a hybrid graded element model for the transient heat conduction problem in functionally graded materials （FGMs）. First, a Laplace transform approach is used to handle the time variable. Then, a fundamental solution in Laplace space for FGMs is constructed. Next, a hybrid graded element is formulated based on the obtained fundamental solution and a frame field. As a result, the graded properties of FGMs are naturally reflected by using the fundamental solution to interpolate the intra-element field. Further, Stefest＇s algorithm is employed to convert the results in Laplace space back into the time-space domain. Finally, the performance of the proposed method is assessed by several benchmark examples. The results demonstrate well the efficiency and accuracy of the proposed method.

Influence of Online Learning Attitude in the Blended Teaching Model on the Final Grades

The combination of online teaching and traditional offline teaching can maximize the advantages of both.Based on the blended teaching of English Reading course,39 students were selected as the research subjects to study the relationship between their online learning attitudes and their grades in the final examination.Judged from the number of times for each student to download teaching resources,the number of assignments submitted online,and the quality of the submitted assignments,each student’s attitude toward online learning was examined comprehensively,and a correlation analysis was conducted through SPSS Statistics 21.0 to explore the influence of online learning attitude on English reading performance.Through data collection and analysis of the online learning attitudes over a 16-week period,a significant positive correlation was found between the online learning attitudes and the English reading grades,indicating that the online learning attitude in the blended learning model plays a crucial role in improving the English reading skill,and students should maintain a positive attitude toward online teaching in blended learning.

A Multiscale Model of Mass-Functionally Graded Beam-Fluid System Under Bending and Vibration Responses

In this paper,a multiscale model is developed for the mass functionally graded(FG)beam-fluid system to investigate its static and dynamic responses based on 3D printed porous beam free vibration tests,which are determined by two aspects.At the microstructural level,the gradient variation is realized by arbitrary distribution of matrix pores,and the effective moduli under specific distribution are obtained using the micromechanics homogenization theory.In the meantime,at the structural level,the mechanical responses of FG porous beams subjected to mass loading are considered in a static fluid environment.Then,the explicit expressions of local finite-element(FE)expressions corresponding to the static and dynamic responses are given in the appendices.The present results are validated against numerical and experimental results from the literature and mechanical tests of 3D printed structures,with good agreement generally obtained,giving credence to the present model.On this basis,a comprehensive parametric study is carried out,with a particular focus on the effects of boundary conditions,fluid density,and slenderness ratio on the bending and vibration of FG beams with several different gradations.

Crashworthiness design of density-graded cellular metals

Crashworthiness of cellular metals with a linear density gradient was analyzed by using cell-based finite element models and shock models. Mechanisms of energy absorption and deformation of graded cellular metals were explored by shock wave propagation analysis. Results show that a positive density gradient is a good choice for protecting the impacting object because it can meet the crashworthiness requirements of high energy absorption, stable impact resistance and low peak stress.

Generalized thermoelastic interaction in functional graded material with fractional order three-phase lag heat transfer

The present work is concerned with the solution of a problem on thermoelastic interactions in a functional graded material due to thermal shock in the context of the fractional order three-phase lag model. The governing equations of fractional order generalized thermoelasticity with three-phase lag model for functionally graded materials(FGM)(i.e., material with spatially varying material properties) are established. The analytical solution in the transform domain is obtained by using the eigenvalue approach.The inversion of Laplace transform is done numerically. The graphical results indicate that the fractional parameter has significant effects on all the physical quantities. Thus, we can consider the theory of fractional order generalized thermoelasticity an improvement on studying elastic materials.

Grade and Tonnage Model of Contact Metasomatic Copper Deposit in China

Grade-tonnage model is one of the research frontiers of systematical exploration theory. Based on the “Reserve Database of Mineral Resources in China (1997)”, this paper establishes the geological model, grade model, tonnage model, grade-tonnage model and tonnage-sequence model of contact metasomatic copper deposits in China. The mathematical properties of these models are described in detail.

Grade-Tonnage Model of Porphyry Copper Deposits of China

A grade-tonnage model is established according to the analysis of 72 porphyry copper deposits recorded in 'The Mineral Resources Data Base of China'. Based on the analysis of frequency histogram, the cumulative frequency distributing graph and the theoretical model with double logarithmic coordinates of copper deposits, the typical mathematical characteristics of grade-tonnage model of porphyry copper deposits are clarified.

Computer Simulation of Distribution of Reinforced Particles in Functionally Graded Materials Fabricated by Centrifugal Casting

A mathematic model is developed with viscosity of molten matrix,centrifugalforce,casting temperature,mold temperature and other parameters taken intoconsideration for preditction of the distribution of reinforced particles during centrifugalcasting of FGM,and the simulation of distribution of reinforced particles and thesolidification process during centrifugal casting is performed with the aid of computergraphics.SiC_p/A356 FGM is fabricated by centrifugal casting.The results of computersimulation of distribution of reinforced particles are in good agreement with experimentalobservations.

Grade-Tonnage, Ore Value-Tonnage, andEnrichment Ratio-Tonnage Models forResource Assessment

According to grade-tonnage diagrams of nickel and zinc deposits, their critical grades are 0.4 % and 3. 4 %, respectively, and hence the former resources can be considered optimistic and the latter pessimistic. The grade-tonnage diagram of gold deposits is convex downwards suggesting that the critical grade is 1 X 10-6 in the low-grade part. The ore value (OV)-tonnage diagram of all deposits In the world consists of three parts: high, middle and low vain f classes. The enrichment ratio (ER)-tonnage diagram of all deposits in the world ho consists of three parts: high, middle and low ratio classes.Nine quality categories defined by ER and OV are characterized by some keywords indicating deposit types as follows: category RH (high ER-high OV: 0. 7 %) by 'unconformity' and 'Mississippi Val-ley', category HM (high ER-middle: OV: 0.7 %) by 'vein', category ML (middle ER-low OV: 0 %) by 'sandstone', 'stockwork' and' dissemination', category LM by 'orthomagmatic',' laterite',komatiite and ' chemical', and category LL by 'porphyry', 'dissemination' and 'placer'. Category MM is not characterized by any keyword. If the commodities of a deposit are defined by both the enrichment ratio and the ore value, the defined commodities are relatively coincident for gold and nickel,but different for copper, silver and zinc, and greatly different for molybdenum and lead. Deposits containing lead and/or zinc are complimentary. If the commodity Ph+Zn is applied, most lead or zinc deposits are classified as Ph+Zu by both definitions. Accessory metals are commonly expected for deposits of kuroko-type zinc, epithermal silver, massive sulfide-type zinc and volcanogenic zinc, but uncommon for deposits of orthomagmatic chromium, chemically precipitated copper and sandstone-type uranium.

The Grade-Tonnage Model for China's Molybdenum Deposits

The authors have studied the statistical characteristics of China＇s molybdenum deposits to establish the grade-tonnage model based on the data updated to the end of 2010. The results showed that each type of China＇s molybdenum deposits complied with Lasky＇s law approximately and the characteristics of grade-tonnage model obey the lognormal distribution. However, there are poor correlations between grade and tonnage respectively. Ultimately, we aimed to fit the grade-tonnage model through the known distribution function, draw the cumulative probability curves, and evaluated undiscovered mineral resources of China＇s molybdenum deposits by means of Monte Carlo simulation integrated in MRAS.

Intelligent prediction model of matte grade in copper flash smelting process

Due to the importance of detecting the matte grade in the copper flash smelting process, the mechanism model was established according to the multi-phase and multi-component mathematic model. Meanwhile this procedure was a complicated production process with characteristics of large time delay, nonlinearity and so on. A fuzzy neural network model was set up through a great deal of production data. Besides a novel constrained gradient descent algorithm used to update the parameters was put forward to improve the parameters learning efficiency. Ultimately the self-adaptive combination technology was adopted to paralleled integrate two models in order to obtain the prediction model of the matte grade. Industrial data validation shows that the intelligently integrated model is more precise than a single model. It can not only predict the matte grade exactly but also provide optimal control of the copper flash smelting process with potent guidance.

Modeling Crash Risk at Rail-Highway Grade Crossings by Track Class

The Federal Railroad Administration (FRA)’s Web Based Accident Prediction System (WBAPS) is used by federal, state and local agencies to get a preliminary idea on safety at a rail-highway grade crossing. It is an interactive and user-friendly tool used to make funding decisions. WBAPS is almost three decades old and involves a three-step approach making it difficult to interpret the contribution of the variables included in the model. It also does not directly account for regional/local developments and technological advancements pertaining to signals and signs implemented at rail-highway grade crossings. Further, characteristics of a rail-highway grade crossing vary by track class which is not explicitly considered by WBAPS. This research, therefore, examines and develops a method and models to estimate crashes at rail-highway grade crossings by track class using regional/local level data. The method and models developed for each track class as well as considering all track classes together are based on data for the state of North Carolina. Linear, as well as count models based on Poisson and Negative Binomial (NB) distributions, was tested for applicability. Negative binomial models were found to be the best fit for the data used in this research. Models for each track class have better goodness of fit statistics compared to the model considering data for all track classes together. This is primarily because traffic, design, and operational characteristics at rail-highway grade crossings are different for each track class. The findings from statistical models in this research are supported by model validation.

Closed-form solution for shock wave propagation in density-graded cellular material under impact

Density-graded cellular materials have tremendous potential in structural applications where impact resistance is required.Cellular materials subjected to high impact loading result in a compaction type deformation,usually modeled using continuum-based shock theory.The resulting governing differential equation of the shock model is nonlinear,and the density gradient further complicates the problem.Earlier studies have employed numerical methods to obtain the solution.In this study,an analytical closed-form solution is proposed to predict the response of density-graded cellular materials subjected to a rigid body impact.Solutions for the velocity of the impinging rigid body mass,energy absorption capacity of the cellular material,and the incident stress are obtained for a single shock propagation.The results obtained are in excellent agreement with the existing numerical solutions found in the literature.The proposed analytical solution can be potentially used for parametric studies and for effectively designing graded structures to mitigate impact.

Free vibration characteristics of sectioned unidirectional/bidirectional functionally graded material cantilever beams based on finite element analysis

Advancements in manufacturing technology,including the rapid development of additive manufacturing(AM),allow the fabrication of complex functionally graded material(FGM)sectioned beams.Portions of these beams may be made from different materials with possibly different gradients of material properties.The present work proposes models to investigate the free vibration of FGM sectioned beams based on onedimensional(1D)finite element analysis.For this purpose,a sample beam is divided into discrete elements,and the total energy stored in each element during vibration is computed by considering either the Timoshenko or Euler-Bernoulli beam theory.Then,Hamilton’s principle is used to derive the equations of motion for the beam.The effects of material properties and dimensions of FGM sections on the beam’s natural frequencies and their corresponding mode shapes are then investigated based on a dynamic Timoshenko model(TM).The presented model is validated by comparison with three-dimensional(3D)finite element simulations of the first three mode shapes of the beam.

Isogeometric analysis for free vibration of bidirectional functionally graded plates in the fluid medium

This paper for first time proposes an isogeometric analysis (IGA) for free vibration response of bi-directional functionally graded (BDFG) rectangular plates in the fluid medium. Material properties of the BDFG plate change in both the thickness and length directions via power-law distributions and Mori-Tanaka model. The governing equation of motion of BDFG plate in the fluid-plate system is formulated basing on Hamilton's principle and the refined quasi three-dimensional (3D) plate theory with improved function f(z). The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to determine the added mass. The discrete system of equations is derived from the Galerkin weak form and numerically analyzed by IGA. The accuracy and reliability of the proposed solutions are verified by comparing the obtained results with those published in the literature. Moreover, the effects of the various parameters such as the interaction boundary condition, geometric parameter, submerged depth of plate, fluid density, fluid level, and the material volume control coefficients on the free vibration behavior of BDFG plate in the fluid medium are investigated in detail. Some major findings regarding the numerical results are withdrawn in conclusions.

Buckling analysis of functionally graded nanobeams under non-uniform temperature using stress-driven nonlocal elasticity

In this work,the size-dependent buckling of functionally graded(FG)Bernoulli-Euler beams under non-uniform temperature is analyzed based on the stressdriven nonlocal elasticity and nonlocal heat conduction.By utilizing the variational principle of virtual work,the governing equations and the associated standard boundary conditions are systematically extracted,and the thermal effect,equivalent to the induced thermal load,is explicitly assessed by using the nonlocal heat conduction law.The stressdriven constitutive integral equation is equivalently transformed into a differential form with two non-standard constitutive boundary conditions.By employing the eigenvalue method,the critical buckling loads of the beams with different boundary conditions are obtained.The numerically predicted results reveal that the growth of the nonlocal parameter leads to a consistently strengthening effect on the dimensionless critical buckling loads for all boundary cases.Additionally,the effects of the influential factors pertinent to the nonlocal heat conduction on the buckling behavior are carefully examined.

Nonlinear stability of advanced sandwich cylindrical shells comprising porous functionally graded material and carbon nanotube reinforced composite layers under elevated temperature

The nonlinear stability of sandwich cylindrical shells comprising porous functionally graded material(FGM) and carbon nanotube reinforced composite(CNTRC)layers subjected to uniform temperature rise is investigated. Two sandwich models corresponding to CNTRC and FGM face sheets are proposed. Carbon nanotubes(CNTs) in the CNTRC layer are embedded into a matrix according to functionally graded distributions. The effects of porosity in the FGM and the temperature dependence of properties of all constituent materials are considered. The effective properties of the porous FGM and CNTRC are determined by using the modified and extended versions of a linear mixture rule, respectively. The basic equations governing the stability problem of thin sandwich cylindrical shells are established within the framework of the Donnell shell theory including the von K’arm’an-Donnell nonlinearity. These equations are solved by using the multi-term analytical solutions and the Galerkin method for simply supported shells.The critical buckling temperatures and postbuckling paths are determined through an iteration procedure. The study reveals that the sandwich shell model with a CNTRC core layer and relatively thin porous FGM face sheets can have the best capacity of thermal load carrying. In addition, unlike the cases of mechanical loads, porosities have beneficial effects on the nonlinear stability of sandwich shells under the thermal load. It is suggested that an appropriate combination of advantages of FGM and CNTRC can result in optimal efficiency for advanced sandwich structures.

Construction of Evaluation Model of Dynamic Stratified Grade Point of Automobile Electrical System Diagnosis Course

The accurate evaluation of a course determines the quality of learning and is the basis for evaluating the students' academic achievements. This paper, on the theoretical basis of grade point, builds a dynamic stratified GPA (Grade Point Average) evaluation model for the course Automotive Electrical System Diagnosis, focuses on the whole process assessment and aims to stimulate the enthusiasm of learning to meet the demands of the training of social skilled talents.