\\n\\n
Dr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\\n\\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\\n\\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\\n\\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\\n\\nThank you all for being part of the journey. 5,000 times thank you!
\\n\\nNow with 5,000 titles available Open Access, which one will you read next?
\\n\\nRead, share and download for free: https://www.intechopen.com/books
\\n\\n\\n\\n
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'
Preparation of Space Experiments edited by international leading expert Dr. Vladimir Pletser, Director of Space Training Operations at Blue Abyss is the 5,000th Open Access book published by IntechOpen and our milestone publication!
\n\n"This book presents some of the current trends in space microgravity research. The eleven chapters introduce various facets of space research in physical sciences, human physiology and technology developed using the microgravity environment not only to improve our fundamental understanding in these domains but also to adapt this new knowledge for application on earth." says the editor. Listen what else Dr. Pletser has to say...
\n\n\n\nDr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\n\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\n\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\n\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\n\nThank you all for being part of the journey. 5,000 times thank you!
\n\nNow with 5,000 titles available Open Access, which one will you read next?
\n\nRead, share and download for free: https://www.intechopen.com/books
\n\n\n\n
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\r\n\tKidney disease is a worldwide phenomenon and affects millions of human beings. Typically for today are increasing figures of patients diagnosed with kidney disease caused by diabetes mellitus or hypertension. However, inherited disorders or autoimmune diseases are often encountered.
\r\n\r\n\tThe mainstay of a nephrologists daily work is and should be prevention and treatment of nephropathies to avoid progression to end-stage renal disease with the need for dialysis treatment. Especially in the field of diabetic nephropathy, with SGLT-2 inhibitors, there are new treatment options available, which will improve the prognosis of diabetic kidney disease.
\r\n\r\n\tAlso for the great group of patients with autosomal dominant polycystic kidney disease, with the availability of tolvaptan, there is hope to slow down the progression of kidney failure and to prolong their lives without dialysis.
\r\n\r\n\tThis book will intend to gather a plentiful of information to help clinicians in their care for patients with nephropathy and to improve the life of our patients.
",isbn:null,printIsbn:"979-953-307-X-X",pdfIsbn:null,doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"8285fdacbd1fc98f1a1cf29464942a1a",bookSignature:"Dr. Thomas Rath",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/8056.jpg",keywords:"Diagnosis, Prognosis, Therapy, Alport-Syndrome, Cystic Kidney Disease, ADPKD, Glomeruolopathies, Arteriosclerosis, Vasculitis, HIV, Drug-Induced Nephropathy, Infectious Disease",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"October 19th 2020",dateEndSecondStepPublish:"November 16th 2020",dateEndThirdStepPublish:"January 15th 2021",dateEndFourthStepPublish:"April 5th 2021",dateEndFifthStepPublish:"June 4th 2021",remainingDaysToSecondStep:"5 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"Dr. Thomas Rath completed his studies at the University of Mainz and became a resident at the Westpfalz-Klinikum in Kaiserslautern, Head of the Department of Nephrology and Transplantation Medicine. He has published more than 25 papers in peer-reviewed journals and more than 150 abstracts and posters at national and international congresses.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"67436",title:"Dr.",name:"Thomas",middleName:null,surname:"Rath",slug:"thomas-rath",fullName:"Thomas Rath",profilePictureURL:"https://mts.intechopen.com/storage/users/67436/images/system/67436.jpg",biography:"Thomas Rath is a doctor of medicine and specialist in internal medicine, nephrology, hypertension, and infectious diseases. He lives in Kaiserslautern, a city of 100,000 inhabitants in the southwest part of Germany. After completing his studies at the University of Mainz, he became a resident at the Westpfalz-Klinikum in Kaiserslautern, a tertiary care hospital with 1300 hospital beds. There, he is Head of the Department of Nephrology and Transplantation Medicine and also responsible for the outpatient clinic for patients with infectious diseases. He is an active member of many national and international societies. During his scientific career he has published more than 25 papers in peer-reviewed journals and more than 150 abstracts and posters at national and international congresses. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"64766",title:"Electrochemical Sensors for Food Safety",doi:"10.5772/intechopen.82501",slug:"electrochemical-sensors-for-food-safety",body:'\nFood safety has become a global public health concern affecting both mostly developing countries and developed countries. In addition, foodborne diseases negatively impact the economy, trade, and industries of affected countries. For example, melamine has been detected in infant formula (milk powder) in China, leading to more than 290,000 infants suffering from severe health problems such as urinary tract stones [1]. Early and accurate detection of food safety is therefore very important for preventing, controlling, and mitigating the impact of potential outbreaks. Many analytic methods, including chromatography methods such as gas chromatography (GC) [2], high performance liquid chromatography (HPLC) [3], gas chromatography-mass spectrometer (GC-MS) [4], and liquid chromatography-mass spectrometer (LC-MS) [5], and immunological detection, such as enzyme linked immunosorbent assay (ELISA) [6] and lateral flow immunoassay [7], have been employed for food safety detection. Although those traditional methods are relatively sensitive and specific, they are expensive, laborious, and time-consuming and require well-trained personnel [8, 9], which make them incompatible for developing countries and areas are lacking equipped facilities and specialists. It is therefore urgent to develop rapid, accurate, sensitive, and online technologies for food safety detection.
\nModern electrochemistry provides powerful analytical techniques for sensors, with the advantages of instrumental simplicity, low cost, and miniaturization, work on-site, and the ability to measure pollutants in complex matrices with minimal sample preparation [10]. Electrochemical sensors and methods are developed as suitable tools for different applications, including bioprocess control, agriculture, and military, and, in particular, for food quality control. Voltammetric techniques, such as cyclic voltammetry (CV) [10], linear sweep voltammetry (LSV) [11], differential pulse voltammetry (DPV) [12], and square wave voltammetry (SWV) [13], have been widely used in food analysis. Among these voltammetric techniques, DPV and SWV are commonly used, as low detection limits and multiplex analysis can be achieved with the two methods. These two techniques involve potential waveforms and their respective current response are shown in Figure 1A. The waveform of DPV consists of pulses of constant amplitude superimposed on a staircase waveform. This method has the highest sensitivity in electrochemistry because the charging current can be ignored against faradaic current, and their ratio is obtained as large. Moreover, SWV consists of symmetrical square-wave pulses superimposed on a staircase waveform. During each square wave cycle, the current is sampled twice, just before the end of each forward and each backward pulse followed by subtraction of the currents. The peak current heights (values) obtained by the two methods are directly proportional to the concentrations of the analyte. Amperometry is another important electrochemical analysis method in which the potential of the working electrode is constant and the resulting current from faradic processes occurring at the electrode is monitored with a function of time. In this method, the current is integrated over relatively longer time intervals, so it gives an improved signal to noise ratio [14].
\nPotential waveforms and their respective current response for (A) differential pulse voltammetry (DPV) and (B) square wave voltammetry (SWV).
Electrochemical sensors can be used as food safety monitoring tools in the assessment of biological/ecological quality or for the chemical monitoring of both inorganic and organic pollutants. In this chapter we provide an overview of electrochemical sensor systems for food safety applications, and in the following sections, we describe the various electrochemical sensors that have been developed for food safety detection.
\nHeavy metals (HMs) are currently defined as metals with a specific gravity greater than 5 g cm−3, which are considered as a serious source for polluting the biosphere throughout the world and causing many healthy and physiological diseases due to their prolonged half-life, non-biodegradability, and potential of accumulation in different parts of the human body [15, 16]. Heavy metals like cadmium, lead, arsenic, chromium, and mercury are considered as hazardous elements even at low concentrations [17, 18, 19]. Therefore, sensitive and selective determination of toxic heavy metals with cost-effective and convenient procedures is of paramount importance.
\nDue to the speed of detection, low cost, high sensitivity, and easy adaptability for in situ measurement [20], electrochemical sensors have attracted great interest in the detection of heavy metal ions for food safety.
\nFor many years, anodic stripping voltammetry (ASV) at the mercury and its modified electrode was extensively applied to the determination of trace metal ions for the extensive cathodic potential range [21, 22]. However, the disposal of the mercury-containing device and the incorrect handling can lead to the formation of mercury vapors that are toxic and represent a significant health and environmental hazard [23]. Therefore, various mercury-free electrodes have been developed in the past few decades. For example, a nanostructured bismuth film electrode (nsBiFE) has been prepared for ASV detection of multiple heavy metals, in which the detection limits of 0.4 and 0.1 μg L−1 are obtained for Cd2+ and Pb2, respectively [24]. Similar to bismuth, antimony nanoparticles have also been proven to be highly sensitive and reliable for tracing analysis of heavy metals [25]. To take into real application, more and more electrochemical sensors based on screen-printed carbon electrode (SPCE) have been fabricated for trace heavy metal detection in food safety as it is inexpensive, portable, and easy for mass production [26, 27].
\nAddition of inedible substances and abuse of food additives are the prominent problems affecting food safety [28]. Typical illegal additives include melamine, clenbuterol, and Sudan I. These illegal actions may pose great threat to human health. For the detection of these chemicals, various nanomaterial-based biosensors have been developed. Various approaches aiming at analyzing specific chemical contaminants and illegal additives have been developed [29, 30, 31]. Li et al. developed a gold nanoparticles (AuNPs)-decorated reduced graphene oxide (RGO) modified electrode for detection of Sudan I in food samples including chili powder and ketchup sauce, demonstrating satisfactory sensitivity, selectivity, and recovery [11]. A sensitive and selective electrochemical sensor based on MIL-53@XC-72 nanohybrid modified glassy carbon electrode (GCE) was also fabricated to determine melamine with a linear range from 0.04 to 10 μM and detection limit of 0.005 μM (S/N = 3) [32]. In addition, the sensor displayed excellent reproducibility, high stability, selectivity, and good recoveries for the determination of melamine in liquid milk. The synergistic effect of nitrogen-doped graphene (NGR) and nitrogen-doped carbon nanotubes (NCNTs) has also been investigated and applied to prepare an electrochemical sensor for simultaneous and sensitive determination of caffeine and vanillin [33]. Electrochemical sensors have also been developed for many other food additives, such as sunset yellow [34, 35].
\nPesticides, including fungicides, herbicides, and insecticides, are widely used in most food production to control pests that would otherwise destroy or reduce food production [36]. In the area of agriculture, the usage of insecticides, herbicides, molluscicides, and fungicides has an increasing importance. However, many pesticides are toxic and can cause many health problems when consumed by animals and humans, such as bone marrow disorders, carcinogenicity, infertility, cytogenic effects, neurological diseases, and immunological and respiratory problems. Hence, pesticide residue detection is very important for food safety [37].
\nTo date, many methods have been applied to determine pesticide residues in food samples. Electrochemical methods provide the elucidation of processes and mechanisms of redox reaction of pesticides and their residues [38]. They are sensitive, reliable, and fast. They can be easily miniaturized and integrated with other analytical methods [39, 40]. A magneto-actuated enzyme-free electrochemical sensor based on magnetic molecularly imprinted polymer was developed, and it showed outstanding analytical performance for the detection of methyl parathion in fish, with a limit of detection of as low as 1.22 × 10−6 mg L−1 and recovery values ranging from 89.4 to 94.7% [41]. Da Silva and coworkers [42] developed an acetylcholinesterase (AChE) biosensor for rapid detection of carbaryl in tomato samples by using electrode modified with reduced graphene oxide (rGO). The electrochemical response increased as the concentration of acetylthiocholine chloride increased, while the response decreased in the presence of AChE inhibitor OPs with a linear response to the inhibition of the thiocholine oxidation process for carbaryl concentrations from 10 to 50 nmol L−1 and 0.2 to 1.0 mol L−1. Compared with AChE, organophosphorus hydrolase (OPH) enzymes catalyze the hydrolysis of organophosphorus pesticides (OPs) with a high turnover rate, can potentially be reused, and are, therefore, suitable for continuous monitoring of OPs [43, 44].
\nVeterinary drugs mainly include antimicrobial drugs, antiparasitic drugs, and growth promoters, which are extensively used for treatment and prevention of diseases in animals, promotion of animal growth, and feed efficiency [45]. But the possible presence of veterinary drugs in animal-derived foods is one of the key issues for food safety, which arouses great public concern. So it is very important to develop quick and accurate methods to detect veterinary drug residues in animal-derived food, and their quantity must be less than the maximum residue limits (MRL) defined in many countries on the basis of food safety [46].
\nElectrochemical sensors have drawn considerable attention in many fields such as food safety, disease diagnosis, and environmental monitoring [47, 48]. Lin et al. [49] developed a hybrid CNT-modified electrode for simultaneous determination of toxic ractopamine and salbutamol in pork samples. Conzuelo et al. developed a novel strategy to construct disposable amperometric affinity biosensors by recombinant bacterial penicillin-binding protein (PBP) tagged by an N-terminal hexahistidine tail that was immobilized onto Co2+-tetradentate nitrilotriacetic acid (NTA)-modified screen-printed carbon electrodes (SPCEs) for the specific detection and quantification of β-lactam antibiotic residues in milk, which was accomplished by means of a direct competitive assay using a tracer with horseradish peroxidase (HRP) for the enzymatic labeling [50]. The sensor showed limits of detection with the low part-per-billion level for the antibiotics tested in untreated milk samples and a good selectivity against other antibiotic residues frequently detected in milk and dairy products. In addition, Wang et al. proposed a simple, rapid, and highly sensitive homogeneous electrochemical strategy for the detection of ampicillin based on target-initiated T7 exonuclease-assisted signal amplification. This biosensor showed a low detection limit of 4.0 pM toward ampicillin with an excellent selectivity, which has been successfully applied to assay antibiotic in milk. Importantly, the sensor system could avoid the tedious and time-consuming steps of electrode modification, making the experimental processes much simpler and more convenient, which has great potential for the simple, easy, and convenient detection of antibiotic residues in food safety field [51].
\nMycotoxins are fungal secondary metabolites that have toxic effects on humans and animals. Generally, mycotoxins can be easily found in agriculture crops, dairy products, including milk and cheese, and alcohols [52]. Mycotoxins enter human or animal bodies through consumption of contaminated animals or industrial food products. Crops and food products that are highly susceptible to mycotoxin contamination include alcoholic beverages, wheat, corn, barley, sugarcane, cottonseed, peanuts, rice, sugar beets, sorghum, and hard cheese [53].
\nMany review articles have focused on mycotoxin detection using different transduction methods [54, 55]. However, only a few review articles have reported on the use of nanomaterials for the electrochemical (EC) sensing of mycotoxins [56]. The present review summarizes the recent developments of nanomaterial-based EC biosensors for mycotoxin detection. It describes the importance of mycotoxin detection and the current progress and necessity of POC analysis of food toxins [56]. Finally, it illustrates the role in mycotoxin detection of EC sensors based on carbon and graphene metal nanoparticles (NPs) combined with different recognition elements, such as aptamers, antibodies, and molecularly imprinted polymers (MIPs) [57, 58]. These sensors exhibited additional analytical merits such as a shortened analysis time with simplified analytical procedures and portability. As such, EC sensors are now acknowledged as promising options for the trace-level identification of mycotoxins in food processing and manufacturing industries.
\nMicrobial pathogens include bacteria, viruses, and protozoa, and failure to detect them can have severe impacts on public health and safety. In the food or water services industries, legislation developed by the appropriate associated regulatory bodies to monitor and control the presence of these microorganisms is vital. Rapid and cost-efficient detection methods, with high-throughput capacity, are essential to implement effective monitoring systems to protect human health [59]. In 2012, the Environmental Protection Agency (EPA) released new Recreational Water Quality Criteria recommendations for protecting human health in waters designated for primary contact recreation [60]. Guner et al. developed an electrochemical sensor for the detection of
Food safety is undoubtedly one of the major global concerns. In this chapter, we summarize some representative electrochemical sensors toward food contaminants such as heavy metals, illegal additives, pesticide residues, veterinary drug residues, biological toxins, and foodborne pathogen. These electrochemical sensors for food safety detection continue to show many advantages including rapid response, field applicability, high sensitivity, high selectivity, and online analysis. Moreover, electrochemical sensors are much cheaper and easier to be miniaturized, which may play a key role on quality control in food processing, improving product quality and safety. However, the stability of the electrochemical sensors is still a challenging problem. Recently, we have developed an electrochemical instrument and a number of electrochemical sensors for the detection of heavy metal ions with excellent sensitivity and reproducibility. The instrument and sensors are being commercialized with satisfactory user feedback.
\nFunctionally graded materials (FGM) have received considerable attention in many engineering applications, since the theory of FGM was firstly introduced in 1984. In general, FGM is made from the volume fractions of two or more material components that have continuous variation of material properties from one surface to another [1]. Nowadays, FGM suits the specific demand in different engineering applications, especially for high temperature environment applications of heat exchanger tubes, thermal barrier coating for turbine blades, thermoelectric generators, furnace linings, electrically insulated metal ceramic joints, space/aerospace industries, automotive applications, biomedical area, etc.
\nThe manufacturing of FGM with fully specified profile of material gradation, however, is very difficult causing significant variability in their mechanical and structural properties. Therefore, proper handling of the randomness in the material properties is required for accurate prediction of structural response for safe and reliable design. The stochastic analysis is a useful analytical tool to predict the response of structures with random material properties. In this direction, there is a reasonable body of recent research on the effect of uncertainties in material properties on the mechanical behavior of FGM. Investigators used stochastic simulation to study the effect of microstructural randomness on stress in FGM [2]. Ferrante et al. studied the effect of non-Gaussian porosity randomness on the response of functionally graded plate [3]. Yang et al. dealt with the stochastic bending response of moderately thick FGM plates [4, 5]. The effect of random material properties on post buckling response of FGM plate are presented in Lal et al. [6].
\nHowever, the above mentioned literatures are for the static analysis. To the best of author’s knowledge, few limited works have been done on the eigen analysis of FGM structures involving randomness in system parameters. Certain efforts have been made in the past to predict the dynamic behavior of structures with randomness. In most of the studies conducted, investigators dealt with the free vibration of functionally graded laminates with random material properties using first-order perturbation technique (FOPT) incorporating mixed type and semi-analytical approach to derive the standard eigenvalue problem [7]. Some of these papers presented the stochastic finite element method (SFEM) to investigate the natural frequency of functionally graded plates based on the higher-order shear deformation theory (HSDT) utilizing first-order reliability method and second-order reliability method [8]. In most cases, Jagtap et al. [9] examined the stochastic nonlinear free vibration response of FGM plate using HSDT with von-Karman kinematic nonlinearity via direct iterative stochastic finite element method. Shegokar et al. investigated the stochastic finite element nonlinear free vibration analysis of FGM beam with random material properties due to thermo-piezoelectric loadings [10]. The above mentioned literatures investigated the free vibration and nonlinear behavior of FGM beam and plate. The material properties, such as Young’s modulus, shear modulus, and Poisson’s ratio of FGM, are modeled as independent random variables.
\nIn this chapter, the stochastic finite element solution is suggested to obtain the variability in the natural frequency of functionally graded material (FGM) beam. The elastic modulus and width of the FGM beam are assumed to vary in thickness and longitudinal directions following the exponential law. The uncertain material properties, such as modulus of elasticity and mass density of the FGM beam, are considered to be a one-dimensional homogeneous stochastic process. The stochastic finite element analysis of FGM beam is performed using the spectral representation method for the description of randomness in conjunction with Monte Carlo simulation (MCS). The response variability of natural frequency due to random elastic modulus and mass density in FGM beam is given. Furthermore, the effect of correlation between the two random parameters is observed as well.
\nAn FGM is defined to be a material which has a continuous gradation through-the-thickness (h). One side of the material is typically ceramic and the other side is typically metal. A mixture of the two materials composes the through-the-thickness characteristics. Let us consider a functionally graded beam shown in Figure 1. The parameters of the model FGM beam are as follows:
Model of FGM beams.
The elastic material properties vary through-the-beam thickness according to the volume fractions of the constituents using power law distribution (as shown in Figure 2).
\nThe variation of elasticity modulus along the corresponding directions.
with
\nwhere
In exponential law, for the material parameter of Young’s modulus
In Eq. (3), \n
The variation of elasticity modulus, mass density, and width along the corresponding directions.
In case of four degree of freedom beam element, as shown in Figure 4, the transverse displacement function may be assumed as a cubic polynomial in \n
Beam element.
The width and mass per unit length of the element are
\nwhere \n
The nodal displacement vector of the element is
\nthen, the displacement field is
\nwhere \n
In this case, the stiffness of the beam \n
where \n
Hamilton’s principle may be a theoretical base for dynamical systems by its nature of integral form in time with Lagrangian density to account for continuous space. In this paper, the analysis of natural frequency of FGM beam is performed using Hamilton’s principle. The strain energy expression \n
The kinematic energy \n
where
\nSubstituting Eq. (7) into Eqs. (9) and (10), the following can be obtained:
\nThe governing differential equations of motion and the related governing equation can be derived using Hamilton’s principle
\nwhere \n
Substituting Eqs. (13) and (14) into Eq. (15), the following can be obtained:
\nhere, \n
For simple harmonic vibration, we assume the displacements to be \n
where \n
In order to model the randomness in the material properties, the modulus of elasticity and mass density along the mid-plane are assumed to vary along its length of FGM beam in a random manner. We can model these variations as one-dimensional univariate (1D-1V) homogeneous stochastic processes. The simple mathematical expressions for the randomly varying modulus of elasticity and mass density can be written as
\nwhere \n
The numerical generation of sample functions of Gaussian zero-mean homogeneous stochastic processes, which describe the randomness in parameters of the structure, is accomplished using the spectral representation method. For a one-dimensional univariate (1D-1V) stochastic process, we have [11]
\nIn Eq. (21), \n
where \n
The uniform random phase angle \n
here, \n
In all examples, the coefficient of variation \n
here,
\nand \n
In order to obtain the response variability in the natural frequency of the FGM beam, we employed the scheme of Monte Carlo simulation (MCS). As a matter of fact, the MCS corresponds to the deterministic analyses on a set of heterogeneous models of the given structure, in which the material properties have different values depending on the position in the domain of the structure.
\nThe generation of heterogeneous random samples is accomplished by the aforementioned spectral representation scheme, and we use 10,000 samples for respective analyses. In particular, we adopt the local average scheme other than the mid-point rule in applying the MCS, with which better results can be obtained especially for the processes with small correlation distance. Figure 5 shows an example plot of the processes employed to model the randomness in the system parameters.
\nExample plot of random process used
The geometric dimensions of the example FGM beam are: \n
FGM beam model.
The results in Figure 7 correspond to the prismatic homogeneous beam since the parameters in exponents, \n
The convergence between exact solution and finite element method.
The discrepancies between exact and finite element solutions for the frequencies for the first three modes are shown in Figure 8. The differences given in percentile tends to zero as the number of finite elements is increased, meaning the results are converging to exact solutions.
\nError depending on mesh refinement (
Figure 9 shows the first three normalized natural frequencies of uniform FGM beams for three cases of modulus ratio (\n
First three normalized natural frequencies of FGM beams for different ratio of Young’s modulus (
Figure 10 shows the COV of response versus the correlation distance (\n
COV of natural frequency as a function of the correlation distance (
The relationship between COV of natural frequency and the COV of stochastic process is shown in Figure 11. The standard deviation of stochastic process is changed from 0.0 to 0.25. As seen in Figure 11, the COV of response shows a slightly nonlinear pattern in all the cases of \n
COV of natural frequency as a function of the standard deviation of stochastic process
The effect of mesh refinement on the COV of natural frequency is shown in Figure 12. The correlation distance log (\n
Effect of mesh refinement on the COV of natural frequency (
Figure 13 shows the variation of the coefficient of variation (COV) depending on the non-uniformity parameter (\n
Effect of non-uniformity parameter (
The overall features of the effect of Young’s modulus ratio on COV of natural frequency are shown in Figure 14. The COV of natural frequency is not affected by the parameter \n
Effect of parameter
It is natural to have preposition that not only the elastic modulus, but also the mass density of the material can have randomness. Therefore, we need to consider the effect of correlation between two random parameters of elastic modulus and mass density. To this aim, we consider three correlation cases: +1.0, 0.0, and −1.0. When the random processes for elastic modulus, \n
As shown in Figure 15, the maximum COV exceeds the input standard deviation of the stochastic process in the case of negative perfect correlation, while it is about 75% in the case of no correlation. However, in case of positive perfect correlation (Figure 15c), the response COV of natural frequency of FGM beam is small enough to be ignored. These results can easily be understood because the stiffness and mass matrix of each elements in Eq. (18) increase or decrease with the same rate in the case of positive perfect correlation. In case of negative perfect correlation, the ratio of random parts in the stiffness and mass are relatively large since the random parts have opposite sign, which makes the response variability large.
\nEffect of the correlation between two random parameters: (a) negative perfect correlation (CC = −1.0), (b) no correlation (CC = 0.0), and (c) positive perfect correlation (CC = +1.0).
In the case, when we take the correlated multiple random material properties into account, we also obtained the slight nonlinear pattern of variation of COV in terms of COV of stochastic fields, as shown in Figure 16.
\nCOV of natural frequency as a function of the standard deviation of stochastic process: (a) negative perfect correlation (CC = −1.0), (b) no correlation (CC = 0.0), and (c) positive perfect correlation (CC = +1.0).
To evaluate response variability due to a single parameter of the random Young’s modulus and multiple uncertain material properties, a formulation in the context of stochastic finite element solution is suggested for the natural frequency of FGM beam. In deriving the formula for the covariance of the response, modified power spectral density and correlation function are defined by using the general formula of random processes. The Monte Carlo simulation is performed employing the statistical preconditioning scheme as a random process generation technique. The local average method is employed instead of mid-point rule in Monte Carlo simulation.
\nIn FGM beam natural frequency, the response COV for correlation between a random of single parameter and two uncertain parameters is observed. The coefficient of variation of natural frequency can only reach about 50% of the input standard deviation of the stochastic process in a single parameter of the random elastic modulus. However, the number of values is increased over 100% of the input standard deviation of the stochastic process in multiple uncertain material properties, when the correlation distance tends to infinity. The results showed that the COV of natural frequency of FGM beam in a single parameter of the random Young’s modulus and multiple uncertain material properties achieved maximum variability for
There is a very small difference between deterministic natural frequency and probabilistic natural frequency of FGM beam for the case of positive perfect correlation. Also, the COV of natural frequency does not depend on the number of elements, Young’s modulus ratio, and the ratio of non-uniformity parameter of FGM beam. The importance of these parameters needs to be studied as a further work.
\nHermite shape functions of beam finite element:
\nStiffness matrix:
\nMass matrix:
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\n\nRESEARCH CHAPTER – A research chapter reports the results of original research thus contributing to the body of knowledge in a particular area of study.
\n\nREVIEW CHAPTER – A review chapter analyzes or examines research previously published by other scientists, rather than reporting new findings thus summarizing the current state of understanding on a topic.
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\n\nPERSPECTIVE CHAPTER – A perspective chapter offers a new point of view on existing problems, fundamental concepts, or common opinions on a specific topic. Perspective chapters can propose or support new hypotheses, or discuss the significance of newly achieved innovations. Perspective chapters can focus on current advances and future directions on a topic and include both original data and personal opinion.
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