Metal ion vs. log βMY values.
\r\n\tIC offer services both at the macro-scale (country) and at the micro-scale (cities). The stability and the protection of these networks on both scales are a significant task, entrusted to the Operators who operate CI in a concession mandate. Due to the high level of interdependency of CI, which exchange services to each other for their functioning, their management and protection cannot be carried out by a "linearized" strategy (each infrastructure managed and protected independently on the others) due to the presence of tight links which connect each other. CI protection through provision of "smart" properties such as resilience, has become a complex task which must be tackled not only by deploying advanced technological means but also by triggering new management strategies, enabling holistic and global management policies.
\r\n\r\n\tThe book aims at providing an overview of the understanding of complex phenomena taking place on interdependent networks and of the advanced technical solutions related to management, risk analysis and resilience enhancement of networks, either from a theoretical and operational (i.e. with solution related to real or realistic cases) points of view. A large emphasis is provided to the capability opened by the use of field and remote sensing tools for monitoring and assessing risks on CI. The use of comprehensive data set, the access to big data is going to open the way to the realization of new tools for supporting the decision making process needed for both daily and emergency management.
",isbn:"978-1-83962-621-0",printIsbn:"978-1-83962-620-3",pdfIsbn:"978-1-83962-628-9",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"7cfcd62bae8c99be207e18bb73e2a7b1",bookSignature:"Dr. Vittorio Rosato and Dr. Antonio Di Pietro",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10256.jpg",keywords:"Complex systems, Interdependence, Resilience, Cascading effects, Event prediction, Emergency management, Decision support, AI, Field sensors, Remote sensing, IoT, GIS",numberOfDownloads:525,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"June 15th 2020",dateEndSecondStepPublish:"July 6th 2020",dateEndThirdStepPublish:"September 4th 2020",dateEndFourthStepPublish:"November 23rd 2020",dateEndFifthStepPublish:"January 22nd 2021",remainingDaysToSecondStep:"8 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"Supervisor and Project Evaluator for EU, for the Italian Ministry of University and Research, and that of Economic Development; Consultant for several Italian Regions and the Italian Ministry of Defense; Coordinator of several National Projects; Co-founder of two SMEs active in software engineering and biotechnology; Author of more than 140 scientific papers in peer reviewed journals and conference proceedings.",coeditorOneBiosketch:"A full researcher at ENEA (Italian Energy, New Technology and Environment Agency) since 2007 and a joined member to the Laboratory for the Analysis and Protection of Critical Infrastructures (APIC) from 2015. Dr. Di Pietro took part in several European and Italian national research projects and acted as an advisor for certain Evaluation Studies commissioned by the EU.",coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"27002",title:"Dr.",name:"Vittorio",middleName:null,surname:"Rosato",slug:"vittorio-rosato",fullName:"Vittorio Rosato",profilePictureURL:"https://mts.intechopen.com/storage/users/27002/images/system/27002.jpg",biography:"Vittorio Rosato received the Laurea degree (M.Sc.) in Physics from the University of Pisa (Italy) and a Ph.D. in Condensed Matter Physics from the University of Nancy (France). He has extensively been working in Computational Physics, particularly in Condensed Matter and Material Science in his positions as Research Assistant at the University College of Wales in Aberystwyth (UK) and at the Centre d'Etudes Nucleaires in Saclay (France). Staff Scientist at ENEA (Italian National Agency for New Technologies, Energy and Sustainable Economic Development) since 1990, he is currently Head of the Laboratory of Analysis and Protection of Critical Infrastructures and Manager of the Italian Node of the European Infrastructure Simulation and Analysis Centre (EISAC.it).\nHis current research activities span from risk analysis to the design of Decision Support Systems for the management of complex technological networks. He acts as Supervisor and Project Evaluator for EU, for the Italian Ministry of University and Research, and that of Economic Development; he is also consultant for several Italian Regions and the Italian Ministry of Defense. He is and has been Coordinator of several National Projects. He is co-founder of two SMEs active in software engineering and biotechnology. He is author of more than 140 scientific papers in peer reviewed journals and conference proceedings.",institutionString:"ENEA (Italian National Agency for New Technologies, Energy and Sustainable Economic Development)",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"0",institution:null}],coeditorOne:{id:"284589",title:"Dr.",name:"Antonio",middleName:null,surname:"Di Pietro",slug:"antonio-di-pietro",fullName:"Antonio Di Pietro",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bReF6QAK/Profile_Picture_1581328351906",biography:"Antonio Di Pietro received the Laurea degree (M.Sc.) in Informatics Engineering from Sapienza University of Rome (Italy) and a Ph.D. in Methodologies for Emergency Management in Critical Infrastructures from Roma Tre University (Rome).\nHe has been working as a full researcher at ENEA (Italian Energy, New Technology and Environment Agency) since 2007 and in 2015 he joined the Laboratory for the Analysis and Protection of Critical Infrastructures (APIC) in the same institution.\nHis research interests include modelling and simulation of critical infrastructures and the development of Decision Support Systems integrating seismic and meteorological natural threat modeling. He is also an Unmanned Aerial Vehicles (UAV) ENAC-certifed pilot to perform critical operations involving aerial photogrammetry tasks for biological and Infrastructure monitoring applications. He took part in several European and Italian national research projects and acted as an advisor in some Evaluation Studies commissioned by the EU. He has also been advisor of several M.Sc. students and also a teacher in several professional courses on Software Engineering and Databases.",institutionString:"ENEA (Italian National Agency for New Technologies, Energy and Sustainable Economic Development)",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"0",institution:null},coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"11",title:"Engineering",slug:"engineering"}],chapters:[{id:"74122",title:"Risk Analysis in Early Phase of Complex Infrastructure Projects",slug:"risk-analysis-in-early-phase-of-complex-infrastructure-projects",totalDownloads:87,totalCrossrefCites:0,authors:[null]},{id:"74493",title:"Flood Risk Analysis for Critical Infrastructure Protection: Issues and Opportunities in Less Developed Societies",slug:"flood-risk-analysis-for-critical-infrastructure-protection-issues-and-opportunities-in-less-develope",totalDownloads:50,totalCrossrefCites:0,authors:[null]},{id:"74123",title:"Resilience in Critical Infrastructures: The Role of Modelling and Simulation",slug:"resilience-in-critical-infrastructures-the-role-of-modelling-and-simulation",totalDownloads:78,totalCrossrefCites:0,authors:[null]},{id:"73984",title:"Validation Strategy as a Part of the European Gas Network Protection",slug:"validation-strategy-as-a-part-of-the-european-gas-network-protection",totalDownloads:35,totalCrossrefCites:0,authors:[null]},{id:"74174",title:"Defects Assessment in Subsea Pipelines by Risk Criteria",slug:"defects-assessment-in-subsea-pipelines-by-risk-criteria",totalDownloads:44,totalCrossrefCites:0,authors:[null]},{id:"74240",title:"Analyzing the Cyber Risk in Critical Infrastructures",slug:"analyzing-the-cyber-risk-in-critical-infrastructures",totalDownloads:85,totalCrossrefCites:0,authors:[null]},{id:"74141",title:"Italian Crisis Management in 2020",slug:"italian-crisis-management-in-2020",totalDownloads:49,totalCrossrefCites:0,authors:[null]},{id:"74668",title:"A Strategy to Improve Infrastructure Survivability via Prioritizing Critical Nodes Protection",slug:"a-strategy-to-improve-infrastructure-survivability-via-prioritizing-critical-nodes-protection",totalDownloads:34,totalCrossrefCites:0,authors:[null]},{id:"74143",title:"Resilience of Critical Infrastructures: A Risk Assessment Methodology for Energy Corridors",slug:"resilience-of-critical-infrastructures-a-risk-assessment-methodology-for-energy-corridors",totalDownloads:66,totalCrossrefCites:0,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"205697",firstName:"Kristina",lastName:"Kardum Cvitan",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/205697/images/5186_n.jpg",email:"kristina.k@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"1591",title:"Infrared Spectroscopy",subtitle:"Materials Science, Engineering and Technology",isOpenForSubmission:!1,hash:"99b4b7b71a8caeb693ed762b40b017f4",slug:"infrared-spectroscopy-materials-science-engineering-and-technology",bookSignature:"Theophile Theophanides",coverURL:"https://cdn.intechopen.com/books/images_new/1591.jpg",editedByType:"Edited by",editors:[{id:"37194",title:"Dr.",name:"Theophanides",surname:"Theophile",slug:"theophanides-theophile",fullName:"Theophanides Theophile"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3092",title:"Anopheles mosquitoes",subtitle:"New insights into malaria vectors",isOpenForSubmission:!1,hash:"c9e622485316d5e296288bf24d2b0d64",slug:"anopheles-mosquitoes-new-insights-into-malaria-vectors",bookSignature:"Sylvie Manguin",coverURL:"https://cdn.intechopen.com/books/images_new/3092.jpg",editedByType:"Edited by",editors:[{id:"50017",title:"Prof.",name:"Sylvie",surname:"Manguin",slug:"sylvie-manguin",fullName:"Sylvie Manguin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3161",title:"Frontiers in Guided Wave Optics and Optoelectronics",subtitle:null,isOpenForSubmission:!1,hash:"deb44e9c99f82bbce1083abea743146c",slug:"frontiers-in-guided-wave-optics-and-optoelectronics",bookSignature:"Bishnu Pal",coverURL:"https://cdn.intechopen.com/books/images_new/3161.jpg",editedByType:"Edited by",editors:[{id:"4782",title:"Prof.",name:"Bishnu",surname:"Pal",slug:"bishnu-pal",fullName:"Bishnu Pal"}],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"}},{type:"book",id:"1373",title:"Ionic Liquids",subtitle:"Applications and Perspectives",isOpenForSubmission:!1,hash:"5e9ae5ae9167cde4b344e499a792c41c",slug:"ionic-liquids-applications-and-perspectives",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/1373.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"}},{type:"book",id:"57",title:"Physics and Applications of Graphene",subtitle:"Experiments",isOpenForSubmission:!1,hash:"0e6622a71cf4f02f45bfdd5691e1189a",slug:"physics-and-applications-of-graphene-experiments",bookSignature:"Sergey Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"371",title:"Abiotic Stress in Plants",subtitle:"Mechanisms and Adaptations",isOpenForSubmission:!1,hash:"588466f487e307619849d72389178a74",slug:"abiotic-stress-in-plants-mechanisms-and-adaptations",bookSignature:"Arun Shanker and B. 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:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"4816",title:"Face Recognition",subtitle:null,isOpenForSubmission:!1,hash:"146063b5359146b7718ea86bad47c8eb",slug:"face_recognition",bookSignature:"Kresimir Delac and Mislav Grgic",coverURL:"https://cdn.intechopen.com/books/images_new/4816.jpg",editedByType:"Edited by",editors:[{id:"528",title:"Dr.",name:"Kresimir",surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3621",title:"Silver Nanoparticles",subtitle:null,isOpenForSubmission:!1,hash:null,slug:"silver-nanoparticles",bookSignature:"David Pozo Perez",coverURL:"https://cdn.intechopen.com/books/images_new/3621.jpg",editedByType:"Edited by",editors:[{id:"6667",title:"Dr.",name:"David",surname:"Pozo",slug:"david-pozo",fullName:"David Pozo"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"41475",title:"Good Practice for Fatigue Crack Growth Curves Description",doi:"10.5772/52794",slug:"good-practice-for-fatigue-crack-growth-curves-description",body:'Fatigue life estimation and crack propagation description are the most important components in the analysis of lifespan of structural components but it may require time and expense to investigate it experimentally. For fatigue crack propagation studying in cases when it is difficult to obtain detailed results by direct experimentationcomputer simulation is especially useful. Hence, to be efficient, the crack propagation and durability of construction or structural component software should estimate the remaining life both experimentally and by simulation. The critical size of the crack or critical component load can be calculated using material constants which have been derived experimentally and from the constant amplitude crack propagation curve, crack size-life data and curve using crack propagation software. Many works in the field of fracture mechanics prove significant development in the numerical analysis of test data from fatigue crack propagation tests.
A simple stochastic crack growth analysis method is the maximum likelihood and the second moment approximationmethod,wherethe crack growth rate is considered as a random variable. A deterministic differential equation is used for the crack growth rate, while it is assumed that parameters in this equation are random variables. The analytical methodsare implemented into engineering practice and are useto estimate of the statistics of the crack growth behavior (Elber, 1970; Forman et al., 1967; Smith, 1986).
Though many models have been developed, none of them enjoys universal acceptance. Due to the number and complexity of mechanisms involved inthis problem, there are probably as many equations as there are researchers in the field. Each model can only account for one or several phenomenological factors - the applicability of each varies from case to case, there is no generalagreement among the researchers to select any fatigue crack growth model in relation to the concept offatigue crack behavior (Kłysz, 2001; Paris & Erdogan, 1963; Wheeler, 1972; Willenborg et al., 1971). Mathematical models proposed e.g. by Paris, Forman, and further modifications thereof describe crack propagation with account taken of such factors as:material properties, geometry of a test specimen/structural component,the acting loads and the sequenceof these loads (AFGROW, 2002; Kłysz et al.,2010a; NASGRO®, 2006; Newman, (1992); Skorupa, 1996). Application of the NASGRO equation, derived by Forman and Newman from NASA, de Koning from NLR and Henriksen from ESA, of the general form(AFGROW, 2002; NASGRO, 2006):
has significantly extended possibilities of describing the crack growth rate tested according to the standard (ASTM E647). The coefficients stand for:
a– crack length [mm],
N – number of load cycles,
C, n, p, q – empiricalcoefficients,
R–stress ratio,
ΔK – the stress-intensity-factor (SIF)range thatdepends on the size of the specimen, applied loads, crack length, ΔK=Kmax-Kmin
ΔKth – the SIF threshold, i.e. minimum value of ΔK, from which the crack starts to propagate:
where:
ΔK0 – thresholdSIFatR→0,
ΔK1 – thresholdSIFatR→1,
Cth – curvecontrolcoefficientfordifferentvaluesofR; equals 0for negative R, equals 1 for R ≥ 0,for some materials it can be found in the NASGRO database,
Kmax – the SIFfor maximum loading force in the cycle,
Kc – critical value of SIF,
f – Newman\'s function that describes the crack closure:
where A0, A1, A2, A3coefficients are equal:
α, Smax/σ0 – Newman\'s empirical coefficients.
Determination of the above coefficients for equation that correctly approximates test data is difficult and causes some singularities described below, when the Least Squares Method (LSM)is used.
Fatigue crack propagation graphs: a) K=f(N); b) a=f(N)and c) da/dN=f(ΔK)
The fatigue crack growth test results provide an illustration of relations such as: specimen stress intensity vs. number of cycles (K=f(N)), crack growth vs. number of cycles (a=f(N)); crack growth rate vs. stress intensity factor range (da/dN=f(ΔK)). These experimental curves can be presented, for example, in the graphical form shown inFig. 1 (for a single specimen, two-stage test: stage I - decreasing ΔKtest, black curve; stage II – constant amplitude test, blue curve).
Specifically, the da/dN=f(ΔK) plots can be obtained directly from the material test machine control software (e.g. by employing the compliance method and by using a clip gauge) or can be obtained by differentiating the a=f(N) curve after correlating it with K=f(N).These plots, for single specimen tests, as well as for tests with multiple specimens under different load conditions (e.g. various stress ratio R values), can be successfully described analytically when appropriate mathematical models and equations are employed.
Fatigue tests for structural components durability analysis can be conducted with the RCT (Round Compact Tension)(Fig. 2a) or with SEN (Single Edge Notch) (Fig. 2b) or other specimens according to the correspondingASTM E647standard.
RCT & SEN specimens for fatigue crack propagation tests
The general formula that describes thestress intensity factor is as follows:
where: P – appliedforce,
B, W – the specimen’s thickness and width,
Y – the specimen’s shape function(ASTM E647, Fuchs& Stephens 1980,Murakami 1987):
for the RCT specimen:
for theSENspecimen:
wherea/Wis a non-dimensional crack length.
The compliance functionto compute the crack length in the RCT specimen has the form:
for theSENspecimen (Bukowski & Kłysz 2003):
where: u – compliance described by the following formula:
E – Young’s modulus,
COD – Crack Opening Displacement.
An example of the F-COD relationship has been plotted in Fig. 3. The plot has been gained from the fatigue crack growth test conducted for the SEN specimen made from constructional steel subjected to constant amplitude loading with overloads (Bukowski & Kłysz, 2003). The records were taken in the course of statically applied 2-cycle overloads of 40% order at subsequent stages of crack propagation. Load base level and overload level were gradually reduced as the crack growth kept increasing and after non-linearity (hysteresis loop) had occurred in the F-COD plot. The objective was to avoid failure of the specimen in a subsequent overload cycle to be able then to continue the crack-propagation test. Any change in the angle of inclination of the rectilinear segment of each of the hysteresis loops (i.e. the F/COD proportion from formula (13)) is a measure of the specimen’s compliance u and proves the crack length in the specimen under examination keeps growing.
Results presented below come from the examination of the 2024 aluminum alloy taken from the helicopter rotorblades (Kłysz & Lisiecki,2009) or from the aircraft ORLIK’s fuselage skins (Kłysz et al., 2010b) and are obtained for three values of test stress ratio R = 0.1; 0.5; 0.8, under laboratory conditions, with loading frequency 15 Hz.The crack length was measured with the CODclip gauge using the compliance method. The crack growthrate was determined using the polynomial method.
Relationship of F-COD recorded in subsequent overload cycles of fatigue crack growth test
Results of fatigue crack growth rate tests for 3 specimens (for R = 0.1; 0.5; 0.8) are presented in Fig. 4 (Lisiecki & Kłysz, 2007).
Fatigue crack growth rates in 3 specimens for different R values – a) test data, b) results of approximation
The NASMAT curve fitting algorithms use the least-squares error minimization routines in the log-log domain to obtain the corresponding constants using the NASMAT module contained within the NASGRO suite of software (NASGRO, 2006). The constants C and n, i.e. the main fit parameters, are determined through the minimization of the sum of squares of errors, where the error term corresponding to the i-th data pair (ΔK, da/dN)i is (Forman etal., 2005):
Values of da/dN are determined using the method of differentiating the dependence a-Nwith the secant or the polynomialmethod applied (AFGROW, 2002; ASTM 647; NASGRO, 2006).
Generally the curve fittingof crack growth data is an iterate process that consists in using established values of various constants (other than C and n), specifying the data sets that typify the material, applying the least-squares algorithm to compute C and n, and plotting the data for various R values with the curve fit of each stress ratio. The process is continued by making slight modifications in the entered values until the best fit to the test data is obtained. In general fitting the NASGRO equation is really a multi-step process involving:
fitting or defining the threshold region;
fitting or defining the critical stress intensity or toughness to be used at the instability asymptote;
making initial assumptions on key parameters such as p and q;
performing the least squares fit to obtain C and n; and finally;
using engineering judgment to adjust the results for consistency and/or a desired level of conservatism.
For the LSMapproximation of test data, analytical description thereof, and determination of coefficients of approximation equations, according to which the criterion used in the analysis is the minimum of the square sum:
of deviations between values of the test data yi and those of the approximated function
in respect of the order of magnitude, value of the sum S increases as magnitudes of approximated values increase, e.g. if values of test data are of the order of magnitude 10, 1000, 1000000, with the scatter of 10%, the summed differences are of the order of magnitude 1, 100, 100000, and hence, dynamic changes in the total value of the sum S depend on values of differences – as a quadratic function it is characterized by a linear function of the derivative, which also means that for differences close to zero (e.g. 10-5, 10-8, etc.) this dynamic change is much smaller than for differences of higher magnitudes, which influences the „flexibility” of the performed approximation;
if the test data significantly differ from each otherin magnitude (e.g. from 1 to 100000 or from 10-8to 10-2), the approximated values near the lower threshold contribute much less to the total sum Sthan approximated values near the upper threshold;this means that, e.g. tens or hundreds of test data with differences in magnitude of 100% from value 1 are less significant in performing the approximation than one or a few data points which differ by 1% from value 100000.
According to the above stated example, the approximation is “asymmetric” since better approximation will be achieved for higher values of test data, neglecting differences around smaller values – an example of such approximation is shown in Fig. 4b, where one can see a good fit of theoretical description of 3 curves for large values of da/dN (over 10-4 mm/cycle) while there is anevident misfit for smallest values (below 10-5 mm/cycle). The presented approximation has been achieved by satisfying the LSM criterion, i.e. the minimum value of the sum S. When the test data are within a wide range of values, e.g. 5 orders of magnitude, i.e. from 10-2to 10-7 mm/cycle,then differences between the highest values and the approximating function will have the largest effect on the square sum S of deviations while differences for small values, sometimes of 2-3 orders of magnitude, do not contribute much to the total sum S.
Hence, the misfit of the approximating function for low values of da/dN, practically for values lower by only 1-2 orders of magnitude than the maximum values of da/dN. Within this range the theoretical description is rather random and has rather no effect on the value of the sum S,which indicates that this criterion is rather useless for this type of analysis.
It seems reasonable to use one of the following criterion modifications, which will allow to remove the above stated problems:
changingthe form of the criterion, or
using logarithmic values of da/dN,
In the present study the first variant has been examined(see section 3.3) due to the fact that it is more general since it does not limit itself only to positive values of predicted yi,which is a requirement in the second variant.In the case of crack propagation test data all the da/dNvalues are positive; therefore the second variant could also be used.
Since the criterion for fitting the theoretical description to the test data in the form of equation (15) or (16), or any other, is closely connected with the number of approximated points (in the case under discussion, coordinates in the graph (da/dNi,ΔKi)), the quality of fit has to depend on:
the distribution of the number of test points among particular curves,
the distribution of test points on particular curves,
not to mention
the scatter of test points and accuracy of finding them.
If the distribution of points among particular curves is not uniform, the approximation will show better fit to the curves with a larger number of points than to those with a smaller number of points – the contribution thereof to the pooled error included in the approximation criterion will be greater; the minimization thereof will occur around the larger data cluster. Similar situation occurs while fitting the description to a given experimentally gained curve – where the data concentration is larger, the approximation will be better than where there is less data, or where the data are only individual points. Therefore, essential to the analysis of test data and to description thereof is the regular distribution of the test data over the whole range to be subject to approximation. Since it is sometimes beyond the reach of researchers while recording the test data directly during the testing work, some modification or recalculation of the test data set may prove indispensable.
As clearly seen in Fig. 4, the number of points in the threshold and critical areas of the scope of the stress intensity factor ΔK is very small, which results from the specific nature of the performed test and data recording.
For crack growth rates lower than 10-6 mm/cycle the increment by 1 mm occurs after approx. 1 million cycles, i.e. the process is a long-lasting one, and the recording of the crack-length increment for instance every 0.01 mm gives 100 points of test data only (while in the case of taking records every 0.005 mm, the number of points will be 200). The testing work for even lower crack growth rates is still more time- and energy-consuming. With as little crack-length increments as these there is practically no chance that in single load cycles any random jump will occur in values of recorded data of the order of 0.01 or at least 0.001 mm (i.e. by approximately 3 – 4 orders of magnitude higher than the crack growth rate under examination). This provides relatively regular recording of crack lengths in the course of the testing work, i.e. for subsequent increments 0.01, 0.02, 0.03, … mm, etc. (even if measurements are taken for crack-length increments by only a fraction of a millimetre, i.e. in a shorter time, which means for the number of cycles lower than the above-mentioned 1 million).
In the range of critical crack propagation, at the crack growth rate higher than 10-3 mm/cycle, the recordings of the crack length increments every 0.01 mm (as above) take place more frequently than every 10 cycles. For load-applying frequencies of 10 – 20 Hz this means 1 s long data-recording intervals in the course of the testing work. The final several millimetres’ crack-length increment occurs as fast as over only several minutes of the testing work, with crack-length increments significantly increasing every cycle. Hence, at the testing rate getting as high, the number of test points remains relatively low and, because of these ever-growing increments, lower than the above-mentioned 100 or 200 points per every 1 mm of the crack length.
In the intermediate area of the graph (10-6 through 10-3 mm/cycle, i.e. covering 3 orders of magnitude of the da/dNvalue) the above-mentioned exemplary crack-length increments every 0.01 mm take place on a regularbasis, however, with random fluctuations typical of the phenomenon under examination – there are no identical data recordings after 0.01, 0.02, 0.03, 0.04, … mm of crack-length increment, since instantaneous readings (variations) from the measuring sensors may cause that the data recording during the test, with the same recording criterion assumed, can occur for increments of, e.g. 0.01, 0.028, 0.038, 0.057, … mm, disturbing at the same time the regular basis of increments in the number of cycles between particular measurements. Fig. 5a illustrates the non-uniformity of such data-recording practice; the arrows point to where such disturbances have occurred, and after which the subsequent record is taken after the higher number of cycles. This, in turn, affects the crack growth rate. Calculation of the da/dNderivative based on the in this way recorded data must also be burdened with a random scatter, Fig. 5b, larger than that resulting from the properties of the material under examination.
To eliminate these incidental disturbances, the experimentally recorded time function may become smoothed by means of interpolation of results on the basis of any linear regression function (with either a straight line or a polynomial). Fig. 5c shows an example of such smoothening: presented with a full line is result of the 7-point regression, i.e. after having interpolated each point (ai;Ni), with account taken of 6 adjacent points: 3 points in front of and 3 points behind a given point (ai;Ni). It is evident that this smoothed curve represents in a reliable (or even better, in a more reliable way) the experimentally recorded dependence between measured quantities. On the other hand, the above-discussed disturbances have been removed from particular measurements.
Calculation of the da/dNi derivative for any point of the plot (ai;Ni) can be carried out on the basis of linear or polynomial regression for e.g. 5, 7, or 9 adjacent points around a given i-th point. Fig. 5d shows result of the 5-point linear regression (2 points in front of the (ai;Ni) point, the (ai;Ni) point, 2 points behind the (ai;Ni) point), of calculations of the da/dNderivative against the unsmoothedplot a-N. What in this case is arrived at from the equation for the line of regression yi\n\t\t\t\t\t= mixi\n\t\t\t\t\t+ ni (and more exactly, ai\n\t\t\t\t\t= miNi\n\t\t\t\t\t+ ni) is:
In the case of linear regression with polynomials of the 2nd (yi\n\t\t\t\t\t= mixi2+nixi+li) or 3rd (yi\n\t\t\t\t\t= mixi3+nixi2+lixi+ki) order, the crack growth rate is calculated from the formulae, respectively:
What becomes evident is a considerable scatter of calculated values of the crack growth rate da/dN, and for points indicated with arrows it can be stated that:
any measurement disturbance results in that the resulting (calculated) value of da/dN at one or two subsequent points is always lower than that for the point in question,
the measurement disturbance is not expected to reflect the accelerated crack propagation, even though in the form of a local maximum, which all the more confirms the correctness of treating this disturbance as a random effect,
where the disturbance occurs in the local-maximum area, it magnifies its value; however, the scale of this increase may prove too large as compared to the actual crack growth rate.
Calculated values of da/dNwith corresponding test data a-N: 5-point linear regression, a) and b) – output data; c) and d) – smoothed data
If calculations are carried out for the smoothed curve a-N (Fig. 5c), the resulting da/dN curve presented in Fig. 5d takes the form of a solid line. The scatter of values of the crack growth rate over the whole range of calculations is much smaller. The local extremes have been maintained, however, slightly scaled down than in Fig. 5b.
In the case the regression used to calculate the da/dN derivative is carried out for a greater number of points adjacent to a given computational point, the corresponding curves look like in Fig. 5a (for 7-point regressions: 3 points in front of the (ai;Ni), the point in question (ai;Ni) and 3 points behind the(ai;Ni)) and Fig. 5b (for 9-point regressions: 4 points in front of the (ai;Ni), the point in question (ai;Ni) and 4 behind the(ai;Ni)). In all the cases the derivative of da/dN has been found from equation (17).
Calculated values of da/dN together with corresponding experimental data a-N: a) 7-point linear regression, b) 9-point linear regression
It is obvious that as the number of points taken into account in the regression analysis increases, the scatter of computational results gets reduced and the curve plotted for unsmoothed data (circles in Fig. 6) ever more resembles the curve plotted for smoothed data (solid line in Fig. 6). It is effected by the fact that the greater number of data accepted for regression brings the result closer to that of regression for smoothed data. There is of course some disadvantage: the greater number of data taken into account in regression analyses, the more reduced number of details referring to, e.g. local changes in value of da/dN are to be seen on the plotted curves. In the extreme, if all the points are subject to regression at once, the smoothed curve a-N would be a straight line and the da/dN curve would run horizontally. Another extreme consists in that the whole curve a-N would be described with only one regression equation, which in turn would provide the reliable mapping of the whole a-N curve; the da/dN derivative could be calculated by means of differentiating this equation. However feasible, it seems unpractical, work-consuming, more of the ‘art for art’s sake’ category. Results presented in Figs 5 and 6 could be considered optimal: they provide good mapping of local changes in the approximated curves and do not require any complicated mathematical apparatus.
Characteristic of these plots (for both the unsmoothed and smoothed data) is that the calculated rates da/dN may be the same for different numbers of cycles N (hence, for different crack lengths a and different values of ΔK). This is the effect of more common, for this range of crack growth rate da/dN, occurrences of changes in the monotonicity of curves a-N than in threshold or critical ranges of da/dN-ΔK. Curves plotted in Figs 5 and 6 correspond to approx. 1-millimetre increment in the crack length (6.6 through 7.5 mm) and cover crack growth rates of 2 ÷ 4.10-6 mm/cycle. At the further stage of the crack growth as the crack length increases, the crack growth rate increases as well, and before the crack reaches the critical growth range the calculated values of da/dN from the range 10-6 through 10-3 mm/cycle will repeatedly appear in the calculations. Hence, the number of measuring points recorded throughout the testing work for this range of da/dN will be higher than for threshold or critical ranges of da/dN-ΔK, what is to be seen also in Fig. 4.
Moreover, in practice, the plotting of a complete crack propagation curve da/dN-ΔK, i.e. starting from critical crack growth rates of 10-8 mm/cycle up to critical ones of 10-2mm/cycle, is not performed in the course of one test only. This is closely related with difference in levels of ΔK for the stage of the specimen’s precracking and the threshold range typical of the rates of 10-8 mm/cycle. The precracking usually finishes at higher values of ΔK, since it cannot proceed with the threshold growth rate. The reason is that it would take much more time than the test itself. Therefore, the test started after the specimen’s precracking stage from the threshold values of the crack growth (change in the loading level from high to lower), would be connected with the crack growth retardation effect, which - in turn - would disturb test results in this area, i.e. it would not allow the researchers to gain the correct curve da/dN-ΔK. Such tests are usually conducted as a two-stage effort – see Fig. 1:
I stage – with exponentially decreasing ΔK (ΔK=ΔK0e-ga), with constant relative gradient, i.e.
The decreasing ΔK, starting from some suitably high value, and the crack length both cause that the crack growth rate becomes reduced to reach then the threshold range of the plot. At this stage, the a-N curve asymptotically approaches the horizontal line as the testing time increases. The testing time depends on the scientifically and economically justified needs of the researcher, although in practice this time much more depends on sensitivity of applied sensors, since both the level of applied loads and the crack opening size decrease for this range to values comparable to electric noise of the testing machine, which usually results in the test being automatically interrupted and the testing machine being stopped for crack growth rates lower than 10-8 mm/cycle. The at this stage obtained curve a-N and the propagation-curve section da/dN-ΔK may look like e.g. those presented in Fig. 7 (to be also seen in Fig. 1b).
II stage – at constant amplitude load (CA test, constant amplitude test) up to the acquisition of the right side within the critical range. The test is carried out at the level of loads higher than the level at which the stage I was completed; as the crack length increases, there is a systematic increase in the ΔK, up to the moment the critical value is reached, at which the specimen fails. The at this stage obtained curve a-N and the propagation-curve section da/dN-ΔK may look like e.g. those presented in Fig. 7 (to be also seen in Fig. 4).
The total result of both the stages has been presented in Fig. 8 – both the curves from Figs 6 and 7 complement one another to full propagation-curve plot a-N and da/dN-ΔK: experimentally found points in the form of circles, curves smoothed in the form of full lines. It is quite clear that the mid section (range) of the da/dN-ΔK curve contains much more experimentally gained points despite the same criterion for data recording in the course of testing work for all three ranges, and also, independently of the fact that both the curves overlap over some specific section common to both of them. Furthermore, the plot presents the above-discussed changes in the monotonicity of how they run, independently of whether the calculations of the da/dN derivative have been conducted for unsmoothed or smoothed data – Fig. 8.
Curves a-N and da/dN-ΔK for the first (I) – a), b), and the second (II) test stages – c), d)
Disturbances in the run, monotonicity of curves da/dN-ΔK as well as different measuring-data density in particular areas of the graph do not serve well any attempts to theoretically describe these curves. As mentioned earlier, the least squares methods better fit regression curves to areas where there is more approximated points, in the case given consideration, in the middle ranges of the da/dN-ΔK curves. To eliminate this effect, application of the Authors’ Method of Regular Curves Mapping (MRCM) to approximate the da/dN-ΔK curves is advisable.
Curves a-N and da/dN-ΔK and how they run at the I and II test stages: – results for data after the curve has been smoothed - a), b), and effect of having applied the MRCM – c)
The MRCM technique of mapping test data consists in fixing, at regular intervals (along axes x or y), the k number of representative points in the data set under analysis (upon the experimentally gained curve). The following actions are to be taken:
determined are selected values of coordinates xi (or yi), for which the above-mentioned points will be fixed (i = 1,2, ….,k),
from the curve under analysis, point x’i (or y’i) is fixed, of coordinate value closest to the assumed value of xi (or yi), and 2m of adjacent points – by assumption, in half these are points of values lower than xi (or yi) and in half - of higher values, (m is equal to, e.g. 2, 3, 4 or 5),
a set of in this way gained data 2m+1,(x’i-m, yi-m) through (x’i+m, yi+m) (or (xi-m, y’i-m) through (xi+m, y’i+m)) – grouped around some selected value of xi (or yi) is subject to regression with any function to determine the approximated value of yi*(or xi*) corresponding to the selected value of xi (or yi),
the point of coordinates (xi, yi*) (or (xi*, yi)) is mapped on the curve under analysis – as the i-th representative data item found on the basis of the assumed criteria,
steps b) through d) are repeated for subsequent k number of values determined in a), until a set of k number of points that represent (map) the curve is obtained.
The effect of the in this way performed mapping of values of da/dN, regularly distributed within particular intervals (orders of magnitude), in selected k = 37 points, for the curve shown in Fig. 8b, ispresented in Fig. 8c. The points in question:
well represent (map) the curve under analysis,
are equidense distributed within the whole range of da/dN variability,
do not show any more or less significant fluctuations/scatter of values resulting from, e.g. random measuring-data dispersions.
Experimentally gained curves da/dN-ΔK for 9 specimens tested at different stress ratios R a) and the same curves having been mapped with points using the MRCM, b) and with extrapolated points that ‘perform’ the mapping according to the MRCM, c) – together with approximation thereof with the NASGRO equation
The set of points that map the curve seems to give good basis, owing to the above described features, for analyses of theoretical description of a given, experimentally gained curve. In the case of nine (9) curves that correspond to tests with three (3) values of the stress ratio R – Fig. 9a, the result of experimentally gained data modification with the MRCM applied (points in the graph) is presented in Fig. 9b, together with the data approximation by means of the NASGRO equation, with the LSM criterion used, according to formula (15).
The MRCM technique also enables, if need be and with scientific correctness maintained, the extrapolation of the mapping points beyond the range of recorded test data, i.e. into the area of crack growth rates lower (the threshold range) or higher (the critical range) than those recorded experimentally, with their tendency to change which is peculiar to those areas, on the basis of regression at boundary (in the graph – lower or upper) points of experimentally gained curves – the extrapolation result has been shown in Fig. 9c. Application of extrapolation to prepare data for the analytical modelling may prove advantageous in the case the particular experimentally gained curves show different ranges of values, and hence, different numbers of mapping points. After correctly performed extrapolation one can arrive at the situation when they are equalized, which means the same ‘power’ of each of the with the regression method approximated curves.
In order to eliminate the approximation misfit as shown in Fig. 1 and to improve the quality of approximation, modification of formula (15) takes the following form:
The fraction in brackets in formula (13), as a relative error, is a measure of deviation independent of the order of magnitude of compared values (approximated and approximating ones), so that the contribution of all the test data is equally "strong" to the total error S, which should have good effect on the approximation within the whole range, since:
each value among test data yihas equal contribution to the sum S*, independent of its magnitude 10-7, 10-2, 1 or 100000 (i.e. it fits in any magnitude range) – always a deviation of e.g. 10-, 50-, 200-percent of approximating valuewill give a component of the sum S* equal to 0.01, 0.25, 4, respectively;
the criterion assures that the achieved approximation is “symmetric”, i.e. the degree of approximation around lower and higher values is the same;
disadvantages of the criterion described with formula (15) are no longer valid.
The criterion described with formula (20) has also some specific property: if the approximating value equals zero (i.e. for the approximation smaller by 100%) or it is twice as big as the approximated value (i.e. for approximation largerby 100%), then independently of the approximated value the component of the sum S* will equal 1.
In order to carry out the approximation of test data it is necessary to calculate coefficients of the approximating equation used to determine
and can be presented in the following general way:
Coefficients biare directly connected with C, n, p andq (b0=log(C), b1=n, b2=p, b3=-q), whereas functions fidepend onΔK and R and include all the remaining coefficients of the NASGRO equation. Coefficientsbiof the approximating equation are calculated from the minimum condition of the equation (20), i.e.:
This leads to the following system of equations:
It is a system of 4 linear equations with 4 unknowns bi, which after transformation takes a form:
and is easily solved by subtracting in the following steps:
eliminatingb0
what gives 3 equations of the general form:
eliminatingb1
what gives 2 equations of the general form:
eliminatingb2
hence:
Hence, coefficient b2can be calculated from one of the formulae (29); secondly, coefficient b1 from one of equations (27),and finally, coefficient b0from one of equations (25).
The in this way found coefficients of the NASGRO equation enable approximation of curves da/dN-ΔK from Fig. 9 to the form shown in Fig. 10a. Considerable improvement in the theoretical (analytical) description for the whole range of plotted curves is evident.
Both criteria (15) and (20) have also some disadvantage consisting in that if the approximating value
Obviously, it is important whether the approximation and behavior of the approximating curve near value yiat the level of e.g. 10-6and lower (i.e. forstrongly decreasing values within the “threshold” range of the graph) take place at the level of 10-8, 10-12or 10-20 (what is not hard to achieve for curves showing strong vertical courses on graphs plotted with the logarithmic scale applied);it is much better when the possible difference between values
Result of approximation of curves from Fig. 9 with the NASGRO equation with the LSM criterion applied: a) by equation (20), b) by equation (32)
Due to dynamic changes around value
Therefore, a modification is proposed to transform the criterion into the following form:
Owing to this for both large values
Component of the sum for the approximation criterion (32)
Component of the sum for the approximation criterion (32a)
Both extremes of the Sifunction for both positive and negative values of
different approximatedvaluesyiequalto 5; 2; 1; 0.25; 0.01; 0.00001,
the same range of variability of
Approximation criterion functions for different approximated values yifor the (-3yi, 3yi) interval – arrow for equation (32a)
All advantages and disadvantages of the above presented LSM approximation criteria can be seen on the graphs above, in particular:
significant dependence of values of components of the sum S (formula (15)) on the approximated valueyi;
invariability of values of components of sums S*(formula (20)) andS** (formulae (32) and (32a))on all the graphs, i.e. for any approximated valueyi;
no response of the components of sums S and S* to the
The only curve that changes in the graphs presented in Fig. 13 is the plot for components of the sum S graph, i.e. for the standard form of the LSM.
Result of approximation with criterion (32a) applied is shown in Fig. 10b – for data sets with no extrapolation points. The same approximation for only 1 specimen tested at different Ris shown in Fig. 14a, and for only 2 specimens tested at different R- in Fig. 14b.
Approximation of da/dN=f(ΔK) data in different variants with the NASGRO equation, LSM modified according to formula (32a)
Exemplary results of approximation for different test data (with slightly smaller scatter between individual da/dN-f(ΔK) curves) is presented in Fig. 14.
Favorable effects of the approximation (in comparison with results showed in Fig. 1) after implementation of the modified LSM criterion can easily be seen. They tend to represent all the test data, within the whole range of data variability, independently of their absolute values, independently of the number of described curves – 3 (Fig. 14a), 6 (Fig. 14b), 9 (Fig. 10b). This effect has been achieved only by modifying the LSM criterion, since the idea underlying the approximation method for all the presented graphs is identical – the minimum of the sum of squared deviations between the approximated test data and the approximating values.
Approximation of curves da/dN-ΔK substantially depends on preset values of parameters Kc and Kth. Hence, it is very important whether they can be determined on the grounds of the test data only (if they cover the whole range of the curve, i.e. 10-7 through 10-2 mm/cycle, which is not always easy to reach), or whether they need any other method/way to be determined, e.g. formula (2), functional dependences of the type ΔKth = f(R) and Kc = f(R), or the above-mentioned extrapolation. The above-discussed results of extrapolation correspond to the case when both the parameters show constant values for all the approximated curves. The plots for the test data show, however, that they depend on the stress ratio R – for each of nine experimentally gained curves parameters ΔKth,i and Kc,i can be estimated and the data gained can then be used to determine dependences ΔKth = f(R) and Kc = f(R), including coefficients for equation (2).
Formulae (2) and (2a) are special cases of a general formula of the following form:
Having re-arranged this formula, the following is arrived at:
and then:
which can be described with the linear-regression equation as:
where
and the corrected value of the threshold range of the stress intensity factor is:
Having found coefficients m0, m1, m2 of the regression equation (36) we can calculate coefficients of equation (33):
at the same time, value of the ΔKth function is calculated from the regression equation by formula:
So, if we have data sets (Ri, ΔKth,i,\n\t\t\t\t\t\tai) – in the case under analysis there are 9 such sets – we automatically can find coefficients by formula (37), thus reducing the number of coefficients of the NASGRO equation to approximate the test data, which we are looking for.
Since there is no similar dependence for the Kc, parameter, the relationship Kc = f(R) can be found in the same way (i.e. using the test data) from the ordinary linear regression Kc\n\t\t\t\t\t\t= mo+ m1R and also use it to describe 9 experimentally gained curves.
Functions ΔKth = f(R) and Kc = f(R) found in this way with the test data applied are shown in Fig. 15, whereas Fig. 16 illustrates effect of approximating curves da/dN-ΔKin the case given consideration.
Functions a) ΔKth\n\t\t\t\t\t\t\t\t= f(R) by formula (33) and b) Kc\n\t\t\t\t\t\t\t\t= f(R) - regression
Evident is good fit of analytical description in both the critical and threshold ranges, whereas worse - in the middle section. The ad hoc accepted linear regression for the experimentally found relationship Kc = f(R) not too precisely describes this relationship (straight line in Fig. 15, correlation coefficient reaches in this case the 0.3 level). Optimisation of values of the Kc coefficient for R = 0.1; 0.5 and 0.8 (here denoted as Kc*) with the LSM method to reach the minimum deviation error (32) results in the da/dN-ΔK curves approximating courses as in Fig. 16b. The curve illustrating the Kc* = f(R) dependence is in this case a broken line shown in Fig. 15b, which – easy to see – considerably strays away from the linear dependence. This proves that, among other things, one cannot ad hoc impose any form upon it. It can be assumed that for a larger number of experimentally gained curves, including the wider scope of values of R, the suggested method of determining the relationship Kc = f(R) will offer better results that better correspond to the actual dependence and will remain useful for approximating the da/dN-ΔK curves. The broken-line curve, as that resulting from the optimisation process, may be described with, e.g. a straight line or a quadratic equation (as in Fig. 17) and used as a component of the theoretical (analytical) description of the test data with the NASGRO equation. In the case of a straight line, the correlation coefficient increases up to approx. 0.78 for the polynomial. Obviously, with three points Kc* the correlation is complete, but if the scope of values of the asymmetry coefficient was greater, i.e. there would be more experimentally gained curves of different values of R (then the number of these points would increase), one should also expect high correlation for the relationship Kc = f(R).
Approximation of curvesda/dN-ΔK with the NASGRO equation, modified by formula (32) LSM, with extrapolated mapping points according to the MRCM: a) with regression applied as in Fig.15, b) with optimisation for values of coefficients Kc*
Linear Kc\n\t\t\t\t\t\t\t\t= f(R) and polynomial Kc*\n\t\t\t\t\t\t\t\t= f(R) functions to optimise theoretical (analytical) description of curves da/dN-ΔK
Curves da/dN-ΔK with coefficients ΔKth and Kc individually fitted to each experimentally gained curve
If we use values of ΔKth i Kc coefficients in forms determined not with the above-mentioned regression and optimisation methods, but as ones individually found for each of the experimentally gained curves (ΔKth,ind i Kc,ind), the theoretical (analytical) description by means of the NASGRO equation - with the above-described methodology of finding other coefficients applied - should give even better result, see Fig. 18.
This variant of the theoretical (analytical) description is of only little practical importance, however, it shows that both the above-described methodology of analysis and the way of finding coefficients of the NASGRO equation result in correct description of experimentally found curves of fatigue-crack propagation and may be applied to this and similar categories of research issues.
Application of the Least Square Method in its classical form to determine coefficients of the NASGRO equation that describes fatigue crack propagation curve is ineffective, since data of the approximated function da/dN=f(ΔK) take values from the range of a few orders of magnitude, measuring points of the curves are irregular and in different numbers distributed in the graph (in threshold, stable-increase, and critical ranges), and subject to approximation are also several curves grouped in several sets (for different values of R).
The paper offers some techniques to modify the LSM criterion to significantly improve approximation results. These include:
modification of the approximation-method criterion,
smoothing of the experimentally gained curves to eliminate slight random disturbances resulting from, e.g. data recording process,
different variants of calculating the derivative da/dN,
regular mapping of the experimentally gained curves in the form of selected points,
regression for points that represent (map) the experimentally gained curves to find coefficients of the crack growth equation,
regression or optimisation of the description of partial dependences of the NASGRO equation as based on experimental data.
Values of parameters to be found as well as quantitative and qualitative results of performed approximations and theoretical (analytical) description are affected by, among other things, the number of tests that produce experimental data, and configurations thereof.
They provide a wider or narrower range of variability of parameters of significance that affect the courses of curves da/dN-ΔK, and also enable determination of accuracy and repeatability of obtained results. Reliability of the theoretical (analytical) description increases and the description itself better characterises properties of the material under examination if there are tens of curves gained experimentally from tests conducted for many (e.g. 5, 7, or 9) levels of the stress ratio R, for a wider range thereof, e.g. 0.2 through 0.9.
The proposed modification of the LSM criterion offers better fit of results of the test data approximation (unachievable with the classical LSM method). These effects are as follows:
the provision of equal “weights” of each of the test data points in the total sum that determines this criterion (i.e. the sum of differences between approximated and approximating values) – independently of the magnitude of difference between values of data subject to approximation and that of difference between the approximated and approximating values,
high effectiveness while approximating single, several, as well as a great number of sets/curves of test data,
it becomes even more precise as the test data from the same (research-testing) groups show smaller scatter,
may be used in other analyses of the same type related with test data regression, since it offers an all-purpose approach not related to propagation curves da/dN-ΔK.
Stability constant of the formation of metal complexes is used to measure interaction strength of reagents. From this process, metal ion and ligand interaction formed the two types of metal complexes; one is supramolecular complexes known as host-guest complexes [1] and the other is anion-containing complexes. In the solution it provides and calculates the required information about the concentration of metal complexes.
Solubility, light, absorption conductance, partitioning behavior, conductance, and chemical reactivity are the complex characteristics which are different from their components. It is determined by various numerical and graphical methods which calculate the equilibrium constants. This is based on or related to a quantity, and this is called the complex formation function.
During the displacement process at the time of metal complex formation, some ions disappear and form a bonding between metal ions and ligands. It may be considered due to displacement of a proton from a ligand species or ions or molecules causing a drop in the pH values of the solution [2]. Irving and Rossotti developed a technique for the calculation of stability constant, and it is called potentiometric technique.
To determine the stability constant, Bjerrum has used a very simple method, and that is metal salt solubility method. For the studies of a larger different variety of polycarboxylic acid-, oxime-, phenol-containing metal complexes, Martel and Calvin used the potentiometric technique for calculating the stability constant. Those ligands [3, 4] which are uncharged are also examined, and their stability constant calculations are determined by the limitations inherent in the ligand solubility method. The limitations of the metal salt solubility method and the result of solubility methods are compared with this. M-L, MLM, and (M3) L are some types of examples of metal-ligand bonding. One thing is common, and that is these entire types metal complexes all have one ligand.
The solubility method can only usefully be applied to studies of such complexes, and it is best applied for ML; in such types of system, only ML is formed. Jacqueline Gonzalez and his co-worker propose to explore the coordination chemistry of calcium complexes. Jacqueline and et al. followed this technique for evaluate the as partial model of the manganese-calcium cluster and spectrophotometric studies of metal complexes, i.e., they were carried calcium(II)-1,4-butanediamine in acetonitrile and calcium(II)-1,2-ethylendiamine, calcium(II)-1,3-propanediamine by them.
Spectrophotometric programming of HypSpec and received data allows the determination of the formation of solubility constants. The logarithmic values, log β110 = 5.25 for calcium(II)-1,3-propanediamine, log β110 = 4.072 for calcium(II)-1,4-butanediamine, and log β110 = 4.69 for calcium(II)-1,2-ethylendiamine, are obtained for the formation constants [5]. The structure of Cimetidine and histamine H2-receptor is a chelating agent. Syed Ahmad Tirmizi has examined Ni(II) cimetidine complex spectrophotometrically and found an absorption peak maximum of 622 nm with respect to different temperatures.
Syed Ahmad Tirmizi have been used to taken 1:2 ratio of metal and cimetidine compound for the formation of metal complex and this satisfied by molar ratio data. The data, 1.40–2.4 × 108, was calculated using the continuous variation method and stability constant at room temperature, and by using the mole ratio method, this value at 40°C was 1.24–2.4 × 108. In the formation of lead(II) metal complexes with 1-(aminomethyl) cyclohexene, Thanavelan et al. found the formation of their binary and ternary complexes. Glycine,
Using the stability constant method, these ternary complexes were found out, and using the parameters such as Δ log K and log X, these ternary complex data were compared with binary complex. The potentiometric technique at room temperature (25°C) was used in the investigation of some binary complex formations by Abdelatty Mohamed Radalla. These binary complexes are formed with 3D transition metal ions like Cu2+, Ni2+, Co2+, and Zn2+ and gallic acid’s importance as a ligand and 0.10 mol dm−3 of NaNO3. Such types of aliphatic dicarboxylic acids are very important biologically. Many acid-base characters and the nature of using metal complexes have been investigated and discussed time to time by researchers [7].
The above acids (gallic and aliphatic dicarboxylic acid) were taken to determine the acidity constants. For the purpose of determining the stability constant, binary and ternary complexes were carried in the aqueous medium using the experimental conditions as stated above. The potentiometric pH-metric titration curves are inferred for the binary complexes and ternary complexes at different ratios, and formation of ternary metal complex formation was in a stepwise manner that provided an easy way to calculate stability constants for the formation of metal complexes.
The values of Δ log K, percentage of relative stabilization (% R. S.), and log X were evaluated and discussed. Now it provides the outline about the various complex species for the formation of different solvents, and using the concentration distribution, these complexes were evaluated and discussed. The conductivity measurements have ascertained for the mode of ternary chelating complexes.
A study by Kathrina and Pekar suggests that pH plays an important role in the formation of metal complexes. When epigallocatechin gallate and gallic acid combine with copper(II) to form metal complexes, the pH changes its speculation. We have been able to determine its pH in frozen and fluid state with the help of multifrequency EPR spectroscopy [8]. With the help of this spectroscopy, it is able to detect that each polyphenol exhibits the formation of three different mononuclear species. If the pH ranges 4–8 for di- or polymeric complex of Cu(II), then it conjectures such metal complexes. It is only at alkaline pH values.
The line width in fluid solutions by molecular motion exhibits an incomplete average of the parameters of anisotropy spin Hamilton. If the complexes are different, then their rotational correlation times for this also vary. The analysis of the LyCEP anisotropy of the fluid solution spectra is performed using the parameters determined by the simulation of the rigid boundary spectra. Its result suggests that pH increases its value by affecting its molecular mass. It is a polyphenol ligand complex with copper, showing the coordination of an increasing number of its molecules or increasing participation of polyphenol dimers used as ligands in the copper coordination region.
The study by Vishenkova and his co-worker [8] provides the investigation of electrochemical properties of triphenylmethane dyes using a voltammetric method with constant-current potential sweep. Malachite green (MG) and basic fuchsin (BF) have been chosen as representatives of the triphenylmethane dyes [9]. The electrochemical behavior of MG and BF on the surface of a mercury film electrode depending on pH, the nature of background electrolyte, and scan rate of potential sweep has been investigated.
Using a voltammetric method with a constant-current potential sweep examines the electrical properties of triphenylmethane dye. In order to find out the solution of MG and BF, certain registration conditions have been prescribed for it, which have proved to be quite useful. The reduction peak for the currents of MG and BF has demonstrated that it increases linearly with respect to their concentration as 9.0 × 10−5–7.0 × 10−3 mol/dm3 for MG and 6.0 × 10−5–8.0 × 10−3 mol/dm3 for BF and correlation coefficients of these values are 0.9987 for MG and 0.9961 for BF [10].
5.0 × 10−5 and 2.0 × 10−5 mol/dm3 are the values used as the detection limit of MG and BF, respectively. Stability constants are a very useful technique whose size is huge. Due to its usefulness, it has acquired an umbrella right in the fields of chemistry, biology, and medicine. No science subject is untouched by this. Stability constants of metal complexes are widely used in the various areas like pharmaceuticals as well as biological processes, separation techniques, analytical processes, etc. In the presented chapter, we have tried to explain this in detail by focusing our attention on the applications and solutions of stability of metal complexes in solution.
Stability or formation or binding constant is the type of equilibrium constant used for the formation of metal complexes in the solution. Acutely, stability constant is applicable to measure the strength of interactions between the ligands and metal ions that are involved in complex formation in the solution [11]. A generally these 1-4 equations are expressed as the following ways:
Thus
K1, K2, K3, … Kn are the equilibrium constants and these are also called stepwise stability constants. The formation of the metal-ligand-n complex may also be expressed as equilibrium constants by the following steps:
The parameters K and β are related together, and these are expressed in the following example:
Now the numerator and denominator are multiplied together with the use of [metal-ligand] [metal-ligand2], and after the rearranging we get the following equation:
Now we expressed it as the following:
From the above relation, it is clear that the overall stability constant βn is equal to the product of the successive (i.e., stepwise) stability constants, K1, K2, K3,…Kn. This in other words means that the value of stability constants for a given complex is actually made up of a number of stepwise stability constants. The term stability is used without qualification to mean that the complex exists under a suitable condition and that it is possible to store the complex for an appreciable amount of time. The term stability is commonly used because coordination compounds are stable in one reagent but dissociate or dissolve in the presence of another regent. It is also possible that the term stability can be referred as an action of heat or light or compound. The stability of complex [13] is expressed qualitatively in terms of thermodynamic stability and kinetic stability.
In a chemical reaction, chemical equilibrium is a state in which the concentration of reactants and products does not change over time. Often this condition occurs when the speed of forward reaction becomes the same as the speed of reverse reaction. It is worth noting that the velocities of the forward and backward reaction are not zero at this stage but are equal.
If hydrogen and iodine are kept together in molecular proportions in a closed process vessel at high temperature (500°C), the following action begins:
In this activity, hydrogen iodide is formed by combining hydrogen and iodine, and the amount of hydrogen iodide increases with time. In contrast to this action, if the pure hydrogen iodide gas is heated to 500°C in the reaction, the compound is dissolved by reverse action, which causes hydrogen iodide to dissolve into hydrogen and iodine, and the ratio of these products increases over time. This is expressed in the following reaction:
For the formation of metal chelates, the thermodynamic technique provides a very significant information. Thermodynamics is a very useful technique in distinguishing between enthalpic effects and entropic effects. The bond strengths are totally effected by enthalpic effect, and this does not make any difference in the whole solution in order/disorder. Based on thermodynamics the chelate effect below can be best explained. The change of standard Gibbs free energy for equilibrium constant is response:
Where:
R = gas constant
T = absolute temperature
At 25°C,
ΔG = (− 5.708 kJ mol−1) · log β.
The enthalpy term creates free energy, i.e.,
For metal complexes, thermodynamic stability and kinetic stability are two interpretations of the stability constant in the solution. If reaction moves from reactants to products, it refers to a change in its energy as shown in the above equation. But for the reactivity, kinetic stability is responsible for this system, and this refers to ligand species [14].
Stable and unstable are thermodynamic terms, while labile and inert are kinetic terms. As a rule of thumb, those complexes which react completely within about 1 minute at 25°C are considered labile, and those complexes which take longer time than this to react are considered inert. [Ni(CN)4]2− is thermodynamically stable but kinetically inert because it rapidly exchanges ligands.
The metal complexes [Co(NH3)6]3+ and such types of other complexes are kinetically inert, but these are thermodynamically unstable. We may expect the complex to decompose in the presence of acid immediately because the complex is thermodynamically unstable. The rate is of the order of 1025 for the decomposition in acidic solution. Hence, it is thermodynamically unstable. However, nothing happens to the complex when it is kept in acidic solution for several days. While considering the stability of a complex, always the condition must be specified. Under what condition, the complex which is stable or unstable must be specified such as acidic and also basic condition, temperature, reactant, etc.
A complex may be stable with respect to a particular condition but with respect to another. In brief, a stable complex need not be inert and similarly, and an unstable complex need not be labile. It is the measure of extent of formation or transformation of complex under a given set of conditions at equilibrium [15].
Thermodynamic stability has an important role in determining the bond strength between metal ligands. Some complexes are stable, but as soon as they are introduced into aqueous solution, it is seen that these complexes have an effect on stability and fall apart. For an example, we take the [Co (SCN)4]2+ complex. The ion bond of this complex is very weak and breaks down quickly to form other compounds. But when [Fe(CN)6]3− is dissolved in water, it does not test Fe3+ by any sensitive reagent, which shows that this complex is more stable in aqueous solution. So it is indicated that thermodynamic stability deals with metal-ligand bond energy, stability constant, and other thermodynamic parameters.
This example also suggests that thermodynamic stability refers to the stability and instability of complexes. The measurement of the extent to which one type of species is converted to another species can be determined by thermodynamic stability until equilibrium is achieved. For example, tetracyanonickelate is a thermodynamically stable and kinetic labile complex. But the example of hexa-amine cobalt(III) cation is just the opposite:
Thermodynamics is used to express the difference between stability and inertia. For the stable complex, large positive free energies have been obtained from ΔG0 reaction. The ΔH0, standard enthalpy change for this reaction, is related to the equilibrium constant, βn, by the well thermodynamic equation:
For similar complexes of various ions of the same charge of a particular transition series and particular ligand, ΔS0 values would not differ substantially, and hence a change in ΔH0 value would be related to change in βn values. So the order of values of ΔH0 is also the order of the βn value.
Kinetic stability is referred to the rate of reaction between the metal ions and ligand proceeds at equilibrium or used for the formation of metal complexes. To take a decision for kinetic stability of any complexes, time is a factor which plays an important role for this. It deals between the rate of reaction and what is the mechanism of this metal complex reaction.
As we discuss above in thermodynamic stability, kinetic stability is referred for the complexes at which complex is inert or labile. The term “inert” was used by Tube for the thermally stable complex and for reactive complexes the term ‘labile’ used [16]. The naturally occurring chlorophyll is the example of polydentate ligand. This complex is extremely inert due to exchange of Mg2+ ion in the aqueous media.
The nature of central atom of metal complexes, dimension, its degree of oxidation, electronic structure of these complexes, and so many other properties of complexes are affected by the stability constant. Some of the following factors described are as follows.
In the coordination chemistry, metal complexes are formed by the interaction between metal ions and ligands. For these type of compounds, metal ions are the coordination center, and the ligand or complexing agents are oriented surrounding it. These metal ions mostly are the transition elements. For the determination of stability constant, some important characteristics of these metal complexes may be as given below.
Ligands are oriented around the central metal ions in the metal complexes. The sizes of these metal ions determine the number of ligand species that will be attached or ordinated (dative covalent) in the bond formation. If the sizes of these metal ions are increased, the stability of coordination compound defiantly decreased. Zn(II) metal ions are the central atoms in their complexes, and due to their lower size (0.74A°) as compared to Cd(II) size (0.97A°), metal ions are formed more stable.
Hence, Al3+ ion has the greatest nuclear charge, but its size is the smallest, and the ion N3− has the smallest nuclear charge, and its size is the largest [17]. Inert atoms like neon do not participate in the formation of the covalent or ionic compound, and these atoms are not included in isoelectronic series; hence, it is not easy to measure the radius of this type of atoms.
The properties of stability depend on the size of the metal ion used in the complexes and the total charge thereon. If the size of these metal ions is small and the total charge is high, then their complexes will be more stable. That is, their ratio will depend on the charge/radius. This can be demonstrated through the following reaction:
An ionic charge is the electric charge of an ion which is formed by the gain (negative charge) or loss (positive charge) of one or more electrons from an atom or group of atoms. If we talk about the stability of the coordination compounds, we find that the total charge of their central metal ions affects their stability, so when we change their charge, their stability in a range of constant can be determined by propagating of error [18]. If the charge of the central metal ion is high and the size is small, the stability of the compound is high:
In general, the most stable coordination bonds can cause smaller and highly charged rations to form more stable coordination compounds.
When an electron pair attracts a central ion toward itself, a strong stability complex is formed, and this is due to electron donation from ligand → metal ion. This donation process is increasing the bond stability of metal complexes exerted the polarizing effect on certain metal ions. Li+, Na+, Mg2+, Ca2+, Al3+, etc. are such type of metal cation which is not able to attract so strongly from a highly electronegative containing stable complexes, and these atoms are O, N, F, Au, Hg, Ag, Pd, Pt, and Pb. Such type of ligands that contains P, S, As, Br and I atom are formed stable complex because these accepts electron from M → π-bonding. Hg2+, Pb2+, Cd2+, and Bi3+ metal ions are also electronegative ions which form insoluble salts of metal sulfide which are insoluble in aqueous medium.
Volatile ligands may be lost at higher temperature. This is exemplified by the loss of water by hydrates and ammonia:
The transformation of certain coordination compounds from one to another is shown as follows:
A ligand is an ion or small molecule that binds to a metal atom (in chemistry) or to a biomolecule (in biochemistry) to form a complex, such as the iron-cyanide coordination complex Prussian blue or the iron-containing blood-protein hemoglobin. The ligands are arranged in spectrochemical series which are based on the order of their field strength. It is not possible to form the entire series by studying complexes with a single metal ion; the series has been developed by overlapping different sequences obtained from spectroscopic studies [19]. The order of common ligands according to their increasing ligand field strength is
The above spectrochemical series help us to for determination of strength of ligands. The left last ligand is as weaker ligand. These weaker ligand cannot forcible binding the 3d electron and resultant outer octahedral complexes formed. It is as-
Increasing the oxidation number the value of Δ increased.
Δ increases from top to bottom.
However, when we consider the metal ion, the following two useful trends are observed:
Δ increases with increasing oxidation number.
Δ increases down a group. For the determination of stability constant, the nature of the ligand plays an important role.
The following factors described the nature of ligands.
The size and charge are two factors that affect the production of metal complexes. The less charges and small sizes of ligands are more favorable for less stable bond formation with metal and ligand. But if this condition just opposite the product of metal and ligand will be a more stable compound. So, less nuclear charge and more size= less stable complex whereas if more nuclear charge and small in size= less stable complex. We take fluoride as an example because due to their smaller size than other halide and their highest electro negativity than the other halides formed more stable complexes. So, fluoride ion complexes are more stable than the other halides:
As compared to S2− ion, O22− ions formed more stable complexes.
It is suggested by Calvin and Wilson that the metal complexes will be more stable if the basic character or strength of ligands is higher. It means that the donating power of ligands to central metal ions is high [20].
It means that the donating power of ligands to central metal ions is high. In the case of complex formation of aliphatic diamines and aromatic diamines, the stable complex is formed by aliphatic diamines, while an unstable coordination complex is formed with aromatic diamines. So, from the above discussion, we find that the stability will be grater if the e-donation power is greater.
Thus it is clear that greater basic power of electron-donating species will form always a stable complex. NH3, CN−, and F− behaved as ligands and formed stable complexes; on the other hand, these are more basic in nature.
We know that if the concentration of coordination group is higher, these coordination compounds will exist in the water as solution. It is noted that greater coordinating tendency show the water molecules than the coordinating group which is originally present. SCN− (thiocynate) ions are present in higher concentration; with the Co2+ metal ion, it formed a blue-colored complex which is stable in state, but on dilution of water medium, a pink color is generated in place of blue, or blue color complex is destroyed by [Co(H2O)6]2+, and now if we added further SCN−, the pink color will not appear:
Now it is clear that H2O and SCN− are in competition for the formation of Co(II) metal-containing complex compound. In the case of tetra-amine cupric sulfate metal complex, ammonia acts as a donor atom or ligand. If the concentration of NH3 is lower in the reaction, copper hydroxide is formed but at higher concentration formed tetra-amine cupric sulfate as in the following reaction:
For a metal ion, chelating ligand is enhanced and affinity it and this is known as chelate effect and compared it with non-chelating and monodentate ligand or the multidentate ligand is acts as chelating agent. Ethylenediamine is a simple chelating agent (Figure 1).
Structure of ethylenediamine.
Due to the bidentate nature of ethylenediamine, it forms two bonds with metal ion or central atom. Water forms a complex with Ni(II) metal ion, but due to its monodentate nature, it is not a chelating ligand (Figures 2 and 3).
Structure of chelating configuration of ethylenediamine ligand.
Structure of chelate with three ethylenediamine ligands.
The dentate cheater of ligand provides bonding strength to the metal ion or central atom, and as the number of dentate increased, the tightness also increased. This phenomenon is known as chelating effect, whereas the formation of metal complexes with these chelating ligands is called chelation:
or
Some factors are of much importance for chelation as follows.
The sizes of the chelating ring are increased as well as the stability of metal complex decreased. According to Schwarzenbach, connecting bridges form the chelating rings. The elongated ring predominates when long bridges connect to the ligand to form a long ring. It is usually observed that an increased a chelate ring size leads to a decrease in complex stability.
He interpreted this statement. The entropy of complex will be change if the size of chelating ring is increased, i.e., second donor atom is allowed by the chelating ring. As the size of chelating ring increased, the stability should be increased with entropy effect. Four-membered ring compounds are unstable, whereas five-membered are more stable. So the chelating ring increased its size and the stability of the formed metal complexes.
The number of chelating rings also decides the stability of complexes. Non-chelating metal compounds are less stable than chelating compounds. These numbers increase the thermodynamic volume, and this is also known as an entropy term. In recent years ligands capable of occupying as many as six coordination positions on a single metal ion have been described. The studies on the formation constants of coordination compounds with these ligands have been reported. The numbers of ligand or chelating agents are affecting the stability of metal complexes so as these numbers go up and down, the stability will also vary with it.
For the Ni(II) complexes with ethylenediamine as chelating agent, its log K1 value is 7.9 and if chelating agents are trine and penten, then the log K1 values are 7.9 and 19.3, respectively. If the metal ion change Zn is used in place of Ni (II), then the values of log K1 for ethylenediamine, trine, and penten are 6.0, 12.1, and 16.2, respectively. The log βMY values of metal ions are given in Table 1.
Metal ion | log βMY (25°C, I = 0.1 M) |
---|---|
Ca2+ | 11.2 |
Cu2+ | 19.8 |
Fe3+ | 24.9 |
Metal ion vs. log βMY values.
Ni(NH3)62+ is an octahedral metal complex, and at 25 °C its log β6 value is 8.3, but Ni(ethylenediamine)32+ complex is also octahedral in geometry, with 18.4 as the value of log β6. The calculated stability value of Ni(ethylenediamine)32+ 1010 times is more stable because three rings are formed as chelating rings by ethylenediamine as compared to no such ring is formed. Ethylenediaminetetraacetate (EDTA) is a hexadentate ligand that usually formed stable metal complexes due to its chelating power.
A special effect in molecules is when the atoms occupy space. This is called steric effect. Energy is needed to bring these atoms closer to each other. These electrons run away from near atoms. There can be many ways of generating it. We know the repulsion between valence electrons as the steric effect which increases the energy of the current system [21]. Favorable or unfavorable any response is created.
For example, if the static effect is greater than that of a product in a metal complex formation process, then the static increase would favor this reaction. But if the case is opposite, the skepticism will be toward retardation.
This effect will mainly depend on the conformational states, and the minimum steric interaction theory can also be considered. The effect of secondary steric is seen on receptor binding produced by an alternative such as:
Reduced access to a critical group.
Stick barrier.
Electronic resonance substitution bond by repulsion.
Population of a conformer changes due to active shielding effect.
The macrocyclic effect is exactly like the image of the chelate effect. It means the principle of both is the same. But the macrocyclic effect suggests cyclic deformation of the ligand. Macrocyclic ligands are more tainted than chelating agents. Rather, their compounds are more stable due to their cyclically constrained constriction. It requires some entropy in the body to react with the metal ion. For example, for a tetradentate cyclic ligand, we can use heme-B which forms a metal complex using Fe+2 ions in biological systems (Figure 4).
Structure of hemoglobin is the biological complex compound which contains Fe(II) metal ion.
The n-dentate chelating agents play an important role for the formation of more stable metal complexes as compared to n-unidentate ligands. But the n-dentate macrocyclic ligand gives more stable environment in the metal complexes as compared to open-chain ligands. This change is very favorable for entropy (ΔS) and enthalpy (ΔH) change.
There are so many parameters to determination of formation constants or stability constant in solution for all types of chelating agents. These numerous parameters or techniques are refractive index, conductance, temperature, distribution coefficients, refractive index, nuclear magnetic resonance volume changes, and optical activity.
Solubility products are helpful and used for the insoluble salt that metal ions formed and complexes which are also formed by metal ions and are more soluble. The formation constant is observed in presence of donor atoms by measuring increased solubility.
To determine the solubility constant, it involves the distribution of the ligands or any complex species; metal ions are present in two immiscible solvents like water and carbon tetrachloride, benzene, etc.
In this method metal ions or ligands are present in solution and on exchanger. A solid polymers containing with positive and negative ions are ion exchange resins. These are insoluble in nature. This technique is helpful to determine the metal ions in resin phase, liquid phase, or even in radioactive metal. This method is also helpful to determine the polarizing effect of metal ions on the stability of ligands like Cu(II) and Zn(II) with amino acid complex formation.
At the equilibrium free metal and ions are present in the solution, and using the different electrometric techniques as described determines its stability constant.
This method is based upon the titration method or follows its principle. A stranded acid-base solution used as titrate and which is titrated, it may be strong base or strong acid follows as potentiometrically. The concentration of solution using 103− M does not decomposed during the reaction process, and this method is useful for protonated and nonprotonated ligands.
This is the graphic method used to determine the stability constant in producing metal complex formation by plotting a polarograph between the absences of substances and the presence of substances. During the complex formation, the presence of metal ions produced a shift in the half-wave potential in the solution.
If a complex is relatively slow to form and also decomposes at measurable rate, it is possible, in favorable situations, to determine the equilibrium constant.
This involves the study of the equilibrium constant of slow complex formation reactions. The use of tracer technique is extremely useful for determining the concentrations of dissociation products of the coordination compound.
This method is based on the study of the effect of an equilibrium concentration of some ions on the function at a definite organ of a living organism. The equilibrium concentration of the ion studied may be determined by the action of this organ in systems with complex formation.
The solution of 25 ml is adopted by preparing at the 1.0 × 10−5 M ligand or 1.0 × 10−5 M concentration and 1.0 × 10−5 M for the metal ion:
The solutions containing the metal ions were considered both at a pH sufficiently high to give almost complete complexation and at a pH value selected in order to obtain an equilibrium system of ligand and complexes.
In order to avoid modification of the spectral behavior of the ligand due to pH variations, it has been verified that the range of pH considered in all cases does not affect absorbance values. Use the collected pH values adopted for the determinations as well as selected wavelengths. The ionic strengths calculated from the composition of solutions allowed activity coefficient corrections. Absorbance values were determined at wavelengths in the range 430–700 nm, every 2 nm.
For a successive metal complex formation, use this method. If ligand is protonate and the produced complex has maximum number of donate atoms of ligands, a selective light is absorbed by this complex, while for determination of stability constant, it is just known about the composition of formed species.
Bjerrum (1941) used the method stepwise addition of the ligands to coordination sphere for the formation of complex. So, complex metal–ligand-n forms as the following steps [22]. The equilibrium constants, K1, K2, K3, … Kn are called stepwise stability constants. The formation of the complex metal-ligandn may also be expressed by the following steps and equilibrium constants.
Where:
M = central metal cation
L = monodentate ligand
N = maximum coordination number for the metal ion M for the ligand
If a complex ion is slow to reach equilibrium, it is often possible to apply the method of isotopic dilution to determine the equilibrium concentration of one or more of the species. Most often radioactive isotopes are used.
This method was extensively used by Werner and others to study metal complexes. In the case of a series of complexes of Co(III) and Pt(IV), Werner assigned the correct formulae on the basis of their molar conductance values measured in freshly prepared dilute solutions. In some cases, the conductance of the solution increased with time due to a chemical change, e.g.,
It is concluded that the information presented is very important to determine the stability constant of the ligand metal complexes. Some methods like spectrophotometric method, Bjerrum’s method, distribution method, ion exchange method, electrometric techniques, and potentiometric method have a huge contribution in quantitative analysis by easily finding the stability constants of metal complexes in aqueous solutions.
All the authors thank the Library of University of Delhi for reference books, journals, etc. which helped us a lot in reviewing the chapter.
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