Alpha-beta model: Difference between revisions

Revision as of 11:11, 13 June 2019

Revision as of 04:49, 19 June 2019

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Originally the Alpha-Beta Model was developed and tested from data generated in tests using commercial grade [[Titanium]] and [[Aluminium Alloy]] 2524-T3 both the structural materials of great interest aeronautical.

This model is applied in two situations: the individual that conforms to the experimental data of a single test and can be compared to [[Paris' law]]; and the generalized one that tries to represent in a bi-parametric way the effects of R - ratio between the tensions intensity, minimum and maximum - for a set of tests in the same material.

In addition to the ease of application, the Alpha-Beta Model allows precise adjustment of the experimental points that do not follow the linearity in region II proposed by the [[Paris' law]], since it is known that some ductile materials and some [[alloy | metallic alloy]] have these linearity deviations<ref>C. A. R. P. Baptista ; ADIBAdib, A. M. L. ; TORRESTorres, M. A. S. ; PASTOUKHOVPastoukhov, V. A. . "Describing fatigue crack growth and load ratio effects in Al 2524 T3 alloy with an enhanced exponential model". ''Mechanics of Materials'' (Print) , v. v.51, p. p.66-7366–73, 2012.</ref><ref>ADIB, A. M. L. ; C. A. R. P. Baptista . "An exponential equation of fatigue crack growth in titanium.". ''Materials Science & Engineering. A, Structural Materials: Properties, Microstructure and Processing'' , v. v.452, p. p.321-325321–325, 2007.</ref><ref>C. A. R. P. Baptista ; PASTOUKHOVPastoukhov, V. A. ; ADIBAdib, A. M. L. ; MADUROMaduro, L. P. . "Avaliação do Comportamento de Trincas de Fadiga na Liga Al2524-T3 por meio de uma Equação Exponencial.". In: Congresso Nacional de Engenharia Mecânica., 2008, Salvador. Anais do V CONEM.. Rio de Janeiro: Associação Brasileira de Engenharia e Ciências Mecânicas., 2008. v. v. 1.</ref><ref>PASTOUKHOVPastoukhov, V. A. ; C. A. R. P. Baptista ; ADIBAdib, A. M. L. . "Simulação de Propagação de Trincas por Fadiga Cíclica: Estudo de Convergência na Integração Numérica de Equações Cinéticas.". In: VIII Simpósio de Mecânica Computacional., 2008, Belo Horizonte. Anais do VIII Simpósio de Mecânica Computacional., 2008. v. v. 1.</ref><ref>ADIBAdib, A. M. L. ; C. A. R. P. Baptista ; PASTOUKHOVPastoukhov, V. A. ; TORRESTorres, M. A. S. . "Describing the Cycle Asymmetry Effects on Fatigue Crack Growth with a Biparametric Exponential Equation.". In: COBEM - 19th International Congress of Mechanical Engineering., 2007, Brasília. Proceedings of Cobem 2007., 2007. v. v. 1.</ref><ref>ADIBAdib, A. M. L. ; C. A. R. P. Baptista . "Descrição da Propagação de Trincas por Fadiga em Regimes de Carregamento Estacionário: Um novo Modelo Cinético.". In: CBECIMAT - Congresso Brasileiro de Engenharia e Ciência dos Materiais., 2006, Foz do Iguaçu. Anais do XVII Congresso Brasileiro de Engenharia e Ciência dos Materiais., 2006.</ref>.The Alpha-Beta Model is given by the following equation: <div style="text-align: center;"><math>\frac{da}{dN}=e^\tfrac{Y}{\Delta K}</math></centerdiv> Where, da / dN is the [[Fatigue (material)|fatigue]] crack growth rate given in units of length per number of cycles. ΔK is the stress intensity factor and the Y parameter is the product between log of the crack propagation rate and the amplitude of the stress intensity factor, as follows:

<div style="text-align: center;"><math>Y=ln\frac{da}{dN}\cdot\Delta K</math></centerdiv>

Representing in a graph Y (ΔK) for any value of R the result obtained is exactly a decreasing straight line, as shown in the figure below:

[[File:Y(∆K).jpg|center|miniaturadaimagem|400x400px|]]

Then Y varies linearly as a function of ΔK and therefore can be written by a line equation: <div style="text-align: center;"><math>Y=\alpha\Delta K+\beta</math></centerdiv> Where the values of the constants α and β are respectively the angular and linear coefficients of the line.

== Individual Alpha-Beta Model ==

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The substitution of Y in the Alpha-Beta Model gives rise to the individual model, used to calculate a single experiment, represented by:

<div style="text-align: center;"><math>\frac{da}{dN}=e^\alpha \cdot e^\tfrac{\beta}{\Delta K}</math></centerdiv>

The constants α and β are calculated from experimental data concerning the assay to be described. In the interval 0 <da / dN <1, they always assume negative values.

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Since the curves of Y (ΔK) behave like parallel lines, this means that a single angular coefficient can be adopted for all the tests. If α is a single value for all the tests it is characterized that the constant β is responsible for the representation of the effect of R. The investigation of the constant β described as a function of R leads to a linear adjustment as in the figure below:

[[File:Beta_log_R.jpg|center|miniaturadaimagem|400x400px|]]

<div style="text-align: center;"><math>\beta=\delta\log(R)+\gamma</math></centerdiv> The substitution of β into Y gives rise to what represents the equation of a bi-parametric plane shown in the below, where this plane is formed from the experimental data.

<div style="text-align: center;"><math>Y=\alpha\Delta K+\delta\log(R)+\gamma</math></centerdiv>

[[File:Plano_Fadiga.jpg|center|miniaturadaimagem|400x400px|]]

The coefficients α, δ, γ are easily found from the experimental data by the mathematical method of first order linear regression in space R ³ known as [[least squares | least squares method]]. Substituting the β equation into the Individual Alpha-Beta Model gives the equation of the generalized bi-parametric model.

<div style="text-align: center;"><math>\frac{da}{dN}=e^\alpha \cdot e^\tfrac{\delta\log(R)+\gamma}{\Delta K}</math></centerdiv>The generalized Alpha-Beta Model is represented by the bi-parametric form, because it describes the [[Fatigue (material)|fatigue]] crack growth for a group of tests carried out on the same material (family of curves) as a function of two loading parameters, ΔK and R.<ref>C. A. R. P. Baptista ; TORRESTorres, M. A. S. ; PASTOUKHOVPastoukhov, V. A. ; ADIBAdib, A. M. L. . "Development and evaluation of two-parameter models of fatigue crack growth.". In: 9th International Fatigue Congress., 2006, Atlanta. Proceedings of 9th International Fatigue Congress.. Oxford: Elsevier, 2006. v. v. 1.</ref>

== ReferênciasReferences ==

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