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#REDIRECT [[Alphabeta (disambiguation)]]
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[[:Category:Engineering]]
The Alpha-Beta Model is a mathematical equation used to describe the velocity of [[Fatigue (material)|fatigue]] crack propagation, da / dN, as a function of a constant amplitude load driving force ΔK where its constants α and β are obtained through a semi-empirical process.
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 ; ADIB, A. M. L. ; TORRES, M. A. S. ; PASTOUKHOV, 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.66-73, 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.321-325, 2007.</ref><ref>C. A. R. P. Baptista ; PASTOUKHOV, V. A. ; ADIB, A. M. L. ; MADURO, 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>PASTOUKHOV, V. A. ; C. A. R. P. Baptista ; ADIB, 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>ADIB, A. M. L. ; C. A. R. P. Baptista ; PASTOUKHOV, V. A. ; TORRES, 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>ADIB, 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: <center><math>\frac{da}{dN}=e^\tfrac{Y}{\Delta K}</math></center> 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:
<center><math>Y=ln\frac{da}{dN}\cdot\Delta K</math></center>
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:
Then Y varies linearly as a function of ΔK and therefore can be written by a line equation: <center><math>Y=\alpha\Delta K+\beta</math></center> Where the values of the constants α and β are respectively the angular and linear coefficients of the line.
== Individual Alpha-Beta Model ==
The substitution of Y in the Alpha-Beta Model gives rise to the individual model, used to calculate a single experiment, represented by:
<center><math>\frac{da}{dN}=e^\alpha \cdot e^\tfrac{\beta}{\Delta K}</math></center>
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.
== Generalized Alpha-Beta Model ==
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:
<center><math>\beta=\delta\log(R)+\gamma</math></center> 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.
<center><math>Y=\alpha\Delta K+\delta\log(R)+\gamma</math></center>
The coefficients α, δ, γ are easily found from the experimental data by the mathematical method of first order linear regression in space R <sup> 3 </sup> 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.
<center><math>\frac{da}{dN}=e^\alpha \cdot e^\tfrac{\delta\log(R)+\gamma}{\Delta K}</math></center>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 ; TORRES, M. A. S. ; PASTOUKHOV, V. A. ; ADIB, 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>
== Referências ==
<references />
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