Alpha-beta model: Difference between revisions

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The '''alpha-beta model''' is a mathematical equation used to describe the velocity of [[Fatigue (material)|fatigue]] crack growth, 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: <div style="text-align: center;"><math>\frac{da}{dN}=e^\tfrac{Y}{\Delta K}</math></div> 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></div>

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|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></div> Where the values of the constants α and β are respectively the angular and linear coefficients of the line.

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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|400x400px|]]

<div style="text-align: center;"><math>\beta=\delta\log(R)+\gamma</math></div> 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></div>

[[File:Plano_Fadiga.jpg|center|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.