Alpha-beta model: Difference between revisions

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{{AFC submission|d|dup|Alpha-Beta Model|u=Anthony Mauriucio Landi Adib|ns=2|decliner=Robert McClenon|declinets=20190612022052|ts=20190604171628}} <!-- Do not remove this line! -->

{{AFC comment|1=Requesting a review at [[WT:WikiProject Engineering]]. [[User:Robert McClenon|Robert McClenon]] ([[User talk:Robert McClenon|talk]]) 02:21, 12 June 2019 (UTC)}}

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'''Alpha-Beta Model'''

{{Commonscat|Topics}}

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[[:Category:Engineering]]

The Alpha-Beta Model is a mathematical equation used to describe the velocity of [[Fatigue (material)|fatigue]] crack propagationgrowth, 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.

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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: <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.

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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|]]

<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>

[[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.