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\title{New Nonlinear Damage Law by Fatigue based on the Curve of Bastenaire}
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             \author[1]{B.  Tikri}

             \author[2]{Nadjitonon  Ngarmaim}

             \affil[1]{  Institut Universitaire Polytechnique de Mongo-Tchad,}

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\date{\small \em Received: 8 December 2013 Accepted: 1 January 2014 Published: 15 January 2014}

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\begin{abstract}
        


This article focuses on the improvement of the nonlinear damage law by fatigue of Chabobe. It is a solution to the dependency problem of parameters of the law vis-à-vis the SN curve of the material used for their production. The determination of the parameters, or wedging of the law, has the drawback of smoothing by linear regression the points of SN curve in a particular space specific to the law, called wedging space. So doing, the law is based on the regression line points, different from the actual SN curve of the material. The evolution of fatigue damage and hence the lifetime are then modified. The main idea is to develop a new damage law model, such as the SN curve generated by this formalism is identical to the one used to describe the fatigue behavior of the material under constant amplitude. In the present case, it?s obviously the SN curve of Bastenaire model.

\end{abstract}


\keywords{uniaxial fatigue; damage law; lifetime; wedging; bastenaire.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
he existence in the literature of a large number of damage laws by fatigue for predicting lifetime of structures subjected to varying loads in service illustrates the difficulties of actually having a universal law to forecast lifetime \hyperref[b1]{[2]}. Although the phenomenon of failure by fatigue of mechanical components subjected to cyclic mechanical strain was discovered more than a century by Wöhler, this type of damage remain still an outstanding event in terms of reliability. However, much progress has been made both in terms of its phenomenological characterization as of understanding of its mechanisms.\par
Among the proposed damage laws since several decades, the law of Chaboche has always seemed promising although its practical application raises some issues. The attractive aspects of this law are multiple: nonlinear evolution and accumulation of damage, influence of "small" cycles (cycles below the endurance limit) as soon as the damage is initiated, sequence effect (influence of the appearance order of cycles) and the effect of mean stress \hyperref[b2]{[3]}. The implementation of Chaboche law faces an important challenge from the material parameters determination point of view. It requires the knowledge of the SN curve of the material which undergoes a linear regression in a reference particular and own to the law. Doing so, it significantly changes the SN curve and therefore deviates from the actual behavior of the material. This work is aimed toward the study and the resolution of this disadvantage. First are presented the issues of wedging of the Chaboche law and its impact on the estimated lifetime. An alternative is then proposed by modifying the formalism of the law in order to clear its sensitivity to the wedging field. 
\section[{II.}]{II.} 
\section[{Presentation of the Chaboche Law}]{Presentation of the Chaboche Law}\par
The implementation of the law of Chaboche is presented here; it helps to highlight the issue faced during its wedging. 
\section[{a) Differential expression of the law}]{a) Differential expression of the law}\par
The increment ?D of damage by fatigue generated by ?N amplitude cycles ?a and of ?m average value is given by:\par
(1) With: -b, ? and M 0 are coefficients specific to the material; b is the slope of Haigh traction diagram, modeled linearly as following: 
\section[{?}]{?}\par
is the amplitude of the fatigue limit of the material under average stress is fatigue limit of the material in alternating symmetrical tensile (R= ?1). ? The coefficient ? is defined by:\par
? For a cycle located above the endurance limit of the material (and called "large" cycle),\par
? For a "small" cycle, b) Integration of differential law\par
The cumulative damage is done cycle by cycle for all cycles encountered during loading. The lifetime of the material is determined by integration of the differential damage law \hyperref[b0]{[1]} knowing that the initial damage is zero (virgin material) and reaches unity when the crack initiation appears. Distinct values of the ? ? ?N ? D) (1 ) m b? (1 o M a ? ? 1 ? D) (1 1 ?D ? ? ? ? ? ? ? ? ? ? ? ? ? A (? m ) = ? -1 (1-b? m ) ? A (? m ) ? m , ? -1 ? a >? A (? m ): a ? m ? Rm ) m (? A ? a ? 1 a ? m ? Rm ) m (? A ? a ? a 1 ? ? ? ? ? ? ? ? ? ? ? a ? a ? ? A (? m ): a = 1\par
parameter ? depending on the type of cycle met ("small" or "large" cycle) generate two different integration formalisms of the damage. 
\section[{i. Integration of damage in case of small cycle}]{i. Integration of damage in case of small cycle}\par
The application of small cycle defined by (?a, ?m), change the damage from Di value to Dj value. The integration of the damage in this case is given by:\par
(2) By putting :\par
(3)\par
The cumulative damage after applying the small cycle (\par
) is given by: (\par
X appears as a functional damage whose characteristic is to cancel or be equal to unity at the same time as D. Note that if the level damage Di is zero initially, it remains after the application of small cycle. Small cycles do not contribute to material damage once the damage is initiated. 
\section[{ii. Integration of damage in case of large cycle}]{ii. Integration of damage in case of large cycle}\par
The application of a large cycle defined by (?a, ?m), moves the material damage from Di value to Dj value. The integration of damage is then given by the following:\par
(5) Under constant amplitude, the integration leads to the SN curve expression within the meaning of the law of Chaboche; it is given by: (6) Where: N f is the number of cycles at crack initiation within the material under constant amplitude loading defined by the (?a, ?m). By repeating, the equation (  {\ref 5}) of the integration of damage becomes:\par
(7) III. 
\section[{Wedging Principle of the Law}]{Wedging Principle of the Law}\par
The damage cumulative observed in the case of a small cycle and a large cycle (equations ( 4), (  {\ref 6} The main problem of wedging of the Chaboche law lays in the fact that the Bastenaire SN curve of the material is not linear in the wedging space, even if we consider as window all the curve or only part of it. No specific guidance is made by the author of the law \hyperref[b0]{[1]}. This will practically generate under uniaxial fatigue with constant amplitude a lifetime prediction significantly different from those projected by the SN curve intrinsic of material. The use under uniaxial fatigue with variable Year 2014 ume XIV Issue IV Version I? ) m b? (1 o M a ? 1 ? ) i D (1 1 1 ? ) j D (1 1 ln 1 ? 1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? a 1 ? ) j D (1 1 j X a 1 ? ) i D (1 1 i X a , ? m ? ? m ? b (1 a ? 1) (? ? o aM e i X j X ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ) b 1 ( M ) 1 ( ) 1 ( ) D 1 ( 1 ) D 1 ( 1 m o a j 1 1 i 1 1 j ? a ? ) m b? (1 K ?) (1 ? o M a 1 f N ? ? ? ? ? ? ? ? ? ? a ? m ? m R ) m (? A ? a ? a ? 1 K ? ? ? ? ? ? ? ? ? ? a 1 ? j j a 1 ? i i ) D (1 1 X et ) D (1 1 X ? ? ? ? ? ? ? ? , fj K i K j N 1 X X j j ? ?\par
material parameters ? and aM 0 -? . This step is the wedging of the law.? m = 0, R = ? ? a ? 1 K ) 1 (? ? o aM 1 f N ? ? ? ? ? ? ? ? ? ? ? f .K) = -?ln? a -ln(aM o -? (? + 1)) (N (9) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? A ) 1 ( aM ln B ln X ) N . K ln( Y 0 a f\par
The SN curve subtended by the damage law of Chaboche is then based on equation [9] a straight line in reference (ln ?a; ln(K.Nf)), reference that is later called the wedging space. To obtain both and ? coefficients used in the damage integration, the points of the SN curve of the material (expressed using Bastenaire model) are placed in the wedging space (Figure \hyperref[fig_3]{1}). A linear regression of all those points or of part of them leads in practice to the determination of and ?.  Proposition of New Damage Law\par
Chaboche law was mainly applied, often successfully, to sequences containing two loading blocks commonly known as sequences High-Low or Low-High depending on the relative intensity of successive levels of stress \hyperref[b0]{[1]}.\par
The disadvantage of the law vis-à-vis its wedging makes the law not useful on its state. Depending on the wedging window chosen, the calculated lifetimes vary very importantly. The failure of the formalism of Chaboche law is in reality due to the fact that the SN curve of the material within the meaning of the law is a straight line in the wedging reference, which does not correspond to the usual SN curves of materials (Bastenaire model used here, but also those of Wöhler, Basquin or Stromeyer \hyperref[b3]{[4]}.\par
In other words, the problem of the sensitivity of the law vis-à-vis the wedging window used is the fact that the actual SN curve of the material does not match the one stipulated by the law, namely a straight line in the wedging space. Based on this observation, the main goal of the work done is to modify the formalism of the law to find the actual SN curve of the material when is realized the cumulative damage under constant amplitude loading.\par
The proposed model uses here also stress as mechanical state parameter, and it ties the increase ?D of damage D to the number ?N of cycles that generated it under the following differential form:  {\ref (10)} The variables used in this law are similar to those used in Chaboche model. The only new variable, denoted ??, represents the Y axis at origin of Goodman diagram passing through the point representative of the cycle analyzed. This variable is given by the expression: ?? = ?a ? b?-1 ?m; the slope b of Goodman line is usually determined from the fatigue limits in symmetrical alternated traction and in repeated traction (?-1 and ?0 respectively). The integration results in the SN curve modeled by Bastenaire expression :(11) ?N c B D ? ) m (? oo ? e D ? ) m (? oo ? . ) m (? A ? a ? ? a ? m R ? D) (1 1 x ? 1 ? D) (1 1 ?D m ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? c B D ? ) m (? oo ? e ) ? ) m (? oo 1)(? (? a 1 fj N ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?\par
B and C are the Bastenaire model parameters calculated by smoothing experimental points used in the determination of the SN curve of the material.\par
V.\par
Comparison of the New Proposed Law with the Chaboche Law Wedged on Several Different Windows and the Milner Law\par
Life Forecasts are made for steel 20MV6 subject to a sequence of variable amplitude loading CARLOS LATERAL used as reference in the automotive industry.\par
The wedging of the new law on the Bastenaire SN curve of the material lead to outstanding forecasts, especially for significant levels of stress, to those obtained by the law of Chaboche. Forecasts obtained from Miner linear law is also shown (Figure 2). The proposed new model provides intermediate life forecasts between those of Miner law and those of Chaboche law in the field of limited endurance.  
\section[{Conclusion}]{Conclusion}\par
A new formalism of non-linear damage law by fatigue has been proposed. It is an alternative to the law of Chaboche because it eliminates the problem of wedging, that Chaboche law faces, on the material data that makes the S-N curve. The new law gives lifetime forecasts coincident with the SN curve of material in uniaxial fatigue under constant amplitude; it keeps the essential characteristics of the Chaboche law: sequence effect, the influence of the mean stress and non-linear accumulation of damage.\begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_1}}\end{figure}
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\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_2}}\end{figure}
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\noindent\textbf{1}\includegraphics[]{image-4.png}
\caption{\label{fig_3}Figure 1 :}\end{figure}
  			\footnote{© 2014 Global Journals Inc. (US)} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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