Optimization of Ultimate Tensile Strength with DOE approach for application FSW process in the aluminum alloys AA6061-T651 and AA7075 -T651

In this paper we aim to treat carefully this process by using the design of experiment method which is an original approach that resides in the possibility of interpretation of experimental results with a minimal effort on the experimental level whichever generates the minimization of the necessary number of experiments and allows a saving in time and in financial cost. For each parameter, the effect of interaction with the rotation speed, the welding speed and the choice of the pin, were determined. A mathematical model between the response and the parameters are modeled, which represents the analytical part of this optimization.


INTRODUCTION
he welding process goes back to the discovery of metals, and since the latter has not stopped evolving as technologically, but also on the methods and working conditions. In 1991 the British Institute (the Welding Institute) invented a new ecological and efficient process especially for aluminum alloys as well as other alloys complicated to weld by conventional methods, this process known as Friction Stir Welding FSW [1][2][3]. This technique, which is currently used in many sectors, consists of assembling two metal sheets by friction and local mixing, using a tool. The latter is placed between the two sheets to create the weld joint, by significant rotational action and a force exerted underneath, causing plastic deformation, we can create an effective weld without ever reaching the melting temperature of the base metal [4][5][6][7]. This process allowed us to use more and more the alloys of non-ferrous metals which offer important mechanical characteristics [9,10], it also contributed to reduce the manufacturing costs [11,12], that is why we find so important, to focus our research on how to optimize the operation at low cost, while controlling the parameters, with the best plausible configuration, which will play a crucial role in the maintenance of structures, like the speed of the tool and welding, but also the tool geometry (Fig. 2). In industrial processes, it is useful to explore the relationships between the key variables (or factors) of the input process and the output performance characteristics (or quality characteristics) [14]. For example, in a metal shotpeening operation, the impact velocity, the processing time that induces the coverage rate, the shot's hardness, can be treated as input variables and the quality of the surface will been considered as performance characteristic so an answer [15]. Design of experiment (DOE) is a robust technique used to explore new processes, gain insights into existing processes, and optimize these processes to achieve world-class performance [16,17]. Many researchers have been involved in the use of DOE since the mid-1990s to promote mathematical and statistical skills [17,18], Yunus et al [19] also reported how Genetic programming can be applied to the FSW process to derive precise relationships between the output and input parameters in order to obtain a generalized prediction model. Though the welding is a controlled process, we aim in this article to get more knowledge and skills by using the DOE, that helps to supervise much more the operation at a lower cost.

OPTIMIZATION BY THE DESIGN OF EXPERIMENT METHOD
n this paper, an experimental design method was used on the FSW friction stir welding process, basically using parameters that effect the process, such as rotational speed, welding speed, and profile. pin which carries a surface ratio (Ss / Sd) representing the type of the pin, where Ss / Sd represents the ratio between the area occupied by the pin in static state and the area occupied by the pin in dynamic state. For example, in the case of a round pin, the surface ratio is equal to 1 and in the case where the pin is square, the ratio 1.57 is found. We focused on the welding area, which consists of two different materials, the AA6061-T651 and the AA7075-T651 elaborated by the friction stir welding method, with decrypting the parameters which are the speed of rotation and welding through the pin profiles on tensile strength of joints. Two types of H13 steel tools have been used to fabricate welded joints with different pin profiles. we have Based on the results of Ravikumar [18], which studied the influence of welding parameter variation on the rupture strength. However, we have well expolited this data by the DOE method to highlight such a deep and profound analysis. Tab. 1 shows the FSW welding parameters and their working ranges of AA6061-T651 and AA7075 -T651. he purpose of the experimental design can be defined as a means of reducing the number of trials with maximum precision. We can use a unique coding relation defined from the bijective transformation defining the value xi from relation (1), which deals also the plans to study the response surface [20]: ui max and ui min being the limits defined by the experimenter and ui the real level given to factor i. The coded factor X, has values in the bounded interval between [-1; 1]. We obtain a complete factorial plan with k factors at two levels by the combination of each two levels of this factor [21].
Our matrix of experiences of three factors is represented by Tab. 2. The Fig. 7 represents the experimental points located at the vertices of a hyper cube which forms a named space, the experimental field of study. The effects and the interaction of the factors given in a matrix of experiences of type 2 k on a well determined response, can be estimated by the values of the coefficients of a mathematical model of the polynomial type of the first degree, which translates the relation between the answer y and the factors Xi. The response y, of a mathematical model in the case of a plane 2 3 is defined by the relation: with: Y: vector of responses represented by a column matrix (2 k , 1), a: vector of the effects of the factors and all the interactions, represented by a column matrix (2 k , 1); these components are the unknowns that we are trying to determine, X: square matrix (2 k , 2 k ) composed of -1 and + 1 according to the values of the levels xi. The relation (10) represents the matrix form of our plane 2 3 system: The orthogonality of our matrix is a very important property because the inverse of X is equal to the transpose of X divided by the number of lines n [22]. Indeed, according to Hadamard, we have the relation (4): with I represents the unit matrix and Xt: is the transposed matrix of X. I: the identity matrix. n: the number of experiments performed (n must be a multiple of 4). From relation (4) and taking account of relation (5), we can calculate the unknown a: X y X X a X y n I a a X y n The elements of a are calculated in the following form: Tab. 3 is used which defines the calculation matrix to gather all the tests carried out. It is made up of several columns; the first column represents the average of the different tests, the other parameters express the state of the coded factors, each column represents a factor. The responses obtained are indicated by the last column [23]. We obtain the following adjusted mathematical model using the least squares method:

EFFECTS ANALYSIS
Principal effect for each factor e are basing on complete factorial plan at two levels; we studied in a first case the effect of each factor separately from each other on the rupture strength, with a simultaneous variation and in an ordered manner balanced (Fig.  4). W Fig. 4 shows the evolution of the rupture strength considered as a response (UTS), influenced by the change of the three parameters, rotation speed (X1), welding speed (X2) and the type of pin profile ( X3), implied from his low to the high level. We can say that the rupture strength increases during the increase of the factors in the welding operation 800 rpm to 1000 rpm, regarding the rotation speed 90 mm / min to 100 mm / min, in addition the circular pin profile shows a significant advantage compared to the square pin, this manifests including the value of the breaking strength which is included between 183 MPa and 164.25 MPa respectively. Otherwise, we observe in Fig. 8

Interaction effect for two factors
To better understand this effect, we have created specific interaction graphs (Fig. 5).   5-left shows us that the rupture strength increases with the increase of the rotation speed. However, a higher influence of the rotation speed during a higher welding speed compared to the low level. In other words, the effect of the speed of rotation depends on the level of the welding speed and especially if the rotation speed has the smallest value. We also see in Fig. 5-right that the influence of the welding speed is greater when the rotation speed is fixed at its higher level.
In an identical way, it can be seen in Figs. 6 and 7 on the left side, that the rupture strength is improved with highest welding speed and higher rotational speed in the case where the round pin's ratio (Ss / Sd) equal to 1 compared with the ratio of 1.57.
We can clearly see from Figs. 6 and 7 on the right side, that the resistance is low for a square pin with high rotation and welding speed values, it remains greater with a round pin. We also observe that the resistance is modest for a large area ratio.

Calculation of residuals
ccording to the calculations made, we can calculate the estimated expenses based on the mathematical model found, we collect all the results in the table indicated below: To calculate the difference between the experimental response and the predicted (estimated) response, it suffices to use the relation (14) which represents the residue of each experiment.
When adjusting a regression model, the most important diagnostic tool consists of two companion parameters R2 and Q2. The R2 parameter is called the quality of the fit, it is a way of measuring where the regression model can be brought to fit the raw data. When R2 is equal to 1, all the points are located on the diagonal of Fig. 9. Consequently, R2 alone, is not a sufficient indicator to probe the validity of a model.
A Also a determination of the parameter of the prediction quality was very important to validate the outcomes of DOE method. Fig. 8 show value of 0.996 which is a more realistic performance indicator, as it reflects the objective of modeling forecasts for new experiences.    We can say that the fit is excellent, because all the points are located near the line according to Fig. 09. However, the estimate of the small deviations between the calculated and measured response values is given by the residual standard deviation.

Signification of effects
We use the Student test given by relation (7) to determine the influence of each variable and each interaction for a chosen risk of 5%. For a given risk, if an effect is different from 0, it can be considered significant on the studied response.
In order to use the Student table, it is necessary to calculate the degree of freedom as follows: n: is the number of experiments carried out, P: is the number of effects including the constant (the average).
To find the signification of an effect:  If ti>tcrit(), the effect is significant.
 If ti<tcrit (), the effect is not significant.

Experimental variance
In order to estimate the common variance of the residuals, it is imperative to neglect at least one effect. Without this action, we cannot calculate the common variance of the residuals   for a complete plan n = p.
It will be useful to pass over the high-order interactions (3 or more) [24]. The estimator of the common variance of the residuals is given by the relation (8): Under these conditions, we can show that all the effects have the same variance given by the relation (9): We collect the residuals and the variances corresponding to each effect in Tab   The calculation of the Student test "t" for each effect is given as follows: