## 3 Phenomenological Model: The MINA Model
### The Use of the Welding Notebook and Grain Growth Rules  
A phenomenological model was conceived from the analysis of multiple welds and by making tests with dedicated mock-ups. It is original in that it uses information contained in the welding notebook and does not develop a complete welding simulation. This model is called MINA, for Modelling anIsotropy from Notebook of Arc welding [60]. Variables, extracted from the notebook, used in this model are the number and the order of passes written down by the welder. Electrode diameters vary with the stacking of passes. This model as developed for shielded metal arc welding (SMAV) and uses metallography analysis of weld cross sections, and so is a two-dimensional model. The thicknesses of passes (height h in Figure 3) are calculated using a proportional rule from electrode diameters, then pass widths (W) are calculated depending of their vertical position and the corresponding weld width. The algorithm has four parts: description of the temperature gradient directions in the weld pool, description of the remelting of passes, description of the incline of passes and calculation of the grain growth. The shape of the weld pool is defined from numerous metallography analyses. This geometry is described with two parabolic curves whose top points are O and C. The positions of points C and D are proportional to h [60]. From this weld pool simplified description, a formula is defined to calculate the direction of the local temperature gradient. The gradient direction comes from point D. The gradient angle measured from vertical is \(\alpha_{g}\). When a new pass fills the weld, laterally or vertically from the previous pass, this operation creates a partial remelting. Two parameters were introduced in the model: \(R_{L}\), which is the lateral remelting rate; and \(R_{V}\), which is the vertical remelting rate. They are the two most important variables to describe the grain structure in the MINA model. The analysis of the welder's gesture shows that the welder has to tilt his electrode during the pass deposit, causing a rotation of the welding pool. A simplified hypothesis for the MINA model is to consider that this phenomenon implies the rotation of the direction of the temperature gradient with no modification in the pass geometry. The analysis of the SMAW welding method combined with all our metallography observations leads us to separate two cases. If a pass is placed and leans on a previous pass, then the temperature gradient is considered to be inclined by an angle \(\theta_{C}\) lower than the second angle, \(\theta_{B}\), which is observed if a pass leans on the sidewall. This angle \(\theta_{B}\) represents in some way the impact of the weld geometry. When a pass fills the weld and leans both on its left and its right, the calculated temperature gradient is symmetric, so \(\theta_{C}=0\). The model MINA provides these two parameters \(\theta_{B}\) and \(\theta_{C}\) for each pass using the position of the pass written in the welding notebook. The fourth part of the algorithm is the calculation of the grain growth. This growth is driven by three physical phenomena: epitaxial growth, influence of the temperature gradient, and selective growth. Selective growth results from the growth competition between grains. Epitaxial growth occurs when the fused metal in the new pass solidifies, keeping the crystalline orientation of the grain that is just below. Grains tend to follow preferentially the direction of the local temperature gradient, which can be considered to be perpendicular to the isotherms. Grain growth occurs at variable speeds. If a grain has its <1 0 0> crystallographic direction parallel to the heat flow direction, this grain grows rapidly, and so stifies the growth of unsuitably oriented ones. When their orientation is close to the heat flow direction, they turn towards this direction thanks to the growth of secondary arms. This phenomenon is the epitaxial: growth when grains make little turns to follow the direction of the gradient \(\alpha_{g}\).