## 2 Material Modelling for Anisotropic and Heterogeneous Welds  
### The Material Issue  
To understand the results of ultrasonic testing, it is of particular importance to know the grain orientation resulting from the material solidification during the welding process. Part 1 in Figure 1 shows a classical micrograph of a multipass weld produced using a metallography analysis. The grain orientation is heterogeneous, with possible strong change in the grain direction as underlined with the purple line in the zoomed part of Part 1. Because of the diameter of the ultrasonic beam, this causes beam deviation and division, as the front wave does propagate in a heterogeneous medium. The methodology is to understand physical mechanisms of grain growth to propose a grain orientation model to be able to propose a complete welded material modelling. Using the formalism detailed in [31], Equation (1) is the elastic wave equation (or elastodynamic wave equation), showing that particle displacement \(u\) depends on the elasticity constants \(C_{ijkl}\). This first equation leads to the known Christoffel equation (Equation (2)). \[\rho\frac{\delta^{2}u_{i}}{\delta t^{2}}=C_{ijkl}\frac{\delta^{2}u_{k}}{\delta x _{j}\,\delta x_{l}} \tag{1}\]  
\[\rho{V_{\varphi}}^{2}\delta_{il}-C_{ijkl}n_{l}n_{k}=0 \tag{2}\]  
In these equations, \(\rho\) is the density and \(V_{\varphi}\) the phase velocity. The Kronecker symbol is \(\delta_{il}\). \(n_{j}\) and \(n_{k}\) are coordinates of \(n\), the propagation direction. In the general case for an anisotropic material, three solutions are calculated for each wave direction. Three group  
Figure 1: Methodology to describe material for UT in anisotropic and heterogeneous welds. velocities (\(V_{g}\)) are related to the propagation of the energy, and the phase velocity \(V_{\varphi}\) is the wavefront velocity. The angle of deviation between these two velocities depends on the wave type, and may reach about 45\({}^{\circ}\) in the case of a vertical transverse wave. These results are schematised in Figure 1 Part 1: depending on the incident angle \(i\) and the Snell law, up to three elastic waves are refracted with various angles \(r\) in the material, and they are transmitted and reflected at any further material interfaces, such as grain interfaces. All the following waves' properties are calculated using the same equations, Equations (1) and (2). Elasticity constants \(C_{ijkl}\) are thus the first data to fix. An austenitic grain is usually considered an orthotropic material. The tensor values of the elastic constants (or stiffness tensor) found in the literature show significant variations depending on the steel composition. Measurements of such material properties are difficult. The grain direction can be obtained by X-diffractometry and elastic constants can be evaluated by ultrasound setups. An inverse method is used with the same equations to calculate wave velocities. Velocities are measured for various angles of incidence. Parallelepipedic samples have been cut in the weld to evaluate the elastic constants with good certainty [32; 33]. These elastic constants correspond to the orthotropic properties related to the longitudinal orientation of a columnar grain. This principal orientation is assimilated to a <1 0 0> crystallographic direction. This direction corresponds to a local coordinate system. This hypothesis is validated with various crystallographic analyses by EBSD and by X-diffractometry. During the welding process, the metal solidification can create diverse grain structures, and thus very different grain orientations. The resulting material is anisotropic, but nearly as importantly, this welded material is heterogeneous. The wave front is transmitted through the same orthotropic material, but as the wave front is not aligned with the grain direction, the apparent elastic constants are modified along the wave propagation direction. The material description becomes as complex as that of a triclinic material.