(a) Macrograph of a weld sample used for the study with inset to show an example of the grain structure; this is a joint between 60-mm-thick plates of 304L austenitic stainless steel, with 308 stainless steel composing the weld; and (b) grain orientation map of the weld inferred from the directions seen in the macrograph using a 1 × 2 mm mesh. 

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... welds in thick-section plates or pipes are nor- mally made using a large numbers of passes, and their solidification results in very complex spatial variations of elastic properties arising from the anisotropic nature of the single crystal stiffness tensor of the austenitic steel. Fig. 2(a) shows a macrograph of a cross-section through a weld sample provided by our industrial partners. The sample contains a V-weld joining two 60-mm-thick 304L austenitic stainless steel plates. A 308 stainless steel con- sumable root insert and filler wire were used to form the weld. Manual tungsten inert gas (TIG) welding was used to form the root of the weld, followed by multiple passes of automated orbital TIG until full-fill. The plates were horizontal when the weld was made. The macrograph was obtained using a Nikon D1X digital SLR camera with a Nikkor 105-mm macro lens (Nikon Corp., Tokyo, Japan), after polishing and etching the specimen. The array of lines was overlaid on the macrograph using image process- ing software (Adobe Illustrator, Adobe Systems, San Jose, CA, USA); each arrow was then manually aligned with the grain structure beneath it. This was intended to be a simple, rapid method of obtaining an initial orientation map. The gray and white streaks in the macrograph indi- cate common directions of the crystal axes. A common way (adopted by all of the cases that will be discussed here) of simplifying the description of the anisotropic material in the weld is to consider it to be transversely isotropic, in which case the plane perpendicu- lar to the direction of the grain growth is considered to be isotropic, with the direction of the grain growth as- sumed to lie within the plane of the welded cross-section. This is known to be strictly incorrect, because the welding wire moves along the weld line (normal to this plane), so the heat flow and solidification direction are tilted out of this plane. There is also an influence of the orientation of the component while the weld is being made, when gravi- tational forces can skew the alignment of the solidifying material; this has a particularly large effect when the weld is vertical. Nevertheless, this simplifying assumption of material symmetry has been shown to be approximately correct from macrograph and electron back scattering diffraction (EBSD) measurements [10], and has been ad- opted widely [3], [5]–[8], [11]. The elastic constants are assumed to be the same everywhere within the weld, with the only variable parameter being the orientation of the unique material property axis in the plane of the cross sec- tion shown in Fig. 2(b). Several models have been developed, each involving only a few parameters to characterize the stiffness map. They have been discussed and compared by Apfel et al . [11], and so will only be briefly reviewed here. Ogilvy [1], [12] used empirical analytical functions to describe the continuous variation in anisotropy across a weld, and this model has been used by other researchers to model the weld [3]. Langenberg et al . [13] further simplified the structure and assumed the grains to be oriented at an angle of 45° to the vertical axis. Spies [14] divided the inhomogeneous region into layers of transversely isotropic material, with the orientation in each layer considered to be the same. Another common approach is to divide the weld geometry into several homogeneous sections, each having fixed orientation of the grains [4], [15], [16]. Most of these models only take into account the boundary ge- ometry of the fusion zone without considering other local geometry such as the grain structure associated with each pass; therefore, they cannot always capture the complex- ity of a heterogeneous structure resulting from multi-pass arc welding. FE models can provide a more precise defini- tion of these details inside the weld, but at cost, because FE models are relatively time consuming to build and solve. Our interest here is to develop a weld model having a small number of key parameters, that is not intended to be strictly precise in local detail, but that is sensibly representative of actual weld material and is quick to run. The MINA model was developed by researchers in France to predict the weld stiffness map for shielded metal arc welding from physical information about the forma- tion of the weld that would typically be documented by the welder [6]; it is now in use in relation to ultrasound inspection of power plant components [17]. It thus has a good physical foundation and validation in its context. A schematic of the MINA model is shown in Fig. 3. The in- formation it takes from the welding procedure includes the dimensions of the weld pool, dimension of the electrodes, the number of layers, and the number and order of passes in each layer. It also considers parameters that affect the direction of grain growth such as the inclination angle of the weld pass toward the weld groove θ B or another weld pass θ C , and two remelting rates which describe the overlapping of weld passes in the vertical ( R v ) and lat- eral ( R l ) directions. The physical phenomena describing the solidification mechanism, which include the influence of temperature gradients in a weld pass and the epitaxy and competition between grains, are then considered it- eratively in the modeling to obtain the global orientation map of the weld. The example weld that we use for the modeling and ex- perimental work in this study was constructed by manual and automated TIG, which leaves it strictly outside the specific context of shielded metal arc welding for which the MINA model was developed. However, it remains at- tractive because it is based on physical phenomena rather than geometric fitting to macrographs, and so has been selected for the inversion task in this work. Of course the proof comes from the comparison of the results, and we have found that this model, with appropriate parameters, is capable of delivering a good representation of both the macrograph geometry and ultrasonic performance. This is perhaps to be expected given the common thermal pro- cesses driving the formation of many kinds of welds and, critically, the fact that we are not pursuing accuracy at fine scale. Thus, using an implementation of the MINA model reported in [6], we have identified the following four MINA parameters θ B = 17.5°, θ C = 0°, R v = 0.15, and R l = 0.335 for the weld of Fig. 2. This was done by an opti- mization process comparing the weld map from the MINA model and the grain orientations inferred from the mac- rograph that are shown in Fig. 2(b). We note that other parameters of the MINA model may also have significant influence on the weld map, for example, Gueudre et al . [18] identified the order of the weld passes to be impor- tant; however, in the interests of simplicity over accuracy of detail, we have limited our study to these four param- eters. Future work could extend our inversion process to additional parameters without difficulty if it is thought to be ...