Full-Field Strain Determination for Additively Manufactured Parts Using Radial Basis Functions

Abstract

Additively manufactured components, especially those produced in deposition welding processes, have a rough curvilinear surface. Strain and surface deformation analysis of such components is increasingly performed using digital image correlation (DIC) methods, which raises questions regarding interpretability of the results. Furthermore, in triangulation or local tangential plane based DIC strain analysis, the principal strain directions are difficult to be calculated at any point, which is due to the non-continuity of the approach. Thus, both questions will be addressed in this article. Apart from classical local strain analysis based on triangulation or local linearization concepts, the application of globally formulated radial basis functions (RBF) is investigated for the first time, with the advantage that it is possible to evaluate all interesting quantities at arbitrary points. This is performed for both interpolation and regression. Both approaches are studied at three-dimensional, curvilinear verification examples and real additively manufactured cylindrical specimens. It is found out that, if real applications are investigated, the RBF-approach based on interpolation and regression has to be considered carefully due to so-called boundary effects. This can be circumvented by only considering the region that has a certain distance to the edges of the evaluation domain. Independent of the evaluation scheme, the error of the maximum principal strains increases with increasing surface roughness, which has to be kept in mind for such applications when interpreting or evaluating the results of manufactured parts. However, the entire scheme offers interesting properties for the treatment of DIC-data.

Application of radial basis functions in strain analysis of digital image correlation

“…( 7) must be satisfied. Depending on the number of DIC points and the number of used center points, a system of linear equations (interpolation) or a linear least-square problem (regression) is solved to determine the unknowns B (n) kj and D (n) lj , see [4] for more details. In the following studies, the same number of DIC points and center points is used so that system of linear equations are obtained.…”

“…The point pattern represents the surrogate DIC-data. It should be noted that the point distance of the point pattern is different from that in the example of [4]. The relative error between the analytically determined principal stretch λ ana k and the principal stretch λ int k resulting from the interpolation at an evaluation point, as well as the mean relative error are defined as Fig.…”

“…2b shows the relative error of the maximum principal stretch at each point on the surface as a function of cylindrical coordinates. Compared to [4], the maximum deviations are lower by two powers of ten and the minimum error by four powers of ten, although a coarser dot pattern was used. The tendency for the deviation to increase towards the edge is still present.…”

“…In the following, the region from −25 • ≤ Φ ≤ 25 • and 30 mm ≤ Z ≤ 50 mm is considered, since the deviation increases towards the edges and to allow a further comparison with the results from [4].…”

“…In this contribution, a global approach is followed, where the deformations are interpolated by radial basis functions (RBF) that are defined over the entire domain of the surface measured by a DIC-system. Based on [4], where RBFs are chosen that tend to zero for increasing radius, a different approach, so-called multiquadric, are examined here for its applicability. RBFs allow full-field strain determination, which means that -in addition to the principal strains -also the principal directions are continuous over the entire region.…”