A plane stress yield surface using Bézier curve interpolation in two directions

Abstract

Current advanced yield criteria, like BBC2005, Yld2000 and Yld2004, offer significant flexibility and are proven to be convex. The number of coefficients of the function is fixed, which limits the number of experimental results that can be fitted independently. A common approach is to define an error function. The optimal anisotropy coefficients are established by minimizing the error function. The Vegter yield criterion is very flexible with respect to experimental results. It is in fact defined as a function that smoothly connects experimental results, while ensuring a continuous function for the derivative. Fitting is therefore a non-issue; the experimental results are the coefficients for the yield criterion. Second order Bézier curves are used for the interpolation between experimental results. Extreme values for these results can cause concavity in the definition of the yield locus. Although rarely an issue, convexity needs to be validated as part of the implementation of the yield criterion. Recently we have seen some cases of concavity that could be attributed to the cosine interpolation in the original Vegter yield criterion definition. Cosine interpolation is used between experimental results at different angles to the rolling direction. An improved method is proposed that uses Bézier interpolation not only between biaxial stress states in one orientation, but also between stress states with a different material orientation. The benefits are greater flexibility in defining experimental results and a coherent approach towards a 3D extension. The benefits of this new yield criterion will be demonstrated in a direct comparison with Yld2000 and Yld2004. The comparison will include the definition of the yield locus, showing actual cases, where in particular biaxial stress states like plane strain and shear are difficult to fit accurately. The comparison will also include improved accuracy in forming simulations.

Implicitization of the Vegter Yield Criterion

“…Although the parametric form of Vegter yield criterion has been used over the years in various sheet metal forming analyses and numerical simulations without much problems at all [10,11,12,13], there are certain advantages when its implicit form is also known. For example, the determination about the onset or switch-off of plastic yielding is crucial in almost all iterative algorithms for analyzing elastic-plastic deformation of a sheet metal.…”

“…Parametric curves are used predominately in computer graphics and Bézier curves are often adapted in computer-aided auto body design and many other applications [19]. A quadratic Bézier curve in a plane (x, y) is specified by three control points (x r 1 , y r 1 ),(x h , y h ) and (x r 2 , y r 2 ) (where the first and third points are called reference (end) points and the second or middle point is called a hinge point by Vegter and co-workers [10,11,12,13]). It may be treated as a linear interpolation of two linear Bézier curves…”

“…In particular, the Vegter yield criterion has a more direct connection to computer graphics and uses multiple second-order Bézier curves to construct the entire closed 2D yield surface. Its methodology has also been called interpolation of biaxial stress states to emphasize its piecewise nature and its flexibility [11,12,13]. In terms of the commonly used terminology in analytical geometry and computer graphics, it belongs to the parametric instead of implicit formulation or representation of yield criteria.…”

BBC05 with non-integer exponent and ambiguities in Nakajima yield surface calibration

“…It is visualized in [14] that the exponent can vary a lot for metallic materials. The utilized material model in [14] is Barlat 2000 [7, 8], equivalent to BBC05 [15]. Practical experience at Volvo Cars also supports exponent values other than 3 and 4.…”

“…Practical experience at Volvo Cars also supports exponent values other than 3 and 4. The variation of the exponent k is also discussed in [15] along with a comparison between the Vegter yield critera and models such as Barlat 2000, andHill (1948). The assumption that k should be 3 or 4 was originally made for an extension of the isotropic von Mises quadratic yield function [16].…”