Interpolation-based plane stress anisotropic yield models
Graphical abstract
Introduction
The anisotropy of metal sheets caused by material processing (such as rolling) cannot be accurately predicted by isotropic yield criteria. It is necessary to choose appropriate anisotropic yield criteria for the yield prediction of anisotropic materials. For decades, various anisotropic yield models have been proposed. Most yield models are phenomenological, and they are formally expressed as specific functional analytic expressions. These functional formulas (with σi,j as variables) adopt specific functional modes (e.g. the sum/difference combination of powers of linear combinations among variables) to characterize specific characteristics of material when yielding, such yield locus under two-dimensional principal stress state is similar to an ellipse. Therefore, the diversity of phenomenological yield functions is often reflected in the order of variables, the combination mode of variables and the type of expressions.
The phenomenological anisotropic yield function can be generally divided into three series. The first series is the Hill series represented by Hill48 [27], which is modified based on the Mises isotropic yield function. It is represented as a quadratic function:where F, G, H, L, M, N are materials coefficients. This criterion is not suitable for the materials with r < 1 (r represents the ratio of transverse strain to normal strain), but it is still widely used in engineering problems due to its simple form. In addition, the series also includes the yield criterion proposed in the following literatures [17], [25], [28], [29], [30], [32], [36], [45]. These yield criteria are essentially proposed based on the expansion or revision of Hill48. The second series is the Hosford series yield criterion represented by Balat89 [5], which is based on the revision of Hosford isotropic yield criterion. It is based on the plane stress assumption, with the stress tensor invariants (K1, K2) as the function variables, expressed as a high-order polynomial:wherea, h, p are material parameters. m takes 6 for BCC (body-centered cubic) materials and 8 for FCC (face-centered cubic) materials. Later, by introducing linear transformation, Barlat and coworkers proposed yield criteria such as Yld2000-2d, Yld2004-13p, Yld2004-18p, etc. [4], [6], [7], [8], [9], [10]. The yield criteria proposed by Karafifillis and Boyce [33], Banabic et al. [12], [13], [14], Aretz [3], Bron and Besson [15], Lou et al. [39] also belong to this series. This series of yield functions usually take the principal values of the stress deviatoric tensor as variables, and implements anisotropy by introducing linear transformations [11], which are mainly applied for the yield prediction of FCC and BCC materials. The last series of yielding criteria is mainly proposed for the issue of tension-compression asymmetry of materials when yielding, represented by CPB06 series yield criteria [21], [41], [42]. CPB06 yield function is express aswhere Σ1, Σ2, Σ3 are the principal values of Σ. The transformed tensor Σ (obtained from a linear transformation on the Cauchy stress deviatoric tensor) is defined as Σ = CTσ, where
The most important feature of these series is that it can characterize the tension-compression asymmetry caused by twins in the yield process, especially for the yield prediction of HCP (hexagonal closed-packed) materials. With regard to asymmetry in tension and compression, Cavaco and other researchers have also proposed a method for constructing a yield function based on stress invariants of stress deviator. It can be traced to the Drucker yield function [26], which is an isotropic yield function represented by the second and third invariants of stress deviatoric tensor. Cavaco and coworkers extended the Drucker yield function to an orthotropic yield function based on stress invariants [18], [19], and began to pay attention to asymmetry of tension and compression [20]. The stress invariants-based yield functions have been widely concerned by many researchers in recent years, and many corresponding yield functions have been proposed [22], [23], [24], [31], [38], [47].
Another alternative method to describe the yield surface is geometric method, specifically interpolation construction method, which was first proposed by Vegter et al. [43]. Vegter et al. [44] took second-order Bézier curves to construct the in-plane yield locus segmentally (see Fig. 1). Some specific biaxial stress yield points are used as interpolation reference points (nodes), and the hinge points are the intersections of tangents (perpendicular to the plastic flow direction) at two adjacent nodes.
By introducing the variable θ (the angle between loading direction and rolling direction), the reference points (yield stresses) and strain ratios can be expressed as periodic functions of θ. Based on the assumption of orthotropy, the periodic changes of the two plasticity indices are represented by even Fourier series, thereby achieving the description of anisotropy. Peng et al. [40] proposed constructing the in-plane yield loci based on Hermite interpolation. In this method, the yield locus is constructed in a polar coordinate system on the quasi-π plane, and the yield locus is the projection of the corresponding deviatoric stress on the plane. However, this method only considers the normal stress components of deviatoric stress tensor, and does not consider the shear stress components, so its application is very limited. Compared with the phenomenological yield functions, the interpolation-type yield functions are no longer continuous but discrete, so the fitting of the yield data is implemented piecewise without global optimization. In addition, since the interpolation curve depends on the interpolation nodes, the corresponding yield locus can be flexibly obtained based on the yield points. Therefore, more experimental data can be considered, and complex anisotropy behavior such as tension-compression asymmetry can be conveniently described. However, there are still some shortcomings in the current interpolation-type models, including that the yield locus is established under the principal stress or normal stress state, and the characterization and description of shear stress are not clear enough; there is no detailed discussion on the tricomponent yield surface under general stress state; there are not enough application examples for yield prediction of various types of materials (BCC, FCC, HCP); there is no detailed explanation for the prediction of flow stresses and r-values, which are often emphasized with phenomenological yield criteria. These problems should be solved urgently in interpolation-type models.
The interpolation yield model proposed in this paper is built in a specific oblique coordinate system. The yield surface is obtained by sequentially constructing three specific “longitude lines” and two sets of quasi-orthogonal “latitude lines” on the yield surface. In this process, "longitude lines" act as the endpoint trajectories of "latitude lines", and the surface to be constructed is formed by the continuous changes of "latitude lines" along the trajectories. The proposed model takes the Bézier curves as the interpolation tools to directly construct the yield surface in a three-dimensional stress apace, and the derivation of physical quantities such as yield function and strain increment are carried out subsequently. It should be noted that we have proposed two adaptive models based on whether or not to consider the asymmetry of tension and compression of the sheets during the yielding process. The two models have certain differences in construction mode and parameter setting. In addition, we also propose simplified models for the two full-parameter models by omitting the parameter determination process of the “latitude lines”. The full-parameter model and the simplified model were taken to predict the yield surfaces, flow stresses and r-values of the three lattice type materials (BCC, FCC, HCP) to illustrate the applicability of the proposed models.
The second section introduces the basic hypothesis and preparation theory of this paper. The third section elaborates the construction method of the new models. The fourth section introduces the model-related derivation and parameter determination issues, and proposes the simplified model based on parameter reduction. In the fifth section, the proposed full-parameter models and simplified models are applied to the yield model constructions of three lattice type materials (five in total).
Section snippets
Tricomponent yield surface and yield locus
The metal sheet is generally considered to be in a plane stress state during plastic deformation, in which the independent stress components include σx, σy, and τxy. The tricomponent (plane stress) yield surface is usually established in σx − σy − τxy rectangular coordinate system, where σx points to the rolling direction and σy points to the transverse direction. The initial yield surface of the sheet is generally considered to be similar to an ellipsoid. In the description of phenomenological
Steps of surface construction
The proposed model is built in a specific oblique coordinate system, which can be obtained from the transformation of original σx − σy − τxy rectangular coordinate system. Specifically, τxy-axis in the oblique coordinate system is unchanged, and σy-axis changes to the direction at an angle of 45° to the rolling direction, denoted as , and σx-axis changes to the tangential direction at the equi-biaxial tensile yield point (σb), denoted as . The two new axes () present a plane
Yield function
The mathematical expression of the yield model proposed above is a vector-type parametric function with u, v as variables. The traditional yield function is based on the concept of equivalent stress, which contributes to the derivation of relevant physical quantities and simulation calculations. Vegter et al. [44] proposed an equivalent stress representation method based on two-dimensional interpolation Bézier curve, and it can be extended to three-dimensional variables to represent the
Applications
In order to illustrate the practicability of the proposed models, we tested five kinds of sheets for the prediction of yield behavior with the proposed models, including two BCC materials (IF-steel and high speed steel), two FCC materials (aluminum alloys of AA2090-T3 and AA6111-T4) and one HCP material (magnesium alloy of AZ31B). To demonstrate the accuracy of the prediction, the uniaxial tensile/compression yield stress prediction curve and the r-value prediction curve along the loading
Conclusion
In this paper, the interpolation-type tricomponent plane stress yield models based on Bézier curve-constructions are proposed, and they are divided into two cases according to whether the tensile and compressive asymmetry of the sheet metals during yielding is considered. Each new yield model is proposed based on the σx − σy − τxy tricomponent yield surface and is built in an oblique coordinate system with the axes of . By sequentially constructing three special "longitude lines" and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
This study was funded by the National Natural Science Foundation of China (grant number 51775335, 51635005).
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