Interpolation-based plane stress anisotropic yield models

Highlights

• Interpolation-type anisotropic plane stress yield models are proposed.

• Tricomponent yield surface is built in a specific oblique coordinated system based on the constructions of two sets of specific quasi-orthogonal yield loci (constructed with Bézier interpolation curves) on the yield surface.

• Two corresponding models are proposed based on whether or not to consider the asymmetry of tension and compression when the materials yield.

• By adjusting the constructors and omitting some parameters, the corresponding two simplified models are proposed.

• The full parameter model can accurately predict the flow stresses and r-values of materials, while the simplified model can reasonably predict the overall trends.

Abstract

In this paper, the interpolation-type plane stress anisotropic yield models are proposed. The models are built in a specific three-dimensional oblique coordinate system by sequentially constructing three specific ``longitude lines" and two sets of quasi-orthogonal ``latitude lines" on the tricomponent yield surface. The three ``longitude lines" act as the trajectories of the endpoints of the ``latitude lines", and the semi-yield surface is formed by the continuous variation of the ``latitude lines" along the trajectories. These curves are constructed by parametric Bézier curves, and different order schemes are adopted for “latitude lines” and “longitude lines”. Some coefficients in the expression of “latitude lines” are represented by specific analytic functions, which are related to the distortion of yield loci with the increase of τ_xy-component. According to whether the asymmetry of tension and compression when yielding is considered, two corresponding models are proposed. We also obtain corresponding simplified forms by omitting the parameter determination of the "latitude lines". The yield loci, flow stresses and r-valus of five kinds of materials with three lattice types were predicted by full-parameter models and simplified models, which showed the accuracy of the proposed models for predicting yield behavior of various materials.

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 σ_i,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:

$$\begin{array}{ccc}\hfill \Phi & =& F{\left({\sigma }_y-{\sigma }_z\right)}^2+G{\left({\sigma }_z-{\sigma }_x\right)}^2+H{\left({\sigma }_x-{\sigma }_y\right)}^2+2L{\tau }_{yz}^2\hfill \\ & & +2M{\tau }_{zx}^2+2N{\tau }_{xy}^2={\overline{\sigma }}^2,\hfill \end{array}$$

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:

$$\Phi =a{\left|K_1+K_2\right|}^m+a{\left|K_1-K_2\right|}^m+\left(2-a\right){\left|2K_2\right|}^m=2{\overline{\sigma }}^m,$$

where

$$K_1=\frac{1}{2}\left({\sigma }_x+h{\sigma }_y\right),K_2=\frac{1}{2}\sqrt{{\left({\sigma }_x-h{\sigma }_y\right)}^2+p^2{\tau }_{xy}^2},$$

a, 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 as

$$\Phi ={\left(\left|{\Sigma }_1\right|-k{\Sigma }_1\right)}^a+{\left(\left|{\Sigma }_2\right|-k{\Sigma }_2\right)}^a+{\left(\left|{\Sigma }_3\right|-k{\Sigma }_3\right)}^a,$$

where 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

$$\begin{array}{ccc}\hfill C& =& \left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& & & \\ C_{21}& C_{22}& C_{23}& & & \\ C_{31}& C_{32}& C_{33}& & & \\ & & & C_{44}& & \\ & & & & C_{55}& \\ & & & & & C_{66}\end{array}\right],\hfill \\ \hfill T& =& \frac{1}{3}\left[\begin{array}{cccccc}2& -1& -1& & & \\ -1& 2& -1& & & \\ -1& -1& 2& & & \\ & & & 3& & \\ & & & & 3& \\ & & & & & 3\end{array}\right].\hfill \end{array}$$

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).