A generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials

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Abstract

In this paper, a generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials is derived. The evolution equation for the active yield surface with reference to the memory yield surface is obtained by considering the continuous expansion of the active yield surface during the unloading/reloading process. The incremental constitutive relation based on the associated flow rule is then derived for a general yield function for pressure insensitive and sensitive materials. Detailed incremental constitutive relations for materials based on the Mises yield function, the Hill quadratic anisotropic yield function and the Drucker–Prager yield function are derived as the special cases. The closed-form solutions for one-dimensional stress–plastic strain curves are also derived and plotted for materials under cyclic loading conditions based on the three yield functions. In addition, the closed-form solutions for one-dimensional stress–plastic strain curves for materials based on the isotropic Cazacu–Barlat yield function under cyclic loading conditions are summarized and presented. For materials based on the Mises and the Hill anisotropic yield functions, the stress–plastic strain curves show closed hysteresis loops under uniaxial cyclic loading conditions and the Masing hypothesis is applicable. For materials based on the Drucker–Prager and Cazacu–Barlat yield functions, the stress–plastic strain curves do not close and show the ratcheting effect under uniaxial cyclic loading conditions. The ratcheting effect is due to different strain ranges for a given stress range for the unloading and reloading processes. With these closed-form solutions, the important effects of the yield surface geometry on the cyclic plastic behavior due to the pressure-sensitive yielding or the unsymmetric behavior in tension and compression can be shown unambiguously. The closed form solutions for the Drucker–Prager and Cazacu–Barlat yield functions with the associated flow rule also suggest that a more general anisotropic hardening theory needs to be developed to address the ratcheting effects for a given stress range.

Introduction

Different hardening rules were proposed in the past to specify the evolution of the yield surface during plastic deformation. In general, these hardening rules allow the yield surface to expand (isotropic hardening), contract (isotropic softening), translate (kinematic hardening) and change shape in the stress space. Under monotonically increasing nearly proportional loading conditions, the isotropic hardening rule appears to give a good approximation of the material plastic hardening behavior. On the other hand, the kinematic hardening rule was proposed to model the Bauschinger effect, which cannot be modeled by the isotropic hardening rule. For the kinematic hardening rule, the movement of the yield surface can be, for example, in the direction of the plastic strain rate (Prager, 1955) or in the direction of the relative stress with respect to the center of the yield surface (Ziegler, 1959). However, under complicated plastic deformation histories, substantial deviations of the actual plastic behaviors from those predicted by both isotropic and kinematic hardening rules are often observed. Different plasticity theories were therefore proposed to model the cyclic hardening behavior, for example, see Armstrong and Frederick, 1966, Mroz, 1967, Dafalias and Popov, 1976, Chaboche, 1977, Chaboche, 1986.

Based on some general observations of the uniaxial cyclic plastic behavior of many metals, Mroz, 1967, Mroz, 1969 introduced the concept of a field of work-hardening moduli corresponding to an infinite number of initially concentric yield surfaces in the deviatoric stress space. In his model, the translation of each surface does not intersect each other but these surfaces are consecutively in contact with each other. Lamba and Sidebottom, 1978a, Lamba and Sidebottom, 1978b conducted cyclic nonproportional strain experiments and showed that the Mroz hardening rule can be used to predict the stress–strain responses of the experiments. Although the Mroz hardening rule gives a good prediction of the material behavior, its numerical implementation has been considered to be inefficient and complex because the positions of all the yield surfaces need to be continuously tracked in the stress space and require a large memory storage.

Due to this computational complexity, the Mroz model was modified by Krieg, 1975, Dafalias and Popov, 1975 where all the yield surfaces in the Mroz model were replaced by an inner yield surface and an outer yield surface with a continuum of intermediate surfaces that are analytically determined. Subsequently, more investigations and plasticity theories were proposed, for example, see Mroz et al., 1981, Tseng and Lee, 1983, Chu, 1984, Chu, 1987, McDowell, 1985a, McDowell, 1985b, Chang and Lee, 1986, Lu and Mohamed, 1987, Takahashi and Ogata, 1991, Lee et al., 1995, Chen and Abel, 1996, Chiang and Beck, 1996, Montans, 2000, Sakane et al., 2002. Specifically, we mention the work of Chu (1984) where a computational model is proposed based on the concept of the Mroz multi-yield-surface model for incompressible isotropic Mises materials. The Chu model has been extended to incompressible anisotropic materials based on the Hill quadratic anisotropic yield function (Hill, 1948) and implemented into a finite element code to simulate sheet metal forming processes by Tang (1990), and the details of the formulation are presented in Tang and Pan (2007). Although the Chu model provides an efficient way in employing the Mroz multi-yield-surface theory by defining and memorizing a specific moving direction of the center of the active yield surface, the center of the active yield surface needs to be updated in each of loading, unloading and reloading steps as in other two-surface models.

Motivated by the need of formulating an anisotropic hardening rule to model the fillet rolling process of cast iron crankshafts (Chien et al., 2005), we developed a computationally efficient anisotropic hardening rule based on the Drucker–Prager yield function. However, when we implemented the anisotropic hardening rule based on the Drucker–Prager yield function to ABAQUS (2004) to simulate the fillet rolling process, we found some very interesting ratcheting behavior that needs to be investigated theoretically further to rule out numerical inaccuracies due to the finite element computations (Choi, 2007). Therefore, we concentrate in this paper on developing an anisotropic hardening rule and closed-form solutions for a general yield function to model the evolution of the yield surface for materials under cyclic loading conditions. The anisotropic hardening rule is applicable to either plastic isotropic or anisotropic materials. The anisotropic hardening rule is also applicable to either pressure-insensitive or pressure-sensitive materials. We choose three commonly used yield functions where the derivatives of the yield functions with respect to the stresses can be explicitly obtained in order to derive closed-form solutions for stress–plastic strain curves under uniaxial cyclic loading conditions. We select the Mises yield function for incompressible isotropic materials, the Hill quadratic anisotropic yield function for incompressible anisotropic materials (Hill, 1948), and the Drucker–Prager yield function for pressure-sensitive isotropic materials (Drucker and Prager, 1952). The Hill quadratic anisotropic yield function is selected for incompressible anisotropic materials although more updated non-quadratic anisotropic yield functions such as those proposed by Barlat et al., 1997, Barlat et al., 2003, Barlat et al., 2005, Cazacu and Barlat, 2004, Cazacu et al., 2006, Hu, 2003, Hu, 2005, Hu, 2007 are available. The anisotropic hardening rule is applicable to these anisotropic yield functions and the detailed constitutive relations based on these anisotropic yield functions can be derived. However, we only present the detailed constitutive relations based on the Hill quadratic anisotropic yield function in this paper.

In this paper, the concepts of the isotropic, kinematic and Mroz hardening rules are first introduced to review briefly the characteristics of the three hardening rules. From the observation of the translation and enlargement of the active yield surface from the current yield surface to the next larger yield surface in the Mroz multi-yield-surface model, a simple evolution equation for the active yield surface, with reference to the memory yield surface, is obtained by considering the continuous expansion of the active yield surface during the unloading/reloading process. The incremental constitutive relation based on the associated flow rule is derived for a general yield function for pressure insensitive and sensitive materials. As the special cases, detailed incremental constitutive relations are derived for the Mises yield function, the Hill quadratic anisotropic yield function and the Drucker–Prager yield function. The closed-form solutions for stress–plastic strain curves are also derived and plotted for materials based on the three representative yield functions under uniaxial cyclic loading conditions. In addition, the closed-form solutions for stress–plastic strain curves for materials based on the isotropic Cazacu–Barlat yield function are summarized and plotted under uniaxial cyclic loading conditions. Finally, some conclusions are made based on the results from the derived anisotropic hardening rule and the one-dimensional stress–plastic strain curves for the four yield functions.

Section snippets

Concepts of hardening rules

Under nearly proportional increasing stressing or straining conditions where the ratios of the stresses or strains remain nearly fixed during the deformation, the isotropic hardening rule is usually used. Fig. 1a schematically shows the evolution of the yield surface according to the isotropic hardening rule. As shown in the figure, when the stresses increase, the stress state will first meet the initial yield surface. As the stresses continue to increase, plastic strain hardening takes place.

Evolution of yield surface and constitutive relations for general yield function

As mentioned earlier, since the Mroz hardening rule follows the isotropic hardening rule under the initial loading condition, the constitutive relation under the isotropic hardening rule is first introduced here. During the initial loading process, the yield surface Φ keeps its center at the origin in the stress space. Here, the yield surface Φ is assumed to have its own constant work-hardening modulus from the concept of the Mroz hardening rule. The yield function Φ can be expressed asΦ(σij,σ¯)

Constitutive relations for three yield functions

The constitutive relations for the initial loading and unloading/reloading processes based on a general yield function in the previous section are adopted here to derive the constitutive relations specifically for materials based on the Mises yield function, the Hill quadratic orthotropic yield function and the Drucker–Prager pressure-sensitive yield function. First, in order to show the difference between the three yield functions, the yield contours are obtained for the three yield functions

One-dimensional stress–strain relations for three yield functions

In this section, the one-dimensional closed-form stress–strain relations based on the three yield functions under cyclic loading conditions are derived in order to identify the root cause for the ratcheting behavior based on the Drucker–Prager yield function from the numerical solutions based on the anisotropic hardening rule. Here, the elastic strain is not considered for convenience. Fig. 6 shows the definitions of the stresses and plastic strains for one-dimensional stress–plastic strain

Conclusions and discussions

In this paper, the concepts of the isotropic, kinematic and Mroz hardening rules are first introduced to briefly review the characteristics of the three hardening rules. From the observation of the translation and enlargement of the active yield surface from the current yield surface to the next larger yield surface in the Mroz multi-yield-surface model, a simple evolution equation for the active yield surface is then obtained by considering the continuous expansion of the active yield surface

Acknowledgements

The support of this work by a DaimlerChrysler Challenge Fund Project and a grant from TRW is greatly appreciated. The support of the Rackham Predoctoral Fellowship of University of Michigan to K.S. Choi is also greatly appreciated.

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