Anatomy of coupled constitutive models for ratcheting simulation

https://doi.org/10.1016/S0749-6419(99)00059-5Get rights and content

Abstract

This paper critically evaluates the performance of five constitutive models in predicting ratcheting responses of carbon steel for a broad set of uniaxial and biaxial loading histories. The models proposed by Prager, Armstrong and Frederick, Chaboche, Ohno-Wang and Guionnet are examined. Reasons for success and failure in simulating ratcheting by these models are elaborated. The bilinear Prager and the nonlinear Armstrong-Frederick models are found to be inadequate in simulating ratcheting responses. The Chaboche and Ohno-Wang models perform quite well in predicting uniaxial ratcheting responses; however, they consistently overpredict the biaxial ratcheting responses. The Guionnet model simulates one set of biaxial ratcheting responses very well, but fails to simulate uniaxial and other biaxial ratcheting responses. Similar to many earlier studies, this study also indicates a strong influence of the kinematic hardening rule or backstress direction on multiaxial ratcheting simulation. Incorporation of parameters dependent on multiaxial ratcheting responses, while dormant for uniaxial responses, into Chaboche-type kinematic hardening rules may be conducive to improve their multiaxial ratcheting simulations. The uncoupling of the kinematic hardening rule from the plastic modulus calculation is another potentially viable alternative. The best option to achieve a robust model for ratcheting simulations seems to be the incorporation of yield surface shape change (formative hardening) in the cyclic plasticity model.

Introduction

As the data base and understanding of ratcheting response (accumulation of strains with cycles) are growing, the number of efforts in developing constitutive models for ratcheting is also increasing (Chaboche et al., 1979, Chaboche, 1986, Chaboche, 1991, Chaboche, 1994, Voyiadjis & Sivakumar, 1991, Voyiadjis & Basuroychowdhury, 1998, Guionnet, 1992, Ohno & Wang, 1993, Hassan & Kyriakides, 1994a, Hassan & Kyriakides, 1994b, Delobelle et al., 1995, McDowell, 1995, Jiang & Sehitoglu, 1996a, Ohno, 1997, Xia & Ellyin, 1997 and others). In many of these models the plastic modulus calculation is coupled with its kinematic hardening rule through the yield surface consistency condition as in the classical model proposed by Prager (1956). These models are referred to as coupled models in this paper. Models proposed by Armstrong & Frederick, 1966, Chaboche, 1986, Chaboche, 1991, Chaboche, 1994, Guionnet, 1992, Ohno & Wang, 1993 belong to this class and are studied along with the Prager model (1956) in this paper.

In another class of models, the plastic modulus might be indirectly influenced by the kinematic hardening rule but its calculation is not coupled to the kinematic hardening rule through the consistency condition. The models proposed by Mroz, 1967, Dafalias & Popov, 1976, Drucker & Palgen, 1981, Tseng & Lee, 1983 and many others belong to this class. These models are referred to as uncoupled models and will be discussed in another paper (Bari and Hassan, 1999a).

Most of the models proposed so far are developed and verified using data from limited or simple experiments. These models have not been tested against a wide variety of ratcheting responses to verify the generality of these models. Consequently, most of these constitutive models might predict a special class of ratcheting responses quite well, but fail to predict a broad class of ratcheting responses (Hassan & Kyriakides, 1994b, Corona et al., 1996).

Most metals cyclically harden or soften up to a certain number of cycles and subsequently stabilize or cease to change the size of the yield surface (Morrow, 1965, Jhansale, 1975, Tuegel, 1987, Ishikawa & Sasaki, 1988). Ratcheting, though, keeps on occurring with cycles even after the material stabilizes (Hassan & Kyriakides, 1992, Hassan et al., 1992, Hassan & Kyriakides, 1994a, Hassan & Kyriakides, 1994b). Hence, the kinematic hardening (translation of the yield surface in stress space) is attributed to be the primary reason for ratcheting. Thus, in order to develop and verify a model for ratcheting, it is essential to study the ratcheting responses of stabilized materials. This, in effect, means that the parameters affecting the isotropic hardening (i.e. yield surface size change) should not be included during the model development for ratcheting. Also, all of the kinematic hardening (i.e. yield surface translation) parameters should be determined using experiments performed on stabilized materials. After achieving a robust model for ratcheting responses of cyclically stable materials, it can easily be extended to cyclically hardening and softening materials following the approach demonstrated by Hassan & Kyriakides, 1994a, Hassan & Kyriakides, 1994b.

A broad set of quasi-static ratcheting data which include uniaxial to complex biaxial ratcheting responses of stabilized carbon steels have been developed by Hassan & Kyriakides, 1992, Hassan et al., 1992, Corona et al., 1996. These data are used in this study to evaluate the performance of the models considered. The cyclic loading histories prescribed in the experiments are shown in Fig. 1. The history in Fig. 1a, which results in uniaxial ratcheting, involves axial stress cycles with a mean stress. The readers are referred to Hassan and Kyriakides (1992) for demonstration of a uniaxial ratcheting response (Fig. 7 in the reference). Two sets of uniaxial experiments are considered in this study. In the first set, the amplitude stresses (σxa) in the experiments are the same while the mean stresses (σxm) vary. Whereas in the second set, the amplitude stresses (σxa) vary while the mean stresses (σxm) are kept constant. Data from these experiments are presented in Fig. 8a and b, where the peak axial strains (εxp) of each cycle are plotted against the number of cycles, N.

The loading history in Fig. 1b involves axial strain cycles in the presence of a constant internal pressure. The history results in circumferential ratcheting as demonstrated in Fig. 1 of Hassan et al. (1992). Two sets of data from this biaxial loading history are also considered in this study as shown in Fig. 8c and d. In these plots, the peak circumferential strains (εθp) of each cycle are plotted against the number of cycles, N. For the set shown in Fig. 8c, the axial strain amplitude (εxc) of the experiments vary with same circumferential stress (σθm) due to internal pressure, whereas in the set of Fig. 8d, the internal pressures of the experiments are different with the same axial strain amplitude.

The bow-tie and reverse bow-tie loading histories in Fig. 1c and d also result in circumferential ratcheting as demonstrated in Fig. 10, Fig. 11 of Corona et al. (1996). The data set from these two biaxial loading histories considered in this study are shown in Fig. 8e and f.

The above mentioned ratcheting data are used to evaluate the performance of several coupled cyclic plasticity models. Also, parameters for each model are determined using experiments on stabilized carbon steels. The same set of parameters for each model are used to simulate uniaxial and multiaxial ratcheting responses. The reasons for the success and failure in simulating these responses by the models are presented through anatomical discussion of the influences of modeling schemes and parameters on ratcheting simulation. Although the Prager, 1956, Armstrong & Frederick, 1966 models are not capable of simulating ratcheting responses satisfactorily, these models are discussed in this study in order to demonstrate the gradual development of different features of plasticity models over time.

Section snippets

Ratcheting models

The plasticity models with the assumption of rate-independent material behavior has the following common features:i.vonMisesyieldcriterion:f(σα)=32(sa)•(sa)120,ii.flowrule:dεp=1Hfσdσfσ,Where, σ is the stress tensor, εp is the plastic strain tensor, s is the deviatoric stress tensor, α is the current center of the yield surface, a is the current center of the yield surface in the deviatoric space, σ0 is the size of the yield surface (constant for a cyclically stable material), and H is

Discussion and conclusions

Five well-known constitutive models are evaluated in this paper in terms of simulations for ratcheting responses from a series of uniaxial and biaxial experiments. Six uniaxial and ten biaxial ratcheting responses on stabilized carbon steels are collected from literature (Hassan & Kyriakides, 1992, Hassan et al., 1992, Corona et al., 1996) for the study. The rate-independent and cyclically stable ratcheting response simulations by Prager, 1956, Mroz, 1967, Armstrong & Frederick, 1966, Chaboche,

Acknowledgements

The financial supports from the Center for Nuclear Power Plant Structures, Equipment and Piping and the Department of Civil Engineering at North Carolina State University are gratefully acknowledged.

References (36)

  • Z. Mroz

    On the description of anisotropic work hardening

    Journal of the Mechanics and Physics of Solids

    (1967)
  • N. Ohno et al.

    Kinematic hardening rules with critical state of dynamic recovery, part I: formulations and basic features for ratcheting behavior

    International Journal of plasticity

    (1993)
  • A. Phillips et al.

    Yield surfaces and loading surfaces. Experiments and recommendations

    International Journal of Solids and Structures

    (1979)
  • Z. Xia et al.

    A constitutive model with capability to simulate complex multiaxial ratcheting behaviour of materials

    International Journal of Plasticity

    (1997)
  • ANSYS, 1998. ANSYS user's Manual, Revision 5.4. ANSYS Inc. Providence, Houston, PA 15342-0065,...
  • Armstrong, P.J., Frederick, C.O., 1966. A mathematical representation of the multiaxial bauscinger effect. CEGB Report...
  • Bari, S., Hassan, T., 1999a. Kinematic hardening rules in uncoupled modeling of multiaxial ratcheting simulation....
  • Bari, S., Hassan, T., 1999b. An improved constitutive model for multiaxial ratcheting. To be submitted to the...
  • Cited by (0)

    View full text