Uniaxial and biaxial ratchetting in piping materials—experiments and analysis

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Abstract

The performance of the Chaboche kinematic hardening model has been evaluated in this paper to predict the ratchetting responses for a broad set of uniaxial and biaxial loading histories. The investigations have been performed with reference to both uniaxial and biaxial experimental data, viz. (a) strain and stress controlled uniaxial tests on tensile specimens; (b) biaxial tests on straight pipes with constant internal pressure and cyclic bending load; and (c) a shake table test on elbow. The parameters of the Chaboche model have been calculated from the uniaxial strain controlled stable hysteresis loop. Amongst the various parameters in the Chaboche model, it has been found that the selection of the value of γ3 plays a crucial role in achieving better simulation. The Chaboche model was observed to predict complete shakedown for γ3=0. On the other hand, the model closely simulated the experimental results for γ3=9. The same parameters have been used to analyze the biaxial loading condition. Ratchetting simulation studies by the Chaboche model have resulted in reasonably good agreement with experiments.

Introduction

Piping networks are often employed in various industrial applications, including nuclear power plants. Generally, piping systems in a nuclear power plant are designed for normal operation loads (pressure) along with cyclic loads, such as earthquake. This cyclic loading on the piping with nonzero mean static stress results in either structural shakedown or ratchetting. With the occurrence of structural shakedown, the dissipated energy in the whole structure remains bounded after initial plastic flow and the structure responds in a purely elastic manner to the applied variable loads. On the other hand, the ratchetting phenomenon is defined as a cycle-by-cycle accumulation of plastic strain with the application of cyclic load characterized by constant stress amplitude with a nonzero mean stress. After a sufficient number of cycles, the total strain (and therefore displacement) becomes so large that the original shape of the structure is altered, thereby making the structure unserviceable. Typical ratchetting and shakedown responses under repetitive loading are shown in Fig. 1.

The ratchetting response of a material is significantly influenced by the stress history, which in turn depends on the external loads as well as on the geometry of a pipe. Also, ratchetting depends on the anisotropic property of the material due to different strain-hardening curves in tension and compression. This differential strain hardening causes structures to ratchet under cyclic loading. The most well known nonlinear kinematic hardening model has been proposed by Armstrong–Frederick [1]. This model includes a kinematic hardening rule containing a ‘recall term’ which incorporates the fading memory effects of the strain path and essentially makes the rule nonlinear in nature. Also, the anisotropy property of the tension and compression curves has been considered in this model, which produces a change in shape between the forward and the reverse loading paths. Therefore, the stress–strain hysteresis loop does not close and the ratchetting phenomenon occurs. However, the stress-strain loop produced by this model deviates significantly from experiment and the ratchetting strain is also overpredicted. Chaboche et al. [2], [3], [4] have proposed a decomposed nonlinear kinematic model, by superposing Armstrong–Frederick hardening rules. Three decomposed hardening rules proposed by Chaboche have been used in the present study to simulate ratchetting. The material constants associated with the Chaboche model can be derived easily from a uniaxial stable hysteresis loop [5].

Garud et al. [6], Hassan et al. [7], Lang et al. [8], Mahbali and Eslami [9], Xia and Elliyn [10] and Yoshida [11] have compared numerical results with experiments under cyclic loadings. Ohno and co-authors [12], [13], [14] have also reported various numerical studies under mechanical and thermal ratchetting.

The theory of the Chaboche nonlinear kinematic hardening model, available in the ANSYS software package [15], is discussed briefly in the present paper. Materials like SA-333 Carbon steel and SS-304 Stainless steel are typically used in Nuclear Power Plants in India. Thus, ratchetting simulation has been performed using the Chaboche model for these materials to understand their behavior under uniaxial and biaxial loading conditions. The data obtained from the following three sets of experiments have been used for comparison.

  • (a)

    Strain controlled and stress controlled uniaxial tests on tensile specimens made of SA-333 Gr.6 carbon steel;

  • (b)

    Three point and four point bend tests on straight pipes made of SA333 Gr.6 carbon steel, subjected to constant internal pressure and cyclic bending load; and

  • (c)

    A shake table test of a pipe elbow of SS-304 stainless steel.

The strain controlled stable hysteresis loop has been used to calculate the Chaboche parameters. The return mapping approach with consistent elasto-plastic tangent moduli has been used in ANSYS [15] for numerical integrations of the constitutive equation.

Section snippets

Chaboche model

The rate independent version of the nonlinear kinematic hardening model proposed by Chaboche [2], [3], [4] has been considered here, which primarily involves superposition of three Armstrong–Frederick kinematic hardening rules. The kinematic hardening rule contains a ‘recall term’, which incorporates the fading memory effects of the strain path. The constitutive equation is based on linear isotropic elasticity, a von-Mises yield criterion and the associated flow rule.

The evolution equation for

Experiments and numerical simulation

Experimental and numerical simulations were performed for uniaxial specimens, straight pipes and an elbow as mentioned in Section 1. Uniaxial experiments have been performed at the material level, where the state of stress is uniform everywhere except near the ends. The cyclic bending and the shake table tests have been conducted at the structural level where the state of stress varies from point to point. While selecting the test specimens for the uniaxial and the biaxial tests, care has been

Conclusions

Performance of the Chaboche model available in ANSYS in predicting ratchetting has been discussed in this paper. The results have been compared with three sets of test-data, viz. (a) uniaxial cyclic tests; (b) three-point and four-point bending tests; (c) a shake table test on a pipe elbow. The Chaboche kinematic hardening model has been used to predict ratchetting under cyclic loading with nonzero mean load. The parameters of the Chaboche model have been calculated from the stable hysteresis

Acknowledgements

The work presented here has been sponsored by the Board of Research in Nuclear Sciences, India (BRNS Grant No. 99KA012). The authors acknowledge with thanks help rendered by Dr Satish Kumar (IIT Madras), Mr M.A. Khan, Mr S.N. Bodele and Mr K.K. Bavu, Bhabha Atomic Research Center (BARC) in conduct of the biaxial pipe test. The authors are grateful to Mr N Gopalkrishanan and other staff at Structural Engineering Research Centre, Chennai for their help in the conduct of the shake table

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