Simulations of isothermal and thermomechanical fatigue of a polycrystal made out of austenitic stainless steels and relation to the Coffin-Manson law

Highlights

• The lifetime is derived from that of the first crack reaching the boundary of the grain of its initiation.

• The calculated lifetime for isothermal fatigue is in agreement by less than 25%.

• The results exhibit a two-parameter power relationship similar to the Coffin-Manson law.

• Its calculated coefficients for five steels are in agreement.

• The Coffin-Manson law and kinetic equation of the crack are not rigorously connected.

Abstract

Based on an experimental result of the literature showing that the crack remains in the grain of its initiation up to about 20% of the lifetime at low cycle fatigue of austenitic stainless steels, the lifetime of a polycrystal made out of these steels undergoing uniaxial isothermal or thermomechanical fatigue from 30 to 340 °C at constant total strain amplitude is calculated. The stresses and strains in the grains and polycrystal are determined in term of mean field with the Hill-Hutchinson model. Dipolar slip markings in the grains are predicted and assumed sites of initiation and propagation of the cracks calculated in terms of critical stress and of shear plastic strain, depth and grain boundary, respectively. There is agreement of the macroscopic stress and plastic strain, but at the residual stress, making the modelling not suitable for thermomechanical fatigue. For isothermal fatigue, the lifetime is in agreement. The lifetime - plastic strain power relationships are addressed to the Coffin-Manson law assumed to derive from the kinetic equation of the crack in a dimensional approach. The calculated constants of the power relationship of five austenitic steels of the literature are in qualitative agreement with those measured of the law. The relationships between the constants of the power relationship, law and kinetic equation are determined. The results and modelling are discussed.

Introduction

Certain components undergo cycling loadings such as isothermal fatigue or thermomechanical fatigue resulting in the generation of cracks. It is essential for the integrity of the component and design of the structure to determine the lifetime of the component. The lifetime will be defined by the number of cycles at the failure of the specimen, $N_f$. The studies referred in this work consider other definitions such as the sudden decrease at a given percentage of the maximum stress of the stress – cycle number plot, i.e. 5% [1], [2], 1% [3] or the start of the drop of the effective modulus [4]. All these definitions will be assumed as equivalent here.

The present work was proposed as a theoretical support to experimental investigations of thermomechanical and isothermal fatigue of austenitic stainless steels used in the nuclear industry and conducted at PSI [1], [2]. The experiments dealt with the generation of cracks observed in components of the primary cooling circuit of light water reactors undergoing transients from 100 to 340 °C, temperatures for which the damage by creep – fatigue interaction and oxidation can be neglected. These steels, of the 316 and 347 types, were uniaxially loaded under constant total strain amplitude, $E_a$. The goal of the experiments was, given $E_a$, to determine $N_f$ and to address its value to that predicted by models of the literature. This value was that of isothermal fatigue given by an equation of the literature available from room temperature up to 400 °C. Results of interest for this work include, given $E_a$, the similar forms of the stress response for the two types of fatigue, at the generation of a residual stress for the thermomechanical tests that affects $N_f$ according to the phase of the cycle [1], [2], and the agreement of the predicted and measured lifetimes with the Coffin-Manson law [2].

The goal of the present study will be the calculation of $N_f$, the comparison of the predicted value, $N_{\mathit{fp}}$, with experimental data and its connection or not to the Coffin-Manson law. The modelling will consider the crystallographic character of the deformation of a metal, which determines also the damage by cracking at stage I of fatigue up to 80–90% of the lifetime for the steels considered here, for example [3]. Relatively simple and validated crystallographic models exist in the literature to calculate the stress and strain fields at the levels of the grain and of the polycrystal through a scale transition and for the initiation and propagation of the crack at the level of the grain. This approach is in the present trend of the literature that promotes physics-based models with finite element schemes [5], [6], [7] rather than constitutive models that include numerous parameters determined by specific tests and result in errors over about one order of magnitude for the prediction of $N_f$ [6], [7], [8].

Magnin et al. report for four different f.c.c. and b.c.c. single-phased materials uniaxially fatigued at the constant plastic strain amplitude of 0.2% at room temperature the stay of the cracks in the grains of their initiation up to about $0.2N_f$; their propagation is transgranular [3]. The materials of interest include the 316L steel and later copper here. It should be stressed that the physical origin of this homologous behaviour for $N_f$ and its degree of generalization were let open. As far as we know, this still remains available. Further experimental investigations of the same laboratory have showed that this behaviour can be extended to two other materials exhibiting a crack propagation of the intergranular or intergranular-transgranular types for values of the plastic strain amplitude from 0.04% to 0.4% [9].

It will be assumed that these observations remain available for the present materials and loading conditions, suggested by Fig. 1, and that the crack is blocked at the boundary of the grain of its initiation at $0.2N_f$. Consequently, the propagation of the crack will be considered only across the grain of its initiation and $N_{\mathit{fp}}$ will directly result from the value of $0.2N_{\mathit{fp}}$. This eliminates from the modelling the crossing of the crack of the grain boundary that is the main obstacle to its propagation and results in a strong reduction of the computing time. A second assumption will be the same mechanisms for the initiation and propagation of the crack at isothermal or thermomechanical fatigue, supported by the above-mentioned experimental results and by observations [10], [11].

The work will be presented as follows. Section 2 will be devoted to the modelling: conception, plasticity of the grains and polycrystal through a scale transition, initiation and propagation of the crack, calculation of $N_{\mathit{fp}}$. Section 3 will define the variables of interest of this work and will check the modelling. It will be stated that the modelling is not suitable for thermomechanical fatigue. Unless otherwise specified, the other sections will concern isothermal fatigue. Section 4 will present mainly the calculated network of the cracks, a description of their initiation and propagation and $N_{\mathit{fp}}$ at isothermal fatigue for four temperatures addressed to an equation of the literature and at thermomechanical fatigue addressed to measurements. Section 5 will deal with the power relationships found between $N_{\mathit{fp}}$ and the macroscopic and microscopic plastic strains in term of the Coffin-Manson law. Cycle $N_{\mathit{fp}}$ and the calculated constants of the power relationships obtained with five austenitic stainless steels of the literature will be addressed to those measured of the law. The power relationships will be theoretically investigated at the microscopic and macroscopic levels for identification or not to the Coffin-Manson law assumed to derive from the kinetic equation of the crack through their respective constants in a dimensional approach. The constants of the power relationships will be discussed in this sense. Section 6 will discuss the modelling for further applications.