Gurson-based incremental damage in fatigue life estimate under proportional and non-proportional loading: Constant amplitude and low cycle regime applications

Highlights

• Gurson-based model in multiaxial fatigue.

• Tension/compression mechanism for damage evolution.

• Multiaxial fatigue life prediction in incremental damage.

• Proportional and non-proportional loading in constant amplitude.

• Shear mechanism in combined tension/compression proportional and non- proportional loading.

Abstract

This contribution proposes an incremental approach to fatigue life estimate of ductile materials subjected to low cycle fatigue. In this approach, the life prediction is based on the evolution of the internal damage variable of a proposed extension for the Gurson model capable of describing the gradual degradation of the material in function of the number of loading cycles. The extension of the model consists in the inclusion of the Armstrong-Frederick kinematic hardening law [4] and the reformulation of the damage evolution law proposed by Gurson [22], assuming the Nahshon and Hutchinson shear mechanism [43] and a term based on the normalized third invariant with the role to promote asymmetric damage growth under axial (tension–compression) cyclic conditions. An implicit integration scheme is developed, and numerical analyzes are done through simulations at one Gauss point. In this context, the incremental damage approach has applied to experimental data extracted from the literature for thin-walled tubular specimens made of two different materials, S460N structural and hot-rolled normalized SAE 1045 steels. Finally, the predictive capacity of the proposed approach is evaluated comparing predicted and experimentally observed results considering six different proportional and non-proportional loading paths.

Introduction

The traditional approach for multiaxial fatigue analysis is based on four steps: a method for obtaining the cyclic stress–strain response present in the component, a method for identifying and counting cycles, a fatigue damage parameter for measuring degradation of material in each cycle and a damage accumulation law. However, the strong dependence of the damage parameter on the experimental data used in its calibration results in some limitations for the application of this type of approach. Socie and Marquis [54] state that the different failure modes, considered in the elaboration of the damage parameters, make that there is not a good correlation between most experimental data, restricting their applications. Another problem with this approach is related to the cycle count method and the law of damage accumulation under complex loading conditions. Often under multiaxial conditions, the task of identifying cycles is not so simple and the use of linear damage accumulation laws such as Palmgren-Miner [42] may not represent the actual behavior of fatigue damage in the material. Alternatively, the use of incremental damage mechanics in multiaxial fatigue analysis for ductile materials has gained strength in recent decades. Desmorat et al. [15] argue that the use of a damage evolution law as an integrated part of the material behavior allows the quantification of material degradation by an internal variable to the constitutive model, eliminating the need for arbitrary methods for cycle counting and accumulation of damage.

Following an incremental methodology, in this type of fatigue analysis, the material degradation process is described by the evolution of the internal damage variable coupled to the constitutive model. At each loading cycle, the model calculates the stress state in the material and the growth of the damage variable. This process repeats until the damage reaches a critical value. Thus, the number of simulated cycles until the moment is assumed as the fatigue life of the material.

The constitutive models with coupled damage variable can be divided into two main groups according to the methodology adopted to define its damage variables. Micromechanical defect-based models rely on nucleation, growth, and coalescence of defects contained in a Representative Elementary Volume (REV) to calculate the degradation level of a material. The best-known model of this group is the Gurson model [22], along with its phenomenological extensions proposed by Chu and Needleman [12] and, Tvergaard and Needleman [56] that resulted in the GTN model. In models based on continuous damage mechanics (CDM), the effects of damage are explained by a proposed internal variable based on the fundamentals of thermodynamics that reduces the stiffness of the material. Within this group, it is important mentioning the models proposed by Krajcinovic [25], Lemaitre [30], Lemaitre and Chaboche [32] and Rousselier [51].

CDM-based damage models are already widely used in fatigue analysis and are applied to both low cycle fatigue life prediction by the approaches proposed by Wang and Lou [57], Chow and Wei [11], Bonora and Nemaz [6], Lopes and Malcher [35] and, Castro and Bemfica [9], regarding high cycle fatigue as presented in the works proposed by Lemaitre et al. [33], Lemaitre and Desmorat [31] and, Araújo et al. [3]. However, the application of models based on micromechanics of defects in cyclic conditions is still little explored.

The first studies on the behavior of micro defects in ductile materials subjected to cyclic loading were performed by Gilles et al. [21]. By finite element simulations of a cylindrical unit cell with a central defect represented by spherical void, Gilles et al. [21] observed that the accumulation of plastic strain near the surface of the defect caused a distortion of its shape, resulting in a gradual increase of its volume with each cycle. This phenomenon was then defined as ratcheting of porosity. Nevertheless, Deuvax et al. [16] remade the simulations of Gilles et al. [21] using a better triaxiality control to confirm definitely the existence of the phenomenon. They also showed that due to the symmetrical character presented by the yield function proposed by Gurson [22], the model predicts a “negative growth” of the damage variable under negative hydrostatic pressure conditions. This makes the Gurson model [22] unable to describe the evolution of ductile damage under cyclic conditions since the growth of damage accounted for during the tension step is accompanied by a similar recovery during the compression step.

As a first alternative, Deuvax et al. [16] suggested the use of the Leblond-Perrin-Devaux (LPD) model proposed by Leblon et al. [28] as an extension of the Gurson model [22] with two internal variables to describe the hardening effects. However, the LPD model (1995) was able to describe the process of accumulation of ductile damage only for a small number of cycles, as evidenced by Brocks et al. [8] and Steglich et al. [55]. Subsequently, many other studies were performed, based on micromechanical simulations of unit cells following the research approach of Gilles et al. [21] and Deuvax et al. [16]. Among them, it is worth mentioning those made by Kuna and Ross [26], Rabold and Kuna [48], Mbiakop et al. [38] and Nielsen et al. [45]. Although they do not propose new models or improvements for the LPD model, Leblon et al. [28], these studies were of great importance in providing a clearer understanding of the micromechanics of defects under cyclic conditions. These studies served as a basis for more recent works, allowing improvements such as those proposed by Klingbeil et al. [24] and Lacroix et al. [27].

Following the approach of Leblon et al. [28], Klingbeil et al. [24] proposed an extension of the GTN model (1984) to finite strain and hyper elasticity able to consider the effects of nonlinear combined hardening (isotropic and kinematic) in describing ductile failure in materials subjected to cyclic loading. However, the focus of the work was the study of crack propagation in specimens of fracture mechanics. In turn, Lacroix et al. [27] proposed the calculation of the two internal variables introduced by Leblon et al. [28] through numerical integration, allowing a better correlation between the results obtained by the model and micromechanical simulations. However, in addition to focusing on simulating a small and moderate number of cycles, they did not consider the effects of kinematic hardening on material behavior.

Furthermore, in a slightly different context, Pirondi et al. [47], Meschke and Hommel [41], Lukhi et al. [36] and Sandoval et al. [52] observed in the phenomenon of cyclic defect growth a powerful tool for predicting the life of metal structures subjected to low cycle fatigue. Using the LPD model, Leblon et al. [28], associated with a strong defect nucleation law, Pirondi et al. [47] simulated in finite elements the macroscopic response and evolution of local damage in notched specimens submitted to cyclic plasticity, and the predicted lives showed good agreement with the experimental lives. In contrast, the model always predicted the initialization of the failure in the center of the specimen, not agreeing with the experimental observations.

Meschke and Hommel [41] proposed a hybrid strategy that combines microscopic analysis using unit cell models with the LPD, Leblon et al. [28], model to predict the life of metal structures subjected to ultra-low cycle fatigue through the growth of damage over the cycles. Then, the results from the unit cell analysis were used to calibrate the model parameters, then validated through fatigue tests on notched specimens. Finally, in order to demonstrate the applicability of the proposed strategy to real structural problems, it was applied by Meschke and Hommel [41] to evaluate a spherical pressure vessel subjected to cyclic loads simulating an earthquake. Recently, Lukhi et al. [36] modeled the low cycle fatigue mechanism in a nodular cast iron matrix from finite element simulations of a unit cell model with a central defect. Lukhi et al. [36] correlated the defect growth at each cycle with the fatigue damage of the material, allowing life predictions following the experimental data. In turn, Sandoval et al. [52] suggested an extension of the Gurson model for cyclic application, regarding the prediction of fretting fatigue life and the study of the influence of the plastic strain field on fretting fatigue for an aluminum alloy.

This paper aims to propose an incremental approach to fatigue life estimation for ductile materials based on the evolution of the internal damage variable of an extension of the Gurson model able to describe the cyclic degradation of the material. The paper is organized as follows:

• Section 2 presents the original Gurson model [22] model and the extension proposed with the role to expand its application to cyclic multiaxial conditions.

• Section 3 presents the calibration process of material and damage parameters used by the proposed approach to analysis of low cycle fatigue through an incremental approach. The experimental data used were extracted from the literature and refer to fatigue tests performed on tubular specimens made with S460N and SAE 1045 steels.

• Section 4 presents a comparison between the results obtained by the proposed approach and those observed experimentally.

• Finally, Section 5 provides a summary of the work done and some considerations about the results obtained.