A strain energy density based life prediction model for notched components in low cycle fatigue regime

Highlights

• A modified plastic strain energy density (PSED)-based model is proposed.

• Fatigue properties under both stress and strain cycling are considered.

• Applicability of the proposed model to low cycle fatigue is verified.

Abstract

Fatigue life prediction of engineering components is essential for safe operation of mechanical systems. Energy based models have attracted much attention due to the combined considerations of stress and strain behavior, especially in low cycle fatigue regime. In this work, a new plastic strain energy density (PSED) based life prediction model for notched components subjected to cyclic loading is proposed, in which cyclic stress-strain responses under stress- and strain-controlled modes are both introduced. Experimental tests on 304 stainless steel smooth and notched plate/bar components are conducted for calibrations and verifications. The prediction capability of the proposed model is verified by experimental results.

Introduction

Many components with geometrical discontinuities, like notches, face low cycle fatigue (LCF) loadings during the service process induced by start-up and shutdown conditions [1,2]. Therefore, LCF life prediction is important for the structural design and the reliability assessment of the component. In the numerical analysis of notch fatigue, precise cyclic responses of the material are essential. Traditionally, cyclic responses with obvious plastic deformation are depicted by the strain-controlled tests in the LCF regime, whereas the high cycle fatigue (HCF) behaviors are characterized by the stress-controlled tests. In the LCF regime, engineering components experience complex stress and strain responses, which are dependent on the local geometrical configuration, loading mode, and so on [3,4]. As shown in Fig. 1, the stress-strain state around the notch root (see Fig. 1b) is quite different than those experienced in the case of stress- (see Fig. 1a) or strain-controlled modes (see Fig. 1c) in smooth specimen, and it seems to be a combination of stress- and strain-controlled modes. Generally, ratchetting (or mean stress relaxation) would occur under stress (or strain) cycling. Even under fully reversed loading condition, a great difference in cyclic plastic deformation response between strain- and stress-controlled tests has been observed in some materials [5,6]. Those studies indicate that the influence of loading mode may depend on the cyclic softening/hardening feature [7]. Therefore, to achieve the accurate fatigue life prediction of the components, especially for materials exhibiting cyclic softening/hardening behaviors, cyclic responses under various control modes should be included in the life model.

In general, fatigue life models of smooth and notched specimens can be divided into three categories: stress-based model, strain-based model and energy-based model. With respect to the stress-based model, various analysis strategies are used, such as the nominal stress model, stress field intensity (SFI) model [[8], [9], [10], [11], [12], [13]], theory of critical plane [[14], [15], [16]] and theory of critical distance (TCD) [17,18]. Those models are based on a simple stress variable or combinations of several stress variables. Meanwhile, the effective stress variable for life prediction is related to the structural response of one single node or the local region. When the strain-based model for fatigue life prediction of components is mentioned, similar analysis strategies are also observed. One example of this type is local strain approach, whose basic assumption is that both smooth and notched specimens have the identical lives if the strain histories at the initiation sites are the same. Then, the fatigue life of the notched components can be calculated by consulting the classical Coffin-Manson law in the LCF regime:

$$\Delta {\epsilon }_p{\left(N_f\right)}^{\beta }=C$$

Different from stress- or strain-based models, energy-based models have been paid much attention because of the simultaneous consideration of stress and strain behaviors, which could achieve a more general and wider description of fatigue behavior, especially under multiaxial stress states. The concept of the energy-based model could be traced to the work of Neuber [19], Morrow [20] et al. The energy-based criteria for fatigue analysis can be mainly divided into three groups in terms of the damage behavior [21]: elastic strain energy density (ESED) based criteria for HCF, plastic strain energy density (PSED) based criteria for LCF [[22], [23], [24], [25]] and total strain energy density (TSED) based criteria [26,27] for both LCF and HCF. Among them, some approaches show the ability to account for the effect of mean stress [24,25,27] and those taking into account the strain energy density (SED) in the critical plane [[28], [29], [30]] seem to be promising especially for multiaxial fatigue. Furthermore, there are no single universal criteria for different loading conditions.

With respect to the SED-based method for notch fatigue analysis, the main analysis routines can be classified into two categories. In view of the great influence of mean stress, stress/strain gradient and size effect in stress concentration regions ahead of notch root, the first category is based on the suggestion that the value integrated in a critical volume and not simply a point near the notch root seems to be a reasonable choice for fatigue life predictions, which can be classified as volume-based energy criteria. Various volume based energy methods have been developed based on different theories dealing with the shape and size of the critical zone, such as the approaches based on TCD, finite-volume-energy based approach based on notch stress intensity factor (NSIF), energy field intensity approach based on SFI theory [31]. A “critical volume approach” has been employed for life predications of notched components within HCF [32] and LCF [33], whose basis is that the fatigue failure necessitates a physical volume (i.e., the effective volume). The effective SED range is an average value in the effective volume. This volume is assumed to be cylindrical and its diameter seems to be determined based on TCD. A general procedure combining a concept of strain energy gradient with TCD has been established for LCF life prediction of notched specimens [34], which relates the fatigue life with the distribution of the strain energy within an effective critical zone. Note that the size of the critical volume based on TCD is material dependent. A “finite-volume-energy approach” based on notch stress intensity factor (NSIF) has been proposed and applied to sharp V-notches [35], and then extended to blunt notches [[36], [37], [38], [39]]. Different from the volume energy approach based on TCD, the radius of control volume is related to material, geometry, loading condition, temperature and so on [40].

Considering the practicability and simplicity of model in engineering application, the second category uses the stress-strain behavior at critical point around the notch for fatigue life assessment, which can be classified as point-based energy criteria. In general, conservative results would be expected when the value at the notch root is directly employed for fatigue life prediction. It needs to be modified by bringing into some relevantly physical quantities that consider the influence of stress concentration regions. Through introducing stress concentration factor K_t [41], a modified relationship between TSED and fatigue life can be expressed as:

$$\Delta W_t{\left(K_t\right)}^{{\beta }_1}=A_1{\left(N_f\right)}^{{\alpha }_1}$$

There is another approach for rapid assessment of fatigue life in terms of energy-based criteria [42,43], which suggests that both smooth and notched specimens have the identical lives if the value of TSED at the initiation sites are same. Then the notch fatigue life can be calculated by consulting a master curve that relates the TSED to the fatigue life of smooth specimens. Note that the value of TSED at the initiation site is calculated on the basis of the line method of TCD. Compared with volume-based energy approach, the point or line-based approach achieves a balanced relationship between computational amount and prediction accuracy, and seems to be more applicable for engineering practice.

In view of the exclusive merits of energy criteria, in this work, a modified PSED model in the frame of point-based energy approach is proposed for rapid assessment of fatigue life in notched components, in which the SED at the notch root is corrected by an empirical formula. Particularly, both the stress and the strain-controlled fatigue data are employed for the calibration of the proposed model. Meanwhile, experimental tests on notched components are carried out for validation of the proposed life model.

The structure of this paper would be arranged as follows. First, Section 2 provides the modified PSED life model in the LCF regime. Experimental procedures are descripted in Section 3. Full reversed LCF experiments under both stress- and strain-controlled modes are conducted on smooth specimens, and asymmetric and symmetric stress-controlled fatigue tests are conducted on notched plate and bar specimens. Numerical strategy for notched components under stress-controlled mode is given in Section 4. Then, results and discussion are provided in Section 5, including parameter calibration and verifications of the proposed model. Finally, concluding remarks are provided in Section 6.