Modeling fatigue crack growth for a through thickness crack: An out-of-plane constraint-based approach considering thickness effect

Highlights

• Empirical formulas for constraint factor distributions are obtained.

• Crack stress state level was evaluated quantitatively by a proposed characteristic parameter.

• A novel cracks growth model considering the thickness effect is built.

• The predicted results have a great agreement with the corresponding experimental results.

Abstract

Though several studies have tried to explain and model the effect of thickness on crack growth behavior, this remains a controversial topic. The present paper proposes an approach to model the fatigue crack growth with different thicknesses, which requires crack growth experimental data with different specimen thicknesses and out-of-plane constraint distribution relations for CT specimens. Empirical formula, for the distributions of two types of constraint factors T_z and T₃₃, were fitted according to the FEA results. The stress state variation induced by the thickness was quantified using the formulas; then, the crack driving force was modified to establish the novel crack growth model. The predictions using this model largely agree with the experimental data. So, the newly developed model was verified. This paper proposes a new perspective on quantifying the effect of crack thickness, and provides an alternative to the traditional method of modeling the crack growth behavior.

Introduction

The prediction of crack growth is critical in the damage tolerance design of engineering components. Varying thickness design in engineering structures is applied more and more widely, such as turbine blades and discs with air-cooling techniques. For these complex geometrical configurations, the thickness effect on crack growth requires to be evaluated to seek for a more accurate and reliable prediction. Several studies have examined this effect with contradictory results. Several scholars [1], [2], [3] proposed that thickness increases the crack growth rate, while Kim et al. [4] stated that thickness has a negligible effect on it. Roughly associating specimen thickness (e.g., the crack thickness for a through crack) with crack growth rate can lead to ambiguous conclusions. Therefore, a more comprehensive explanation of the mechanism behind thickness is needed to develop an accurate and reliable crack growth model. Yu et al. [5] proposed that the microcosmic mechanism about the thickness effect on nickel-base superalloy's creep and crack behavior was associated with the depth of γ’ depleted region. Many studies [6], [7], [8], [9] have similarly suggested that the stress state associated with the thickness has a significant effect on crack propagation. However, the traditional crack propagation theory framework is limited when considering the high stress triaxiality close to the crack, as a quantitative description of the relationship between the stress state and thickness is lacking.

Conventional quantitative models mainly employ the crack closure mechanism to explain the effect of thickness on fatigue crack growth by introducing the parameter of thickness B into the crack closure correction factor U. The non-dimensional load ratio parameter U is defined in Eq. (1), often used to represent the effects of the R ratio (R = F_min/F_max) on crack closure:

$$U=\Delta K_{eff}/\Delta K$$

where ΔK_eff is the effective stress intensity factor (SIF) range and ΔK (= K_max - K_min) is the traditional stress intensity factor range. Novelman et al. [10] employed the parameter α, a mathematic parameter based on fitting the experimental data of crack growth with different thicknesses, in his crack closure formula to describe the thickness effect. As this model can conveniently be applied in fracture mechanics analyses, further studies [11], [12], [13], [14] used it and built on it. Roychowdhury and Dodds [15] and Kotousov et al. [6] proposed another type of model that introduces the governing parameter $\eta =K_{max}/{\sigma }_f\sqrt{B}$ into the crack closure correction factor formula U = f(η, R). However, all these crack growth models rely on the concept of crack closure. Although thickness affects crack closure, these two effects are not synonymous. When the closure effect is weak, such as under a high stress ratio R, these crack growth models cannot invalid to account for the thickness effect.

Another common method to explain the thickness effect involves to examining the size of the plastic zone around the crack tip [6,[16], [17], [18]]. Variation of the crack growth rate is attributed to change in the plastic zone size, induced by thickness variation. Plane stress or plane strain is assumed to calculate the plastic zone size [17]. However, the stress state of an actual crack in engineering structures cannot be simply determined. Moreover, it is difficult to detect the plastic zone of an actual crack, and there are no practical formulas to apply the size of the plastic zone in crack growth models. Therefore, this method is generally limited to qualitative or theoretical analyses.

Current theories rely on the assumption of plane stress or plane strain condition. However, actual cracks in engineering structures, even simple straight-through ones, are always under a triaxial stress state, as illustrated by the phenomenon of a straight crack front tending to become a bowed curve during crack propagation [19]. The presence of cracks introduces complex triaxial stress states in the region surrounding the crack tips, and the stress state varies from the surface to the internal region in the transverse direction [20]. The region close to the surface is less constrained, so the plane stress state dominates, while the internal region is highly constrained by both sides, so the plane strain state dominates. The stress state transition is highly affected by crack thickness [20,21]. Therefore, the stress state should be part of the crack growth model to reflect the mechanical principle of the thickness effect.

In the last decades, the crack constraint parameters were employed to characterize the effect of thickness on the fracture behavior. The constraint parameters have been introduced into the fracture toughness expressions for different thicknesses [22,23]. The influence of the constraint on the creep crack growth were evaluated [3,24,25]. Guo et al. [24] used the constraint factor T_z to define a new creep SIF. Xu et al. [25] proposed an in-plane constraint Q* to quantify the crack tip constraint level under creep conditions, and stated that the thickness has a significant effect on the crack constraint.

The present paper focuses on developing a methodology to propose a fatigue crack growth model that considers the thickness effect for through cracks. A characteristic parameter to quantify the stress state variation depending on thickness is defined based on the crack constraints, then, it was introduced into crack growth models to account for the thickness effect. The paper is organized as follows. Section 2 gave a brief background of two out-of-plane constraints used in this paper. Section 3 described the outline and working flow of the approach to illustrate the procedure of model development. In Section 4, three finite element (FE) models of compact tension (CT) specimens are set up according to an experiment. In Section 5, parameter identification of the formulas used in the approach are carried out. Section 6 is the discussion of the approach. In summary, the out-of-plane constraint factors are used to characterize the crack stress state, and a novel crack propagation model is set up by introducing the crack stress state level.