Effect of specimen thickness on fatigue crack shape evolution for aluminum alloy

The investigation of fatigue crack propagation and fracture mechanisms in high-thickness metal plates holds pivotal importance in the advancement of a three-dimensional damage tolerance assessment methodology. This research also bears significant implications for ensuring the safe operation of large-scale mechanical equipment. In this study, we examine the influence of specimen thickness on the evolution of fatigue crack shapes in an aluminum alloy. Fatigue crack growth tests were conducted on single-edge notch tension specimens to explore the impact of specimen thickness on the shape of fatigue crack growth. Additionally, we propose a calculation model for crack growth shape based on an energy model in this paper. Both experimental and analytical findings indicate that specimen thickness does indeed exert an influence on fatigue crack shape. Specifically, as the specimen thickness increases, the crack shape transitions from resembling a “fingernail” to a “saddle” shape.


Introduction
Heavy equipment, commonly employed across various industries including machinery, transportation, and energy, often boasts a lengthy service history.Ensuring the dependable estimation of the service life of structural components prone to cracking is a highly effective means to enhance equipment utilization and the management of potential failure incidents.Premature retirement of equipment leads to substantial wastage and economic losses, while continued operation necessitates sound life prediction methodologies to ensure safety.
As the allocation and investment in metal materials have witnessed a notable upsurge, particularly in significant load-bearing components of large-scale equipment like high-speed railways and ocean-going vessels, a pressing question arises: how can the safe operation of these mammoth machinery and equipment throughout their service life be guaranteed?
In recent years, the evolution of fatigue crack shapes has emerged as a topic of considerable interest.Numerous research studies have been undertaken to investigate crack shapes [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], and the collective findings suggest that finite thickness specimens exhibit a phenomenon known as the "tunneling effect."Specifically, in specimens with an initially straight crack that eventually penetrates, the central crack front advances ahead of all others as the loading cycles progress.Subsequent crack fronts follow suit, resulting in the formation of a distinctive "fingernail" shape at the crack tip, a phenomenon termed the tunneling effect (see Figure 1).However, it is noteworthy that the existing body of research has primarily focused on specimens with a thickness of less than 20 mm, leaving a gap in our understanding of how thickness, particularly exceeding 20 mm, influences crack shape evolution.This paper seeks to address this knowledge gap by examining the impact of specimen thickness on the evolution of fatigue crack shapes in 7050 aluminum alloy.

Materials and specimen
Specimens were meticulously crafted from alloy 7050-T7451, and the selected specimen type for the study was the single edge notch tension (SENT) specimen, as illustrated in Figure 2.These specimens commenced with an initial crack length of 5 mm.To investigate the influence of specimen thickness, two distinct thicknesses were utilized: 20 mm and 60 mm.Notably, each thickness category underwent testing with three individual specimens, ensuring robust and representative data collection.

Experimental procedure
Fatigue crack propagation tests were conducted using the MTS 100kN machine, maintaining a controlled environment at room temperature with a relative humidity ranging between 55% and 65%.The testing regimen adhered to a stress ratio (R) of 0.06. Figure 3 illustrates the specific loading mode employed for the fatigue load during these experiments.

Experimental results and discussions
Figure 4 depicts the fracture morphology observed in specimens with thicknesses of 20 mm and 60 mm.It is noteworthy that in the case of the 20 mm thickness specimen, the fracture morphology at the crack tip exhibits a distinctive "fingernail" shape.Here, the central crack front takes precedence in growth, followed by the progression of other crack fronts, resulting in the characteristic tunneling effect.Conversely, the 60 mm thickness specimen displays a fracture morphology at the crack tip resembling a "saddle" shape, notably devoid of the tunneling effect.This particular crack shape is referred to as the reverse tunneling effect.
The test results unequivocally indicate a noteworthy transformation in crack shape as the specimen thickness increases, transitioning from the "fingernail" shape to the "saddle" shape.In order to gain deeper insights into the fatigue crack growth shapes exhibited by specimens of varying thicknesses, we employed a scanning electron microscope (SEM) to examine the fracture morphology.The specific location of the SEM analysis is indicated in Figure 4 (in the "A area").The SEM findings are visually presented in Figure 5.As depicted in Figure 5, clear and discernible fatigue striations are evident in specimens of varying thicknesses.This observation suggests that the crack tip experiences an opening mode plain strain state.In particular, the fatigue striations in the 20 mm thickness specimen exhibit a well-defined and smooth pattern, with the normal direction nearly perpendicular to the initial crack.Conversely, in the case of the 60 mm thickness specimen, the normal direction of the fatigue striations forms an angle of approximately 115° relative to the initial crack.This discrepancy in the orientation of the fatigue striations implies that, upon close examination of these features, the fatigue crack growth rate in the 20 mm thickness specimen is discernibly faster than that in the 60 mm thickness specimen.

Calculation model of crack growth shape
The correlation between the energy release rate (G) and the stress intensity factor (K) for a threedimensional (3-D) stress state has been a subject of extensive research over the years.The effective energy release rate (Geff) can be mathematically expressed as per reference [19].
2 ( 1) where ΔKeff is the effective stress intensity factor amplitude, E is the elastic modulus, υ is the Poisson's ratio and Tz is the tri-axial stress constraint.
Based on the finite element analysis, the tri-axial stress constraint Tz for through crack is given [20] 0.58 where p r is the average plastic zone size when θ=0°, B is the thickness of specimen and n is the Ramberg-Osgood hardening index, n=33 for aluminum alloy.
The effective stress intensity factor amplitude eff K △ can be calculated as follows max ( 1) where U is the crack opening ratio, Rs is the stress ratio, σmax is the maximal stress, σy is the yield strength, σu is the ultimate strength and α is the constraint factor.
If the thickness effect is not taken into account, then the expression of stress intensity factor is [21] 1 2 ( sec ) 2 a Ka W   = (10) where W is the width of specimen.
The crack tip near free-surface is in the stress state for the 3-D through crack [22], the constraint factor α is 1, so the crack opening ratio Us for crack tip near free-surface can be calculate by Eqs.( 4)- (10).For the through crack, the peak value Ksmax0 of stress intensity factor for crack tip near free-surface can be calculated by Eq. (10).After thinking about the curve shape of through crack front, the stress intensity factor Ksmax for crack tip near free-surface can be thought as max max 0 s s K CK = (11) where C is the constant.Therefore, the effective stress intensity factor for crack tip near free-surface is max 0 (1 ) For the plane stress state, the tri-axial stress constraint Tz is 0, so the effective energy release rate Geffs for crack tip near free-surface is The effective energy release rate is equal to the crack growth resistance R when the crack grows steadily, that is The crack growth resistance R is the material constant, it can be thought that the effective energy release rate for every point of crack front is the same value when the crack grows steadily.The effective energy release rate Geffi for the i point of crack tip can be calculated by Eq.( 1)-( 10), and Geffi is equal to Geffs, so Furthermore The stress intensity factor at the crack tip can be computed using finite element analysis software, such as FRANC3D.To determine the appropriate value of C, a trial-and-iteration method is employed.The objective is to adjust the value of C until the calculated ratio of the stress intensity factor at the crack tip to C equals value of Kimax/C as calculated by Eq. (17).Once this equilibrium is achieved, the crack shape can be accurately calculated.

Crack growth shape analysis
The three-dimensional model of a single-edge cracked plate is illustrated in Figure 6.In this representation, "a" represents the length of the through crack, "W" denotes the width of the specimen, "B" stands for the thickness of the specimen, and "σ" represents the tensile stress.The three-dimensional growth shapes of through cracks in single-edge crack plates with varying thicknesses can be determined using Eq.(17).The resulting crack growth shapes for plates of different thicknesses are visually depicted in Figure 7.The crack shape calculated using Eq. ( 18) closely aligns with the observed crack shape from the fatigue test, as depicted in Figure 7.This congruence suggests the practical viability of the crack growth shape calculation model based on the energy model.Moreover, it substantiates the occurrence of the reverse tunneling effect in crack growth shape as the specimen thickness increases.

Conclusions
(a) The morphology of the crack front undergoes changes with increasing specimen thickness, gradually diminishing the "tunnel effect."When the specimen thickness reaches approximately 40 mm, the crack front's morphology gradually exhibits a "saddle shape." (b) The practicality of the crack growth shape calculation model based on the energy model is substantiated.
(c) Fatigue striations in the 20 mm thickness specimen appear well-proportioned and smooth, with the normal direction nearly perpendicular to the initial crack.In contrast, the normal direction of fatigue striations in the 60 mm thickness specimen forms an angle of approximately 115° with the initial crack.This observation suggests that the fatigue crack growth rate in the 20 mm thickness specimen is faster than that in the 60 mm thickness specimen, as discerned from the fatigue striations.

Figure 2 .
Figure 2. Geometry and dimensions of the specimens tested (mm).
(a) 20 mm thickness (b) 60 mm thickness Figure 5. SEM results for A area.

Figure 6 .
Figure 6.Three dimensional model of single edge crack plate.

Figure 7 .
Figure 7. Crack shape and Kimax/C.The crack shape calculated using Eq.(18) closely aligns with the observed crack shape from the