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Experimental Investigation of Thickness Effects on Mixed-mode I/II Fracture of an Aluminum Alloy

Experimental investigations on the effect of thicknesses on mixed-mode I/II fracture are performed with an aluminum alloy with thicknesses of 2, 4, 8 and 14 mm. It is found that under pure-mode I loading condition, the loading capacity per unit thickness monotonically decreases with increasing thickness. However, under mixed-mode loading condition, the dependence of loading capacity on the thickness becomes very weak. Although the measured crack initiation angles from the mid-plane of the fractured specimens coincide well with typical existed criteria in the thin and thick specimens under all the mixed-mode conditions except pure-mode II, exceptional low crack initiation angles are found in the 8mm-thick specimens. Although ductile dimples are the dominated microscopic features in all the specimens, the size of the final voids varies significantly with specimen thickness and loading mode. The ductile dimples in the crack initiation zone become larger and shallower as the specimen thickness decreasing.


Introduction
Fracture mechanics has long been playing an important role in integrity assessment of engineering structures, and experimental data of materials obtained by using standard or simple specimens in laboratory have been proven to be the necessary fundament. However, the loading condition, stress states and geometry of the specimens in laboratory are far from that of the structural components in engineering. Most of the cracks in engineering structures usually present three-dimensional (3D) mixed-mode in spite of lots of studies focused on the single mode crack. In the last decade, 3D fracture theory [1][2][3][4][5][6][7][8] along with constraint theories [9][10][11][12][13][14] were developed to explain the discrepancy in fracture toughness caused by changes in geometry and stress state. As the most typical 3D problem, the effect of thickness on fracture has been intensively studied theoretically, numerically and experimentally [15][16][17][18]. It is shown that thickness has strong influence on the crack-tip fields as well as the fracture toughness of materials. However, most of the 3D studies aforementioned are confined to pure mode I crack. In fact, crack tunneling, curving and slanting are inevitable during crack growth even in an initially pure mode I cracked body [19,20]. Therefore, 3D mixed-mode fracture is much more important in engineering and need extensive study.
Since the maximum tangential stress (MTS) criterion was proposed by Erdogan and Sih [21] for mixed-mode crack initiation, mixed-mode fracture has long been studied and many mixed-mode fracture criterions have been put forward including the minimum strain energy density criterion [22], the maximum stress triaxiality criterion [23], and the crack-tip opening displacement (CTOD) based fracture criterion [24].
However, the limited available experimental data exhibits large scatter, and does not support a certain criterion for different loading conditions [25,26]. The most important potential reason may be the influences of specimen thickness and geometry used in different experiments. It is deserving that the constraint effect on mixed-mode fracture has been taken into account in some studies [27][28][29][30][31]. Recently Khan and Khraisheh [32] proposed a new criterion for mixed-mode fracture initiation based on the crack-tip plastic core region. The proposed criterion states that the crack extends in the direction of the local or global minimum of the plastic core region boundary depending on the resultant stress state at the crack-tip. We know that the stress state near the crack-tip depends on the thickness of the cracked plate, so it can be expected that there may be some relations between the thickness of specimen and crack initiation under mixed-mode loading conditions. However, to the best knowledge of the authors, no systematical experimental investigation has been conducted on the simplest 3D effect on any materials up to now.
For a better understanding of mixed-mode fracture, not only the corresponding 3D or constraint theories are highly needed, but also the most fundamental 3D phenomena have to be elucidated experimentally. The main aim of the present work is to investigate experimentally the effect of thickness on mixed-mode fracture of materials. Systematical experimental investigations are performed on mixed-mode fracture with compact-tension-shear (CTS) specimens of a LC4-CS aluminum alloy with combinations of four thicknesses from 2 to 4 mm and seven loading angles from pure mode I to pure mode II being considered. Exceptional thickness effects on some important features such as the load versus crack opening displacement curve, macroscopic and microscopic fractographs as well as the crack initiation direction are revealed the first time. Some existing criterions for mixed-mode crack initiation are discussed against the background of the test data and the problems needed further study are proposed.

Material and specimens
The material used was a high-strength aircraft structural aluminum alloy LC4-CS.
The typical chemical composition for the alloy is shown in Table 1. The yield stress and ultimate tensile stress of the material are 485 MPa and 510 MPa, respectively.
The CTS specimens were chosen for all fracture testing and fabricated in the T-L orientation. The specimens with four thicknesses of 2, 4, 8, and 14 mm are designed to reflect the variation in three-dimensional constraint near the crack-tip. The same in-plane geometry is used to all specimens to avoid change in the in-plane constraint.
All of the specimens were machined from a single 14 mm thick mother plate to avoid lot-to-lot variation. The microstructure of the material was found to be homogeneous along the thickness. For the 14 mm thick specimens, only final surface machining was required in the thickness direction. To produce specimens with thickness less than 14 mm, the outer surfaces of the 14 mm plate were machined by mechanical milling to reduce the plate to the desired thickness. During this thickness machining process, care was taken to alternate the material removal side of the plate to minimize possible out-of-plane warping. An initial 2 mm wide V-rooted notch in each CTS specimen was mechanically cut to depth of a n =25 mm, with notch depth to specimen width ratio of a n /w=0.5.
Before the fracture tests, the notched specimens were fatigue precracked under constant amplitude loading with a stress ratio slightly above zero. The final total length of the notch and the fatigue crack, or the initial crack length is about a 0~2 7 mm (a 0 /w is about 0.54). The detailed in-plane geometry of the CTS specimen is shown in Fig.1. In every specimen, the fatigue crack had a tunneling curved front and the initial crack length a 0 is taken as the average along the thickness in the way recommended by ASTM standard E399-83. As the fatigue precracking process was controlled by examining the fatigue crack length from the surfaces of the specimens, slight larger average a 0 /w is obtained for thicker specimens for the tunneling crack front in the interior.

Experimental fixture and procedures
The special loading fixture designed by Richard [33], shown in Fig.2, was employed. The loading angle φ is defined to be the angle between the initial crack direction and the direction of the applied loading. The appropriate selection of the loading holes in the loading fixture results in an entire range of mode I/II loading from pure mode I (φ=90°) to pure mode II (φ=0°). 6 All experiments were performed under laboratory condition with temperature of 17°C and humidity of 52% RH. Each specimen was loaded in a displacement-controlled 100 kN MTS 810 testing machine at a constant cross-head rate of 0.4 mm/min.
During the test, a COD clip-gauge was mounted across the notch at one side of the specimen for measurement of the crack mouth opening displacement (COD).
Traces of load versus COD were recorded automatically by the test system every 0.05 mm of displacement.
A zoom-field microscope, a CCD camera, and commercial hardware/software for facilitating the image monitoring and acquisition were used for crack initiation measurements. The CCD camera output was routed through the digital image acquisition system and into a computer for monitoring and storage. Figure 3 shows the crack-tip appearances of a 2 mm thick specimen under 60° loading angle during the fracture test process. Figure 3(a) shows the original precracked crack with initial length of a 0 . With increasing load, the crack becomes wider. When the load reaches a critical value, the original crack will initiate to grow, as shown in Fig.3(b). Certain crack-tip opening and slipping displacements can be observed, and a small crack-tip plastically deformed zone (the shiny zone) appears in front of the crack-tip in certain angle to the initial crack plane. As shown by Fig.3(c), near the maximum load level, notable crack growth can be observed and the crack-tip opening and slipping displacements become significant and the plastically deformed zone becomes larger.
The confined plastic zone ahead of the crack-tip and plastic wake left by the crack growth show that the ductility of the tested material is relatively week even in the thinnest specimens.
After fracture, all specimens were sectioned into two halves at the mid-thickness by the electric spark wire-cutting method. Crack initiation directions were measured with a universal bevel protractor at the middle plane as crack initiation always occurs first at the center of the specimen. Measurement at the middle plane can also avoid the influence of the plastic shear lips formed in the crack wake near the free surfaces of the specimens. Fracture surfaces of the fractured specimens were examined by means of macroscopic observation and a scanning electron microscope HITACHI S-2700.

Experimental results and analyses
3.1. Load-crack opening displacement curves

Effect of thickness on the load-crack opening displacement curves
The load-crack opening displacement (load-COD) curves of various thicknesses under each mixed-mode loading condition are drawn together to find out the thickness effect on the load-COD curves. Because the in-plane geometry of the specimens remains unchanged except the slight difference in a 0 , the loading capacity of a specimen is represented by the load normalized by thickness, as shown in Fig.4. loading condition (φ = 90°). As can be seen clearly, the maximum load-carrying capacity per unit thickness decreases significantly with increasing specimen thickness.
The capacity of the 14 mm thick specimen is only 59% of that of the 2 mm specimen. and 30°, respectively. It is interesting to find that the loading capacity per thickness of the 4 mm thick specimens is the maximum in most mixed-mode loading conditions, and the capacity of the 2 mm, 8 mm and 14 mm specimens are almost the same except of the 14 mm curve in Fig.4(e). We check the data and find that the fatigue precrack length of this 14 mm specimen (a 0 /w=0.570) is obviously larger than that of the rest specimens (a 0 /w=0.535, 0.531 and 0.524 for the 2, 4 and 8 mm specimens respectively). The reason why P max /B of the 2 mm specimens is not the maximum may is partially due to the out-of-plane buckling, especially for smaller loading angle φ, as the initial crack length of all the 2mm specimens is nearly the same or slightly smaller than that of the rest sets of specimens. As can be seen in Fig.4(e), the influence of thickness on the loading capacity becomes even weaker at φ=15 o . The slight difference in loading capacity is due to the slight difference in the initial crack length and uncertainties in the experiment. Under pure mode II loading, it has been shown by detailed 3D finite element analyses that the crack-tip stress and strain fields, as well as the stress state, do not change along the thickness [34]. As can be expected, with the loading angle decreasing from 90 o , the mode II component will increase and the thickness effects will decrease and finally disappear at φ=0 o . As the COD gauge cannot be fixed well in the pure mode II test, no complete load-COD curve is recorded and we present the maximum loading capacity in this case in Fig.5 against the thickness. It can be found that the test data are nearly independent of thickness in overall.
To clarify the influence of the slight difference in the initial crack length, we represent the load-COD curves for φ=90 o to 15 o in Fig.6 by use of the effective stress intensity factor evaluated at the initial crack length a 0 2 II 2 I (1) Where I K and II K are the components of the stress intensity factor for mode I and mode II loading as shown latter by equation (4) following the 2D theory. It should be pointed out that such a K eff is not the real stress intensity factor at given COD as no crack growth is taken into account and no 3D distribution is considered. We can simply use the K eff as an alternative measure of the normalized load level, which can partially eliminate the influence of the slightly difference in initial crack length.
Comparison of Fig.6 and Fig.4 shows that the influence of initial crack length does not change the general trend. An exceptional case is the 14 mm thick specimen in Fig.6(c). This specimen has initial crack length of a 0 /w=0.573, much higher the crack length in the rest cases in Fig.6(c), with a 0 /W≤0.534 and this difference leads to a change in the K eff as high as 18%. It is important that Fig.6 confirms again that even at φ=75 o , with II K of about 12% I K , the strong thickness dependent loading capacity at pure mode I has nearly disappeared completely. Therefore, the thickness effects on the fracture of mixed-mode I/II cracked specimens cannot be estimated by some simple superposition of the effects in mode I and mode II conditions. is that the loading capacity decreases with increasing φ, although a few curves do not obey the general trend due to the slight difference in initial crack length and other uncertainties in the experiments.

Effect of loading angle on the load-crack opening displacement curves
In the 2 mm specimens, things become much more complicated mainly because of the buckling and out-of-plane twisting whose degree depends on the loading angle.
For φ≤60 o , the overall trend is similar as that of the 4 and 8 mm specimens, or the loading capacity decreases with increasing φ. However, when φ≥60 o where the mode I loading becomes more dominated and the out-of-plane buckling and twisting may become some less possible with increasing φ, a reverse trend is observed. This reverse trend cannot be explained by the difference in the initial crack length as all the 2 mm specimens tested with φ≥60 o is quite stable, with 0.527≤a 0 /W≤0.529. Consequently, the dependence of the loading capacity on the mixed-mode loading condition is dependent upon the thickness of specimen.

Macroscopic fracture surfaces
Different fracture mechanism may be excitated by variations in loading conditions and stress states. So analyses of the morphology of the fracture surfaces may provide better understandings of the complexity of mixed-mode fracture of finite thickness specimens. As illustrated in Fig.8(a), a typical fracture surface under mode I loading condition consists of four distinct regions: fatigue pre-cracking, flat fracture, flat-to-slant transition, and slant fracture [18,35]. However, there is wide variability in the macroscopic fractographs for the mixed-mode fracture of various thickness specimens in this study. As shown in Fig.8 For thin specimens (such as B=2 and 4 mm), the size of the plastic zone is large and failure occurs on the plane containing the maximum shear stress, so slant fracture forms. In addition, for thick specimens namely when the plane strain state dominates, the plastic zone is much smaller and the fracture surface is oriented nearly at 90° to the loading direction. Since plane stress prevails at the free surfaces of a specimen, shear lips are always present in mixed-mode fracture, albeit small in some instances.
The flat-to-slant transition is really a highly three-dimensional process. The 8 mm thick specimen ranks between thin and thick, so the macroscopic fracture is more complex. Under pure mode II loading condition, all specimens fracture in an in-plane shear mode and thickness has no effect on the macroscopic fracture surface morphology. ridges. It is very interesting that the shear-type voids shown in Fig. 11(d) and

Microscopic fractographs
tensile-type voids shown in Fig. 11(e) coexist in the 2 mm thick specimen. In general, figures 9-11 show that voids become larger in thinner specimens for all the mode I, mode II and mixed-mode I/II loading conditions.
The fractographic characterizations demonstrate that the fracture of LC4-CS aluminum alloy is ductile, i.e. consists of microvoid nucleation, growth and coalescence. Thickness has significant effect on the microscopic fractographs of mixed-mode fracture. In thicker specimens, the ductile voids cannot grow larger and the ductility is relatively lower. As thickness decreases, both tensile-type dimples and shear-type voids stretch larger and shallower with ductile deformation can be better developed. Consequently, both the loading angle and specimen thickness have significant effects on the fracture mechanism.

Crack initiation angles
Two dimensional mixed-mode fracture investigations assume that from the practical point of view, the elastic solutions are sufficient for predicting the crack propagating direction, even for very ductile materials [36,37]. As shown by the confined plastic zone in Fig.3, small scale yielding assumption can be adopted for the present tests. In the following sections, the crack initiation angle will be predicted by some respective approaches based upon the elastic solutions and validated against our experimental results. As shown in Fig.12, the crack initiation angle, θ 0 , for the  specimens is close to that in the 14 mm thick specimens. However, it is slightly higher in the 4 mm thick specimens and significantly lower in the 8 mm thick specimens.
Such an exceptional thickness dependent mixed-mode crack initiation angle has not reported for any kind of material and loading condition.

Discussions
The crack growth direction along the specimen thickness is exceedingly complex where D = tanβ eq , and β eq is the so called loading mixity parameter, or the equivalent crack angle defined by II I 1 eq Where where P is the applied load, a is the crack length, w is the width of the specimen, B is the thickness of the specimen, and f 1 (φ,a/w) and f 2 (φ,a/w) are geometry factors which can be found in Ref. [33].
The relationship between φ and β eq can be obtained from the above formulas, and the angles used in the present work are listed in Table 2 where, µ = ) , σ x , σ y , σ z and σ xy are the stress components.
Another criterion which tries to consider the effect of triaxial stress states on cracking under mixed-mode loading is the maximum stress triaxiality criterion (M-criterion) [23]. It states that the direction of crack initiation should be coincided with the direction of the maximum stress triaxiality (the ratio of the hydrostatic stress to the equivalent stress) along a constant radius around the crack-tip. Mathematically, the M-criterion can be obtained from the elastic solution as 0 2 We call it an 8mm-phenomenon for the tested LC4-CS alloy.
We believe that the 8mm-phenomenon is a compositive effect of thickness It must be realized that the transition region of thickness may shift from material to material, mainly associated with the ductility and plasticity of the materials. For example, in very brittle materials the plane strain domination will sustain to extremely thin plate, while in materials with high ductility the plane stress domination can sustain to much large thickness. The fractographic investigations have shown us that the micro mechanism of fracture of the LC4-CS alloy is ductile dimples dominated, but the overall ductility is relatively low. This should be the main reason for the 8mm-phenomenon shown in this study.
Recently, more and more deep investigations show the subtile effects of slight shear stress or deformation on material strength [38,39]. Such fundamental researches are mainly confined to two-axial loading situations [40], and the complete 3D picture is far from clear up to now. The present experiments just contribute one additional example for the complex picture, and we are unable to give an exact solution to it in the present paper. Further efforts are needed from not only material and mechanical aspects, but also physics level.
For pure mode II crack initiation angle, data are only obtained from the 14 mm and 8 mm specimens. It is about 18 o~2 0 o , much lower than all the theoretical predictions, but not down to zero as expected by the pure shear fracture theory. As deformation is inevitable, in fact, pure mode II loading condition is hard to be guaranteed in the present CTS tests. For the 4 and 2 mm thick specimens, plastic deformation is too severe in mode II loading to obtain meaningful initiation angle.

Conclusions
A systematical experimental investigation is performed on the effect of thickness on mixed-mode fracture of the low ductile LC4-CS aluminum alloy with compact-tension-shear specimens of thicknesses ranged from 2 to 14 mm and loading angle ranged from pure mode I to pure mode II in the ambient condition. Loading versus crack opening displacement curves are recorded, in-site crack growth tracing by a CCD camera is adopted. After the test, macro-and micro-fractographic analyses, and careful crack initiation angle measurements at the mid-cutting plane of the specimens are conducted. Comparison with selected theoretical criterions is also made. The following conclusions can be drawn from the study. This strange 8mm-phenomenon should be a combinatorial effect of thickness induced variation in the 3D constraint or stress state near the crack front, the ductility and damage mechanism of the materials, the confined plastic deformation and the mixed-mode loading, but it is requires deeper interdisciplinary study.    Table 2 The equivalent crack angle β eq versus the loading angle φ (degree)