Stress Intensity Factor Equations for the Evolution of Surface and Corner Cracks to Through Cracks

Part-through surface or corner 2D cracks are commonly found in structural components, even because practically all fatigue cracks tend to start this way. It is a reasonable hypothesis to model them assuming the shape of their 2D fronts can be approximated by an elliptical arc, as supported by many fractographic observations. However, their transition to a 1D throughcrack, an important issue in many practical applications, is normally not properly addressed in fatigue life predictions. Although experimental results reveal that the frontiers of surface cracks essentially retain their elliptical shape as they gradually grow into an 1D through-crack, it is usual to assume they are immediately transformed into an 1D through-crack when their depth reaches the cracked component thickness. This crude approximation may create a large jump in stress intensity values, leading to excessively conservative fatigue crack growth predictions; or else, the crude shape jump hypothesis might induce false overload events that can much affect fatigue crack growth retardation models, leading to inadmissible non-conservative life predictions. To minimize such problems, an improved model to describe the transition of 2D surface cracks to 1D through-cracks is proposed and verified by crack propagation tests in two different materials, 4340 steel and polycarbonate (PC). Moreover, fatigue life predictions based on this improved model are compared with experimental results obtained with these two materials.


INTRODUCTION
atigue cracks tend to nucleate on surfaces, usually at a notch tip whose stress concentration effects cannot be neglected in fatigue analyses. Most fatigue cracks initiate and initially grow in 2D with a crack front that can be approximated by an elliptical arc, see e.g. the ASM Fractography Handbook for plenty of evidence to support this F claim [1]. In fact, even when fatigue cracks initiate from a far-from-elliptical notch tip, they quickly tend to an approximately elliptical front, as shown by the crack that departs from an elongated rectangular notch illustrated in Fig. 1. Figure 1: Crack that initiated from a rectangular notch, but changed to an elliptical-like front within a short distance from it [2].
Neglecting short crack issues (which, however, may be important in many practical cases, as studied e.g. in [3][4]), fatigue crack growth (FCG) lives can usually be well estimated by integrating the FCG da/dNK curve of the material, where da/dN is the FCG rate caused by the stress intensity factor (SIF) range K. Such calculations apply in simple memory-free constant amplitude loading cases. Since FCG curves can be properly measured following standard ASTM procedures, then well-known numerical integration techniques can be used to perform fatigue life predictions of cracked components once the appropriate K is known for the problem in hand, no matter how complex the SIF expression is. Most fatigue cracks tend to grow in three distinct phases. They usually initiate at a surface as a 2D crack and grow in 2D up to reaching the piece thickness. Then they make a gradual transition from their initial 2D shape to a through 1D crack, and eventually propagate until the final fracture with a crack front shape that can be treated as 1D (albeit they cannot have a straight front, as discussed in [5]). The weak modeling link in this process is the 2D to 1D transition, since there are no standard SIF solutions for this problem, even though there are plenty of handbook expressions to evaluate SIFs for many geometries in which the cracks grow in a single 1D direction, as well as various classic solutions for 2D cracks that propagate maintaining elliptical crack fronts. For practical reasons, these solutions assume an elliptical crack front shape to avoid having to model the free surface singularity [6]. There are classic analytical expressions for quantifying the SIF of elliptical 2D cracks under combined tension and bending, most obtained by proper fitting 3D finite element (FE) solutions. These expressions are complex and difficult to use for quick estimates, but they can be relatively easily integrated by standard numerical methods. For instance, if these cracks occur in a plate of width w (or 2w, if the subsequent through-crack has two tips) and thickness t, their SIF range K is a function of the stress range , of the ratios a/c, a/t and c/w, and of an angle  defined in Fig. 2, because their SIF values vary from point to point along their fronts [7][8][9][10]: K = (a)f  (a/c, a/t, c/w) (1) where f  is a crack shape function and a and c are the ellipsis semi-axes. It usually is a quite reasonable approximation to assume that real 2D crack fronts have a near elliptical shape, but this is not enough to model their FCG behavior and to make FCG life predictions. It is also necessary to assume that the successive fronts of such 2D cracks retain the elliptical shape but may change their a/c ratio at every load cycle, because their SIFs vary along the crack front. Based on the maintenance of an elliptical geometry, the basic modeling idea for 2D FCG predictions is to couple the crack growth in the depth (a) and surface width (2c for surface and c for corner cracks) directions at every load event, as studied e.g. in [10]. However, the transition behavior from a corner or surface flaw into a through-crack usually is not even addressed when modeling them. For example, consider a surface crack as in Fig. 2 (left). If the cracked plate material is tough enough and if 2w >> 2c, this surface crack can propagate until reaching the back surface of the plate. Then there is a transition region until the 2D crack can be considered a 1D through crack, whose traces are nearly equal both on the front and on the back face of the plate. There are authors that claim that, for fatigue life estimation purposes, it can be assumed that surface cracks immediately transform into through-cracks as their depth a reaches the thickness t at the back surface of the cracked component [11]. An example to justify this simplification in the FCG transition is presented by Grandt et al. [12][13]. They studied a crack that grew on a double-T cross-section beam under bending. Theirs initially 2D quarter-elliptical crack propagated twodimensionally through the lower corner of the section until it almost penetrated the inner rear face of the section at 60,923 cycles. Then, the crack increased rapidly and cut the entire leg in only 80 more cycles. Hence, the 2D to 1D transition cycles could be neglected based on a supposed "catch up effect" [12], assuming the crack front at the back surface experiences a much higher SIF (and so a much faster FCG rate) than at the front surface, tending to straighten its profile. However, this approach creates a discontinuity in the calculated SIF values, often causing the crack growth prediction to be excessively conservative, because such a simplification can behave like if it was an overload that causes subsequent FCG delays in most numerical FGC prediction codes, see e.g. [10,14]. Moreover, other experimental results reveal that quasi-elliptical surface 2D flaws may not immediately transform into through-cracks. Fawaz e.g. studied the shape of fatigue crack fronts that started from a hole in Al 2024-T3 Alclad thin plates with thickness t  1.6mm, loaded under combined tension and bending [15]. He found that his cracks essentially retained their elliptical shape as they gradually grew into through-cracks. Even though this detailed experiment indicates a need for it, no closed-form SIF-solution is available for through-cracks with such oblique fronts. Some approximations have been suggested based on the boundary element method [8], however they do not account for the component width effect, a major limitation on practical applications. A simplified model for the 2D-1D crack front transition was proposed by Johnson a long time ago [16]. According to him, after the crack depth reaches the specimen thickness t, the crack can be assumed to keep its elliptical shape in the transition zone. Johnson used an adaptation of the surface (2D) stress intensity factor (SIF) expressions described by Hall et al. [17] to model the surface crack transition. However, Johnson's original approach may be inaccurate, since it uses overly simplified the SIF expressions for surface cracks, without considering the effect of the specimen width 2w and the variation of the front surface effect (assumed constant and equal to 1.1). In addition, it does not guarantee continuity of the SIF expression KI(c) in the width direction between the transitioning period and the 1D crack growth regimen. This problem is particularly deceiving when FCG retardation models are considered in fatigue life calculations. If the considered SIF equation calculates a smaller value for the through crack than for the transitional range, the retardation model may consider the larger SIF as an overload, delaying its subsequent 1D growth. Therefore, when considering load interaction models, Johnson's approach may predict FCG retardation even under constant amplitude loading, resulting in non-conservative fatigue life predictions.
Castro and Meggiolaro [10] proposed in their 2016 book improved equations for 2D-1D crack transitions, which are inspired on Johnson's approach but use much improved Newman-Raju's solutions for the 2D crack SIFs [18][19][20][21][22]. However, their proposed equations have not been properly checked by experimental data. To verify them, in this paper 2D-1D FCG transitions are measured in two different materials, 4340 steel and polycarbonate (PC), in order to present suitable crack size and shape data, and the corresponding numbers of cycles spent in the 2D-1D transition zones, as discussed in the following. These studies were done only for plate-like geometries. Some authors have worked with different geometries, like the pipes studied in [23][24], for example.

IMPROVED MODELS FOR THE TRANSITION FROM SURFACE TO THROUGH CRACKS
ewman-Raju's equations are adapted to model the transition from surface semi-elliptical or corner quarter elliptical 2D cracks to through-the-thickness 1D cracks. In addition to the effects of the back face magnification and crack shape parameter, Newman-Raju's expressions also model the specimen width and front surface effects. To model the 2D/1D transition zone, the basic hypothesis of ellipsoidal geometry preservation is assumed. Therefore, as illustrated in Fig. 3 [10], Johnson's criterion c'  0.9c for the end of the 2D/1D transition zone is equivalent to In other words, the 2D/1D transition zone starts when the crack depth equals the specimen thickness (a'  a  t), and is assumed to end when the imaginary crack depth a' reaches 2.3t. Using this result, continuity of the width SIF expressions for the transitioning period, K I (c), and for the 1D crack growth regimen, K I,1D , can be achieved replacing all occurrences of c/t by a parameter where  is the value of c/t that satisfies KI(c)  KI,1D.
Note that r'  c/t in the beginning of the 2D/1D transition zone (when a'  t), and r'  a when the transition ends and the crack finally becomes a 1D through-the-thickness crack with a'  2.3t. The application of this approach for the transition from surface (semi-elliptical) cracks to through-cracks is presented next.
Transition from 2D semi-elliptical surface cracks to 1D through-cracks Newman and Raju modeled the SIF on the width and depth directions of surface cracks under uniaxial tension , respectively KI(c) and KI(a), by where Q is the crack shape parameter, F s,w is the specimen width effect, M s is the back face magnification factor, and F s,c and F s,a are respectively the front surface effects on the width and depth directions, given by For the 1D crack growth regimen, Tada [7] lists the SIF of a center-cracked plate as where 2c and 2w are the through-crack and plate widths. The last polynomial term in Eq. (10) improves its precision from 2.6% to better than 0.2%. Therefore, comparing Eqs. (4) and (10), and noting that the only term in Eq. (4) that depends on c/w is F s,w , a modification for Eq. (6) is proposed Eqn. (11) improves Newman-Raju's SIF solution to better model the plate width effect. The transition from a part-through 2D surface crack to a 1D through center crack is then modeled under uniaxial tension using Eqs. (4-5), (7)(8), and (11), considering also F s,a  1.1 to account for the back face becoming a free surface at a  t, resulting in where the prime (') symbol denotes the expressions for the 2D/1D transition from part-through 2D to through 1D cracks (for t  a'  2.3t), and thus To guarantee continuity of the K I (c) expression, Eqs. (10) and (12) should be equivalent at the end of the 2D/1D transition  20) and the SIF during the transitioning period is then modeled replacing c/t by r' in Eqs. (12)(13)(14)(15)(16)(17)(18). When the imaginary crack depth a' reaches 2.3t, Eq. (15) is then used to model the subsequent 1D crack growth. Fig. 4 plots the ratio between the transition and the 1D SIFs calculated using the proposed approach, to show their smooth transition. Two main improvements are achieved using this approach. First, the effect of c/t on K I (c) is much better modeled by Newman-Raju's equations than by the expressions used by Johnson. Second, continuity in the KI(c) function is guaranteed for the threshold from the transitional range to 1D growth through the use of Eq. (20). Notice in Fig. 4 that the function r'(c/t, a'/t) only starts influencing significantly the calculated KI(c) when c'/c > 0.5. Fig. 5 plots the normalized stress intensity factor (also known as the geometry factor) in the width direction, calculated using the proposed approach. Notice the smooth transition between the surface and through-crack growth regimes, especially under high c/w ratios. The only remaining discontinuity in this model is due to the front surface effect on the depth direction, F s,a . However, since this discontinuity predicts a larger stress intensity factor as the crack enters the 2D/1D transition zone, false FCG retardation effects cannot cause calculation problems in fatigue life predictions. In the next section, the proposed approach to model the 2D/1D transition from part-through to through cracks is applied to corner (quarter-elliptical) cracks.

Transition from 2D corner quarter-elliptical cracks to 1D through-cracks
where Q is the crack shape parameter defined in Eq. (4), Fq,w is the specimen width effect (modeled by Newman and Raju using the same function they used for surface cracks), M q is the back face magnification factor, and F q,c and F q,a are respectively the front surface effects on the width and depth directions, given by where c and w are the through-crack and plate widths.
This expression is precise within 0.5% for any a/c ratio, since it models the significant bending moment caused by the presence of the single edge crack, which exists even under pure uniaxial tension. Analogously to the surface crack modeling, a modification for Eq. (23) The transition from 2D to 1D crack growth is then modeled considering a  t in Eqs. (21)(22), (24)(25)(26), and (28). Like for the surface cracks, the ratio c/t is replaced by a function r'(c/t, a'/t) that guarantees continuity between Eqs. (21) and (27).
For corner quarter-elliptical cracks, the value of  in Eq. (3) The 2D/1D transition from part-through corner to through single-edge cracks (t < a' < 2.3t) is then modeled under uniaxial tension by Therefore, as long as c' is much smaller than c (which happens in the beginning of the 2D/1D transition zone), Eq. (37) implies that the back face may experience a much larger growth rate than the front surface. This result is well in agreement with the "catch up effect" described by Grandt et al. [12], which is reproduced by the proposed interpolated equations.

EXPERIMENTAL RESULTS
his section presents data from FCG transition 2D-1D tests for two different materials: polycarbonate (PC) and 4340 steel. PC is a transparent and relatively tough polymer, so it can be used to obtain images from crack fronts while the fatigue crack grows. However, since it has long been recognized that its da/dNK behavior is very dependent on hard-to-control environmental parameters [25][26][27][28][29][30], the PC tests are considered here qualitatively only, since they could not yield reliable quantitative FCG rate data. Quantitative data are obtained by testing 4340 steel specimens. This material presents a well-defined da/dNK curve, thus such tests can also be used to validate fatigue analyses.

Polycarbonate (PC)
365x47x10mm plate PC specimens are tested under R   min / max  0.1 tension loads. First, a small initial triangular notch is machined on the corner of each specimen with a size between 1 and 2mm, to initiate and propagate a fatigue crack. This T notch is easily made manually with a watchmaker's saw. Two digital cameras and one microscope are used to measure the FCG behavior during the tests. The first set of tests is performed under constant loads. However, the cracks propagated only in their 2D phase with a/t < 1, so they did not enter the 2D-1D phase transition, as the fracture toughness was reached before that. Indeed, their SIFs increase relatively fast as the cracks grow and, at the same time, despite its relatively high toughness, the SIF range in the da/dNK PC curve is still very narrow (between 0.5-5MPam). Specimens used in this first set of tests are named S1, S2, and S3. The fatigue crack surfaces of these specimens after they broke are shown in Fig.   8. The load range values used in theses PC tests are ΔP 1  5130N, ΔP 2  6840 N and ΔP 3  6000 N, respectively.
To avoid this undesired behavior, a second set of PC specimens, named S4, S5, and S6, is tested under decreasing load ranges. However, even this precaution was not enough to eliminate experimental problems. Indeed, although their cracks did transition from 2D to 1D, high plasticity occurred on these specimen surfaces, creating non-negligible tunneling effects [5,31], as shown in Fig. 9. Note that it is impossible to precisely define the crack sizes with such severely tunneled crack fronts. So, a third set of PC tests is performed to properly simulate the desired 2D-1D crack transition, maintaining the SIF ranges low and approximately constant, K I (a)  K I (c)  1MPam, gradually reducing the load applied on the specimens as the cracks grew to obtain the desired well behaved 2D-1D transition. This testing strategy was successful and allowed the clear identification of the growing crack fronts. Fig. 10 shows the resulting crack surfaces for the set of specimens named S7, S8 and S9. Fig. 11 is a mosaic that contains about 140 photomicrographs of successive crack fronts observed in specimen S9. Such well-defined beach marks helped to track the crack front lines as the crack gradually grew by fatigue. Notice in Fig. 11 that even though the initial 2D notch front is relatively far from a quarter-circular or elliptical shape, the following crack fronts quickly converge to such shapes, an indication that supports the claim stated in Fig. 1 that fatigue cracks apparently do like this well-behaved elliptical arc shape. Fig. 12 shows a sequence of images with different crack positions obtained in specimen S8. This test started with a frequency of 4 Hz, which was reduced to 1 Hz and then to 0.1 Hz.

Alloy Steel
Due to the experimental difficulties faced when trying to reproduce smooth 2D-1D crack transitions in the PC specimens, additional FCG tests have been performed on similar 20021.510mm plate specimens made of annealed SAE 4340 steel. Initial triangular notches are also machined on the corner of these specimens, with initial sizes ranging from 2 to 3 mm. Like in the PC tests, microscopes are used to monitor and measure the crack traces on the different faces of the opaque steel specimens. FCG tests are performed at frequencies between 10 and 20Hz in a 100kN computer-controlled servo-hydraulic testing machine, with a baseline SIF range between KI  20MPam and KI  60MPam, under R 0.1. These SIF ranges lie in phase II of the 4340 da/dNK curve, which can be well modeled by the Paris rule, avoiding issues with crack closure and short crack behaviors, see Fig. 13.
Its da/dNK data, obtained under R  0.1 and measured following ASTM E 647-99 procedures, see Fig. 13, can be well fitted by a Paris equation da/dN  2.510  9 K 2.48 mm/cycle.
Two 4340 alloy steel specimens are tested, named S10 and S11. To avoid early rupture during 2D-1D FCG transition, the applied load is gradually decreased to maintain a quasi-constant SIF range K  24MPam.
Tabs. 1 and 2 show the load amplitudes applied on the specimens, the measured number of cycles and the corresponding crack lengths, according to the nomenclature adopted in Fig. 11. In these tables, the accumulated number of cycles (#cycles) is relative to the event where the crack depth first reached the thickness of the plate specimen, i.e. when a/t  1.

NUMERICAL FATIGUE ANALYSES
o compare the experimental data reported above with numerical fatigue life predictions, the proposed 2D to 1D transition equations are implemented on the software ViDa (which means life in Portuguese). This versatile software has been developed to automate all traditional local approach methods used in fatigue design, namely the SN (including IIW and DNV recommended practices for welded structures) and the N for modeling crack initiation, and the da/dN for crack propagation [14]. In particular, two methods can be used to simulate 1D and 2D FCG problems under variable amplitude loads: the so-called Krms and the cycle-by-cycle methods (but only the latter can properly consider load sequence effects). For completeness purposes, these two methods are briefly discussed next. 16.00 Table 1: Measured data for the S10 plate specimen, starting to count the #cycles when the crack reaches its thickness Table 2: Measured data for the S11 specimen, starting to count the # cycles when the crack reaches its thickness. Figure 14: Successive crack fronts for the steel specimens S10 and S11.

The Krms method for 2D cracks
The principal characteristic of 2D cracks is a non-homologous FCG behavior in two directions. Thus, although retaining the basic elliptical-like form, in general the 2D crack fronts tend to change shape from cycle to cycle, because their SIFs vary from point to point along their fronts. That is the main reason why 2D elliptic FCG problems can be a convenient approximation for many actual surface, corner, or internal cracks, since fractographic observations as those presented above support the idea that the successive crack fronts tend to achieve an approximately elliptical form and to stay approximately elliptical during their FCG. Therefore, it can be quite reasonable to assume when modeling such 2D cracks that their FCG just changes the shape of their fronts (given by the a/c ratio between the ellipsis semi-axes, which quantifies how elongated the cracks are), while preserving their basic ellipsoidal geometry. Since any ellipsis is completely defined by its two semi-axes, to model and predict the FCG behavior of such 2D elliptical cracks, including their shape changes, it is enough to calculate at each load cycle the lengths of the ellipsis axes a and c, solving the coupled da/dN and dc/dN propagation problems. Although laborious, this is not a particularly difficult task [10]. Hence, the simplest way to simulate the growth of 2D elliptical cracks, either under constant or under variable amplitude loadings, is to calculate their equivalent root mean square (rms) stress range rms along the entire load history. The load history is then treated as a constant amplitude loading, a simplification that certainly cannot be used in the presence of rare but significant overloads, which can cause load interaction effects like FCG retardation or even arrest. Anyway, in this rms approach, the SIF range Krms is initially calculated at the initial crack size (a0, c0) by Consequently, the number of cycles N 0 the crack takes to grow from a 0 to a 0 + a is given by where a is a small crack increment that must be specified by the fatigue analyst (e.g. 10 to 50m, numbers of the order of the resolution threshold of practical crack measurement methods in fatigue tests, can be a good choice both from the physical and from the numerical points of view). The associated SIF in the width direction Krms(c0) can then be calculated from the proposed equations, to get the corresponding growth c0 in the direction of the semi-axis c, given by , where c1  c0  c0, and then starts over again. The calculation precision can be adjusted by the chosen a value. As these cracks usually have different values for K(a) and K(c), there are four distinct 2D propagation cases under constant amplitude loading: 1) K(a0) and K(c0) > Kth: the crack propagates in both directions, changing shape at each i-th load cycle depending on the ratio K(a i )/K(c i ). 2) K(a 0 ) > K th and K(c 0 ) ≤ K th : the crack grows only in the a (depth) direction, until its size is large enough to make K(c0) > Kth, when the problem reverts to Case 1.
The crack growth history is then calculated by the simultaneous solution of a i and c i . As the crack increments a i and ci depend on both semi-axes ai and ci, the coupled 2D growth is well characterized. Contrary to the Krms method, which as any statistics loses sequence effects, the cycle-by-cycle method can deal with load order effects, as long as appropriate crack retardation (or acceleration) models are introduced in Eqs. (42-43), see [10] for details.

Comparison between Experimental and Numerical Results
Figs. 15 and 16 compare experimental data and numerical predictions for the FCG behavior of the 4340 alloy steel specimens S10 and S11, respectively. Experimental data points plotted in these figures are listed in Tabs. 1 and 2, while the numerical results are represented as continuous lines. The numerical simulations are obtained by the cycle-by-cycle fatigue life prediction method outlined above, considering the load applied on each specimen using the proposed corner crack transition equations, the actual da/dNK curve measured for the material as well as its other properties, and the specific geometry of the specimens. The initial values of c adopted in these simulations are equal to the measured ones in the beginning of the transition phase, for a fair comparison neglecting possible previous accumulated errors in the 2D FCG predictions, since only the 2D/1D transition equations are evaluated here. Moreover, since all load decrease steps are very small during the tests, no FCG retardation models are needed in these simulations either.  The S11 4340 steel specimen is tested until failure, which happened 453,594 cycles after the beginning of the 2D/1D transition. The simulated front and back face crack sizes after such applied cycles are c 16.0 mm and c'  14.40 mm, respectively a difference of 1.23% and 10.0% with respect to the experimental results, see Fig. 16. For the S10 specimen, the final simulated front and back face crack sizes are c  12.71 mm and c'  8,06 mm, respectively a difference of 12.41% and 38.0% with respect to the experimental results, see Fig. 15.

CONCLUSIONS
he transition of 2D part-through cracks (such as surface or corner flaws) to 1D through-cracks was modeled based on Newman-Raju's SIF equations for 2D elliptical cracks. The proposed interpolation equations were applied to describe the fatigue crack growth behavior of both surface and corner cracks on rectangular plates. The resulting continuous expressions were shown to better model not only the back face magnification and crack shape effects, but also the influence of the specimen width and front surface on the growth of part-through cracks. In addition, the proposed equations guarantee continuity of the SIF expressions between the transitioning period and the through-crack growth regime, as well as being able to reproduce the observed "catch up effect." Comparison with experimental results showed that the proposed equations are able to reasonably represent the global behavior of the 2D-1D crack transitions.