Effect of a hard artificial asperity on the crack closure behavior in an annealed SAE 1015 steel

Engineering Fracture Mechanics 72 (2005) 2106–2127

www.elsevier.com/locate/engfracmech

Effect of a hard artificial asperity on the crack closurebehavior in an annealed SAE 1015 steel

M. Okayasu a, Z. Wang a,*, D.L. Chen b

a Department of Materials Science and Engineering, University of Toronto, 184 College Street, Toronto, Ont., Canada M5S 3E4b Department of Mechanical and Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, Ont., Canada M5B 2K3

Received 3 March 2004; received in revised form 16 November 2004; accepted 16 December 2004

Available online 24 May 2005

Abstract

The load–crack opening displacement (COD) curves and deformation characteristics in the vicinity of a hard arti-

ficial asperity in an annealed SAE 1015 steel were studied. The artificial asperity was found to have a significant effect on

the trend of the load–COD curves. The lower portion of the load–COD curves in the unloading phase exhibited a con-

vex shape without the asperity, but a concave shape with the asperity. The concave shape, signifying the acceleration in

the COD decrease, was further verified by varying the size of the asperity, conducting special compression tests and

elastic–plastic load–COD tests. The plastic deformation in the vicinity of both asperity and crack tip was studied via

microhardness tests, etching techniques, and finite element analysis. Based on the experimental observations, a modified

crack closure process model was proposed, where three stages of the unloading curve was defined: (i) the asperity does

not contact the upper crack face, (ii) a process where both the asperity and the specimen material deform elastically, and

the elastic-wedge model is applicable, and (iii) the plastic deformation of the specimen material adjacent to the asperity

occurs, thus resulting in the concavely shaped load–COD curves. An equation was proposed to estimate the COD val-

ues, in which the plastic deformation both at the crack tip and at the asperity was considered. The residual COD cal-

culated from the proposed equation was found to be consistent with the experimental results.

� 2005 Elsevier Ltd. All rights reserved.

Keywords: SAE 1015 steel; Crack closure; Plastic deformation; Crack opening displacement (COD); Artificial asperity

1. Introduction

Since the crack closure behavior was first observed by Christensen in 1963 [1] and defined by Elber in

1970 [2], fatigue crack growth tests have been performed to identify the effect of crack closure on the

0013-7944/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfracmech.2004.12.006

* Corresponding author. Fax: +1 416 978 4155.

E-mail address: [email protected] (Z. Wang).

mailto:[email protected]

M. Okayasu et al. / Engineering Fracture Mechanics 72 (2005) 2106–2127 2107

da/dN–DK relationship. In particular, numerous studies have demonstrated that the extrinsic resistance of

fatigue crack growth was attributed to the roughness-, oxide- and/or plasticity-induced crack closure

behavior.

The plasticity-induced crack closure arises from the situations, where the stretched material in the crack

wake leads to a premature contact of crack faces [3–19]. This closure mechanism has been observed usingan etch-pit etching technique [3], a digital photo-elastic technique [4], and finite element analysis (FEA) [5–

15]. Specifically, the FE analysis has been extensively utilized by many researchers to examine this closure

mechanism under different conditions. Very recently, based on their previous FE analysis data, several

researchers have further calculated the crack growth behavior more accurately [14,15]. Although a number

of investigators have studied the mechanism of plasticity-induced crack closure, the fundamentals of this

closure mechanism have been challenged by Vasudevan et al. [16–18]. They reported that there is no sig-

nificant contribution to crack closure by the plasticity behind the crack tip, as the permanent deformation

takes place just in front of the crack tip. However, their claims were countered by the work of Pippan andRiemelmoser [19], who developed mathematical and dislocation models for the fracture surface contact

mechanism.

Roughness-induced crack closure mechanism has also been studied experimentally and numerically [20–

29]. Rough crack surfaces are considered to be produced by the following process: the low restraint on cyc-

lic slip will primarily promote crack extension along a single slip system at the microscopic scale, resulting

in serrated or zig-zag fracture paths [20], in which the maximum plastic zone size at the crack tip is typically

less than the grain size [21]. On the basis of this process, a geometrical crack closure model (zig-zag model)

was presented by Suresh and Ritchie [20]. Parry et al. [22] studied roughness-induced crack closure behaviorusing FE analysis, in which the calculated crack closure level under a plane strain condition was found

similar to those results obtained via the zig-zag model. However, this closure model has been recently ques-

tioned by Wang et al. [23] and Wasen and Karlsson [24] due to its over simplification. In Ref. [23], the mod-

el of fracture surface mismatch has been extended using dislocation theory; the effective stress intensity

range for the crack growth predicted with their dislocation model was found to be superior to the zig-

zag model. However, such a dislocation model has also been countered. Parry et al. [22] stated that ‘‘the

predictive power of their dislocation model [23] is limited due to the inclusion of unknown proportionality

constants, which must be fitted to experimental results’’.To understand the crack growth behavior associated with crack closure more clearly, several researchers

[30–36] made attempts to use artificial wedge/asperity in their fatigue tests. Vecchio et al. [30] measured the

crack growth rate influenced by a needle tip positioned adjacent to the crack tip in a 2024 aluminum

alloy. They concluded that ‘‘the crack growth rate changed marginally despite the dramatic increase

in apparent closure levels’’. In the same manner, Hertzberg et al. [31] evaluated the effect of the artifi-

cial asperities—shims of different sizes—on the level of crack growth rate and the effective stress intensity

range, where the conventional effective stress intensity ranges measured with CMOD gauges underesti-

mated the observed crack growth rate. In addition, artificial asperities made of epoxy with 0.2 lm and0.5 lm alumina particles were used to evaluate the effect of infiltration induced crack closure on crack

growth retardation by Sheu et al. [32], who concluded that ‘‘although the crack growth increment is affected

by the size of reinforcing particles, no distinct trend was detected in the crack growth rate at the maximum

retardation’’.

The experimentally obtained crack growth behavior affected by the artificial wedge has been further

assessed by theoretical crack closure model. There are several theoretical models [37–43] for characterizing

the crack closure involving the artificial asperity. Suresh and Ritchie [37] and Maiti and Paramguru [38]

proposed rigid crack closure models, in which the wedge is assumed to be completely stiff as shown inFig. 1(a). In this case, Stage II, i.e. after the surface contact occurred, is represented by a vertical line.

In Chen et al.�s elastic-wedge model [39], an elastically deformable asperity is placed into crack surfaces;

the applied stress–COD behavior in Stage II exhibits a gradual and linear variation, as shown in

σmax

Stage II

δ

σop

δmaxo

Stage I

δmin

σ(a) (b)

Stage II

δ

σop

σmax

δmaxo

Stage I

δmin

σ

δop

σclosed

σmin

σmin, act

Stage II

δ

σmax

δmaxo

Stage I

δmin

σ

δop

(c)

Nonlinear portionσ1

σ2, or σop

Fig. 1. Schematic illustrations of typical crack closure models: (a) rigid-wedge model [37,38]; (b) elastic-wedge model [39] and (c) crack

closure model associated with crack surface interferences [40].

2108 M. Okayasu et al. / Engineering Fracture Mechanics 72 (2005) 2106–2127

Fig. 1(b). Very recently, Tada and Paris [40] proposed a crack closure model, where a nonlinear unloading

portion between Stage I and II is introduced due to the effect of crack closure or surface interference be-

tween the asperity and the crack face, shown in Fig. 1(c). In this model, Stage II can be defined by simple

elastic analysis models. Song et al. [33] calculated the crack growth rates using both the rigid-wedge model

[37] and the elastic-wedge model [39], and made a comparison with their experimental results obtained froma 4130 steel specimen with electroless nickel artificial asperities. They concluded that ‘‘the elastic-wedge

model predicted more accurately a post-plating crack growth rate than the rigid-wedge model’’. In the same

way, Rehman and Thomason [34] analyzed the crack growth rate using the rigid-wedge model presented by

Maiti and Paramguru [38], in which four different artificial asperities (electroless nickel, electroplated nick-

el, ethylcyanoacrylate adhesive and low-melting point solder) were used. However, the model predicted well

only for the softer asperity materials, such as adhesive and solder.

Although the artificial wedges into the crack surfaces have been simply designed and employed in their

studies to understand the stress intensity level and the crack growth rate, the fundamental aspects of such aclosure mechanism have not been considered nor even substantiated. The fundamental aspects, involving (i)

deformation characteristics in the wedge and specimen, and (ii) the related COD value, are especially

important for understanding clearly the load–COD relations and the associated crack growth behavior.

In addition, even though these theoretical closure models are considered valid for particular reasons, there

is apparently a lack of experimental data for their validation.

The aim of this paper is, therefore, to identify the crack closing mechanism involving an artificial asperity,

and to understand the underlying load–COD relations. Moreover, an attempt is made to link the theoretical


and numerical modeling of closure to the experimental measurement of the load–COD curve. The experi-

mental work is carried out using a harder artificial wedge inserted into softer crack surfaces, since such a

material condition is considered to be similar to the crack closing behavior occurring naturally [44–54],

e.g., harder and round shaped asperities (such as reinforcing Si particles [44–47], primary Si [48,49] parti-

cles and inclusions, Al2O and SiO2 particles [50–52]) coming into contact with relatively softer crack sur-faces. It should be pointed out that, although the size of wedges selected in the present experiment is

much larger compared to the aforementioned particles, the selection of such larger and harder artificial

wedges should be appropriate and constructive in facilitating a more accurate examination of the crack clos-

ing behavior. As a matter of fact, the results obtained have revealed, indeed, a rather indicative crack closure

behavior.

2. Materials and experimental procedure

2.1. Material and specimens

The material selected in this investigation is a commercial hot rolled low carbon steel, SAE 1015, sup-

plied in the form of 12.7 mm thick flat bars. The chemical composition of the alloy is (wt%): 0.14 C, 0.64

Mn, <0.01 P, <0.01 S and balanced with Fe. The alloy was annealed at 650 �C for 2 h to remove potential

residual stresses. The mechanical properties of the alloy tested at room temperature are: ry = 259 MPa,

rUTS = 417 MPa, elongation = 34%, and average Vickers hardness = 173 HV.Three point bending specimens with two different thicknesses, B = 8.0 mm or 12.0 mm, were used in this

study. The specimen surfaces were ground with 600 emery paper to facilitate the observation of the pre-

crack. A pre-crack of approximately 3 mm in the specimen was produced through fatigue using an elec-

tro-servo-hydraulic system with 100 kN capacity under load control at room temperature. The pre-cracking

test was conducted with a sinusoidal waveform at a frequency of 10 Hz and load ratio of R = 0.1. The crack

length during the cyclic loading was continuously monitored using a traveling light microscope with a res-

olution of 0.01 mm. After the crack had been produced, a semi-circular hole on the crack was machined

through electro-discharge machining (EDM) to facilitate the insert of a high carbon steel wire (piano wire)between the crack surfaces as shown in Fig. 2. In this study, different sizes of piano wires, B0.75 mm and

B0.90 mm, were used, with mechanical properties of ry = 1470 MPa, rUTS = 1950 MPa, elongation

<0.02%, and average Vickers hardness = 550 HV [55].

2.2. Load–COD tests

The crack closing behavior, both with and without the hard asperity, was evaluated via the load–COD

curves, which were monitored by a data acquisition system in conjunction with a computer through a stan-dard load cell and extensometer. At a maximum value of approximately 85% or 110% of the load corre-

sponding to the bending yield load of the three point specimen, the piano wire was tightly inserted into

the semi-circular hole. It should be pointed out that the bending yield load is defined as the load level,

at which a 0.01 mm proof-COD value is measured. The load–COD tests were then executed at a displace-

ment rate of 0.5 mm/min.

2.3. Deformation zone examination

To reveal the deformation zone in the specimen after the load–COD tests, the work-hardened area was

examined by means of the microhardness, a newly developed etching technique [56] and FE analysis. In the

(b)Inserted piano wire

Three point bending specimen

W =12mm

l = 80mm

B = 8mm or 12mmS = 53mm

(a)

Semi-circular hole a1=2.5mm

Pre-crack

ai = 5mm

0.1mm

Equivalent height

Fig. 2. Dimensions of three point bending specimen and a semi-circular hole used to facilitate inserting piano wires.


microhardness tests, two regions surrounding the piano wire and the crack tip, respectively, as shown in

Fig. 3, were studied using a Vickers microhardness tester. A load of 98 N was used on the ferrite grains,

including grain boundaries. The etching procedure based upon Fry etching method [57] has been briefly

described as follows. First, an annealed specimen was strained, and the surface for observation was ground

with emery papers down to 2000 grade. Then the sample was heated at 550 �C for 2 h, and the surface was

polished to a mirror level. Finally, the polished surface was etched with a solution of 50 ml hydrochloric

acid, 30 ml ethyl alcohol, 50 ml water and 10 g CuCl2 under ultrasonic vibration. After about 10 min etch-

ing, the etched surface was lightly polished again by a cloth. In this etching technique, the plastic zone wasrevealed by different degrees of etching between the plastically deformed and nondeformed regions; the

plastic zone was found to be only slightly etched, whereas the other region was deeply etched due to

their different morphologies in the microstructure. Moreover, due to the surface offset after etching, the

Piano wire

Crack tipB

A

Fig. 3. Schematic illustration of the locations for the microhardness measurement: A, adjacent to the piano wire and B, around the

crack tip.


deformation zone became observable at a low magnification. Details of the mechanisms behind such

etching technique can be found in Ref. [56].

2.4. Finite element analysis

Finite element analysis was employed to analyze both the load–COD relation and the deformation pat-

tern in the areas surrounding the piano wire and the crack tip. In this analysis, two-dimensional finite ele-

ment simulation with eight-noded quad elements was used. The FE model was designed based on the

specimen geometry shown in Fig. 2, and the mesh size adjacent to the asperity and the crack tip was

0.02 and 0.05 mm, respectively. In this calculation, the case of bilinear kinematics hardening was selected,

where the initial slope of the stress–strain curve is taken as Young�s modulus of the material, and after

yielding the curve continues along the second slope defined by the tangent modulus. The following material

properties were selected: Young�s modulus of E = 210 GPa, a tangent modulus was defined to be equal 10%of the Young�s modulus and, hence, a value of 21 GPa was obtained [58], yield stress ry = 259 MPa, and

Poisson�s ratio m = 0.3. A friction coefficient of 0.1 between the asperity and the specimen was assumed

in this model.

2.5. Deformation displacement examination

The displacement value during the deformation process due to the contact behavior between the annealed

SAE 1015 and the piano wire was specifically investigated. The following special experiment was designed,where the rectangular annealed steel samples were slowly compressed against a piano wire with the electro-

servo-hydraulic system shown in Fig. 4. The contact condition between the piano wire and the rectangular

specimen material was similar to that in the load–COD tests as shown in Fig. 2(b), where the piano wire

was inserted into the semi-circular hole. In addition, to confirm the experimentally obtained load–displace-

ment relation, FE analysis was also performed. Fig. 5 shows the FE model based on the experimental

set-up in Fig. 4, where the mesh size in the region surrounding the B0.90 mm piano wire was selected to be

0.01 mm. The analysis was carried out under plane strain condition to evaluate the displacement change.

The required material properties were the same as those described in Section 2.4.

Piano wire

Load

SAE 1015 steel

8mm

Clip gauge

High hardness steel (HRC55)

High hardness steel (HRC55)

8mm 10mm

B=12mm

SAE 1015 steel

Fig. 4. An experimental set-up used to determine the displacement around the piano wire.

Fig. 5. A FE model used to determine the deformation displacement value (DL).


3. Results and discussion

3.1. Effect of piano wire on the crack opening behavior

Fig. 6 shows the load–COD curves obtained by the application of a load up to approximately 85% yield

load of the specimen. The three plots in Fig. 6 are for the following conditions: (a) without any artificial

asperity, (b) with B0.75 mm piano wire, and (c) with B0.90 mm piano wire. The loading and unloading

curves exhibited apparent hysteresis-like loops, when the asperities were inserted. However, even without

the artificial asperity, a weak hysteresis behavior could also be detected. This may be attributed to the

localized yielding occurred at the crack tip. The maximum difference in the COD between the loading

and unloading phases, indicated by Ddmax, of the samples displays an obvious trend in proportion to the

size of the asperity, i.e., the larger the asperity size, the greater value of the Ddmax.It is of particular interest to note from Fig. 6 that there exists an inflection point on all the unloading

load–COD curves although the trend between Fig. 6(a)–(c) is totally different. The lower portion of the

unloading curve exhibits a convex shape in Fig. 6(a) without any artificial asperity, but a concave shape

in Fig. 6(b) and (c) with an artificial asperity. It should be pointed out that the convex shaped load–

COD curve has been reported in almost all the related publications, as a consequence of the normal crack

closure effect (e.g., near-tip fracture surface roughness-induced closure, oxide-induced closure, etc.). The

theoretical basis for the existence of such a convex shaped load–COD curve has been recently proposed

[43], whereas the concave shaped load–COD curves and the residual COD with the plastic deformationof the specimen material (dRP) will be discussed in the later sections of this report.

To verify the above observations, the load–COD curves were also determined under loading up to 110%

of the bending yield load. The obtained results are shown in Fig. 7: (a) without any artificial asperity; (b)

withB0.75 mm piano wire and (c) with B0.90 mm piano wire. Unlike the results shown in Fig. 6, the upper

portion of load–COD curves obtained in the loading phase (Fig. 7) are clearly curved due to the yielding of

the specimen. This also results in wider hysteresis-like loops shown in Fig. 7. Again, the lower portion of the

unloading curves is very different between the samples with and without artificial asperities, an observation

similar to that seen in Fig. 6.The unloading portion of the load–COD curves was further examined via FE analysis.The load–COD

results calculated with 85% and 110% yield load are shown in Figs. 6(b) and (c) and 7(b) and (c), respectively.

El

0

500

1000

1500

2000

2500

3000

3500

0 0.1 0.2 0.3 0.4 0.5

COD, mm

Loa

d, N

Real zero COD

max

Unloading

Loading

Linear portion

Measurement zero

0

500

1000

1500

2000

2500

3000

3500

0 0.1 0.2 0.3 0.4 0.5

COD, mm

Loa

d, N

max

PR

Linear portion

Unloading

Loading

R

Experiment× FEA

Elastic wedge model [39]

0

500

1000

1500

2000

2500

3000

3500

0 0.1 0.2 0.3 0.4 0.5

COD, mm

Loa

d, N

max

Linear portion

Unloading

Loading

PR

R

Experiment× FEA

astic wedge model [39]

(b)(a)

(c)

Fig. 6. Load–COD curves for an annealed SAE 1015 steel loaded at a maximum value of about 85% of the load corresponding to the

yield load: (a) without any asperity; (b) with a B0.75 mm piano wire and (c) with a B0.90 mm piano wire. (Specimen size:

12.0(W) · 12.0(B) · 80 mm(l).)


Similar to the experimental results, the obtained FEA load–COD data exhibited a concave change with

decreasing the load. While the FEA data in the case of 85% yield load are somewhat shifted to the right-hand

side, the FEA data for 110% yield load are in excellent agreement with the experimental results.

The upper unloading portion shown in Fig. 6(b) and (c) may be understood using the elastic-wedgemodel [39], which was developed with an elastic wedge located adjacent to the crack tip in an elastically

deformed material. In the elastic-wedge model the residual crack opening displacement, dR, after the

applied load is removed, can be expressed [39]

dR ¼ kh11þ k

ffiffiffiffiffiaia1

r; ð1Þ

with

k ¼ 8E1W 1

ph1E0 ; ð1aÞ

E0 ¼ E for plane stress; ð1bÞ

0

500

1000

1500

2000

2500

3000

3500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

COD, mm

Loa

d, N

Unloading

Loading

Linear portion

Real zero COD Measurement zero

0

500

1000

1500

2000

2500

3000

3500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

COD, mm

Loa

d, N

Unloading

Loading

Linear portion

δPR

δR

Experiment× FEA Elastic wedge model [39]

0

500

1000

1500

2000

2500

3000

3500

0 0.1δ

δ

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

COD, mm

Loa

d, N

Unloading

Loading

Linear portion

PR

R

Experiment FEA

Elastic wedge model [39]×

(a) (b)

(c)

Fig. 7. Load–COD curves for an annealed SAE 1015 steel loaded at a maximum value of about 110% of the load corresponding to the

yield load: (a) without any asperity; (b) with a B0.75 mm piano wire and (c) with a B0.90 mm piano wire. (Specimen:

12.0(W) · 8.0(B) · 80 mm(l).)


E0 ¼ E1� m2

for plane strain; ð1cÞ

where h1 is the height of the asperity, W1 is the width of the asperity, al is the distance from the center of

the asperity to the crack tip, ai is the distance of the measuring location from the crack tip, m is Poisson�sratio, E1 and E are Young�s modulus of the asperity and the specimen, respectively. Since the piano wire

was tightly inserted at the maximum load shown in Fig. 6(b) and (c), it would contact the fracture surface

immediately during the subsequent unloading process. This corresponds directly to Stage II in the elastic-

wedge model [39]. It should be noted that, the elastic-wedge model was originally adapted for the asperity

height smaller than or equal to the COD value at the location of the asperity under the maximum load,

although our wedge height was higher than the COD value as shown in Fig. 2(b). Based upon the elastic-wedge model, the equivalent height of our asperity was obtained from the wedge height, or the distance,

prior to the EDM machining between the two points on the top and bottom fracture surface, as shown in

Fig. 2(b). The height for the B0.75 mm and B0.90 mm piano wires was determined to be 0.275 mm and

0.35 mm, respectively. Since the piano wire has a circular shape it is difficult to determine accurately the

Fig. 8. Micrographs of the annealed SAE 1015 steel adjacent to the piano wire after the load–COD test, showing: (i) contact between

the piano wire and specimen and (ii) plastic deformation in the specimen around the piano wire.


value of W1. However, as a first approximation, W1 might be taken from the contact width between the

asperity and crack surface, since W1 is always equal to the contact width in the case of rectangular shapedwedge used [39]. Fig. 8 shows a micrograph of the specimen in the vicinity of the B0.90 mm piano wire

after the load–COD test, where the specimen was loaded to 110% of the yield load. The micrograph in

Fig. 8 indicates clearly that the wire is in contact with the permanently deformed specimen surface, where

the contact width is about 0.18 mm. Based on this observation, W1 could be considered to be approxi-

mately one fifth of the asperity height. If a plane strain condition of the three-point bending specimens

is considered, E1 = E = 210 GPa for both the high strength steel piano wire and the annealed SAE

1015 steel, and m is taken as 0.3, k becomes a constant, i.e., k = 1.6 · (1 � m2)/p = 0.464 according to

Eq. (1a). As shown in Fig. 2, a1 = 2.5 mm and ai = 5.0 mm, the residual crack opening displacement(dR) calculated in terms of Eq. (1) is dR = 0.123 mm and dR = 0.157 mm for the B0.75 mm and

B0.90 mm piano wires, respectively. The dR value could also be obtained from the load–COD curves

shown in Fig. 6(b) and (c). It corresponds to the value from the real zero COD, obtained on the basis

of linear elastic fracture mechanics [39], to the intersection of a straight line passing through the linear

portion of the upper unloading curve with the abscissa. The experimentally determined dR value was

about 0.122 mm and 0.161 mm for the B0.75 mm and B0.90 mm piano wires, respectively, as shown

in Fig. 6(b) and (c). These values are in good agreement with the above theoretical prediction according

to Eq. (1).In addition, the upper unloading portion shown in Figs. 6(b) and (c) and 7(b) and (c) may also be

understood using another theoretical model proposed by Tada and Paris [40]. In this model a nonlinear

portion between Stage I and II is indicated with a convex shape connecting two load points, r1 and r2, asshown in Fig. 1(c). The model is about a center crack partly or wholly filled with a wedge. The range of

this portion between r1 and r2 is dependent on the ratio of the asperity width W1 to the crack length a,

e.g., this range diminishes with decreasing W1/a ratio. From Figs. 6(b) and (c) and 7(b) and (c), it is seen

that no Stage I was present due to the fact that the piano wire was tightly inserted into the crack surfaces

as described above, the potential nonlinear portions based on Ref. [40] were also not observed in theupper unloading portions. This may be due to the fact that the ratio of the asperity width to the crack

length was very small, i.e., W1/a = 0.03 and 0.036 for the B0.75 mm and B0.90 mm piano wires, respec-

tively. It should be noted that, since three point bending specimens were used in this study, the crack

length was taken as ai, i.e., from the bottom edge of the specimen to the crack tip, as indicated in

Fig. 2(a).


3.2. Deformation zone

As seen in Fig. 8, the specimen surface was permanently deformed by the contact action, in which the

surface was clearly dented along the outline of piano wire. Fig. 9 illustrates, at larger magnifications, the

microstructure of the same sample as shown in Fig. 8, where two typical features, i.e. delamination andplastic flow, can be seen. Apparently, more severe plastic deformation occurred in the specimen materials,

while no clear deformation was observed in the piano wire. Such a deformation feature could be eas-

ily understood based upon the following reasons: (i) the yield strength and hardness of the piano wire

are approximately three times higher than those of the specimen material, and (ii) a larger strain occurs

in the local area of the specimen around the asperity due to the contact on the round surface of the piano

wire.

To further clarify the deformation characteristics in the same sample, the work-hardened areas were

examined via the microhardness test and the etching technique. Fig. 10 presents the microhardness mappingindicated by contour lines around the piano wire and the crack tip before and after the load–COD test. To

identify the hardened zones more easily, the contour lines are defined by the ratio of the hardness in the

deformed region to that in the nondeformed material. As can be seen from Fig. 10, before testing

the work-hardened zones, e.g., characterized by the ratio above 1.2, are observed to be restricted only in

the small regions surrounding the fatigue crack and very close to the crack tip. The hardened zones were

obviously expanded after testing, and they are indicated in these figures by aforementioned ratio.

To further reveal the hardened zones, the special etching procedure described earlier was applied. Fig. 11

shows a typical macrograph after etching, where the etching makes the hardened regions darkened. Thus,the plastic zone profiles are clearly seen in the region around the piano wire and the crack tip, respectively.

Fig. 9. Micrographs of the annealed SAE 1015 steel adjacent to a piano wire after the load–COD test, showing: (a) delamination and

(b) plastic flow.

Fig. 10. Microhardness mapping before and after the load–COD test: (a) and (b) adjacent to the piano wire, (c) and (d) around the

crack tip.

Fig. 11. Work-hardened zone revealed by an etching technique indicated by the dotted curve.



In comparison to the hardened zones obtained via the microhardness test, the darkened areas around the

crack tip and the piano wire revealed by the etching technique are more clearly visible.

3.3. Finite element analysis

As mentioned earlier, the deformation areas were further analyzed by FE analysis. Fig. 12 presents von-

Mises plastic strain distribution calculated under plain strain condition: (a) around the asperity and (b)

around the crack tip. It is seen that higher plastic strain areas were found in areas close to the crack tip

and the piano wire; their profiles seem similar to those obtained experimentally. To make a comparison

between the numerically and experimentally obtained work-hardening profiles, the results of the micro-

hardness and etching technique coupled with the FE analysis under the same magnification are shown in

Fig. 13: (a) around the asperity and (b) around the crack tip. It should be pointed out that the two pairs

of comparison, i.e., hardness vs. FEA and etching vs. FEA, were all obtained from the same location andthen illustrated with symmetry image, respectively. From the comparison, the overall plastic strain fields

obtained numerically are similar to the experimental results, in which the etching results are especially in

good agreement with the FEA results.

The results described above clearly indicate that the plastic deformation, induced during the load–COD

testing, is largely produced around the artificial asperity and the crack tip in the sample. Such a plastic

deformation would lead to an acceleration in the reduction of the measured crack opening displacement,

as demonstrated by the concavely curved unloading portion and the smaller value of the residual COD,

dRP [Figs. 6(b) and (c) and 7(b) and (c)]. It is, therefore, concluded from the above results that the yield

Fig. 12. von-Mises plastic strain distribution: (a) adjacent to the piano wire and (b) around the crack tip.

Fig. 13. Deformation zones revealed by the microhardness test, etching technique, and FE analysis: (a) adjacent to the piano wire and

(b) around the crack tip.


strength and hardness of, and their differences between the asperity and the specimen materials as well as

the resultant stress/strain level around the asperity all play important roles in the load–COD characteristics.

The plastic deformation in the vicinity of a harder asperity could result in a significant reduction in the mea-

sured COD value.

4. Fatigue crack closure process with a hard asperity and COD estimation

There are several models [37–43] so far for characterizing the crack closure involving the artificial

wedges. As shown in Fig. 1, either a rigid- or an elastic-wedge is artificially placed in the vicinity of a crack

tip to stimulate the obstruction of crack closure; their stress–displacement relations in Stage II exhibit linear

or linear plus nonlinear variations. However, as seen from the above experimental/numerical results shown

in Figs. 6(b) and (c) and 7(b) and (c), when the stress level around the asperity and the strength ratio of the

asperity over the specimen material are high, the crack closure behavior is strongly affected. Such an influ-

ence on the load–COD relations has not been considered in the earlier models. By combining the presentresults with the crack closure model [39], a modified applied stress–COD process is presented in Fig. 14.

If the applied stress is high, the plastic deformation at the crack tip would occur. This causes a convex

curvature of the applied stress–COD relationship in the loading phase towards the right-hand side, as

shown in Fig. 14. In the unloading phase, the first stage corresponds to the situation where the asperity does

not contact the upper crack face. At rop, the asperity just touches the upper crack face. In Stage II, both the

Stage II

Stage Iσop

δmaxo

σmin

σmax

δ (COD)

δminStage III

σ

σpl

δII/III δop

σII/III

σclosed

σmin, act

Stage I

P1

Stage II

COD Stage III

P1

Fig. 14. Schematic illustrations of the proposed crack closure process including the plastic deformation in the specimen material at the

asperity.


asperity and the specimen material (near the asperity) deform elastically. Therefore, the equation derived in

the elastic-wedge model [39] is applicable, which is also verified by the present experimental results, as

shown in Figs. 6(b) and (c) and 7(b) and (c). At rII/III, the onset of the plastic deformation occurs. From

this point on, plastic deformation of the specimen material in the vicinity of the asperity occur, i.e., Stage

III, since the annealed steel is considerably softer than the piano wire. This is also verified by the observa-

tions of the deformation behavior, examined both experimentally and numerically, as demonstrated in Figs.

8–13.It should be pointed out that even if the strength of the asperity is similar to that of the base material,

localized plastic deformation is still inevitable, as there exists a strong stress concentration near the contact

area between the asperity and the specimen.

4.1. Determination of COD

It can be seen from Fig. 14 that the COD value at zero load should be associated with the plastic defor-

mation behavior in the specimen around the asperity and at the crack tip. To estimate the COD values withconsideration of the plastic deformation, a schematic drawing is shown in Fig. 15. In the figure, COD1 (or

Dc) is the crack opening displacement contributed by the crack tip blunting, which could be approximated

using Irwin�s formula [59,60], whereas COD2 is the crack opening displacement associated with the artificial

wedge with considerations of both elastic and plastic deformation in the vicinity of the wedge. Hence, the

crack opening displacement may be expressed as,

COD ¼ COD1 þ COD2; ð2Þ

COD1 ¼ Dc; ð2aÞ

COD2 ¼ aiLA � Dc � DA

a1

� �; ð2bÞ

where LA is the height of the asperity from the lower crack surface, ai is the distance of the measuring loca-

tion from the crack tip, a1 is the distance of the asperity from the crack tip, and Dc and DA are the displace-

ment at the crack tip and at the asperity, caused by the plastic deformation, which are required to be

determined.

COD

ai

a1

DCLADA

Plastic zone(around crack tip)

2 ry

W

BRound asperity

Crack face

COD1

(DC)

COD2

Plastic zone(around asperity)

Fig. 15. Schematic illustrations of a model proposed to estimate the crack opening displacement caused by the plastic deformation

both at the crack tip and at the asperity.


4.2. Determination of the displacement or blunting value at the crack tip

To assess an approximate value of the displacement caused by the plastic deformation at the crack tip,

Dc, Irwin�s plastic zone correction modified by Dugdale is used here. Then the displacement at the crack tip

can be described as follows [59–61]:

Dc ¼4rE0

ffiffiffiffiffiffiffiffiffiffi2airy

p; ð3Þ

where r is externally applied stress, E 0 is given in Eqs. (1b) and (1c), ai is the distance of the measuring loca-

tion from the crack tip, and ry is half of the plastic zone size at the crack tip.

It should be pointed out that the externally applied stress in this equation is considered for a center

cracked tension specimen. As a first approximation, let us assume that Eq. (3) is also applicable for thethree point bending specimen with the following applied stress:

r ¼ 3PS

2BðW � aiÞ2; ð4Þ

where P is the externally applied load, S is a supporting span, B and W are the specimen thickness and

width, respectively. Substituting Eq. (4) into Eq. (3) leads to an expression for the displacement at the crack

tip, caused by the plastic deformation, in the three point bending specimen as follows:

Dc ¼6PS

BE0ðW � aiÞ2ffiffiffiffiffiffiffiffiffiffi2airy

pð5Þ

with

ry ¼1

2pKry

� �2

for plane stress; ð5aÞ

ry ¼1

6pKry

� �2

for plane strain; ð5bÞ


K ¼ 3YPS

2BW 2

ffiffiffiffiffiffiffipai

p; ð5cÞ

Y ¼ 1.09� 1.735nþ 8.20n2 � 14.18n3 þ 14.57n4; ð5dÞ

n ¼ ai=W ; ð5eÞ
where K is the stress intensity factor for the three point bending specimen, ry is the yield strength of the
specimen, and Y is the geometric correction factor [62].

4.3. Determination of the displacement at the asperity

To obtain the displacement value, DA, the experiment involving the annealed SAE 1015 rectangular sam-

ples compressed slowly against a piano wire was conducted, Fig. 4. The load–displacement curves, obtained

with B0.75 mm and B0.90 mm piano wires, respectively, are shown in Fig. 16. The two curves display

essentially a similar tendency. However, the displacement obtained with theB0.75 mm piano wire is slightly

larger than that obtained with B0.90 mm piano wire in the region with more than 2000 N due to the higher

stress concentration. It is of particular interest to note that the tendency of the curves is similar to the lowerportion of the unloading load–COD curves as shown in Figs. 6(b) and (c) and 7(b) and (c). Thus, the defor-

mation displacement at the asperity, DA, in the load–COD tests may be estimated using the load–displace-

ment behavior shown in Fig. 16. Based on the experimental data, an empirical polynomial equation for the

load vs. displacement behavior in the two cases may be expressed as follows:

DA ¼ 2� 10�11P 3 � 6� 10�8P 2 þ 1� 10�4P � 4.1� 10�3 for £0.75 mm piano wire; ð6Þ
or
DA ¼ 9� 10�12P 3 � 3� 10�8P 2 þ 9� 10�5P þ 3.2� 10�3 for £0.90 mm piano wire; ð7Þ
where P is the externally applied load [N].
To verify the above load–displacement behavior, FE analysis was also performed. The obtained load vs.

displacement curve is shown in Fig. 17, which in general is consistent with the experimentally obtainedload–displacement results.

By combining the above Eqs. (2)–(7) one may express the crack opening displacement as

COD ¼ Dc þaia1

ðLA � Dc � DAÞ; ð8Þ

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0

Loa

d, N

Displacement, mm (∆L)

∅0.90mm

∅0.75mm

Fig. 16. Load–displacement curves for the annealed SAE 1015 steel against a B0.75 mm and B0.90 mm piano wires.

0

500

1000

1500

2000

2500

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7COD, mm

Loa

d, N

Eq. (8)

Experimental data

Unloading portion in Fig. 6(b)

0

500

1000

1500

2000

2500

3000

3500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8COD, mm

Loa

d, N

Eq. (8)

Experimental data

Unloading portion in Fig. 6(c)

(b)

(a)

Fig. 18. Comparison between the theoretical estimates from Eq. (8) and the unloading portion of the load–COD curves in an annealed

SAE 1015 steel loaded up to 85% of the yield load: (a) with a B0.75 mm piano wire and (b) with a B0.90 mm piano wire.

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0Displacement, mm (∆L)

Loa

d, N

Fig. 17. A load–displacement curve analyzed by means of FE analysis (B0.90 mm piano wire).



where Dc is given by Eq. (5) and DA by Eq. (6) or (7) for B0.75 mm and B0.90 mm piano wires,

respectively.

To check the validity of the model described above, the COD values with B0.75 mm and B0.90 mm pia-

no wires were calculated using Eq. (8). The required parameters for the three point bending specimen were

taken from Fig. 2. The specimen thickness of 8 mm was chosen. The externally applied load of P = 2700 Nwas taken from the load–COD curves shown in Fig. 7(b) and (c). If a plane strain condition is assumed for

the three point bending specimen, the ry value is approximately 0.342 mm according to Eq. (5b). Since the

equivalent height of LA would be 0.275 mm and 0.35 mm for the B0.75 mm and B0.90 mm piano wires,

respectively, the corresponding CODs at zero load are 0.089 mm and 0.275 mm. These two values are about

54% and 96% of the experimentally determined residual crack opening displacements, dPR = 0.163 mm and

dPR = 0.285 mm after the plastic deformation or the concavely curved portion is taken into consideration,

as shown in Fig. 7(b) and (c). The proposed equation may be, therefore, used to predict the approximate

crack opening displacement value in Stage III, where the plastic deformation occurs.To further confirm the validity of the proposed equation for Stage III, the theoretical curve calculated by

Eq. (8) was compared to the experimentally obtained lower portion shown in Figs. 6(b) and (c) and 7(b)

and (c). The required geometrical parameters were the same as those described in Fig. 2. It should be noted

that, in the case of loading up to 85% of the yield load, Dc value might be neglected due to the tests carried

0

500

1000

1500

2000

2500

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

COD, mm

Loa

d, N

Eq. (8)

Experimental data

Unloading portion in Fig. 7(b)

(a)

(b)

0

500

1000

1500

2000

2500

3000

0 0.2 0.4 0.6 0.8 1

COD, mm

Loa

d, N

Eq. (8)

Experimental data

Unloading portionin Fig. 7(c)

Fig. 19. Comparison between the theoretical estimates from Eq. (8) and the unloading portion of the load–COD curves in an annealed

SAE 1015 steel loaded up to 110% of the yield load: (a) with a B0.75 mm piano wire and (b) with a B0.90 mm piano wire.


out in the elastic region. Fig. 18 shows the theoretical and experimental load–COD curves, loaded up to

85% of the yield load, (a) with a B0.75 mm piano wire and (b) with a B0.90 mm piano wire. The trend

of the theoretical estimate is similar to that in the lower portion of the load–COD curve, although the val-

ues of the theoretical estimate are positioned at the right-hand side of the experimentally obtained curve.

However, in the case of loading up to 110% of the bending yield load, as shown in Fig. 19, the theoreticalestimate based on Eq. (8) is in good agreement with the lower portion of the experimentally-determined

load–COD curve for both B0.75 mm and B0.90 mm piano wires.

5. Conclusions

The effect of a harder artificial asperity on the crack closure behavior in annealed steel samples has been

studied experimentally and numerically. The following conclusions can be drawn.

1. In the elastic load–COD tests, an inflection point can always be observed on the unloading portion of all

load–COD curves, regardless of whether or not there is an artificial asperity. However, the trend of the

inflection is different in cases with and without any asperity. In the case where the asperity is not inserted,

the lower portion of the unloading curves exhibits a convex shape, whereas a concave shape is found for

samples with an artificial asperity.

2. The concave nature of the unloading curve indicates acceleration in the COD reduction. The change of

the curve shape from convexity to concavity induced by the addition of a hard asperity has been verifiedby different experimental observations such as (a) the effect of asperity size, (b) the load–displacement

behavior via pressing a piano wire against an annealed steel sample, and (c) the elastic–plastic load–

COD tests.

3. The width of the loading–unloading hysteresis-like loops and the residual COD values in the elastic–

plastic tests are both larger than those in the elastic tests. This indicates that a large plastic deformation

or blunting occurs at the crack tip. However, in all the load–COD tests, the plastic deformation in the

vicinity of the piano wire similarly occurred, which has been confirmed via the microhardness tests, etch-

ing techniques, and finite element analysis.4. A modified model on the crack closure process in the case of hard asperities in soft material samples has

been proposed. During unloading three stages of crack closure can be defined: in the first stage, the

asperity does not get in touch with the upper crack face; in the second stage, both the asperity and

the specimen material deform elastically, and thus the elastic-wedge model is applicable; in the third

stage, plastic deformation of the specimen material close to the asperity occurs, leading to the observed

concave shape on the load–COD curves.

5. A semi-empirical equation has been proposed to estimate the COD value, in which the plastic deforma-

tion at both the crack tip and the asperity is taken into consideration. The residual COD value calculatedfrom the proposed equation is found to be in fair agreement with the experimental result.

Acknowledgements

The authors would like to express the appreciation to Prof. M.N.James for helpful comments and sug-

gestions. This work was performed under the financial support of the Natural Sciences and EngineeringResearch Council of Canada (NSERC), Materials and Manufacturing Ontario (MMO). The work was

technically supported by ANSYS JAPAN LTD, ISOYA LTD, JIT Automation Inc. and ROI Engineering

Inc. M.Okayasu acknowledges financial support from the University of Toronto, the Ontario Graduate


Scholarship Award (OGS) and the Award of Association of Japanese Canadian Businesses and Profession-

als. D.L.Chen would also like to thank the financial support received from the Premier�s Research Excel-

lence Award (PREA), and Canada Foundation for Innovation (CFI).

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Effect of a hard artificial asperity on the crack closure behavior in an annealed SAE 1015 steel

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