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International Journal of Solids and Structures 113–114 (2017) 108–117
Contents lists available at ScienceDirect
International Journal of Solids and Structures
journal homepage: www.elsevier.com/locate/ijsolstr
Surface stress concentration factor via Fourier representation and its
application for machined surfaces
Zhengkun Cheng
a , Ridong Liao
a , Wei Lu
b , ∗
a School of Mechanical Engineering, Beijing Institute of Technology, Beijing 10 0 081, PR China b Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA
a r t i c l e i n f o
Article history:
Received 20 September 2016
Revised 12 January 2017
Available online 16 January 2017
Keywords:
Surface topography
Stress concentration factor
Analytical solution
Digital image correlation
Finite element analysis
a b s t r a c t
An analytical solution to the stress concentration factors (SCFs) for slightly roughened surfaces is derived
and validated by Digital Image Correlation (DIC) experiment as well as finite element analysis. Surface
topography is considered as a superposition of numerous cosine waves by means of Fourier transform.
The Airy stress function of a semi-infinite half plate with superposed surface topography under tension
loading is proposed. It is found that the perturbations of stress concentrations obey the superposition
principle under the limitation that the surface topography is shallow. Moreover, the proposed analytical
expression is applied to predict the SCFs of real machined surface topographies, and the prediction is
validated by finite element analysis. The comparisons show that the proposed analytical expression is
feasible in calculating the SCFs of real machined surface topographies. A quantitative relation between
the root-mean-square (RMS) value of SCFs and RMS of the surface profile slope is given.
© 2017 Elsevier Ltd. All rights reserved.
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1. Introduction
Stressed solids often show inhomogeneous stress distribution
in the surface layer as a result of surface topography. This stress
inhomogeneity is characterized by a smaller stress at the peak
of the surface topography and a larger stress at the valley. It is
known that stress concentration caused by surface topography dur-
ing loading can significantly influence surface instabilities such as
fatigue failure ( Taylor and Clancy, 1991 ), resistance to corrosion at-
tack ( Burstein and Vines, 2001 ), and stress-driven surface evolution
( Gao, 1994 ).
The impetus to undertake the present work comes from an
ongoing effort to evaluate the effect of machined surface topog-
raphy on fatigue behavior of high strength steel. In engineering
design practice, the impact of surface quality on fatigue limit is
commonly characterized using empirical reduction factors, which
modify the endurance limit of the material ( McKelvey and Fatemi,
2012; Stephens et al., 2001 ). However, this empirical method es-
tablished by time-consuming and expensive fatigue tests is conser-
vative in predicting the fatigue strength of a structure and lacks
sound scientific basis ( Suraratchai et al., 2008 ). Micro-geometrical
irregularities are known to influence the fatigue performance,
which promote crack initiation through local stress concentrations.
∗ Corresponding author.
E-mail address: weilu@umich.edu (W. Lu).
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http://dx.doi.org/10.1016/j.ijsolstr.2017.01.023
0020-7683/© 2017 Elsevier Ltd. All rights reserved.
he magnification of the bulk stress at the valleys of surface to-
ography plays an important role in triggering the nucleation of
islocations and cracks ( Gao, 1991a ).
Important progresses have been made to quantitatively esti-
ate the stress concentrations imposed by machined surface to-
ography. Early works mainly treated machined surface topogra-
hy as successive adjacent notches and attempted to establish a
elationship between the roughness parameters and the stress con-
entration factor (SCF) of machined surface topography. The Neu-
er rule ( H.Neuber, 1958 ) was considered to be the first expres-
ion for evaluating the SCF of surface topography. Based on the
euber rule, Arola and Ramulu (1999 ) suggested another model
o predict the SCF. These two models were derived from Inglis’s
ork on stress concentration due to an elliptical hole in a plate
Inglis, 1913; Medina et al., 2014 ). In the case of AISI 4130 CR steel
Arola and Williams, 2002 ), the Arola-Ramulu model provided bet-
er estimation of the fatigue stress concentration factor than the
euber rule. The height parameters and the effective valley radius
f surface topography used in these two models are regarded as
he most critical parameters in determining the SCF of machined
urface topography. However, the empirical models based on geo-
etrical average parameters can fail to describe important charac-
eristic of the full stress distribution along the machined surface.
ith the ever increasing computer power, finite element descrip-
ion of measured surface profile was used to analyze the stress dis-
ribution imposed by machined surface topography ( ̊As et al., 2008,
005; Suraratchai et al., 2008 ). Simulations have also been used to
Z. Cheng et al. / International Journal of Solids and Structures 113–114 (2017) 108–117 109
Fig. 1. A semi-infinite thin elastic plate with arbitrary surface topography whose slope is small everywhere.
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Fig. 2. Shallow cosine-shaped surface topography.
g
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tudy complex random rough surfaces ( Medina et al., 2014; Pida-
arti et al., 2009; Turnbull et al., 2010 ). These are expected to pro-
ide more accurate fatigue prediction compared with the empirical
odels. However, the numerical method is time-consuming when
finite element model needs to be built for each rough specimen.
Gao, (1991a,b ) obtained an analytical solution of the SCF in-
uced by a sinusoidal shallow surface. He employed elastic Green’s
unctions for a surface which was considered to be perfectly
at and treated the sinusoidal surface as being perturbed from
hat referential plane to develop the first-order solution of the
tress concentration factor. Based on Gao’s work, Medina ( Medina,
015 ) derived a stress-concentration-formula-generating equation
SCFGE) for arbitrary shallow surfaces, which can be applied to
arious cases including semi-elliptical notches, undulating surfaces,
arabolic notches and Gaussian notches, for the plane stress con-
ition to a first-order approximation. Medina and Hinderliter pro-
osed an analytical solution of the SCF for slightly roughened ran-
om surfaces ( Medina and Hinderliter, 2014 ). In their work, the
urface topography was assumed to have a Gaussian distribution of
eights and auto correlation length (ACL). Gao’s first-order pertur-
ation method, the Hilbert transform and the energy conservation
rincipal related to the Parseval theorem were combined to derive
formula, which showed that the root-mean-square value of SCF
s a function of the RMS-roughness to ACL ratio.
Despite effort s made by these researchers, an analytical solution
or the stress concentration induced by machined surface topogra-
hy is still lacking. In this study, the machined surface topography
as simulated by superposing a series of cosine components using
ourier transform. In addition, a first-order perturbation approach
o analyze undulating surfaces was explored. The analytical solu-
ion of random surface topography was derived by employing the
iry stress function. Furthermore, the analytical solutions of the
tress concentration induced by random surface topography and
achined surface topography were validated by digital image cor-
elation experiment and finite element analysis. For the machined
urface topography modeled by Fourier representation, the root-
ean-square (RMS) value of SCFs was derived as a function of the
MS of the surface profile slope.
. Analytical solutions of SCF induced by shallow surface
opography
Consider the surface profile in Fig. 1 , which is slightly perturbed
rom a flat surface. The profile f ( x ) is a real continuous function
hat satisfies the Hölder condition within its domain. The profile
( x ) can be described by the superposition of numerous cosine
omponents using Fourier transform.
For the discussion in this section, it is assumed that the slope
f the surface is small everywhere, at all points on the surface. The
oundary condition which must be enforced on the wavy surface
s that the traction is zero. If σ ij is the stress field evaluated at
point on the surface and n j is the outward unit normal vector,
he boundary condition is given by σ ij n j =0. To the first order of
he surface slope, the vector n j has components of n x ≈ −y ’, n y ≈. Here y ’ denotes dy / dx . The tangential direction on the surface is
iven by the unit vector s i , which has components of s x =n y and
y = −n x . The normal traction on the surface is σ n =n i σ ij n j . Simi-
arly, the shear traction is σ s =s i σ ij n j . Both σ n and σ s must van-
sh on y = f ( x ) because the surface is free of applied load. When
xpanded to the first-order of the surface slope, these conditions
ecome
yy − 2 y ′ σxy = 0 , (1)
′ ( σxx − σyy ) − σxy = 0 . (2)
When the surface is flat, i.e. prior to perturbation, an equilib-
ium stress σ 0 i j (x, y ) exists in the plate. The stress field with a per-
urbed surface has the form of ( Gao, 1991a )
i j (x, y ) = σ 0 i j (x, y ) + σ h
i j (x, y ) , (3)
here σ 0 i j (x, y ) is the bulk stress with a flat surface in response to
he applied load σ m
. Based on Eq. (3) , the boundary conditions for
he additional stress field, σ h i j (x, y ) , due to perturbation of the free
urface to the shape of y = f ( x ), are given by
h yy − 2 y ′ σ h
xy = 0 , (4)
′ ( σm
+ σ h xx − σ h
yy ) − σ h xy = 0 . (5)
.1. Effect of shallow cosine-shaped surface topography on SCF
To make the discussion more concrete, suppose that the surface
hape is cosine in the x -direction with a wavelength λ and an am-
litude a, or y = a cos (2 πx / λ), as shown in Fig. 2 . The restriction of
mall surface slope implies that a / λ � 1.
For a cosine perturbation and a uniform initial stress field, the
dditional elastic stress is also expected to be cosine in x . The
tress field has the appropriate symmetry if it is derived from an
iry stress function of the form ( Freund and Suresh, 2003 )
(x, y ) = f (y ) cos
(2 πx
λ
), (6)
here f ( y ) is to be determined. The stress function A ( x, y ) must
atisfy the biharmonic equation, which ensures that the stress field
s in equilibrium and the associated strain field is compatible. Fur-
hermore, all stress components must vanish as y → −∞ , which im-
lies that
f (y ) =
(c 0 + c 1
y
λ
)e 2 πy /λ, (7)
110 Z. Cheng et al. / International Journal of Solids and Structures 113–114 (2017) 108–117
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where c 0 and c 1 are constants to be determined by the boundary
conditions. The surface stress components σ h i j (x, y ) derived from
Eq. (6) can be written as
σ ( h ) xx =
∂ 2 A
∂y 2 = f ′′ ( y ) cos
(2 πx
λ
)
=
((2 π
λ
)2
c 0 +
4 πc 1 λ2
)cos
(2 πx
λ
),
(8)
σ ( h ) yy =
∂ 2 A
∂x 2 = −
(2 π
λ
)2
f ( y ) cos
(2 πx
λ
)
= −(
2 π
λ
)2
c 0 cos
(2 πx
λ
),
(9)
σ ( h ) xy = σ ( h )
yx = − ∂ 2 A
∂ x∂ y =
(2 π
λ
)f ′ ( y ) sin
(2 πx
λ
)=
2 π
λ
(2 π
λc 0 +
c 1 λ
)sin
(2 πx
λ
).
(10)
Substituting these stress components into Eqs. (4) and (5) , we
get
c 0 = 0 , c 1 = −aλσm
, (11)
To the first-order in a / λ, the corresponding stress components
along the surface are
σxx = σm
+ σ (h ) xx = σm
− 4 πa σm
λcos
(2 πx
λ
), (12)
σyy = σ (h ) yy = −
(2 π
λ
)2
c 0 cos 2 πx
λ= 0 , (13)
σxy =σ ( h ) xy =
2 π
λ
(2 π
λc 0 +
c 1 λ
)sin
(2 πx
λ
)= −2 πaσm
λsin
(2 πx
λ
).
(14)
The perturbation solutions have been developed by several re-
searchers. Gao derived the solutions using two different meth-
ods based on the stress Green’s function for the elastic half-plane
with a slightly perturbed surface ( Gao, 1991a ) and on the Muskel-
ishvilli’s complex variable representation ( Gao, 1991b ), respectively.
Related expressions can also be found in the work of Srolovitz
( Srolovitz, 1989 ).
The stress concentration of each surface point is given by
K t (x ) =
σxx
σm
= 1 − 4 πa
λcos
(2 πx
λ
). (15)
It can be seen that the maximum stress concentration fac-
tor K t max = 1 + 4 πa /λ occurs in the valley. Note that this shallow
cosine-shaped surface is a special case of the Hölder-continuous
surface function, and the SCFGE proposed by Medina ( Medina,
2015 ) can be used to get the same result.
2.2. Effect of surface topography superposed by numerous cosine
waves on SCF
Surface topography can be regarded as a stationary stochastic
process, and the true machined surface topography can be simu-
lated by superposing a series of cosine components through spec-
trum analysis ( Aono and Noguchi, 2005 ). The true surface topogra-
phy can be expressed as
y (x ) =
n ∑
i =1
a i cos
(2 πx
λi
+ θi
), (16)
where θ i is the phase angle.
The restriction on random surface topography is that the slope
of each surface point should be small, or | y ’| � 1, For a random
erturbation and a uniform initial stress field, the Airy stress func-
ion can be written as a superposition of the Airy stress function
f each cosine component,
( x, y ) =
n ∑
i =1
(c i 0 + c i 1
y
λi
)exp
(2 πy
λi
)cos
(2 πx
λi
+ θi
). (17)
The stress function A ( x, y ) fully satisfies the biharmonic equa-
ion. Furthermore, all stress components must vanish as y → −∞ .
he components of stress σ h i j (x, y ) derived from Eq. (17) can be
ritten as
(h ) xx =
∂ 2 A
∂ y 2 =
n ∑
i =1
((2 π
λi
)2
c i 0 +
4 πc i 1
λ2 i
)cos
(2 πx
λi
+ θi
), (18)
(h ) yy =
∂ 2 A
∂ x 2 = −
n ∑
i =1
(2 π
λi
)2
c i 0 cos
(2 πx
λi
+ θi
), (19)
( h ) xy = − ∂ 2 A
∂ x∂ y =
n ∑
i =1
2 π
λi
(2 π
λi
c i 0 +
c i 1 λi
)sin
(2 πx
λi
+ θi
). (20)
Substituting these stress components into Eqs. (4) and (5) , we
btain
i 0 = 0 , c i 1 = −a i λi σm
. (21)
The corresponding stress components along the random surface
re
xx = σm
+ σ (h ) xx = σm
−n ∑
i =1
4 a i πσm
λi
cos
(2 πx
λi
+ θi
), (22)
yy = σ (h ) yy = −
n ∑
i =1
(2 π
λi
)2
c i 0 cos
(2 πx
λi
+ θi
)= 0 , (23)
xy = σ ( h ) xy = −
n ∑
i =1
2 πa i σm
λi
sin
(2 πx
λi
+ θi
). (24)
Eqs. (22) –(24) indicate that the perturbation stress obeys the
uperposition principle, whose value is the sum of the would-be
erturbation stress corresponding to each cosine surface. The SCF
s given by
t (x ) =
σxx
σm
= 1 − 4 πn ∑
i =1
a i λi
cos
(2 πx
λi
+ θi
). (25)
The shallow surface superposed by numerous cosine waves
eets the Hölder continuous condition. The SCFGE ( Medina, 2015 )
an also be used to calculate the SCF. The derivation of the
CFGE was based on Gao’s boundary perturbation approach and
he Hilbert transform. Under the condition of shallow surface to-
ography, the SCFGE can be applied to any first-order Hölder con-
inuous surface function. After applying the equation, the SCF is
ound to be
t (x ) = 1 − 2 H
(y ′ (x )
)= 1 − 4 π
n ∑
i =1
a i λi
cos
(2 πx
λi
+ θi
), (26)
here H stands for the general Hilbert transform. Using the gener-
ting equation gives the same result of SCF.
The autocorrelation function R ( τ ) of the superposed surface to-
ography is as follows
(τ ) = lim L →∞
1
L
L ∫ 0
y (x ) y (x + τ ) dx =
n ∑
i =1
a 2 i
2
cos
(2 π
λi
τ). (27)
Z. Cheng et al. / International Journal of Solids and Structures 113–114 (2017) 108–117 111
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Specimen II:
Specimen III:
Specimen I:
Fig. 3. The configurations of specimens.
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The power spectrum density (PSD), P ( ω), which is the Fourier
ransform of the autocorrelation, is given by
( ω ) =
1
2 π
∫ + ∞
−∞
R ( τ ) exp ( − jωτ ) dτ
=
n ∑
i =1
a 2 i
4
(δ(ω − 2 π
λi
)+ δ
(ω +
2 π
λi
)), (28)
here δ( x ) is the Dirac-delta function and ω is the angle frequency
f the superposed surface topography.
The moments of PSD is defined as ( Nayak, 1971 )
n =
∫ + ∞
−∞
ω
n P (ω) dω . (29)
Accordingly, the zeroth and second PSD moments of surface to-ography can be written as
0 =
∫ + ∞
−∞
P ( ω ) d ω =
∫ + ∞
−∞
n ∑
i =1
a 2 i
4
(δ(ω − 2 π
λi
)+ δ
(ω +
2 π
λi
))d ω
=
n ∑
i =1
a 2 i
2 , (30)
2 =
∫ + ∞
−∞
ω
2 P ( ω ) dω =
∫ + ∞
−∞
n ∑
i =1
a 2 i
4
(δ(ω − 2 π
λi
)+ δ
(ω +
2 π
λi
))ω
2 dω
=
n ∑
i =1
a 2 i
2
(2 π
λi
)2
. (31)
They can be rewritten in the following form by Parseval’s theo-
em
0 = R
2 q =
1
L
∫ L
0
y 2 (x ) dx, (32)
2 = 2 q =
1
L
∫ L
0
(dy
dx
)2
dx, (33)
here R q is the root mean square (RMS) roughness of the surface
rofile and q is the RMS of the slope of the surface profile.
Similarly, the autocorrelation function of SCFs, or Eq. (25) , is
( τ ) | SCFs = lim
L →∞
1
L
∫ L
0
K t ( x ) K t ( x + τ ) dx
= 1 + 8 π2 n ∑
i =1
a 2 i
λ2 i
cos
(2 π
λi
τ). (34)
The PSD of SCFs is given by
( ω ) | SCFs =
∫ + ∞
−∞
R ( τ ) | SCFs exp ( − jωτ ) dτ
= δ( ω ) + 4 π2 n ∑
i =1
a 2 i
λi 2
(δ(ω − 2 π
λi
)+ δ
(ω +
2 π
λi
)).
(35)
Accordingly, the zeroth PSD moment of SCFs can be written as
m 0 | SCF s =
∫ + ∞
−∞
P (ω) dω = 1 + 8 π2 n ∑
i =1
a 2 i
λi 2 . (36)
Substituting Eqs. (31) –(33) into Eq. (36) , the RMS of SCFs in-
uced by the superposed surface topography is given by
R q | SCF s =
√
1 + 4 m 2 =
√
1 + 42 q . (37)
The RMS of SCFs, R q | SCFs , represents the average stress concen-
ration level. Eq. (37) shows that R q | SCFs is closely related to the
econd moment of surface topography m 2 . This relation provides a
eans of estimating the overall stress concentration level by using
he surface roughness parameters.
. Experimental validation
To validate Eq. (25) , Digital Image Correlation (DIC) experiments
ere carried out. There are various contact and non-contact tech-
iques in the field of experimental mechanics for the measure-
ent of surface deformation and strain. Direct measurement tech-
iques include strain gauge method while non-contact methods in-
lude Moiré interferometry ( Post, 1983 ), holography ( Dudderar and
orman, 1973 ), and speckle interferometry ( Jacquot, 2008 ). Among
hem, DIC is the most popular one. DIC has been extensively used
or displacement and strain field estimation in various applications
uch as material characterization, structural health monitoring, fa-
igue crack growth and high temperature testing. With the ad-
ancement in computational capabilities, more robust algorithms
ave emerged for tracking the material points to estimate whole
eld displacements and strains. With advancements in the image
apturing technology, the DIC technique enables using microscopes
nd high speed cameras to estimate displacement and strain from
he captured images.
Various commercial software are available for 2D DIC to obtain
isplacements and strain fields. Ncorr ( Blaber et al., 2015 ) is an
pen source 2D DIC code based on MATLAB. It is capable of calcu-
ating displacement and strain fields from speckle images ( Harilal
nd Ramji, 2014 ). This section shows the displacements and strain
elds generated by Ncorr using experimental speckle images col-
ected from various experiments with different rough specimens
nder tensile loading.
.1. Specimen geometry and experiment set up
Three specimens with different surface topographies were ma-
hined by a CNC laser machine from a rubber plate of 1.5 mm
hickness. The effective length and width of each specimen were
00 mm and 15.5 mm, respectively. The images of the specimens
re shown in Fig. 3 . The surface topographies for the three speci-
ens are as follows (unit in mm):
Specimen I : y ( x ) = 0 . 5 cos
(2 π
20
x − π
2
);
Specimen II :
y ( x ) = 0 . 5 cos
(2 π
20
x − π
2
)+ 0 . 4 cos
(2 π
16
x − π
2
);
Specimen III :
y ( x ) = 0 . 5 cos
(2 π
20
x − π
2
)+ 0 . 3 cos
(2 π
16
x − π
2
)+ 0 . 2 cos
(2 π
10
x − π
2
)+ 0 . 2 cos
(2 π
8
x − π
2
);
(38)
The surface of the specimens was first coated with a thin layer
f black acrylic paint. A white paint was then sprayed on the black
urface, creating random black and white artificial speckle pat-
erns. A 16 megapixel Olympus PL5 with an Olympus 14–42 mm
112 Z. Cheng et al. / International Journal of Solids and Structures 113–114 (2017) 108–117
1. Camera 2. LED diffusion Light 3. Specimen4. Actuator 5. Instron control panel
2
3
4
5
1
Fig. 4. Experimental set up of 2D DIC measurement.
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b
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ment load.
f 3.5–5.6 lens were used to capture images of the random pattern
throughout the experiment. The base sensitivity (ISO 200) was set
for the camera to minimize the sensor noise and the aperture was
adjusted to f/10. These camera settings required the use of power-
ful light sources. Two LED diffusion lights were equipped to pro-
vide soft and even light. The shutter speed of the camera was set
to its maximum sync speed. A remote shuttle control was used to
focus and capture pictures. An Instron 5900 series machine was
used to apply the tensile load. The experimental setup is shown in
Fig. 4 .
3.2. Experimental procedure
We first performed material testing with a rectangular speci-
men under axial loading, and found that the stress-strain relation
was linear to at least 5% strain. We then loaded the specimens
with curved surfaces from 0 to 5 N and collected a sequence of im-
ages. The stress concentration factor can be obtained by comparing
any one of the images with a reference image. We made sure that
the nominal strain and the strain at the valley were both in the
linear regime. The maximum strain (which was at the valley) in all
the images that we used was about 3%.
Fig. 5. v displacement contour and εyy strain contour for specimen I under tensile lo
wavelength of 20.
The images collected during the experiments were processed
sing Ncorr to calculate the displacement and strain fields. The re-
ion of interest (ROI) is the field to calculate the displacements
nd strains, which need to contain the curved boundary and can
e extracted by Photoshop. For computing displacements by DIC, a
ubset was chosen from the reference image and its corresponding
ocation was tracked in the deformed image. Ncorr was equipped
ith circular subset and its radius was set to 40 with a subset
pacing of 3. Ncorr uses the Inverse Compositional Gauss-Newton
IC-GN) nonlinear solver which is fast, robust and accurate in dis-
lacement measurement compared to classical Newton Raphson or
orward Additive schemes ( Blaber et al., 2015 ). For strain calcula-
ion, Ncorr uses a least squares plane fit on a contiguous circular
roup of displacement data. The radius was set to 3 pixels prior to
train computation.
For sinusoidal deformation with increasing strain gradient and
mposed noise, strain measurement with Ncorr DIC shows a mean
train deviation no more than 7.1 × 10 −4 for different subset radii
nd different strain windows ( Blaber et al., 2015 ). The error is
uch smaller for homogenous uniaxial strain and rigid body ro-
ation. In our experiments we chose the state where the strain at
he valley was about 3%. With this strain level, the estimated strain
ccuracy was better than 2.4%.
. Results and discussion
.1. Specimen I: single cosine-shaped surface
Fig. 5 shows the v displacement contour and the εyy strain con-
our obtained by Ncorr using the DIC technique for specimen I.
his specimen has single cosine-shaped surface topography. The v
isplacement and εyy strain correspond to the displacement and
train along the loading direction.
To examine the results more closely, the εyy strain values along
he cosine-shaped surface topography were extracted. SCFs along
he curved boundary were obtained by dividing the strain over the
verage strain in the ROI. To check the accuracy of the DIC results,
e also carried out finite element analysis of the specimen using
BAQUS. Quadrilateral elements with quadratic interpolation were
sed for the mesh. Uniform displacement load was applied on the
eft and the right ends of the finite element model. Fig. 6 shows
he finite element mesh of specimen I and the distribution of the
xial strain at a particular displacement loading. The SCF was ob-
ained by dividing the strain on the surface to the applied strain.
his ratio is independent of the magnitude of the applied displace-
ading. The curved surface consists of one wave with an amplitude of 0.5 and a
Z. Cheng et al. / International Journal of Solids and Structures 113–114 (2017) 108–117 113
Fig. 6. Finite element mesh of specimen I and distribution of the axial strain under tension.
Fig. 7. Comparison of SCFs along the surface obtained from experiment, finite element simulation and analytical solution. The surface, y (x ) = 0 . 5 cos ( 2 π20
x − π2 ) , consists of
one wave.
Fig. 8. v displacement contour and εyy strain contour for specimen II under tensile loading. The curved surface consists of two waves with amplitudes of 0.5, 0.4 and
wavelengths of 20, 16, respectively.
o
a
4
w
c
t
a
q
e
s
4
w
c
t
a
q
e
s
e
a
p
n
p
Fig. 7 shows the comparison of SCFs along the curved boundary
btained from the DIC technique, from the finite element analysis,
nd from the analytical solutions. They agree well with each other.
.2. Specimen II: surface superposed by two cosine waves
The surface of specimen II was superposed by two cosine
aves. Fig. 8 shows the v displacement contour and the εyy strain
ontour obtained by Ncorr using the DIC technique. Fig. 9 shows
he finite element mesh of specimen II and the distribution of the
xial strain at a particular displacement loading. Fig. 10 shows the
uantitative comparison of SCFs along the surface obtained from
xperiment, finite element simulation and analytical solution. They
how good agreement.
.3. Specimen III: surface superposed by four cosine waves
The surface of specimen III was superposed by four cosine
aves. Fig. 11 shows the v displacement contour and the εyy strain
ontour obtained by Ncorr using the DIC technique. Fig. 12 shows
he finite element mesh of specimen III and the distribution of the
xial strain at a particular displacement loading. Fig. 13 shows the
uantitative comparison of SCFs along the surface obtained from
xperiment, finite element simulation and analytical solution. They
how good agreement. There is slight difference between the finite
lement and the analytical result at the edge. This is because the
nalytical solution is derived under the condition of a semi-infinite
late while the finite element calculation is for a model with a fi-
ite size. However, the difference is small because the surface to-
ography is shallow.
114 Z. Cheng et al. / International Journal of Solids and Structures 113–114 (2017) 108–117
Fig. 9. Finite element mesh of specimen II and distribution of the axial strain under tension.
Fig. 10. Comparison of SCFs along the surface obtained from experiment, finite element simulation and analytical solution. The surface, y (x ) = 0 . 5 cos ( 2 π20
x − π2 ) +
0 . 4 cos ( 2 π16
x − π2 ) , consists of two waves.
Fig. 11. v displacement contour and εyy strain contour for specimen III under tensile loading. The curved surface consists of four waves with amplitudes of 0.5, 0.3, 0.2, 0.2
and wavelengths of 20, 16, 10, 8, respectively.
Fig. 12. Finite element mesh of specimen III and distribution of the axial strain under tension.
Z. Cheng et al. / International Journal of Solids and Structures 113–114 (2017) 108–117 115
Fig. 13. Comparison of SCFs along the surface obtained from experiment, finite element simulation and analytical solution. The surface, y (x ) = 0 . 5 cos ( 2 π20
x − π2 ) +
0 . 3 cos ( 2 π16
x − π2 ) + 0 . 2 cos ( 2 π
10 x − π
2 ) + 0 . 2 cos ( 2 π
8 x − π
2 ) , consists of four waves.
Fig. 14. Amplitude and frequency of the machined surface topography.
Fig. 15. Machined surface topography and simulated surface topography.
5
5
t
r
a
w
b
w
S
p
p
f
q
t
f
c
b
m
t
s
t
c
t
s
2
l
(
√
w
l
l
c
o
v
t
a
|
. Application
.1. Stress concentration of machined surface topography
In this section the analytical solution of the stress concentra-
ion factor for arbitrary surface topography is extended to analyze
eal machined surface topography. The validity of the approach to
ddressing real surface topography was examined by comparing
ith finite element results. A 42CrMo steel bar was first machined
y turning, and then the surface topography of the machined bar
as measured by a TR300 stylus roughness measuring instrument.
ince surface topography can be regarded as a stationary stochastic
rocess, the true surface topography can be simulated by super-
osing a series of cosine components by means of Fourier trans-
orm. Fig. 14 shows the relation between the amplitude and fre-
uency of the machined surface topography. Here we denote the
rue machined surface topography by R ( x ) and the simulated sur-
ace topography by W ( x ). The simulated surface topography W ( x ) is
loser to R ( x ) when more wave components are used.
If the superposed surface topography is shallow, Eq. (25) can
e used to calculate the stress concentration factor no matter how
any wave components to be used. However, it is unnecessary
o take into account very high frequency components of the true
urface topography. In terms of metal fatigue, study has shown
hat with a decrease of surface topography and an increase of in-
lusion size or grain size, the positions of fatigue crack initiation
ransfer from the valleys of the surface profile to the subsurface
uch as persistent slip bands or grain boundaries ( Novovic et al.,
004 ). Surface roughness has a size effect on the fatigue limit. The
ower limit of the size, which is denoted by √
are a c , is given by
Murakami, 2002 )
are a c =
[1 . 43(Hv + 120)
1 . 6 Hv
]6
, (39)
here Hv is a micro Vickers hardness. For surface topography, the
ower limit of defect depth, c min , is given approximately by the fol-
owing equation ( Murakami, 2002 )
min =
√
are a c √
10
. (40)
Because a defect with depth smaller than c min has no effect
n the fatigue limit, surface topography can be coarse-gained. The
ery high frequency components are therefore removed from the
opography profile by meeting the following equation ( Miyazaki et
l., 2007 )
R (x ) − W (x ) | < c min . (41)
116 Z. Cheng et al. / International Journal of Solids and Structures 113–114 (2017) 108–117
Fig. 16. All surface components are rather flat.
Fig. 17. Surface topography and the corresponding SCFs calculated with Eq. (25) .
Fig. 19. SCFs induced by machined surface topography.
t
a
s
(
t
c
5
s
t
w
r
F
e
a
T
1
o
l
F
e
t
o
a
f
Based on Eqs. (39) –(41) , we extracted 73 surface components
from the machined surface topography to simulate the surface to-
pography. Fig. 15 shows the comparison. The solid line represents
the machined surface topography, while the dash line represents
Fig. 18. Finite element m
he simulated surface topography. Multiplying the amplitude, a i ,
nd frequency, f i , of each surface component, we found that each
urface component is rather flat, as shown in Fig. 16 . Therefore, Eq.
25) can be applied to calculate SCFs of the true machined surface
opography. Fig. 17 shows the machined surface topography and
orresponding SCFs calculated using Eq. (25) .
.2. Comparison with finite element simulations
We calculated the stress concentration factors of the machined
urface topography with finite element simulations and compared
hem with the analytical solutions. The model is a square plate
ith a side length of 4 mm. The top is the simulated surface topog-
aphy. The finite element analysis was performed using ABAQUS.
ig. 18 shows the finite element mesh of the model. Quadrilat-
ral elements with quadratic interpolation were used for the mesh
nd mesh refinement was carried out near the surface topography.
he smallest element size near the surface topography was about
.3 μm, which satisfied the requirement of numerical convergence
f stress distribution. A uniform displacement load was applied on
eft and right side of the finite element model.
The SCFs induced by the surface topography are presented in
ig. 19 . The solid line indicates the results calculated by finite el-
ment simulation, which the dash line indicates the results ob-
ained by the Eq. (25) . They are in good agreement with each
ther. Therefore, the proposed analytical formula can be used with
ssured confidence to calculate the SCFs induced by machined sur-
ace topography.
esh used to SCFs.
Z. Cheng et al. / International Journal of Solids and Structures 113–114 (2017) 108–117 117
6
f
t
t
o
P
f
t
a
m
u
F
a
g
c
A
f
c
f
R
A
A
A
A
A
B
B
D
F
G
G
G
N
H
I
J
M
M
M
M
M
M
N
N
P
P
S
S
S
T
T
. Conclusion
In this paper an analytical solution of the SCFs of shallow sur-
ace topography based on Fourier analysis is presented. The solu-
ion was validated by DIC experiments and finite element simula-
ions. It was shown that the perturbations of stress concentrations
bey the superposition principle. We calculated the surface profile
SD moments to connect surface topography parameters and sur-
ace SCFs. It was found that the RMS of SCFs induced by surface
opography is a function of the RMS of the surface profile slope.
We applied the Fourier representation of the surface SCFs to an-
lyze a true machined surface topography. The criterion for deter-
ining the appropriate cut-off frequency was described, since it is
nnecessary to take into account very high frequency components.
inite element simulations were performed and compared with the
nalytical results of SCFs. The two agree well with each other, sug-
esting that the formula can be used with assured confidence to
alculate the SCFs induced by machined surface topography.
cknowledgments
The authors are thankful to the developer of Ncorr, Justin Blaber
rom Georgia Institute of Technology for his assistance in dis-
ussing the setup of DIC experiments. The authors are also thank-
ul to the financial support by the China Scholarship Council.
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