international journal of solids and...

International Journal of Solids and Structures 113–114 (2017) 108–117

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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: [email protected] (W. Lu).

W

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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


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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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2

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Fig. 2. Shallow cosine-shaped surface topography.

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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)


t

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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


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.

u

g

a

b

s

l

w

s

(

p

F

t

g

s

i

s

a

m

t

t

a

4

4

t

T

d

s

t

t

a

w

A

u

l

t

a

t

T

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


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.


Fig. 9. Finite element mesh of specimen II and distribution of the axial strain under tension.


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.



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)


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.


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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