DIGITAL IMAGE CORRELATION FOR
DEFORMATION MEASUREMENTS
NEAR A CRACK
By
SAMPATHKUMAR PERIASAMY
Bachelor of Engineering
Government College of Technology
Coimbatore, India
1999
Submitted 10 the Faculty
of the Graduate College of
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
December, 2002
DIGITAL IMAGE CORRELATION FOR
DEFORMATION MEASUREMENTS
NEAR A CRACK
TheSiSA~~
Thesis Ad'vi,sor
7~~~
~DeanotUraduateCollege
11
ACKNOWLEDGEMENTS
First and the foremost, I would like 10 thank Dr. Hongbing Lu, for his
ex.cellent guidance, inspiration and friendship without which Ihis work would not
have been possible. I am grateful for your patience and support throughout my studies
in OSU.
I extend my sincere gratitude to the members of my advisory committee, Dr.
Ranga Komanduri and Dr. Gary Young for finding time in their busy schedule. I
would like to thank Dr. Bo Wang for his helpful suggestions and ideas whenever
needed.
Further, I wish to thank my parents and my sister for their understanding, love
and support throughout my slay away from home.
I would like 10 thank Mr. Harish Viswanathan, Mr. Gyuha Kim and all other
weblab people for helping me in my research, by creating a pleasant working
atmosphere. Finally, 1 would like to thank the School of Mechanical and Aerospace
Engineering for supporting me during my M.S program.
III
Chapter
TABLE OF CONTENTS
Page
I.INTRODUCTION 1
I. I. Introduction to Digital Image Correlation I
1.2.0vervicw of Digital Image Correlation Method 2
1.2.1.Applicalion of Speckles 3
1.2.2.lmagc Acquisition 4
1.2.3.Correlation Method 5
1.3.Thc.<;is Outline 8
2.THEORY BEHIND IMAGE CORRELATlON 10
2.I.Effects of Mapping Parameters on Subset Around Node 10
2.2.Bicubic Spline Interpolation 12
2.3.Correlalion Method 13
2.3.1. Least Squares Correlation Coefficient 13
2.3.2. NewtonRaphson Method 14
2.4.Processing of Subsequent Nodes 17
3.DIGITAL IMAGE CORRELATION FOR DEFORMATION MEASUREMENTS
NEAR A CRACK 18
3.I.Solutions for Problems Encountered When Specimens had Cracks 19
3.1.I.Crack Detection by Connected Component Labeling 19
3.1.2.Procedure 20
iv
Chapter Page
3.1.3.Conversion of 8bit grayscale image to binary image 20
3.1.4.Thresholding an image 21
3.1.5.ldentifying Connected Components on a binary image 23
3.2.Modification to Lhe Grayscale Interpolation and Least Square Correlation
Algorithm LO Account for Cracks 24
3.3.DeformcdlmageMaker Program 26
3.4.Result~ from ew Algorithm 27
4.COMPUTATION OF IINTEGRAL 29
4.1.Jintegr<11 29
4.2.Calculation of Jintegral Around Crack Tips 31
4.3.Theorics Used for the Calculation of Stress 33
4.3.I.Linear Elastic Model 34
4.3.2.RambergOsgood Model 34
4.3.3.Bilinear Model 36
4.4.Validalion of J integral Algorithm Using Ideal Images 38
4.5.J integral for Actual specimens 42
4.5.I.Uniaxial Tensile Test on Rubbery Particulate Composite 42
4.5.2. Test results from materials which obey bilinear model 44
4.5.2.I.Particulate Composite Specimen Tested at 20% Prestrain 44
4.5.2.2.Particulate Composite Specimen Tested at 15% Prestrain 46
5.VALIDATION OF ALGORITHM USING NANOCOMPOSITE SPECIMENS .48
5.I.Background on nanocomposites 48
5.2.Specimen preparation 49
v
Chapter Page
5.3.Experimental setup 50
5A.Experimental procedures 52
5.5.Results and Discussion 52
6.CONCLUS IONS . 56
REFERE CES 57
APPE DIXES 59
APPE D1X A  WinDIC_LS User's Guide 60
APPENDIX B  Flow of Code for WinDIC 74
APPENDIX C  Flow of Code for Jintegral Computation 76
APPENDIX D  DeformedlmageMaker User's Guide 77
APPENDIX E Images Used in Uniaxial Tensile Test on Particulate Composites 79
APPENDIX F  Images of Particulate Composite Used in 20% Pre·Strain Test.. 81
APPENDIX G ~ Images of Particulate Composite Used in 15% PreStrain Test 83
VI
Table
LIST OF TABLES
Page
4.1.Results for theoretical images..... . , _ .40
4.2.J integral (KPa m) values for Uniaxial Tensile Test on Particulate Composite .42
4.3.S1ress Intensity Factor Values for Uniaxial Tensile Test on Particulate Composite ..43
4.4.Jintegral (Kpam) values for 20% Prestrain Test .45
4.5.Jintcgral (KPam) values for 15% Pre·slrain Test .47
S.I.Fracture Toughness Values of samples ...................................•.................53
VII
LIST OF FIG RES
Figure Page
1.I.Naturally occurring speckle pattems 4
1.2.Speckle patterns generated by spraying paint . 4
1.3.lmage acquisition setup...... . 5
1.4.Simplified demonstration of ole 6
I.S.Grid of nodes to be placed on reference image 7
1.6.Reference image with reference grid on it 7
1.7.Defonned image with grid on it 8
2.I.Bicubic Spline Interpolation for grayscale data 12
3.I.lmage of a specimen with crack 18
3.2.lmage with unconverged nodes around crack 19
3.3.lnput image 21
3.4.Binary image threshold = 69 . 21
3.5.Binary image threshold = 25 22
3.6.Binary image threshold = 106 22
3.7.Identification of independent regions 23
3.8.Cracked specimen with grid of nodes 25
3.9.Nodes lying in the crack had been removed 25
3.10.Bicubic Interpolation for modified correlation 26
3.II.Theoretical image used for computation 27
VIII
3.12.Theoretical image with nodal grid placed on it 28
3.13.Theoretical image with nodes falling in 10 the crack removed 28
4.1.Jinlegral contour. 30
4.2.Direction of right outward normal 32
4.3.Direclion of left oUlward normal 33
4.4.Reference ideal image 41
4.5.Deformed Ideal image 4I
4.6.Plot of K Vs Distance for uniaxial tensile tcst on particulate composile 44
4.7.Stressstrain curve for 20% pre~strain Icst 45
4.8.Stressstrain curve for IS% preslrain lest 46
S.I.Specimen dimensions 49
S.2.Threepoint bending specimen 50
5.3.lnstron machine setup 51
5.4.Reference image 52
5.5.Deformed image 52
5.6.Strain distribution nanocomposite specimen with 3% nanoclay loading 54
ix
Uo , Vo
Uy , Vy
IV
p
s
x,y
x,Y
NOMENCLATURE
translation mapping parameters; u(x,y), v(x..v).
first order mapping gradients; au (x,y), dv (x,y). ax ax
first order mapping gradients; all (x,y), av (x.y).
ay ay
second order mappm. g grad·lents; a'2" (X,y ), a2', (X,y ). ax ax
second order mappm. g grad·tents; a'",(x,y), a'~v (.t.y).
OJ OJ'
second order mappl.llg grad·lents; a'"(x, y), a'v (x. y).
axay axay
height or grayscale value offset.
represents all thirteen mapping parameters.
represents all points in subset around a node.
represents a single point in the subset at location (x,y).
subset point location in reference image.
Subset point location in defonned image.
dislance from node 10 subset point
x
C, filC, filfilCleast square correlation coefficient, gradient, and Hessian (second gradient).
distance from current node to next node.
principal strains.
XI
CHAPTER I
INTRODUCTION
1.I.Introduction to Digital Image Correlation
Digital Image Correlation (Ole) is a method for measurements of surface
displacements and displacement gradients in materials under deformation. It is a noncontact,
full field deformation measurement method. that can be used in a variety of
situations that include high temperature situations. It is also appropriate for relatively soft
materials where strain gauges are not appropria!c for deformation measurement The
method is applicable for both infinitesimal and large deformations. It was originally
proposed by a group of researchers at the University of South Carolina (Peters and
Ranson, 1982, Sutton, et ai, 1983 and Bruck. ct ai, 1989). Later work has been done by
Knauss and Vendroux (1998), to refine and optimize the basic algorithms for use in inplane
and outof plane deformation measurements.
Recently Lu and Cary (2000), extended Ihe previous DIC methods by accounting
for the second order displacemenl gradients. Ganesan (2001), has included third order
terms in Ihe deformation mapping. This higher order approximation provides a better
measurement of infinitesimal slrain in relatively nonlinear deformation situations. It has
also provided a method to measure first and second order strain gradients. However
problems were encountered in obtaining deformation data near a crack. The reason
behind this is the inability of the present algorithm to converge near the crack tip.
Sutton et a!.. (1991), have reported a reduced accuracy in detemlining J+inlegral
around a crack. They auribute this reduced accuracy in determining the path independent
Jintegral to the inability of finding the displacements close to the crack. Knauss and
Gonzalez (1998) have reported the same shortcoming when using DIC to study the
inhomogeneity in particulate composites.
This thesis work describes the methods to overcome this difficulty. The algorithm
will first detect the crack edges and then take into account, the presence of the crack in
deformation measurements through image correlation. The crack is detected by the
method of Connected Component Labeling. After the crack is detected the nodes falling
into the crack region are excluded in ole for displacement computations. An algorithm
for finding the Jintegral for linear elastic materials, materials obeying RambergOsgood
model, and bilinear materials are also proposed. The proposed methods are validated by
performing correlation on theoretical images generated using Deformed image maker
program. Results obtained for rubbery particulate composites are also presented.
Experimental images acquired in three·point bending tests on Epoxy nanocomposite
specimens with cracks are also presented to validate the Ole.
1.2.0verview or Digital Image Correlation Method
There are three steps involved in carrying out ole method to correlate two
images, one undeformed and one deformed to determine deformation field. They are:
(I). Application of speckles to the specimen, it is noted that for specimens already
possessing surface gray scale patterns. no surface treatment is required. In micro· and
nanoscale deformation measurements, the polished surface followed by chemical
etching produce sufficient gray scale patlem for Ole.
2
(2). Image Acquisition. On a macroscopic scale a digital camera can be used to
acquire Images. On microscale an optical microscope or an SEM can be used to
acquire micrographs. On nanoscale AFM or TEM or other microscopy instrument
can be used to acquire images representing surface topology or surface gray scale.
(3). Correlation of reference and deformed images. Using the code developed from an
appropriate algorithm. in this study, a code named WinDle developed in Dr.Lu's
laboratory is used. In the sequel a detailed description of these steps is provided.
I.2.t.Application of Speckles
For computing the deformations some random texture or paucrns are needed on
the specimens. These are called speckles. Some specimens have naturally occurring
speckles so thai images of the specimen surface can be used for correlation directly.
Figure 1.1 (a) shows some naturally occurring speckle patterns on rubbery particulate
composites and Figure l.1(b) is that of nanocomposites. Some specimens do not have
naturally occurring speckle pattern. Then the speckles for such specimens are to be
applied manually. The speckles can be generated by spraying paint by holding the paint
can at a distance. Figure 1.2 shows the speckle patterns generated by spraying painl. Care
should be laken to see that the painl creates speckles with random gray scale pattern. The
speckle size should be optimized so that each speckle has a characteristic length of about
a few pixels. The density of the speckle pauem should be high enough for the accurate
correlation process of a gray scale pattern around a point. Locating a subset of points
found on the reference image, in the deformed image, carries out the mapping of a point
on the reference image on to the deformed image. This is the reason for having a random
pattern of points on the specimen surface.
3
(a) Rubbery particulate Composite. (b) Nanocomposite.
Figure 1.1: Nalurally occurring speckle pauems.
Figure 1.2: Speckle patterns generated by spraying pain!.
1.2.2.1mage Acquisition
The image acquisition system used in this study consists of a KodakES camera
mounted on a tripod. The camera is connected to a computer, which has the software for
acquiring images of the specimen as the test proceeds. A Techniquip Corp, R 150 A2
fibre optic light source is used to illuminate the specimen surface so as to enable the
camera to capture the image. Figure 1.3 shows the image acquisition set up for capturing
images of specimen during test.
The software used for capturing images is called "CAPTURE". This software has
provisions for capturing the images at regular intervals of time. A total of 99 images can
be captured for one timer file. The software starts capturing the images once the trigger is
4
given by striking a key. Once the timer is set, the images will be acquired automatically.
Thus the images for the image correlation procedure are captured.
Camera
Figure 1.3: Image acquisition SClUp
1.2.3.Correlation Method
Data
acquisition
computer
Light
source
DIC method uses two images of the specimen under deformation for correlation.
The images are the reference and defonned images. The reference image is the image
acquired in the undeformed state. Deformed image is the image acquired after the
specimen has undergone deformation. Figure 1.4 illustrates how the correlation of two
images is carried out. Let A be a point on the reference image, which moves to point B in
the deformed image. The displacements in the X and Y directions are U and V
respectively. These displacements are for one particular point on the specimen. If the
deformation along the surface of the specimen is to be observed a number of points in the
fonn of a grid can be placed on the reference image. Each point on the reference grid can
be mapped from reference image to the deformed image to get the deformation along the
surface of the specimen. So the reference grid placed on the reference image defonns to
different shapes for a deformation value of 1.0 for different defonnation gradients.
5
tV
o~
A_
B•
.
Reference Image Deformed Image
•••••••••••••• •••••••••••••• •••••••••••••• •••••••
Reference grid
•••••••••••••• •••••••••••••• •••••••••••••• •••••••
Uy =1.0
•• •••••••••••• •• •••••••••••• •• •••••••••••• •••••••
U... =1.0
•••• •••••••••• • •••••• :..::::.::: •••
v... = 1.0
•••••••
•••••••
•••••••
••••••• ••••••• •••••••
Vy = 1.0
Figure 1.4: Simplified demonstration of DIe.
DIC is carried out by comparing the defonned image with the reference image.
There exists a distinct grayscale pattern around a point where deformation will be
computed. The grayscale pattern can shrink, expand or rotate, but the basic shape does
not change in the deformed image afler the deformation. By tracking (he movement of
the unique grayscale pattern on the defonned image the deformation can be determined.
This process is performed by a code using DIC algorithm described in Chapter 2.
6
A set of poinls, or equivalent nodes, is needed on both images (0 carry Ou(
correlation. A grid of nodes is placed on the reference image and a mapping is performed
on each of Ihe~e nodes (0 capture the same point on the deformed image. Figure 1.5
shows a grid of nodes that is placed on the reference image to carry out Ihe correlation on
(he deformed images.
•••••••••••••• •••••••••••••• •••••••••••••• •••••••
Figure 1.5: Grid of nodes [0 be placed on reference image.
The number of nodes in Ihe grid can be changed according to the extenl of the
correlation to be carried out on the specimen. Figure 1.6 shows the reference image with
the grid on it before Ihe correlalion is carried out. The grid of nodes have to be placed on
Ihe reference image in such a way Ihat the grid is totally on the image of Ihe specimen.
Figure 1.6: Reference image with reference grid on it.
Figure 1.7 shows the deformed image with the deformed grid after the correlation
has been carried out. A sel of points around a node in Ihe grid is considered.
7
Figure 1.7: Deformed image with grid on it.
Using these random points a particular node is identified on the deformed image.
This is repeated for each and every node on the grid. The fundamental level of calculation
is at the node level, so the whole algorithm is for each node.
1.3.Thesis Outline
In Chapterl, an introduction to the DIC and a simplified demonstration of the
correlation process has been provided. In Chapter 2, a detailed discussion of the Iheory
behind DIC is provided. Discussion on least squares correlation coefficient; NewtonRaphson
method and bicubic spline interpolation method is provided. Chapter 3 presents
a discussion on the problems experienced with DIC when used for specimens with cracks
and a method to overcome the problems associaled with such specimens. A detailed
explanation of the modified correlation method is also provided. Chapter 4 discusses the
Jintcgral calculation approach and the results obtained with DIC for particulate
composites in various lest conditions. Chapler 5 discusses the results obtained from
threepoint bending tests conducted on Epoxy nanocomposites to validate the improved
algorithm for deformation measurements are provided. From the results it was found that
8
the algorithm could be used for nanocomposile materials for evaluation of strain. Chapter
61islS the conclusions based on the current study.
9
CHAPTER 2
THEORY BEHIND IMAGE CORRELATIO
This chapter deals with the theory for Ole. The interpolation techniques and the
correlation using NewlonRaphson method, the Iwo major sleps in ole. are explained in
detail.
2.1.Effects of Mapping Parameters on Subset Around Node
A subset of points around a node is used for measurement of displacements for the
node. Let us consider a subset of points around a node thaI is mapped from the reference
image to the deformed image. Each of these subset points is located in the reference
image at (x, y) and is mapped to the deformed image at location (x, y) using
x= x+U(x,y),
y = y+V(x,y)
where U is the displacement in the x direction and V is the displacement in the y
direction of each subset point. The terms U and V when approximated using first order
Taylor series expansion around a poinl (xoo Yo) leads to the following mapping functions
x= x+Uo+U"L1x +U... L1y,
y= y +Vo +V"L1x +V,.Ay
where
Ax=xxo•
Ay=yyo
10
Ux. V•. UI • VI' are in.plane components of the fir~1 order displacement gradient.
The two terms Uoand Voare displacement components of ('o.Yo)' There are seven
mapping parameters involved in the first order Taylor series expansion they are
Uo. Vo.V x' V•. U" V,. and w. Likewise there are mapping parameters calculated for second
order and third order Taylor series to compute the deformation of the specimen.
The second order Taylor series approximation was carried out by Lu and Cary.
2000. The terms U and Vwhen approximated using second order Taylor series expansion
around a point (xo' Yo) leads to the following mapping functions.
 I 2 I •
x = x +U0 +U~LJx+UyL1y +"2Uut1x +"2U "LJy +U"LJxLJy.
1 2 I 2 y=y+Vo+V.LJx+V,.LJY+"2VaLJx +"2V,.,.L1Y +V",L1xL1y
There are thirteen mapping parameters involved in the second order Taylor series
expansion, they are Uf), Vo, U¥, Vx, U).• V)., U.n. V.u, U).)"> V).,., Uno, V,n.and w.
Likewise the third order Taylor series approximation carried out by Ganesan
(2001), leads to the following mapping functions,
There are twenty three mapping parameters involved in the third order Taylor
series expansion, they are Vo, Vo• Ux , Vx, Up V.I. UXX, VU • V.,.. V)')"> U¥)" V¥),. Uxxx. Vxxx, U)')')'.
II
2.2.Bicubic Spline Interpolation
The images are interpolated using bicubic spline interpolation to achieve sub·
pixel accuracy. Figure 2.1 shows a bicubic spline surface, Sutlon, Turner et al (1991),
that has been interpolated through a set of image pixel grayscale values. The interpolation
technique uses a third order polynomial 10 detennine grayscale values al locations
between the pixel locations; the third order polynomial has a C2 cOntinuolls gradients of
the grayscale value field.
The grayscale value at any location in the interpolated region of the reference
image can be calculated using the equation,
3 3
g(x.y) = L:J.:a•.x· y'
"'=0 ....0
where a_ are Ihe coefficients of Ihe bicubic spline interpolation. They are
determined by using the grayscale values of the comer points of any rectangular surface
patch on the bicubic surface.
• Pixel Location
Figure 2.1: Bicubic spline interpolation for image grayscale data
12
The thineenth mapping parameter IV comes in the form of a grayscale value offset
of the bicubic spline interpolation for the deformed image. This is introduced to
compensate for the errors introduced by changes in the light intensities when the image~
are acquired for image correlation.
J J
lz(x,y,P) = ILP......r·y .. +w
...:0,,:0
For laler simplification, lei P represent a vector wilh all thineen mapping parameters
as its components (in case of second order Taylor series being used). The number of
mapping parameters depends on the order of Ihe Taylor series used to carry out the
approximation. For second order Taylor serried approximation,
The gradients are needed to carry out the correlation process, where the
minimization of the least squares correlation coefficient is carried out to find the mapping
parameters.
2.3. Correlation Method
2.3.1.Least Squares Correlation Coefficient
The values of the mapping parameters are 10 be determjned to obtain the
deformation values in the sample. This is carried out by Least squares correlation. Least
squares correlation method works by minimizing the sum of the squares of the errors.
Here in this case the difference between the grayscale values of the reference and the
deformed image is taken as the error. When this difference tends to zero the point on the
reference image can be reached on the deformed image. Let S represent all the points in
the subset and Sp represent any single point in the subset. Let g(Sp) represents tbe
13
grayscale value of a particular poim in the subset on the reference image and h(Sp)
represents the grayscale value of the same point in the subset on the defonned image.
Then correlation coefficient is defined as
L: !g( 5,)  "(S"S)}'
C = S~d
The summations are perfonncd over all the points of the subset region. From the
equation the range of values for C is from lO, 00], where the optimum value of C is
rcached when g = h which leads to c ".0. The set of P, which minimizes the correlation
coefficient C, are in fact the parameters of the mapping caused by the deformation in the
sample.
To find the minimum of C all the gradients of C must converge to zero. The
thirteen gradients of Care
The NewtonRaphson method can be used to solve for the roots of these gradients
simultaneously.
2.3.2.NewtonRaphson Method
NewtonRaphson method is an iterative procedure for finding the roots of an
equation. It requires initial guesses for variables that are to be detennined. A set of linear
equations must be solved using the initial guess for equations of the form,
x =x _ !(x.)
.1 "/(X.)
14
f(X~)is the value of the gradients(VC) at the current location. {(X
n
) is the
value of VVC at the current location. User supplies the initial guess for the mapping
parameters of the first node. The Xn+l are the incremelH values for the next location. In
the present case the roots are being detennined for the gradient of the correlation
coefficient, 'Vc. The VVC is called the Hessian matrix of the correlation coefficient.
The Crout Lu (Gerard, Wheatley, 1994) decompo~ition method is used to solve
the set of linear equations. Subsequent nodes use the resuh~ from the previous node as the
guess values. The Hessian matrix of the correlation coefficient is given by,
vvc= a'c
ap,ap,
: 'I ah(Sp, P) ah(Sp, P)
'Ig(Sp)s,.s ap, ap,
S~cS
g(x,Y) "'" h(x,y,P) when the initial guesses for P are close to the actual solution then we can
make the following approximation,
Then the Hessian matrix of the correlation coefficient can be approximated as,
vvc= 2 'I ah(Sp,p) ah(Sp,p)
'Ig'(Sp)s,.s ap, ap,
s~"s
The approximate Hessian matrix is sufficient in this case because we assume that
the initial guesses for P are close to the actual solution. The partial derivatives of the
grayscale values are functions of both the displacement mapping and the bicubic
15
interpolation of the deformed image pixel values. The partial derivati\'e~ can be evaluated
using the chain rule,
ail(S,.p) ail(x.Y.P) ax(S,) ail(x,y,P) ay(S,) ail(x. v,P) ':"= . + . + .
a~ ax a~ ay a~ a~
The cqLlat.lon,s lor caIeuIat'mg the partJ.aIden'vatl.vcs for a,,(s,..P.)u.smg second
a~
order deformation gradient, for each point Sp are as follows,
a" ail = au ax
ail ail = av ay
ail ail =.1x au, ax
ail ail =LlxL1y
au" ax
ail ail =.1x avo ay
The equations for calculatI.ng the part.Ial d.env.atlves for ail(S"p. u)S.ing th.'Ird
ap,
order deformation gradient, for each point Sp are as follows,
.l!L =1 ail Llx' ...Q!L =1 ail Llx' .l!L =1 a~ .1y' ...Q!L=lail Ll/
au= 6ax av= 6ay aUm 6ax aVm 6ay
~=!~'Lk'4 ~=! <:v. £1<'4 2L=! ill. L!t4" ~=! ilIl £I< 4" au", 2Cl< av", 2<ry au,,, 2Cl< aV;,. 2<ry
~=LlT ~=Lly ail =I
OW;r aW, aW
16
The and daYh lerm~ 0fht ":''': pam·aIdenV·a·llves are the gradients of the bicubic spline
interpolating polynomial from lh..: ddonned image,
aaixl= p21+ pnY+ p11.,"+ p!~Y, + 2P31 X + 2P12 xy+2P3Jxy,·+2P\.IX.\', +3pw\"'·
+ 3P42X2y + 3!3wr2 y2 + 3{3.1.1;2 y',
~ = Pll +2f313Y +3PI~y2 +{322; + 2Pnxy +3P24XY2+ finx! +2{31j;2 y +3f3~x2y2
+P.I2X, +2P.UX,Y +3P+IX'y '•.
2.4.Processing of Subsequent Nodes
After the minimization of the correlation coefficient is done, the process is
repeated using the next node location on the grid of nodes. The values obtained in the
previous calculations are used as the initial guesses for the next node, but the previous
values of the displacements Uo and Vo are adjusted for the next node's guesses using the
deformation equations
I 2 1 2 U. =U.+U 6.x:+U !:iy+U Ar +U...,l1y +UnArl1y,
/I A '·2>;1: 2N ,
where
!i.x: = xxo'!i.y = yYo·
More closer the guess as to the location of the next node in the deformed image
can be made if the above adjustment can be added into the next node's initial translation
parameter values.
17
CHAPTER 3
DIGITAL IMAGE CORRELATION FOR DEFORMATION
MEASUREMENTS NEAR A CRACK
When ole is used on specimens without cracks the resuhs for displacement
gradients are accurate. All the nodes placed on the deformed images can normally
converge very well. Problems arose when the correlation was used on specimens with
cracks on them. The nodes do not converge and the results obtained are inaccurate.
Figure 3.1 shows an image of the specimen with a crack on it The specimen used is
rubbery particulate composite. Resolution of the image is 512x486 pixels. The width of
the actual specimen is I inch. The image has 166 pixels per meter which is the
conversion factor used for the calculation of lintegral.
Figure 3.1: Image of a specimen with crack
The following problems are ofleo encountered in ole in the vicinity of a crack.
18
1) The nodes surrounding the cracks did not converge and so the deformation results
could not be obtained in those regions. Figure 3.2 shows the image of the specimen with
the deformed nodal grid on it. There are a number of unconverged nodes surrounding the
crack. This is because the present algorithm has not considered the nodes in the crack
area and is unable to converge to solutions for the nodes near the crack edges.
Unconverged
odes
around the
crack
Figure 3.2: lmage with uncooverged nodes around crack
2) The calculation of Jintegral around cracks was affected because of the nonconvergence
of nodes around crack, as the Jintegral computation depends upon the
displacement results for each node that lies along a path chosen for Jintegral
computation.
3.I.Solutions for Problems Encountered When the Specimen had Cracks
3.I.I.Crack Detection by Connected Component Labeling
The aim of this process is to identify and mark or label independent connected
regions. Connected components labeling is a method by which an image is scanned and
pixels are grouped into components based on pixel connectivity. The pixels in a
component share similar pixel intensity values. After grouping the pixels, each of the
pixels is labeled with either of the 1\\10 colors, black or white. Once the grouping is done
19
the regions with black pixels are extracted. This forms a connected componenl and that is
the crack in this case. The method used to detect the crack relies on the fact that the crack
appears as a region of either higher or lower grayscale as compared to its surrounding
area. The image is first converted from a grayscale image to a binary image by the
operation of thresholding the grayscale of an image. All pixels with a grayscale value
above a certain threshold are made white and the ones with lesser grayscale black. Thus
leading to connected components.
3.1.2.Procedure
An image is scanned pixelby·pixel from top to bottom and left to right. so as to
identify the connected regions. Connected regions are the adjacent pixels. which share the
same intensity value v. For a binary image v ={ I It whereas for a grayscale image v can
take a range of values for the different gray scales.
Connected component labeling method needs a binary image as input. In the case
of DIC an 8bit gray scale image is used. So it is to be converted into binary image by
taking the black and white pixels on the image into consideration.
3.t.3.Conversion of 8bit Grayscale Image to Binary Image
The method used to detect the crack relies on the fact that the crack appears as a
region of either higher or lower grayscale as compared to its surrounding area. The image
is first converted from a grayscale image to a binary image by the operation of
thresholding the grayscale the image. All pixels with a grayscale values above a certain
threshold are made white and the ones with lower grayscale values black. Thus black and
white regions are identified on the image. The black and white regions are used to form
20
independent connected regions on the image, which will serve as the input for the
Connected Component labeling.
3.1.4.Thresholding an Image
To threshold an image the user provides a particular graysc.t1e value as the
threshold value. A binary image has only two components of color namely black and
white. Based on the threshold '·alue the black and white components in an image are
identified. The pixels with grayscale values less than the threshold values are taken as
black and the pixels with grayscale values greater than the threshold value are taken as
white. So when all the pixels on the image are replaced using this procedure, it results in
an image with just two intensity values namely black and white. Thus a binary image is
created from a grayscale image. Figure 3.3 shows an 8bit grayscale image of the epoxy
nanocomposite. The image caplUred by the camera is converted into 8·bit grayscale
image with the help of Paint Shop Pro software. Figures 3.4  3.6 show the conversion of
a grayscale image into binary images when different threshold values are chosen .
Figure 3.3: Input image
21
•
. ..
'_:.'~....
Figure 3.4: Binary image threshold =69
_1'
Figure 3.5: Binary image threshold == 25 Figure 3.6: Binary image threshold = 106
Figure 3.4 shows the resulting Image of Connected Component Labeling when the
threshold value chosen is 69. There are some black spots left on the image. As there are
many pixels on the image with grayscale values less than 69. many dark spots are formed
on the image. Figure 3.5 shows an image that results when the threshold value is 25. It
can be seen from the image that as the threshold value decreases the crack detection the
resulting image has lesser number of black spots on it. Figure 3.6 shows an image with
threshold value of 106. It can be seen thai as the threshold value is high the image
obtained does not result in good crack detection. The image shows a lot of black spots,
which have to be removed in order (0 obtain images on figure 3.4 or figure 3.5. This can
be achieved by either doing a rerun with different threshold value or with the erase option
in the DIe software to remove the spots. An optimum value of threshold value must be
chosen to obtain good results, which can be done by running the crack detection with
different threshold values and selecting the value, which produces an image with lesser
number of spots.
22
3.1.5.Identifying Connected Components on a Binary Image
The aim of this process is to identify and mark or label independent connected
regions. A connected region is made of adjacent pixels that share lhe same intensity
value. In this process, each pixel of the picture is assigned a number. also called a label.
All pixels in a connected region are assigned the same label. The process works by
scanning the image pixel by pixel, left to right. top to bottom. When a dark pixel p is
encountered, two of its neighbors. one to the left of p and the one above p are examined.
If both neighbors are white, a new label is assigned to p. If only one neighbor is dark, its
label is assigned to p. If one or more of the neighbors are dark, one of the labels of the
two neighbors is assigned to p and a note is made of the equivalence of labels of the two
neighbors. After completing the scan, the equivalent label pairs are processed and a
unique label is assigned to each connected region.
• White •
White • ." Pixel being
analyzed
• • New label
• Dark •
White•
Pixel being
analyzed
• • Dark
Figure 3.7: Identification of independent regions
23
The dark pixels. which are clustered in the fashion shown in the Figure 3.4 lead to
many independent connected regions. The area of each independent connected region is
determined by counting the number of pixels with the same label.
With this count of area for each label, all the pixels in the image whose label has
an area less than a particular threshold area value are identified and removed. This will
remove all the dark speckles. which are lesser in area compared to the crack, as the crack
will have the largest independent connected region area, producing an image with just the
crack
3.2.Modification to the Grayscale Interpolation and Least Square Correlation
Algorithm to Account for Cracks
The presence of cfClck in the subset region affects the mapping of the subset points
on the reference image to that of the deformed image. As the crack is not a part of the
material and as it does not deform just like any other material point in the subset, the
algorithm does not converge for nodes that fall into the crack region. As the crack is not a
part of the material, the nodes in the crack region will nOt have any deformation values.
As the nodes in the crack do not dislocate to the same extent as those of the specimen, the
algorithm does not converge to give accurate results. So they should not be taken into
consideration for deformation computations. Modified correlation works by excluding all
the points that lie in the crack region at each stage of iteration.
Figure 3.7 shows an example of a specimen with computation nodal grid. Note
that there are some nodes into the crack region. In computation, the crack surface does
not contribute to the defonnation field. So the nodes in the crack region should be
removed from the grid such that they are not taken into consideration of deformation
24
calculation. Figure 3.8 shows the specnncn with the nodes lying In the crack region
removed. So the present interpolalion algorithm should be changed to account for the
nodes removed from the subset.
:;•~~•j'•~~'~'~~'•~'•~'•~;'j•'~•'~~ • .•.•.•.•.••.•.•.
• ••• • • • • • • •• • •••
•'"•'''•'~•'''•'''.'. .• .. • ;'~•:'~•';•'~;''';'~•:'•~'~~r
• • • • • • • • • • • • •••••
·• ..•.••.•.•.• •.•.••..• ••••• ~".:..,.".:..:...:.~
Nodes on the
material
Sllrf:'lC:~
Crack edge
• • • ••••
• ••••••••
• •••••••
•
• •••••••••
• •••••••••
• •••••••••
• •••••••••
• ••••••••
• •••••••••
• ••••••••••
• ••••••••••
•• ••••••• •
• • • • • •• • ••
• •
• •
• •• •
• ••••
• ••••
• ••••
• •
•
• ••••••
• ••••••••
• ••••••
•• • ••••••
• •• • ••••
• •••••
• ••••
• ••
• • • •
• ••
• • • • • •• •
• •••••••
Nodes outside • •
the material
Sll rfac.~
Figure 3.8: Cracked specimen with grid of nodes
• •••••••••••••••••••••
• • • •• • •••••••••••••••
• •••••••••••••••••••••
• •••••••••••••••••••••
• • • • • • •• • ••••••••••••
• •••••••••••••••••
• •••••••••••••••
Grayscale values
of these poinls
are used for
correlation
• ••
• ••••
• • •••••••
• ••••••••••
• ••••••••••••
• ••••••••••••
••
•
• ••••••
• ••••••••
• •••
• •••
• •••••• • • • • • • • • • • • • • •
..•...•...••..•.: •• • • ••••••••••• ..;.;.:.~.~.~;.j.~.~;.. . ..... • •••••••••••••
• • • • •••••••
• ••••
• ••••
• ••••
• ••••
• ••••
Figure 3.9: Nodes lying in the crack had been removed
The new subset for image correlation is then
SNow
where,
25
S~~f represents the set of point:. in crack region in reference image.
S~~J represents the set of points in crack region in deformed image.
S is the subset of pOints.
• Pa~d Locacion • Pu;el LOCUlOll.
•
Figure 3.10: Bicubic Interpolation for modified correlation
The correlation coefficient is [hen given by
c
Crack images generated by the connected component labeling mcthod are used to find
SNew at each itcration.
3.3.Deformedlmagemaker Program
Deformed Imagemaker program is a Visual Basic program, whjch takes the
deformed image as input and produces the reference image with the given input
parameters. The input parameters are the displacement in the x direction, displaccment in
the y direction and the first order and second order gradients. It is tracking backwards
from the deformed image to the reference image. The User's guide for this program is
provided in Appendix G.
26
3.4.Results from New Algorithm
Theoretical images are used to validatc the new algorithm to compute
defonnations near the crack region. A theorctical image of 512x486 pixels was
generated using Defonnedlmagemaker program and used in the WinDle program. Figure
3.9 shows the theoretical image used for validation of the new algorithm. The image
shows the presence of a crack on the specimen. The input parameters for this image are
E =200 Gpa, K =1.40 G pa . .r,;; and v =0.3 .
Crack in the
specimen
Figure 3.11: Theoretical image used for computation
Figure 3.10 shows the theoretical image with the nodal grid for computation.
Figure 3.1 t shows the theoretical image with the nodes that falls into the cracked region
removed, after computation. Moreover the nodes thal are ncar the edge of the crack have
also converged. This serves as the validation for the algorithm to take the presence of crack
in to account.
27
Nodal grid
on the image
Figure 3.12: Theoretical image with nodal grid placed on it
Nodes falling into
the crack region
have been removed
from computation
Figure 3.13: Theoretical image with nodes falling into the crack removed
28
CHAJYfER4
COMP TATIONOF JINTEGRAL
4.I.Jintegral
Jintcgral IS a key parameter that has been developed 10 define the fracture
conditions in a component experiencing both elastic and plastic defonnation. A
mathematical expression for the characterization of a line of the local stressstrain field
around a crack fronl. The J~inlegral expression for a twodimensional crack in the xy
plane with the crack front parallel to the y·axis is the line integral:
R
au.) J= WdyT;'ds
r ih'"l
where W is the strain energy density. 5 is the line coordinate, T is the traction vector and
II is the displacement vector.
Figure 4.1 shows the crack and the path along which the lintegral is calculated. It
is noted that Jintegral is related to the energy in the vicinity of a crack in the presence of
plastic deformation. Fracture of the specimen occurs when the Jintegral reaches a critical
value. The line S is the path along which the Jintegral is 10 be computed. Then ds is a
small segmenl of the path. n is the direction of the normal 10 the path. The frame along
with the contour shows the coordinate system used with respect to the crack.
29
From figure 4.1 it can be seen that the path starts from the lower edge of the crack
and ends on the upper edge. This is how lht; palh should be chosen for Jinlegral
compulation.
n
ds
A
CraCk~
Figure 4.1: Jilllcgral contour
r
Jintegral is formulated for non linear elastic materials. Non linear elastic
materials are similar to plastic materials during loading. Only restriction is that unloading
is not permitted on these materials.
Jintegral plays an important role in the assessment of fracture and fatigue
performances of struclures. II is used to measure the stress intensity factor around crack
tips, which can be used to correlate the initiation of crack propagation. Jintegral can be
used to characterize the initiation of crack in a specimen.
Rice has shown that Jintegral is pathindependent. The strain energy
determination using J at crack tip is valid as long as there is no unloading. Jintegral
cannot be defined under cyclic loading conditions also. As such Jintegral has
significance in terms of defining the stress and strain conditions for crack initiation under
30
monotonic loading conditions and also in pre.<>enc(' of a limited amount of stable crack
extension.
J is related to rate of change of potential energy with respect to change in crack
size (Ashok Saxena., 1998). This interpretation i!lo u~eful in showing that under linear
elastic conditions. J=g, the Griffith's crack exten:.ion force. For either linear or nonlinear
elastic conditions, J is the energy made a\'aibble at the crack tip per unit crack
extension. The relation between J and K, the stress intensity factor is,
K'
J =  for plane stress and
E
K 1 (I_v1 )
J = for plane strain. where v is the Poisson's ratio.
E
Jintegral uniquely characterizes the crack tip stress fields. As it is path
independent. it can be measured at points remote from the crack tip. This property is very
important when the calculations involve numerical solutions, as they are often not
accurate in the immediate vicinity of the crack tip but increase in accuracy as one moves
away from the crack tip.
4.2.Calculation of Jintegral Around Crack Tips
The image correlation process gives the displacements in the corresponding
directions namely Ux• U" Vx and V,.. The strains in the x, y and xy directions can be
computed as
31
E :.!.(V +U )
" 2' ,
(Xl' Y. ) is the coordinate of the current node.
(x2 , )'2) i~ the coordinate of the next node.
The unit nonnals can be calculated by using the following forms.
Case 1: For outward normal to be on the right hand side
Figure 4.2: Direction of right outward normal
Path along which J.
intcgml is calculated
Normal
Direction
Crack
Case 2: For outward normal to be on the left hand side
fI = (Y2  Y.)
~~(x,x,) 2 +(y,y,)2
32
Path along
which J
integral i~
calculated
Crack
Figure 4.3: Direction of left outward normal
The traction components arc in the following form:
then
T, =(yuTlJ+(Ynll~
Ty = Un·fl.• +Unll,
As can be seen from the fonnula for traction components, the stress components
need to be calculated. The computations of stress components from strains are described
for several materials as follows.
From the values of traction and stress the Jintegral can be calculated along the
path. The area under the stress~strain curve can be used as the rough verification for the
Jintcgral value obtained.
4.3.Theories Used for the Calculation of Stress
In this pan, the computations of stress components are described, these stress
components are used in calculating stress. Jintegral was calculated using the above fonn
based on three theories for calculatjng stress. The three theories used were:
33
4.3.l.Linear Elastic Model
The equation for calculating the stress components using linear elastic model i:o.
where ), is the Lame's constant. it can be computed in tenns of Young's Modulus E and
Poisson's ratio v, Cij are the components of strain, 0ij is the Kronecker Delta.
,1= vE
(I + v)(I 2v)
where J1 is the shear modulus. it can be computed by
4.3.2.Ramherg·Osgood Model
E
/1=.
(I +v)
This model can be used for materials that show a non linear behavior when the
material is under deformation.
The power law relation for this model in one dimensional stress state is given by
U
£ =.(J < (Jo.
E
where, f is the strain. 0' is the stress in psi, a is the hardening coefficient, n is the
nonlinearity index, E is the Young's modulus of the material and 00 is the yield stress.
So =££0' where£o is the yield strain.
34
Let p= 23c'+(c"+, r,+: 2c'n) .
e3= P.
ec=IOO.
Steps I to 4 arc carried out till «ec  P) fee) reaches a value less than tolerance
(0.OOOOOOOO I).
(5 5)
3) 53 =51 + 2 I .(e3el)
(e2  el)
5
Let es=_' ,
p
I
Let A=.
es
B =.'C +.!(!..._2), £ 2 es E
35
Then the components of stress are given by,
I
<122 =(8£. + A£,)
F
Thus the components of stress are evaluated from strains usmg the Ramberg·
Osgood model. The validation of this algorithm .done is also carried out separately. The
strain values are given as input and stresses are calculated. Then the reverse calculation is
done to get the strain values.
4.3.3.Bilinear Model
The governing equation for a bilinear material is given by,
where
v is the Poisson's ratio, Q.=EJE, Es is the secanl modulus, oij is the Kronecker
delta, Sij are the deviatoric stress components defined by,
1
S 0 <1 d ij  'I 3 I'Ut II
. .. ~a.
The stress at the crack tip reaches mfimty. So the value of . approached unity.
a,
36
Based on the assumption the equation reduces to
I [ 3 1 1 C .. ::: (I+V)CT.. I"LCT +(a l)s.'1
I) £ IJ IJm" 2 11J
The method for obtaining stress vaJues from strain~ i:. c~plaincd with respect to bilinear
model. The following are the steps involved in arriving at the final equalion for stress.
Let i =j =I.which yields,
Therefore.
let E
P~I
E,
Therefore.
Similarly for i = j =2
and when i =j =3
37
Let
P
D=l'2
when the above ~I of equations are written in matrix form we get.
[
to" 1 1[(I + P)  D  D [U" 1
Ell =ED (I+P) DJ U"
£1\) 0 D (I+P q33~
The symbolic form of the above equation is given by,
I kl=IM(u]
£
From this the stress matrix can be found using the following relation,
4.4 Validation of J integral Algorithm using Ideal Images
Ideal images were generated using Deformed image maker with the following
input parameters:
Stress intensity factor K =: lAO G pa . .r;;; ,
Poisson's ratio::: 0.3,
Young's Modulus E::: 200G po.
38
With the above value" for parameters a displacement field was prescribed to the
images using Deformedlmagcmaker program. The theoretical displacement fields is
given by the asymptotic displacement field in mode I crack problem.
U=K Ji,;(I+v{(2KI)cos(0 )co{30)~.
2£ 2JT 2 2 J
v=K Ji,l;l+v)[(2K+I)sln(0 )Sin(30)~•
2£ 2JT 2 2 ~
where, U is the displacement in the x direction and V is the displacement in lhe y
direction,(r,B) is the position wilh respect to the crack tip, K is the slress intensity factor.
K=34v
3~v
for plane strain. K=(
I + v)
for plane stress, v is the Poisson's ratio, in this case
plane strain condition is considered, £ is the Young's modulus.
Figure 4.2 shows the reference image that was generated using the deformed
image, which is shown in Figure 4.3. The J integral was calculated using the constitutive
law of the linear elastic material. The results are shown in Table I. The fracture
toughness of a material and the J integral are related by the following equation
K'
}=£
The K values for lhe theoretical images were calculated from the abovementioned
equation. The ideal images actually does not show the presence of a crack, but
when the images are examined it can be found that reference image looks like it has been
shrinked by holding the specimen along its edges, and the deformed image looks like it
has been expanded, thus accounting for the crack. The prescribed value of K, the stress
intensity factor is 1.4 Cpa·;;; .
39
Table 4.1 shows the results that were obtained for the theoretical images with
WinDle. The results show that the prescribed fracture toughness (K) values are
recovered. The maximum error is 12.40%, which suggests that the J integral algorithm is
validated using ideal images. The first path is the path that is close to the crack tip. where
the J integral values obtained is not very accurate. As the path move outwards the error
involved in the recovery of K is reducing and it remains almost a constant.
Table 4, I: Results for theoretical images
Path J integral (pa' m) K (po . .,J;;) Error in K (%)
7519639.061 1226347346 12.403
2 8445931.041 1299686965 7.165
3 8962927.645 1338874725 4.366
4 8921503.58 1335777196 4.587
5 8906648.316 1334664626 4.666
6 8852798.987 1330623838 4.955
7 8811407.305 1327509496 5.177
8 8757930.825 1323475034 5.466
9 8708797.780 1319757385 5.731
10 8670696.449 1316867226 5.938
II 8607571.163 1312064874 6.281
12 8874825.444 1332278157 4.837
13 8523293.326 1305625775 6.741
40
Figure 4.4: Reference ideal image
Figure 4.5: Deformed Ideal image
41
4.5. J integral for Actual Specimens
4.5.1. Uniaxial Tensile Test on Rubbery Particulate Composite
lmages were obtained from tests performed on actual specimens with cracks.
The materials for these specimens were rubbery particulate composites. The test
conducted on the specimens is the uniaxial tension test. The images obtained were
transformed into uncompressed 8bit grayscale images for using in WinDlC. The images
used for the test are shown in the APPE 'DIX C. The tests results obtained from this test
are shown in table 4.2. The results for the last 4 images are shown in the Table 4.2. The
material properties are, £ = 474 IKPa , v =0.499 '£'0 =0.092.
Table 4.2: J inlegral (KPam) values for uniaxial tensile lest on particulate composite.
Distance
(m) 04.tif 05.tif 06.'if 07.'if 08.'if 09.tif 10.tif 12.lif
0.030 0.216 0.404 0.183 1.655 0.662 2.105 1.802 2.500
0.038 0.237 0.476 0.898 1.934 1.715 2.357 2.268 2.698
0.046 0.246 0.571 1.028 1.684 2.029 2.499 2.295 2.942
0.053 0.231 0.491 1.061 1.842 2.069 2.527 2.595 3.122
0.061 0.255 0.493 1.077 1.895 2.016 2.619 2.549 3.308
0.068 0.270 0.541 1.119 1.867 2.149 2.543 2.611 3.191
0.076 0.287 0.610 1.144 1.778 2.163 2.621 2.716 3.332
0.084 0.292 0.539 1.180 1.883 2.300 2.644 2.904 3.440
0.091 0.292 0.607 1.184 1.952 2.273 2.719 2.898 3.468
Then K values for the specimen is calculated using the J integral values that
were obtained from the WinDIC test. Table 4.3 shows the K values for different images.
42
The values suggest that the variation in K values for a particular image is less. which
suggests that the J integral is path independent. Thus the path independent propeny of J
integral is verified. Moreover the critical value for K occurs at the time of propagation of
the crack in the specimen, which in the present case occurs from image 5 to 6.
The critical value of K was found to be 64KPa· J;;; from the force at the time
when the crack initiates during the test. The values of K for image 5(K5) and image 6{K6)
suggest that the average of the K lies in the required value region. Thus the K values
have been recovered and at the same time the Jintegral values are constant which
suggests that the algorithm is validated. Due to the inhomogenity of the paniculate
composite material a variation of 20% is allowed in the value of JintegraJ.
Table 4.3: Stress intensity factor ( KPa· .j;;; ) for uniaxial tensile lesion particulate composite.
Distance
(m) K4 K5 K6 K7 K8 K9 KIO K12
0.030 31.967 43.781 29.453 88.566 56.009 99.890 92.432 108.854
0.038 33.526 47.492 65.226 95.754 90.159 105.693 103.687 113.095
0.046 34.139 52.005 69.812 89.354 98.066 108.827 104.305 118.083
0.053 33.065 48.263 70.921 93.446 99.022 109.451 110.905 121.643
0.061 34.766 48.340 71.445 94.778 97.757 111.421 109.917 125.224
0.068 35.789 50.622 72.838 94.083 100.916 109.788 1I 1.256 122.990
0.076 36.897 53.756 73.637 91.797 101.261 I 11.454 113.473 125.681
0.084 37.200 50.548 74.772 94.475 104.409 I I 1.959 117.325 127.686
0.091 37.200 53.640 74.911 96.190 103.791 113.524 117.208 128.218
43
"B
75
u
.!!
~oc
SO
.!!
of;
oo~
25
en
Figure 4.6 shows the plot of fracture toughness versus path number for
differenl images in leSI 10. The plot shows thai the variation of fracture toughness for
different images after inilial paths is less. The initial paths are Ihe palh~ Ihat are very
close to the crack tip where the J integral calculations are not so accur;.l.le because of some
unconverged node near Ihe crack tip.
". .
DE.co 100
'0" '"
o 04.tif
A OS.tit
'i7 06.lit
(] OT.lit
~08.li1 e 09.li1
 lO.til
1~~n!<....'~n!;;;'~~'n!< ...~"'::~';=~'~'~"J 0"
0.04 0.06 0.08 0.1
Distance (m) from crack tip
Figure 4.6: Plot of K Vs Distance for uniaxial tensile test on particulate composite.
4.5.2.Test Results from Materials Obeying Bilinear Model
4.5.2.I.Particulate Composite Specimen Tested at 20% Prestrain
Images from tests carried out on bilinear materials are used in WinDIC for
obtaining results based on bilinear model. Figure 4.7 shows the stressstrain curve for
uniaxial tension test with 20% prestrain, which obeys bilinear model. The dashed line
shows the actual curve. The first linear region has a modulus of 2034 KPa. The second
region's modulus led the secant modulus (Es) is 1330 KPa. These values are supplied to
the WinDIC program with the help of the user interface.
44
 e: 2034 KPe (Flrstlin... ~)
70
60
50 . 0
" 40 ~
~ 30 en
20
'0
~
~
00 0.05 0.1
Strain
,I
I
,I ,,
Ie,. 1330 I(P.
/ (Second Wnnr
I region) , ,
0.15
Figure 4.7: Stressslrain curve for 20% prestrain lest
Table 4.4: Jinlegral (KPam) values for 20% prestrain lest
Distance (m) 10 II 12 13 14
0.030 0.018 0.098 0.300 0.342 0.440
0.038 0.025 0.172 0.368 0.413 0.495
0.046 0.033 0.239 0.455 0.557 0.495
0.053 0.038 0.233 0.427 0.539 0.556
0.061 0.044 0.257 0.467 0.591 0.616
0.068 0.037 0.228 0.424 0.546 0.614
0.076 0.036 0.258 0.439 0.553 0.627
0.084 0.037 0.277 0.460 0.554 0.616
0.091 0.040 0.293 0.482 0.580 0.646
45
The Jintegral values obtained from WinDle for 20% prestrain test is shown in
Table 4.4. The images, which were used in this compUlation. are shown in APPENDiX
D. The crack propagation in this tesl stans on image l2.tif. The Jintegral value obtained
on this image serves as the critical value of J.
4.5.2.2.Particulate Composite Specimen Tested at 15% Prestrain
Next images from another set of images of material obeying bilinear model were
used 10 compule Jinlegral values. The stress strain curve for second set of malerial
obeying bilinear model is shown in figure 4.8. The modulus for the first linear region is
J889 KPa and thai of the second linear region is 1510 KPll. Table 4 shows the Jintegral
values oblained from WinDle on 15% prestrain tesl images. The images, which were
used in this computation. are shown in APPENDIX E. The crack propagation for this test
starts on image J4.tif. The Jinlcgral obtained for this image serves as the critical value of
J for this lest.
 __  e .. 1889 KP.(Flrstfinel,regionj
70
60
..50 0
~ 40
~e
 30 '"
20
'0
00 0.05 0.1
Strain
I
I
I
I
I
I
I
I
/
/
/ e... 1Sl0KP.
(Second U"..r
region)
0.15
Figure 4.8: Stressstrain curve for 15% preslrain test
46
Table 4.5: Jintegml (KPam) values for 15'k preslmin lest
Distance
(m) 12 14 15 16 17 18
0.030 0.054 0.20\ 0.308 0.368 1.243 0.928
0.038 0.099 0.356 0.629 0.872 0.9\2 0.974
0.046 0.130 0.374 0.664 0.901 0.866 0.927
0.053 0.\46 0.4\8 0.720 0.910 0.951 1.056
0.06\ 0.154 0.433 0.771 0.947 1.049 1.117
0.068 0.\68 0.469 0.816 1.003 1.100 1.\53
0.076 0.196 0.498 0.859 1.041 1.\56 1.213
47
CHAPTERS
VALIDAnON OF ALGORITHM USING NANOCOMPOSITE SPECIME 'S
The improved algorithm for measuring deformation near a crack lip is evaluated
using test performed on bending specimens. The material chosen for the test is epox.y
nanocompositc. A series of threepoint bending tests were conducted on nanocomposite
specimens. The deformation fields were observed and convergent results of deformations
were obtained near crack lips. The nodes falling in to the crack region have also been
removed from deformation computation. Thus a validation of the improvements to ole
was carried out using nanocompositc specimens.
S.J.Background on Nanocomposiles
Nanocompositcs are an emerging class of mineralfilled plastics thai contain
relatively small amounts (s 10%) of nanometersized clay particles. The definition of
nanocomposite material has broadened significantly to encompass a large variety of
systems such as onedimensional. twodimensional, threedimensional and amorphous
materials. made of distinctly dissimilar components and mixed at the nanometer scale.
The class of nanocomposite materials is a fast growing area of research.
Significant effort is focused on the ability to obtain control of the nanoscale structures via
innovative synthetic approaches. The properties of nanocomposite materials depend on
both the properties of their individual components and also on their morphology and
interfacial characteristics.
48
5.2.Specimen Preparation
The epoxy resin Epon82. a bisphenolA derivative made by Shell. is first heated to
600e to lower the viscosity. since lower viscosity promOles enhanced dispersion. The
nanoclay particles (MMT) are then hand mixed IOgcther wilh the resin for about 20
minutes. Adding a few drops of the mixture onto a glass laboratory slide and pressing it
together with another slide check dispersion of the nanoclay. A light source is used
against those slides to check for solid particles. If Ihe slides are found to be a clear
mixture then the dispersion is complete.
When the epoxynanoclay mixture is ready, curing agent and accelerator are added
into the resin. Once this mixture has been thoroughly mixed, the mixture is then degassed
and cured on a mold accordingly. Thus the nanocomposite specimens are prepared.
The prepared materials are then machined into standard threepoint bending
specimen. The specimen geometry is shown in Figure 5.1. ASTM E399 was used 10
design the sample. The basic dimension, namely the width is laken to be 17mm.
1+8.5
171..~_1
1,D0,t1

52 I D
7.62
L357
I 17
All dimensions are in millimeters
Figure 5.1: Specimen dimensions.
Samples were machined and sharp cracks were introduced for each specimen by a
new razor blade. Annealing is carried out to relieve the residual stresses present in the
49
specimen due to machining and various proce~!>es carried Ollt on the specimen. The glass
transition temperature of the epoxy nanocompo,)ilcs is 130°C. The specimens were
annealed at 125°C. Specimens were prehealed to a temperature of 100°C before being
subjected to a temperature of 125°C. The specimen,) were maintained at 125°C for 4
hours so that structural rearrangements can lake place. Then the temperature was broughl
to room temperature. After annealing the specimens were held in dessicator at 50± 2 %
relative humidity level for a period of 3 days. All specimens had approximately the same
aging times prior to testing.
Figure 5.2: Threepoint bending specimen.
5.3.Experimental Setup
The loading setup is shown in Figure 5.3. An MTS threepoint bend fixture is
placed on the lnstron machine. Care is taken to sec lhal the fixlure is placed in a rigid
manner to prevent any shaking of the fixture, to avoid inaccurate results.
In this test the crosshead moves down as the load is to be applied on lhe specimen.
The magnitude of compressive force range is SCi from 0 N to 1120 N as the specimen can
break between these loads. The crosshead speed is set to 2 mm/min. The crosshead
moves downwards until the specimen breaks. The centertocenter span between the
supporting rollers is set to 68mm according to ASTM standards
LabView is used for capturing the data from the Instron machine. Data such as load
and extension are read from the machine. The time inlerval between each data acquisition
50
can be controlled ll<;ing the provIsions in the soflwarc. The Data Acquisition (DAQ)
board has provisions for capturing the data from the machine to the computer.
Before the dala was acquired the load cell was calibrated with DAQ using deadweight
to facilitate in getting the correct data from the Instron machine to the computer.
The calibration can be used to update the equation in the .vi diagram in the LabView file.
The data such as load applied to the specimen and the displacement of the specimen can
be acquired from (he LabView software.
Threepoint
bending
specimen
Threepoint
bending
fixture
Figure 5.3: Inslron machine setup.
The data acquisition setup is made ready before the crosshead of the Instron
machine is moved down. Image acquisition intervals are set and the LabView data
acquisition is also made ready. As the crosshead moves down the images are acquired
until the specimen breaks. The load al which the specimen breaks is obtained from the
data file generated by the LabView software. The length of the crack is measured from
the fractured surface after the test to get accurate value of the length of the crack. The
experiments were repeated for epoxy nanocomposite specimens with different set of
loading of nanoclay particles.
51
5.4. Experimental Procedures
The reference and the deformed image used in the calculation of Jintegral are
shown in figures 5.4 and 5.5 respectively. The reference image has the reference nodal
grid on it. Figure 5.5 shows that the nodes that fall into the crack region have been
removed. Most of the nodes surrounding the crack have all converged to give results of
strain near the crack edge.
.. ~., ..·
"
•'
. . '
•
'.
·,
,
,"
;".,'t: ~.' .
',.
, ~..
"· '.'. ~.... · , '
~~": ...~ , .,·..,... .. ' .
,.,'.,._...._..". ... ~...... . ' . .. _~..",
4': :.. ~r".
,,,';.,,,,,.. , . , . . .....,:_.~~.~.;:J...". . 't... ,......." ~ .... ' ' .. ,. ~.' ~. '"do' , .... ·"")".A.· '. ,.s.
...~ ."" lIrt·,,.,,. n"tl_"".
Nodes falling
into the crack
region have
been removed
Figure 5.4: Reference image. Figure 5.5: Deformed image.
Digital image correlation is used to find the strain energy rates (G(e) and compare
the values with the fracture toughness value obtained from the fracture toughness tests.
G =Kic
IC E
where KIe is the fracture toughness and E is the young's modulus.
5.5. Results and Discussion
The fracture toughness of the threepoint bending specimen is determined using
the relation by Lee and Yee (2000),
Y =1.93  3.07(a IW) + 14.52(a IW)'  25.11(al W)' + 25.80(al W)'
52
where Y is the shape factor, P is the load at failure. S is the length of the span (68 mm). a
is the crack length and W is the width of the specimen (17 mm).
Table 5.\: FraclUre Toughne~" Values of samples
Nanoclay loading in Nanocomposites (%)
o
3
5
7
Fracture Toughness (MP" j;;;)
2.52
1.53
1.39
0.90
Table 5.1 shows the fracture toughness for nanocomposites with different
nanoclay loadings between 0% and 7%. The fracture toughness values are seen 10 be
decreasing with the increase in the percentage of the nanoclay panicles in the epoxy
resin. The brittleness of the specimen appears to increase with the increase in the addition
of nanoclay particles to the neal resin.
Figure 5.6(a) shows the distribution of strains in the X direction. The strains were
plotted using Teeplot 9.0. The contours show the distribution of strain along the specimen
near the crack region. These strain values were obtained from each node on the nodal grid
to be ploned by Tecplo!. From the figure it can be seen thaI the strain values very near to
the crack can be evaluated with the present algorithm. Thus it can be seen that the nodes
falling into the crack region have been removed from computation. The strain values are
very low as the images acquired were well before crack propagation in the specimen. The
crack origination and breakage of the specimen takes place within a fraction of a second.
As the specimens were very brittle, crack propagation cannot be seen on the images.
53
Strains in the region
around crack have
been evaluated
Crack in the
specimen
Slrains in the region
around crack have
been evaluated
Crack in the
specimen
(b) E,.,
Strains in the region
around crack have
been evaluated
Crack in the
specimen
Figure 5.6: Strain Distributions for Nanocomposite specimen with 3% nanoclay loading.
54
Figure 5.6 ~how, the contour plOl for the strains in the specimen during the test.
The strain values are nOl 'ymmetric about the crack region.
55
CHAPTER 6
CONCLUSIONS
I.An improved algorithm for taking the cracks into account is proposed and validated.
The algorithm can take the presence of crack into account for computing Ihe
deformations in Ihe specimen. The validation was carried out using ideal images
generated with the Deformed Image maker program. This shows that Ihe WinDle can be
used (0 take any type of cracks into accounl. As the crack could be detected on the
specimen the deformation values near the crack edges can also be detected. Thus the ole
can be used 10 obtain deformation values near a crack, which was not possible with the
old algorithm.
2.The nodes that fall into the crack region are removed from computation, and are not
taken into account for calculation of deformation. In the previous algorithm there was no
provision for removing the nodes that fall into the crack region, from defonnatioll
computation. This problem has been overcome with the present algorithm.
3.A method for computing Jintegral using ole is proposed and validated. The property
of path independence of Jintegral can be seen from the results obtained for the test
shown. The results obtained for the tests are almost constant with very little variation.
This variation can be attributed the numerical algorithms that are used in the
computations.
56
REFERENCES
Bezerr3. L.. and Medeiros, J.M.S.. "Using boundary elements and Jinlegral for the
detenninalion of K] in fracture mechanics," IABEM (2002).
Bouchard, P.J., GoJdthorpe, M.R., and Proney. P.. "Jintegral and local damage fracture
analyses for a pump casing containing large weld repairs." International Journal of
Pressure Vessels and Piping, 78, 295305 (2001).
Bruck, H.A.. McNeil, S.R., Sutton, M.A., and Peters. W.H.. "Digital Image Correlation
using NewtonRaphson method of panial differential correction," Experimental
Mechanics, 29 (3), 261267 (1989).
Chevalier, L., Calloch, S., Hild. F., and Marco. Y.. "Oigital Image Correlation used to
analyze the multiaxial behavior of rubberlike materials:' Eur. 1. Mech. A, 20, 169187
(2001).
George.E.Dieter., "Mechanical Metallurgy", 3rd Edition. McGrawHilllnc (1986).
Gerald, C. F., and Wheatley, P.O., "Applied Numerical Analysis", AddisonWesley
Publishing Company, Reading, Massachusetts, (1994).
Hussain, M., Nakahira, A., Nishijima, S., and Niihara. K.. "Fracture behavior and fracture
toughness of particulate filled epoxy composites:' Materials Leiters 27.,2125 (1996).
Haddi, A., and Weichert, D., "On the computation of the Jintegral for threedimensional
geometries in inhomogeneous materials," Computational Materials Science, 5,143150,
(1996).
Kim, Y.J., Shim, D.1., Choi, J.B., and Kim, YJ., "Approximate J estimates for tensionloaded
plates with semielliptical surface cracks:' Engineering Fracture Mechanics, 69,
14471463 (2002).
Kim, Y.J., "Experimental J estimation equations for singleedgecracked bars in fourpoint
bend: homogeneous and bimaterial specimens," Engineering Fracture Mechanics,
69, 793811 (2002).
Knauss, W.G., Gonzalez. 1., "Strain inhomogeneity and discontinuous crack growth in a
particulate composite", Journal of Mechanics and Physics of Solids, 46(10),19811995
( 1998).
57
Lee.J., and Vee.A.F.. "Role of inherent matrix toughness on fracture of glass bead filled
epoxies", polymer 41.8375·8385 (2000).
Lu, H., Cary. P.D.. "Deformation measurements by Digital Image Correlation:
implementation of second order displacement gradient". Experimental Mechanics. 4(4).
393·310 (2000).
Lu, H., Ganesan Balaji., and S.Hariharan., "Deformation measurements by Digital Image
Correlation: implemcntation of third order displacement gradient". Creative Component
report, (2000).
McNeill, S.R., Peters, W.H., and Sutton, M.A.. "Estimation of ~tress intensity factor by
Digital Imagc Correlation," Engineering Fracture Mechanics, 101 I 12 (J 987).
Peters, W.H.. and Ranson, W.F., "Digital imaging techniques in experimental stress
analysis," Opt. Eng., 21 (3),427·432 (1982).
Rahman. S.. ·'Prob'lbilistic fracture mechanics: Jestimation and finite clement methods,"
Engineering Fracture Mechanics, 68,107125 (2001).
Saxena.A.. "Non linear fracture mechanics for engineers". CRC Press LLC (1998).
Sun, Z., Lyons. J.S., and McNeill, S.R., "Measuring microscopic deformations with
digital image correlation," Optics and Lasers in Engineering. 27. 409428 (1997).
Sutton, M.A., Wolaters, W.J., Peters, W.H., Ranson, W.F.. and McNeill, S.R.,
"Determination of displacements using an improved digital image correlation method,"
Image Vision Computing, I (3), 133·139 (1983).
Sutton, M.A., Turner. J.L., Bruck, H.A., and Chae, T.A., "Fullfield representation of
discretely sampled surface deformation for displacement and strain analysis",
Experimental Mechanics, 31 (2), (168·177) (1991).
SUllon, M.A.. Cheng, M., Peters, W.H., Chao, Y.J., and McNeil, S.R., "Application of an
optimized digital image correlation method to planar defonnation analysis," Image
Vision Computing, 4 (3), 143·150 (1986).
Vendroux. G. and Knauss, W.G., "Submicron defonnation field measurements: part 2.
improved digital image correlation," Experimental Mechanics, 38, 8691 (1998).
Vendroux, G., "Correlation: A Digital Image Correlation program for displacement and
displacement gradient measurements," GALCrr Report SM9019, California Institute of
Technology, (1990).
58
APPENDIXES
59
APPENDIX A
WINDICLS USER'S GUIDE
The WinDIC_LS program was developed to provide an efficient, user.friendly
means if using the large deformation refinements to the digital image correlation method.
There are eight basic modes of the program. The tabbed option box allows immediate
access to the each of the five modes. Note that the origin (0,0) of the images is located at
the lower left corner following the standard Cartesian coordinate system. Most other
image manipulation programs set the origin at the lefl comer.
The FILES menu mode allows the user to Open the existing projects. save the
current projects, start a new projeci and exit the WinDIC~LS program. Also in this mode,
the user can add image files to be used in the digital image correlation. The list box
shows the name of each of the images in the current project. The user must arrange the
images in Ihe order in which they will be correlated using the up and down bunoos. Only
8bit (256 level) grayscale uncompressed TLFF images are recognized with the current
version of WinDIC_LS. If you want to use files that are stored in another file fonnat, you
must use a paint program Paint Shop Pro or some other software to convert the images in
to the proper file fonnal. Once the images have been selected then the user can proceed to
the initial guess mode.
60
,
,
1"" '''''''I..,u ....1.....1""'I '....J
Or.oOl:Poqoot... ,....1Ir~
S_I*:~ OJ EIIi""'v
Mode I: FILE MENU MODE
In most of the modes the tool bar is present for manipulating the displayed images.
There are three zoom levels, Full scale, Zoom to fit the image the windows and Triple
zoom. Also the arrows provide a way to switch between all the sets of images. At any
lime, the images can be panned around by holding down the right mouse button in one of
the image windows and dragging the mouse around.
The initial guess mode is where the user will establish the initial guesses the DIe
solver will use when starting the correlation process. These initial guesses are calculated
by the program from reference marks the user, located on each image. The reference
marks can be either a single point or a vector. Select which type to be used by selecting
61
the appropriate option box. Points are represented by a red dOt on the image. To place a
reference point click the POINT option box, then simply click with the left mouse button
where the reference point is to be located. This red reference poim should be close to
where you wam the ole solver to begin correlating the images. Vectors are represented
by a red location dOt and a green direction point.
u.......l..I.l.I._.............l.D.l.I.&..
~a.t._o.,g ....."'"
Mode 2, INITIAL GUESS MODE
,....
SIIl0l3 SMtIDFI
~~ ,..
To place a vector reference, click the VECfOR option box then press and hold
down the left mouse to place the red location dot and then drag the mouse to locate the
green direction dot and finally release the left mouse bUllon.
62
In situations where the deformation does not have significant rotation. then the
POINT type reference will allow the program to simply compute guesses for the initial
offset parameters. All the other mapping parameters will be given an initial guess value
of zero. In situations where the deformation has large rotation angles, the VECTOR
reference type will allow the program to compute better initial guesses for both the
location and first order gradients. The program will assume the higher order mapping
terms are zero. After the reference marks have been positioned, the program will be ready
to position the grid of nodes in the reference image.
Mode 3: Grid Menu mode
63
The Grid Menu mode is where the grid of nodes is generated and located in the
reference image. The first reference image is always displayed in the left image window.
Use the mouse to locate the grid by pressing the left mouse button and dragging the grid
to the desired position. The grid parameter~ can be modified by changing their values in
the corresponding boxes. The grid will automatically change when !.he box that the grid
parameter has been changed loses keyboard focus. The grid can be generated in several
different patterns. The pattern determines the sequence in which the nodes are analyzed.
In the grid mode you will notice a dashed magenta box around the grid of nodes and also
in the right image window. This represents the approximate area of the images that will
be used in the ole procedure. The user will want to make sure that this region does not
fall outside the boundary of the actual image it the solver will most likely generate an
error.
The SOLVER menu mode provides the final set of solver settings. The SUBSET
RADIUS set's !.he radius around a node thai will be used to correlate the two images
together. The smooth radius is used to smooth the image grayscale values. Using a value
of zero will tum the image smoothing off. The iteration limit sets the maximum number
of iterations each node will be allowed to converge on a solution. The tolerance is the
maximum amount each of the mapping parameters can change before the solution is
considered converged.
There are several option boxes so that different orders of displacement gradient
terms can be calculated.
Mode 4: SOLVER MENU MODE
First order reference only: This option is used (0 make the computations with the
first image as reference when there are more than two images. This makes the
computation to be a cumulative one, so the displacement results are cumulative resuhs.
Linear terms only: when this option is turned on the displacement calculations
will be based on the first order terms only. The second and third order terms will not be
computed.
Linear and second order: When Ihis option is turned on the displacement
computations will be based on the first and the second order terms together.
No grayscale offset: The grayscale offset will not be taken in to consideration.
65
Look for cracks: When the images that are used in the correlation process have
cracks present on them the cracks can be taken in to consideration. i.e .. crack detection
process can be done.
Fresh Crack detection every time: If the correlation stops for some rcason and if it
is to be started again without detecting cracks again, this option is to be turned on.
Crack detection menu mode is used to delecl Ihe crack on the specimen. When
user clicks Start solver bunon on the solver menu mode the software enters in (a the crack
detection menu mode. Here the user can draw a rectangle around the crack to detect the
crack.
Mode 5: CRACK DETECTION MENU MODE
When the detect cr3ck bullan is pressed the crack detection process slarts, after
this detect edge crack can be pressed (if the image used has edge crack in it). The crack
66
detected appears ali black and white region on right hand side of the screen. as can be
seen in the figure below.
The threshold value to be used during crack detection can be chosen using the
slider bar present in this menu mode. The number of pixels in the crack can also be varied
by using the edit box as can be seen on the figure. Editing of the crack detected can be
done using some options present on this form. The size of the brush can be changed lIsing
the brush size edit box. The user can enter the desired value for the brush size. There are
options for erasing and drawing on the screen. If there are unwanted parts on the cnlck
detected they can be removed by using the eraser option. If some pixels are to be added to
the crack detected that can be done using the marker option. The brush size edit box
controls the brush size for both these options.
There is another option called the material pointer. This can be used to show the
material to the software. After the crack is detected the user can point to the material by
switching on this option and clicking on the material side of the crack. This will aid in
showing the material to the software and there by improve the speed of calculation.
The user can traverse between images using the next image and previous image
buttons. Fresh selection button can be used to do a new crack detection process. The stop
detection button can be used to terminate the crack detection process. If the crack
detection is not satisfactory the user can detect the crack after pressing the "fresh
selection" bUllon. After the crack is detected the user can go back to the original screen
by pressing the '"Resume solver". Then the solving process starts and the results are ready
for some time. The user can go to the results menu mode to view the displacement and
deformation results.
67
'L_I _,..1 I_I
='"li CiooIt.o I  ....1 ~~....!:J ...
Mode 6, CRACK DETECTION ME U MODE WITH THE DETECTED CRACK
Importam thing to be taken care of during the crack detection step is that the area
selected should be small and closer to the crack, so that the crack detection will take
place soon. As can be seen from Mode 6. the rectangle drawn surrounding the crack is
close enough to the crack region. This allows the crack to be detected soon and the
modifications if any can be done soon. The nodal grid placed on the reference image is
not visible during this mode.
The results menu mode allows an easy way to look at the results of correlation.
The left pane shows (he complete set of results for a single node. To select another,
simply select that node in the image display on the right and its results will be displayed
in the information window.
68
EJ
Mode 7(.), CRACK DETECTION MENU MODE WITH THE CRACK AFTER
ERASING UNWANTED PARTS OF THE CRACK.
Mode 6(a) shows the crack afler the unwanted parts of the crack have been
removed using the eraser option in the crack detection screen. As can be secn from the
figure. the crack alone is detected and laken. This helps in improving the algorithm by
taking only lhe crack region in to account and not the part of the specimen.After the crack
is detected the solver can be started to perform the computation of deformation for the
specimen.
69
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Mode 8: RESULTS MENU MOOE
The program can also output a tecplot data file containing either the incremental
or the cumulative results of the image shown in the image display. Incremental results are
just the results of the deformation from the previous image. Cumulative results are the
results from the first image to the selected image. The write raw function outputs a
commaseparated list of all the data generated by the WinDIC_LS program. This raw file
can be easily read into a spreadsheet program like Excel, or used as inputs to other
programs.
in the results mode the user can generate files with various kinds of outputs
namely nodal displacement file, nodal definition file and nodal files.
70
D./X D.!V ...
ll!ISl IS152 ,
nl15 ISI!i' 5
nun IS1" 1
"",,*,.OAn.t. lJoWo PootiI._ •••_
o HiI'
1~_\I1'
~.Uil'
90_7_1.11
AoiIX ,,",V
ltl 151
ZltSJ 15152 nus lSI!i'
,,_p_.r,~..~ f •• OocIo< ... ~Sec:andOoclo<.
VU·v,Urv,u.v..1Jl.
·1 oIQ3 051Sl'9 (IOXl!lol Oo:nlll (1l1li" 00XlXl 00DXl nlDXXl nl
·2lt5lill OSJCltl 01XOl2 000ll5i OIlDla> 0(Dl10 00DXl ocom 01
·2!1&151 0lil5117 000S!l1 Olll1:1l (Il101D OlDl'J 00DXl naxm 0.1
_INOlEJol[HT",AESlt.TS_
Looo...w. Slo....,I._,.,. .n It tWWW'l .n ,v.. n .."" • 10............,*9
Mode 9, PLOT MENU MODE
After the results are computed the strain values can be plotted on the images as
bands or as lines in the plot menu mode. The user can choose the number of levels for the
band and line plots. Based on the number of levels the plots will be made in the fonn of
color lines or bands on the images. The plot lines and plot bands will appear to the extent
of the nodal grid placed as the results are available only for that region of the specimen
surface. When the plot is displayed on the image a scale will appear on the side 10 show
the range of values in a scale. The user can traverse to differenl images with the buttons
provided on the interface.
71
• T",** I .....
Mode 10, JlNTEGRAL MENU MODE
The Jintegral menu mode allows the user to calculate the Jintegral after the
computation of the deformations for the nodes on the grid are over. There are provisions
for entering the material properties such as Young's modulus (E), alpha, n the hardening
coefficient and yield strain. The conversion factor should also be entered. It is the number
of pixels per millimeter. The user has the option of choosing which the theory by which
the stresses are to be calculated. The choices are linear elastic, bilinear as can be seen on
the screen. If none of these two are chosen RambergOsgood model will be chosen for
calculating the stresses. When Jintegral is computed for square paths then the crack tip
location that is in terms of pixels should be entered.
72
After entering the necessary paramelers the.: user can choose the particular path
along which the J~integral is to be computed. The path appears as red line as the user
starts clicking On the grid of nodes on the image. After choosing the palh the user has 10
click on the Compote Jintegral button to gellhe Jimegral value for that particular palh.
1••liljA_*,II.iiP
Mode II: JINTEGRAL ME U WITH PATH CHOSEN
This program has been complelely developed using Microsoft Visual Basic 6.0.
73
APPENDIX B
FLOW OF CODE FOR WINDIC
The following flowchart describes the flow of the WinDle code for calculating
the strain values from the images provided as inpul. The input parameters are the
parameters such as the guess localion, the location of the nodal grid placed on the
reference image. Basic assumption is thai the images are related to one anOlher, in such a
way thai the grayscale paucm from the reference image to the dcfonncd image is present
Slart Solver Bullon
click
,l.
Gel Number of parameters
Linear = 8
Linear and Second order = 14
Third order =22
,l.
Gel Input parameters
(Subset radius, iterations,
smooth radius, tolerance)
1
Get Guess parameters
(U.Ux,Uy,V,Vx,Vy)
,l.
LsDICsolve
~
( * )
Read Input images
Crack
Present?
YES
Remove the nodes falling NO
in to the crack region
Find lhe Slartnodeindex
(Work from the slart node through all
(he nodes in the grid)
74
( A+
Get Subset (calculates the
positions of subset points
around node)
Inlcrpsmooth zone
(Interpolate reference image in the
zone of the subset)
..
Ca1cRcfvalues (interpolate gray values of
the subset points) • Optimization
(Calculate all Hessians and gradients and
get the value to increment the parameters
using Newlon raphson)
~
Do the iteration till the iteration
limit
~
Write the results to the
results array
Calculate strains from
results.
75
APPENDIX C
FLOW OF CODE FOR JINTEGRAL COMPUTATION
The following nowchart describes the now of code for the Jinlegral computation.
The input parameters for the J computation arc the data of the nodes that are on the path
chosen along which J is nceded 10 be calculated. The input parameters arc first order
gradients of U and V namely. Ux. Vx. Uy and Vy
InpUI parameters such as
young's modulus,
hardening coefficient.
Poisson's ratio etc.
1
Select the path along which
the Jinlcgral is to be
calculated. (Can be done
using mouse clicks)
1
Capture the nodes along
which the path chosen
passes through.
1
The values such as strains along
X. Y and XY directions, first
order displacement components
such as Ux, Uy, Vx and Vy arc
taken.
e*A)
(A
Depending on the type of model
chosen for stress calculation,
stresses are calculated.
1
The traction vectors are
calculated with respect to the
directio.n of the path between
Add all the J values obtained
for the nodes lay along the
palh chosen.
The values calculated above
are substituted in to the Jintegral
equation to obtain J
for a oarticular node.
76
APPENDIX D
DEFORMED IMAGEMAKER USER'S GUIDE
The DeformcdImagcMaker is a Visual Basic program used to generate a reference
image from the given defomlcd image. It is used in developing ideal images. which can
be used for validation purposes. The user can get a reference image from a given
deformed image. The figure below shows the user interface of the program.
Deformed image
!N..5"" jj,j_W""ttii,;
Figure 1 : User Interface of Deformed ImageMaker program with deformed image
17
The "Open Tiff Image" button can be used to gel Ihe deformed image as input in to the
program. The user can input the parameters l<>llch a... U. V and the first order and second
order gradients in to the software to gel Ihe cone,ponding reference image. The calculate
button is used to start the program 10 get the reference image from the deformed image.
After the calculations are over the reference image gets displayed in the left hand
side 3..<; shown in Figure 2. The reference image can be saved using thc "Savc Raw
Image" bulton. The raw image can be converted 10 TIFF fomlat to be later used in
WinDle program. Thus the DefonnedlmageMaker can be used to get a reference image
from a given deformed image.
x
Reference image
y. 0..1 U 0 II 0 0
w· •
Deformed image
Figure 2: After calculation.
78
APPENDIX E
IMAGES USED IN THE UNIAXIAL TENSILE TEST ON PARTICULATE
COMPOSITES
Following are the images that were used in the uniaxiallcnsion tcst on particulate
composites. The images show that the specimen has natura) texture on its surface. which
shows thai there is sufficient grayscale pattern required for conducting Digital image
correlation. From the images it can be seen that the crack starts to propagate from image
05.tiL The variation of Jintegral values obtained can vary at about 20% from the original
value because of the inhomogenity in the pm1iculate composite material.
Image 01.lif
Image 03.tif
79
Image 02.tif
Image 04.tif
Image 05.tif
Image 07.tif
80
Image 06.tif
Image OS.tiC
APPENDIX F
IMAGES OF PARTICULATE COMPOSITE USED IN 20% PRESTRAI TEST
Following are the images that were used in the uniaxial tension test on particulate
composites. The images show that the specimen has natural texture on its surface. which
shows that there is sufficient grayscale pattern required for conducting Digital image
correlation. From the images it can be seen thai the crack starts to propagate from image
12.li1.
Image 01 .tif
Image 11.tif
81
Image IO.tir
Image 12.tif
Image 13.tif
82
Im:lgc 14.lif
APPENDIX C
IMAGES OF PARTICULATE COMPOSITE USED IN 15% PRESTRAIN TEST
Following are the images that were u~('d in the uniaxiallcnsion test on particulate
composites with 15% prestrain. The crack prop<lg~ltion is from Image 12.liflO 14.tif.
Image 08.lif
Image 14.tif
Image J8.lif
S3
Image 12.1if
Image J5.tif
Image J9.tif
VITA fL
Sampathkumar Pcriasamy
Candidate for the Degree of
Master of Science
ThesIs: DIGITAL LMAGE CORRELATION FOR DEFORMATION
MEASUREMENTS NEAR A CRACK
Major Field: Mechanical Engineering
Biographical:
Education: Received Bachelor of Engineering in Mechanical engineering from
Government College of Technology. Coimbatore, India in June 1999.
Completed the requirements for the Master of Science degree with a major in
Mechanical Engineering at Oklahoma State University in December 2002.
Experience: Employed by Oklahoma State University as graduate research
assistant, from October 2000 10 July 2002 and subsequently as graduate
tcaching assistant. from August 2002 to December 2002.
Professional Memberships: American Society of Mechanical Engineers.