Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 2, pp. 609-626, Warsaw 2012

50th Anniversary of JTAM

MICROMECHANICAL MODELLING OF DIAMOND
DEBONDING IN COMPOSITE SEGMENTS

İsmail Ucun

Afyon Kocatepe University, Technical Education Faculty, Afyonkarahisar, Turkey

e-mail: iucun@aku.edu.tr

Kubilay Aslantaş

Afyon Kocatepe University, Technology Faculty, Mechanical Enginering Department,

Afyonkarahisar, Turkey; e-mail: aslantas@aku.edu.tr

İ. Sedat Büyüksaği̇ş

Afyon Kocatepe University, Engineering Faculty, Mining Engineering, Afyonkarahisar, Turkey

e-mail: sbsagis@aku.edu.tr

Süleyman Taşgetiren

Afyon Kocatepe University, Technical Education Faculty, Afyonkarahisar, Turkey

e-mail: tasgetir@aku.edu.tr

Diamond debonding in composite segments used in cutting of natural
stones was investigated in this study. The finite element method was
used for numerical solutions, and the problem was modelled as two di-
mensional. The problem was investigated under linear elastic fracture
conditions. Stress intensity factors and strain energy release rates were
obtained. Numerical solutions were performed for different diamond si-
zes and diamond heights. According to the obtained results, the most
effective factor for growth of the interface crackwas considered to be the
tangential force acting on the diamond.

Key words: composite segment, diamond debonding, stress intensity fac-
tor, finite element method

1. Introduction

Diamond cutting discs used in cutting of natural stones are composed of two
parts including the disc body and composite segments. The segments are atta-
chedbybrazing on thediscbodymanufactured fromsteel. Thecuttingprocess
of natural stones is performed by the diamonds within the composite matrix.



610 İ. Ucun et al.

Cutting forces vary depending on the type of natural stone subjected to the
cutting process, diamond concentration, diamond size and cutting parameters
(Büyüksağiş and Göktan, 2005; Ersoy and Atıcı, 2004; Luo and Liao, 1993;
Taşgetiren and Ucun, 2004; Ucun et al., 2008). While the cutting process is
performed through diamond, composite matrix holds the diamonds together.
As thenatural stones donothave homogenous structures, cutting forces acting
on the disc during the cutting process are variable. Shock forces may cause
separation of the diamond from thematrix or fracture of the diamond.

The segment is subjected to abrasive and erosive wear together with in-
crease of the cutting volume. Height of the diamond from the matrix surface
increases due to this wear and, consequently, the diamond is subjected to hi-
gher cutting forces.Thismaybea result of debondingof the diamond fromthe
matrix or the diamond pull out. Failures such as fracture and flat (polish) are
also seen in diamonds. Because the cutting process becomes very complicated,
it is quite difficult to predict these failures. Studies about composite segments
generally focus on the formation mechanism of these failures (Jennings and
Wright, 1989; Luo, 1997; Luo and Liao, 1995; Ucun et al., 2008; Wright and
Cassapi, 1985;Yu andXu, 2003; Zhan et al., 2007). If the diamond is hard and
it becomes subjected to high shock forces during the cutting process, fracture
of diamonds considerably increases.When the contact temperature on the dia-
mond becomes around 1000◦C during the cutting process, some deformations
may occur on the diamond surface. The surface of the diamond is flatted due
to the effect ofmechanical forces (Ucun, 2009). Debonding and pull out of the
diamond from the composite matrix is not a desired situation among failures.
The matrix is subjected to more wear by increasing of diamond pull out and
therefore the forces on the segment and power consumption increase. As a
result, the cutting efficiency considerably decreases. There is a chemical bond
between the diamond and the composite matrix. This bond occurs during the
sintering process and it is very important for effective lifecycle of the segment.
The diamond may be separated from the matrix due to resulting weak inter-
face bond and forces on diamond during the cutting process, (Karagöz and
Zeren, 2001; Zeren andKaragöz, 2007). As the number of diamonds separated
from the segment increases, the contact area of thematrix with the stone gets
larger. This enlarged contact area causes a rise of friction and cutting forces.

In this study, diamond debonding in composite segments used as a cut-
ting tool in natural stone industry was examined. The finite element method
was used for numerical solutions and the problem was considered as two-
dimensional. Cutting forces acting on the segment were experimentally deter-
minedand forces on eachdiamondwere calculated usingan analyticalmethod.



Micromechanical modelling of diamond debonding... 611

An interface crack was defined between the diamond and thematrix. Besides,
stress intensity factors and strain energy release rateswere obtaineddepending
on growth of the crack, diamond size and diamond height.

2. Diamond cutting disc

Cutting discs are generally made of alloy steels such as 75Cr1 (AISI 1075)
and 80CrV2 (Tönshoff et. al, 2002) and composite segments are placed on
discs using the brazingmethod. The segments are composite materials manu-
factured through the technique of powder metallurgy. Diamonds (in varying
sizes and quantities) are embedded into these segments. There are two various
diamond types including natural and synthetic diamonds. Synthetic diamonds
are generally used in segments and they have a cubic-octahedral structure.
The diamond cutting disc, composite segment and diamond used in cutting of
natural stones are given in Fig.1.

Fig. 1. Diamond cutting disc, composite segment and diamond

3. Material and methods

3.1. Calculation of stress intensity factors

The stress intensity factor is used in order to define stresses that occur in
the crack region of linear elastic materials. In general, three different stress
intensity factors (KI,KII,KIII) are defined at the edge of crack depending



612 İ. Ucun et al.

on the type of loading applied to homogenous and elasticmaterials. Therefore,
the stress intensity factor applied to the edge of crack is directly associated
with the type of load applied to thematerial.When the problem is considered
in a two-dimensional way, the openingmode (KI) and the shearmode (KII)
are the two different stress intensity factors available at the edge of the crack.
Although various methods are used in order to calculate KI and KII stress
intensity factors that occur at the edge of the crack, the node displacement is
one of the most important methods. The method gives more accurate results
than other methods and, therefore, it is widely used in numerical solution
methods such as the finite element method and boundary elements method.
KI and KII stress intensity factors can be calculated based on the displace-
ment values obtained from 1, 2, 3 and 4 (Fig.2) nodes on the surface of the

Fig. 2. Location of the nodes used to calculate the stress intensity factor

crack.The relation between thedisplacement values on the surface of the crack
and the stress intensity factors are determined from (Aslantaş andTaşgetiren,
2004; Tan andGao, 1990)

KI =
G

κ+1

√

2π

La
[4(h1−h3)+(h4−h2)]

KII =
G

κ+1

√

2π

La
[4(u1−u3)+(u4−u2)]

(3.1)

and

κ=











3−4v for the plane stress

3−v

1+v
for the plane strain

(3.2)

where, ui and hi are nodal displacements in the x and y directions, respec-
tively (Fig.2). La is the element length, ν is Poisson’s ratio, G is the shear
modulus and κ is determined for the plane stress and strain conditions.



Micromechanical modelling of diamond debonding... 613

3.2. Calculation of the strain energy release rate

The strain energy release rate is used in order to examine fracture beha-
viour of two differentmaterials being interrelated or bonded.As thematerials
on both surfaces of the crack will be different, differences between mechani-
cal features of both materials are expressed as α and β, and they are called
the Dundurs parameters. These parameters are calculated by (Madani et al.,
2007)

α=
E′1−E

′

2

E′1+E
′

2

β=
µ1(1−2ν2)−µ2(1−2ν1)

µ1(1−2ν2)+µ2(1−2ν1)
(3.3)

where E′1 and E
′

2 are the elasticity moduli of material 1 and 2, respectively.
µ and ν are the shear modulus and Poisson’s ratio of the materials, respec-
tively. The strain energy release rate as a function of crack length along the
interface is explained as (Kim et al., 2009; Leblond and Frelat, 2004; Madani
et al., 2007)

Gd =
1

4cosh2πε

(1−ν1
µ1
+
1−ν1
µ2

)

(K2I +K
2
II) (3.4)

where KI and KII are the stress intensity factors inMod I andMod II, ε is
a function of the material constant and is explained in the following equation

ε=
1

2π
ln
1−β

1+β
(3.5)

3.3. Geometric model of the composite segment

Aspecific number of segments on the cutting disc contact with the natural
stone depending on the depth of cut and these segments are subjected to
cutting forces. In this study, the forces occurred during the cutting process of
the granites called Blue Pearl and Nero Zimbabwe were measured through a
three dimensional dynamometer (ESIT). A computer-controlled block cutting
machinewas used during the cutting process.Themeasuredmaximumnormal
and tangential forces were 610N and 180N, respectively (Ucun, 2009). The
cutting experiments were performed with various parameters in this study.
These parameters were depth of cut (ap) 40mm, circular velocity (Vc) 30m/s
andcutting speed (Vs) 0.6m/min.Lengthof the contactwith thenatural stone
was calculated in an analytical way depending on the depth of cut (40mm).
The number of active cutting segments and diamonds was obtained by the
length of contact. The diamonds on each segment were determined using a
USB microscope with a size of 250X. The normal and tangential forces on a



614 İ. Ucun et al.

single diamond were determined by the following formulas (Yu et al., 2006;
Zhang et al., 2008)

fn =
Fn

As
ft =

Ft

As
(3.6)

where, Ft and Fn are the tangential and normal forces acting on the disc,
respectively. As represents the number of active cutting diamonds on the com-
posite segments. As a result of the analytical solutions, the maximum normal
force acting on the diamond was obtained to be 1.135N and the maximum
tangential force was obtained to be 0.27N.The forces were applied by distri-
bution on a diamond as shown in Fig.3. Geometrical features and boundary
conditions of the diamond and the composite matrix used in themodelling of
finite elements were given in Fig.3.

Fig. 3. Geometrical model and boundary conditions of the diamond and the
matrix [mm]

Sizes of the matrix structure were selected in such a way that they would
not affect the stress distribution around the diamond. Besides, image analysis
software called MICROCAM was used in order to define sizes of the matrix.
SEM image was obtained under cutting process conditions taken into account
for determination of the twodimensionalmodel.The image in the cross section
of a diamond is shown in Fig.4.
Indeed, the height of each diamond from thematrix h1 and its size d are

variable. For this reason, diamond size and diamond height were considered
to be variable in the analyses. Geometrical features of the segment and the
diamond are given in Table 1. Sizes of the diamond in the segments used in
the study were taken into account as the diamond size. Sizes of the diamond
within the composite segments used in the study are 40-50 USMesh. There is
a difference of 0.280 and 0.375mmbetween these sizes and actual sizes of the
diamond.The segmentmatrix and the diamonds are subjected towear due to
the forces acting on the segments. Because thematrix is subjected to a higher



Micromechanical modelling of diamond debonding... 615

Fig. 4. Image in the cross section of the diamond

level of wear, the diamond rises to the surface. Solutions were prepared for
different heights and sizes of diamond h1 by considering the level of wear for
the matrix in this study.

Table 1.Geometric properties of the diamond

Diamond heights Diamond sizes
h1 [mm] d [mm]

0.03 0.06 0.09 0.12 0.15 0.18 0.25 0.3 0.35 0.4

Twodifferentmaterials were defined such as the diamond and the segment
matrix in the finite element model. Mechanical properties of these materials
are given in Table 2. Mechanical properties of the segment matrix preferred
in the analyses were obtained according to EDS results.

Table 2. Mechanical properties of the segment matrix and the diamond (Hu
et al., 2003, 2004; Upadhyaya, 1998)

Elasticity modulus Poisson’s ratio Density
E [GPa] ν [kg/m3]

Diamond 1150 0.0691 3500

Segment matrix 560 0.25 8100

3.4. Finite element model of the diamond and the segment matrix

The finite elementmethodwas used to solve the problem under linear ela-
stic fracturemechanical theory. Thefinite elementmodel is done byFranc2DL



616 İ. Ucun et al.

software program (Franc2DL, 2011). The finite elementmodel of two different
materials such as diamond and segment matrix is given in Fig.5. Triangular
and rectangular elementswere used together in the finite elementmodel.Whi-
le triangular elements with six nodeswere used in the diamond, isoparametric
rectangular elements with eight nodes were preferred at the corner points of
the diamond.The problemwas considered in the light of two dimensional pla-
ne stress conditions. As shown in Fig.5, a more intense mesh structure was
created for the corner points, connecting the diamond and thematrix, because
a considerable stress intensity occurred in the interface of the diamond and
the composite matrix. As a result, possible debonding areas and cracks will
start in this area.Thefinite elementmodel of the problem is composed of 5346
nodes and 2259 elements.

Fig. 5. Finite element model of the diamond and the composite matrix

4. Results and discussion

4.1. Stress analysis

The surface area of the segment varies depending on the depth of cut. The
number of active cutting diamond also differs due to this surface area. Cutting
forces acting on the segment are actually compensatedby thesediamonds.Due
to the effect of these forces, considerable stresses occur in the diamond and
thematrix that keep it intact. The equivalent stress (Von-Mises) distribution
on the diamond and thematrix is shown in Fig.6.Maximum stresses occur in
the interface of the diamond and thematrix. Besides, this critical stress point
is the initial point of debonding.



Micromechanical modelling of diamond debonding... 617

Fig. 6. Equivalent stress distribution on the diamond and the composite matrix

Variation of equivalent stresses obtained depending on the diamond size
and the diamond height is given in Fig.7. These stresses were obtained from
critical areas, where the diamond and thematrix were integrated and thema-
ximum stress integrity occurred. As the diamond size increases, it is observed
that the stresses considerably decrease. Maximum stresses were obtained at
0.25mm,where the diamond sizewas the lowest. As the diamondheight incre-
ases, the stresses in the interface increase in a linear way and this rise occurs
in two stages. Firstly, the diamond rises to the surface due to abrasive wear of
thematrix (Fig.8). Secondly, the area around the composite segment becomes
hollow due to the effect of erosive wear around the diamond depending on the
cutting direction (Fig.8). For both types ofwear, forces acting on the diamond
and the stresses will increase. As for the resulting stresses, it is observed that
the diamondheight has amore important effect than that of the diamond size.
While themaximum stress increases by nearly 20%with a rise in the diamond
size, it increases by nearly 40% in the case of the rise in the diamond height.

In composite segments, an interface bondoccurs between the diamond and
the matrix. Resistance of this bond varies depending on element compounds
within the matrix, element types and sintering conditions. This bond should
be resistant especially against shear stresses. The bond formed between the
diamond and the matrix is continuously subjected to shear stresses with the
effect of tangential and normal forces, and cracks occur in the interface ac-
cordingly. Variation of shear stresses depending on the diamond size and the
diamond height is given in Fig.9. Similar to equivalent stresses, shear stresses
considerably decrease with an increase in the diamond size. Minimum shear



618 İ. Ucun et al.

Fig. 7. Variation of the maximum equivalent stress at diamond height and size

Fig. 8.Wear occurred on the diamond segment: (a) abrasive, (b) erosive

stresses were obtained when the diamond size was the maximum. In parallel
with the resulting equivalent stresses, shear stresses considerably increase as
the diamond size increases.

Fig. 9. Variation of shear stress at diamond height and size



Micromechanical modelling of diamond debonding... 619

4.2. Fractography

SEM (scanning electron microscopy) analysis was performed on segment
surfaces after the cutting process. The observations showed that the initial
point of the crack was the interface of the diamond and the matrix. It was
observed that the cracks generally occurred in the interface of the diamond
and the matrix in these analyses. The crack grows after a specific period of
time and the diamond pulls out without completing its life.

A strong bond between the matrix and the diamond extends the lifecycle
of the segment. It alsominimizes the diamondpull out. However, forces acting
on the segment during the cutting process may result in debonding of the
diamond from the matrix. If the forces acting on the diamonds during the
cuttingprocess arenotdispersedhomogenously, andthese forces are repetitive,
compressive and tensile stresses occur in the interface of the matrix. These
stresses play an important role in debonding of the diamond from thematrix.
The failuremechanism in a diamond is given in Fig.10. Before pull out of the
diamond, micro cracks are formed in the critical stress point (Fig.10a). The
crack grows due to the effect of the cutting forces and the diamond debonds
from thematrix and pulls out when the critical value is reached (Fig.10c). It
is suggested that debonding in the interface of the diamond and the matrix
occurs based on the cutting forces. The diamond debonds from thematrix as
the cutting forces are repetitive and the diamond is subjected to impact forces
when it contacts with the natural stone for the first time.

Fig. 10. Failure mechanism in the diamond

4.3. Numerical modelling of debonding

In the finite element solutions performed depending on fractography ana-
lyses, a numericalmodelwas created for debonding that occurred between the
matrix and the diamond.Debonding in the interface was defined as a crack in
the finite element analysis. The crack grows with the effect of forces when the



620 İ. Ucun et al.

stress intensity factor at the edge of the crack reaches the critical value. Be-
cause the problemwasmodelled in a two-dimensional way, two different stress
intensity factors (KI and KII) were obtained at the edge of the crack. Varia-
tion of KI with debonding length at various diamond sizes is given Fig.11.
As the length of debonding increases, KI increases a little. After the length of
crack reaches 0.05mm, KI stress intensity factor is subject to a sharp decline.
In a sense, KI value decreases down to the corner point of the diamond. It is
fixed at very low values after this point.

Fig. 11. Variation of KI with debonding length

Variation of the crack length in the interface of the diamond and the ma-
trix is given in Fig.12. The opening mode (Mode I) is more important for
debonding of the diamond up to a certain point (the 3rd state in Fig.12). In
this case, it is estimated that the tangential forces are more effective during
the cutting process when compared with the normal forces. While the maxi-
mum stress intensity factor was acquired at the smallest diamond size, the
minimum stress intensity factor was obtained at the largest diamond size. It is
suggested that a diamond with a small size bears more risks when compared
to a larger diamond in terms of debonding. After a certain length of debon-
ding, KI decreases to negative values. It goes up to positive values again after
a certain length (see Fig.12). This rise in KI shows that Mode I starts to
become effective again.
KII value obtained in Mode II depending on different diamond sizes are

given inFig.13. Contrary to values of KI,KII does not have any effectiveness
up to a certain length of debonding. However, it was observed that through
the corner point of the diamond (the 4th state in Fig.12) the crack changed
direction and became more effective when compared to Mode I. A possible
debonding between the matrix and the diamond occurs with the effect of
Mode II after reaching the corner point in the 3rd state (Fig.12). In other



Micromechanical modelling of diamond debonding... 621

Fig. 12. Variation of debonding occurred between the diamond and the composite
matrix

Fig. 13. Variation of KII with debonding length

words, debonding occurs in the shear mode after reaching the corner point
of the diamond. The opening and shear zones are shown in Fig.14. While
possible failure in the diamond zone occurs as an opening up to the 3rd state
(Fig.12), it occurs as a shear after a certain length of debonding (the 3rd to
6th states in Fig.12).



622 İ. Ucun et al.

Fig. 14. Deformation region in the maximum debonding length

Fig. 15. Variation of the strain energy release rate with debonding length

As known, the strain energy release rate is generally taken into account
for determination of possible failure for two different bonded materials. The
strain energy release rates were obtained depending on the stress intensity
factors obtained in the analyses (Fig.15).While the strain energy release rate
decreases a little down to a certain length of debonding for all diamond sizes,
this rate considerably increases after the diamond reaches the corner point
(the 3rd state in Fig.12). This rise continues up to a certain length (the 5th
state in Fig.12). Themost critical point for failure risk is the point, where the
maximum strain energy release rate is achieved. As the debonding continues
after this area, the strain energy release rate tends to decrease. As the strain
energy release rate is low in the beginning, debonding occurs due to the effect
ofMode I (openingmode).After a certain length of opening, a rapid rise in the
strain energy release rate shows that Mode II (shear mode) is quite effective.
A decline in the rate after the maximum value is reached indicates that the



Micromechanical modelling of diamond debonding... 623

shear effect is eliminated. Size of the diamond used within the matrix has
an important effect for debonding. As the diamond size increases, the strain
energy release rate considerably decreases.

5. Conclusions

In this study, debondingbetween the diamond and thematrixwas determined
in terms of experimental and theoretical methods. In the experimental study,
diamond segments were investigated through SEM analysis as a result of the
cutting process. Linear fracture analysis was performed using the finite ele-
mentmethod and stress intensity factors, and strain energy release rates were
obtained in the theoretical study. The following conclusions were reached as a
result of this study:

• It is observed that debonding between the diamond and the matrix oc-
curs with the effect of the cutting forces. Repetitive and variable forces
acting on the diamond are considered to be the reason for debondingdu-
ring the cutting process. However, it is highly possible that the diamond
debonds from the matrix with the effect of scrape when the diamond
segment contacts with the natural stone for the first time during the
cutting process.

• Maximum equivalent and shear stresses were obtained in the corner po-
ints between the diamond and the matrix. While the diamond height
increases with the matrix wear, the equivalent and the shear stresses
also increase. Unlike this trend, the stresses decrease a little as the dia-
mond size increases. Maximum stresses were obtained at small diamond
sizes.

• Stress intensity factors inMode I andMode IIwere determinedwith the
crack defined between the diamond and the matrix. Results of the frac-
ture analysis show that the crack occurs in the openingmode (Mode I)
and KI increases with the start of debonding.However, while KII valu-
es increase after reaching a certain crack length, KI decreases. In other
words, the crack grows in the shearmode (Mode II) after this length. It
was observed that the diamond size had an important effect for deter-
mination of KI and KII. The stress intensity factors (KI and KII) in
Mode I andMode II increased upon a decline in the diamond size.

• The strain energy release rate explains debonding between the diamond
and thematrix. The rate considerably increased upon a rise in the crack



624 İ. Ucun et al.

length. The maximum strain energy release rate is the most important
point in terms of failure, and this rate was obtained when the diamond
had the smallest size. As the diamond size increases, the strain energy
release rates decrease. It is suggested that possible failure will occur as
a result of shear rather than debonding.

Acknowledgement

This study was supported by TUBITAK project (106M189 number).

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Modelowanie mikromechaniczne zjawiska odpadania kryształów diamentu
od kompozytowego fundamentu

Streszczenie

W pracy omówiono problem odpadania kryształów diamentu od kompozytowej
matrycy narzędzi do cięcia naturalnych kamieni. Zagadnienie sformułowano dwuwy-
miarowo, a do uzyskania wyników symulacji numerycznych wykorzystano metodę
elementów skończonych. Zjawisko badano dla warunków liniowego sprężystego pę-
kania. Obliczono współczynniki koncentracji naprężeń oraz tempo uwalniania energii
sprężystości. Symulacje przeprowadzonodla różnychwymiarówkryształówdiamentu.
Zgodnie z otrzymanymi rezultatami, stwierdzono, że głównym czynnikiem odpowie-
dzialnym za wzrost powierzchni pękania jest styczna składowa siły działającej na
diament.

Manuscript received September 9, 2011; accepted for print November 16, 2011