Crack arrest in Si
_{3
}N
_{4
}based layered composites with residual stress
M. Lugovy
^{a
}, V. Slyunyayev
^{a
}, V. Subbotin
^{a
}, N. Orlovskaya
^{a
}, G. Gogotsi
^{b
}
^{a
}Institute for Problems of Materials Science, National Academy of Sciences of Ukraine, 3 Krzhizhanovsky Str., Kiev 03142, Ukraine
^{b
} Institute for Problems of Strength, National Academy of Sciences of Ukraine, 2 Timiryazevskaya str., Kiev 01014, Ukraine
Received 28 May 2003; received in revised form 17 February 2004; accepted 19 February 2004
Available online 12 April 2004

COMPOSITES
SCIENCE AND
TECHNOLOGY

Abstract
Effect of macroscopic residual stresses on fracture resistance and crack arrest in nonsymmetric Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–
x
TiN (
x
= 20, 30 wt%) layered specimens was investigated in this study. One of the aims of the work was also an examination of compliance technique to study
R
curve effect as applied to layered specimens. A special attention was paid to the development of an analytical method to calculate fracture resistance – crack length dependence in layered structures having different elastic moduli of layers. The validity of the method and its application were examined by calculating stress intensity coefficients for edgecracked layered specimens and comparison of the results with mechanical characteristics obtained from bending test.
© 2004 Elsevier Ltd. All rights reserved.
Keywords: A. Layered structures; B. Fracture toughness; Modelling; C. Crack; Residual stress
1. Introduction
Multilayered ceramic matrix composites (MCMC) have a wide variety of applications in modern technology. Layers comprising ceramic materials are extensively used in engineering structural components with the objective to improve the mechanical, thermal, chemical and tribological performance. Recent research and development in the area of MCMC seek to utilize such materials in such diverse applications as surface coatings, thermal barrier protection for turboengines, valves in reciprocating engines for automobiles and cutting tools.
Despite of many attractive properties such as high hardness and high temperature stability, MCMC have the major disadvantage of lacking reliability and sensitivity to surface contact damage. The last factor can lead to strength decreasing and even to catastrophic failure.
A number of strategies have been developed in recent years to design tough and strong MCMC [
1
]. These include designing weak interfaces for crack deflection [
2
], using residual compression in surface layers [
3
], promoting crack bifurcation effect in compressive layers [
4
], controlling the frontal shape of the transformation zones in zirconia ceramics [
5
]. These mechanisms should provide an arrest of cracks in layered structure, increasing consequently its reliability. The reliability of the MCMC can be improved also by controlling the size of flaws introduced into the material during processing. This may be achieved by dispersion of a slurry of the designated power and by its passing through a filter. As a result only heterogeneities with sizes smaller than a critical size can pass through, depending on the filter fineness. Drawback of this procedure is its expensiveness and that such material is still subject to damage during machining with the reliability degraded accordingly.
In multilayered materials with strong interfaces the differences in the coefficients of thermal expansion (CTE) between dissimilar materials or phase transformation in layers inevitably generate thermal residual stresses during subsequent cooling [
6
]. The essential feature of residual stress distribution in a layered structure is that it arises on a macroscopic scale. The relative thickness of different layers determines the relative magnitudes of compressive and tensile stress, while the magnitude of the strain mismatch between the layers dictates the absolute values of the residual stresses. Control of the thermal stresses and the accompanying changes in structure are important to ensure the structural integrity of the layered component during manufacturing and in service.
A key feature that imparts good mechanical properties in the multilayer systems is the ability to be toughened significantly by placing their surfaces in residual compression and to arrest crack. It was shown in [
3
] that a residual surface compression of ~500 MPa in a surface layer of threelayered alumina–zirconia specimen can increase its fracture toughness by a factor of 7.5 (up to 30 MPam
^{1/2
}) for edgecrack lengths of the order of the surfacelayer thickness. The toughening derived from macroscopic surface compression was in fact a crack shielding phenomenon and the fracture toughness increasing was equivalent to a crack growth resistance (
R
) behavior [
7
]. The
R
behavior is often connected with bridging mechanism. The mechanism is associated with the closure stress field that acts behind the tip of the advancing crack [
8
]. However, there are some differences in Rbehavior resulted from bridging mechanism (this is typical for nonlayered ceramics) and from the shielding phenomenon in layered structures. First, while bridging mechanism gives rise to dependence of fracture resistance only on crack length increment, the shielding effect results in the dependence of fracture resistance on overall crack length [
3
,
7
,
9
]. Second, the bridging mechanism promotes an increasing fracture resistance with crack advance whereas the shielding effect can induce both increasing and decreasing of fracture resistance depending on crack tip location in tensile or compressive layer.
Actually layered specimen fracture resistance measured experimentally is the apparent fracture toughness. This is due to superposition of different effects like residual stress shielding and structure inhomogeneity. In fracture mechanics one usually includes stresses in the crack driving force; however it is sometimes useful to consider residual stresses as part of the crack resistance. Thus the higher resistance to failure for layered structure with residual stress is obtained from a reduction of the crack driving force rather than an increase in the intrinsic material resistance to crack extension [
9
].
Despite of numerous experimental and theoretical studies of fracture resistance of MCMC, systematic researches of
R
behavior and of crack arresting in layered composites are very scarce. A great number of publications are dealing with symmetrical layered structure. This is an idealized situation. Really laminates are characterized by some dissymmetry of their architecture due to random deviations in fabrication process. Besides specific nonsymmetrical layered structures are important in some engineering applications. Conventional analytical consideration of shielding effect in laminate also neglects the difference between elastic moduli of layers [
3
,
7
]. However, effect of different moduli on fracture resistance of laminates is not so negligible. The influence of elastic modulus variation across a layered sample on
R
curve behavior is investigated in [10]. It was shown that the elastic moduli difference affects residual stress distribution and has consequently a significant influence on the measured
R
curve behavior. But detailed analysis of conditions of crack arresting, its stable/nonstable growth was not carried out in [
10
].
The effect of macroscopic residual stresses on fracture resistance and crack arresting in nonsymmetric Si
_{3
}N
_{4
}based layered structures fabricated in the form of singleedgeVnotchbend (SEVNB) specimens is investigated in this study. One of the aims of this work is also an examination of compliance technique to study
R
curve effect as applied to layered specimens. A special attention is paid to the development of an analytical method to calculate fracture resistance – crack length dependence in layered structures having different elastic moduli of layers. The validity of the method and its application are examined by calculating stress intensity coefficients for edgecracked layered specimens and comparing the results with the mechanical characteristics obtained from bending test data.
2. The model

Fig. 1. Schematic of the twocomponent multilayer specimen.

Fig. 1
shows a schematic of the twocomponent multilayer specimen analyzed in this study. Parameter
t
_{i
}
designates thickness of layer number
i
. The total thickness of specimen of rectangular crosssection is
w
, its width is
b
, the total number of layers is
N
. Choice of coordinate system is important for further consideration. It is the most appropriate to put the coordinate origin on tensile surface of bending specimen. The geometry of the multilayered material analyzed here is such that the problem can be reduced to one dimension, and that analytically tractable solutions can be used. Here, the parameters of interest in the study of mechanical behavior depend only on
x
coordinate.
It was shown in [
3
,
11
] that the stress intensity factor,
K
_{1
}
, due to an arbitrary stress distribution in the prospective path of the crack in the absence of the crack
σ
(
x
), can be obtained as

(1)

where
x
is the distance along the crack length measured from the surface for an edge crack, a is the crack length, a α=
a/w
,
w
is the specimen thickness (
Fig. 2
). Note that the distance along the crack length coincides with
x
coordinate in coordinate system chosen. For edgecracked specimens, Fett and Munz [
11
] have developed the following weight function:

(2)

The values of the coefficients
A
_{νμ
}
and the exponents
ν
and
μ
in (
2
) are listed in
Table 1.
Table 1.
Values of coefficients
A
_{νμ
}
in Eq. (2) [11]
ν

μ
= 0

μ
= 1

μ
= 2

μ
= 3

μ
= 4

0

0.498

2.4463

0.07

1.3187

3.067

1

0.54165

5.0806

24.3447

32.7208

18.1214

2

0.19277

2.55863

12.6415

19.763

10.986


Fig. 2. Schematic of analyzed crack location in layered specimen.

In a case when deformation is a function of coordinate
x
only, it follows from strain compatibility [
12
] that overall deformation
ε(x)
must be linear for elastic material:

(3)

Here
ε
_{0
}
is the deformation at
x
=0,
k
is the specimen curvature. An equal biaxial stress state is known to be the most appropriate approximation to describe stress state in real layered specimens [
13
]. This is the case for infinite dimensions along
y
 and
z
directions, but with finite value of specimen thickness. In the equal biaxial stress state we have:
ε
(
x
) =
ε
_{zz
}
=
ε
_{yy
}
,
σ
(
x
) =
σ
_{zz
}
=
σ
_{yy
}
, where
ε
_{zz
}
,
ε
_{yy
}
,
σ
_{zz
}
,
σ
_{yy
}
are strain and stress components along
z
 and
y
axis, respectively. Edge effects (occurrence of threedimensional stresses near the edges of layered composite over a distance from the edge which is approximately equal to the layer thickness [
14
]) can be neglected due to their highlocalized character. Then

(4)

where

(5)

In Eqs. (
4
) and (
5
),
E
(
x
),
ν
(
x
) are the elastic modulus and Poisson ratio distributions along
x
axis.
is the strain nonassociated with stress. It is associated with thermal expansion or/and with a volume change due to a crystallographic phase transformation.
The static balance conditions [
12
] in chosen coordinate system result in a system of linear equations with unknown values
ε
_{0
}
and
k
:

(6)

where
F
_{a
}
is the applied axial force, Ma is the applied bending moment. Solution of the system is [
13
]

(7a)


(7b)

where

(8)


(9)

Note that superposition principle is valid for this problem. It permits to express the stress variation along the path of the crack in a specimen as

(10)

where
σ
_{a
}
(
x
) is the bending stress in the prospective path of the crack in the absence of any residual stress,
σ
_{r
}
(
x
) is the residual stress distribution.
In [
3
] the bending stress
σ
_{a
}
(
x
) was expressed as follows:

(11)

where
σ
_{m
}
is the applied stress on tensile surface of bending specimen. It is well known that

(12)

Here
P
is the critical load (applied bending load corresponding to specimen failure),
s
is the support span. However, the differences in the elastic moduli of the layers were not taken into account in [
3
]. Really different elastic moduli of the layers result in specific distribution of applied stress along
x
direction. Elastic material demonstrates continuous linear distribution of applied strain under bending. This promotes piecewiselinear distribution of applied stress, shown schematically in
Fig. 3
. To derive applied stress distribution under bending we can use expressions (
3
), (
4
), (
7a
), (
7b
) and (
12
), taking into account that in this case
F
_{a
}
= 0. We can take
= 0 if applied stress acts only. Then it follows that applied stress acting in the layer with number
i
is

(13)

Here
x
_{i
}
is the coordinate of upper boundary of
i
th layer.
E'
_{i
}
=
E
_{i
}
/ (1
ν
_{i
}
),
E
_{i
}
and
m
_{i
}
are the elastic modulus and Poisson ratio of layer number
i
, respectively.
I
_{Lj
}
can be obtained from expression (
8
) accounting for layered structure (
Fig. 1
)

(14)

Residual stress distribution can be found from Eqs. (
3
), (
4
), (
7a
) and (
7b
), taking into account that
F
_{a
}
= 0,
M
_{a
}
= 0 (
Fig. 3
):

(15)

where
J
_{Lj
}
can be obtained from the expressions (
9
) accounting for layered structure

(16)


Fig. 3. Schematic of residual and applied stress distribution in layered specimen.

Here
is the strain of ith layer nonassociated with stress. The thermal expansion or/and a volume change due to a crystallographic phase transformation can be the source of this strain, however, the case of phase transformation is out of the scope of this paper. In case of thermal expansion
where
β
_{i
}
(
T
) is thermal expansion coefficient of
i
th layer at temperature
T
;
T
_{0
}
,
T
_{j
}
are actual and joining temperature, respectively. Joining temperature is determined as the temperature at which the layers constituting the material are joined. If
β
_{i
}
(
T
) is a linear function,
= (
β
_{i
}
)
ΔT
, where
ΔT
=
T
_{j
}

T
_{0
}
,
is the average value of thermal expansion coefficient in temperature range from
T
_{0
}
to
T
_{j
}
.
Using the condition of crack growth (
K
_{1
}
=
K
_{1c
}
,
K
_{1c
}
is the fracture toughness of the layer which embeds the crack tip) and (
1
), (
10
), we obtain

(17)

Using Eq. (
13
) the first integral in (
17
) can be expressed for a layered material as

(18)

where
n
is the number of layers broken by the crack (or notch) completely (
Fig. 2
). Using Eq. (
15
), the second integral in (
17
) for a layered material takes the form

(19)

K
_{r
}
is the stress intensity due to the residual stresses.
The following formula is given for the stress intensity of an edge crack in the specimen under bending as being accurate to ±0.2% in the range
α
= 0 to 1 [
15
]:

(20)

where ƒ
_{0
}(
α
) is a nondimensional stress intensity coefficient given by the following expression [
15<>]:

(21)

Taking into account Eqs. (
12
) and (
21
), expression (
20
) can be transformed to the form

(22)

where

(23)


Fig. 4. Optical photograph of layered specimen with typical notch tip.

It was shown in [
3
] that Eqs. (
20
) and (
22
) can be successfully used to determine fracture toughness of ceramic matrix layered materials. However, it should be noted that as applied to inhomogeneous (particularly, layered) materials the equations give socalled apparent fracture toughness. In bending test this is the fracture toughness of some effective homogeneous specimen that meets the following conditions: (1) to have the same dimensions as real layered specimen; (2) to have notch depth equal to that of a real layered specimen; (3) under the same loading conditions to demonstrate the same critical load as that for real layered specimen. In spite of relativity of this value it is useful characteristic allowing contributions of such factors as residual stresses and material inhomogeneity to be accounted for. Thus experimental value of apparent fracture toughness of layered specimen can be found using expression (
22
):

(24)

It follows from Eqs. (
17
) – (
19
) and (
24
) that apparent fracture toughness of layered composite
K
_{app
}
can be written as

(25)

where
K
^{(i)
}
_{1
c
} is the fracture toughness of ith layer material.
3. Experimental
The choice of composition for Si
_{3
}N
_{4
} based ceramics laminates is determined by the coefficient of thermal expansion and Young’s modulus of the compounds. Three compositions of composite layers were used: (1) Si
_{3
}N
_{4
} (M11, Starck, Germany); (2) Si
_{3
}N
_{4
}–20 wt% TiN (grade C, Starck, Germany); (3) Si
_{3
}N
_{4
}–30 wt% TiN (grade C, Starck, Germany). Young’s moduli, Poisson ratios and average values of coefficients of thermal expansion of the components are presented in
Table 2
. The more detailed information on thermomechanical parameters as well as joining temperature
T
_{j
}
for materials under investigation is given in [
16
]. Mean values of intrinsic fracture toughness of monolith materials are evaluated in the work to be approximately the same for all layer compositions, being 5 MPa m
^{1/2
}. Note that the intrinsic fracture toughness corresponds to the fracture toughness of layer material.
Table 2.
Thermoelastic parameters of components of layered structure
Composition

Elastic modulus
(GPa)

Poisson ratio

Average value of thermal expansion coefficient
(× 10
^{6
} K
^{1
}

Si
_{3
}N
_{4
}

308

0.25

3

Si
_{3
}N
_{4
}–20 wt% TiN

316

0.25

3.82

Si
_{3
}N
_{4
}–30 wt% TiN

323

0.25

4.28

Milling of mixtures of certain compositions were done in the ball mill for 5 h. The formation of a thin ceramic layer is of specific importance, as the sizes of residual stress zones (tensile and compressive) are directly connected with the thickness of layers. Green tapes were manufactured with rolling. Rolling permits to control the thickness of green layers, to obtain high green density and a rather low amount of solvent and organic additives in comparison with other methods such as a tape casting [
17
]. However there is a problem to produce thin tapes, <100 μm, with a small amount of plasticizer and sufficient strength and elasticity to handle the green layers after rolling.
Crude rubber (4 wt%) was added to the mixture of powders as a plastisizer through a 3% solution in petrol. Then the powders were dried up to the 2 wt% residual amount of petrol in the mixture. After powders were sieved with a 500 μm sieve, granulated powders were dried up to the 0.5 wt% residual petrol. A roll mill with 40 mm rolls was used for rolling. The velocity of rolling was 1.5 m/min. Working pressure varied from 10 MPa for relative density of tapes 64% to 100 MPa for 74% density. The thickness of green tapes was either 0.4–0.5 or 0.8–1.0 mm, the width is 60–65 mm. Samples of ceramics were prepared by hot pressing of tapes stacked together. The hot pressing was performed at the temperature 1780–1820 °C, the duration 20 min and the pressure 30 MPa.
Green tapes were stacked together to form the desirable layered structure. Graphite dies were used for the hot pressing without protective atmosphere. After hot pressing, the thickness of the Si
_{3
}N
_{4
} layers was in the range of 160–960 μm, and the thickness of the Si
_{3
}N
_{4
} layers with TiN additive varied from 160 to 480 μm. Geometrical characteristics of layered specimens are listed in
Table 3
.
Table 3.
Geometrical characteristics of Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–X wt% TiN layered materials.
Layer number

Layer composition

Layer thickness (μm)



X
= 20
(specimens of type 1)

X
= 20
(specimens of type 2)

X
= 30
(specimens of type 3)

1

Si
_{3
}N
_{4
}

295

275

955

2

Si
_{3
}N
_{4
}–X wt% TiN

315

380

355

3

Si
_{3
}N
_{4
}

260

205

175

4

Si
_{3
}N
_{4
}–X wt% TiN

465

400

345

5

Si
_{3
}N
_{4
}

205

180

185

6

Si
_{3
}N
_{4
}–X wt% TiN

370

285

320

7

Si
_{3
}N
_{4
}

200

230

180

8

Si
_{3
}N
_{4
}–X wt% TiN

330

280

315

9

Si
_{3
}N
_{4
}

190

175

160

10

Si
_{3
}N
_{4
}–X wt% TiN

370

355

380

11

Si
_{3
}N
_{4
}

180

200



12

Si
_{3
}N
_{4
}–X wt% TiN

390

390



13

Si
_{3
}N
_{4
}

235

295



14

Si
_{3
}N
_{4
}–X wt% TiN

195

175



15

Si
_{3
}N
_{4
}



175



The specimens for mechanical tests were prepared from hot pressed plates. SEVNB specimens were used for testing. The test data have confirmed that the SEVNB method can be easier applied in practice and can be used for the majority of advanced ceramics and ceramic particulate composites [
18
]. The Vnotches with tip radii of an order of 10–15 μm were made in the specimens. Optical photograph of typical notch tip is presented in
Fig. 4
. The dimensions of specimens were 45 mm × 5 mm × 4 mm (types 1 and 2), 45 mm ×5 mm × 3.37 mm (type 3). The depth of the notches was about 60–80% of the specimen thickness.
A stiff load cell ensuring the rigid loading of specimens under threepoint bending with a 16 mm span was used for mechanical tests. This cell is equipped with specific rigid dynamometer providing an ultimate load of 2000 N with a specimen deflection measuring system using a deflectometer suspended on the specimen. The testing machine used is designed only for the displacement of a loading crosshead and control of its speed. To study
R
curve effect, a compliance technique was used. Notched specimen was placed into the hard load cell. Then loading of the specimen was made up to crack growth onset followed by unloading. Besides recording loaddeflection diagrams, after each unloading of specimen, its polished lateral surface was examined by an optical microscope (× 1000) to measure crack length. After measurement of crack length the next loadingunloading cycle was made. The operations were repeated up to total failure of specimen. Apparent fracture toughness was calculated by using of the expressions (
12
), (
23
) and (
24
).
4. Results and discussion
Asymmetric structure of layered specimens under study results in linear variation of residual stresses within each layer. The residual stress values calculated from thermoelastic parameters and specimen geometry using Eq. (
15
) are listed in
Table 4
. One can see that the greatest stress gradient is in laminate specimen Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–30 wt% TiN.
Table 4.
Calculated residual stresses in Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–X wt% TiN layered materials.
Layer number

Layer composition

Residual stress
^{a
} (MPa)



X
= 20
(specimens of type 1)

X
= 20
(specimens of type 2)

X
= 30
(specimens of type 3)

1

Si
_{3
}N
_{4
}

131.7/135.9

157.7/156.6

31.3/131.6

2

Si
_{3
}N
_{4
}–X wt% TiN

128.0/123.4

106.8/108.5

283.9/244.8

3

Si
_{3
}N
_{4
}

140.4/144.1

154.9/154.1

168.9/187.3

4

Si
_{3
}N
_{4
}–X wt% TiN

119.6/112.8

109.4/111.1

225.5/187.5

5

Si
_{3
}N
_{4
}

150.8/153.7

152.4/151.6

223.5/243.0

6

Si
_{3
}N
_{4
}–X wt% TiN

109.8/104.3

111.9/113.1

167.1/131.9

7

Si
_{3
}N
_{4
}

159.0/161.8

150.4/149.4

276.6/295.5

8

Si
_{3
}N
_{4
}–X wt% TiN

101.4/96.6

114.1/115.3

112.1/77.3

9

Si
_{3
}N
_{4
}

166.5/169.2

148.3/147.5

328.6/345.4

10

Si
_{3
}N
_{4
}–X wt% TiN

93.8/88.4

116.1/117.6

59.7/17.9

11

Si
_{3
}N
_{4
}

174.5/177.1

146.0/145.2



12

Si
_{3
}N
_{4
}–X wt% TiN

85.8/80.0

118.5/120.2



13

Si
_{3
}N
_{4
}

182.7/186.0

143.5/142.3



14

Si
_{3
}N
_{4
}–X wt% TiN

76.6/73.7

121.5/122.2



15

Si
_{3
}N
_{4
}



141.5/140.8



^{a
}Residual stresses are presented in the format ‘‘stress on low interface’’/‘‘stress on up interface’’; compressive stresses are negative, tensile stresses are positive.

The critical issue to analyze fracture behavior of laminates is a choice of coordinate system. Calculated values of apparent fracture toughness
K
_{app
}
in layered specimens under study are analyzed depending on crack length parameter
, where
=
Y
(
α
)a
^{1/2
}. The crack length parameter
is the most appropriate to demonstrate critical conditions of crack growth. One of advantages of this parameter using is that stress intensity coefficient of an edge crack for fixed value of applied stress
σ
_{m
}
is depicted in coordinate system
K
_{app
}
—
as a straight line from the coordinate origin. Indeed, it follows from (
22
) that
K
_{1
}
=
σ
_{m
}
, therefore, the slope of straight line equals to applied stress
σ
_{m
}
. The conditions for unstable crack growth in internal stress field are as follows [
9
]:
K
_{1
}
(
σ
_{m
}
,
a
) =
K
_{app
}
(
σ
_{m
}
,
a
); d
K
_{1
}
(
σ
_{m
}
,
a
)/d
a
≥ d
K
_{app
}
(
σ
_{m
}
,
a
)/a
a
;. Using parameter
, these conditions become:
σ
_{m
}
=
K
_{app
}
(
);
σ
_{m
}
≥ d
K
_{app
}
(
)/d
. The last two conditions can be reduced to

(26)


Fig. 5. Condition of unstable crack growth in internal stress field.

It follows from Eq. (
26
) and
Fig. 5
[
9
] that unstable crack growth occurs as the slope of straight line, corresponding to stress intensity factor at constant applied stress, is no less than the slope of tangent line to fracture resistance curve at the same point.
Fig. 6
shows dependence of apparent fracture toughness on crack length parameter
in laminate Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–20 wt% TiN, specimens of type 1 (solid curve). The areas corresponding to compressive and tensile layers are shown in gray and white, respectively. The fracture toughness of layer material is shown as horizontal straight line. The dependence of apparent fracture toughness on
is nonmonotonous. The apparent fracture toughness increases in compressive layers and decreases in tensile layers. The peak values of Kapp correspond to interfaces between layers. The apparent fracture toughness of the layered composite varies from 2 to 10 MPa m
^{1/2
} depending on crack length. Initial notch tip is in tenth layer that is under residual tension. Mean value of measured apparent fracture toughness corresponding to initial notch is 5.57 MPa m
^{1/2
} that is in accord with calculated value. Note that each experimental point in
Figs. 6–8
corresponds to average value for three to five specimens. Unloading was made after small advance of crack from initial notch. Crack arrest occurred in 12th layer of specimen. The length of arrested crack was measured. Then the next loading resulted in the total failure of specimens. Mean value of measured apparent fracture toughness corresponding to arrested crack is 7.42 MPa m
^{1/2
} that is also in accord with calculated value.

Fig. 6. Dependence of apparent fracture toughness on crack length parameter
in laminate Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–20 wt% TiN (specimens of type 1). Areas of compressive layer are shown in grey. Solid curve is the calculated dependence, horizontal line is the fracture toughness of layer material. Dashed line is the stress intensity factor at constant applied stress of crack growth onset. Open circle corresponds to initial notch, filled circle corresponds to arrested crack.



Fig. 8. Dependence of apparent fracture toughness on crack length parameter
in laminate Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–20 wt% TiN (specimens of type 3). Designations are the same as in
Fig. 6
.

Fig. 7
shows dependence of apparent fracture toughness on crack length parameter
in specimens of type 2 of laminate Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–20 wt% TiN. Designations are the same as in
Fig. 6
. The dependence of apparent fracture toughness on crack length parameter is nonmonotonous also. The fracture toughness behavior in compressive and tensile layers in specimens of type 2 is qualitatively similar to that in specimens of type 1. However, difference of specimens geometry results in some difference of apparent fracture toughness range. Specifically the apparent fracture toughness of specimens of type 2 varies from 3 to 11 MPa m
^{1/2
}. Initial notch tip in the specimen is also in tenth layer that is under residual tension. In this case mean value of measured apparent fracture toughness corresponding to initial notch is 6.39 MPa m
^{1/2
}. That is in accord with calculated value too. After unloading crack was arrested in 12th layer like specimens of type 1. Specimens of type 2 with arrested crack demonstrate the apparent fracture toughness value of 6.27 MPa m
^{1/2
}. This is in good accord with calculated value.

Fig. 7. Dependence of apparent fracture toughness on crack length parameter
in laminate Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–20 wt% TiN (specimens of type 2). Designations are the same as in
Fig. 6
.

Fig. 8
shows dependence of apparent fracture toughness on crack length parameter ~a for the laminate Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–30 wt% TiN (specimens of type 3). Designations are also the same as in
Figs. 6
and
7
. In this case we have nonmonotonous dependence too. Apparent fracture toughness range in
Fig. 8
is more wide and different from range of Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–20 wt% TiN laminate due to differences of composition and geometry. The apparent fracture toughness of Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–30 wt% TiN laminate varies from 0.1 to 12 MPa m
^{1/2
}. Initial notch tip in the specimen is in eighth layer that is under residual tension. The mean value of measured apparent fracture toughness corresponding to initial notch is 5.31 MPa m
^{1/2
}. After unloading crack was arrested in the same (8th) layer. Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–30 wt% TiN specimens with arrested crack demonstrate the apparent fracture toughness value of 5 MPa m
^{1/2
}. In contrast to Si
_{3
}N
_{4
}/Si
_{3
}N
_{4
}–20 wt% TiN layered specimens, there is essential difference of experimental and calculated apparent fracture values. Specifically, mean apparent fracture toughness, corresponding to initial notch, exceeds the calculated value by 2.1 MPa m
^{1/2
}. The mean apparent fracture toughness, corresponding to arrested crack, exceeds the calculated value by 4 MPa m
^{1/2
}. The difference can be associated with existence of stable crack growth in compressive layer following the layer containing notch tip. An explanation of the difference between experimental and theoretical values of fracture toughness of layered specimens studied is considered elsewhere [
16
].

Fig. 9. Stages of crack arresting in layered specimen: (a) crack is arrested in tensile layer; (b) crack is arrested in compressive layer.

Let consider some features of crack arresting in our experiments. Open circle A in
Fig. 9
designates the initial state of testing: initial notch without loading. Open circle B corresponds to the crack growth onset at some critical applied stress. Open circles depict the initial notch with length
a
_{0
}
. Filled circle C corresponds to the onset of unloading. Note that the applied stress increases permanently during loading stage (from A to C in
Fig. 9
), while the crack begins to grow only as applied stress intensity factor becomes more than apparent fracture toughness (from B to C in
Fig. 9
). The crack growth under unloading can vary depending on the rate of applied stress decreasing. Schematically it is shown in
Fig. 9
by two ways of crack development (C–D–E–F,
Fig. 9(a)
; C–D'–E'–F', Fig. 9(b)). The curve C–D'–E'–F' corresponds to greater rate of applied stress falling. Filled circles D and D' characterize current positions of moving crack tip under unloading. Filled circles E and E' designate the crack arresting when applied crack intensity factor becomes less than apparent fracture toughness. Filled circles F and F' depict the final state of crack with length
a
_{f
}
(or
a'
_{f
}
) after full unloading. One can see from
Fig. 9
that different conditions of unloading can result in various distance passed by the crack. If C–D–E–F way is realized, the crack is arrested in next tensile layer (
Fig. 9(a)
). If C–D'–E'–F' way is realized the crack is arrested in the nearest compressive layer (
Fig. 9b
). In general case, unloading conditions can result in many different final positions of crack tip. It can be either in layer with initial notch tip or in any more remote layer. Returning to our experimental data we note that rather C–D–E–F way than C–D'–E'–F' is realized in the laminate specimens containing 20% TiN in tensile layers. At the same time the specimens with 30% TiN in tensile layers demonstrate C–D'–E'–F' way at unloading, which may be associated with existence of stable crack growth in compressive layer following the layer containing notch tip. The problem will be analyzed elsewhere [
16
].
In such a way we have two stages of loading process and three stages of crack behavior. First stage of loading process is the applied stress increasing to some maximum value (from A to C in
Fig. 9(a)
). Second stage of loading process is the applied stress decreasing up to zero (from C to F in
Fig. 9(a)
). First stage of crack behavior is the absence of crack growth until applied stress intensity factor is less than apparent fracture toughness (from A to B in
Fig. 9(a)
). Second stage of crack behavior is the crack growth (from B to E in
Fig. 9(a)
). Third stage of crack behavior is the absence of crack growth as applied stress intensity factor is less than apparent fracture toughness again (from E to F in
Fig. 9(a)
).
The condition for stable crack growth in residually stressed layers can be obtained from (
26
). The stable crack growth can occur when
K
_{app
}
(
)/
< d
K
_{app
}
(
)/d
. As the condition is fulfilled load decreasing results in indispensable crack arresting. The crack arresting in stable growth condition differs from features of crack arresting being in our work. This is due to the fact that unstable crack growth was observed for the layered specimens investigated. Crack arresting does not depend upon unloading rate in conditions of stable crack growth. Crack will be arrested in any case as applied load does not increase. At the same time crack arresting for unstable crack growth is dependent strongly on stress decreasing rate. The unloading rate is determined mainly by stiffness of loading cell. The more is the stiffness the less is crack path before arresting. It is evident that stable crack growth will lead to strengthening and an insensitivity of strength to the initial flaw size. In a layered material design process it is necessary to determine the range of stable crack growth and strengthening if the range of flaws in layers is known. Usually technological flaws are relatively small in laminates. Only rising dependence of apparent fracture toughness on crack length is not enough to obtain desirable strengthening and toughening. The dependence is effective in imparting flaw tolerance only if the slope of the apparent fracture toughness curve is steep at short crack lengths. Obtaining high residual compressive stress is an effective way to provide high toughness at small crack lengths, thereby ensuring flaw tolerance and surface damage resistance.

Fig. 10. Stresses required for crack growth vs crack length. 1 is the crack growth stress in monolithic material; 2 is the fracture stress of the layered material; 3 is the superposition of the crack growth stress in monolithic material and residual stresses in layers.

A key issue of the mechanical behavior of layered composites is an effect of their apparent fracture toughness on strength properties and reliability. As was shown above the crack instability criterion involves the point of tangency between the applied (crack driving) stress intensity factor and the apparent fracture toughness curve. It is only beyond the tangency point that unstable crack propagation occurs, leading to catastrophic failure. A unique material toughness parameter,
K
_{1c
}
, no longer controls the applied stress at which fracture occurs; it is the exact form of the
K
_{app
}
curve,
K
_{app
}
(
a
), that determines the fracture stress of a layered material. The residual stresses distribution is superimposed upon the strength behavior of layered composites. Residual stress as well as initial cracks have essential effect on fracture stress. For instance, it was shown in [
19
,
20
] that the strength of the threelayer Al
_{2
}O
_{3
}–15 vol% ZrO
_{2
} composite with residual stress was about 500 MPa greater than those of monolithic ceramic. One can suppose that, in case of crack growth, residual stress affects laminate strength in more complicated manner as compared to simple superposition of stresses.
It is of interest to compare how the stress required to grow a crack of a given length differed from what would be required to grow the crack in the monolith minus the internal stress in the layer where the crack tip lies. For example, calculation results to compare the behavior of these stresses in specimens of type 2 depending on relative crack length a=w are shown in
Fig. 10
. Curve 1 corresponds to the crack growth stress in monolithic material calculated as fracture toughness of the material divided by crack length parameter. Curve 2 is the fracture stress of the layered material calculated from (
24
). Curve 3 is the superposition of the crack growth stress in monolithic material and residual stresses in layers. One can see that in first layer the difference of the stresses depicted by curves 2 and 3 is small. In other words, for the first layer the difference of fracture stress for the monolith and that for the laminate practically equals absolute value of residual stress in this layer.
In the next layers the stresses corresponded to curves 2 and 3 differ substantially. Note that the fracture stress magnitudes depicted by curve 3 can be negative for tensile layers. This is due to the fact that tensile residual stress is more than the fracture stress for monolithic material if crack is too long. It means that simple superposition is not valid for long cracks crossing several layers.
5. Conclusions
The toughening of ceramic matrix asymmetric laminates with elastic inhomogeneity has been studied both analytically and experimentally. Applied and residual stress distributions are determined for arbitrary alteration of different layers. Expression for apparent fracture toughness of residually stressed asymmetric layered material is obtained. The most appropriate coordinate system to analyze fracture conditions of laminar composite was shown to be the system where apparent fracture toughness is depicted depending on crack length parameter
=
Y
(
α
)
a
^{1/2
}
. The dependences of apparent fracture toughness on crack length parameter are calculated for the specimens tested. Conditions of crack arresting, stable and unstable crack growth in layered structure are analyzed. It was shown that the crack path before arresting can vary depending on stiffness of loading cell. Experimental values of apparent fracture toughness were measured with compliance technique. They are in good agreement with calculation data.
Acknowledgements
The work was supported with European Commission. It is part of the Project ‘‘Silicon nitride based laminar and functionally gradient ceramics for engineering application’’, Proposal No. 1CA2199910109, Contract No. 1CA2CT200010020, the program Copernicus – 2. The authors thank also V. Galenko and B. Ozersky for their help during the course of this work.
References
[1]

Chan M. Layered ceramics: processing and mechanical behavior. Annu Rev Mater Sci 1997;27:249–82.

[2]

Clegg WJ, Kendall K, Alford NM, Button TW, Birchall JD. A simple way to make tough ceramics. Nature 1990;347:455–7.

[3]

Lakshminarayanan R, Shetty DK, Cutler RA. Toughening of layered ceramic composites with residual surface compression. J Am Ceram Soc 1996;79(1):79–87.

[4]

Lugovy M, Orlovskaya N, Slyunyayev V, Gogotsi G, Kuebler J, SanchezHerencia AJ. Crack bifurcation features in laminar specimens with fixed total thickness. Compos Sci Technol 2002;62:819–30.

[5]

Marshall DB, Ratto JJ, Lange FF. Enhanced fracture toughness in layered microcomposites of CeO–ZrO
_{2
} and Al
_{2
}O
_{3
}. J Am Ceram Soc 1991;74(12):2979–87.

[6]

Lugovy M, Orlovskaya N, Berroth K, Kuebler J. Macrostructural engineering of ceramicmatrix layered composites. Compos Sci Technol 1999;59:1429–37.

[7]

Blattner AJ, Lakshminarayanan R, Shetty DK. Toughening of layered ceramic composites with residual surface compression: effects of layer thickness. Eng Fract Mech 2001;68:1–7.

[8]

Evans AG. Perspective on the development of hightoughness ceramics. J Am Ceram Soc 1990;73(2):187–206.

[9]

Sglavo VM, Larentis L, Green DJ. Flawinsensitive ionexchanged glass: I, theoretical aspects. J Am Ceram Soc 2001;84(8):1827–31.

[10]

Moon RJ, Hoffman M, Hilden J, Bowman K, Trumble K, Roedel J. Weight function analysis on the Rcurve behavior of multilayered alumina–zirconia composites. J Am Ceram Soc 2002;85(6):1505–11.

[11]

Fett T, Munz D. Influence of cracksurface interactions on stress intensity factor in ceramics. J Mater Sci Lett 1990;9:1403–6.

[12]

Timoshenko SP, Goodier JN. Theory of elasticity. 3rd ed. New York: McGrawHill; 1970.

[13]

Giannakopoulos AE, Suresh S, Finot M, Olsson M. Elastoplastic analysis of thermal cycling: layered materials with compositional gradients. Acta Metall Mater 1995;43(4):1335–54.

[14]

Sergo V, Lipkin DM, De Portu G, Clarke DR. Edge stresses in alumina/zirconia laminate. J Am Ceram Soc 1997;80(7):1633–8.

[15]

Srawley JE. Wide range stress intensity factor expressions for ASTM E 399 standard fracture toughness specimens. Int J Fract 1976;12:475–6.

[16]

Lugovy M, Slyunyayev V, Orlovskaya N, Blugan G, Kuebler J, Lewis M. Apparent fracture toughness of Si
_{3
}N
_{4
}based laminates with residual compressive or tensile stresses in surface layers. Acta Mater [submitted].

[17]

Hyatt T. Electronics: tape casting, roll compaction. Am Ceram Soc Bull 1995;74:56–9.

[18]

Gogotsi GA. Fracture toughness studies on ceramics and ceramic particulate composites at different temperatures. In: Salem JA, Quinn GD, Jenkins MG, editors. Fracture resistance testing of monolithic and composite brittle materials ASTM STP 1409. West Conshohocken (PA): American Society for Testing and Materials; 2002. p. 199–212.

[19]

Hansen J, Cutler RA, Shetty DK, Virkar AV. Indentation fracture response and damage resistance of Al
_{2
}O
_{3
}–ZrO
_{2
} composites strengthened by transformationinduced residual stresses. J Am Ceram Soc 1988;71(12):C501–5.

[20]

Cutler RA, Bright JD, Virkar AV, Shetty DK. Strength improvement in transformation toughened alumina by selective phase transformation. J Am Ceram Soc 1987;70(10):714–8.

