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Chapter 12 Toughening in DZC by crack trapping
395
Chapter 12
Toughening in DZC by Crack Trapping 12.1
Introduction
In the preceding Chapters we considered toughening mechanisms in which the toughening resulted from the shielding of a macrocrack front (tip) by a zone of transformation or microcracks. These two mechanisms are sometimes grouped under a single category called the process zone mechanism. Another category of toughening mechanisms relies on the inhibition of propagating cracks by the presence of a second phase which is here understood in a very broad sense. It could for instance refer to the second phase particles in T T C or the external reinforcement by small particles (as in metal-ceramic composites) or by continuous fibres (as in whisker reinforced ceramics or ordinary resin-based composites). We shall only describe this category with respect to second phase particles which are intrinsic to the T T C material, as for example t-ZrO2 in a c-ZrO2 matrix (PSZ) or t-ZrO2 in an A1203 matrix (DZC). This toughening mechanism has the added advantage over the process zone mechanism in that it is temperature insensitive. The restraining effect of the second-phase inclusions on an advancing crack front which is the basis for the reduction of crack-front stress intensity factor (i.e. for toughness increment) may be visualized in two ways. First, it can be presumed that parts of an advancing crack in the ceramic matrix are pinned together by unbroken second-phase inclusions over a certain distance behind the crack tip.
396
Toughening in DZC by Crack Trapping
The length of the bridged crack portions and the consequent toughness increment depend on the breaking strength of the inclusions. Figure 12.1 shows how parts of an advancing crack are pinned in Mg-PSZ, resulting in bridging over the uncracked ligaments (the crack is advancing from left to right in this optical micrograph). Secondly, it may be viewed as the trapping of a crack front against forward advance by contact with an array of obstacles. The toughness increment in this viewpoint is determined by the ease (or difficulty) of cutting through or circumventing the obstacles. We shall however find that these two viewpoints overlap. We shall present below models based on both viewpoints. The latter viewpoint has the added advantage in that it permits us also to describe (by analogy with the two-dimensional model) the three-dimensional model to the first order. For clarity of presentation, we shall first consider the models in isolation from phase transformation, that is we shall assume that just this mechanism of toughening is operative. We shall then present the interaction effects between the crack-bridging and transformation toughening mechanisms, by assuming that the obstacles are transformable precipitates which not only transform to monoclinic form but offer resistance to crack propagation by trapping it.
12.2
Small-Scale Crack Bridging
Several equivalent approaches are available for studying the mechanism of toughening due to the bridging of some portions of an advancing crack by second phase particles. In all approaches the bridged portions are modelled by a continuous distribution of springs between crack faces. These springs may be linear or non-linear, although the non-linear spring model does not lend itself to easy mathematical analysis, with just one exception considered by Nemat-Nasser & Hori (1987). Three equivalent approaches have been proposed by Nemat-Nasser & Hori (1987), by Rose (19875), and by Budiansky, Amazigo & Evans (1988). The first two approaches stress the mathematical viewpoint, whereas the third the physical viewpoint at the expense of some mathematical rigour. We shall take this last approach and draw heavily from the above paper. In all three approaches, the implementation of the crack bridging model involves two distinct steps. First, a stress analysis must be performed for an assumed constitutive equation for the reinforcing spring (i.e. assumed force-displacement law). Secondly, an appropriate forcedisplacement law for the springs must be established, either theoretically
12.2. Small-Scale Crack Bridging
397
F i g u r e 12.1: An optical micrograph of a bridged crack
or experimentally. Significant simplifications in the first (analysis) step result from the assumption that the bridged length is small relative to crack length and specimen dimensions and that the reinforcing springs have a linear force-displacement characteristic. This small-scale bridging assumption is akin to the small-scale yielding assumption made in the Dugdale model for metals with limited plasticity (e.g. high-strength steels). We shall bring out more clearly this similarity between smallscale bridging and yielding later in this Section. Following Budiansky et al. (1988), and Rose (1987b) we assume in the analysis step that the spring stress tr(x) is linearly proportional to the crack-face displacement in the bridged zone (Fig. 12.2) a(x) = kE'v(x)
(12.1)
Here, E' - E (plane stress) or E' - E / ( 1 - t, 2) (plane strain), k is the spring stiffness which we shall calculate later (in the second step), and 2v(x) is the crack-face displacement. We now use an argument based on simple energy considerations (also follows from the J-integral formalism adopted in w to relate the energy input to the energy consumed by the snapping of last spring and by the crack growth. The energy input is provided by the applied field K appz whereas the energy consumed by the crack growth is related to the critical stress intensity factor Kc for crack growth in the matrix (Kc has the same meaning as in transformation toughening mechanism), and the energy loss caused by the fracture of
Toughening in DZC by Crack Trapping
398
F i g u r e 12.2: Crack bridging by second phase particles and equivalent reinforcing spring model
the last spring (which is required for crack growth to occur) is related to the spring breaking strength a(L) - a~. The above energy balance condition at the instant of imminent crack growth may be written as (Kappt) 2
EI
Kc 2
=
EI
o.y 2
,
(12.2)
EEI
In other words, crack propagation will occur when the toughening ratio A - KaPPZ/Kr attains the value
A -
~/1-~- kK-----~r~ ~
(12.3)
The analysis problem therefore reduces to determining the relationship between A and the bridge length L, for which the spring stress distribution a(x) is required. This is because the stress intensity factor at x - L may be written as K appz less the contribution from the spring stress over 0 < x < L. This last contribution is known from handbooks (see, e.g. Tada et al., 1985), and so Kc may be written as
Kc
-
K appl -
foL
~
dx
(12.4)
Instead of calculating a(x) from (12.4), one can first calculate v(x) from the same argument,
12.2. Small-Scale Crack Bridging
v(x)-
4Ii~PPZv/7 E'vf~
399
2 fo L
7rE'
dx'
cr(x') log
(12.5)
and then calculate a(x) from (12.1). Whichever route one takes, the solution of a singular integral equation is unavoidable. We shall not reproduce the solution steps here, but give just the final result. The interested reader will find the details in the paper by Budiansky et al. (1988). It transpires that 1
A -
(12.6)
V / 1 - 2g2(c~) where g(s) - ~r(s)/[KaPPtx/~], s - 4kx/Tr, and a - 4kL/Tr. The results for the non-dimensional spring stress g(a; s) are shown in Fig. 12.3 as a function of the non-dimensional distance (a - s) from the last spring, for a = 2, 5 and 20. The limiting case c~ = c~ (Rose, 1987b) is shown for comparison. g(s;a)
~ I 0.6
0.4
0.2 0~=oo
0.0
0
5
I
I
10
15
~20 2O
O~-s
F i g u r e 12.3: Variation of spring stress along the bridging zone as a function of spring stiffness We now proceed to the next step, that of estimating the effective spring constant k. It depends on the shape, size, distribution, and content of the second-phase particles in the matrix. For spherical particles,
Toughening in DZC by Crack Trapping
400
Budiansky et al. (19881 estimate k to be 2c)
E"
(12.7)
where E-~ is the effective elastic constant (it should actually be two constants, E-~ and P, in plane strain situation) of the composite material consisting of the ceramic matrix, and a random distribution of spherical particles (radius a) and volume fraction c. The effective elastic constant may be calculated on the basis of Hill's self-consistent method (see w The parameter ~ which also depends on c can be approximated by fl ,~ (1 - c)(1 - x/~). Because the above estimate is obtained on the assumption that the spherical particles are "smeared" out in the thickness direction, it is appropriate to replace ~ry by cS (where now S is the strength of one particle at fracture) and to account for the reduction in the length of advancing crack by (1 - c) in (12.2), so that it now reads
(Koppt)2
K~
~ S 2ac(1 - c)(1 - v ~)
Elm (1 - c ) + ~
E~
(12.8)
Then, for the case of particles that break elastically, the modified toughening ratio (cf. (12.311
A -
_ ~
~/'7(1 - c)
r S2ac(1 - x/~) 1+ ~
K~
(12.9)
where 7 - - ~ / Z ~ . It is clear from (12.9) that the modified toughening ratio depends on the strength, size and concentration of the second phase particles (see Fig. 12.4). To conclude this section, we assume that the springs are perfectly rigid plastic in the spirit of ideal plasticity inherent to Dugdale-type models for small-scale yielding in metals. In other words, we assume that the energy loss caused by the fracture of the last spring is simply equal to 2cSv/ (the last term in (12.8)), where 2vl is the relative displacement of crack faces at fracture. In this idealized case, (12.81 gives the modified toughening ratio as
A
-
-
K appl K~V/7(1 - c)
_ ~
~C 1 + (1
2E~Sv/ c)
K2
(12"10/
401
12.2. Small-Scale Crack Bridging
lO-
appl K
h
/K c
- - .
~/~'(l-c) 2 E( 1-v m ) 2
Em(1-v ) c L P = (1-c)(1-~c) Ta 0
0
1
I
I
I
10
20
30
40
I
50 P
F i g u r e 12.4: Toughening ratio for bridging by elastic spherical particles
In Dugdale-type models one often defines the fracture toughness K:, i.e. the critical stress intensity factor corresponding to the critical crackface displacement 2vl at fracture. K/ is related to 2v: as follows
K:
-
~JE-4S(2v/)
(12.11)
Note that both K: and v: refer to the composite. In metallurgical terminology K] is the fracture toughness and 2v! the critical crack opening displacement of an ideally-plastic material (yield strength S) exhibiting only small-scale yielding. In analogy with Dugdale-model, the modified toughening ratio (12.10)simplifies to give Is appI
KC
-
i(
1-c)+c
gr
(12.12)
Equation (12.12) is nothing but the law of mixtures on the work of fracture. For small amounts of second phase particles (c < < 1) it reduces to the law of mixtures on the toughnesses K ~ppz K: = (1 - c ) + c Kr K~
(12.13)
Toughening in DZC by Crack Trapping
402
We can continue the analogy with Dugdale model and express A =
KaVPZ/Kc in terms of the bridge length Lc at fracture. This is readily given by (12.4) after substituting or(x) = S and integrating from 0 to
Lc.
12.3
Crack Trapping by Second-Phase Dispersion
We now adopt the second viewpoint in which the toughness increment is thought to result from the trapping of a crack front against forward advance by contact with an array of obstacles. We shall in particular identify the conditions under which this viewpoint coincides with the crack-bridging viewpoint, as well as with the three-dimensional crack trapping model that will be described subsequently in this Section 12.3.1 Two-Dimensional Crack Trapping Model
F i g u r e 12.5: An idealized representation of crack front trapping by second-phase inclusions We follow the analysis by Rose (1987c, 1987d) and consider a planar crack (in the plane y = 0) with an initially straight front coinciding with the z-axis and growing in the direction of x-axis. Figure 12.5 shows an idealized representation of the crack front which is held up by second-phase inclusions, indicated by shaded circles. The idealization is intended to suggest that the major portion of the crack front can be regarded as having progressed to a distance 2A ahead of the line of inclusions but is trapped by unbroken ligaments stemming from
12.3. Crack Trapping by Second-Phase Dispersion
403
-..-.----__.._~~
a) J
Lr
7
Z
C
"
~
~,
~
~.
~
~,
F i g u r e 12.6: (a) Crack bridging represented by springs over the segment ]z] < A, (b) a section along x - 0, and (c) the unbroken ligaments between cracks replaced by springs
the inclusions. We assume for simplicity that these deep crack front perturbations (A > > A) are periodic in nature and approximate this configuration by regarding the crack to extend to x = A, but with a uniform distribution of springs acting between the crack faces over the segment Iz[ < A, as in Fig. 12.6a. These springs (which we shall assume, as before to be linear) represent the restraining action of the unbroken ligaments. In other words, we use the linear spring stress-displacement relationship (12.1), but for clarity affix appropriate suffices to the stress and displacement o'uy = E'kuy (12.14)
Toughening in DZC by Crack Trapping
404
To obtain the appropriate spring constant k, we consider a thin slice taken along x - 0, as shown in Fig. 12.6b. The section through this slice is nothing but a periodic array of through cracks consisting of cracks of length 2s separated by unbroken ligaments of length 2r along the zaxis. Its response to a mode I stress field prescribed by K appl c a n be determined analytically (see, e.g. Tada et al., 1985), with the net y-axis displacement between y = 0 + and y = 0- given by 2uu(z )
- 4cruu~ log cos ~ + v/cos 2 ~- - cos 2 ~ 7rE'
The average crack opening (2Uy(Z)) over the gap Izl < , uy(z) - 0 over z covering the particles)
(2uu(z))-~
(12.15)
cos
s 2uy(z)dz-
7rE' log sec -~-
is (note
(12.16)
so that the average spring constant k is given by (12.14) and (12.16) to be
k~ - ~'/{21og (ser
}
(12.17)
)~ = (2r + 2s) is the spacing between the inclusions. It may now be shown using the previously described distributed spring model (12.4) that the ratio of the actual stress intensity factor K tip at the crack tip x = A to the nominal factor K appz depends only on the non-dimensional ratio kl, provided 2A(= l) is less than the crack and specimen dimensions,
Ktip Kappl
= Fl(kl)
(12.18)
where
Fl(x) -
i
1 + 0.355x 1 + 2.90x + 2.23x 2
(12 19)
The maximum spring stretch is given by (vide (12.5))
6m,~ - 2uv =
4K appl E'
Vl(kl)
(12.20)
12.3. Crack Trapping by Second-Phase Dispersion
405
where
VI(x) -
i
1 + 0.656x
i + 2.68x + 1.67x 2
(12.21)
FI(x) and VI(x) are interpolating functions constructed so as to reproduce the correct asymptotic behaviour for soft springs (kl < < 1) and for hard springs (kl > > 1). These functions were obtained by Rose (1987b). We mention en passant that there is a misprint in the corresponding equation in the paper by Rose (1987c) from which we have drawn much of the material in this Section. We now need to specify the dynamical conditions for quasi-static (steady-state) crack growth. We assume that failure occurs at a value of the ap.plied stress field K appl when K tip achieves the critical value for the material If/. As before, we shall assume the corresponding value of the maximum spring stretch 8ma~: to be 2vy (see (12.10)). It is convenient to specify the failure criterion at the failure of the spring (at x = - A ) in terms of a limiting stress intensity factor KI, rather than a limiting spring stretch 2yr. The connection between these two specifications (i.e. K/ at x = A and 2v/ at x = - A ) is established via the collinear crack array model mentioned above. For such an array (see, e.g. Tada et al., 1985) K/
-
crvv A tan -~ - E'kv!
tan -~-
(12.22)
From (12.20) and (12.22), we obtain the following equation for describing the onset of fracture
If/ _ 2 V l ( k l ) i k / t a n ~ ) K~PPz log(sec - -
(12.23)
Ky may be regarded as the critical stress intensity for the lateral growth of the crack front, that is for increasing s with A fixed, because the growth in the x-direction (increasing A for fixed s) occurs only when the spring at x = - A is stretched to its maximum value 2vy. It should be obvious that this K/ (at x = A), as well as v! (at x = - A ) are different in physical interpretation and magnitude from the Ky and vy used in the previous model (12.11). It is also convenient to express the critical value of K tip for the composite (i.e. KF) in terms of the intrinsic matrix toughness Kr and the toughness K / f o r lateral crack growth. K r
Toughening in DZC by Crack Trapping
406
which should now be regarded as the critical stress intensity factor of lateral and forward growth of the crack front is chosen to be related to K l and I~'~ by a law of mixtures on the work of fracture (cf. (12.12)) in proportion to the contents of matrix and inclusions
-
/2s
2 + 2r K}
(12.24)
Then from (12.18) and (12.24) we get the following toughening ratio
K ~ppt
V/(1 -
c) +
c(Kf/K~) 2
(12.25)
where, in analogy with (12.12), we have denoted by c = 2r/,~ the area (volume) fraction of inclusions, such that 2s/,~ = ( 1 - c) is the area (volume) fraction of matrix. It is worth stressing again that the K] in (12.25) has a different physical interpretation from the Kf appearing in (12.12). Eliminating K appl between (12.23) and (12.25), one obtains an equation for determining kl in terms of the material parameters Kf/K~, and c
4- v (kt)
(l-c)+c
log
7r(12"-c )
(12.26)
Solving (12.26) for kl and substituting into (12.25), one obtains the toughening ratio K appz/Kc. The corresponding equation in the paper by Rose (1987c) is unfortunately incorrect casting doubts on his results and conclusions. The correct solution of (12.26) for several values of Kf/Kc and 0 <_ c _ 1 is shown in Fig. 12.7. The corresponding values of the toughening ratio (12.25) are shown in Fig. 12.8. For comparison, we have also shown (by broken lines) the toughening ratio Kappz/K~ predicted by a simple law of mixtures (i.e. by the numerator of eqn (12.25)). It is evident that by allowing the broken segments of the crack front to grow laterally (in the z-direction) before they advance in the x-direction, the toughening effect is amplified. We may therefore regard 1/Fl(kl) > 1 as the amplification factor in (12.25). The lateral growth of broken segments of the crack front without their forward progress is akin to an increase in the volume (area) fraction of the matrix traversed by the segments and a corresponding decrease in the effective area fraction of the inclusions. We may therefore introduce
12.3.
Crack Trapping by Second-Phase Dispersion
407
kl 0.8
12
0.6
0.4
0.2
t
I
I
1
l
0.0
0.2
0.4
0.6
0.8
0.0
Figure
-~._1
1.0 c
12.7: Spring stiffness for various values of
K//Kc
appl
K
/Kc Kf/K =12
12 10 ..--""""
9
8
6 4
4
2
3 2
0 0.0
Figure
l 0.2
i 0.4
t 0.6
i 0.8
1 2 . 8 : T o u g h e n i n g ratio for various values of
i 1.0 c
KI/Kc
Toughening in DZC by Crack Trapping
408
the effective area fraction of inclusions through f (which will now depend on current position in the z-direction), so that the area fraction of the matrix is (1 - f), and rewrite (12.25) as follows
-
where f - { 1 - ( 1 - c ) / F l ( k ' ) }
(1 - / )
+ /
(12.27)
and K ; -
K! {1 + ( F ? i k l ) - 1 ) / f } l , 2 .
We can now identify K~ with the K l appearing in (12.12), so that the latter equation is identical in form to (12.27). The above argument may be viewed as an attempt by the broken segments of crack front to join up (because it is easier to grow in the matrix even under the influence of inclusions) before making forward progress. This is quite likely to happen when A > > A, but not so when A < < A because of the strong influence of the inclusions. In the latter case, the law of mixtures (numerator in (12.25))is more likely than (12.27) to be closest to the toughening ratio, as we shall see in the next Section where we consider this case. 12.3.2
Three-Dimensional ping
Small-Scale
Crack
Trap-
The discussion in the preceding Section was based on a two-dimensional plane strain configuration, although there are elements of three-dimensionality implied in the lateral growth of the broken crack segments. Here we present a three-dimensional analysis of limited validity (A < < A) based on the first-order perturbation of the stress intensity factor distribution along the front of a half-plane crack, when the location of that front differs moderately from a straight line. We shall use this perturbation solution to the configuration of the front of a planar crack that is trapped against forward advance by contact with a periodic array of closely spaced obstacles. For this description we borrow heavily from a paper by Rice (1988). Consider a half-plane crack in the plane y = 0, growing in the direction z and having a straight front along x = a0 (Fig. 12.9). For fixed loading the stress intensity factor along the straight crack front parallel to the z-axis, K~ a0], may be obtained from (9.1) with c~ = I and x replaced by (z - a0). If the crack front is not straight but lies along the arc x = a(z)
12.3. Crack Trapping by Second-Phase Dispersion
409
F i g u r e 12.9: (a) A half-plane crack with a straight front at x = a0, (b) a half-plane crack with a moderately curved front at z = a(z), and (c) crack front trapped by impenetrable obstacles
(Fig. 12.9b) in the plane y = 0, then Rice (1988) has shown that, provided a ( z ) i s small such that in an average sense (on a large scale) the crack front is still straight and the variation of K~ a0] with a0, i.e. OK~ is much smaller that K ~ itself, the stress intensity factor K[a(z), 0, z] (denoted simply by K(z)) along the moderately curved crack front is
K(z) - K~ K~
a0]
a0]
1 / = ~ da(z')/dz'
= 2-~
co (z' - z) dz'
(12.28)
to the first-order deviation of a(z) from a0, i.e. to the first order in da(z)/dz. The singular integral in (12.28)is to be understood in the sense of Cauchy principal value. Next, consider the situation depicted in Fig. 12.9c. The crack front is trapped by impenetrable obstacles of some given distribution. Then a(z) is known along the contact zones Ltrap but K(z) is unknown there. Conversely, K(z) = Kc is known along the penetration zones Lpe,~ (i.e. the matrix part) but the depth of penetration a(z) is unknown there. These two conditions together reduce (12.28) to the singular integral equation
K~
ao] - Kc K~ ao]
1 [ da(z')/dZ'dz , 27r JLt,.~p (z - z')
Toughening in DZC by Crack Trapping
410
= 1/L
da(z')/dZ'dz'
(z-z')
(12.29)
for all z included in Lven. Once (12.29) is solved for a(z) along Lv,=, K(z) along Lt,-ap can be found from (12.28). In fact, the solution of (12.29) can be lifted by analogy from known solutions in two-dimensional crack theory. To explore the analogy consider a two-dimensional medium in (y, z) plane containing a single crack or an array of cracks. The medium is loaded remotely in mode I such that the stress avy = cravPZ corresponding to K appl and the opening gap (i.e. net y-direction displacement) between y = 0 + and y = 0- is 2uy(z) = 5(z). Then cryy(z) along z-axis is obtained as
E' f ? 5(z')/dz' aY~ - ~raPP'+ -~r oo ( z ' - z) dz'
(12.30)
For a single (finite or semi-infinite)crack (12.30) can be readily derived from (4.21), (4.24), and (8.14) or (8.3). Equation (12.30) may be rearranged to coincide exactly with (12.28) if one makes the identifications 2[cryy(z) - ~,pvt] .-..+ [K(z) - K ~
E'
K~ 5(z) ~ a ( z )
(12.31)
In this analogy, along Lt,-av the opening displacement is 6(z) = a(z), whereas along Lven the crack faces sustain the stress ayy = crappz + E ' ( K c - K~ ~ Note that cryu will be less than crappz, since K ~ > Kc. Thus, the crack faces will open with the opening displacement 5(z) cor-
F i g u r e 12.10: Periodic array of impenetrable obstacles in the path of a half-plane crack
12.3. Crack Trapping by Second-Phase Dispersion
411
responding to the crack front penetration a(z) in the three-dimensional trapping problem. Rice (1988) extends the analogy to obstacles which are not completely impenetrable, but we shall here limit ourselves to impenetrable obstacles and consider the periodic array shown in Fig. 12.10. The impenetrable obstacles with centre-to-centre spacing 2L (= A used in the preceding Section) have a gap 2H(= 2s of the preceding Section) between them, into which the crack front can penetrate. We have already used the solution for this periodic configuration in the preceding Section (see (12.15), (12.16)). So with substitutions for avv and $(z) identified above, (12.15)and (12.16) read (for - H < z < H)
a(z) = 2Uy(Z)
_( 4L
1-
log
7r
(ap~,> -
K~
{
COS~_L" + r
4L2(
(2uv(z)> - -~ff
2 ~'z
,rH
~-T - c~ cos xH 2--X-
I<~
~-
(rH)
1 - KO) log sec ~
}
(12 32)
(12.33)
Substituting (12.32)into (12.29) one obtains along Lt,-~p (H < ]z I < 2 L - H) the following stress intensity factor sin K(z) =
~z
(12.34)
+ (K ~ -
~//COS 2 ~ ~'H - - c~
~'z 2---E
which is singular, as expected, at the borders of the trap zone. The average value of K(z), denoted (K(z)) over the trap zone, (H < [z I < 2 L - H) is
(K(z)) -
LK ~ - HKc L-H
(12.35)
Denoting by f - ( L - H)/L the fraction of contact, so that (1 - f) is the fraction of penetration, eqn (12.35) can be written as
K~ (K(z)> /~ = ( l - f ) + f Kc
(12 36)
The limiting value of K ~ denoted as before K appl corresponds to the instant at which the crack front just breaks through the obstacles.
Toughening in DZC by Crack Trapping
412
This value can be calculated exactly. For the small perturbation approximation considered here (i.e. amax < < 2L, equivalent to A < < ~ of the preceding Section), it is necessary that
(K~ 2 - {(K(z))} 2
(12.37)
This is obvious from the observation made earlier that the assumption of small-scale deviation of crack front from straightness is akin to the assumption that the function a(z) fluctuates in z about a mean value so that, in an average sense, the crack front is still straight. K ~ is indeed the stress-intensity factor for the straight crack front a(z) = ao. Since K(z) is known everywhere at breakthrough (it is equal to Kc o n - H < z < H and Kp o n H < z < 2 L - H , where Kp > K ~ is the stress-intensity factor for circumventing the particles), eqn (12.37) gives the exact value of K~ K appz) at breakthrough
IiaPPZI(c
1 - f) + f
~
(12.38)
When (Kp - Kc)/Kc < < 1, (K(z)) ~ Kp, so that at the instant of breakthrough (12.36) reduces to
=(1-y)+y
(12.a91
which also follows from (12.38), as it should. It should be stressed that the linear perturbation theory is in fact applicable only to the case when ( K p - gc)/Kc < < 1. A comparison of (12.38) with (12.13), and with (12.27) quickly establishes the connection, at least for small crack front penetrations, between the three-dimensional and two-dimensional models of crack trapping on the one hand, and between the crack trapping and crack bridging models, on the other. All that is required for establishing this connections is an appropriate interpretation of K.t , K~, and lip, as discussed above.
12.4
Crack Trapping by Transformable Second-Phase Dispersion
We now consider the situation when the crack is trapped by second-phase dispersed precipitates which can also transform to monoclinic phase. The
12.4. Crack Trapping by Transformable Second-Phase Dispersion 413 ceramic matrix in such ZTC is toughened not only by the phase transformation of the tetragonal precipitates but also because the precipitates impede the progress of the macrocrack and trap it due to the mismatch in elastic properties. The length of the trapped zones is determined by the size, volume fraction and phase transformation characteristics of zirconia precipitates which we shall assume, for simplicity, to be periodically distributed. These parameters will therefore also determine the spring stiffness which will vary along the bridging zone. In this Section, we shall assume in the spirit of Dugdale (1960) and Bilby, Cottrell & Swinden (1963) model that the transformation zone has no thickness and is coplanar with the discontinuous macrocrack fragments, as shown in Fig. 12.11. We shall follow closely the paper by Jcrgensen (1990) to determine the spring stiffness in (12.1). The tetragonal precipitates that have impeded the progress of the macrocrack and have fragmented it will transform into monoclinic phase because of the very high stresses at the tips of the fragments, thus reducing the stress intensity factor at these tips. The discontinuous crack front cannot therefore grow laterally until the external loading is increased to overcome the shielding effect due to phase transformation. It will be assumed that the transformation is accompanied by dilatation alone so that the transformation zone at each crack tip in the periodic array (Fig. 12.11) can be regarded as planar to which the BCS model with the modification by Rose & Swain (1988) is applicable. In this modified BCS formulation, the stress distribution in the transformation zone at each crack tip in the array is obtained by superposition of two fields. The first stress field corresponds to a stress intensity factor equal to the fracture toughness of the matrix, Kc. The second field is due to a stress intensity factor (K appt- Kc), where K appt corresponds to the applied stress crappz, together with a constant stress (aappz- or*) acting across the transformation zone length I. The cohesive stress a* due to transformation-induced dilatation has to be determined from the dynamic condition for transformation, according to which the total stress O'zz at the end of transformation zone must equal the characteristic value a0 for a tetragonal precipitate, or0 is in turn related to the critical mean stress crm for tetragonal to monoclinic transformation via 3 fro = 2(1 + v)tr~ (12.40) Now following Bilby et al. (1964), and Rose and Swain (1988) it can be shown that or* and l are related to the loading and transformation
414
Toughening in DZC by Crack Trapping
F"
Vl
Figure 12.11: Crack trapping by periodic array of transformable precipitates
12.4. Crack Trapping by Transformable Second-Phase Dispersion 415 characteristics as follows 2 /Trs K~ o'* - o'o
1+
~" (1
l -
-1
~--(Is s
'~ KaPPz K~ ~/ rs K.ppz ) tan -~-
(12.41)
- Is
(12.42)
tan 7r___ss
8(~*) 2
Next we need to calculate the opening of each of the cracks in the array. For this we again follow Bilby et al. (1964), and calculate the average opening over each crack (with its transformation zones)
2 [(s+l)
I
fy
'~ H(y')dy'dy'
(12.43)
I
where H(y') is given by
g(r
- ~
~osh -~
~'(r - a')
(r162
(12.44)
c' - sin ( ~ - ~ )
(12.45)
_ ~osh-~ ((a')~ +
and y' - sin (~~-~) , a'-- sin
Now assuming that the crack face displacement at section x of the macrocrack (Fig. 12.11) is given by the average opening of the lateral crack array with the transformation zones at this section, eqn (12.1) may be rewritten as
k(x)
-
6r a p p l Et(uz(x)),
"
-o "appl
~)i I~[ appl
7r8
tan-~-
(12.46)
Figure 12.12 shows an example of the variation of k(x) with K.ppz/Kc
Toughening in D Z C by Crack Trapping
416
"~ 1.O o t~
,I--4
E
0.8
0
z
0.6 0.4 0.2 0.0
0
I
I
i
I
I
I
I
1
2
3
4
5
6
7
appl
K
/K c
F i g u r e 12.12: Normalized spring stiffness
at a given instant (i.e. given x). k(x) is normalized by k corresponding to a linear spring model in the absence of phase transformation (12.17) which is equivalent in the present formulation to the condition K appl <_ Kc. It is clear that in the presence of a phase transformation the springs are softer. Having calculated the spring stiffness, we determine the macrocrack opening displacement using (12.5). v(x) - (uz(x)) in this equation is approximated by a second-order spline, and the resulting non-linear equation is solved iteratively to calculate ~r(x) for various values of Aa. The stress intensity factor at the tip of the macrocrack is then calculated from (12.4). The length of the bridging zone Aa is finally established from the dynamic conditions for growth of the macrocrack. The latter will grow when K tip (left hand side of (12.4)) reaches the intrinsic fracture toughness of the matrix Kc, while simultaneously the spring at x - Aa stretches to its limit and snaps. The limiting value of the spring extension at x - Aa is in turn determined by the dynamic condition for unstable crack growth in the lateral direction (z-direction). The unstable lateral growth occurs when the unbroken ligaments between the cracks join to regain the continuous front that pertained prior to its trapping. This is equivalent to saying that the phase transformation capacity of tetragonal precipitates has been exhausted and there is no further crack shielding available from them to prevent unstable lateral crack growth. At this instant, the applied stress
12.4. Crack Trapping by Transformable Second-Phase Dispersion
K
417
appl
/Kc 1.5-
1.4 ~ 1.3
~
1.2
~
~,-2s = 2001xm 1501xm lO0~m 50lam
1.1 1.0
1
J 5
I 9
i
I
13
17
I
21 L/2s
F i g u r e 12.13: Toughening ratio as a function of the size and volume fraction of transformable precipitates
intensity factor K appt will equal the fracture toughness of ZTC in which the only toughening mechanism is the phase transformation. Having satisfied the two dynamic conditions for the growth of the macrocrack (in the x-direction), one can estimate the toughening resulting from the joint action of the phase transformation of dispersed tetragonal precipitates and crack trapping, using eqn (12.4). The resulting toughening ratio Kappt/K c is shown in Fig. 12.13 as a function of the volume fraction of precipitates (A/2s) for several values of the precipitates size ( A - 2s). It is interesting to note that in non-transformable ceramics this ratio is independent of precipitate size (cf. Fig. 12.8). This is explained by the fact that the toughening ratio is controlled by two distinct mechanisms. The contribution from the phase transformation to the toughening effect is most pronounced for small (A/2s), but diminishes with increasing (A/2s) when the bridging mechanism progressively takes over. It would appear that the transition in dominant mechanisms occurs at A/2s ~ 3. In applying the above results to ZTC it is worth remembering that the model assumes a small transformation zone compared to crack size and ignores any transformation of tetragonal precipitates in the x-direction. It is for this last reason that the model is likely to be more accurate for small rather than large values of (A/2s) because of the assumption that
Toughening in DZC by Crack Trapping
418
the opening displacement of the macrocrack at any x equals the average opening of the crack array in the y-direction for this section, i.e. the fracture toughness of precipitates themselves is not very different from that of the matrix. We now remove the restriction that the transformation zones are coplanar with the macrocrack fragments and allow them to grow out in the z-direction as far as required by the critical mean stress criterion (Fig. 12.14). For this we utilize the R-curve analysis for a collinear array of internal cracks in a TTC from w The exposition below follows closely the paper by M011er & Karihaloo (1995).
F i g u r e 12.14: Collinear array of plane cracks in a transforming ceramic As before, the effect of crack front trapping by transformable particles is modelled in two dimensions by a shielding stress, ~r' (x) acting over the shielded length Aa , as depicted in Fig. 12.15. The magnitudes of ~r8(x) and Aa are related to the volume fraction and size of transformable particles and to the intrinsic toughness of the matrix material, as shown below. The relative displacement of the crack faces for the above problem is again given by (12.5). Consider the point of instability of the macrocrack under a monotonically increasing load. At this point, the displacement of the crack faces at x = Aa is assumed to have reached a critical value, u(Aa)= (uc) so that a further increase in load would result in breakdown of the shielding mechanism. This dynamic condition for macrocrack growth gives the first governing equation
8K avpzv / ~
4 f a,
E'x/
Jo
o" (t) log
dr-(u
) = 0 (12.47)
12.4. Crack Trapping by Transformable Second-Phase Dispersion
419
Aa F i g u r e 12.15: Tip of macrocrack shielded due to crack front trapping
The crack will not however grow in a catastrophic manner unless the effective stress intensity factor at its tip simultaneously reaches the intrinsic toughness of the matrix material. The effective stress intensity factor for the trapped crack K tip is given by (12.4) with L replaced by Aa. At catastrophic growth K tip = Kc, which forms the second governing equation written in normalized form as
KavvZ
KC
r~ /o t' a
o's(x)
dx - 1 -
0
(12.48)
The macrocrack resumes its continuous front when the two dynamic conditions (12.47) and (12.48) are simultaneously satisfied. To relate cr~(z) and ur to the volume fraction and size of transformable particles, let us consider a cut t - t as indicated in Fig. 12.11. It is assumed that the relative displacement of the crack faces of the macrocrack at instability is equal to the average opening (ur of the array of cracks at peak applied stress ~rp. This stress can also be thought of as the apparent strength of the material. By introducing the length parameter L (7.28) and the critical average crack face displacement (re)
E/~, (vc) -
( u c ) 1 2 r ( 1 - v)
eqn (12.47) can be written in normalized form as
(12.49)
Toughening in DZC by Crack Trapping
420
~/Aa ~Kc L
K appl O m
1
+ 7rIi~ 2vrff~"
fo
crs(t) log
v/S-d_ e7
dt
(vc) L
(12.50)
Equations (12.48) and (12.50) constitute the governing equations for the crack trapping problem in the two unknowns Aa and K appl. r and (re/ are determined from the analysis of the array of cracks in the transverse y-direction as discussed in Chapter 8 (w The critical average crack face displacement at x = Aa is obtained by integrating the dislocation density function
-
dx
(12.51)
D* ( x 0 ) d z 0
~r'(t) along Aa is approximated by a linear or quadratic function and the two governing equations are solved for Aa and K appl using the standard iterative procedure, described in w
( appl YY
0.24
p~-
0.23 0.22
0.21 0.20 0.0
0.5
i 1.0
I
1.5
J
2.0 (C-Co)/L
F i g u r e 12.16: Normalized peak applied stress during quasi-static crack growth, w - 10, AlL - 20. The normalized initial crack length is
coil - 5
12.4. Crack Trapping by Transformable Second-Phase Dispersion
x
a)
-
[-F"
-
421
~
Aa
P t3
__________=
b)
[-r
Aa
"
F i g u r e 12.17: Crack tip zone shielded by (a) linearly decreasing stress, (b) quadratically decreasing stress
The applied stress at which the transformation capacity of the particles is exhausted is shown in Fig. 12.16 for a given initial crack length. As foreshadowed above, the shielding stress will be approximated over Aa by a linear or quadratic function, as shown in Fig. 12.17. The applied stress intensity factor at the tip of the trapped crack K appt and the shielded length Aa are determined after the values of (uc) and a p have been calculated for different particle sizes s and area (volume) fractions A / f r o m the analysis of the array of cracks
mI -
(12.52)
The logarithm of the normalized shielding length is shown in Figs.
Toughening in DZC by Crack Trapping
422
log Aa L 2 1
0 -1 -2 -3 -4
-5 0.0
I 0.2
t 0.4
I 0.6
I 0.8
I
1.0
Af
F i g u r e 1 2 . 1 8 : Shielding l e n g t h as a f u n c t i o n of a r e a f r a c t i o n for linear shielding stress; s/L - 2, 5, 50
log Aa L
0 -1 -2 -3 -4
-5 0.0
i 0.2
i 0.4
i 0.6
t 0.8
I
1.0
Af
F i g u r e 1 2 . 1 9 : Shielding l e n g t h as a f u n c t i o n of a r e a f r a c t i o n for q u a d r a t i c shielding stress; s/L - 2, 5, 20
12.4.
Crack Trapping by Transformable Second-Phase Dispersion
423
appl
K
IKc_
1.16 1.12
0
5
1.08
1.04 1.00 0.0
I
I
I
l,
0.2
0.4
0.6
0.8
I
1.0 A, J
Figure 1 2 . 2 0 : Applied stress intensity factor as a function of area fraction for linear shielding stress; s / L - 2, 5, 50
appl
K
IKc
1.20 -
1.16 1.12 5 1.08
1.04 1.00 L_ _ _
0.0
0.2
1
0.4
_ _ L
_ _
0.6
I
0.8
__1
1.0 ,4, ./
F i g u r e 1 2 . 2 1 : Applied stress intensity factor as a function of area fraction for q u a d r a t i c shielding stress; s/l - 2, 5, 50
424
Toughening in DZC by Crack Trapping
12.18-12.19 as a function of area fraction of transformable particles for three normalized particle sizes under the linear and quadratic approximations to crS(z), respectively. The corresponding applied stress intensity factors are shown in Figs. 12.20-12.21. For a given transformable particle size s the shielded length decreases with an increase in the volume (area) fraction. This is a consequence of the behaviour of the critical crack face displacement (vc) obtained from the analysis of the array of cracks. Low volume (area) fractions of a given particle size, s correspond to large values of ~ (12.52), i.e. long cracks between the particles. This leads to a large average displacement before instability sets in, even though the applied load is relatively low. The large critical displacement of the macrocrack at x = Aa results in a large shielded length for the low volume (area) fractions, and vice versa. This is in agreement with the results of Budiansky et al. (1988) for tough non-transforming particles in a brittle matrix. As expected the quadratically varying shielding stress distribution results in a higher level of toughening and longer shielding lengths than does the linear distribution. The behaviour described above is reflected in the behaviour of K appl in Figs. 12.20-12.21 since the applied stress intensity factor is calculated at the tip of the shielded crack. All curves in Figs. 12.20-12.21 peak at moderate volume (area) fraction, but the magnitude of K appt increases with increasing particle size. The same trend is forecast by the theoretical analysis of Rose (1987c) who also found agreement of his results with experimental observations for an epoxy matrix containing a dispersion of alumina trihydrate inclusions (Lange & Radford, 1971). There is also experimental evidence (Lange, 1982) to support that peak toughness of an alumina ceramic toughened by yttria stabilized tetragonal zirconia particles is attained at a moderate fraction of dispersed zirconia particles. For a given set of material parameters the above analysis permits determination of the toughness of a brittle matrix ceramic toughened by dispersed transformable particles. As the peak toughness is attained at moderate volume (area) fractions of transformable particles, the model is useful for tailoring the microstructure of these materials to suit specific applications.
Chapter 12
Toughening in DZC by Crack Trapping 12.1
Introduction
In the preceding Chapters we considered toughening mechanisms in which the toughening resulted from the shielding of a macrocrack front (tip) by a zone of transformation or microcracks. These two mechanisms are sometimes grouped under a single category called the process zone mechanism. Another category of toughening mechanisms relies on the inhibition of propagating cracks by the presence of a second phase which is here understood in a very broad sense. It could for instance refer to the second phase particles in T T C or the external reinforcement by small particles (as in metal-ceramic composites) or by continuous fibres (as in whisker reinforced ceramics or ordinary resin-based composites). We shall only describe this category with respect to second phase particles which are intrinsic to the T T C material, as for example t-ZrO2 in a c-ZrO2 matrix (PSZ) or t-ZrO2 in an A1203 matrix (DZC). This toughening mechanism has the added advantage over the process zone mechanism in that it is temperature insensitive. The restraining effect of the second-phase inclusions on an advancing crack front which is the basis for the reduction of crack-front stress intensity factor (i.e. for toughness increment) may be visualized in two ways. First, it can be presumed that parts of an advancing crack in the ceramic matrix are pinned together by unbroken second-phase inclusions over a certain distance behind the crack tip.
396
Toughening in DZC by Crack Trapping
The length of the bridged crack portions and the consequent toughness increment depend on the breaking strength of the inclusions. Figure 12.1 shows how parts of an advancing crack are pinned in Mg-PSZ, resulting in bridging over the uncracked ligaments (the crack is advancing from left to right in this optical micrograph). Secondly, it may be viewed as the trapping of a crack front against forward advance by contact with an array of obstacles. The toughness increment in this viewpoint is determined by the ease (or difficulty) of cutting through or circumventing the obstacles. We shall however find that these two viewpoints overlap. We shall present below models based on both viewpoints. The latter viewpoint has the added advantage in that it permits us also to describe (by analogy with the two-dimensional model) the three-dimensional model to the first order. For clarity of presentation, we shall first consider the models in isolation from phase transformation, that is we shall assume that just this mechanism of toughening is operative. We shall then present the interaction effects between the crack-bridging and transformation toughening mechanisms, by assuming that the obstacles are transformable precipitates which not only transform to monoclinic form but offer resistance to crack propagation by trapping it.
12.2
Small-Scale Crack Bridging
Several equivalent approaches are available for studying the mechanism of toughening due to the bridging of some portions of an advancing crack by second phase particles. In all approaches the bridged portions are modelled by a continuous distribution of springs between crack faces. These springs may be linear or non-linear, although the non-linear spring model does not lend itself to easy mathematical analysis, with just one exception considered by Nemat-Nasser & Hori (1987). Three equivalent approaches have been proposed by Nemat-Nasser & Hori (1987), by Rose (19875), and by Budiansky, Amazigo & Evans (1988). The first two approaches stress the mathematical viewpoint, whereas the third the physical viewpoint at the expense of some mathematical rigour. We shall take this last approach and draw heavily from the above paper. In all three approaches, the implementation of the crack bridging model involves two distinct steps. First, a stress analysis must be performed for an assumed constitutive equation for the reinforcing spring (i.e. assumed force-displacement law). Secondly, an appropriate forcedisplacement law for the springs must be established, either theoretically
12.2. Small-Scale Crack Bridging
397
F i g u r e 12.1: An optical micrograph of a bridged crack
or experimentally. Significant simplifications in the first (analysis) step result from the assumption that the bridged length is small relative to crack length and specimen dimensions and that the reinforcing springs have a linear force-displacement characteristic. This small-scale bridging assumption is akin to the small-scale yielding assumption made in the Dugdale model for metals with limited plasticity (e.g. high-strength steels). We shall bring out more clearly this similarity between smallscale bridging and yielding later in this Section. Following Budiansky et al. (1988), and Rose (1987b) we assume in the analysis step that the spring stress tr(x) is linearly proportional to the crack-face displacement in the bridged zone (Fig. 12.2) a(x) = kE'v(x)
(12.1)
Here, E' - E (plane stress) or E' - E / ( 1 - t, 2) (plane strain), k is the spring stiffness which we shall calculate later (in the second step), and 2v(x) is the crack-face displacement. We now use an argument based on simple energy considerations (also follows from the J-integral formalism adopted in w to relate the energy input to the energy consumed by the snapping of last spring and by the crack growth. The energy input is provided by the applied field K appz whereas the energy consumed by the crack growth is related to the critical stress intensity factor Kc for crack growth in the matrix (Kc has the same meaning as in transformation toughening mechanism), and the energy loss caused by the fracture of
Toughening in DZC by Crack Trapping
398
F i g u r e 12.2: Crack bridging by second phase particles and equivalent reinforcing spring model
the last spring (which is required for crack growth to occur) is related to the spring breaking strength a(L) - a~. The above energy balance condition at the instant of imminent crack growth may be written as (Kappt) 2
EI
Kc 2
=
EI
o.y 2
,
(12.2)
EEI
In other words, crack propagation will occur when the toughening ratio A - KaPPZ/Kr attains the value
A -
~/1-~- kK-----~r~ ~
(12.3)
The analysis problem therefore reduces to determining the relationship between A and the bridge length L, for which the spring stress distribution a(x) is required. This is because the stress intensity factor at x - L may be written as K appz less the contribution from the spring stress over 0 < x < L. This last contribution is known from handbooks (see, e.g. Tada et al., 1985), and so Kc may be written as
Kc
-
K appl -
foL
~
dx
(12.4)
Instead of calculating a(x) from (12.4), one can first calculate v(x) from the same argument,
12.2. Small-Scale Crack Bridging
v(x)-
4Ii~PPZv/7 E'vf~
399
2 fo L
7rE'
dx'
cr(x') log
(12.5)
and then calculate a(x) from (12.1). Whichever route one takes, the solution of a singular integral equation is unavoidable. We shall not reproduce the solution steps here, but give just the final result. The interested reader will find the details in the paper by Budiansky et al. (1988). It transpires that 1
A -
(12.6)
V / 1 - 2g2(c~) where g(s) - ~r(s)/[KaPPtx/~], s - 4kx/Tr, and a - 4kL/Tr. The results for the non-dimensional spring stress g(a; s) are shown in Fig. 12.3 as a function of the non-dimensional distance (a - s) from the last spring, for a = 2, 5 and 20. The limiting case c~ = c~ (Rose, 1987b) is shown for comparison. g(s;a)
~ I 0.6
0.4
0.2 0~=oo
0.0
0
5
I
I
10
15
~20 2O
O~-s
F i g u r e 12.3: Variation of spring stress along the bridging zone as a function of spring stiffness We now proceed to the next step, that of estimating the effective spring constant k. It depends on the shape, size, distribution, and content of the second-phase particles in the matrix. For spherical particles,
Toughening in DZC by Crack Trapping
400
Budiansky et al. (19881 estimate k to be 2c)
E"
(12.7)
where E-~ is the effective elastic constant (it should actually be two constants, E-~ and P, in plane strain situation) of the composite material consisting of the ceramic matrix, and a random distribution of spherical particles (radius a) and volume fraction c. The effective elastic constant may be calculated on the basis of Hill's self-consistent method (see w The parameter ~ which also depends on c can be approximated by fl ,~ (1 - c)(1 - x/~). Because the above estimate is obtained on the assumption that the spherical particles are "smeared" out in the thickness direction, it is appropriate to replace ~ry by cS (where now S is the strength of one particle at fracture) and to account for the reduction in the length of advancing crack by (1 - c) in (12.2), so that it now reads
(Koppt)2
K~
~ S 2ac(1 - c)(1 - v ~)
Elm (1 - c ) + ~
E~
(12.8)
Then, for the case of particles that break elastically, the modified toughening ratio (cf. (12.311
A -
_ ~
~/'7(1 - c)
r S2ac(1 - x/~) 1+ ~
K~
(12.9)
where 7 - - ~ / Z ~ . It is clear from (12.9) that the modified toughening ratio depends on the strength, size and concentration of the second phase particles (see Fig. 12.4). To conclude this section, we assume that the springs are perfectly rigid plastic in the spirit of ideal plasticity inherent to Dugdale-type models for small-scale yielding in metals. In other words, we assume that the energy loss caused by the fracture of the last spring is simply equal to 2cSv/ (the last term in (12.8)), where 2vl is the relative displacement of crack faces at fracture. In this idealized case, (12.81 gives the modified toughening ratio as
A
-
-
K appl K~V/7(1 - c)
_ ~
~C 1 + (1
2E~Sv/ c)
K2
(12"10/
401
12.2. Small-Scale Crack Bridging
lO-
appl K
h
/K c
- - .
~/~'(l-c) 2 E( 1-v m ) 2
Em(1-v ) c L P = (1-c)(1-~c) Ta 0
0
1
I
I
I
10
20
30
40
I
50 P
F i g u r e 12.4: Toughening ratio for bridging by elastic spherical particles
In Dugdale-type models one often defines the fracture toughness K:, i.e. the critical stress intensity factor corresponding to the critical crackface displacement 2vl at fracture. K/ is related to 2v: as follows
K:
-
~JE-4S(2v/)
(12.11)
Note that both K: and v: refer to the composite. In metallurgical terminology K] is the fracture toughness and 2v! the critical crack opening displacement of an ideally-plastic material (yield strength S) exhibiting only small-scale yielding. In analogy with Dugdale-model, the modified toughening ratio (12.10)simplifies to give Is appI
KC
-
i(
1-c)+c
gr
(12.12)
Equation (12.12) is nothing but the law of mixtures on the work of fracture. For small amounts of second phase particles (c < < 1) it reduces to the law of mixtures on the toughnesses K ~ppz K: = (1 - c ) + c Kr K~
(12.13)
Toughening in DZC by Crack Trapping
402
We can continue the analogy with Dugdale model and express A =
KaVPZ/Kc in terms of the bridge length Lc at fracture. This is readily given by (12.4) after substituting or(x) = S and integrating from 0 to
Lc.
12.3
Crack Trapping by Second-Phase Dispersion
We now adopt the second viewpoint in which the toughness increment is thought to result from the trapping of a crack front against forward advance by contact with an array of obstacles. We shall in particular identify the conditions under which this viewpoint coincides with the crack-bridging viewpoint, as well as with the three-dimensional crack trapping model that will be described subsequently in this Section 12.3.1 Two-Dimensional Crack Trapping Model
F i g u r e 12.5: An idealized representation of crack front trapping by second-phase inclusions We follow the analysis by Rose (1987c, 1987d) and consider a planar crack (in the plane y = 0) with an initially straight front coinciding with the z-axis and growing in the direction of x-axis. Figure 12.5 shows an idealized representation of the crack front which is held up by second-phase inclusions, indicated by shaded circles. The idealization is intended to suggest that the major portion of the crack front can be regarded as having progressed to a distance 2A ahead of the line of inclusions but is trapped by unbroken ligaments stemming from
12.3. Crack Trapping by Second-Phase Dispersion
403
-..-.----__.._~~
a) J
Lr
7
Z
C
"
~
~,
~
~.
~
~,
F i g u r e 12.6: (a) Crack bridging represented by springs over the segment ]z] < A, (b) a section along x - 0, and (c) the unbroken ligaments between cracks replaced by springs
the inclusions. We assume for simplicity that these deep crack front perturbations (A > > A) are periodic in nature and approximate this configuration by regarding the crack to extend to x = A, but with a uniform distribution of springs acting between the crack faces over the segment Iz[ < A, as in Fig. 12.6a. These springs (which we shall assume, as before to be linear) represent the restraining action of the unbroken ligaments. In other words, we use the linear spring stress-displacement relationship (12.1), but for clarity affix appropriate suffices to the stress and displacement o'uy = E'kuy (12.14)
Toughening in DZC by Crack Trapping
404
To obtain the appropriate spring constant k, we consider a thin slice taken along x - 0, as shown in Fig. 12.6b. The section through this slice is nothing but a periodic array of through cracks consisting of cracks of length 2s separated by unbroken ligaments of length 2r along the zaxis. Its response to a mode I stress field prescribed by K appl c a n be determined analytically (see, e.g. Tada et al., 1985), with the net y-axis displacement between y = 0 + and y = 0- given by 2uu(z )
- 4cruu~ log cos ~ + v/cos 2 ~- - cos 2 ~ 7rE'
The average crack opening (2Uy(Z)) over the gap Izl < , uy(z) - 0 over z covering the particles)
(2uu(z))-~
(12.15)
cos
s 2uy(z)dz-
7rE' log sec -~-
is (note
(12.16)
so that the average spring constant k is given by (12.14) and (12.16) to be
k~ - ~'/{21og (ser
}
(12.17)
)~ = (2r + 2s) is the spacing between the inclusions. It may now be shown using the previously described distributed spring model (12.4) that the ratio of the actual stress intensity factor K tip at the crack tip x = A to the nominal factor K appz depends only on the non-dimensional ratio kl, provided 2A(= l) is less than the crack and specimen dimensions,
Ktip Kappl
= Fl(kl)
(12.18)
where
Fl(x) -
i
1 + 0.355x 1 + 2.90x + 2.23x 2
(12 19)
The maximum spring stretch is given by (vide (12.5))
6m,~ - 2uv =
4K appl E'
Vl(kl)
(12.20)
12.3. Crack Trapping by Second-Phase Dispersion
405
where
VI(x) -
i
1 + 0.656x
i + 2.68x + 1.67x 2
(12.21)
FI(x) and VI(x) are interpolating functions constructed so as to reproduce the correct asymptotic behaviour for soft springs (kl < < 1) and for hard springs (kl > > 1). These functions were obtained by Rose (1987b). We mention en passant that there is a misprint in the corresponding equation in the paper by Rose (1987c) from which we have drawn much of the material in this Section. We now need to specify the dynamical conditions for quasi-static (steady-state) crack growth. We assume that failure occurs at a value of the ap.plied stress field K appl when K tip achieves the critical value for the material If/. As before, we shall assume the corresponding value of the maximum spring stretch 8ma~: to be 2vy (see (12.10)). It is convenient to specify the failure criterion at the failure of the spring (at x = - A ) in terms of a limiting stress intensity factor KI, rather than a limiting spring stretch 2yr. The connection between these two specifications (i.e. K/ at x = A and 2v/ at x = - A ) is established via the collinear crack array model mentioned above. For such an array (see, e.g. Tada et al., 1985) K/
-
crvv A tan -~ - E'kv!
tan -~-
(12.22)
From (12.20) and (12.22), we obtain the following equation for describing the onset of fracture
If/ _ 2 V l ( k l ) i k / t a n ~ ) K~PPz log(sec - -
(12.23)
Ky may be regarded as the critical stress intensity for the lateral growth of the crack front, that is for increasing s with A fixed, because the growth in the x-direction (increasing A for fixed s) occurs only when the spring at x = - A is stretched to its maximum value 2vy. It should be obvious that this K/ (at x = A), as well as v! (at x = - A ) are different in physical interpretation and magnitude from the Ky and vy used in the previous model (12.11). It is also convenient to express the critical value of K tip for the composite (i.e. KF) in terms of the intrinsic matrix toughness Kr and the toughness K / f o r lateral crack growth. K r
Toughening in DZC by Crack Trapping
406
which should now be regarded as the critical stress intensity factor of lateral and forward growth of the crack front is chosen to be related to K l and I~'~ by a law of mixtures on the work of fracture (cf. (12.12)) in proportion to the contents of matrix and inclusions
-
/2s
2 + 2r K}
(12.24)
Then from (12.18) and (12.24) we get the following toughening ratio
K ~ppt
V/(1 -
c) +
c(Kf/K~) 2
(12.25)
where, in analogy with (12.12), we have denoted by c = 2r/,~ the area (volume) fraction of inclusions, such that 2s/,~ = ( 1 - c) is the area (volume) fraction of matrix. It is worth stressing again that the K] in (12.25) has a different physical interpretation from the Kf appearing in (12.12). Eliminating K appl between (12.23) and (12.25), one obtains an equation for determining kl in terms of the material parameters Kf/K~, and c
4- v (kt)
(l-c)+c
log
7r(12"-c )
(12.26)
Solving (12.26) for kl and substituting into (12.25), one obtains the toughening ratio K appz/Kc. The corresponding equation in the paper by Rose (1987c) is unfortunately incorrect casting doubts on his results and conclusions. The correct solution of (12.26) for several values of Kf/Kc and 0 <_ c _ 1 is shown in Fig. 12.7. The corresponding values of the toughening ratio (12.25) are shown in Fig. 12.8. For comparison, we have also shown (by broken lines) the toughening ratio Kappz/K~ predicted by a simple law of mixtures (i.e. by the numerator of eqn (12.25)). It is evident that by allowing the broken segments of the crack front to grow laterally (in the z-direction) before they advance in the x-direction, the toughening effect is amplified. We may therefore regard 1/Fl(kl) > 1 as the amplification factor in (12.25). The lateral growth of broken segments of the crack front without their forward progress is akin to an increase in the volume (area) fraction of the matrix traversed by the segments and a corresponding decrease in the effective area fraction of the inclusions. We may therefore introduce
12.3.
Crack Trapping by Second-Phase Dispersion
407
kl 0.8
12
0.6
0.4
0.2
t
I
I
1
l
0.0
0.2
0.4
0.6
0.8
0.0
Figure
-~._1
1.0 c
12.7: Spring stiffness for various values of
K//Kc
appl
K
/Kc Kf/K =12
12 10 ..--""""
9
8
6 4
4
2
3 2
0 0.0
Figure
l 0.2
i 0.4
t 0.6
i 0.8
1 2 . 8 : T o u g h e n i n g ratio for various values of
i 1.0 c
KI/Kc
Toughening in DZC by Crack Trapping
408
the effective area fraction of inclusions through f (which will now depend on current position in the z-direction), so that the area fraction of the matrix is (1 - f), and rewrite (12.25) as follows
-
where f - { 1 - ( 1 - c ) / F l ( k ' ) }
(1 - / )
+ /
(12.27)
and K ; -
K! {1 + ( F ? i k l ) - 1 ) / f } l , 2 .
We can now identify K~ with the K l appearing in (12.12), so that the latter equation is identical in form to (12.27). The above argument may be viewed as an attempt by the broken segments of crack front to join up (because it is easier to grow in the matrix even under the influence of inclusions) before making forward progress. This is quite likely to happen when A > > A, but not so when A < < A because of the strong influence of the inclusions. In the latter case, the law of mixtures (numerator in (12.25))is more likely than (12.27) to be closest to the toughening ratio, as we shall see in the next Section where we consider this case. 12.3.2
Three-Dimensional ping
Small-Scale
Crack
Trap-
The discussion in the preceding Section was based on a two-dimensional plane strain configuration, although there are elements of three-dimensionality implied in the lateral growth of the broken crack segments. Here we present a three-dimensional analysis of limited validity (A < < A) based on the first-order perturbation of the stress intensity factor distribution along the front of a half-plane crack, when the location of that front differs moderately from a straight line. We shall use this perturbation solution to the configuration of the front of a planar crack that is trapped against forward advance by contact with a periodic array of closely spaced obstacles. For this description we borrow heavily from a paper by Rice (1988). Consider a half-plane crack in the plane y = 0, growing in the direction z and having a straight front along x = a0 (Fig. 12.9). For fixed loading the stress intensity factor along the straight crack front parallel to the z-axis, K~ a0], may be obtained from (9.1) with c~ = I and x replaced by (z - a0). If the crack front is not straight but lies along the arc x = a(z)
12.3. Crack Trapping by Second-Phase Dispersion
409
F i g u r e 12.9: (a) A half-plane crack with a straight front at x = a0, (b) a half-plane crack with a moderately curved front at z = a(z), and (c) crack front trapped by impenetrable obstacles
(Fig. 12.9b) in the plane y = 0, then Rice (1988) has shown that, provided a ( z ) i s small such that in an average sense (on a large scale) the crack front is still straight and the variation of K~ a0] with a0, i.e. OK~ is much smaller that K ~ itself, the stress intensity factor K[a(z), 0, z] (denoted simply by K(z)) along the moderately curved crack front is
K(z) - K~ K~
a0]
a0]
1 / = ~ da(z')/dz'
= 2-~
co (z' - z) dz'
(12.28)
to the first-order deviation of a(z) from a0, i.e. to the first order in da(z)/dz. The singular integral in (12.28)is to be understood in the sense of Cauchy principal value. Next, consider the situation depicted in Fig. 12.9c. The crack front is trapped by impenetrable obstacles of some given distribution. Then a(z) is known along the contact zones Ltrap but K(z) is unknown there. Conversely, K(z) = Kc is known along the penetration zones Lpe,~ (i.e. the matrix part) but the depth of penetration a(z) is unknown there. These two conditions together reduce (12.28) to the singular integral equation
K~
ao] - Kc K~ ao]
1 [ da(z')/dZ'dz , 27r JLt,.~p (z - z')
Toughening in DZC by Crack Trapping
410
= 1/L
da(z')/dZ'dz'
(z-z')
(12.29)
for all z included in Lven. Once (12.29) is solved for a(z) along Lv,=, K(z) along Lt,-ap can be found from (12.28). In fact, the solution of (12.29) can be lifted by analogy from known solutions in two-dimensional crack theory. To explore the analogy consider a two-dimensional medium in (y, z) plane containing a single crack or an array of cracks. The medium is loaded remotely in mode I such that the stress avy = cravPZ corresponding to K appl and the opening gap (i.e. net y-direction displacement) between y = 0 + and y = 0- is 2uy(z) = 5(z). Then cryy(z) along z-axis is obtained as
E' f ? 5(z')/dz' aY~ - ~raPP'+ -~r oo ( z ' - z) dz'
(12.30)
For a single (finite or semi-infinite)crack (12.30) can be readily derived from (4.21), (4.24), and (8.14) or (8.3). Equation (12.30) may be rearranged to coincide exactly with (12.28) if one makes the identifications 2[cryy(z) - ~,pvt] .-..+ [K(z) - K ~
E'
K~ 5(z) ~ a ( z )
(12.31)
In this analogy, along Lt,-av the opening displacement is 6(z) = a(z), whereas along Lven the crack faces sustain the stress ayy = crappz + E ' ( K c - K~ ~ Note that cryu will be less than crappz, since K ~ > Kc. Thus, the crack faces will open with the opening displacement 5(z) cor-
F i g u r e 12.10: Periodic array of impenetrable obstacles in the path of a half-plane crack
12.3. Crack Trapping by Second-Phase Dispersion
411
responding to the crack front penetration a(z) in the three-dimensional trapping problem. Rice (1988) extends the analogy to obstacles which are not completely impenetrable, but we shall here limit ourselves to impenetrable obstacles and consider the periodic array shown in Fig. 12.10. The impenetrable obstacles with centre-to-centre spacing 2L (= A used in the preceding Section) have a gap 2H(= 2s of the preceding Section) between them, into which the crack front can penetrate. We have already used the solution for this periodic configuration in the preceding Section (see (12.15), (12.16)). So with substitutions for avv and $(z) identified above, (12.15)and (12.16) read (for - H < z < H)
a(z) = 2Uy(Z)
_( 4L
1-
log
7r
(ap~,> -
K~
{
COS~_L" + r
4L2(
(2uv(z)> - -~ff
2 ~'z
,rH
~-T - c~ cos xH 2--X-
I<~
~-
(rH)
1 - KO) log sec ~
}
(12 32)
(12.33)
Substituting (12.32)into (12.29) one obtains along Lt,-~p (H < ]z I < 2 L - H) the following stress intensity factor sin K(z) =
~z
(12.34)
+ (K ~ -
~//COS 2 ~ ~'H - - c~
~'z 2---E
which is singular, as expected, at the borders of the trap zone. The average value of K(z), denoted (K(z)) over the trap zone, (H < [z I < 2 L - H) is
(K(z)) -
LK ~ - HKc L-H
(12.35)
Denoting by f - ( L - H)/L the fraction of contact, so that (1 - f) is the fraction of penetration, eqn (12.35) can be written as
K~ (K(z)> /~ = ( l - f ) + f Kc
(12 36)
The limiting value of K ~ denoted as before K appl corresponds to the instant at which the crack front just breaks through the obstacles.
Toughening in DZC by Crack Trapping
412
This value can be calculated exactly. For the small perturbation approximation considered here (i.e. amax < < 2L, equivalent to A < < ~ of the preceding Section), it is necessary that
(K~ 2 - {(K(z))} 2
(12.37)
This is obvious from the observation made earlier that the assumption of small-scale deviation of crack front from straightness is akin to the assumption that the function a(z) fluctuates in z about a mean value so that, in an average sense, the crack front is still straight. K ~ is indeed the stress-intensity factor for the straight crack front a(z) = ao. Since K(z) is known everywhere at breakthrough (it is equal to Kc o n - H < z < H and Kp o n H < z < 2 L - H , where Kp > K ~ is the stress-intensity factor for circumventing the particles), eqn (12.37) gives the exact value of K~ K appz) at breakthrough
IiaPPZI(c
1 - f) + f
~
(12.38)
When (Kp - Kc)/Kc < < 1, (K(z)) ~ Kp, so that at the instant of breakthrough (12.36) reduces to
=(1-y)+y
(12.a91
which also follows from (12.38), as it should. It should be stressed that the linear perturbation theory is in fact applicable only to the case when ( K p - gc)/Kc < < 1. A comparison of (12.38) with (12.13), and with (12.27) quickly establishes the connection, at least for small crack front penetrations, between the three-dimensional and two-dimensional models of crack trapping on the one hand, and between the crack trapping and crack bridging models, on the other. All that is required for establishing this connections is an appropriate interpretation of K.t , K~, and lip, as discussed above.
12.4
Crack Trapping by Transformable Second-Phase Dispersion
We now consider the situation when the crack is trapped by second-phase dispersed precipitates which can also transform to monoclinic phase. The
12.4. Crack Trapping by Transformable Second-Phase Dispersion 413 ceramic matrix in such ZTC is toughened not only by the phase transformation of the tetragonal precipitates but also because the precipitates impede the progress of the macrocrack and trap it due to the mismatch in elastic properties. The length of the trapped zones is determined by the size, volume fraction and phase transformation characteristics of zirconia precipitates which we shall assume, for simplicity, to be periodically distributed. These parameters will therefore also determine the spring stiffness which will vary along the bridging zone. In this Section, we shall assume in the spirit of Dugdale (1960) and Bilby, Cottrell & Swinden (1963) model that the transformation zone has no thickness and is coplanar with the discontinuous macrocrack fragments, as shown in Fig. 12.11. We shall follow closely the paper by Jcrgensen (1990) to determine the spring stiffness in (12.1). The tetragonal precipitates that have impeded the progress of the macrocrack and have fragmented it will transform into monoclinic phase because of the very high stresses at the tips of the fragments, thus reducing the stress intensity factor at these tips. The discontinuous crack front cannot therefore grow laterally until the external loading is increased to overcome the shielding effect due to phase transformation. It will be assumed that the transformation is accompanied by dilatation alone so that the transformation zone at each crack tip in the periodic array (Fig. 12.11) can be regarded as planar to which the BCS model with the modification by Rose & Swain (1988) is applicable. In this modified BCS formulation, the stress distribution in the transformation zone at each crack tip in the array is obtained by superposition of two fields. The first stress field corresponds to a stress intensity factor equal to the fracture toughness of the matrix, Kc. The second field is due to a stress intensity factor (K appt- Kc), where K appt corresponds to the applied stress crappz, together with a constant stress (aappz- or*) acting across the transformation zone length I. The cohesive stress a* due to transformation-induced dilatation has to be determined from the dynamic condition for transformation, according to which the total stress O'zz at the end of transformation zone must equal the characteristic value a0 for a tetragonal precipitate, or0 is in turn related to the critical mean stress crm for tetragonal to monoclinic transformation via 3 fro = 2(1 + v)tr~ (12.40) Now following Bilby et al. (1964), and Rose and Swain (1988) it can be shown that or* and l are related to the loading and transformation
414
Toughening in DZC by Crack Trapping
F"
Vl
Figure 12.11: Crack trapping by periodic array of transformable precipitates
12.4. Crack Trapping by Transformable Second-Phase Dispersion 415 characteristics as follows 2 /Trs K~ o'* - o'o
1+
~" (1
l -
-1
~--(Is s
'~ KaPPz K~ ~/ rs K.ppz ) tan -~-
(12.41)
- Is
(12.42)
tan 7r___ss
8(~*) 2
Next we need to calculate the opening of each of the cracks in the array. For this we again follow Bilby et al. (1964), and calculate the average opening over each crack (with its transformation zones)
2 [(s+l)
I
fy
'~ H(y')dy'dy'
(12.43)
I
where H(y') is given by
g(r
- ~
~osh -~
~'(r - a')
(r162
(12.44)
c' - sin ( ~ - ~ )
(12.45)
_ ~osh-~ ((a')~ +
and y' - sin (~~-~) , a'-- sin
Now assuming that the crack face displacement at section x of the macrocrack (Fig. 12.11) is given by the average opening of the lateral crack array with the transformation zones at this section, eqn (12.1) may be rewritten as
k(x)
-
6r a p p l Et(uz(x)),
"
-o "appl
~)i I~[ appl
7r8
tan-~-
(12.46)
Figure 12.12 shows an example of the variation of k(x) with K.ppz/Kc
Toughening in D Z C by Crack Trapping
416
"~ 1.O o t~
,I--4
E
0.8
0
z
0.6 0.4 0.2 0.0
0
I
I
i
I
I
I
I
1
2
3
4
5
6
7
appl
K
/K c
F i g u r e 12.12: Normalized spring stiffness
at a given instant (i.e. given x). k(x) is normalized by k corresponding to a linear spring model in the absence of phase transformation (12.17) which is equivalent in the present formulation to the condition K appl <_ Kc. It is clear that in the presence of a phase transformation the springs are softer. Having calculated the spring stiffness, we determine the macrocrack opening displacement using (12.5). v(x) - (uz(x)) in this equation is approximated by a second-order spline, and the resulting non-linear equation is solved iteratively to calculate ~r(x) for various values of Aa. The stress intensity factor at the tip of the macrocrack is then calculated from (12.4). The length of the bridging zone Aa is finally established from the dynamic conditions for growth of the macrocrack. The latter will grow when K tip (left hand side of (12.4)) reaches the intrinsic fracture toughness of the matrix Kc, while simultaneously the spring at x - Aa stretches to its limit and snaps. The limiting value of the spring extension at x - Aa is in turn determined by the dynamic condition for unstable crack growth in the lateral direction (z-direction). The unstable lateral growth occurs when the unbroken ligaments between the cracks join to regain the continuous front that pertained prior to its trapping. This is equivalent to saying that the phase transformation capacity of tetragonal precipitates has been exhausted and there is no further crack shielding available from them to prevent unstable lateral crack growth. At this instant, the applied stress
12.4. Crack Trapping by Transformable Second-Phase Dispersion
K
417
appl
/Kc 1.5-
1.4 ~ 1.3
~
1.2
~
~,-2s = 2001xm 1501xm lO0~m 50lam
1.1 1.0
1
J 5
I 9
i
I
13
17
I
21 L/2s
F i g u r e 12.13: Toughening ratio as a function of the size and volume fraction of transformable precipitates
intensity factor K appt will equal the fracture toughness of ZTC in which the only toughening mechanism is the phase transformation. Having satisfied the two dynamic conditions for the growth of the macrocrack (in the x-direction), one can estimate the toughening resulting from the joint action of the phase transformation of dispersed tetragonal precipitates and crack trapping, using eqn (12.4). The resulting toughening ratio Kappt/K c is shown in Fig. 12.13 as a function of the volume fraction of precipitates (A/2s) for several values of the precipitates size ( A - 2s). It is interesting to note that in non-transformable ceramics this ratio is independent of precipitate size (cf. Fig. 12.8). This is explained by the fact that the toughening ratio is controlled by two distinct mechanisms. The contribution from the phase transformation to the toughening effect is most pronounced for small (A/2s), but diminishes with increasing (A/2s) when the bridging mechanism progressively takes over. It would appear that the transition in dominant mechanisms occurs at A/2s ~ 3. In applying the above results to ZTC it is worth remembering that the model assumes a small transformation zone compared to crack size and ignores any transformation of tetragonal precipitates in the x-direction. It is for this last reason that the model is likely to be more accurate for small rather than large values of (A/2s) because of the assumption that
Toughening in DZC by Crack Trapping
418
the opening displacement of the macrocrack at any x equals the average opening of the crack array in the y-direction for this section, i.e. the fracture toughness of precipitates themselves is not very different from that of the matrix. We now remove the restriction that the transformation zones are coplanar with the macrocrack fragments and allow them to grow out in the z-direction as far as required by the critical mean stress criterion (Fig. 12.14). For this we utilize the R-curve analysis for a collinear array of internal cracks in a TTC from w The exposition below follows closely the paper by M011er & Karihaloo (1995).
F i g u r e 12.14: Collinear array of plane cracks in a transforming ceramic As before, the effect of crack front trapping by transformable particles is modelled in two dimensions by a shielding stress, ~r' (x) acting over the shielded length Aa , as depicted in Fig. 12.15. The magnitudes of ~r8(x) and Aa are related to the volume fraction and size of transformable particles and to the intrinsic toughness of the matrix material, as shown below. The relative displacement of the crack faces for the above problem is again given by (12.5). Consider the point of instability of the macrocrack under a monotonically increasing load. At this point, the displacement of the crack faces at x = Aa is assumed to have reached a critical value, u(Aa)= (uc) so that a further increase in load would result in breakdown of the shielding mechanism. This dynamic condition for macrocrack growth gives the first governing equation
8K avpzv / ~
4 f a,
E'x/
Jo
o" (t) log
dr-(u
) = 0 (12.47)
12.4. Crack Trapping by Transformable Second-Phase Dispersion
419
Aa F i g u r e 12.15: Tip of macrocrack shielded due to crack front trapping
The crack will not however grow in a catastrophic manner unless the effective stress intensity factor at its tip simultaneously reaches the intrinsic toughness of the matrix material. The effective stress intensity factor for the trapped crack K tip is given by (12.4) with L replaced by Aa. At catastrophic growth K tip = Kc, which forms the second governing equation written in normalized form as
KavvZ
KC
r~ /o t' a
o's(x)
dx - 1 -
0
(12.48)
The macrocrack resumes its continuous front when the two dynamic conditions (12.47) and (12.48) are simultaneously satisfied. To relate cr~(z) and ur to the volume fraction and size of transformable particles, let us consider a cut t - t as indicated in Fig. 12.11. It is assumed that the relative displacement of the crack faces of the macrocrack at instability is equal to the average opening (ur of the array of cracks at peak applied stress ~rp. This stress can also be thought of as the apparent strength of the material. By introducing the length parameter L (7.28) and the critical average crack face displacement (re)
E/~, (vc) -
( u c ) 1 2 r ( 1 - v)
eqn (12.47) can be written in normalized form as
(12.49)
Toughening in DZC by Crack Trapping
420
~/Aa ~Kc L
K appl O m
1
+ 7rIi~ 2vrff~"
fo
crs(t) log
v/S-d_ e7
dt
(vc) L
(12.50)
Equations (12.48) and (12.50) constitute the governing equations for the crack trapping problem in the two unknowns Aa and K appl. r and (re/ are determined from the analysis of the array of cracks in the transverse y-direction as discussed in Chapter 8 (w The critical average crack face displacement at x = Aa is obtained by integrating the dislocation density function
-
dx
(12.51)
D* ( x 0 ) d z 0
~r'(t) along Aa is approximated by a linear or quadratic function and the two governing equations are solved for Aa and K appl using the standard iterative procedure, described in w
( appl YY
0.24
p~-
0.23 0.22
0.21 0.20 0.0
0.5
i 1.0
I
1.5
J
2.0 (C-Co)/L
F i g u r e 12.16: Normalized peak applied stress during quasi-static crack growth, w - 10, AlL - 20. The normalized initial crack length is
coil - 5
12.4. Crack Trapping by Transformable Second-Phase Dispersion
x
a)
-
[-F"
-
421
~
Aa
P t3
__________=
b)
[-r
Aa
"
F i g u r e 12.17: Crack tip zone shielded by (a) linearly decreasing stress, (b) quadratically decreasing stress
The applied stress at which the transformation capacity of the particles is exhausted is shown in Fig. 12.16 for a given initial crack length. As foreshadowed above, the shielding stress will be approximated over Aa by a linear or quadratic function, as shown in Fig. 12.17. The applied stress intensity factor at the tip of the trapped crack K appt and the shielded length Aa are determined after the values of (uc) and a p have been calculated for different particle sizes s and area (volume) fractions A / f r o m the analysis of the array of cracks
mI -
(12.52)
The logarithm of the normalized shielding length is shown in Figs.
Toughening in DZC by Crack Trapping
422
log Aa L 2 1
0 -1 -2 -3 -4
-5 0.0
I 0.2
t 0.4
I 0.6
I 0.8
I
1.0
Af
F i g u r e 1 2 . 1 8 : Shielding l e n g t h as a f u n c t i o n of a r e a f r a c t i o n for linear shielding stress; s/L - 2, 5, 50
log Aa L
0 -1 -2 -3 -4
-5 0.0
i 0.2
i 0.4
i 0.6
t 0.8
I
1.0
Af
F i g u r e 1 2 . 1 9 : Shielding l e n g t h as a f u n c t i o n of a r e a f r a c t i o n for q u a d r a t i c shielding stress; s/L - 2, 5, 20
12.4.
Crack Trapping by Transformable Second-Phase Dispersion
423
appl
K
IKc_
1.16 1.12
0
5
1.08
1.04 1.00 0.0
I
I
I
l,
0.2
0.4
0.6
0.8
I
1.0 A, J
Figure 1 2 . 2 0 : Applied stress intensity factor as a function of area fraction for linear shielding stress; s / L - 2, 5, 50
appl
K
IKc
1.20 -
1.16 1.12 5 1.08
1.04 1.00 L_ _ _
0.0
0.2
1
0.4
_ _ L
_ _
0.6
I
0.8
__1
1.0 ,4, ./
F i g u r e 1 2 . 2 1 : Applied stress intensity factor as a function of area fraction for q u a d r a t i c shielding stress; s/l - 2, 5, 50
424
Toughening in DZC by Crack Trapping
12.18-12.19 as a function of area fraction of transformable particles for three normalized particle sizes under the linear and quadratic approximations to crS(z), respectively. The corresponding applied stress intensity factors are shown in Figs. 12.20-12.21. For a given transformable particle size s the shielded length decreases with an increase in the volume (area) fraction. This is a consequence of the behaviour of the critical crack face displacement (vc) obtained from the analysis of the array of cracks. Low volume (area) fractions of a given particle size, s correspond to large values of ~ (12.52), i.e. long cracks between the particles. This leads to a large average displacement before instability sets in, even though the applied load is relatively low. The large critical displacement of the macrocrack at x = Aa results in a large shielded length for the low volume (area) fractions, and vice versa. This is in agreement with the results of Budiansky et al. (1988) for tough non-transforming particles in a brittle matrix. As expected the quadratically varying shielding stress distribution results in a higher level of toughening and longer shielding lengths than does the linear distribution. The behaviour described above is reflected in the behaviour of K appl in Figs. 12.20-12.21 since the applied stress intensity factor is calculated at the tip of the shielded crack. All curves in Figs. 12.20-12.21 peak at moderate volume (area) fraction, but the magnitude of K appt increases with increasing particle size. The same trend is forecast by the theoretical analysis of Rose (1987c) who also found agreement of his results with experimental observations for an epoxy matrix containing a dispersion of alumina trihydrate inclusions (Lange & Radford, 1971). There is also experimental evidence (Lange, 1982) to support that peak toughness of an alumina ceramic toughened by yttria stabilized tetragonal zirconia particles is attained at a moderate fraction of dispersed zirconia particles. For a given set of material parameters the above analysis permits determination of the toughness of a brittle matrix ceramic toughened by dispersed transformable particles. As the peak toughness is attained at moderate volume (area) fractions of transformable particles, the model is useful for tailoring the microstructure of these materials to suit specific applications.