International Journal of Fracture 44: 233-258, 1990. ©1990 Kluwer Academic Publishers. Printed in the Netherlands.
233
Crack propagation in monolithic ceramics under mixed mode loading M. ORTIZ and A.E. G I A N N A K O P O U L O S Division of Engineering, Brown University, Providence, RI 02912, USA Received 20 October 1988; accepted in revised form 30 March 1989
Abstract. Finite element calculations are presented for a semi-infinite crack in a brittle solid undergoing microcracking normal to the maximum tensile direction. Microcracks are presumed stable and a saturation stage is postulated wherein the effective elastic moduli attain steady state values. Mode I, mode II and mixed mode loading conditions are investigated. In these two latter cases, the method of analysis employed allows for cracks to grow out of their initial planes. The mixed mode loading case investigated corresponds to taking equal values of the remote mode I and II stress intensity factors. Contrary to what is observed in the mode I case, no appreciable R-curve behavior is found under mode II or mixed mode conditions.
1. Introduction
The micromechanical origin of toughness in ceramics is currently the subject of intense investigation. Despite recent progress in the area, a number of fundamental questions remain unresolved. For instance, it is not known with certainty why, the fracture of ceramics being mostly intergranular (Fu [1]; Evans and Fu [2] and [3]), there is a discrepancy of roughly an order of magnitude between the measured surface energy of grain boundaries and the overall toughness of the polycrystalline material. Several plausible scenarios have been proposed to account for this discrepancy. For instance, it has been argued that stable microcracks shield crack tips from the remotely applied loads, thereby effectively increasing the toughness of the material (Evans [4]). Indeed, extensive microcracking is observed to develop in the vicinity of the tip of a crack (Hoagland et al. [5]; Claussen [6]; Wu et al. [7]). For stationary cracks, a number of studies (Hoagland and Embury [8]; Kachanov [9]; Charalambides [10]; Gong and Horii [11]; Hutchinson [12]; Ortiz [13]; Rodin [14]; Ortiz and Giannakopoulos [15]) have demonstrated the potential of microcracking for reducing the magnitude of the near-tip stress intensity factor under mode I loading, and, to a lesser extent, under mode II and mixed mode conditions (Ortiz and Giannakopoulos [16]). However, microcrack shielding can only account, at best, for modest reductions in the near-tip stress intensity factor. Furthermore, this reduction cannot be directly construed as a toughness enhancement. In fact, the microcracks immediately adjacent to the crack tip provide a weak path for crack extension, thus reducing the intrinsic resistance to fracture of the solid. The deleterious effect of microcracking on toughness has been estimated by Ortiz [17] to almost exactly offset the gains due to shielding. Thus one concludes that microcracking should have little or no effect on crack growth initiation. Charalambides and McMeeking [18] have shown that, following growth initiation, resistance to crack extension may build up as a result of the wake of microcracked material left
234
M. Ortiz and A.E. Giannakopoulos
behind by the advancing crack. Their calculations are based on a model of isotropic damage and concern growth under mode I conditions. Modest gains in toughness of up to 40 percent are reported. In this work we investigate the effect of microcracking on toughness for cracks propagating under conditions ranging from mode I to mode II loading. The calculations are based on a model of anisotropic damage in which microcracking is assumed to take place primarily in the direction of maximum tension. In stationary cracks, this highly polarized form of damage results in somewhat higher extents of shielding than the isotropic model of Fu [1] and Evans and Fu [2 and 3]. In fact, it may be shown that microcracking normal to the maximum tensile direction results in maximal shielding (Ortiz and Giannakopoulos [15]). Our results for mode I loading essentially reproduce those of Charalambides and McMeeking [18]. However, we show that no appreciable shielding is present in cracks growing in mode II and in mixed mode with KI]~/K~ = 1. In these two cases, the crack path is allowed to deflect out of its initial plane. These results cast some doubt on the effectiveness of microcrack shielding as a toughening mechanism even for growing cracks. At any rate, it would appear that microcracking alone cannot account for the discrepancy between toughness and grain boundary surface energy alluded to above. Instead, it seems more likely that the toughness of ceramics is the result of a combination of toughening mechanisms operating in conjunction with shielding, such as crack deflection (Faber and Evans [19 and 20]) and bridging (Swanson et al. [21]).
2. Damage model Microcracks develop in ceramics mainly at grain boundary facets as a result of residual stresses generated during cooling and of applied tensile stresses [1]. Once the microcracks are formed they tend to remain stably confined to their facets. As the number of microcrack nucleation sites is exhausted the elastic moduli attain a steady state value and a saturation stage ensues, Fig. 1. In addition to rendering the material more compliant, microcracking has the effect of relieving grain-to-grain residual stresses generated during fabrication. O-
/~
/
SATURATION STAGE
yes
~T
Fig. 1. Stress-strain behavior of monolithic ceramics,
C r a c k propagation in monolithic ceramics under m i x e d mode loading
235
The magnitude and statistical properties of these residual stresses have been computed analytically by Ortiz and Molinari [22]. At the macroscopic level, the stress relaxation process manifests itself as the development of permanent or transformation strains [12]. Thus, upon removal of the loads, the strains do not reduce to zero but rather attain a residual value, Fig. 1. Transformation strains have been shown to exert a significant toughening effect on growing cracks; by Budiansky et al. [23] for transforming second-phase particles, and by Charalambides [10] for microcracking. Throughout the present work, the microcracked solid is idealized as a homogenized continuum undergoing distributed damage. The particular damage model adopted here conforms to the general framework formulated by Ortiz [24]. The stresses o-~jand strains e4j in the solid are assumed to be linearly related according to T + eu,
eij=
(1)
where the elastic compliances Cijk~ are regarded as internal variables characterizing the current state of microcracking. In the applications pursued here, microcracks are subjected to highly nonproportional stress histories. Under these conditions, microcrack closure becomes a distinct possibility. Within the framework of homogenized damage theories, closure may be characterized analytically as follows [24]. The strain tensor ~d contributed by the microcracks is given by 4
=
(cijk, , -
o T C~ikl)akz + e~j,
(2)
where C Oare the elastic compliances of the intact material. Evidently, since ed is the result of the opening of microcracksl it must define a stretch in all directions. In terms of the principal values e~ of ~d, this condition requires that d e~>~0,
e=
1,2,3,
(3)
i.e., ed must be positive definite. As befits the contact nature of microcrack closure, the closure condition (3) takes the form of a unilateral constraint. Failure by ~d to satisfy condition (3) signals the occurrence of microcrack closure. Under these conditions, the effective moduli C need to be adjusted so as to restore compliance with the closure constraint [24]. Physically, this adjustment entails removing from C the contribution arising from microcracks undergoing closure and retaining only that from the active microcracks. Thus, a double dependence is introduced into the elastic moduli: on C, which now takes the significance of the effective moduli that would be observed under conditions resulting in the opening of all microcracks present in the solid; and on the state of stress, which determines the current fraction of active microcracks [24]. Consideration of this double dependence considerably compounds the formulation and implementation of the model. However, in the analyses discussed subsequently, it is found a posteriori that no microcrack closure takes place anywhere in the solid and, thus, the closure constraint (3) is never activated. In view of this simplifying circumstance, here we omit details on the implementation of the closure constraint. In this work, we confine our attention to the development of damage in tension. A wealth of observational evidence exists on microcrack formation in ceramics at elevated temperatures
M. Ortiz and A.E. Giannakopoulos
236
(see Suresh and Brockenbrough [25], for a recent review). Under these conditions, microcracks appear to be mainly intergranular and to form preferentially normal to the tensile axis. Similar observations have been made by Hayhurst [26] and Hayhurst and Leckie [27] for creep rupture of metals under multiaxial states of stress. Hayhurst [26] conducted biaxial tension creep rupture tests on an aluminum alloy and obserwed that in all cases grain boundary cracks had grown on planes which were at an angle of 9.0 deg to the direction of maximum tension. By way of contrast, similar data for ceramics at low temperatures does not seem to be as yet available in the literature. Microcrack observations in rocks are mainly confined to compressive states of stress (see, e.g., the review of Kranz [28]). It seems reasonable to assume, however, that, as in the high temperature range, microcracks nucleated at low temperatures in multiaxial tension will also tend to be preferentially aligned normal to the maximum principal stress. We shall further presume that the transformation strains are collinear with the nucleating microcracks and are, thus, also aligned with the direction of maximum tension. Damage and transformation rules consistent with these assumptions are
~jkt =
(1 - o:)~ninjnknl,
'T ei) = ~Jcninj~;1,
(4)
where the unit vector n points in the direction of maximum tension o-~, 2 is a scalar measure of cumulative damage, and ~ is an adjustable parameter taking values in the range [0, 1]. Two limiting cases of (4) are noteworthy. The choice ~ = 0 has the effect of suppressing the transformation strains. Setting ~ = 1, on the other hand, results in constant elastic moduli. A similar parametrization of the transformation strains relative to the variations in the elastic moduli has been employed by Brockenbrough and Suresh [29]. For a stationary crack a n d , = 0, the damage rule (4) has been shown by [15] to maximize the extent of crack-tip shielding. More precisely, of all possible arrangements of a given microcrack density, maximum shielding is obtained when all microcracks develop normal to the direction of maximum tension, as required by (4). In order to have a closed set of constitutive equations, a rule for computing the evolution of the cumulative damage parameter 2 is required. This may be accomplished by postulating a damage criterion of the form O1
=
Oc,
(5)
with the understanding that incremental damage can be sustained only if (5) is satisfied. The critical tensile stress o-c is itself a function of the state of damage, e.g., through the cumulative damage parameter 2. The dependence of G on 2 may be determined simply from the uniaxial tension test. In the calculations that follow, we have assumed the trilinear law depicted in Fig. 2. The transition zone following the elastic limit % and preceding the saturation stress o-s is characterized analytically by the relation G. =
ao + Er[(2 + 1/Eo)G-- eo], °o ~< G ~< G,
(6)
Crack propagation in monolithic ceramics under mixed mode loading
237
o"
o-s
%
)f,,"
Fig. 2. Assumed trilinear stress-strain law. (a) No transformation strains, ~ = 0, (b) nonvanishing transformation strains, 0 < c~ ~< 1.
where E0 is the initial Young's modulus of the material, E r the transition modulus, and eo = ao/Eo. The saturation value esr of the transformation strains is given by
er-
1~
Es
10)
as.
(7)
A stress-strain dependence of this nature has been generally adopted in the majority of the studies on microcrack shielding to date. Under proportional stressing, it may be shown [13] that the above constitutive response becomes indistinguishable from that of a nonlinear elastic material and may thus be formulated as a deformation theory of damage. In the saturation stage, the governing complementary energy potential is homogeneous of degree two. Under these conditions, an argument of Rice [30] shows that the asymptotic fields arising at the tip of a stationary crack are square root singular.
3. Crack growth criterion Next, we formulate a fracture criterion for a crack in a microcracking solid subjected to arbitrary mixed mode loading. For the class of materials under consideration here, the existence of a saturation stage implies that the near-tip asymptotic fields are square root singular. Under these conditions, a pair of stress intensity factors, Kltip and K;~p, c a n be defined in the usual manner, K; ip =
lim 2,~aoo(r, 0), r~0
(s) K~Ip =
lim 2x~o-,.0(r, 0), r~0
where (r, 0) denote polar coordinates centered at the crack tip, with 0 measured from the plane of the crack, and or00and ~r0 are the corresponding polar components of the stress tensor. It seems reasonable to expect that the conditions for crack growth be expressible in terms of K~~pand K~t~p . For instance, a model of mode I growth devised by [13] yields the critical
M. Ortiz and A.E. Giannakopoulos
238
CRACK
xl
I~
LXa
(a)
,I
'--7 CRACK
_
h
I
Xl
(b) ~.,,~h ~
Fig. 3. (a) Comparison of two notchedbodies at two neighboringconfigurationsof the notch. (b) Finite element model of a growingcrack. value of g ~ ip a s a function of the toughness of the uncracked material and of the saturation microcrack density. For linear elastic solids, the assumption of a critical energy release ffc furnishes a crack growth criterion applicable under arbitrary mixed mode conditions, namely, ] m ~2
E
(K2 + K2~) =
if""
(9)
Here, we seek to derive a similar criterion for the growth of a notch in a microcracking solid. The convenience of considering a notch instead of a sharp crack will become apparent in the formulation of numerical procedures. Energy release rates for sharp cracks differ little from those of notches of comparable length and position, so long as the ratio a/r t between the semi-length of the crack a and the root radius r t exceeds, say, 4 [30]. In numerical computations we shall adopt aspect ratios well beyond this reference value, thus essentially reproducing crack-like conditions. We formulate the crack growth criterion by recourse to energetic considerations. Following [30], consider two configurations of the notched body corresponding to neighboring stages of the growth of the notch, Fig. 3. The extended configuration may be thought of as the result of removing a volume of material A V from the root of the initial notch. As a result of this process of erosion, a new surface AS is created. The body is assumed to contain numerous interacting microcracks, which may themselves grow as a result of the extension of the notch. Both configurations of the body are loaded by identical boundary tractions Ti applied on a portion of the boundary Sr and by identical imposed displacements ui applied on the remaining part of the boundary Sv. We shall assume that, for the processes of interest here, the distribution of microcracks is amenable to homogenization, resulting in an effective stress-strain law of the type (1). In this context, aij and eij are to be interpreted as the macroscopic stresses and strains of the homogenized continuum. Let a~j + 6a~j and e~j + 6e~jdenote the microscopic stresses and strains of the microcracked body, respectively. Thus, 6ai; and 6e~j are the microscopic fluctuations of stress and strain, respectively, introduced by the microcracks. The choice of
Crack propagation in monolithic ceramics under mixed mode loading
239
macroscopic and fluctuation fields is rendered unique by the subsidiary condition that
IB &~jdV = O, IB & r J V
=
0,
(10)
where B is any neighborhood of the body satisfying the conditions of being large with respect to the microstructural dimensions but small with respect to the characteristic length of variation of the macroscopic fields. Equation (10) simply embodies the requirement that the fluctuation stresses and strains average to zero. Since the body is microscopically elastic, we can introduce the potential energy P through its usual definition, P
:
; v 1 D ~jklteU O +
6eij)(a~t + 6ekl) dV -- f& Tiu i dS,
(11)
where D O = (C °) 1 are the elastic moduli of the solid in the absence of microcracks. Equivalently, we may express P as
P = fv½D~jk, aijek, d V -
f&Tiu~dS,
(12)
where D = C i are the current effective moduli of the homogenized solid. Equations (11) and (12) can be manipulated to take a revealing form. Expand the first integrand in (11) and apply (10) to obtain P =
J( r :1D°i,k,ei,e~, dV + fv½D°k, gmi,&k, d V -
f, Tr, u, dS.
(13)
On the other hand, insert the identity
Dijkl = D°kt + D~kt
(14)
into (12) to obtain f 1D°~/k,e~jek,dV + fv½D~;~,e~iekzd r P = Jr:
fsvTiuidS .
(15)
Here D,}kt denotes the variation in the elastic moduli due to the presence of the microcracks. Comparison of (13) and (15) reveals that
fv ½D~klaiJ~k' dV = Iv ½D°k'&iJ (Sak' dV.
(16)
Thus, one concludes that the second term in (15) represents the strain energy of the microscopic strain fluctuations introduced by the microcracks. Next, we estimate the first order variation in the strain energy of the microcracked solid which results from the extension of the notch. Let o-ij + Aa~j, e~j + Ae~j,denote the stress and strain fields after the extension. Following [30], the corresponding increase in potential
M. Ortiz and A.E. Giannakopoulos
240
energy is found to be the expressible as
-AP
fay ½D~kte~jekt dV + fv-Av (a!/ + Aa~)Aeij dV -- fv Av [½(DiJk' + ADijkl)(eiJ + Ag~)(ek' + Aek') -- ½Dijk'eiJek'] dV,
(17)
where AD = AD ~ is the variation in the effective elastic moduli due to microcrack growth. Expanding the integrands, (17) may be reduced to
-AP
:
fav½DijkieijekzdV-
fv_av½AD!lklg6gkldV + fv
avAauA% dV"
(18)
Finally, a virtual work transformation enables (18) to be recast as
-AP
= favlDijk, eije~,dV - fv Av ½ADijk'e'ijek'dV + fas ATiAuidS"
(19)
N o w suppose that the two notches being compared differ by a small extension. Then, all terms in (19) are found to be of first order except for the last, which is of second order. Thus, to within first order terms, the potential energy decrease is
-AP
= fav ½Dijk,e~jek, d V - fv_avlADijk, e~jgk, dV + h.o.t.
(20)
Let a be a measure of the length of the notch. Then, (20) may be expressed in rate form as
AP _ 1 ( 1 fv 1ADijkl Aa AaJAV2 DijkteijektdVAv2- - Aa eij ekt d V
+ O (ka) = f ¢ e + fad + O (Aa),
(21) where one writes fae =
lim A1a fAv 2Di/klSi/ekl 1 d V, Aa+O
1 ADijkl fa~ = Aa~01im-- fv-Av ~ A----d- eije~l dV.
(22)
In view of identity (16), fad may be interpreted as the energy release rate due to microcrack growth. On the other hand, fae is, evidently, the energy release rate due to an extension of the notch at constant effective moduli, i.e., in the absence of microcracking. It is seen that, to within first order terms, fae coincides with the strain energy of the eroded material. On the other hand, fad is determined by the variation in the effective moduli. If, as an additional constitutive restriction, we suppose the effective moduli D to remain positive definite at all times, and the material to soften in compliance with the inequality ADijk/eij ekl ~< 0
(23)
Crack propagation in monolithic ceramics under mixed mode loading
241
then, both terms in (21) contribute positively to the energy release rate. In the absence of microcracking, ADijkt = 0 and (21) reduces to the well-known result of [30]. Energy release rates for cracks follow from (21) by reducing the width of the notch to zero [30]. From definitions (22), it is apparent that Ne is entirely determined by the conditions existing at the tip of the crack. Conversely, for the square root singular near-tip fields which arise in the materials under consideration here, Ne may be thought of as a scalar measure of the corresponding stress intensity factors. It seems reasonable to expect the conditions for crack growth to be expressible in terms of the near-tip stress intensities. In keeping with this presumption, we postulate a fracture criterion of the form .~e =
"(fie
(24)
for some material constant No. For simplicity, in subsequent calculations fgc is assumed to remain constant as the crack advances. In the absence of microcracking, (24) reduces to the familiar critical energy release rate criterion (9). It bears emphasis that, for a microcracking solid, fqc does not define a critical energy release rate for the entire body. In fact, during crack growth, a combination of (21) and (24) reveals that AP Aa -
1 ADijkl ~c - fv-Av 2 A ~ ~ij~kl d V + O(Aa).
(25)
Since the last term in (25) is positive for materials satisfying (23), crack growth generally occurs at overall energy release rates in excess of fqc.
4. Finite element implementation The fracture criterion formulated above is amenable to a particularly convenient finite element implementation. Of particular interest to the analyses sought here is the ability to handle simply cracks growing along arbitrary paths. All aspects of the method can be conveniently implemented at the element level, with no regard for element connectivity or mesh topology. The method applies without conceptual modification to both 2D and 3D problems, and to any materials possessing a well-defined near-tip energy release rate in the sense of (22a). We shall idealize a cracked solid as one with a missing string of elements, Fig. 3, not necessarily aligned with the grid directions. Strictly speaking, the crack is thus modelled as a one element wide notch. Element removal as a procedure for advancing cracks through a finite element mesh has been employed by Tvergaard and Needleman [31] for the analysis of cup-cone failure in porous metals. In their calculations, the elements were eliminated as the material attained its ultimate bearing capacity. Thus, the advance of the crack was directly governed by the constitutive behavior. By way of contrast, here we seek to remove elements in such a way as to satisfy fracture criterion (24). Consider for simplicity, a square finite element mesh of grid size h, Fig. 3, containing a preexisting crack. Let f~r denote the domain of a generic element, with r ranging over all elements in the mesh. We estimate the near-tip energy release rate %" due to the
242
M. Ortiz and A.E. Giannakopoulos
removal of element f~r by means of (22a), which in the present context yields
qJ~ ~ h
(26)
r ~Dijk~ei;ektdfL
The criterion adopted here for eliminating an element from the mesh is that ~f >~ No.
(27)
For isoparametric quadrilaterals and exahedra, the integral in (26) may be further approximated by a one point quadrature rule as ~ ~ W~h,
(28)
where W~ signifies the strain energy density 1 W = ~Di;kleijekt
(29)
at the centroid of the element. The behavior of (28) upon successive mesh refinements is readily established. For simplicity, consider a cracked elastic body discretized into a regular square mesh. For cracks modeled as one element wide notches, increasing refinement of the mesh results in near-tip asymptotic finite element fields independent of the geometry of the body and the details of the loading. Under these conditions, dimensional considerations alone suffice to establish that, on the element adjacent to the crack tip, Wr "" 1/h as h ~ 0. Then, it follows from (28) that (¢~ ,~ O(1) as h ~ 0. Thus, as h is decreased the computed energy release becomes independent of the mesh size, which demonstrates the objectivity of the results. As a simple validation test, consider the case of a double cantilever specimen, Fig. 4. The material is assumed to be linear elastic and to be constrained to deform in plane strain. Loading is applied in the form of symmetric displacements u prescribed at the extremities of
×2
×1
i_
Fig. 4. Double cantilever beam specimen.
t:
Crack propagation in monolithic ceramics under mixed mode loading
243
the beams. A Poisson ratio of 0.27 was employed in the calculations. Using elementary beam theory, the energy release rate may be estimated as [30] 3 -
E
16 1 -
d3 v2a 4u'
(30)
where E is the Young's modulus, d the depth of the beams, and a the crack length. In the finite element calculations, the specimen is discretized into a regular mesh of square four noded isoparametric elements. The energy release rates computed from (28) are found to agree closely with the estimate (30) beyond the crack length to mesh size aspect ratios a/h of, roughly, 20. This demonstrates the convergence of (27) with successive mesh refinements. Figure 5 shows the computed dependence of the energy release rate on the extent of crack advance Aa for an aspect ratio a/h = 20. Also shown for comparison are the predictions of estimate (30). Both methods are seen to yield ostensibly similar results. It bears emphasis that the criterion (27) may be checked entirely at the element level, without need for knowing the current location of the crack tip. In the actual implementation of the method, the loading is applied incrementally. After each load step, a loop is performed over the elements, and condition (27) is tested. Those elements found to meet the criterion are removed from the mesh, in the sense that subsequently they do not contribute to the stiffness and out-of-balance force arrays. The load increment is chosen such that one single element fractures at a time. By virtue of the stress concentrations introduced by the crack itself, the critical element is assured to be always adjacent to the crack tip. This enables condition (27) to be tested element by element, which greatly facilitates the implementation of the method. It is also noteworthy that initially planar cracks are not in any way constrained to grow within their plane. As a result, the crack is free to follow the path that maximizes the crack-tip energy release rate.
t~
BeamAnal,ysis
40
20 ~o
0.0
1,0
Aa
2.0
Fig. 5. Comparison of energy release rates for the double cantilever beam specimen computed from elementary beam theory and finite element method.
244
M. Ortiz and A.E. G&nnakopoulos
5. Numerical results
The finite element mesh employed in the mode I calculations is shown in Fig. 6. The mesh is identical to that used by Charalambides and McMeeking [18] except for an extra row of elements used to accommodate the initial crack. A total of 1291 four node isoparametric elements are used in the discretization. Crack growth and microcracking are mainly confined to an inner core of 18 x 36 square elements, Fig. 6. The size of the finer elements relative to the radius of the outer boundary is of 4 x 10 -4. Tractions consistent with an elastic mode I field are applied on the remote boundary. The material parameters adopted in the calculations are v0 = 0.25, Eo/E, = 2, o-o/Eo = 3.7 x 10 -4, and o-~/o-0 = 1.13. Figure 7 shows the level contours of maximum tensile stress o-1 for e = 0, and various crack extensions. For this choice of e the transformation strains are suppressed. The region contained within the contour o-1/a 0 = 1 corresponds to the zone of material undergoing incremental damage, or active damage zone. As may be seen, the contours remain ostensibly self-similar as the crack grows, except for some degree of expansion. Thus, for instance, the width of the active damage zone is observed to increase to a maximum of roughly 30 percent above its initial value. The effect of transformation strains is illustrated in Fig. 8, which corresponds to a value of c~ of 0.5 or e~ = 1.3 x 10 -4. It is seen that the presence of transformation strains results in a modest increase in the size of the damage zone ahead of the crack tip, an indication of the occurrence of higher stress levels in that region. Deformed meshes for two different crack extensions are shown in Fig. 9. As expected from the mode I symmetry of the problem, the crack grows within its plane. Similarly to what is found for the stress fields, the opening profile appears to remain roughly self-similar as the crack propagates. Contours of the cumulative damage parameter 2 are shown in Figs. 10 and 11. Note that E02 attains a maximum value E0/(1 - e)E, - 1 at saturation. The region contained within the innermost contour in Fig. 10 corresponds to the portion of material having attained saturation, or saturation region. It is observed that the contours of damage are drawn out as the crack grows. The leading front of the contour 2 = 0 coincides with the boundary of the active damage region o-1/ao = 1. The portion of damaged material lying outside the active damage zone defines a wake region. As the wake is drawn out, it also
~!i!12:1111::11:11111:1111:1111111111111111',1
I
I
I~l
] I
I I I IM-~+~ ' '............... ' ' ~ ' ' ' ' ~ ' ' ' ' ' ' ' , ',l',ll',l[l'.',',ll,q IJ II II Iq Ilrllll[lllilllllllllJ]lalllllJplllllll
\ _
II
Fig, 6. Detail of mesh employed in mode I calculations showing the inner core of square elements surrounding the crack tip.
Crack propagation in monolithic ceramics under mixed mode loading
o.3!
245
MODE Z
A
a=O
Aa= 0
/ 0.2
/
0.8
0.1 1.2
1
I
I
I
I
I
I
0
0.1
0.2
0.3
0.4
0.5
B
Eo ~c Ao=0. 12 - -I-ug o-g
0.3
0.2
)
0.1
0
0.3
/
I
I
I
I
I
]
0
0.1
0.2
0.3
0.4
0.5
C
Aa=0 36 E--2-° ~c I-vg o-g
0.2-
OI
]
[
I
I
I
0
0.1
0.2
0.3
0.4
0.5
Eo ~ X
/
-
-
[-u2
0
- -
o" z
o
Fig. 7. Level contours of maximum tensile stress ~ normalized by the elastic limit 0 0. Mode I, ~ = O.
246
M. Ortiz and A.E. Giannakopoulos MODE Z :05 Aa : 0
A 0.5
/
02
\
0.8
J
0.1
I
I
J
0
01
02
I
B
0.:5
I
05
I
04
Aa:OO86
0.5
E---Q-° ~._._£c 2 I-ug o- 0
02
~1~ o
OI
0
03
I
I
I
I
I
I
0
0.1
0.2
0.5
0.4
0.5
C
Eo #c Aa:O.25--I_vo2 o-o2
02
OI
0 I
I
0
0 I
I
02
Eo
X/--
I
I
I
0.:5
0.4-
0,5
~o
--
I-~g ~o~ Fig. 8. L e v e l c o n t o u r s o f m a x i m u m t e n s i l e stress o-1 n o r m a l i z e d b y t h e e l a s t i c l i m i t ~0. M o d e I, c~ = 0.5.
Crack propagation & monolithic ceramics under mixed mode loading
247
Fig. 9. Mode I opening profiles at various stages of crack extension, a = O.
experiences a certain expansion in width commensurate with that of the active damage zone. For sufficiently large crack extensions, the contours of damage in the wake tend to become parallel and a steady state is approached. As in the case of the contours of o-t , the effect of transformation strains is to broaden somewhat the width of the wake. As microcracks get deeper into the wake region, the possibility of closure arises. As discussed in Section 2, microcrack closure may be characterized as a unilateral constraint requiring that the strains (2) contributed by damage be positive definite. Interestingly, for the range of parameters and crack extensions investigated, no closure is detected in the wake or elsewhere in the solid. The variation of the applied stress intensity factor K~ with crack extension, or R-curve, is depicted in Fig. 12. In order to isolate the toughening derived as a result of crack growth from that due to microcrack shielding, which is also operational for the stationary crack, we plot the increment of Ki~ over its critical value K~. at crack growth initiation. Modest toughness gains of the order of 12 percent, for ~ = 0, and 19 percent for e = 0.5, are computed. As expected, the introduction of transformation strains results in slightly higher levels of toughness. We also note, for further reference, that most of the toughness gains are
248
M. Ortiz and A.E. Giannakopoulos A
0.2
0.1
~"~o.~
MODEZ e=O
Aa=O I
I
0
0.2
I
0.1
F
0.2
I
0.5
B
I
0.4
0.5
Aa=0.12 E° ~c I-%2 O-oZ
o
~2
'~o
0.2
0.1
I
I
I
I
I
0
0.I
0.2
0.3
0.4
I
0.5
io6 ?
C
~o.2
L0
EO
or
I
0
I
0.1
~c
I
I
I
I
0.2
0.3
0.4
0.5
Eo qc
x/----
l-u2 o-2 o o
Fig, 10. Level contours of cumulative damage parameter 2 normalized by the initial Young's modulus E 0. M o d e I , ~ = 0.
obtained during the early stages of the analysis, as the crack propagates over a distance of four or five elements. The above results are in qualitative agreement with those of [18]. For instance, for a saturation microcrack density of es = 0.3, roughly equivalent to our choice of parameters Eo/E = 2, Charalambides and McMeeking [18] report a toughness increase of the order of 10 percent, comparable to the value of 12 percent obtained here for the case ~ = 0. However, we show next that the observed R-curve behavior is peculiar to mode I loading and does not in general carry over to other loading conditions. Our remaining numerical calculations extend the above mode I analysis to mixed mode
Crack propagation in monolithic ceramics under mixed mode loading 02 _A
249
MODE I ~:0.5 Aa=O
0.
0.2
I
I
i
0
0.1
0.2
B
05
I
I
0.4
0.5
E o ~c Au--0.086 - --
o
o ~o ,,,i
O.
0
0.2
c
0.1
0.2
0.3
0.4
,~o=o.23 Eo 4/0
0.5
--o.2-.
~06~
0.1
0 0
0.1
0.2' '
0.5
0.4
0.5
Eo --~¢
X / m
I-u~ %a
Fig. 11. Level contours of cumulative damage parameter Z normaiized, by the initial Young's moduius E0. Mode I, ~ = 0.5.
and mode II remote loading. Owing to the lack of symmetry of the problem, a full finite elenent mesh is required in the calculations, Fig. 13. A total of 2646 four node isoparametric elements are used in the discretization. The resolution of the inner core of elements, as well as the overall dimensions of the mesh, are kept identical to those in the foregoing computations. Tractions consistent with mode II and ]nixed mode elastic fields are applied on the remote boundary. In the latter case, the remote stress intensity factors, K [ and K~ are chosen to be of equal magnitude. The material parameters adopted in the calculations are as in the mode I analysis.
250
M. Ortiz and A.E. Giannakopoulos 1.20
MODE - i Es/E o =0.5 Vo:O 25 cz : 0 . 0
O
~ 1,10
xX x x x x X X X ÷ + + + ' { ' + + +
0 0 []
0 0 /', K / K c ':5 0 , 0 0 8 0 0.004 X 0.002 + 0.0004
[] 1.0£] 0.00
1 0.09
I
I
I
0.18
0.27
0.36
AO / (KC/o-o)z
1,20
++++++
00XX 00
MODE- 3: 0
Es/E o = 0.5
0
Uo:0.25 e=0.5 0 I,I0 0 0
&K/K c 0 0.004 X 0.002 + 0.0004 1.00 o 0.00
I 0.09
I 0.18
I
I
0.27
0.36
Aa/(KC/GO )z
Fig. 12. R-curves in mode I (a) ~ = 0. (b) ~ = 0.5.
The path followed by the growing crack under mode II loading is depicted in Fig. 14. As may be seen, the crack grows at 90 deg to the plane of the initial crack. The preference for the 90 deg path exhibited in the computations may be, to some extent, influenced by the limited angular resolution of the mesh. The corresponding level contours of maximum tensile stress and cumulative damage for ~ = 0 and 0.5 are shown in Figs. 15-18. As may be seen, damage is confined to the tensile region above the crack plane. Interestingly, as the crack grows, crack-tip fields develop which are reminiscent of those obtained in the previous mode I analysis. The present fields, however, are rotated so as to be aligned with the growing branch. Thus, the crack appears to choose a path resulting in the vanishing of the mode II crack-tip stress intensity factor and in the development of mode I fields locally about the tip.
Crack propagation in monolithic ceramics under mixed mode loading
251
Fig. 13. Detail of mesh emplo3~ed in mode II calculations showing the inner core of square elements surrounding the crack tip.
Jl
(J [I
Fig. 14. Path followed by crack under remote mode II loading.
M. Ortiz and A.E. Giannakopoulos
252 04
A
04 03
02
02 OI O O, 02 -03
-04~
/',0 =0 03
I 02
I -01
I 0
I OI
I~ 02
I 03
I 04
~o =0
-04
1
03
-012
-011
i
0
02
-03
02
OI
0
OI
02
03
02
013
014
Z
03
ii
:
011
J
04
-03
-02
-0.1
0
OI
02
03
04
:I\t J2/ ~ ~ ~
o
o~Uo? 03
.~.~
Eo
02
-01 X/ E°
\
/ 0
OI
02
03
04
~e
03
-02
-01 x/
0 OI FO ~e
02
03
04
E~o2 %2
Fig. ]5. Level contours of m a x i m u m tensile stress a t
Fig. 16. Level contours of m a x i m u m tensile stress a~
normalized by the elastic limit a0. M o d e II, c~ = O.
normalized by the elastic limit a 0. M o d e I I , ~ = 0.5.
Crack propagation in monolithic ceramics under mixed mode loading
253
0.4 05 04
0.2
~-~XO
6 08 MODE Z[ e=O ~a=O
°I 0
-03
-0!2
0.4
~
I
I
-0[
0
OI
012
0'3
0¸4
I
e.>~I'~o
~flS_o o~ ~. >'
~I~~!~~ Eo
O[
~ =ol; I-~ ~g
0 -05
04
-0.2
-0.1
0
t
0.1
02
0.3
04
2 06
031
~o
o8
o
~o ~'o
22 - -
I-~,~ 7o~
I
-03
-02
l__
I
-0 I
Eo 0
X/~
~
0 I
I
I
02
03
~
04
~~-
%' ~
Fig. 17. Level contours of cumulative damage parameter 2 normalized by the initial Young's modulus E0.
Mode II, e = 0. Contrary to the mode I case, the introduction of transformation strains results in a contraction, rather than an expansion, of the region of active damage and the width of the wake. In addition, no R-curve behavior is detected. Thus, at the maximum crack extension considered in the analysis, only a negligible increase in K~]°"of about 0.5 percent is required to sustain crack growth. In view of the fact that, in mode I, most of the R-curve behavior develops during the fracturing of the first few elements, here it seems equally unlikely that further crack growth should result in additional toughness gains of any significance. Similar conclusions can be made extensive to the mixed mode case. Figure 19 shows the path followed by the growing crack, which subtends an angle of 45 deg to the plane of the crack. This feature of the solution is in close agreement with the experimental observations of Shetty et al. [321, who measured a branching angle of the order of 43 deg. As in the mode
254
M. Ortiz and A.E. Giannakopoulos A 0.4
~
03
?"~os 0.2 OI 0 _01.2
-013
I
Oil
MODE Tie =0.5 Ao =0
0 II
0
I
I
I
02
03
0.4
0.4 03
~1~ o
oE o 02 :"
0.1
=0.11
l-u02 o-02
0 I
-0.3
I
-0.2
I
I
I
-O.I
0
0[
I
I
I
O.2
0.5
0.4
0,41- C
03 I
0.2 OI 0 -03 I
Eo ~ I -02
I -01
I O
01 ]
! 02
I 05
I 04
X/---- Eo
r-~J %2
Fig. 18. Level contours of cumulative damage parameter 2 normalized by initial Young's modulus E0. Mode II, = 0.5.
II case, the level contours of maximum tensile stress, Figs. 20 and 21, and of cumulative damage, Figs. 22 and 23, reveal a tendency towards the development of mode I crack-tip fields aligned with the growing branch of the crack. No expansion of the active damage zone or the wake region are observed when transformation strains are accounted for. In addition, the computed R-curve is almost perfectly flat. In conclusion, our results indicate that the modest R-curve effect which is observed under mode I conditions may not be present in cracks growing in mixed mode or mode II. For stationary cracks, previous studies have demonstrated that microcracking has little or no effect on the toughness of ceramic materials. From our present results it appears that, for general loading conditions, similar conclusions can be extended to growing cracks as well.
Crack propagation in monolithic ceramics under mixed mode loading
~"I i , I~'~,, ,
~~
,~11111t
,
~ 1- ~~ ~ f , ~!,~,~ • !,~,, ~
,~
I
....
~,~,
~
L~l~ll~ll ~,1
:-~lhllll
~ ~
I,
Fig. 19. P a t h followed by crack u n d e r remote m i x e d m o d e loading.
255
M. Ortiz and A.E. Giannakopoulos
256 A
MIXED MODE a;O
0.2
I AMJXEDMODE
o,
0.1
0
o
o/
o L/
02
I
0
-01
~
i
o4
I
OI
-"1-
0.2
-OI
0
OI
02
0 I
0
01[
012
OI
02
B
eL C
C
O~
°°
Aa=o o89 E°
~c
-02Oil
O
OI
Eo ~
-Of
0 Eo ~e
x/--
--
I ~2oo-~
Fig. 20. Level c o n t o u r s o f m a x i m u m tensile stress a~ n o r m a l i z e d by the elastic limit a0. Mixed mode, ~ =
O.
Fig. 21. Level c o n t o u r s o f m a x i m u m tensile stress a] n o r m a l i z e d by the elastic limit G0. Mixed m o d e , = 0.5.
Crack propagation in monolithic ceramics under mixed mode loading
257
B
02
081 A ~1%°
O~ ~ ° i
~ @
~12° ~ ~ < ~ ~
MIXED MODE ~=0 Ao:O
O6 0
~
I
0.1
02
"-
~
/,,/,/;"
o~ Eo ~#c Aa=O056 I-~
0
0
0I
0~2
o.21 c
Ol
~
o
o
:
O
0
089 0.1
Eo #c --,-dd
02
Eo ~ X/----
,-~g~g
Fig. 22. Level contours of cumulative damage parameter Z normalized by initial Young's modulus E 0' Mixed mode, c~ = O.
0.2
0¸2
QI
@ 0
O]I
~>
~]~os
o~
.~ ~
MIXED MODE a=05 ~a:O
Eo ~c ~a:O 0 5 6 - - - '~o~ d
o
02~
0
OI
OI2
0.[
0
i
£a=-Q089 E° ~c I-uoa Ooa O x/
0.1 Eo ~c
O2
I-~o~ ~# Fig. 2#. ~eve] co~ou~s o£ cumulative da~ase parameter £ ~o~al~zed by ~k~a] ~oun@~s modulus £0, ~ e d mode, ~ = 0.5,
258
M. Ortiz and A.E. Giannakopoulos
Acknowledgements The support of the Office of Naval Research through grant N00014-85-K-0720 is gratefully acknowledged.
References 1. Y. Fu, Ph.D. thesis, University of California, Berkeley (1983). 2. A.G. Evans and Y. Fu, Acta Metallurgica 33 (1985) 1515-1524. 3. Idem, ibid. 33 (1985) 1525-1531. 4. A.G. Evans, in Defect Properties and Processing of High-Technology Nonmetallic Materials, North-Holland (1984) 63-80. 5. R.G. Hoagland, G.T. Hahn, and A.R. Rosenfield, Rock Mechanics 5 (1973) 77-106. 6. N. Claussen, Journal of the American Ceramic Society 59 (1976) 49-51. 7. C.C. Wu, S.W. Freiman, R.W. Rice, and J.J. Melcholsky, Journal of Materials Science 13 (1978) 2659-2670. 8. R.G. Hoagland and J.D. Embury, Journal'of the American Ceramic Society 63 (1980) 404-410. 9. M. Kachanov, International Journal of Fracture 30 (1986) R65-R72. 10. P.G. Charalambides, Ph.D. thesis, University of Illinois, Urbana-Champaign (1986). 11. S. Gong and H. Horii, General Solution to the Problem of Microcracks near the Tip of a Main Crack, Report 87-12, Department of Civil Engineering, University of Tokyo (December 1987). 12. J.W. Hutchinson, Acta Metallurgica 35 (1987) 1605-1619. 13. M. Ortiz, Journal of Applied Mechanics 54 (1987) 54-58. 14. G. Rodin, International Journal of Fracture 33 (1987) R31-R35. 15. M. Ortiz and A.E. Giannakopoulos, Journal of Applied Mechanics 56 (1989) 279-283. 16. M. Ortiz and A.E. Giannakopoulos, International Journal of Solids and Structures, to appear. 17. M. Ortiz, International Journal of Solids and Structures 24 (1988) 231-250. 18. P.G. Charalambides and R.M. McMeeking, Mechanics of Materials 6 (1987) 71-87. 19. K.T. Faber and A.G. Evans, Acta Metallurgica 31 (1983) 565-576. 20. Idem, ibid. 31 (1983) 577 584. 21. P.L. Swanson, C.J. Fairbanks, B.R. Lawn, Y.W. Mai and B.J. Hockey, Journal of the American Ceramic Society 70 (1987) 279-289. 22. M. Ortiz and A. Molinari, Journal of the Mechanics and Physics of Solids 36 (1988) 385-400. 23. B. Budiansky, J.W. Hutchinson and J.C. Lambropoulos, International Journal of Solids and Structures 19 (1983) 337-355. 24. M. Ortiz, Mechanics of Materials 4 (1985) 67-93. 25. S. Suresh and J.R. Brockenbrough, Acta Metallurgica 36 (1988) 1455-1470. 26. D.R. Hayhurst, Journal of the Mechanics and Physics of Solids 20 (1972) 381-390. 27. D.R. Hayhurst and F.A Leckie, Journal of the Mechanics and Physics of Solids 21 (1973) 431 446. 28. R.L. Kranz, Tectonophysics 100 (1983) 449. 29. J.R. Brockenbrough and S. Suresh, Journal of the Mechanics and Physics of Solids 35 (1987) 721-742. 30. J.R. Price, in Fracture 2, Academic Press (1968) 191 311. 31. V. Tvergaard and A. Needleman, Acta Metallurgica 32 (1984) 157-169. 32. D.C. Sheny, A.R. Rosenfield, and W.H. Duckworth, Journal of the American Ceramic Society 69 (1981) 437-443.
R~sum~. On pr6sente des calculs par 616ments finis pour une fissure semi-infinie dans un corps fragile comportant une micro-fissuration normale par rapport 5. la direction des tensions principales. On suppose que les microfissures sont stables et on postule un stade de saturation au cours duquel les modules d'61asticit~ atteignent des valeurs constantes. Les conditions de sollicitation en Mode I, et Mode II et en mode mixte sont 6tudi6es et, dans les deux derniers cas, la mdthode d'analyse utilis6e autorise les fissures ~i croitre hors de leur plan initial. Le mode mixte de mise en charge 6tudi6e revient/t prendre des valeurs 6gales pour les facteurs d'intensit6 des contraintes agissant ft. distance selon les Modes I e t II. A 1'inverse de ce que l'on observe dans le cas du Mode I, on ne trouve pas de comportement significatif scion une courbe R pour les conditions en Mode II et en mode mixte.
