International Journal of Plasticity, Vol. 6, pp. 263-280, 1990
Printed in the U.S.A.
0749-6419/90 $3.00 + .00
Copyright © 19~) Pergamon Press pie
FINITE ELEMENT ANALYSIS OF STATIC AND DYNAMIC
FRACTURE OF BRITTLE MICROCRACKING
SOLIDS
P a r t 2: S t a t i o n a r y a n d G r o w i n g M a c r o - C r a c k s U n d e r Static L o a d i n g
Y. Tol and S.N. ATLURI
Center for Computational Mechanics, Georgia Institute of Technology
A b s t r a c t - T h e continuum constitutive modeling for rate-dependent fracture of brittle
microcracking solids, which was described in Part 1 of this paper, is applied to the finite element analysis of stationary and nonstationary macrocracks in a single-edge-notched three-point
bend specimen under a static loading. The microcrack damage zones near the macro-crack-tip
can be classified as being "large-scale." The microcrack toughening effect in stationary cracks
is observed through the behavior of the crack tip energy-flux parameter, the T*-integral. The
numerical results of the nonstationary crack analysis are compared with the available experimental results for a silicon carbide specimen.
I. INTRODUCTION
In this paper, the continuum constitutive modeling for rate-dependent fracture of brittle
microcracking solids, which was described in the Part 1 of this paper (Toi & ATLURI
[1990]), is applied to the finite element analysis of stationary and nonstationary macrocracks in a single edge notched beam under a static three-point bending load.
The present constitutive modeling has been derived by transforming the hyperelastic
type continuum model employed by C ~ m E S
and MCMEEK~6 [1987] into a rate
form, and extending it to the rate-dependent problem, in which the elastic modulus reduction due to microcracking is introduced, as in EVANSand Fu [1985] and Fu and
EVANS [1985], by the self-consistent formulation of BUDL~'qSKYand O'CoN~r~LL [1976]
through a single internal state variable. The additional effect of viscoplasticity is also
included in the present modeling; however, it is not considered in the numerical analyses of this report because of lack of experimental informations on the viscoplastic
deformation in microcracking solids.
The primary purpose of numerical studies here is to discuss the so-called "microcrack
toughening effect" in static stationary and nonstationary crack problems when the
microcrack damage zone near the macro-crack-tip can be classified as being "largescale." The first question in the discussion of the microcrack toughening effect is that
of an appropriate crack-tip parameter to describe the severity of the crack tip under
large-scale damage. Under conditions of small-scale damage near the crack-tip, the notions of a near-tip elastic stress-intensity factor and a "far-field" elastic stress-intensity
factor are applicable and lead to results which are easy to physically understand (BuDW'qSKY, HtrrcnmsoN, ~ LA~mROPOtn.OS[1983]; EVANS• Fu [1985]; C ~ m E S
& McMEEKmG [1987]; RUnL~ et al. [1987]; O~TSZ [1988l). However, in general, the
integral-type crack-tip parameter T* (ATLUm [1986]), which is the generalization of the
conventional J, can be considered to be more appropriate, because T* is applicable to
263
264
Y. To~ and S.N. ArLL;Rr
any type of materially nonlinear behavior, and unloading, in the presence of large-scale
microcrack damage.
The outline of the two-dimensional constitutive modeling employed in the present report is described in Section II, with some remarks on the finite element calculations. In
Section III, the static stationary crack analysis is conducted for a single edge notched
beam under three-point bending, and the microcrack toughening effect in stationary
macrocracks with large-scale microcrack damage zones is observed through the behavior of the crack-tip parameter, the T*-integral. In Section IV, this analysis is extended
to take into account the quasi-static crack propagation, and the calculated result, which
is based on the experimentally determined material constants, is compared with the experimental result for the silicon carbide specimen given in HAGGAOet al. [1987]. Section V contains concluding remarks.
II. C O N S T I T U T I V E M O D E L I N G A N D FINITE E L E M E N T C A L C U L A T I O N S
II. I. Constitutive modeling
In this section the two-dimensional constitutive modeling of brittle microcracking
solids is summarized. Here, the rate effect on microcracking is taken into account; however, the plasticity is not included.
The relation between the microcrack density rate ~ and the equivalent stress o e [ (aO~ii) t/z] is given by the following three equations, depending on the equivalent stress
and the microcrack density values:
= 0
when ae < Oc
= (l/~)[[ae/(ac+~/,~)]
= 0
- II
when ae_> [o~+ (~/A)]
(1)
when ~s -<
where oc, A, and ~, are the microcracking critical stress, the microcracking rate with
stress, and the saturated value of microcrack density, respectively.
The self-consistent theory of BUDLA.~SKYand O'CONr~ELI. [1976] gives the following
relation between the reduced elastic moduli and the microcrack density:
E / E = O/v = [l - (16/9)~] = l / f
(2)
which leads to the rate o f f :
f=
[144/(9 - 16~)21~.
(3)
By replacing the elastic constants in two-dimensional stress.strain relations of isotropic linear elastic bodies with the reduced moduli given in eqn (2), the following expressions can be obtained:
eo = [ ( f + v ) / E } [aij - J'*okk6ij]
( i , j , k = 1,2)
(4a)
Analysis of static and dynamic fracture
265
where
v* = v / f
(plane strain)
v* = v / ( f + v)
(4b)
(plane stress).
(4c)
The differentiation of eqns (4) with respect to time leads to the following rate-form
constitutive relationship:
~ij : Cijkl( ~)~kl + ~iT( aij,~)
= { ( f + v)/Eleij - { ( v / E ) ( l + u/f)lbkkSij
+ I(.f/E)la e + {(//E)(P/f)2lOkk6e
(plane strain)
~ij = Cijkt(~)Okl + ~g(a;j, ~)
=
{(f+ v ) / E l d i j -- ( V/El(Tkk~ij
+ {//Eloij
(plane stress).
(5)
Equations (1)-(5) give a complete set of equations for two-dimensional constitutive modeling of brittle microcracking solids to be used in the present study.
II.2. Finite element calculations
The constitutive modeling described in the preceding section has been implemented
in the existing two-dimensional elasto-viscoplastic code (OwEN ~ HINTON [1980]) in
which four-node bilinear quadrilateral isoparametric elements are employed with 2 x 2
Gaussian quadrature rule for the numerical integration of stiffness matrices and force
vectors.
The unloading was not taken into account in Part 1 of this paper, however, it must
be considered in the crack propagation analysis to be conducted in this paper. The unloading is treated in the following way in the present analysis. The unloading takes place,
when the following condition is satisfied in the microcracked region:
o, < [(G/A) + Orc]
(6)
where ae and oc are the equivalent stress and the critical stress for microcrack initiation
respectively, and A is the microcracking rate with stress. During unloading the reduced
elastic constants J~ and ~ have the following values:
E / E = ~/v = 1 - (16/9)~
(7)
where ~ is the current value of microcrack density. The above-mentioned treatment has
no firm experimental basis, however, it is presumed to be reasonable for the purpose
of qualitative discussion of the fracture behavior of brittle microcracking solids.
To quantify the energy flux to the macro-crack-tip, which is surrounded by a non-
266
Y. Tol and S.N. ArLURI
Fig. I. Nomenclature for a plane crack.
linear material zone o f microcracking material, the T*-integral, which is defined as follows (refer to Fig. l), is used.
T* = fr~ [Wnl - ti(Oui/Oxl)]dI"
(8)
where
x; = a system o f Cartesian coordinates such that xt is along the crack axis, x2 normal to the crack axis
F, = an arbitrary small loop surrounding the crack tip
t,. = the components in the xi direction o f the traction on the contour F~
ui = the components in the xj direction o f displacements
n~ = the c o m p o n e n t along the xi direction o f a unit outward normal to the contour
W = the total stress-working density per unit volume defined by
W=
a~jdE~j.
(9)
f0 E/j
If the integrand on the right-hand side o f eqn (8) has a singularity of the order (l/r) near
the crack tip, for a F, o f a circular geometry with radius ~, the integrand behaves as
(1/~) at ~ where dI" = ~ dO. Thus, T* remains finite even in the limit as ~ --, 0. On the
other hand, if the integrand has a weaker singularity than ( l / r ) , T* tends to zero in the
limit as ~ --, 0. However, in this latter case T* has a finite value when evaluated on a
finite-sized ~ , say o f a circular path o f a small, but finite sized radius E. In the present material model, with microcracking near the crack-tip, when microcracking saturates
asymptotically close to the crack-tip, the material asymptotically near the crack-tip behaves linearly, with reduced elastic moduli, as in eqn (2), wherein ~ = ~s. Thus, asymptotically close to the crack tip, the linear elastic type stress and strain singularities m a y
be presumed to exist. Thus, for a stationary crack-tip, the integrand in eqn (8) is o f the
order ( l / r ) and T* has a finite value in the limit as ~ --, 0.
Analysis of static and dynamic fracture
267
The T*-integral is calculated at each loading step in the finite element analysis by
using the following domain integral expression (NIKXSrXKOVa ATLURI [1987]) (refer to
Fig. 1):
(10)
T* = J ( S ) + J ( W )
where
I"
J(S) = -I
J(W) = -(
J,4 --A~
JA --a~
{ w ( o S / O x , ) - oij ( Oui/Oxi ) ( OS/Oxj )] dA
(11)
[OW/Ox~ - oiy(Oeiy/Oxl)}SdA.
(12)
The function S in eqns (11) and (12) is an arbitrary continuous function such that
S= 1 onF~andS=0onIf.
(13)
In the numerical calculation the following expression for J ( S ) is employed (refer to
Fig. 2):
J ( S ) = ~a [J(S)lelem = ~] R r S r
elem
elem
(14)
where
R K = - f + l f_+l [ W ( O N r / O x l ) - aij(o3NM/OxI )(ONX/Oxj)ui ~ } (det J) d~ d~
I
ui = N X u ( ,
1
S = NxS x
(isoparametric representation, i = 1,2; K = 1 - 4) (15)
All of the quantities contained in eqn (15) are already calculated in the routine of the
finite element code, so it is easy to estimate J ( S ) by using eqns (14) and (15). As for
J(W), the following explicit expression is given in NtKISnKOV and A~trRx [1987]:
4
2
Fig. 2. A four-noded isoparametric element.
268
Y. Tol and S.N. ArLual
J(W)
=-(,~/'~){W
(1) -
W(3))(X 2 - X
4) + ( W (4) -
W(2))(x~ - x 2 3)
- ab°)t(e~) ) - eb3')(x~ - x ~ ) + (eb 4' - eb2')(x~ - x3)] }SO (16)
where a~°) and S Oare the average values of a o and that of S K at the center of the element, respectively. It is worth remembering that superscripts here imply mode numbering; and superscripts in parentheses imply numbering of integration points (see Fig, 2).
III. ANALYSISOF A STATIONARYCRACK IN A SINGLE EDGE NOTCHED BEAM
III.1. P r o b l e m d e s c r i p t i o n
Figure 3 shows the dimensions of the beam specimen with a single edge notch, used
in the three-point bending test in HAGGAGet al. [1987], where S denotes the span between two simply supported points. The edge notch exists uniformly through the breadth
of the beam, so this problem is treated as a two-dimensional plane strain problem.
The purpose of this analysis of a stationary crack is to observe the behavior of the
crack-tip energy-release parameter, T*-integral (ATLUm [1986]), and to obtain some
quantification of the microcrack toughening effect. Thus the present analyses are carried out both for unmicrocracking and microcracking beams in order to compare the
results. The assumed material constants are as follows, which are almost irrelevant to
the actual material values employed in the next section:
Young's modulus: E = 0.161 x 10 t2 (N/m 2)
Poisson's ratio: v = 0.225
Critical stress for microcrack initiation: ac = 0.226 x 108 (N/m 2)
Microcracking rate with stress: A = 0.15417 × 10 -6 (mZ/N)
Saturated value of microcrack density: ~s = 0.37
Viscous coefficient for microcrack density: rl = 0.1 (I/see).
The plastic deformation was neglected by assuming extremely large yield stress value in
the input data. For these values of material constants, it can be seen from Fig. 1 of Part
l of this paper that the stress-strain curve is linear until tr = Oc when microcracking initiates; is significantly nonlinear in the region 0 < ~ < ~s, and again becomes linear for
>_ ~s (i.e., when the microcrackiag level reaches its saturation). Also, note the signifi-
oTI'
S
v~
......
D
W= 5, B-IO, a : 2 ,
--I
.....
S=40.6,
i
D = 5 0 [mrn]
Fig. 3. Three-point bending of a SENB.
Analysis of static and dynamic fracture
269
cant inelastic behavior (i.e., the significant difference between the loading and unloading
paths in the stress-strain space of the material which has not yet reached saturation levels of microcracking). Note also that the material, in which microcracking has reached
a saturation, also behaves linearly with loading and unloading occurring on a straight
line of lesser slope than for the virgin uncracked material.
III.2. Results of finite element analysis
Figure 4 shows the assumed mesh subdivision for the left half span of the beam in
Fig. 3:
Total number of elements = 240
Total number of nodes = 279
Total number of degrees of freedom = 544.
Figure 5 and Fig. 6 show, respectively, the calculated load-displacement curve and the
process of the development of the microcracked process zone in front of the notch. In
Fig. 6, the horizontal length of the plotted zone is the actual beam depth, and the height
in Fig. 6 corresponds to the 4 rows of smallest elements near the crack as shown in
Fig. 4. In Fig. 6, the microcracking density is plotted for each element referring to the
~0
Z
L
SO
ACK INITIAHON
!
f
I
I
O,Oe
CJAQ( TIP
Fig. 4. Finite element mesh subdivision.
CaS~ACEMiNT U [m.]
Fig. 5. Load-displacement curves for the stationary notch.
Y. ToI and S.N. Arru~
270
00000000000000000000
00000000000000000000
00000000000000000000
00000000~00000000000
00000000~00000000000
00000000000000000000
00000000000000000000
00000000000000000000
00OO000000000OOO0000
00OOOOOOOOO000000000
0000000~0000000000
O000000ltlO000000000
0000000•~I0000000000
0000000~1~'~0000000000
00000000000000000000
00OO0000000000OO0000
P=IO.OEN]
00000000000000000000
00000000000000000000
00000000000000000000
~ ~ O O O O 0 0 0 O O 0 0
00000000000
00000000000000000000
00000000000000000000
00000000000000000000
00000000000000000000
0000000~000000000
O00000~11~0OOOOOOO0
~ 2 ~ 0 0 0 0 0 0 0 0 0
O00~O--O~Q2~O00000000
000000~11~000000000
0000000~000000000
00000000000000000000
P=34.2[N]
00000000000000000000
0OOO000000OOOO000OO0
0000000~0~0000000000
~
2~0000000000
2~0000000000
0000000~0~0000000000
00000000000000000000
00000000000000000000
P = 68.4 {~]
o0oo0o0~0o00oo00o0o
o0o000~00000ooo
o00000~2221~0o000ooo
0~.~2~000o0o0o
OOOO000gOB2~O00000OO
0o0o00~2221~00000o0o
000000~00000000
0000000~000000000o0
P=79.8CN]
P = 45.6 [N]
0.0 < ~ < 0.1 (initiation)
1
0.1 =~ ,~ < 0.2
2
0.2~ ~ <0.3
3
0.3 ~ ~ < 0.37
•
OOOOOOR~R~OOO00OO00
00OOOO~1111~000OOOO0
000000104~2~00000000
~ 3 1 0 O O O O 0 0 0
O000000gqQ3100000000
0000001311~2~00000000
000000~1111~00000000
000000~000000000
~ = 0.37 (saturation)
P = 91.2 Is]
Fig. 6. Development of the zone o f microcracks near the notch.
maximum value on four Gaussian integration points within each element. The microcracking initiates at the load P = 19.0 (N) and the size of the process zone at P = 91.2
(N) is considerably large, nearly one third of the beam depth as shown in Fig. 6. However, even at this large-scale microcracking there cannot be observed a severe nonlinearity in the overall load-displacement curve (see Fig. 5). Beginning at a load level of
P = 57 (N), the microcracking in the material closest to the macrocrack-tip reaches a
saturation level. Note from Fig. l(a) of Part 1 of this paper that the material with a saturated level of microcracking again starts behaving linearly, with a lowered elastic moduli. Thus in the load range 57 (N) < P < 91 (N), the material near the crack-tip, as seen
in Fig. 6, is behaving linearly.
In the same load range 57 (N) ___P _.<91.2 (N), the core of microcrack-saturated material (indicated by solid circles) near the crack tip is surrounded by microcracking material (indicated by 1 , 2 , 3 . . . and solid stars) that has not yet reached saturation levels
of microcracking. This second layer of material, as seen from Fig. 1 of Part 1, exhibits
significant inelastic behavior with radically different loading and unloading paths. This
inelastic layer of material, as seen from Fig. 6 for 57 (N) _< P _< 91.2 (N), is again surrounded by unmicrocracked material which behaves again linear elastically, with the elastic moduli of the virgin material. Note however, that the uncracked material (the third
layer) is stiffer than the inner-most core of microcrack-saturated material. These three
layers of distinctly different material behavior, involving significant nonhomogeneous
Analysisof static and dynamicfracture
271
materiai behavior near the crack-tip, is the source of microcrack toughening as explained
later in this paper.
Figure 7 shows the equivalent stress distributions near the notch at the load level
P = 91.2 (N), both for the cases when a microcracking zone is present as in Fig. 6, as
well as when microcracking is suppressed in the constitutive model and the material is
assumed to behave as a homogeneous linear elastic material with the moduli as in the
virgin state. Note that at P = 91.2 (N), the zone of material with saturated microcracking
is quite large, as seen from Fig. 6, and in this region, the material again behaves like
a linear elastic solid, with reduced moduli. In Fig. 7, X is a coordinate measured from
the bottom of the beam, and X = 2(ram) denotes the crack-tip. From Fig. 7, it is seen
that the stress distribution in the region 2(mm) < x _< 2.5(ram), wherein the material has
saturated microcracking, the stress variation is almost of the ( l / , f f ) type as in the material model with no microcracking at all; however, the magnitude of stress in the
microcracked zone is considerably smaller than when microcracking is assumed to be
absent. From Fig. 6, it can be seen at P = 91.2 (N), the material with saturated
microcracking (which behaves linearly with reduced moduli) is surrounded by material
with varied degrees of microcracking (which behaves like a nonlinear inelastic solid), and
this in turn is surrounded by unmicrocracked material (which again behaves linear elastically with the moduli of the virgin material). The stress reduction in the microcracksaturated material right near the crack-tip due to elastic moduli reduction right near the
crack-tip, and due to the presence of a nonhomogeneous and nonlinear material zone
immediately surrounding this region, can be labeled as the shielding of a macrocracktip by the damaged material surrounding it. However, the earlier arguments in BUDZA~SKY, HUTCHINSON,& LA~IBROPOULOS[1983], EVAZ~Sand Fu [1985], CHAR.~LAMmDESand
McMEEKINO [1987], RUHLE et al. [1987], and ORTLZ [1988] concerning microcrack-
1.0
I/
•
z
- - - - - - UNMICROCEACKED
(7, - KIA/-r
, ~!
,x.
Og
I,-
0.5
MICROCRACKED
1\i\--
/
3
0ua
0
I
t
l
2
3
4
x Cmm]
Fig. 7. Equivalent stress distributions near the notch.
272
Y. Tot and S.N. ATLURI
toughening relate to the case when the microcracking near the crack-tip is of a sufficiently small-scale such that the elastic asymptotic singular stress solution (Kj~:~) still
dominates at the boundary of the microcracked-zone. However, as seen from Fig. 7,
where the asymptotic singular solution (Kt/4?) for the unmicrocracked material is also
shown, the solution (Kt/4-f) is not dominant at the microcrack-zone boundary. (Here
Kt is the stress-intensity imposed on the crack tip in a specimen of the present geometry, if the material had been homogeneous, isotropic and linear elastic, as computed
from eqn (17) below.) Thus the present results can be considered to correspond to "largescale microcracking."
The computed stress distribution in the microcracking process zone, near the cracktip, at P = 91.2 (N) as shown in Fig. 7 is fitted with a (K)iP/vT) type function. The
value of K: i° extracted from this fitting is 35 N/(mm) 3/2. That the computed K] ip is
smaller than the Kz calculated to be 65,5 N/(mm) 3/z from eqn (17) below, is in itself an
indication of the microcrack toughening effect.
Assuming that the material model precludes any microcracking and that the material
is homogeneous, isotropic and linear elastic throughout the loading, the stress-intensity
factor Kt imposed at the crack-tip is estimated as (KoBAYASHI[1973]):
Kt2 = [62.50(M/W)2]/(B2W)
(17)
M = (PS/4).
(18)
At the load of P = 91.2 (N), the value of K / f r o m eqn (17) is 65.5 N/(mm) 3/z. From a
comparison of the (Kt/4-f) curve, and the actual numerically computed stresses in the
unmicrocracked solid plotted in Fig. 7, it may be seen that the finite element mesh in
Fig. 4 is not fine enough to capture the imposed value of Kt [of 65.5 N/(mm) 3/z]
through curve-fitting the computed stresses near the crack tip, ignoring microcracking
altogether. (The finite element mesh underestimates the imposed KI as compared to
eqn 07).) For the same reasons, it may be concluded that the K) ip estimated from the
near-tip stresses in the microcracked solid, in Fig. 7, is smaller than the hypothetical exact value. ~ However, for the given finite element mesh, the ratio of the computed K] it'
to the computed Kt is perhaps closer to its hypothetical exact value.
The energy-flux to the crack-tip, under the crack-tip stress-intensity of the magnitude
given in (17), depends on the elastic material properties E and ~. For the purposes of
the ensuing arguments we consider two sets of elastic moduli (E, p) : (a) in the first set
we take E, t, to be those of the unmicrocracked virgin material, labeled here as (E~ and
uu); (b) in the second set we take (E, z,) to be those of the microcrack-saturated material near the crack-tip labeled here as (E, and ~,), respectively. The crack-tip energyflux for these two sets of moduli are labeled, respectively, as T~ and T~*. Thus,
T~ = (1 - Uu)2KZt
Eu
(19)
qn AppendixA of this paper results for a finer finite element mesh, for which the K-factor for the hypothetical unmicrocrackedsolid agrees with the K~ [of 65.5 N/(mm)3/2] from eqn (17), are presented, to further validate the arguments related to microcrack-toughening,presented here.
Analysis of static and dynamic fracture
T; =
273
(1 - ~s)2K2
E'~
(20)
The energy fluxes T~ and T~* are plotted as functions of load P in Fig. 8, and since
K~ ~ P, these curves vary quadratically with P. Also plotted in Fig. 8 is the actually
computed value of T* [labeled as T : ] , using its definition as in eqn (8), for the
microcracked material with its zone represented as in Fig. 6.
Given that the load-displacement curve, for the material model with microcracking,
is nearly linear as shown in Fig. 5, it is not surprising that T~, and 7"* are nearly equal
at all load levels as shown in Fig. 8; it is only at higher levels of P that T* is somewhat higher than T: for a given R However, at higher values of P, microcracking is
saturated near the macrocrack-tip, and hence the material there behaves again linear
elastically with moduli Es and ~s. If the effect of the nonlinear (inelastic) and nonhomogeneous material zone surrounding the microcrack-saturated material near the cracktip is ignored (i.e., the entire specimen is hypothetically homogeneous and isotropic, and
is assumed to have linear elastic material properties of £'~ and ~s), the energy-flux to the
crack-tip per unit of crack-growth in the microcrack saturated zone asymptotically close
to the crack-tip, in the limit as 4 a -, 0, may be estimated from eqn (20). That the actual energy flux T* (as computed from eqn (8)) is substantially less than this hypothetical T; implies that the energy dissipated in microcrack formation (in the damage zone
near the macro-crack tip as shown in Fig. 6, especially in the nonlinear inelastic material region wherein microcracking has not yet reached saturation) when included in a
global-energy-balance relation for crack-growth, serves to substantially reduce the
amount of energy that is otherwise available for fracture (i.e., creation of new cracksurfaces). This, then, is an explanation of the so-called "microcrack toughening" or the
"shielding" of the macro-crack tip by a zone of inelastic material wherein microcrack-
10
~'
~'
2
~x
s
T'm ,, calculated for mkrocracking
material
- - - - - - T'. = [ 1 - ,.Z]KI2/E .
/
--'-T', - [ 1 - ~,Z]KI=/~ $
/"
/
0
[ 1 - pz2][Kltip]Z/E$
/
/
/
<
•0
/
0
•
50
90
LOAD P IN]
Fig. 8. Variation of 7"*, T~, and T~*with load for the stationary notch.
274
Y. Tol and S.N. ATLURI
ing has yet to saturate. It is also interesting to note that at P = 91.2 (N), the K) 'p extracted from Fig. 7, is 35 N/(mm)3/2; and the energy release rate corresponding to this
1(: ip is given by : g = [(1 - ~2)K2/Es] = 2.2 × 10 -5 (N.m), which agrees quite well
with the computed T,~ at the same load level. (The discrepancy between these values for
g and T,~ stems from the coarseness o f the finite element grid as explained earlier.)
This confirms that even in the case o f large-scale microcrack damage near the crack-tip,
the material asymptotically close to the crack-tip behaves essentially linear elastically,
and that the concept o f a near-tip elastic stress intensity factor K) ~p is a valid one.
IV. ANALYSIS OF QUASI-STATIC CRACK-GROWTH IN A
SINGLE EDGE NOTCHED BEAM UNDER 3-POINT BENDING
IV. 1. Problem description
The same beam as treated in the preceding section is again analyzed here, taking into
account the propagation o f the major crack. The dimensions of the beam are shown in
Fig. 3 with the loading and the supporting conditions.
The material constants assumed are described below.
Young's modulus: E = 0.123 x 1012 ( N / m 2)
Poisson's ratio: u = 0.185
Critical stress for microcrack initiation
oc = 0,810 x 10 s ( N / m 2) . . . . . . . . . . .
[FEI]
ac = 0.718 x l0 s ( N / m 2) . . . . . . . . . . .
[FE2]
Microcracking rate with stress: A = 0.12333 x 10 - 6 ( m 2 / N )
Saturated value o f microcrack density: ~s = 0.37
Viscous coefficient for microcrack density: rt = 0.1 (l/sec)
Critical crack-tip T* value
To* = 1.660 x 10 -5 (N.m) . . . . . . . . . .
[FEI]
Tc = 0.928 x 10 -5 (N.m) . . . . . . . . . .
[FE2].
The purpose of the analysis in this section is to compare the calculated result with that
o f the three-point bending test for the silicon carbide specimen conducted in HAC~AC
et al. [1987]. Therefore some attention has been paid to the determination of the abovementioned material constants. Through private communication ( E p s a ~ [1988]), the authors obtained the material test results based on the measurement o f the velocity o f the
supersonic wave transmitted in the tested bar specimen (the so-called pulse echo method),
however, it became clear after a trial analysis that the Young's modulus determined by
this method ( E = 0.395 x 10 ~2 (N/m2)) is valid only for an extremely small strain range
(e -~ 10 -6) and it is not necessarily applicable (which is too high) to materials such as
ceramics which exhibit material nonlinearity even in a small strain range. Therefore in
the present analysis Young's modulus E has been determined so that the initial slope in
the load-displacement curve calculated by the finite element analysis agrees with the experimental result. The Poisson's ratio u is the value obtained by the material testing.
The critical stress for microcracking, ac, and the critical value o f T*-integral for
crack propagation, Tff, both o f which are assumed in two different ways referred as
[FEll and [FE2], respectively, have the following experimental basis (HAC_~AG et al.
[1987]):
Analysis of static and dynamic fracture
[FEll
[FE2I
oc:
T;:
oc:
T;:
275
microcrack initiation load determined from AE data
macrocrack-growth initiation load determined from AE data
10070 smaller than o, in [FEll
microcrack initiation load determined from AE data.
In [FELL both critical values concerning microcracks are based on the result of the AE
measurement conducted in HAC~A~ et al. [1987], while in [FE2] the major crack starts
propagating at the loading level corresponding to a, in [FEll and a¢ is assumed to be
10070 smaller than that in [FEll.
The assumed saturation value of microcrack density, ~s, is the value of initial porosity measured in TEM HAC-~AOet al. [1987]. The microcracking rate with stress, A, and
the viscous coefficient for microcrack density, 7, have no experimental basis. The assumed A gives 0.030 x 108 (N/m 2) as the difference between the stress for microcrack
saturation and that for microcrack initiation, and the assumed ~ has been determined
so as to cause little rate effect by referring to the numerical results contained in Part 1
of this paper.
IV.2. Results o f finite element analysis
The assumptions for the finite element analysis are the same as those in the preceding section, including the mesh subdivision given in Fig. 4.
The crack propagation is treated in a standard way, that is, by releasing the crack tip
node when T* reaches the assumed critical value, and releasing the nodal-restraining
force gradually in order to release the cohesive stress on the newly formed free surface.
Figure 9 shows calculated load-displacement curves using two different assumptions
on material constants for microcracking, [FE 1] and [FE2], and the experimental result.
The finite element results are plotted, exactly tracing the actual computational process,
which involves an increase in external loading as well as the crack-tip nodal-force release, both of which are incremental processes and do not take place at the same time.
The result of [FE1], which is based on the actual material constants for microcracking,
agrees relatively well with the experimental curve; however, the following two differ-
150
a
SO
0
I
I
I
I
0.01
0.02
0.03
(X04
O~
OlSPLAC|MENT [mm]
Fig. 9. Load-displacement curves ([TEll, (FE2], and experiment).
276
Y. To[ and S.N. ATLURI
ences between the calculation and the experiment can be pointed out. At first, the finite element results have no maximum loads in the calculated range where the crack
propagation was always slow and stable, while the maximum load, 95 (N), was obtained
in the experiment. This difference is due to the effect of the stiffness of the testing machine which is inevitable in the actual experiment, and the neglect of the deterioration
effect of microcracking on the crack tip toughness in the finite element analysis. The
second difference is that the nonlinearity, which can be observed in the experimental
curve between microcrack initiation and major crack propagation, does not appear in
the finite element result. This difference suggests that there is room for improvement
in the constitutive modeling adopted in the present analysis.
Figure 10 and Fig. 11 show the equivalent stress distribution near the notch and the
crack opening profiles respectively at three different loading steps, from which it can
be seen that the present analysis successfully simulated the crack propagation process
accompanied by the unloading in the process zone wake. (Note that prior to any propagation, the crack-tip is at X = 2 (mm).)
Figure 12 shows the microcracked process zone and its wake. The finite element mesh
assumed in the present analysis is not so fine that the saturation of microcracks did not
occur even at the nearest Gaussian integration point from the crack tip.
Figure 13 shows the contours of the displacement in the direction normal to the crack
surface, which are compared with the experimental Moire fringe patterns (HAGGAG
et ai. [1987]). Figure 13(a) is at the state immediately before the crack propagation starts,
and Fig. 13(b) is at the experimental maximum load. In numerical results the range between the maximum value and the minimum value is equally subdivided into twenty
smaller ranges, and the numerals 0-9 assigned to smaller ranges alternately are plotted
at each node. Although the mesh near the crack tip is a little too coarse, the concentration of strains near the crack tip, as observed in the experimental Moire fringes, can
be also observed in the finite element result.
2.0
•
F e = 9s.o I:N]
I
1.5
P =92.1 ~N] -~
X
i
t
1.0
I.
0
1.5
,
2.0
I
I
I
2.5
3,0
3.5
,
x r,~-]
Fig. 10. Equivalent stress distribution near the notch ([FEI]).
Analysis of static and dynamic fracture
I
277
f P - 9 ~ IN]
0
0.5
1.0
1.5
2.0
2.5
X [mm]
Fig. ! 1. Crack opening profiles ([FEI]).
00000000000000000000
00000000000000000000
00000000000000000000
0000001100000000000
0000001 I00000000000
00000000000000000000
00000000000000000000
00000000000000000000
~
P=92.1[N]
0OOO00OOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOO
O00~O~StO000000000
~y'O-O--=O'~IStO000000000
00000000000000000000
00000000000000000000
00000000000000000000
P=~.I[N]
00000000000000000000
00000000000000000000
00000000000000000000
OOOOOOO~11OOOOOOOOO
00OOOOO-~11OOOOOOOOO
OOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOO
P : 96.0 [N]
St
1
0.0 < ,~ < 0.1 (initiation)
0 . 1 S ,~ < 0 . 2
2
3
•
0.2=i ,~ <0.3
0.3 :i ~ <0.3?
~ = 0.37 (saturation)
•~
unl~ulin8
Fig. 12. Development of the zone of microcracks near the notch ([FEI]).
278
Y. Tot and S.N. ATLURI
r
¢~ tN Oi t ~
t",d ¢~1 i'M t'M
e'~ ¢"1
t'N
tN
~I,D
~O~
F
t'~ t'ql
tO ~D ',D
~
O~
tD tO ff~
b
Fig. 13. Contours of
Uy near
the notch ([FEll) ( a ) / ' = 92.1 (N), (b) P = 95.0 (N).
V. C O N C L U D I N G R E M A R K S
The analyses of stationary and nonstationary macrocracks in a single edge notched
beam under three-point bending have been conducted by the finite element method. The
obtained results can be summarized as follows:
1. The crack tip energy-flux integral T* (or J) should be used as the parameter
describing the severity of the crack tip in the discussion of the microcrack toughening
effect, because it is applicable to any type of materially nonlinear behaviors, including
unloading.
2. In problems of large-scale microcracking near a macro-crack-tip, the following situation may exist in general: the material closest to the crack-tip reaches a saturation level
for microcracks (and hence behaves linear elastically with reduced elastic moduli); and
this material zone is surrounded successively, first by a microcracking material zone
(wherein the microcrack density has not yet reached a saturation level), which behaves
inelastically; and then by an unmicrocracked material zone which behaves linear elastically with the initial elastic moduli. The loss o f energy in the irreversible process of
microcracking in the above first zone of inelastic microcracking material (which has not
yet reached saturation levels), when included in a global energy balance of crack-growth,
effectively reduces the energy that is otherwise available for crack-propagation.
Analysis of static and dynamic fracture
279
3. T h e results o f the finite e l e m e n t analysis o f q u a s i - s t a t i c stable c r a c k g r o w t h in a
single edge n o t c h e d b e a m , in which m a t e r i a l c o n s t a n t s for m i c r o c r a c k i n g b a s e d o n A E
m e a s u r e m e n t s are e m p l o y e d , has c o r r e s p o n d e d r e a s o n a b l y with the e x p e r i m e n t a l result.
Acknowledgements--The support of this work by the U.S. Army Research Office, under the Balanced Technology Initiative, is gratefully acknowledged. It is a pleasure to thank Ms. Deanna Winkler for her assistance
with this manuscript.
REFERENCES
1973
1976
1980
1983
1985
1985
1986
1987
1987
1987
1987
1988
1988
1990
KOBAYASm,A.S. (editor), "Experimental Techniques in Fracture Mechanics, Society for Experimental
Stress Analysis Monograph, No. 1" (published jointly by the Iowa State University Press and Society for Experimental Stress Analysis).
BtrDL~NSgY,B. and O'CONsELL, R.J., "Elastic Moduli of a Cracked Solid," Int. J. Solids Structures,
12, 81-97.
OwEs, D.R.J. and HINTON,E., "Finite Elements in Plasticity; Theory and Practice," Pineridge Press
Ltd.
BtrDL~,~SKY,B., HUTCHINSON,J.W., and LAMBROPOLq.OS,J.C., ~Continuum Theory of Dilatant Transformation Toughening in Ceramics," Int. J. Solids Structures, 19, 337-355.
EVANS,A.G. and Fo, Y., "Some Effects of Microcracks on the Mechanical Properties of Brittle
Solids-il. Microcrack Toughening," Acta Metall., 33, 1525-1531.
Fo, Y. and Ev^ss, A.G., "Some Effects of Microcracks on the Mechanical Properties of Brittle
Solids-i. Stress, Strain Relations," Acta Metall., 33, 1515-1523.
ATLORI,S.N. (editor), "Computational Methods in the Mechanics of Fracture, Vol. 2 of Computational Methods in Mechanics, First Series of Handbooks on Mechanics and Mathematical Methods,"
North-Holland.
CF,AgAL~atDES, P.G. and McMEEgXNG, R.M., "Finite Element Method of Crack Propagation in
a Brittle Microcracking Solid," Mech. of Materials, 6, 71-87.
HAGG^O,F.M., EPSTEIN, J.S., REUTER, W.G., and DsAsos, V.A., "Fracture Toughness Testing of
Silicon Carbide," Proceedings of the 1986 SEM Fall Conference on Experimental Mechanics, 130-137.
Nn<iSlUCOV,G.P. and A~b'g~, S.N., "An Equivalent Domain Integral Method for Computing CrackTip Integral Parameter in Non-Elastic, Thermo-Mechanical Fracture," Engr. Fracture Mech., 26, 6,
85 i-867.
RUHLE,M., Ev^Ns, A.G., McM~EKING, R.M., CKARALAMBIDES,P.G., and HUTCHINSON, J.W.,
"Microcrack Toughening in Aiumina/Zirconia," Acta Metall., 35, 2701-2710.
EPSTEIS,J.S., Private communication.
OgTIZ,M., "Microcrack Coalescence and Macroscopic Crack Growth lnitation in Brittle Solids," Int.
J. Solids Structures, 24, 231-250.
Tol, Y. and ATtURI, S.N., "Finite Element Analysis of Static and Dynamic Fracture of Brittle
Microcracking Solids (Part 1: Formulation and Simple Numerical Examples)," Intl. J. Plasticity, 6,
169-188.
Institute of Industrial Science
University of Tokyo, 7-22-1
Roppongi, Minato-ku
Tokyo 106
Japan
(Received 25 October 1988; in final revisedform 20 April 1989)
APPENDIX A
Analysis o f a stationary crack with a finer mesh
In o r d e r to f u r t h e r v a l i d a t e t h e d i s c u s s i o n o n t h e m i c r o c r a c k - t o u g h e n i n g p r e s e n t e d
in Section III, t h e s a m e p r o b l e m for a single edge n o t c h e d b e a m u n d e r t h r e e - p o i n t b e n d ing has been a n a l y z e d with a finer m e s h here. F i g u r e A1 f o r t h e finite e l e m e n t m e s h ,
Fig. A 2 for t h e e q u i v a l e n t stress d i s t r i b u t i o n n e a r the n o t c h , a n d Fig. A 3 f o r t h e v a r i a tion o f T* integrals with l o a d c o r r e s p o n d to Fig. 4, Fig. 7, a n d Fig. 8 in Section III, re-
280
Y. Tol a n d S.N. ATtt:R.~
i
I
....
'
×
i" ~
z" o.s)
0 ['
~ ~ --
UNMICROCIIA~CKED
- -
MICROCIIACKEO
-- K,tiP/,~ -
. . . . .
i\~'~, \
L,
r.
3
2
.
.
.
Z ~
4
X [men]
Fig.
A2.
Equivalent
,,
J ~
~
stress distributions
T'm
r
¢olculm~
rnmemd
near
f ~ mlc,ocrackln R
T"u : [1 - V.I] KI~IEu
----
o
x
the notch.
r,. [, - ~,'] v.,~,
C,- ~,,]CK ,iR]~i,
/
/ /
/
/
]
:
I
S
,(
g
CRACK 'liP
Fig. AI. Finite element
mesh subdivision.
/
/
z
O
50
/s
[
. / " ..4
9O
tOAO e [N]
Fig. A3. Variation of T* integrals with load
for the stationary notch.
spectively. The number of smallest elements near the notch in Fig. AI is about four times
as many as that used in the former analysis shown conducted in Section III. It can be
seen in Fig. A2 that the calculated stress distribution asymptotically close to the macrocrack tip in the unmicrocracked material almost captures the imposed stress intensity
value (Kt = 65.5 N/(mm)3/2). The value for Ki ip, for the microcracked solid, extracted
from the curve fitting in Fig. A2 is 45 N / ( m m ) 3/2, and the energy release rate corresponding to this KJip is calculated as 3.7 x 10 -5 (N-m), which agrees quite well with the
computed T,~ at the same load level, as shown in Fig. A3.