International Journal of Fracture 49:305-315, 199 I. © 1991 Kluwer Academic Publishers. Printed in the Netherlands.
305
The sensitivity of fracture location distribution in brittle materials ROBERT C. WETHERHOLD Department of Mechanical and Aerospace Engineering, State University of New York, Buffalo, New York 14260, USA Received 30 October 1989; accepted 16 May 1990
Abstract. The Weibull weak-link theory allows for the computation of distribution functions for both the fracture location and the applied far-field stress. Several authors have suggested using the fracture location information from tests to infer Weibull parameters, and others have used the predictive capabilities of the theory to calculate average fracture locations for brittle bodies. By a simple set of example calculations, it is shown that the fracture location distribution function is distinctly more sensitive to perturbations in the stress state than the fracture stress distribution function is. In general, the average fracture location is more subject to stress perturbations than the average fracture stress. The results indicate that care must be exercised in applying fracture location theory.
1. Introduction
The traditional Weibull formulation for the probability of fracture (distribution function) as a function of some characteristic far-field stress o-,, is given by
F,m(am)
=
1 - exp
[(~m/j~)m],
where/~ is the resultant of both an intrinsic strength parameter and the integration of the stress state over the entire volume; m is an intrinsic shape parameter. In a classic paper, Oh and Finnie [1] demonstrate how this Weibull analysis may be extended to predict information about the fracture location. Using the examples of a 3-point beam bending and a spherical indenter, they predict and experimentally measure the cumulative fracture stress distribution and its average along with the cumulative fracture location distribution and its average. In general, the predictions for fracture stress are fair, and predictions for fracture location are moderately good. The predictive methods of Oh and Finnie have been applied by a number of authors. Cassenti [2], performing an identical formulation, compares prediction and experiment for three-point beam bending, with fair results at best. He also predicts four-point bending results. Aoki et al. [3] use the analysis in predicting the time-dependent failure. In general, the time-to-failure predictions are satisfactory, while the fracture location predictions are somewhat less satisfactory. Other authors [4-6] have predicted aspects of fiber failure and pull-out in composites. The usual application is for brittle fibers which debond and pull-out, with an assumed stress profile which is linear in distance from the crack plane. All of these analyses are theoretical in nature, and usually find the average fracture location. Another proposed use of the Oh and Finnie analysis is to provide extra information for inferring Weibull parameters from experimental strength data [7, 8]. This seems appealing
306
R.(.
getherhold
in that it utilizes all intormation from the fracture tests. In general, the experimental data lit the inferred distribution for strength reasonably well, and fit the inferred distribution for location somewhat less well. Note that all of the above-mentioned analyses assume that the stress state is well known. Yet we know that the actual stress state may not be completely known or may be different from our predictions. In a finite element analysis, the calculated stress state is an approximation whose values converge only for infinitely small elements. When beam samples are molded or cut, the surfaces are never completely fiat and smooth due to the surface waviness caused by cutting tools. Since the calculations involved in the theory all rely upon stress values taken to large powers, it behooves us to question the effect of these stress deviations on the predictions. In specific, we should know if the predictions for fracture location are more or less sensitive to stress deviations than the predictions for fracture stress. This will also be useful in judging the quality of data to be used in statistical inference.
2. Theory The development for the predictions of the density functions and distribution functions for fracture stress and fracture location is based on a weak link theory [1]. This development is summarized here using event statements, which has not been previously done. Consider an elastic body whose stress state can be described as proportional to a single causative load or reference stress, a,,,. Divide the body up into N elements, and define the ith element as having volume Af,, centroid ~, and stress state a, = a,(a .... ~). The probability of fracture of the element is defined by P[element A~/fails for reference stress ~< a,,,]
where ~bi is the probability of failure per unit volume for the stress state in element i. The notation P[E] is understood to define the probability of event E. The probability P[element A~ survives for reference stress ~< (7,,,] is simply the complement of (1), namely I GA<, (a .... ~,, A~,). The sensitivity to the reference stress may be found by P[element A~ fails for reference stress 8(a .... or,,, + da,,,)]
-
- -
G%(a
{?~/ (v,,,,
.... ¢i,
A~i) do-,,,
~i)A~i da,,,.
(2)
Fracture location distribution
307
The most useful forms come from combinations of the above events. Consider the hazard function P[element A~ fails for reference stress S(am, a,,, + da,.)/element survived up to a,,,] P[element A~ fails for reference stress ~(a,., am + da,,,) c~ element survived a,.] P[element survived to reference a,,,] _
OGA~, dam~[1 - Ga~(am, ~i, A~i)] -l, Oam
(3)
where the equality (3) follows due to Bayes' theorem. Since failure of the elements is assumed independent, we have a simple product for the probability P[element A~g fails for reference stress E(a,., a,. + da,.) c~ all other elements survive up to reference stress am]
0Ga~, -
(4)
[1 - Gado,., G a O ]
c~o.,,;damfi[lj=, - ~ : ~ i , A ~ i ) ] "
Assuming that the number of elements N ~ ~ in such a way that the norm of all volume elements vanish ]IA~iH ~ 0, we may pass to the integral form of (4), to within a factor of exp {Oi A~i}. (This employs the usual expansion for the products of (1 - xi), xj small [1].) P[element d~ fails for reference stress e(a,., am + da,.) c~ all other elements survive up to reference stress am]
(5) = h(a,., ~) dam d~,
where h (a,,, ~) = ~¢/aam exp { - ~,, ff dv}. The primal form (5) may be integrated two ways to give the density and distribution functions that we need. Given the independence of failure of the infinitesimal volume elements, the probability of the body's failing is the (integral) sum of the probability of any element's failing. We may define a density function g for the reference stress at fracture P{body fails for reference stress e(a,., am + da,,,)} = g(a.,) dam,
(6)
where g(am)
= f~h(am, 4) d~ --
- 0/c~a,,,exp[- f ~ g t d ~ l .
If we then wish to know the distribution function or average for the reference stress, this would be given by F..,(a) = fo g(am) dam
(7)
308
R.('. Wetherhoht
and [~ o.,,,g(o.,,,) do-,,,
(8)
respectively. The companion integration of (5) to obtain the fracture location density ~b would be P{element d~ fails for reference stress e.(O, % ) ¢~ all other elements survive up to reference stress a,.} = q~(~) d~
(9a)
where q~(¢) = f.., h(o.,., ~) do.,..
(9b)
Note that although the integration (9b) is usually performed over all values of a,, on (0, crc), it could also be used to determine a failure location density for a given range of applied reference stresses. The location distribution function F: and average location are given by /~)(a)
=
) ~b(~)d~
(10)
and =
IOL ~q~(~) d~
(11)
respectively. (L is the specimen length). Note that (10)-(11) are sensibly posed only under conditions where the stress state is a function of one spatial dimension and the geometry is linear. Otherwise, the average fracture location is a vector whose coordinates ~,,, would be given by
~2
:
q~(~l' ~2' ~3) dr.
~2
(12)
1"
In keeping with the definition of distribution functions, the integral F~m(~ ) of (7) and the integral ~ (oo) of (10) must be normalized to 1, if this has not already been done. Using the a~sumptions of Oh and Finnie, which coincide with general Weibull form assumptions, we will choose a simple power form for the probability of failure function per unit volume under uniaxial stress.
{( ) O.
qJ(o.)
=
--
O-.
O- >/ O- ,
o-o
0
(13) o- < o-.,
where o.,,, %, and m constitute the standard three parameter Weibull model.
Fracture location distribution
309
3. Example calculations 3.1. Three-point beam bending
In order to provide a comparison with the work of Oh and Finnie, we choose the three point beam geometry, with the weak-link integration being carried out over the surface. The stress state on the beam surface is piecewise linear
~-a(x) =
f
x s [0, xm],
Xm
x
(14)
-
x e [xm, LI,
where Xm is the beam midspan, equal to L/2 and a,, is the stress at the midspan. To calculate items dealing with fracture location, we need only consider half the beam. We are interested in the distribution of the fracture location away from the beam midpoint, and the result is symmetric about this point. (Indeed, the average fracture location for the entire beam is always at the midpoint.) For calculations involving fracture stress, the entire beam area must be considered. Symmetry may still be utilized for these calculations, so we normally consider only the interval x ~ [0, x,,] for numerical evaluation, then make the reliability correction for volume using the factor 2 1/,,. Consider two deviations from the ideal stress state of (14). In case I, the linear stress state is approximated by its Fourier series.
a~(x) = a,.{~
~Z 2
4 ~
7Z.=1,3....
cos(n2gx/L)
n2
~'
(15)
This simulates the numerical approximation nature of many stress calculations, such as finite elements. For case II, we consider a deviation which might result from the waviness or thickness variation caused by a cutting tool.
x( j
all(x) = a m - -
Xm
(16)
with thickness h(x) =
h0(1 + C lsin2rcm---~x)L'
(17)
where h0 is the nominal thickness. In addition, an alternative is considered for case II using the same thickness variation (17), but delayed or offset by one-half cycle. h(x) =
hot1 - C l s i n 2 L m x )
We chose the number of periods per beam length m as 10.
(18)
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R.C. Wetherhold
Table 1. Material strength parameters
Material
(7,, MPa (psi)
~r,, MPa (psi)
m
Standard Model
34.5 (5000) 34.5 (5000)
24.1 (3500) 24.1 (3500)
3.2 7
Table 2. Fracture results for beam
Average failure location 2/x,,, Standard Model material material
Average failure load 6,,, MPa (103psi) Standard material
Model material
Exact (14)
0.892
0.944
70.00 (10.16)
65.85 (9.557)
Fourier expansion (15) 1 term 2 terms
0.863 0.886
0.906 0.928
71.10 (10.32) 70.14 (10.18)
68.56 (9.957) 66.55 (9.659)
Thickness variation (16, 17) C~ = 0.02 C~ = 0.05
0.903 0.897
0.950 0.953
67.49 (9.795) 69.24 (10.05)
64.49 (9.360) 61.67 (8.957)
Thickness variation, offset (16, 18) C~ = 0.02 (7'~ = 0.05
0.886 0.874
0.932 0.906
70.48 (10.23) 70.42 (10.22)
66.69 (9.679) 66.56 (9.676)
T w o materials are considered for the Weibull function of (13): the standard values o f Oh and Finnie, and a model material with less strength variability. See Table I for material parameters. The b e a m length, width, thickness dimensions were 1 0 3 m m x 4 . 7 8 m m x 20.6 m m (4.063 in. x 0.188 in. x 0.813 in.). The integrations were carried out numerically using the I M S L routines Q A N D and Q D A G . The results for average fracture loads and locations are given in Table 2, and the distribution functions in Figs. 1-6. N o t e that the shapes of the location distribution functions are significantly distorted by the stress deviations, while the stress distribution functions keep their shape. As expected, the distribution functions for the model material are n a r r o w e r for b o t h location and stress, since that material is less variable than the standard material.
3.2. Spherical indenter
Again following the example o f Oh and Finnie, we choose the example of a spherical indenter on a flat plate. Letting the reference stress a,,, be the p e a k tensile stress, we have the form
r e (0, a),
rk, ~(r)
=-
kl rh ~
(19)
,
area k~r; - k~r 2
re(a,
~),
Fracture location distribution
1.00
0.750 x
Exact:: 2 Term F.S. A 1 Term ES. A
,,× 0.500
0.250
0 ~
0.5
0.6
0.7
0.8
0.9
1.00
X/Xm Fig. 1. Fracture location distribution for standard material, Fourier series expansion.
1.00(5.0),
0.750 U_tE 0.500
(10.0),
Exact Stress 2 Term ES. Approx. (Indistinguishable from Exact Stress)
LI
, __....----(15"0)
1 Term ES. Approx.
0.250
0I
I
40
~
50
I
I
I
]
I
60
70
80
90
100
Fracture Stress O'm MPa (10 3 psi) Fig. 2. Fracture stress distribution for standard material, Fourier series expansion.
311
312
R.C. W e t h e r h o l d 1.00
0.750 x
,,×
Exact Stress - 2 Term F.S. Approx. - 1 Term F.S. Approx. - -
0.500
0.250
0
0.5
0.6
0.7
0.8
0.9
1.00
X/X m Fig. 3. Fracture location distribution for model material, Fourier series expansion,
1.00(5.0),
0.750
bE
(10.0), ..--.--
(15.0),
Exact Stress
v
0.500 2 Term F.S. Approx. 1 Term F.S. Approx. 0.250
0 ~
40
50
60
70
80
90
100
Fracture Stress O m MPa (10 3 psi) Fig. 4. Fracture stress distribution for model material, Fourier series expansion.
Fracture location distribution 1.00
0.750 x u_×
0.500
Actual ~ Sine Variation, C 1 Sine Variation, C1
0.250
0
0.5
I
'
0.6
0.7
~
I
0.8
0.9
1.00
X/X m Fig. 5. Fracture location distribution for standard material, offset thickness variation.
1.00(6.0)
'
'
'
(lO.O) II
y
//
0.750
Actual Stress Sine Variation, C1 = 0 . 0 2 ~ Sine Var 0.500
0.250
i ~
50
I
I
I
60
70
80
Fracture Stress (3"m MPa (10 3 psi) Fig. 6. Fracture stress distribution for standard material, offset thickness variation.
313
314
R.('. 14"'ctherhohl
TaMe 3. Elastic properties for indenter problem
lndentor ball Plate
1:" GPa (lff' psi)
~
radius r~, mm (in.)
207 (30.0) 82.7 (12.0)
0.3 0.3
6.35 (0.25)
where rt, is the indenter ball radius, a is the peak stress location, and k~ and k~ are related to the elastic constants of the ball and plate.
k,
=
0.75
(1-v 1-v )
k:
-
1 -
2v,,
+
'
(20)
2=
a
--
k I rh 05. k~
The values of these parameters are given in Table 3. A modified stress profile is considered with a superimposed sinusoidal variation with essentially the same period for the two intervals r < a and r > a. The peak stress is still reached at r = a, with an even number of periods of the perturbation of the nominal stress o-(r) from (19).
o-(r)
1 - C~sin
a~(2~r),]~
)'~
r ~ ( 0 , a),
c,m(r ) =
(21) o"09 {l
C:sin((6~,._
a)/2C3]j
re
(a, rre,-).
The reference radius rre~,is chosen as 1.27 mm (0.05 in.) to reflect a suitable upper limit to the integrations. The volume (surface) element of Section 2 will be given by d V = 2~r dr. The number of periods inside the linear stress region is selected as C 3 = 1. The results for location and stress averages are given in Table 4. Tahh, 4. Fracture results for indenter
Average failure location
/:/r;, 10
Average failure load ?,,,; MPa (103psi)
Standard material
Model material
Standard material
Model material
Exact (19)
61.8
25.6
284,5 (41.29)
133.5 (19.38)
Varied (21 ) ('~ 0.10 (', = 0.18
60.1 59.0
25.1 24.7
275.9 (40.04) 270.2 (39.22)
130.6 (18.95) 127.9 (18.57)
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315
4. Discussion and conclusions The calculations for the distribution functions for fracture stress and fracture location may be rigorously derived, assuming that we know the state of stress. Due to the exponential and power law forms involved, there can be substantial variability in the distribution functions caused by deviations from the ideal stress state. These perturbations appear to affect the fracture location distribution more than they affect the fracture stress distribution. The average values of fracture location and fracture stress are more stable with respect to stress perturbations, but even here the location average generally shows more deviation than the stress average. In the case of the Fourier series beam stress approximation, the distribution function and average converge more quickly for stress than for location. For surface waviness in the beam, the location distribution function is clearly more perturbed than the stress distribution function, while the location average is only generally more perturbed than the stress average. The differences generated by a half-period offset in thickness variation clearly demonstrate the sensitivity problem. The results for the spherical indenter are analogous to those for the beam. The results of this study have impact on the potential users of this theory. The types and magnitudes of variability which are to be expected in a particular application must be assessed for their impact on the results of interest. It appears that the use of averages for fracture stress and location is reasonable, but the use of distributions is rather problematic. Clearly the use of location data to infer Weibull parameters [7, 8] is less desirable, since it is based on lower quality information. This may explain the relative poorness of fit to data for fracture location as opposed to fracture stress.
Acknowledgements This work was made possible by Grant NAG3862, NASA-Lewis. The author thanks Gregory Popp for assistance with the numerical calculations.
References 1. 2. 3. 4. 5. 6. 7. 8.
H.L. Oh and I. Finnie, International Journal of Fracture Mechanics 6 (1970) 287-300. B.N. Cassenti, AIAA Journal 22 (1984) 103-110. S. Akoi, I. Ohta, H. Ohnabe and M. Sakata, International Journal of Fracture 21 (1983) 285-300. M. Sutcu, Journal of Materials Science 23 (1988) 928-933. M. Sutcu, Acta Metallurgica 37 (1989) 651-661. M.D. Thouless and A.G. Evans, Acta Metallurgica 36 (1988) 517-522. B. Schultrich and M. Fahrmann, Journal of Materials Science Letters 3 (1984) 597-599. K. Trustrum, Journal of Materials Science Letters 6 (1987) 1351-1352.
