Statistical theory of fragmentation Page: 4 of 6
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Jo!.n K. Di.:nes
for c < :t and
N + -N -kt - (c-cO)c(lib)
for c > Ct.
We still need a lat for Q. In the SRI approach
it was taken to be dependent on the stress level, but
we propose a different line of attack based on the
argument th-it cracks are nucleated near the tips of
active crneke, Nucleation is expected to take place
in the rxisyvrnetric region exterior to the two-sheet-
ed cone centered at the center of symiretry of the
crack, as illustrated in Fig. 2. Thus, it is natural
to net the number of cracks formed by a crack of ra-
dius c proportional to its volume, i.e.,(12)
y - Be3
where
e3 3 -kr
Qn - J dT T Q(t - r) ae
3uwhere in the last integral we have set c - cT.
It is'
of Laplace
grals. It
forms areparticularly convenient to use the method
transforms in dealing with these inte-
is straightforward to show that the trans-- 6AM -2' -e2
. 00 -3 + C- --
i +k + a+k (s4k)2+ 3)
(s+k)36N C
00
5and
- 6Bc3 1.4
(s4k)4 2
B - irutl + cx) h tan 8 .(13)
Here 8 in the angle of the cone that bounds the frac-
tured region, indicated in the figure. a is the frac-
tion by which the fr;:ctur- zene exceed, the' crack ra-
dius, and h in the number of nucleated cracks per
unit volume in the region of high stress. We esti-
mate that a is typically .3 and tan 0 is on th order
of 1/4. ThenQ sc IdNI - 0c IdN0I
(14)
Combining, we find by use of Laplace transforms that
(14) can be put in the form_ 6011 P3a
- -.S
5 (s 14 -3
(.4k) - iAc swhere P3(c) denotes the third order polynomial
P3(a) - 61%3(9+k)3 + c2c(a'k)2 + c c2(s+k)3 3 4
+c - c (+k) .(20)
(21)
is the total number of nucleated crack- and I de-
notes the initial dir.tribution of cracks. The dis-
tribution consists of two parts,(15)
where Q denotes the nuuubnr of cracks nucleated by
the initial population and Q in the number of
cracks nucleated by other nucleated crackn. Speci-
ficelly.3
It can be rhown thnt if Pc exceeds a critical value
thr.t two of the roots are unstable.
RfDUCI:D MODULUS
The crack density can be used in cnniunction
with the 1'rdale-:unihelishvili approach to crack
growth (2,3) to obtain a reduced modulus for a crack-
ed material under relatively simple conditio-r.. To
do this we write for one-dincrnsional stronsm f Pc3 -(c-ct)/c-kt 0dc/c - 6r;3
Qi t fc a" NOd/ (r(16)
. - C + cf
indicating that the strain in a fractured mnterinl in
tension In the sum of a part di.' to cracking, , and
a part due to linear elastic behavior, c. The con-
tribution due to cracking can be writtenC - env t
c o-- -- - - -
- -
- ,- -where. is denotes the number of cracks per unit area,
v denotes the dluplaceneut at the center of the
crock, C denotes the hal f-length of tho crack, and
it it. ann,;umred that the crack 'us an clliptic cr',ts
section.
Tua' ro~ntribuutilou Oise to solid elasticit! L isrim-ply
Pig. 2. ]lustri ion of ths r(-irus:. mn ur a ri. I i*
(17)
(18)
(19)
and
(22)
.- , - .- :- -
(23)
3
Q - Qi+ Qn,
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Dienes, J. K. Statistical theory of fragmentation, article, January 1, 1978; New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc1052533/m1/4/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.