During the past thirty years, a tremendous amount of research was done on the development of crazing in polymers. The phenomenon of crazing was recognized as an unusual deformation behavior associated with a process of molecular orientation in a solid to resist failure. The craze absorbs a fairly large amount of energy during the crazing process. When a craze does occur the surrounding bulk material is usually stretched to several hundred percent of its original dimension and creates a new phase. The total energy absorbed by a craze during the crazing process in creep was calculated analytically with the help of some experimental measurements. A comparison of the energy absorption by the new phase and that by the original bulk uncrazed medium is made

Chern, S. S.

Hsiao, C. C.

Pang, S. S.

Zhang, Z. D.

English

NASA Technical Reports Server

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https://ntrs.nasa.gov/search.jsp?R=19840008267 2020-03-20T23:35:01+00:00Z
fl
(NASA ­CR-173275) ENERGY ABSORPTION BY 	 N84-16035
POLYMER CRAZING (Minnesota Univ.) 23 p
HC A02/MF A01	 CSCL 11E
Unc las
H3/27 15286
ENERGY ABSORPTION BY POLYMER CRAZING*
S. S. Pang, Z. D. Zhang, S. S. Chern and C. C. Hsiao
`	 Department of Aerospace Engineering and Mechanics
University of Minnesota, Minneapolis MN 55955 USA
The total energy absorbed by a craze during its develop-
ment in creep is analyzed and calculated based upon a timo dependent
theory of crazing. Experimental measurements of the craze length
have been utilized in the energy calculations. For polystyrene the
initial energy absorption in the craze region is found to be several
wg	 hundred times that in the uncrazed medium. This ratio reduces
sharply in a short period of time to about 50 to 1 and less and
remains so afterwards. For polycarbonate, a somewhat similar
behavior has been found.	 The initial strain energy absorption
by crazing is about 200 times that in the uncrazed region.	 The
3	 energy ratio reduces rapidly to about 55 to 1 and rends to level
off thereon. However, in general, the amount of strain energy,
a
prt
4)
r	 absorbed does increase as a function of time as it should.
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1.
I. Introduction
During the past thirty years, a tremendous amount of
research has been done on the development of crazing in
polymers [ 1-31. The phenomenon of crazing has been recognized
as an unusual deformation behavior associated with a process of
molecular orientation in a solid to resist failure. The craze
absorbs a fairly large amount of energy during the crazing
process. When a craze does occur the surrounding bulk material
is usually stretched to several hundred percent of its original
dimension and creates a new phase.
Since the energy content in the crazing process from time
to time is helpful in understanding the craze formation, an
effort is made to obtain the energy absorption by a craze.
This interest has stemmed from the principal role of crazing
as an energy absorption process associated with micro molecular
orientation in glassypo.lymers. This has important consequences
in further studies of craze initiation and propagation. Both
the deviatoric and volumetric energies during the formation
of a craze affect its nucleation. The ability of the changing
system to absorb energy from time to time governs the propaga-
tion behavior of the craze.
In this paper, the total energy absorbed. by a craze during
the crazing process in creep has been calculated analytically
with the help of some experimental measurements. A compari-
son of the energy absorption by the new phase and that by the
original bulk uncrazed medium is made in this report.
I
I
(±Y
2.
II. Energy Absorption by Crazing
The energy balance concept has been considered one of
the cornerstones of fracture mechanics in application to
solids [4,5,6] for many years. From the global balanc es of
energy a conservative viscoelastic system ender nonisothermal
process is describable by the following rate equation:
W (t) = I  (t) + Db (t) + Kb (t) + 4H6 (t)	 (1)
where a dot is used as a total time devivative,
W(t) is the mechanical work done by external forces,
Db (t) is the dissipative energy of uncrazed bulk medium,
Ib (t) is the internal energy of uncrazed bulk medium,
Kb (t) is the kinetic energy of uncrazed bulk medium, and
Hc (t) is the energy absorption b,
	
adrantal craze.
Due to inhomogene.ties (flaws, inclusions, surface grooves,
cracks etc.) the stress is not uniformly distributed throughout
	
r
kk:.
the medium. Crazes nucleate, in general, from regions where
	 t
0
stress concentrations occur„ Under such a critical state, the 	
9	 t
x
original configurational structure (parent phase) is no longer	 i
stable. A new phase (craze) consisting of highly oriented
domains of fibrils and voids is created which can continue to
propagate by absorbing more energy from its surroundings.
Y
I't
I	 ^ ^
3.
F
	
	 :Vow let us center our attention on the total absorbed
energy by a single craze. Referring to a rectangular coordinate
system (0-xyz), an idealized symmetrical craze of length c(t)
at time t is placed in a constant simple tensile stress field
ao (t) as shown in Fig. 1. In the craze region, oriented molec-
ular bundles (fibril domains) and voids are formed. We define
ac (x,t) as the craze envelope stress which is an average engi-
neering stress taking fibril domains and voids together into
consideration. At any section x, the half distance between the two
top and bottom craze surfaces is called the craze opening displace-
ment and denoted by w (x, t) . Based upon an earlier the-pry (7 1
the energy absorption. rate H c (t) for any craze length c(t)
can be obtained by performing integration of the product of the craze
envelope stress ac (x,t) and the craze opening velocity aw(x,t)
at
along the envelope of the new phase boundary, i.e.
C W
Hc (t) =	 f a (x,t) aw(X,t) dx	 (2)
at
0
According to the previous result [7-11), in the case of creep
ao (t) =ao a (t)	 where u (t) is a unit step function, it has been
shown that the use of the two-step stress distribution results in
good appxoximatign as compared with experimental findings:
al (x,t) = aa o ,	 for o < x < a(t)
	 (3)
a2 (x,t) = $a 0 1 	 for a(t) < x < c(t) ,
where a and ^ are material constants, a o
 is a constant represent-
ing the external stress as shown in Fig. 1. According to experi-
mental results for polystyrene [91 the stress a 1 (t) is about 10%
below the applied stress along the majority of its length and
about 15% above the applied st;-ess near the craze tip. That is
R
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URIGINAL PAGE IS
	
4.
OF POOR QUALITY
a 2 (t) = 115% a 0 0	 It was found that a(t) must satisfy the
following relationship:
a(t)	 = c(t)cos t	 8 - 2 l-a
	
(4)
The craze opening displacement field for polymers having
a constant natural draw ratio I n is
^
fw(x,t)	 _ [Cb(0)4^(x,t)	 + 
	
C^(t-T)
	 $ (x, T)	 dT)
	 (5)
^nln	 ^0
where
t	 k
C (t)
( x , t )	 =	 ( > 	do	 ,	 (6)+(n-X }
T
andr^
i
n
CIO
 -CS	 (x, t)
g(n ,t)	 _	
dx	 (7)(n2-x2)
0
9
Here Cb (t) is the time dependent compliance function represent- 'I
ing the constitutive beh.,vior of the uncrazed bulk material..
For most glassy polymers, the strain in the uncrazed portion of E'
the polymer is usually about 1%.
	 Hence the strain ratio or the 1
so called Poisson's ratio v may be assumed constant [12].
	 Under
this situation Cb (t) may be reduced to: I,U
Cb (t)	 2(1-v2)J(t),
	
(8)
where J(t) is the uniaxial creep compliance function of the
bulk medium.
t
N
,r
• 4t1
iA ,
z5.
ORIGINAL PAM k
OF POOR nUALW Y
Combining	 (2) - (7) one obtains: (see Appendix)
, a 1t)
Swat, ^ dx = 
X^—
n
-Z (R-a) [C A (0) (2b,-b,))c(t)+Cb(0)blc(t)]aoc(t) ^ (9)
o 
c(t)
8w(x,t) dx - n— (^-a) [ C (0) (2b +b )c(t) +C (0)b c(t) ]Q c(t) ,(10)
a 
(t)at 	 - a n -1	 b.	 3 2
	 b	 3
S,
^k
t
k^
G
V
^	 4
k	 t
R
r
where
bi =	 cos 2 6 ((^ - Z) tan g + lnsece),
	 (11)
b2 =	 cos 2 e lnsece,	 (12)
b3 = 22 cos 2 e (2t an g - insece).	 (13)
Substituting (3) and (4) into (2) gives
fa(t)	 c(t)
Hc(t) = Q o oc
	
aw(Y,t) dx + R
	aw(x,t) dx]
	 (14)at	 —at
°	 a(t)
Now putting (9) and (10) into (14) yields:
Iic (t) _
	 Xn ^(R-a)6o{(f3b3+ab1)Cb(0)c2(t)
n
+ [0(2b 3 +b 2 )+a(2b 1 -b 2 ) ]Cb(0)c(t)c(t) j
	 (15)
O F POOR QUALITY
Equation (15) represents the energy absorption rate by a craze
at any time. Theref,are the to :al energy absorbed by a craze
is;
ft
(t) = 	 He (t) dt
0
t
_	 n	 '
X -1 ($-a)a 2 ((ab^+ab l )C b (0)	 c1(t)dtn	 o
t
+(0(2b 3 +b z ) ta(2b1..;,2)
 IC b (0)	 c(T)	 (T)dt	 (16)
 j 
0
In the following section, the a'no,.re equation will be utilized
together with experimental. results of the craze length c(t) for
calculating the total absorbed energy. It will be seen
that the integrals in (16) have to be properly represented before
programing and computing.
r.
9
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a
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7.
III. Results and Discussion
In order to obtain c(t) experimentally, creep tests of
polystyrene have been performed (8]. A typical curve for the
changing craze length as a function of time is shown in
Fig, 2. A fairly good approximation for the crazu length c(t) is
obtainable using a function of the form
c(t) = c(0) + Atn ,	 (17)
where c(0) is associated with the initial flaw size
and A and n are material constants obtainable by curve
fitting techniques.
The following-integrals are rePresented as:
t
fo c(t)c(t)dt = Ac(0)to +0.5 A 2 t2n
	 (18)
f
t
 
c 2 (t)dt = c 2 (0)t + nZ— c(0)A to+l + 2ri 1 A2t2n+1 	 (19)
0
Then (16) was calculated and the result is shown in Fig. 8.
For the unoriented medium the energy density is:
ft
(t) = 
	 a(T)E(T)dT ,
	
(20)
0
where
a is the simple tensile stress,
E is the simple tensile strain.
In the case of creep a = a., with a' o as a constant
Since the strain
E(t) = a0 j(t)	 (21)
so the energy density may be written as:
e(t) _ a 2j ( .t)	 (22)
r
x
i	 d
i
.	 t
g
z
a
F	 ^
i
i
r	 ,
i
w
4
I
w^
'R
k
h'	 {
^y
l
ORIGINAL IPACiU V.0'
OF POOR QUAt-1IN
8.
where J ( t) is the creep compliance of the unoriented material,,
which can be obtained through a creep test and using curve
fitting technique.
Based upon crazing model (Fig. 1) the craze envelope surface
initiation rate is defined as follows:
C(t)
S (t) =	 aw(x, t)  dx.	 (23)
By substituting Eqs. (9) and (10) into (23), we have:
S (t)	 4^ nr (a-a,) (b l+b 3 ) [2Cb (0)c(t)+Cb (0) c(t) l Q pc(t) .
n
So the area (^f the craze ,region at any time is:
S (t) = f S (t) dt ---X- (^ = a,) ( b 1 +bIS, n0
tft[2Cb (0) j c(t) C(t)dt + Cb (0)	 c2(L)dt)c0
	
0	 0
Then the energy contained in a comparable volume with respect
to a craze in the original polymer is as follows:
H 0 (t) = e(t) S (t)
x n	
ft^-^ (S-^a) (b l+b 3 )c^J(t) [ Cb(0)	 c(t)' C•(t)dt
0
f
c
t
+ Cb (0)	 2(t)dtl
0
which is plotted in Fig. 4.
Fig. 5 shows the ratio of H e ( t) and H 0 ( t) as a f unction of
time. It is evident that the energy absorbed by the craze is
much greater than that in the original uncrazed medium. Initially
several hundred folds of energy absorption took place. Later
on a ratio of about 50 to 1 is maintained for the propagation , of
s
rl	 l
^,s_
(24)
(25)
t
K"
G	 ?
(26)
A	
^^
r	 j
9.
the craze.
For polycarbonate a somewhat similar behavior in energy
absorption has been found. 'ii l ig. 6 shows the time dependent
craze length variation obtained analytically. Figs. 7 and
8 show respectively the energy absorbed by crazed and
uncrazed polycarbonate. Fig. 9 gives the ratio of the
energies absorbed for comparison.
The absorbed energy has several functions in developing
the craze. At first the fibril domains and possibly some
voids near the craze tip are formed. r)uring this process
some energy will dissipate, parts will be stored and new
surfaces are created. These energies may be calculated.
One of the important features in crazing formation is
the variation of the temperature field near thr neckdown
region where the original polymer transfers from the original phase
to a new crazed phase. This temperature field may be
obtained by solving an appropriate heat equation satis-
fying the craze modelling. Dissipated energy may be
subsequently obtained. This work is being carried out now
and will be reported in the future elsewhere.
tl ^i
10.
'0'31C3tjr';'.
OF
Appendix
By substituting (2) into (6) and (7), one gets: 	 d
Ox, t) _ 1 (R-a) c oc (t) [ (X+g) cosh
-1 it—X% - (X-g) cosh-1 '-Xg ] , (AlX+g
where
X = x/c (t) ,	 (A2)
g = cos6 .	 (A3)
From (4) by differentiation one obtains,
R
a(t) = t (t)cos8.
	
(A4)
Then the following expressions can be derived
f (a, t) _ [7r22 (S-a) cocos6lnsece ] c (t) , 	 (A5)
f
a(t)
f (x, t) dx = 22 ($-a)
 
ci ocos 2 6 (Mane- 2tan6 -lnsec6) c 2 (t) 	 (A6)
o a(t)
^ (x, t) dxQOcos 2 8 (Ita n6-6tan6-2lnsec8) c(t) c (t) 	 (A7)
c(t)
1 ^ (x, t) dx = (0-a) Qocos 2 6 (Ztan6-lnsec6) c 2 (t) ,	 (A8)
a(t) i
C(t)
dt ^ (x, t) dx= (S-a) Qocos 2 e (6tane-2lnsec6) c (t) c C-t) , 	 (A9)	 k
E
•	 1
1
From (1) , we have	 }
a(t)	 a(t)	
F	 i
w(x,t)dx = Xn=l Cb(0){- A(t)d)[a(t),t]+^ ^(x,t)dx}	
Y1
n
l0	 X	 a(t)	 a(t)	 O
+ Ann 
{Cb (0) ^(x,t)dx +	 [f^b(t-T)O(x,T)dT]dx}, (A10)
^
o
}
A
{I^
i
E `
ORIVIAL r R ^P t
c 
f	 ) OF POOR QUA°►) Irf
	 c f ^)
x
w(x^t)dx ^
T
n-	 c:'b (0)
n
aW4[a(t),t •rid-^.-	 fx,b)dxI
^^ m (t) a ( t )
c(t)	
c(OL.
•r
^1 1
l
•Cb (0) f c;) ^.., L) dx	 ^r	 i j	 Cb ( t^-'T') ep (x T) c1,'1 dr f
	(All
o	 fn
a(t)	 a(t)A
F4, Since Cb ( t)	 is rather small at all times, it can be nerlec per .
Nor if we substitute (	 ) -- ( A9)	 into	 (Al0)	 and	 0111)
' respectively, the following ea-nations may be obtained:
s
{ a (t)
^7(x,t)dx = ^n l ($-a) [ Cb (0) ( 2b^-b 2 )c(t)•+cb(0 )blc(t))aoc(t)	 (Al2n
0
c (t)
( J^
Jw(x,t)dx	 ^n 1 (0-a)[Cb (0) (2b 3 r1a 2 )c(i,) rCb (0)b 2c(t)	 0C (t)	 (A13 .
	
t
.a()
n	 ^
	
A	
^^
where	 t
	
n	 i
b	 ? cos 2 0 [ (1-©) tan0+lnse08 ]
	 (,la )1 r Tr	 4 2
b2 =	 cosz©lnsecO	 (A15)
b3 _	 os26(2tan9--lnsec6)
	 (A16)
(9
12.
References
1. C. C. Hsiao and J. A. Sauer. J. Appl. Phys. 21, 1071 (1950).,
2. S. Rabinowitz and P. Beardmore, Crit. Rev. Macromol.
Sci. 1.1 (1972).
3. R* P. Kambour, J. Polymer Sci. D7, 1 (1973).
4. A. A. Griffith, Phil Trans. Roy Sac. (Lond)• A221 0 163
(1921);	 The Theory of Rupture, Proc. First Int.
Congress Appl. Mechs. DelA. 55, (1924).
5. E. Orowan, "Fundamentals of Brittle Behavior in Metals"
Proc. Symposium on Fatique and Fracture of Metals.
ed. by W. M. Murray, John Wiley and Sons, 139, (1950).
6. R. S. Rivlin and A. G. Thomas, "Rupture of Rubber",
J. Polymer Science, 10 (3), 291, (1953).
7. S. S. Chern and C. C. Hsiao, J. Appl. Phys. 53, 6541 (1982).
8. Z. D. Zhang, Si S. Chern and C. C. Hsiao "Propagation of
Crazing in Viscoelastic Media" J. Appl. Phys. 54, (1983)
9. 2a. D. Lanternasser and E. J. Kramer, Phil. Mag. A. 39, 469,(1979).
10. L. Bevan, J. Polymer Sci. Polymer Phys. 19, 1759 (1981).
11. L. Bevan, J. Appl. Polymer Science, 27, 4263 (1982).
12. Z. Rigbi, J. Appl. Polymer Sci. Symp. 5, I f (1967).
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FIGURE ATTACHMENTS
Fig. 1 Quadrantal Craze and a Two Step Craze Envelope
stress
Fig. 2 Time Dependent Craze Propagation in Polystyrene
Fig. 3 Energy Absorbed by a Craze in Polystyrene
Fig. 4 Energy Absorbed in a Craze Volume in Uncrazed
Polystyrene
Fig. 5 Ratio of Absorbed Energies by Crazed and Uncrazed
Polystyrene
Fig. 6 Time Dependent Craze Propagation in Polycarbonate
Fig. 7 Energy Absorbed by a Craze in Polycarbonate
Fig. 8 Energy Absorbed in a Craze Volume in Uncrazed
Polycarbonate
Fig. 9 Ratio of Absorbed Energies by Crazed and Uncrazed
Polycarbonate
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Energy absorption by polymer crazing

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