Integrodifferential criterion of the strength of aging polymers
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| Datum: | 1985 |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
1985
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| Schriftenreihe: | Проблемы прочности |
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| Zitieren: | Integrodifferential criterion of the strength of aging polymers / V.A. Buryachenko, B.D. Goikhman // Проблемы прочности. — 1985. — № 8. — С. 1138-1141. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1828892022-01-23T01:27:17Z Integrodifferential criterion of the strength of aging polymers Buryachenko, V.A. Goikhman, B.D. Scientific-technical section 1985 Article Integrodifferential criterion of the strength of aging polymers / V.A. Buryachenko, B.D. Goikhman // Проблемы прочности. — 1985. — № 8. — С. 1138-1141. — Бібліогр.: 14 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/182889 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Scientific-technical section Scientific-technical section Buryachenko, V.A. Goikhman, B.D. Integrodifferential criterion of the strength of aging polymers Проблемы прочности |
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Article |
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Buryachenko, V.A. Goikhman, B.D. |
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Buryachenko, V.A. Goikhman, B.D. |
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Buryachenko, V.A. |
| title |
Integrodifferential criterion of the strength of aging polymers |
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Integrodifferential criterion of the strength of aging polymers |
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Integrodifferential criterion of the strength of aging polymers |
| title_fullStr |
Integrodifferential criterion of the strength of aging polymers |
| title_full_unstemmed |
Integrodifferential criterion of the strength of aging polymers |
| title_sort |
integrodifferential criterion of the strength of aging polymers |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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1985 |
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Scientific-technical section |
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http://dspace.nbuv.gov.ua/handle/123456789/182889 |
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Integrodifferential criterion of the strength of aging polymers / V.A. Buryachenko, B.D. Goikhman // Проблемы прочности. — 1985. — № 8. — С. 1138-1141. — Бібліогр.: 14 назв. — англ. |
| series |
Проблемы прочности |
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AT buryachenkova integrodifferentialcriterionofthestrengthofagingpolymers AT goikhmanbd integrodifferentialcriterionofthestrengthofagingpolymers |
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2025-07-16T02:08:43Z |
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2025-07-16T02:08:43Z |
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1837767562089201664 |
| fulltext |
INTEGRODIFFERENTIAL CRITERION OF THE STRENGTH
OF AGING POLYMERS
V. A. Buryachenko and B. D. Goikhman UDC 539.4
In the general case, failure is a multi-stage topochemical reaction activated by the
temperature T and the stress o and coinciding in outline with the reaction of thermal de-
gradation [i, 2]. With constant temperatures and loads, failure is determined by a small
number of effective constants of speeds which, on the assumption that the principle of
quasisteadiness of the concentration of intermediate products of the reactions is correct,
are expressed through the constants of the speeds of the elementary stages. Under these
conditions the phenomenological approach is used in engineering practice with sufficient ac-
curacy; this approach is based on the introduction of the concept of damageability [3-9],
whose physical meaning is determined by the method of indication [i0]. Here, a change of
damageability is described by the equation
tt__~ = F1 (~, T, ~), (1) dt
where F~ takes into account the degree to which the process is concluded and the increase of
true stress on junctions with increasing damageability. Failure of polymers is a topochemi-
cal reaction concentrated at the apex of growing defects and in the bulk of the material
when nuclei of destruction [2] originate. During aging (under the effect of thermal, thermo-
oxidative, and photo degradation, mass transfer, etc.) the molecular characteristics in the
bulk of the material change, and the constants of the rate of aging are activated by stress
[i, 4]. According to the model of failure, the function F~ is determined by the growth rate
of the defects [2] or, in the final analysis, by the kinetics of the reactions at the crack
tips, and that means, by the depth 0 at which various physicochemical processes of destruc-
tion, structuring, etc. occur: FI(o,T, ~ = Fl(o, T, O, ~).
In view of the strong dependence of the constants of the speeds on stress a and the
stress concentration at the tip of the flaw, it may be taken in the first approximation
that 0 is independent of a [5]. In that case we have an explicit dependence of Ft on time.
In bulk failure [i], we have another special case where FI is explicitly independent of
time.
Since it is difficult to identify several stages of mechanical destruction from stand-
ard experiments for determining the time to failure ro with constant values of oo, To, 00,
it is often taken that F~(o, T, 0, ~) = fl(o, T, o)'f=(~) (we will call this process of
failure simple). When in this case Eq. (i) is used for predicting endurance with variable
o, T, and 0 = 0o = const, it causes the history of change of a, T to be ignored. Correct
prediction based on revealing the main stages of the process [i] requires a much larger
body of experimental data.
Moreover, when this last approach is put into effect it gives rise to fundamental dif-
ficulties connected with the incorrectness of the principle of quasisteady conconcentrators
with variable stresses [i]. In engineering practice, the history of change of the load is
therefore taken into account with the aid of methods adopted in strain and fracture mechanics
of solids and using integral operators with convolution-type kernels [6-9]. These criteria
do not take the increase of true stress on junctions with increasing damageability into ac-
count; they postulate the simplest regularities of change ~ = ~(t) with constant values of
o, T, 0, which is not always in agreement with the experiments [ii, 12].
We attempted obtaining an integrodifferential criterion of strength which would be free
of the above shortcomings.
Central Research Institute of Scientific and Technical Information (TsNIINTI), Moscow.
Translated from Problemy Prochnosti, No. 8, pp. 88-91, August, 1985. Original article sub-
mitted November i0, 1983.
1138 0039-2316/85/1708-1138509.50 �9 1986 Plenum Publishing Corporation
We take it that with constant o, T, 0, the kinetics of change of ~ is described by Eq.
(i). In the fairly general case with variable 0, T, 0 the accumulation of damage can be
represented (in analogy with [9]) by the integral equation
t
r (3) -= 1' Q (t - - ~) dp (o, T, O, ~), (2)
0
where the effect of ~ on the function p(o, T, O, ~) describes the increase of true stress on
the junctions with increasing n; the function p is discontinuous with a value equal to zero
for ~ = 0. In accordance with the physical sense, r and p(o, T, 0, ~) are strictly mono-
tonically increasing functions of their arguments. When the load is constant, the solutions
of (i) and (2) coincide. To find the functions ~, p we assume that Q(t) = t ~
Let us first examine the case when ~ = i. Using the integral (2) in parts, we obtain:
t
(~) = ~ p (a, T, p, ~) d~. (3)
0
Differentiating equality (3) with respect to time, we write:
d t , = k ~=) p (a, T, 9, ~). (4)
By c h o o s i n g c o r r e s p o n d i n g v a l u e s o f ~, p ( e . g . , r = ~ ) , we can a t t a i n t h a t the s o l u -
t i o n s o f ( 1 ) , (4) c o i n c i d e , i . e . , f o r ~ = 1 Eq. (2) c o n t a i n s a r b i t r a r y schemata o f m e c h a n i c a l
destruction described by the dependence (I).
In the case ~ ~= 1 we make the additional assumptions that with o, T, p = const, the
process of failure is simple and p(~, T, p, ~) = p,(q, T, P)P2(Z).
Without loss of generality it may be assumed that p,(+0) = i, p=(+O) = i, p,(0) =
p=(O) = 0. Then, with ~, T, p = const, it follows from (i), (2) that
tf~ (~, T, p) = j t, (~)
0
t (5)
r (~) .t" (tP~'/~' (a, T, p) - - ~p[/~' (0, T, p)"dp, (~ (~)).
0
With the adop ted a s s u m p t i o n s t h e s o l u t i o n s o f (1 ) , (2) c o i n c i d e i f and on ly i f p* /a
(0, T, p) = af,(~, T, 0) (a = const, not depending on o, T, 0, ~). Without loss of gen-
erality, we take it that a = I. Then
r (~) = S ~(~) -- f(~))=dP2 (~)" (6)
0
With the examined constraints on the form of the functions Q, p, criterion (2) encom-
passes the known integral criteria [3-9]. For instance, with ~(~) = ~, p=(~) = 1 and p(t) =
1 we obtain Bart's criterion [8]. With the additional assumptions that a = i, px (0, T, i) =
I from ~2) follows Baily's criterion [4],andwithpl=oandpx =I/Y(q), ll'yushin's
I:(o,T, I) '
and Moskvitln's criteria [6, 7]. For aging materials with p ~= const we adopt r = ~,
I
p=(~) = i. Then with ~ = i, p, (0, T, p) - ~(o.T,p) we obtain Baily's generalized criterion
for aging bodies, and with px (0, T, p) = ~8(~, T, p) the criterion suggested previously
[9].
Let us dwell more in detail on the practical utilization of criterion (i), (2). For
elastomers we may take as f,(~, T, 0)- (for O = i) Aexp I~- ~ ~, for brittle and oriented
polymers Aexp( ,-U + Y") RT . According to data of [5], for many elastomers to a depth of trans-
formation of 30-50% �9 = B0o-mexp ---~ . Then f~(~, T, O) = A ~ omexp ~-~-.
In the general case, to find f,(o, T, 0) it is necessary to investigate endurance on
specimens preliminarily aged to different depths of transformation [5, ii] or to use the de-
1139
pendences ~ = ~(t) obtained in physical experiments for ~, T, p = const. The kinetic regu-
larities of change of ~ (the form of the function f=(~)) may be determined from the known
criteria of strength for o, T, p = const.
For instance, in Kachanov's criterion [3] f2(~) = ~r(l--~)T, in Bai!y's criterion [4] f=~:~)=
i, in Ii'yushin's criterion [6] f=(w) = T-B, which qualitatively corresponds to the kinetics
of accumulation of damage in oriented polymers [12]. For nonoriented polymers [13] it is
apparently possible to take f2(~) = ~ + (B-~)Y. When there is experimental information ~ =
(o, T, p, t) obtained with the aid of physical methods (e.g., electron paramagnetic resonance,
small-angle scatter of x-rays, etc.), we need not confine ourselves to specifying a p~o~
the form of the function fa. Differentiating the extremal curves ~ = ~(o, T, D, t) by
numerical methods, we transform them in coordinates ~ ~ ~ (a similar procedure is used in non-
isothermal kinetics). Then f2(~) is specified in the form of a table, which is not diffi-
cult with a modern computer, and fl(o, T, p) can be found, e.g., by the method of transforma-
tion.
Then we determine analytically or numerically the monotonically increasing functions
f(~) and Pl(o). From the physical prerequisities it follows that with a > 0, p~(~) increases
monotonically. Then ~(~) from (6) always exists, it is unique and is a monotonically in-
creasing function. We find ~(~) from (6) according to the known values of f(~), p2(~) ana-
lytically, which is usually impossible in elementary functions, or else with the aid of the
known numerical methods of solving integral equations. In the simplest case f2(~) = (i-~) -B
[1-3], the contribution of damage ~ to stress concentration on a single defect can also be
determined by the methods of the self-consistent field [14]. Since individual parameters
of (2) (e.g., =) cannot be found with ~, T, 0 = const, we determine them in dynamic change
of ~, T, p, as for the criteria suggested in [7-9].
Example. Let ~=B~ -m, p= I, f1(o)----- m~-l B o m, f~(~)----(I- ~)-~. We postulate that the de-
pendence PI(o), llke in ll'yushin's criterion, is linear. Then
p ( ( L ~ ) = [ m + l l Vm ~ . o~=__1 ;
1 - - ~ ' rn
q) (~)= ~ [(l--g)m+~--(i--~)m+~l' / '~d]__g �9
0
We compare the different criteria of strength on the basis of evaluations of the safety
factors 11 at the instant of failure in the regime of" cyclic change of ~, and also in load-
ing and unloading (regimes i, 2, 3, respectively) [8]. For the random sample q~, ~, ...
the evaluations of the sample mean m~ and of the sample dispersion o~ according to Baily's,
ll'yushin's, and Moskvitin's criteria and (2), (7) (criteria I, II, III, IV), respectively,
are equal to 0.963; 0.953; 0.97; 0.99 and 1.42"i0-=; 0.53"i0-~; 7.84"i0-~; 2.89-10 -~. The
parameter of nonlinearity m' = 18.8 in criterion III is determined in regime i. In cal-
culations of m~ ~ the possibilities of prediction of criterion III in regimes 2, 3 were
evaluated. It was found that m' does not lie in the range of permissible values m'<~. m- i =
13.51. From a comparison of mq, ~ for different criteria the advantage of criterion IV
can be seen.
Thus the integrodifferential criterion of strength of aging materials suggested in the
present article generalizes the previously known criteria [3-9]. The criterion takes into
account the history of the change of load, temperature, aging processes, and also the in-
creases of the true stress in the material with increasing damageability, and it describes
any desired kinetic schema of simple failure for constant ~, T, p.
LITERATURE CITED
i. E. V. Deyun, G. B. Manelis, E. B. Polianchik, and L. P. Smirnov, "Kinetic models in
the prediction of endurance of polymer materials," Usp. Khim., 19, No. 8, 1574 (1980).
2. B. D. Goikhman, and V. A. Buryachenko, "Topokinetic model of failure of solids," Probl.
Prochn., No. 2, 28-31 (1980).
3. L. P. Kachanov, Fundamentals of Fracture Mechanics [in Russian], Nauka, Moscow (1974).
4. J. Baily, "Attempt to correlate some tensile strength measurements on glass," Glass
Ind., 20, Nos. 1-4, 26-28 (1939).
5. B. D. Goikhman, "Accumulation of damage and time dependence of the strength of solids
under conditions of physicochemical transformations," Fiz.-Khim. Mekh. Mater., ii, No.
3, 65-69 (1975).
1140
6. A.A. ll'yushin, "One theory of long-term strength," Mekh. Tverd. Tela, No. 3, 21-35
(1967).
7. V. V. Moskvitin, The Resistance of Viscoelastic Materials [in Russian], Nauka, Moscow
(1972).
8. Yu. Ya. Bart, V. P. Trifonov, A. B. Kozachenko, and N. I. Malinin, "Generalized cri-
terion of long-term strength of viscoelastic materials," Mekh. Polim., No. 5, 791
(1975).
9. B. D. Goikhman, ~. A. Buryachenko, R. A. Cheperegina et al., "Prediction of the change
of characteristics of composite materials in long-term storage under natural conditions,"
Mekh. Kompozitn. Mater., No. 5, 941 (1981).
I0. A. K. Malmeister, V. P. Tamuzh, and G. A. Teters, The Resistance of Tough Polymer
Materials [in Russian], Zinatne, Riga (1972).
ii. G. I. Sarser, B. D. Goikhman, N. G. Kalinin, and A) N. Tyannii, "Evaluation of the
stability of the properties of products made of SKEPT based rubbers by the method of
accelerated thermal aging under tensile stresses," Fiz.-Khim. Mekh. Mater., No. i,
89-92 (1975).
12. V. P. Tamuzh and V. S. Kuksenko, Fracture Micromechanics of Polymer Materials [in
Russian], Zinatne, Riga (1978).
13. A. Ya. Gol'dman, V. V. Shcherbak, and S. Ya. Khaikin, "The ~netics of accumulation of
damage in polymers under conditions of creep under long-term loading," Mekh. Polim.,
No. 4, 730-734 (1978).
14. V. M. Levin, "Stress concentrations on inclusions in composite materials," Prikl. M~kh.,
41, No. 4, 735-743 (1977).
FAILURE OF ORGANIC GLASS AFTER ALTERNATING LONG-
TERM AND CYCLIC LOADING
R. A. Arutyunyan, L. I. Doktorenko,
V. V. Drozdov, and V. M. Chebanov
UDC 539.376:4
The Palmgren--Meiner hypothesis of accumulation of damage, or the rule of linear summing
of damage, was suggested for describing failure under conditions of fatigue [I]. For the
case of two-stage loading by stresses o, and o~ this hypothesis can be formulated as follows.
First the specimen is loaded by stress o, for N, cycles, then it is tested at stress
~ for N2 cycles. At the level of the stress a~ the tests are continued up to failure of
the specimen. We assume that N, R and N2R are the numbers of cycles under stresses a, and a~,
respectively, and then N,/N, R and N2/N~ R are the proportions of damageability in the process
of the first and second loading, respectively. According to the Palmgren--Mainer hypothesis
NI + N, =l. (i)
N; R N2R
An analogous hypothesis for static testing (creep) was formulated by Robinson [2].
Within the time t, the specimen is tested under stress ~,, then the tests are continued as
stress as to failure within time tl. If we denote by t,/t, R and tm/t2R the proportions of
damageability in the first and second loading, respectively, then in accordance with Robin-
son's assumptions
G + ~ = I, (2)
t ~ R t2 R
where t,R, tsR are the time to failure under stresses a, and a2, respectively.
When long-term and cyclic stresses alternate, the hypothesis of accumulation of damage
is written in the following form:
t N
+ w7 = ~, (3)
Research Institute of Mathematics and Mechanics, Leningrad State University. Trans-
lated from Problemy Prochnosti, No. 8, pp. 91-94, August, 1985. Original article sub-
mitted December 20, 1983.
0039-2316/85/1708-1141509.50 ~ 1986 Plenum Publishing Corporation 1141
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