Mechanics of Composite Materials, Vol. 35, No. 5, 1999

ESTIMATION OF THE LONG-TERM STRENGTH OF POLYMER MATERIALS UNDER ARBITRARY LOADING HISTORIES

Yu. Potapov and S. Yermolov

Keywords: polymer materials, long-term strength, damage accumulation, changing load, cyclic load 94 nonlinear version of the phenomenological theory of long-term strength of polymer materials (viscoelastic bodies) is proposed. It is based on the introduction of a function accounting for the damage accumulation connected with changes in the load intensity. Theform of thisfunction may be determinedfrom the results of testing the material with a load changing with time in a certain way, for instance, periodically. As a parameter, thefunction contains the rate of the changing load or the frequency for periodic loads. For a quasi-isotropic material, the basic relationships of the theory proposed are generalized to the case of combined stresses. The durability (failure time) calculations o f the material based on this theory are compared with ~rperimental datafor a number ofpolymer and composite materials in a wide range of loading modes.

The phenomenological approach is widely and efficiently used for estimating the long-term strength (durability) of materials and structures. It is based on the notion that a material subjected to a long-term load fails duo to damage accumulation, quantitatively characterized by a scalar or tonsorial quantity (which, at the moment of failure, reaches a certain limiting value). For viscoelastic materials (including polymeric ones), this approach is presented, in the most general form, in the well-known paper of A. A. II'yushin [1]. Some studies in this feld (see, for example, [2-4]) are mainly dedicated to the application of the general II'yushin's criterion in estimating the durability of some particular materials under different modes of loading and in selecting the respective effective functions (kernels) and nonlinear stress functions, as well as in formulating the limiting conditions of failure under combined stress states using different strength theories. The results of these studies provide a necessary basis for reliable estimation of the durability of polymer materials under constant loads and, in general, under all loads whose intensity varies only slightly with time. At significantly variable and particularly periodic (cyclic) loads, the durability of materials calculated by the known procedures, as a rule, proves to be overestimated as compared to its experimental values. This is apparently explained by the fact that the constitutive relations of the phenomenological theory of long-term strength of viscoelastic bodies, constructed by a formal analogy with the hereditary theory of viscoelasticity, do not reflect quite adequately some specific features of the process of damage accumulation in the material (fast of all, its essential irreversibility) and allow us to take them into account only indirectly ~ in the form of additional corrections. In particular, the accelerated failure of the material in cyclic loading is usually associated with its "softening" due to self-heating [2]. However, the same regularity is also observed for thermostable materials (for example, reinforced plastics based on thermosetting resins) at rather low loading frequencies when the effect of self-heating is practically absent. This allows us to assume the existence of a specific mechanism of damage accumulation in the materials which is associated with the changes in the load intensity with time. In what follows, we propose a variant of the phcnomenological theory of long-term strength taking into account the above-mentioned fe,atures. The damage of the material is represented by a symmetric second-order tensor H(d# ), the material is quasi-isotropic, and the moment (gradient) effects are neglected. The temperature of the medium is assumed constant (fixed). In the case of uniaxial tension--compression, the damage is written as d(~)ffi-dH ( ~ ) = I g (~-n)[Ln +(Z~ - L o )P(L~ +Lq )] drl,

(1)

o

Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 5, pp. 585-594, September-October, 1999. Original article submitted September 29, 1998; revision submitted May 7, 1999.

0191-5665/99/3505-0389522.00 9 1999 Kluwer Academic/Plenum Publishers

389

where L~ =L[s(vl)], L~ =L[s(~)], s=la/a,[, a is the olmmting stress,o, is the short-turin("instantaneons") strength,11 =~/to, = t/to,%is the current time, t is the fixed time, to is the "base" time (a constant oftbe material),and K(~-~)is the influence function (a viscoelastickernel).lThe function L(s)mflects the nonlinr dependence of damage accumulation on the stress,whereas the absolute value of the stressis the monotonity condition of the process of damage accumulation when the sign of the stresschanges. The function P(L)takes into account the damage accumulation due to changes in the load intensitywith time; as will be shown later, this function includes the loading rate (forcyclic loading-- the frequency) as a parameter. It is assumed that the failure of the material occurs in a time ~ = ~ : on condition that d(~/)ffi 1, where d is the largest value of the principal components o f the tensor H(dr ) or the greatest difference of these components. The functions L(s) and K(x)can be determined directly from the results of testing the material at separate stress states un= der the action of constant stresses of differeot magnitude. For example, in uniaxial tension, the function of long-term strength for some polymer materials is satisfactorily approximated by the equation [5, 6] o
or $-$0

(2)

l-s

where so = at/o, and ~t is the long-term strength limit("threshold,"or"safe," stress).In thiscase, we obtain the failurecondition from Eq. (I)

d(~: )= L(s)S K(~.f -~) eq = t

(3)

o

On comparing Eqs. (2) and (3),we have 2

IS--S O

L(S)=~ 1-s ' S>So'

L O,

X ( ~ - ~ ) = a ( ~ -~)a-l-

s<-so,

The shape of the function P(L)can be found, for example, from the results of testing the material by regular cyclic loads of different frequency and intensity. Let us consider a "two-step" sign-constant cyclic loading with a period T where, during a time yT within a cycle, the material is subjected to a stress a I and, during a time (1-T)T, to a stress ~2 < ol (upon loading with a "rest," c 2 -0). Assuming that L~ =Lmax =L(s I ) EL', we can obtain from Eq. (1) the expression of damage in n complete cycles: d(nO)=L" "-~ Z kffiO

(k+T)o

f K(~-II)dI3+[L"+(L'-L'~(L" +L")]~ kO

where 0 = Tit o and L ~ = L(s 2 ); in the particular case a 2 < r

j" ~(~-,q)aq, (k+z)e

k=O (k +7)0

L" ffi O. This expression can be transformed to the form

d(nO)=[cp(nO)-q)(O)]{L'-(1-m)(L"-ff )[I-P(L"+ L')][

(4)

B

where r

•fK(x)dx,

m= ~

k=l

r

~nO)--~O)

The quantity m with an increased number of cycles n grows monotonically and tends to y, at K (x) = ax a-I, r

= x a , and 0 < a < 1

(see above), the approximate formula m = T-I( 1-T)a - (l_T)l/n a can be used with a maximum relative error (at small a and y) not exceeding 10-15%. Formula (4) can be extended to sign-constant cyclic loadings with any form of cycles by assuming that I f =Lr~x, L" = Lmin , and T = 1 - A / ( L " - L " ) , where Lmx' mln are the largest and smallest values of L per cycle, respectively, A = L " - M , and o

M m L~ = 0 -1 ~ L(~)dq is the mean value e l l per cycle. The quantities M and A can be regardad as analogs of the mean stress and 0 amplitude in the linear variant of the theory. At a "sufficiently great" number of cycles n, when m = y, from Eq. (4) it follows that IThe constants of the material and the influencefunctions can br different for different stress states, includingtension and compression; in Eq. (1) these distinctions were neglected (for brevity of repr~ntation). 2The same form of the nonlinear stress function is assumed in [7].

390

d(nO) = [q)(nO) - q~(0)][M + AP(L" + L")].

(5)

Assuming in Eqs. (4) or (5) the failure condition d(nfO) =-d(~f ) = 1, where nf and ~f are the number of cycles and the time to failure (durability), respectively, it is possible, for the given loading mode and level, to calculate the value of P(L) from the known durability or, with the known form of the function P(L), to find the durability of the material. In particular, from Eq. (5) at ~x) =x a , we obtain

P(L" + L"),=(~-fa -M)/A,

(6)

~f = [M + AP(L" + L')] -I/a.

(7)

For a complex stress state, the formula (1) can be generalized as follows3:

dO.(~)=I K(~-T1)[Ln ~iJ!TI!+(L~ -I.~ ),%. (~)-%.0])~-,~ o,, +L~ )1dq, where L~ = L[Gi (~)], Ln = L[ai (Yl)], ~i = (3sr s~j/2) V2 is the stress intensity, and c U and s~ are the stress tensor and its deviator (here, they are dimensionless quantities referred to a s). For synchronous biaxial tension, in the former "two-stage" cyclic mode (GH = r components of the tensor H can be presented, similar to Eq. (5), in the form

G22 ----G2, and G33 ----0), the principal

dta (k)---at (~)= [(p(~)-q~(0)][?~ k +AkP(L'+L") ], k ffi l, 2,

(8)

where Mk =yL9_--Z;-+ k (1-y)L inOk `4k~= (1-y)(L'-L") a - a , , , L'= L(~:. ~ L" = L(aT), (~i9 =((~2 + a~2 - a . 9a 92 )'/2, ai 9 "~-'" r -ai

~7=(~;2 +~: _~;~ ),/2, and a~ and ff~ are the largest and smallest cyclic stresses, respectively. Assuming the maximum value of the principal component of the tensor rl as a measure of damage, we find the durability from the condition d I (~y)ffi 1 or d 2 ( ~ f ) = 1. To determine the function P(L)and its parameters, we used the results of the studies containing adequate experimental data on the long-term strength of some polymer and composite materials both under constant and variable loads - - mainly cyclic with different loading frequencies (we should note that such studies on the problem of long-term strength are very few in the literature). For each of the materials considered, the values of constants of the function of long-term strength (at a constant load) were first obtained in the form (2) and then used for determining the values of the function L(s) for different stress levels. Next, for cyclic loadings, the values of M and.4 were found taking into account the shape and parameters of the cycle, and then the values of the function P at the corresponding values of the "reduced" time before failure, ~f, were calculated from the approximate formula (6). For all the materials and modes of cyclic loading examined, the function P is satisfactorily approximated by the general dependence p ( r ) = p ( f t 0 )~ r.-~,

(9)

where f = ~/T is the loading frequency, and (z, 13,and p are positive constants. It follows from Eq. (9) that the values of the function P(L) and therefore the damage accumulated for a fixed time of loading increase with increased frequency of loading (which is proportional to the average in absolute value or to the root=mean-square rate of change in the load intensity), while the durability of the material decreases correspondingly in this case. In what follows, the results of calculating the long-term strength (durability) of the above polymer and composite materials using the procedure proposed and their comparison with the experimental data are given. All the materials were tested in tension at room temperature. 3See footnote 1.

391

70 ~-~ MPa 6O 50

u

,~

4O 3O

I

\oo

20 .

%.

"~_~176

1

2

3

4

'x

-"

[]

lg tf, s 5

6

7

8

9

I0

Fig. 1. Experimental (points) and the calculated long-term strength of SAN copolymer in tension: 1, 9 - - constant load; 2, 9 - - cyclic load at a frequency of 10 "3 HZ; 3, A 0.1Hz; 4, n - - 12 H_z.

60 ~.G, MPa

~ 301

65

\ ~.~N~.N~

20

"~ 1

2

3

4

5

lg tf, s 6

7

8

9

I0

Fig. 2. Experimental (points) and the calculated long-term strength of PMMA in tension under a constant load and a cyclic load with "rest". I, 9 - - constant load; 2 - - 9 - - cyclic load (cycle time/loading time): 2, <3 - - 24/5 h; 3, [] - - 240/20 min; 4, A - - 60/5 min; 5, 9 - - 11/1 min; 6, 9 630/30 s; 7, 9 - - 120/60 s; 8, C~- - 90/60 s; 9, 9 - - 75/60 s.

SANcopolymer [8] (Fig. I). The constants of the function of long-term strength were o s = 68 MPa, o I = 18 MPa, t o = 6.46 and a = 0.182. The material was tested at loading frequencies 10"3, 10"l (zero-reference rectangular cycle), and 12 Hz (zero-reference sinusoidal cycle). The constants of the function P(L)were a = 0.157, [3= 0.909, and p = 0.625. 91 0 6 s,

Po~nethylmethacrylate (PM) (acrylic plastic) [9] (Fig. 2). The constants of the function of long-term strength were o~ = 58 MPa, o I = 18 MPa, t o = 6.3.105 s, and a = 0.200. The material was tested by rectangular loading impulses with complete unloading ("rest") at ratios between the cycle and loading times 24/5 h, 240/20 min, 60/5 rain, 11/1 min, and 630/30 s, as well as 120/60 s, 90/60 s, and 75/60 s (for one stress level of 40 MPa). The constants of the function P(L)were tx = 0.221, [3= 0.850, and p = 0.500. The "inversion" of the long-term strength curves 7-9 and of the respective experimental points is explained by the fact that the relative growth of the loading time per cycle influences the durability more significantly than a small decrease in the cycle time (an increase in the loading frequency).

Glass-reinforced plastic KACT-B (hot-molded sheet material based on fiberglass fabric and phenolformaldehyde resin) [9] (Fig. 3). The constants of the function of long-term strength were o s = 400 MPa, o I = 175 MPa, t o = 2.24.103 s, and a = 0.279. The test mode was similar to the previous one, at the following ratios of the cycle and loading times, s: 1260/60, 630/30, 315/I 5 and 90/60, 75/60, 65/60 (for one stress level of 280 MPa). The constants of the function P(L)were r 0.547, [3= 0.410, and p = 0.481. As in the above case, an "inversion" of the long-term strength curves 5-7 and of the respective experimental points is s~n.

392

I G, MPa

380 I 1

340

t~ti~"x

30O

56 ~ ,

~ ~ ~ 4

32

260 220

180

0

i

I

i

i

i

i

r

1

2

3

4

5

6

7

lg tf, s

Fig. 3. Experimental (points) and the calculated long-term strength of glass-reinforced plastic KACT-B in tension under a constant load and a cyclic load with "rest": 1, 9 - - constant load; 2 - - 7 - - cyclic load (cyclic time/loading time, s): 2, O - 1260/60; 3, [] - - 630/30; 4, A - - 315/15; 5, 9 - - 90/60; 6, 9 ~ 75/60; 7, 9 65/60. Arrows show the end of the 1st cycle.

~, MPa

120

lg v, Mea.s "1

10~

80

6O 10-2

4O 20 I

I

-4

-2

0

I

I

I

I

I

t lg ~, h

2

4

6

8

10

12

10 -4

0

I

I

2

4

Ilg tf, s 6

Fig. 4 Fig. 5 Fig. 4. Long-term strength function of glass-reinforced plastic GUP-M in tension under a constant load (1 m experiment). Fig. 5. Experimental (points) and the calculated (line) long-term strength of PMMA in tension at a constant loading rate.

GUP-Mattelaminat (material based on polyester resin reinforced by knitted mats of glass filaments) [I0], (Fig. 4 and Table 1). The constants of the function of long-term strength were a s = 108 MPa, ~t = 6.5 MPa, t o = 1.47.10 7S, and a = 0.187. The material was tested at loading frequencies of 16.6 and 1 Hz by rectangular cycles with partial unloading, including the tests in biaxial tension (on tubular specimens); in the latter case, the durability was calculated from formula (8) on condition that max(d1, d 2 ) = t The constants of the function P(L)were r = 0.0467, [3= 0.910, and p = 2.911. The procedure for calculating the long-term strength was also checked by the results of testing the polymethylmethacrylate in tension with a different growth rate of the stress v = ~ [9] (Fig. 5). The constants of the function of long-term strength for the material examined were c s = 46 MPa, fit = 18 MPa, t o = 1.63.105 s, and a = 0.470. The function P was assumed in the form similar to that obtained for cyclic loading: P(L) = p(Vto/a s )a L-~. The constants of the function were determined from the experimental values of v and ~f using formula (1) and the corresponding numerical methods: a ffi 0.485, [3= 0.804, and p ffi 1.459. 393

TABLE 1. Experimental and Calculated Durability (the Number of Cycles to Failure) of Glass-Reinforced Polyester Resin in High-Frequency Cyclic Tension with Partial Unloading

lgnf ~l'Iz

sn~

~n

16.6

0.5

0.4

16.6

0.5

0.3

16.6

0.5

0.2

16.6

0.5

m

16.6 16.6

0.6 0.6

0.4

0.5

Ignf

experi- [ calculament [ tion 7.00 5.80 5.08

5.61 4.92

Hz

Sm~

Stain

6.39 5.63 5.17

16.6 16.6 16.6

0.8 0.8 0.8

0.5 0.4 0.3

3.53 3.29 3.11

3.59 3.28 3.06

--

16.6

0.8

0.2

2.85

2.89

5.63

5.40

4.53

4.75

4.21

4.22

3.91

3.78

f,

5.63

.

4.94

1.0

.

.

16.6 16.6

0.6 0.6

0.3 0.2

4.45 4.00

4,50 4.21

. 1.0

.

16.6 16.6 16.6 16.6

0.7 0.7 0.7 0.7

0.5 0.4 0.3 0.2

4.42 4.00 3.64 3.34

4.28 3.87 3.58 3.38

. 1.0 . 1.0

.

*Biaxial tension: numerator m sN and denominator I s22.

.

experi- T calculament / tion

.

0.55*

0.34*

0.16

0.10

. 0.56* 0.17

. 0.19" 0.055

.

.

. 0.19 . 0.23

.

0.58 .

. 0.64

.

-

As is obvious from a comparison of the experimental and calculation data, the variant of the phenomenological theory proposed ensures a reliable estimate of the long-term strength (durability) of polymer and composite materials under loads (including cyclic ones) whose intensity varies significantly with time; the estimation error in essence does not exceed the scatter of the experimental data. The function P(L), which takes into account the "additional" accumulation of damages under variable loads, has the same form at different loading modes and contains the change rate of the load (for cyclic modes - - the loading frequency) as a parameter.

REFERENCES I. A.A. II'yushin, "On a theory of long-term strength," Inzh. Zh. Mekh. Tverd. Tela, No. 3, 21-35 (1967). 2. V.V. Mnskvitin, Strength of Viscoelastic Materials [in Russian], Nauka, Moscow (1972). 3. B.S. Boborov, Problems of the Theory and Calculation of Strength and Deformability of Polymeric Constructions [in Russian], LGU, Riga (1978). 4. Yu. P. Zezin and N. L Malinin, "Method of describing the deformation and strength properties ofhighly filled polymer systems," Mech. Compos. Mater., 16, No. 4, 405-411 (1980). 5. S.B. Yermolov and Yu. N. Potapov, "Estimating the durability of polymers over an extensive temperature-time range," Polym. Mech., 12, No. 4, 673-676 (1976). 6. YtL S. Urzhumtsev, Prediction of Long-Term Strength of Polymer Materials [in Russian], Nauka, Moscow (1982). 7. S.A. Shesterikov, "Some problems of long-term strength and creep," in: Nonlinaar Models and Problems of the Mechanics of a Deformable Solid Body [in Russian], Nauka, Moscow (1984), pp. 180-189. 8. W. Mayer, M. Schlagenhauf, and S. Wintergerst, "Zeitfestigkeit yon SAN-Copolymerisat bei statischer, schwingender und intermittierenden Beanspruchung," Kanststoffe, 64, No. 4, 197-200 (1974). 9. Yu. G. Korabel'nikov, Investigation of Durability of Structural Plustics at Different Modes of Static Loading and Rest. PhD Thesis, Moscow (1967). 10. A. Pawlow and H. Opitz, "Festigkeits-, Verformungs- und Scl~dignngsverhalten von GUP-Mattelaminaten unter nichtstation~en Langzeitbelastungen," Plaste Kautschuk, No. 6, 350-353 (1982). 394