A Theory of Crack Initiation and Growth in Viscoelastic Media !II

Home , Stress intensity factor, Viscoelasticity

International Journal of Fracture, Vol. 11. No 4 August 1975 Noordhoff International Publishing Leyden 549 Printed in The Netherlands

A theory of crack initiation and growth in viscoelastic media !II. Analysis of continuous growth*

R. A. SCHAPERY

Civil and Aerospace Engineerin9 Departments, Texas A & M University, College Station, Texas 77843, U.S.A. (Received April 19, 1973; in revised form February 21, 1975)

ABSTRACT The theory of crack growth developed in Parts I and II is used to predict crack velocity and failure time for an elastomer under simple uniaxial and biaxial stress states. Included are consideration of the effect of specimen size on failure time for initially small cracks, experimental determination of fracture properties, the effect of strain level on crack propagation, and the validity of the plane strain assumption. The second half of the paper examines the effect upon crack velocity of nonlinear viscoelasticityof failing material at the crack tip in media which is otherwise linearly viscoelastic. The dependence of fracture behavior on environmental changes and aging is also considered.

I. Introduction

In this paper we examine the validity of the opening-mode crack growth theory developed in Parts I [1] and II [2] by comparing predictions of crack velocity and failure time with experimental data obtained by others. Also, certain practical implications of the theory are explored. In Section 2, experimental and theoretical results for the time required for complete failure of centrally cracked elastomeric sheets (Solithane) are compared and found to be in good agreement over a range of constant applied loads and temperatures. Similar agreement is shown in Section 3 for the crack velocity in long, narrow strips of the same elastomer under constant applied deformation; the same mechanical and fracture properties are used in the analysis of both geometries. At relatively large overall strains some discre- pancy between theory and experiment is noted, and a strain-modified stress intensity factor is found to remove this error. In both of these problems, stress intensity factors for plane stress are used in a crack growth equation which is based on the assumption of plane strain at points close to the crack tip; justification for this procedure is given in Section 4. For the particular elastomer investigated, fracture energy and a measure of the strength of the failing material at the crack tip are taken to be constant. However, the underlying theory developed in Parts I and II is not restricted to constant properties. In Section 5 we examine the influence of nonlinear viscoelasticity of the failing material on crack growth behavior in viscoelastic media. Effects of environmental changes and aging are briefly described in Section 6. For the prediction of crack velocity and failure time of the elastomer, we shall use a result obtained in [2, Eqn. (56)]:

da/dl=(TC/2) F C2 ][/m ~I/m{]~2(l+i/m,,/~2 12 1_8F(1-K?/K?,)_] "~m ,--t ,, ~,.-, (1) This expression for crack velocity da/dt is based on the use of a creep compliance Cv(t) which is written in the form of a generalized power law : Cv(t) = Co --}- C2 t m (2) where Co=C~(0); C2 and m are positive but not necessarily constant. For a constant * Parts I and II are published, respectively, in Vol. 11, No. 1 (1975) 141-159, and Vol. 11, No. 3 (1975) 369-388.

Int. Journ. of Fracture, 11 (1975) 549-562 550 R.A. Schapery

Poisson's ratio v, this compliance is simply related to the creep compliance D (t) under uniaxial stress [1, Eqn. (38)],

c (t) = 4(1 -,,2)o (t) (3) where D (t) is the axial strain for a unit uniaxial stress applied at t = 0. Additionally, F = fracture energy, am= maximum stress within the failing material near the tip, I x =integral measure of the shape of the stress distribution in the failing material (0 < 11 =< 2, with I1 = 2 when this stress is uniformily distributed) and 2~/"=parameter depending only on m (:1/,,~ 1 for O

K, = lim~,~o (2n¢1) ~ a ° (4) where ayo= ao (~1) is the normal stress acting across the plane of crack prolongation and ~1 is the distance ahead of the crack tip; by definition, a ° is the singular stress corresponding to a mathematically sharp crack tip without a finite zone of failing material. Also,

K,g =_ (st~Co) ~ (5) is identical to the critical stress intensity factor for an elastic body having a constant compliance Co. The compliance parameters, Co, C2 and m, and the fracture properties, F and am I1, will be assumed constant in Sections 2 and 3.

2. Crack growth and failure of a sheet under constant load

Consider a centrally cracked plate, such as shown in Fig. 1. Assuming the instantaneous crack length, 2a, is large compared to sheet thickness but small compared to the in-plane dimensions, the stress intensity factor is approximately [e.g., 3],

K, = (an) ~ a (6) where a is the applied stress. Substituting Eqn. (6) into Eqn. (1), assuming a is applied at t = 0 and is constant thereafter, and integrating yields

t = (2/n 2) (a~ I21/a 2) (Co/2,, C2) 1/rn Ial/a°[(Oy/O0) ') -- 1]l/m(d~}/) ') (7a) where 2a o is the initial crack length, and a o is defined as

a o =-- 8F/nCo a 2 . (7b)

When the central crack reaches length 2ag (assuming 2a o < 2ao) complete failure of the sheet occurs; this point follows from Eqns. (1), (5), (6) and (7b) which imply fi--*~ as a--*ag. The time at which a = a 9 is called the failure time, t/, which, after changing the integration variable in Eqn. (7a), becomes

t r = (2/n2)(a2 12/a2) (C0/2," C2)1/,, J'~o/o,(1 - u) 1/m U -(1 + 1/m, du. (8)

The integral can be expressed in terms of incomplete Beta functions. It is of interest to notice that when ao/a o >> 1 and 0_< m < 1 the integrand in Eqn. (7a) goes quickly to zero as 7 increases. In fact, the current time is already 90 ~o of t f when

Int. Journ. of Fracture, 11 (1975) 549 562 Theory of crack initiation and gru~th m ui.s~,ul, t,tiu media. III 551

a/a o -~ I(Y". (9)

One can easily show that if the assumption ao/a o >> 1 is removed, the crack growth at t = 0.90 t/will be even less than given in Eqn. (9). These observations are important because they tell us if the stress intensity factor, Eqn. (6), is valid over most of the time period 0_< t _< t/ or if the effect of finite sheet width must be included when predicting failure time. For three typical values of m, Eqn. (9) yields

10 , m= 1 a/a° ~- i 3.3, m=0.5 (10) ( 1.6, m=0.2.

Surprisingly, most of the time required for failure is consumed while the crack is still relatively small, especially for 0< m< 0.5. As a point of comparison, if the bracketed term in the integrand in Eqn. (7a) is replaced by (ag/ao7) TM for all 0_< t< t/, the failure time is derived by setting a/ao= oo to find

tf= [(2ma2~I~)/Tz(2+ 1/"'](8r/2mCzao)t/ma 2,1 + l/m) (11)

which brings out very simply the effect of stress on failure time. This is the same result as would be obtained by neglecting (KZ/KZg) in Eqn. (1) at the outset of the analysis.

///////// ~ 0.25"

13 0 ° C "~l t° • 5 ° C ~eO I - 0 I0 ° C ~Z~e ./--The°ry' -- O • 15 ° C ~/ (..9 O A 25 ° C ._1 • :30 ° C 0 40" C

'%. I I I \l O( 2 4 6 8 LOGio tf/aT, SEC.

Figure 1. Experimental and theoretical failure time t~. as a function of the applied gross stress a for Solithane 50/50. Theory: Eqn. (13). Experimental data after Knauss [3].

Equations (8) and (11) will be compared to experimental data on Solithane 50/50, which is a crosslinked, amorphous, polyurethane rubber. The experimental data for failure time, which are shown in Fig. 1, have been normalized with respect to the stress cry, where ao~ =- (8FE~c/3~zao) ~ (12)

Int. Journ. of Fracture, 11 (1975) 549 562 552 R. A. Schapery is the critical stress for the onset of crack growth in an elastic plate having Poisson's ratio ~' = ½ and Young's modulus Eoo = 4 (1 - v2)/Cv (~) = 3/C~ (~); thus, no crack growth occurs for log (a/aoo)2< O. Also, all data have been reduced to a common temperature of 0°C by recognizing that the material is essentially thermorheologically simple [3]; according to Section 6, the major effect of temperature can be introduced by replacing t/with t//aT in Eqn. (8). Since log (t/aT)---- log t-- log aT, the theory, if valid, implies experimental failure data obtained at different temperatures can be superposed by means of horizontal translations of magnitude log aT SO as to form a single curve; the failure data at the temperatures indicated in Fig. 1 were shifted to the data at 0°C. (One can interpret the reduced failure data as being the results of tests conducted entirely at 0~ C.) We should add that the uniaxial creep compliance for this material, which is shown in Fig. 2, was formed in the same way; in fact, the values of log a T used to shift the failure data are those obtained by shifting the creep compliance. That a single creep compliance curve was obtained is a check on the thermo- rheological simplicity assumption; the fact that the failure data superpose quite well (as ~cen in Fig. 1) using the same values of log a T helps to substantiate the assumption that I mid tr,, 11 are constant, and checks the underlying fracture theory itself.

-2 I I I I I I I I I

D(t) = I0 -4-8 + I0 "1'7 t °.5 /

T -3 J --2"-- - - = '

9 o O -4 .J

TEMPERATURE 0 ° C

-5 I I [ i I I I I I -tO -e -6 -¢ -2 0 LOGIo t/at, MIN. Figure 2. Creep compliance in uniaxial tension for Solithane 50/50. Experimental data, -, after Mueller and Knauss [5].

The generalized power-law fit shown in Fig. 2 implies m = 0.5 ; we carry out the integration in Eqn. (8) to find

t/---- (2/~z2) (a21 ]/a z) (C O/2,, C2) z {~z+ ½(ao/ao) z - 2 (ao/ao) + In (ao/ao) } (13) where ag depends on stress through Eqn. (7b). When ag/ao ~ 30 this equation reduces to Eqn. (11) after setting m=½. The latter equation shows the theory is a straight line with slope of -½ when log (cr/cr~o)2 is plotted against log t/; it is observed in Fig. 1 that this is indeed the slope of the experimental data in the low stress level range ((cr/tr~) 2 ~ 10°"75). We evaluate the constants C O and C 2 in Eqn. (13) by assuming v=½ and using Eqn. (3) together with the constants shown in Fig. 2 for the generalized power-law fit to D (t); although the compliance data do not obey the power law at long times, we will show later that this fit is valid over most of the failure time range covered in Fig. 1.

Int. Journ. of Fracture, 11 (1975) 549-562 Theory of crack initiation and 9rowth in viscoelastic media. III 553

Referring to Eqn. (11), we see that the influence of the fracture properties on the relation between a and t: in the straight line region is through the combination a,,212 F1/m. Had the fracture data been given in terms of a in [3], rather than a/tro~, this combination of properties could have been found by matching the theory and experiment at any point in the straight line range. (Note that a,,I 1 and F appear by themselves in the high stress level range of Eqn. (13); if the theory is applicable in this range one can then determine amI 1 and F separately.) Knauss [3] obtained the fracture energy itself from other experiments on swollen rubber, in which state the material is essentially elastic, and then used the corresponding value ofao~ to normalize the failure data. He found F = 2.41 × 10- 2 lb./in., which corresponds to a~ = 8.40 psi. (Knauss actually reported the value of F = 3.21 x 10-2 lb./in. [-3] ; but this value was obtained by using the plane stress equation for critical stress in an elastic sheet. Later it will be found that ~ is much less than sheet thickness, which means the plane strain version should have been used. In order to correct the reported F value we have multiplied it by (1 - y2) = k.) Now, express Eqn. (11) in terms of (a/ao~) and solve for o-,,11 by noting that the straight line in Fig. 1 intercepts the stress axis at t:= 108 see.= 106.22 rain. ; hence o-,,11 = 172 × 103 psi. By taking into account the fact that 0< 11 < 2, a lower bound to the maximum failure stress is obtained : am > 86 × 103 psi, where the equality is used if the stress distribution is constant throughout the failure zone. The ideal theoretical strength based on failure of a regular arrangement of carbon atoms is approximately thirty-times this value [4] ; however, Considering the irregular nature of real networks, the presence of shear forces at the tip (see Fig. 3 in [1] ) and that the above value for tr,, is the lower bound and not necessarily o-,., this numerical result does not seem unreasonable. Inasmuch as both u,,l 1 and F are now known, we can plot in Fig. 1 the remainder of the solution as given by Eqn. (13). While having the correct shape, the theory underpredicts failure time in the high stress level range above log (tr/uo~)2 - 0.8, say ; this stress corresponds to an overall strain of approximately 5 ~. Since t:~ KI 6 for m----½, even a small amount of nonlinearity, such as blunting of the crack tip caused by large rotation of the near-tip crack faces, could have a significant effect. In fact, we find the theory agrees with all of the experimental results if the stress intensity factor in Eqn. (6) is replaced with

K, = (1 + 2e) -1 (a~z)½ u (14)

(It is to be noted that the effective modulus of the sheet is a constant 430 psi since the time scale for crack growth in Fig. I is far beyond the viscoelastic range in Fig. 2 ; therefore, both overall stress and strain are constant during the tests.) Even though the stress at the tip is very high, the normal strain is not excessive because practically equal triaxial tension exists. Using a typical value of bulk modulus, 2 x l0 s psi, t he Yo ung's modulus of 430 psi, ax= O'y, and plane strain (as = v (ax + at)) we find e x = e r-~ 21 ~. Moreover, because ay=a,,+O(¢~) near the tip [1], the strain decays rapidly along the line of crack prolongation. Of course very high shear strains may exist on either side of the failure zone. Recalling that the above predictions are based on the use of a generalized power law to represent the creep compliance, as shown in Fig. 2, we must now determine if the time range for which this fit is valid (t ~ 10-2 min. at 0° C) covers the range of the effective time which controls crack growth. According to the theory in [2], this latter time is ~/3d, where ~t is the length of the failure zone (i.e., the distance over which the material near the tip dis- integrates). Thus, let us first calculate the range of ct, where [2, Eqn. (2a)] :

= ~rcKl/u.,I1 2 2 21 (15)

For the lowest and highest stresses at which experimental data are shown in Fig. 1 we find, respectively,

Int. Journ. of Fracture, 11 (1975) 549-562 554 R. A. Schapery

t0,6 ~, t =0 + (16) (i) log(a/aoo)2=0.2: c~ = [2.0A, t 0.9t I

[10A, t=0 + (17) (ii) log(a/~oo)2 = a.6: = = [32 A, t 0.9 t I where Eqn. (14), rather than Eqn. (6), has been used to calculate stress intensity factor. Although the accuracy of the theory in predicting failure zone lengths comparable to inter- atomic spacing is suspect, it is believed the scale is at least correct in view of the agreement between experiment and theory in the low stress level range in Fig. 1 ; the slope is somewhat sensitive to ~ in that if c~ were a constant, for example, a slope of (-½), rather than (-½), would be predicted for this range. (In an attempt to bracket the actual failure zone size, stationary and moving cracks were observed with the TAMU scanning electron microscope. Figure 3 shows a tip which was moving at approximately 10-3 in./min, at 77°F. Cracks at 10 x the magnification of this figure were also studied visually but the failure zone at the tip was still too small to be observed; because of excessive discharging, with and without a coating, the resolution of the crack boundary was not as good as in Fig. 3.) The instantaneous value of e/3~i is found from Eqns. (1) and (15) to be a decreasin9 function of K~. Therefore, the largest e/3ci for the entire set of experiments occurs at t= 0 + under the lowest stress. We find e/3ti (max) __- 10-1.3 min. Referring to Fig. 2, we see (e/3fi) (max) is slightly outside of the range of the generalized power law creep compliance; however, since (c~/3ci)~ K~- 4, the net error in failure time ~I will be very small except possibly at the lowest stress level. Another possible source of error is due to the fact that the velocity does not initially obey Eqn. (1), even if the load could be applied instantaneously and inertia effects are not considered. A small amount of time elapses before propagation actually begins (which is the so-called "fracture initiation time", tl). However, from [-2], the maximum fracture initiation time is approximately (c~/36)(max) and therefore tl is negligible relative to the range of t I in Fig. 1.

5 x i0 -~ in.~

Figure 3. Crack in Solithane 50/50.

Int. Jourll. q/' Fracture, 11 (1975) 549-562 Theory of crack initiation and growth in ciscoelastic media. III 555

Finally, as required by the underlying theory, we find the change in the crack tip velocity over each time increment ~/~i is small for the entire experimental range of behavior; this observation follows from the above estimate of ~/3fi (max) and the fact that the increase in K~ during most of the propagation time is small, according to Eqn. (10).

3. Crack velocity in a long strip under constant deformation

The previously determined creep compliance and fracture properties of Solithane 50/50 will be used in conjunction with Eqn. (1) to predict constant velocity crack propagation in the long strip shown in Fig. 4. Provided a > 1.5b, the stress intensity factor in a strip clamped to rigid grips is [5],

K, = [b(1-v2)]~a (18)

where a is the stress far from the crack which must be applied to the clamps to produce the overall strain e. Although this applied stress and strain could vary with time, the experimental results are for the case in which the strain is held constant and the strip is in a fully relaxed state outside the neighborhood of the tip; thus a=Eoo~/(1- v2), which when substituted into Eqn. (18) yields

K, = E~ (4b/3) ~ e (19) where we have set v=½. Theoretical upper and lower bounds on strain for which stable growth exists in the Soli- thane are obtained from I-2, Eqn. (52)]* ; there results - 1.89 < log e < - 0.81. These bounds are drawn with dashed lines in Fig. 4. 0 I I I I STRIP THICKNESS c =1/32 in. STRIP WIDTH 2b =1 3/8 in. K z = Klg ,.~ ooooo

/ E <~ K I = K le Pt" o 0 ° C 2b~ I-- -2 03 IO ° C [3 20° C Theory TI ~-----2a--~,~ ' / 30 ° C 2b A 40 ° C _L //// I / /I / // -3 50 ° c I I I I -3 -2 -I 0 I CRACK VELOCITY, LOGIo ~, IN./MIN.

Figure 4. Experimental and theoretical crack velocity as a function of the applied gross strain e for Solithane 50/50. Theory: Eqn. (1). Experimental data after Mueller and Knauss [5].

Except at high strains, the predicted velocities are seen to be in very good agreement with the experimental data. (If the stress intensity factor in Eqn. (19) is divided by (1 + 2e), as in Eqn. (14), the predictions at the high strains are likewise found to agree.) Whenever K~/K~g can be neglected in the denominator in Eqn. (1) we find ~ ~g 6, and therefore the theory plots as straight lines in Fig. 4 with slope of ~; the upper strain limit for this behavior is found to be at log e,-~ - 1.2.

* This reference gives the upper and lower bounds on stress intensity factor; the former bound is KIg , Eqn. (5), while the latter one, Kl~, is identical to Eqn. (5) except C,,(oo) replaces C o.

Int. Journ. of Fracture, 11 (1975) 549 562 556 R.A. Schapery

For the range of experimental data in Fig. 4, Eqn. (15), with K~ corrected for finite strain as mentioned above, yields 0.70 A < ~ < 53/k. The largest value of ct/3& after converting to T= 0 ° C, is 0.9 × 10-3 min; this result implies, of course, that the generalized power law in Fig. 2 is valid for the entire experimental range. As noted earlier, the theoretical predictions in Fig. 4 were made using the same material properties (viz. D (t), a T, F and a~/1) as used for the constant-load problem of Fig. 1. Mueller and Knauss [5] give a value of F which is approximately three-times that used here and reported in Knauss' constant-load study [3]. In an even more recent study [6], which was brought to this writer's attention by Dr. Knauss after completing the analysis at hand, a value of F = 0.014 lb./in, was obtained, which is nearly one-half that used here. Experimental factors, batch-to-batch variability, and aging all apparently contribute to these differences in fracture energy [6, 7] ; indeed, F seems to diminish with time in that the smallest value (0.014 lb./in.) is for specimens which were intentionally aged for one year, while the largest reported value is for material tested approximately one month after manufacture. With reference to the present theory, changes in F affect the upper and lower limits, Klg and Kle, respectively, for which stable crack growth exists [2]. Also h vs. e and t: vs. a in the power law region of behavior depend ,,-.... ,,2 ,,,a- t2r~/,,. , thus, changes in F would affect values of a,,I~ which are inferred from data in this region.

4. Comments on the state of strain near the crack tip

Crack velocity, Eqn. (1), is based on the assumption of plane strain in the neighborhood of the crack tip. In contrast, stress intensity factors, Eqns. (6) and (18), are based on the assumption that plane stress exists throughout the sheets (although the former equation is valid for plane strain as well). In this section we shall argue that these two different assumptions can be used together because ~ is small compared to sheet thickness and the thickness is, in turn, small relative to sheet height and crack length. For this purpose, it will be helpful to focus our attention on the material which is inside a small imaginary circle centered at one of the crack tips and with a radius significantly larger than the sheet thickness; this radius should be large enough that the material outside of it is (approximately) in a state of plane stress. Now, imagine the inside material as a free body with tractions acting on the circular boundary whose values are equal to the internal reactions before this material is removed; these tractions may vary slightly through the thickness, but this variation is neglected. Next, suppose that a normal traction distribution, az, is applied to the plane faces of this circular sheet, and is adjusted so as to produce a state of generalized plane strain ; viz., such that the stresses and strains are independent of thickness location, z. It is a simple matter to show, by examination of the equations of linear iscoelasticity, that the in-plane stresses (~.~, ~v, r~y) throughout this small circular shcet are independent of material properties and are identical to those predicted from plane stress analysis of the complete sheet. This result is, of course, contingent upon the tractions which act on the circular boundary being essentially equal to those calculated from plane stress analysis of the full sheet; if the thickness is small compared to the radius of the imaginary circle, this equality will certainly exist. Thus, the stress intensity factors in Eqns. (6) and (18) apply to this modified problem. These conclusions are valid regardless of the magnitude of the thickness strain gz. We can, in fact, let e= take on the value which causes the tOtal load due to az to wmish. This value is small compared to the strains in the neighborhood of the tip (where the size of this neighborhood is on the order of ~), and therefore material points near the tip will be essentially in a state of plane strain. Let us now apply a surface traction distribution equal to -az and thereby return to the original situation. The circular sheet will then fit back into the larger sheet, and the only matter left to argue is that the crack tip neighborhood remains in a state of plane strain

Int. Journ. of Fracture, 11 (1975) 549 562 Theory of crack initiation and growth in viscoelastic media. III 557 with the stress intensity factor unchanged, except possibly in thin layers near the plane surfaces. Recall that the disintegrating material near the crack tip is represented by its reaction tr s over the length ~ on the adjacent linear continuum [1, Figs. 1-3]. The maximum value of trs is denoted by am, and is much larger than the stresses far from the crack tip. Since trm can be reasonably expected to act very close to or at the tip, ~1 = 0, the normal stresses in the linear continuum, a~ and try, are essentially equal to trm at the tip and fall off rapidly with distance from the tip as measured relative to ~ [1] ; indeed, I~trdOGI = IOa~/d~ ~1 = ~ at ~ =0 apart from certain unusual cases [1, Appendix A], and as trm--"~, Cc-~O and

trx = try --~Ki/(27r~l) ½ (20) for ~ close to the tip [1, Appendix B]. The stress G in the generalized plane-strain problem is easily shown to behave in the same way. Thus, the internal stresses resulting from superposition of tractions - tr= decay rapidly with depth from the plate surfaces as measured relative to ct. Thus, we conclude that the neighborhood of the tip remains essentially in a state of plane strain except for surface layers of material whose thickness is on the order of ~. Also, the crack opening displacement is approximately that for plane strain close to the tip (relative to ~) and it approaches that for plane stress at points far from the tip (relative to thickness). Since the classical singular solution is recovered when ~0, Eqn. (20) applies to this case where K I is the stress intensity factor for plane stress. Although K~ itself does not var~ through the thickness of the plate, the largest value of ¢~ for which Eqn. (20) is a good approximation becomes smaller the closer the stresses are to the free surfaces of the sheet.

5. Effects of nonlinear viscoelastic behavior of the failing material

The basic equation governing continuous crack growth [2, Eqn. (47)] and the resulting Eqn. (1) used above were derived without having to introduce any explicit representations of the stress distribution, trl, in the failing material next to the crack tip. In this section we shall examine possible effects of loading history on stress trI and fracture energy F for this continuous growth case. Following [2], we continue to assume that the current crack tip velocity, ci, is essentiallyconstant during the time it takes for the tip to move the distance ~. First, it should be noted that the assumed narrow zone of failing material probably limits us to functions trI which are essentially non-increasing functions of ¢, where ~ is the distance from the crack tip [1]. For if trs increased appreciably with ~ (such as for a metal with significant strain hardening), much of the continuum near the failure zone may be subjected to stresses outside of its range of linear viscoelasticity, thereby invalidating a basic assumption of the theory. This limitation is not believed to be a serious one for most filled and unfilled polymers; certainly the theory will be suspect for some materials, such as natural rubber in which stress- and time-dependent crystallization forms a strong, anisotropic crack tip reinforcement [8]. Although it is not essential in the following arguments, on the basis of these observations we shall assume that a s has its maximum value (with respect to ~) at ~ = 0; as before, we denote this maximum by trm. In order to gain some insight into the nature of the failure stress distribution, trs, and fracture energy, F, consider a rectangular material element centered on the line of crack prolongation and having initial height (in the y-direction) equal to tip thickness d~ [ 1, Fig. 3] and cross-sectional area dA (in the x-z plane); in [1] as well as here we assume d~ ~. As the crack tip approaches, the normal stresses (trx, trr, tr,) acting on the faces of this element approach (a., tr,., try) [1], where

trz = ~o v(t- z) [~3 (trx + try)/&] dr + J~o E(t- Q(&J&)dz. (21)

Int. Journ. qf Fracture, l I (1975)549-562 558 R. A. Schapery

(v (t) and E (t) are Poisson's ratio and modulus, respectively, for a uniaxial tension relaxation test.) We see that a~ generally depends on stress and strain history. However, considering the insentivity of Poisson's ratio to time for many polymers [9], and since the second integral is small compared to the first one in view of the discussions in Section 4, the three-dimensional state of stress as the tip approaches becomes defined by the instantaneous value of a~. It is of interest to recall that there is actually a concentrated shear stress at the tip whose region of influence is on the order of d i [1] ; therefore, strictly speaking, the above observations on the state of stress apply before the crack tip is within the distance d~ from the material element under consideration. The way in which degradation and failure of the material element under discussion eventually occurs can, in principle, be expressed in terms of its constitutive properties and the time history of its boundary displacements. While the element is still in its linear range, we calculate these displacements in terms of the stress history using linear viscoelastic constitutive equations. From [1, Eqn. (17)]:

a~ = u, = (~/n) I~ ~- ½(4 + ~l)-i as` (~) d~ (22) where crz is given by the first integral in Eqn. (21) and d~ < ~ < e; the distance in front of the tip 41 can be written as (tl - t)fi, in which t~ is the time at which the tip reaches the element. Neglecting the stress history outside the distance c~ from the tip and using the previously introduced assumptions [2] of the approximate time-independence of ci and as` during a time increment comparable to e/~, we find easily from linear viscoelastic theory that the time-dependent displacements over d~ < ~ < ~ depend on the current values of ci, e, and an average of as` (4) over 0 < ~ < ~. After the tip passes through the element, its height is given by 2v (~)+ di, where the cusp- shaped opening displacement 2v (4) is predicted in [2, Eqn. (32)]. The distribution of stresses throughout the failure zone (and o-s`(~) in particular) is a function of the time-dependence of this opening displacement and of the cross-sectional area dA at the failure zone-linear continuum interface over the entire failure zone ; however, strains due to changes in dA (~) are negligible compared to those due to v (4) because strains in the adjacent linear medium are small. In addition, the stresses in the failure zone, especially those very close to the tip, may depend on the time-dependence of the element's boundary displacements just before it starts to fail, which displacements were discussed above. Thus, for the element under consideration, we obtain

us`=fntn([as`]~, [v]~, a, ~, ~, d,) (23a) where [ ]8 represents the set of all values of the quantity in square brackets over 0 < ~ < c~. The initial element height di is included since it defines the volume of the material element which participates in the failure process, and the element's displacements just before the tip arrives are affected by the value ofd~. Also, the opening displacement ( x ½) from the analysis in [2, Eqn. (32) or Eqn. (3)] is

v =fntn([as`];, & e, ~). (23b)

When v from Eqn. (23b) is substituted into Eqn. (23a) one obtains an equation from which a s can be found. If, for example, we were to approximate the equation by replacing [v]~ and [-as`]~ with the displacements and stresses at a finite number of points ~1, ~2 .... , ~n, say, a set of n algebraic equations for as` at the n points would result. In turn, Eqn. (23b) yields the solution for v. Thus, we can write

u I = fntn (ci, c~, ~, dl) (24) lnt. Jourl~. ~fFracture, 11 (1975) 549 562 Theory of crack initiation and growth in viscoelastic media. III 559

v = fntn (& ~, ~, di). (25)

The fracture energy F for a given material element is defined fundamentally as [1, Eqn. (44)],

F = [~o"aldv (26) where v,,-v at ~=e. An alternative form is [1, Eqn. (46)],

F = ,[~o(Ov/~?~)crld~ (27) where the differentiation and integration are carried out for one infinitesimal material element. Substituting Eqns. (24) and (25) into Eqn. (27) yields

F =fntn (4, ~, d/). (28)

Recall that ~ is defined such that a I is zero [1], or nearly so [2], when ~ > e. Thus, we set a I = 0 at ~ = e in Eqn. (24) and solve for tip thickness d~,

d, = fntn (~, 4). (29)

In turn, substitution of Eqn. (29) into Eqns. (24) and (28) yields the failure stress distribution,

a I =fntn (4, c~, ~) (30) and fracture energy,

F =fntn (f, e). (31)

The functions a I and F can be substituted into the equations governing crack growth developed in [2]. Specifically, from [2, Eqn. 47],

F = ~ Cv(~,)K? (32) where ~=---2,~/"c~fli and n = fntn (a/d). Also, imposing the condition that the crack tip stress is finite yields [2, Eqn. (2)],

= (re/2) (Ki/o-m I,)2 (33a) where ffmi1 = ~1 o.fr/-~dt/ (33b) and r/= ~/':~. After substituting Eqns. (31 ) and (30) into Eqns. (32) and (33), respectively, the latter ones may be solved for the two unknowns, ~ and & in terms of the current value of the stress intensity factor, K I. It is therefore concluded that the instantaneous crack tip velocity depends on only the instantaneous stress intensity factor; neither the history of this factor nor the history of the stresses in the continuum enter. This last observation has an important practical implication. Suppose that the function KI= h(~i) has been obtained experimentally over the velocity range of interest. Then to predict crack growth due to a stress intensity factor which has different time-dependence than used in the experimental determination of h(fi), one does not have to introduce any new functions or explicit fracture theory. Rather, all one need do is integrate the first order differential equation K~ = h(6), where K~ may possibly depend on a as well as other time-

Int. dourn, of Fracture, 11 (1975) 549 562 560 R. A. Schapery dependent parameters. Of course, use of the previously derived theoretical relationships may greatly facilitate the evaluation of h(~i) from experimental data (especially if F and i,,I 1 are constant) and simplify the determination of its dependence on such important factors as chemical aging and temperature. Incorporation of these factors will be discussed shortly. For the elastomer studied in this paper, it was found that F and ffmll can indeed be taken as constants; the theory then provided a simple analytical representation of the data in terms of creep compliance. Finally, it should be clear that if only the function h(~i) is known, it is not possible to determine the separate fracture properties, F and am I1, appearing in the theory. For example, we could arbitrarily assume am/1 is constant and select any constant value for a,~ and still adjust the theory, Eqn. (32), so as to fit the function h(d); this fitting would be accomplished by absorbing as much velocity dependence in F as necessary by setting KI = h and combining Eqns. (32) and (33) so as to find F from the equation,

F = ~C~(~,)h 2 (34) Also, e/it = ½re (hZ/gw~, 12). (35)

Unless imI 1 is actually constant and its value correctly chosen, F as given by Eqn. (34) cannot be interpreted as true fracture energy. On the other hand, if both h and e are determined experimentally as functions of fi, Eqn. (34) yields the actual fracture energy after substituting these two functions. Then Eqn. (35) can be used to find if,nit in terms of ft. We are thus led to an important conclusion concerning fracture experiments. If it is desired to use these experiments to establish the rate dependence of F and imI1, if indeed any exists, one must measure both crack velocity and failure zone length as functions of K1. It is believed the above discussion provides a general framework within which one could theoretically develop explicit expressions for i z and F, given some constitutive model for the failing material. Indeed, the representation of o-i, Eqn. (23a), and governing equation for crack growth, Eqn. (32), contain as a special case the fracture theory of Barenblatt et al. [10], which is based on the application of rate process theory to the failure zone in an otherwise linearly elastic material. Their special form of a I is derived from a rate equation for bond failure which is probably over-simplified [8] and also predicts the physically incorrect result of a non-zero tip velocity for a vanishingly small stress intensity factor; while such behavior could exist in non-crosslinked polymers above their glass-transition temperaturc. these materials do not satisfy the elasticity assumption made by Barenblatt et al. [10] when ~'t~0.

6. Effect of environmental changes and aging

The creep compliance C v and fracture properties F and a,,I 1 which appear in the basic fracture Eqns. (32) and (33) can be expected, in general, to vary with the environment through, for example, dependence on temperature and relative humidity. These properties may also vary directly with age due to such phenomena as post cure reactions and oxidation. As long as the resulting changes in Cv, F, and im I1 are relatively small in the generic time interval ct/& we can incorporate this time variation in properties directly into Eqns. (32) and (33), as well as Eqn. (1), without having to alter the form of the equations; integration of the appropriate equation, taking into account the time-varying properties, will yield instantaneous crack size as a function of environment and aging histories. A study of the possible ways in which the fracture properties may change is beyond the scope of this paper. But we will comment briefly on temperature dependence of the creep compliance as a large amount of experimental data exists on this property. For many

Int. Journ. qfFracture, 11 (1975) 549 562 Theory of crack initiation amt growth in viscoelastic media. III 561 crystalline and amorphous polymers, below and above the glass-transition temperature, To, temperature dependence of the compliance is characterized under isothermal conditions by the simple equation [9],

o (t, T) = Oo + AD (O/a (36) where : D O= D O(T) = initial compliance; AD (0 =- (D- Do)a G ; ~ = t/ar = reduced time ; a~ = ao (T) = compliance factor and ar = at(T) = time-scale factor. Thus, three functions of temperature (Do, aG, at) plus one function of reduced-time (AD(~)) define the compliance D(t, T). It should be added that most of the data used to establish the form of Eqn. (36) are from uniaxial tensile tests ; however, this form should be quite accurate for Cv as well since CV-~ 4(1- v2)D, according to the quasi-elastic method [1, Eqn. (38)], and the factor (1- v2) varies only slightly with time compared to D. Now, replace D by C in Eqn. (36) and assume the common power law form for AC(~),

AC( 0 = CRY" (37) where CR and m are constants. Upon comparing Eqns. (2) and (37) we see that C2 is a function of temperature; viz.,

C2 = CR/a~rao. (38)

This coefficient, as well as C O(T), can be substituted directly into Eqn. (1) in order to predict crack growth under different constant or transient temperatures. When amorphous polymers are in their glass-to-rubber transition range one can neglect the temperature dependence of a~ and C o, and therefore write

Cv = Cv(¢) (39) where ~ = t/a r. Any material whose temperature dependence enters entirely through the time scale, as in Eqn. (39), is called "thermorheologically simple" [9].

7. Conclusions

The theory of crack propagation developed in [1 and 2] has been successfully applied to an elastomer under the assumption that the fracture properties F and o',,I1 are constant. Some influence of finite far-field strain on crack velocity was noted, in that the actual crack velocity falls below the linear theory prediction as the strain is increased above 5 ~. A simple ad hoc modification for the effect of finite strain was used which brought the theoretical and experimental results together at all strain levels. The underlying fracture theory [1, 2] is not limited to constant fracture properties in that the behavior of the failing material at the crack tip may be highly nonlinear and viscoelastic. However, even with such complex behavior, the theory was shown to predict that the instantaneous crack velocity ~ depends on the instantaneous stress intensity factor K~ and not on past values of this factor or other parameters related to the stress history; in establish- ing this result, it was assumed that the velocity is essentially constant during the time needed for the crack to grow an amount equal to the length of the small failure zone, ~. Consequently, once the function relating ti and K~ has been established from laboratory tests, one can use this same function to predict crack growth in geometries and under loading histories which are different from those used in the experiments.

Part IV of this series of four papers will deal with the analysis of crack propagation under

lnt. Journ. of Fracture, 11 (1975) 549-562 562 R. A. Schapery complex loading histories, especially for composite materials.

Acknowledgment

The author is indebted to Professor J. S. Ham for elucidating molecular aspects of polymer fracture and to Mr. N. Conrad for providing Fig. 3. This work was sponsored by the Office of Naval Research under Contract No. N00014- 68-A-0308-0003 with Texas A&M University.

REFERENCES

[1] R. A. Schapery, Int. J. Fracture, 11 (1975) 141 159. [2] R. A. Schapery, Int. J. Fracture, 11 (1975) 369-388. [3] W. G. Knauss, lnt. J. Fracture Mechanics, 6 (1970) 7 20. [4] M. K. Davis, Electron Paramagnetic Resonance Studies of Molecular Fracture in Oriented Polymers, Ph.D. Dissertation, Texas A&M Univ. (1970). [5] H. K. Mueller and W. G. Knauss, J. Applied Mechanics, 38 Series E (1971) 483-488. [6] W. G. Knauss, On the Steady Propagation of a Crack in a Viscoelastic Sheet: Experiments and Analysis, in De/'ormation and Fraeture of Hi~jh Polynwrs, H. H. Kausch, J. A. Hassell and R. Jaffee, Eds.. Plenum Press (1974) 501 541. [7] W. G. Knauss, (private communication). [8] W. G. Knauss, Applied Mechanics Reviews, 26 (1973) 1-17. [9] R. A. Schapery, Viscoelastic Behavior and Analysis of Composite Materials, in Composite Materials, 2, G. P. Sendeckyj Ed., Academic Press (1974) 85-168. [ 10] G. I. Barenblatt, V. M. Entov and R. L. Salganik, Some Problems of the Kinetics of Crack Propagation, in Inelastic Behavior ~f Solids, Mat. Sci. Eng. Series, M. F. Kanninen, W. F. Adler, A. R. Rosenfield and R. 1. Jaffee, Eds., McGraw-Hill (1970) 559-585.

RI~SUMI~ On utilise la th6orie de croissance de la fissure d6velopp6e dans les parties 1 et I1 pour r6duire la vitesse de fissuration et le temps de rupture d'un 61astom6re soumis ~ des 6tats de contraintes uniaxiales et biaxiales. Le m6moire comporte des consid6rations sur l'effet de la dimension des 6prouvettes sur le temps de rupture au d6part de petites fissures, la d6termination exp6rimentale des propri6t6s de la rupture, l'61ude de I'effct du niveau de d6formation sur la propagation des fissures, et de la validit6 d'hypoth6ses d'6tat plan dc d6formation. La seconde partie de l'article comporte l'examen de l'effet d'une visco61asticit6 non lin6aire dans la matit)re qui se rompt il la pointe d'mlc li~surc, laquelle est par ailleurs sise dans un milieu caract6Hse par tree visco- 61asticite lin6aire. On consid~re 6galement la mani~re dont le comportement ~ la rupture d6pend du vieillissement et des changements de l'environnement.

Int. Journ. ofFracture, ll (1975)549 562