Viscoelastic Creep Crack Growth: a Review of Fracture Mechanical Analyses

Mechanics of Time-Dependent Materials 1: 241–268, 1998. 241

c 1998 Kluwer Academic Publishers. Printed in the Netherlands.

Viscoelastic Creep Crack Growth: A Review of Fracture Mechanical Analyses

W. BRADLEY1, W.J. CANTWELL2 and H.H. KAUSCH2 1Department of Mechanical Engineering, Texas A&M University, College Station, TX, U.S.A.; 2Materials Science Department, EPFL, DMX-LP, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland

(Received 8 February 1997; accepted in revised form 2 October 1997)

Abstract. The study of time dependent crack growth in polymers using a fracture mechanics approach has been reviewed. The time dependence of crack growth in polymers is seen to be the result of the viscoelastic deformation in the process zone, which causes the supply of energy to drive the crack to occur over time rather than instantaneously, as it does in metals. Additional time dependence in the crack growth process can be due to process zone behavior, where both the ﬂow stress and the critical crack tip opening displacement may be dependent on the crack growth rate. Instability leading to slip-stick crack growth has been seen to be the consequence of a decrease in the critical energy release rate with increasing crack growth rate due to adiabatic heating causing a reduction in the process zone ﬂow stress, a decrease in the crack tip opening displacement due to a ductile to brittle transition at higher crack growth rates, or an increase in the rate of fracture work due to more rapid viscoelastic deformation. Finally, various techniques to experimentally characterize the crack growth rate as a function of stress intensity have been critiqued. Key words: fracture, fracture mechanics, polymers, viscoelastic crack growth

1. Introduction The subject of slow, stable crack growth in polymeric materials is growing in importance. The use of plastic pipe for natural gas and water distribution and for transport of raw sewage has become quite common during the past 15 years. Such applications are made with the expectation of service lifetimes of at least 30 years. However, such lifetimes under service conditions of constant stress, due to ground loading and internal pressure, will require polymeric materials with a high resistance to slow, stable crack growth, sometimes called static fatigue. In metals at above half of their melting temperature (expressed in degrees K), time dependent plastic deformation at the tip of cracks can result in stable, time dependent crack growth, usually referred to as creep crack growth. For most metals, neither creep deformation nor creep crack growth is significant at ambient temperature. On the other hand, many polymers experience considerable creep at room temperature, especially for long term service. This is a consequence of the fact that ambient temperature is a significant fraction of the glass transition temperature (again, expressed in degrees of absolute temperature) for most polymeric materials. This creep, which results from the viscoelastic character of polymeric 242 W. BRADLEY ET AL. materials, can also give creep crack growth, or more appropriately, viscoelastic creep crack growth. The usual mechanical properties that are measured in materials characterization such as loss modulus, storage modulus, tensile yield strength and ultimate elongation may not be very useful in predicting a polymer’s resistance to viscoelastic crack growth. Thus, new approaches are being developed to better predict a polymer’s resistance to viscoelastic creep crack growth for engineering applications and materials development The susceptibility of polymers to viscoelastic creep crack growth has resulted in some rather expensive lessons in service. For example, it has been common in the United States for polyethylene pipe used for city distribution of natural gas in a metropolitan area to be pinched clamped to stop gas flow and allow repairs to be made. This practice makes it possible to install natural gas distribution systems in metropolitan areas with a minimum of valving compared to distribution systems made with cast iron pipe. It has been assumed that the polyethylene can be pinched clamped without introducing any damage to the pipe. However, it has been found that pinch clamping may introduce damage in the form of surface cracks at the inside diameter of polyethylene pipe that will subsequently have viscoelastic creep crack growth over a period of 5–10 years before causing leaks (Jones and Bradley, 1987). The use of plastic pipe in connection with metal fittings with sharp edges to connect city water service to residential customers resulted in the initiation and propagation of viscoelastic creep crack growth, causing leaks to develop. However, the viscoelastic creep crack growth that developed where the plastic pipe was bent around the sharp corners of the fittings took several years to produce leaks. By the time the problem was identified, the improperly designed fitting had been widely used and several thousand such applications subsequently leaked, resulting in enormous replacement costs. Metal crimps used in conjunction with plastic fittings for plastic pipe for water service in residential homes have been found to produce viscoelastic creep crack growth leading to leaks, but again only after several years of service. Thus, the problem did not “surface” until the use of the crimped fittings had become widespread. Buried PVC pipe has been found to be susceptible to viscoelastic creep crack growth when the pipe is of inferior quality and/or the installation leads to excessive ground loading (U.S. Department of Transportation, Research and Special Programs Administration, 1994). However, failures in service may not occur for several years and conventional mechanical properties tests such as tensile or internal pressure tests do not always identify pipe that is susceptible to viscoelastic creep crack growth (Jones and Bradley, 1993; Richard et al., 1959; Kausch von Schmelig and Niklas, 1963). It is clear that applications of polymers involving long term loading resulting in either constant or intermittent stresses can cause viscoelastic creep crack growth. Thus, a better understanding of how to evaluate a materials resistance to viscoelastic creep crack growth and how to produce polymeric materials with a high degree of resistance to such cracking is essential to the successful retention of some existing markets and expansion into some new markets for polymeric materials. VISCOELASTIC CREEP CRACK GROWTH 243

The application of fracture mechanics to viscoelastic media goes back to the middle 1960s (Williams, 1965; Vincent and Gothan, 1966; Retting and Kolloid, 1966). Extending Grifﬁth’s work to linearly viscoelastic materials Williams (1965) found that the crack initiation criterion depends on the loading history. Vincent and

Gotham (1966) and Retting and Kolloid (1966) were among the first to note that _ the work of fracture in polymers, 2, was a function of the crack growth rate, a. Kostrov and Nikitin (1970), following the lead of Dugdale (1960) and Barenblatt (1962) for time independent materials, were the first to note that a failure zone needs to be introduced ahead of the crack if the time dependence of the fracture process is to be properly modeled. Various approaches to model the viscoelastic fracture in the process zone have been taken, depending on the assumed geometry of the process zone ahead of the crack tip and the constitutive properties for the material in the process zone. If a process zone, or failure zone, is assumed to be finite, then the constitutive properties of the material in the process zone need to be specified (Wnuk and Knauss, 1970; Knauss 1993). Alternatively, the failing material in the process zone can be rep- resented by a stress-displacement relationship for a zone of zero thickness as was done by Knauss (1970). Knauss (1970) solved the Griffith problem for an assumed finite thickness process zone assuming a linear, viscoelastic constitutive behavior for the material in the process zone and used his model to analyze center-cracked panels of Solithane 50/50. Subsequently, Mueller and Knauss (1971) studied crack propagation in a linearly viscoelastic strip. Knauss (1974) also modeled the steady- state crack propagation in a viscoelastic sheet. Thus, by the mid-seventies, Knauss and co-workers had established a framework for the linear viscoelastic fracture of polymers through the application of cohesive crack models. At the same time Knauss and co-workers were developing their theories of viscoelastic fracture mechanics, Williams and Marshall (1972–1975) put forth the idea that fracture mechanics for viscoelastic materials could be treated with an approach that is similar to the traditional fracture mechanics developed for metals by simply replacing the time independent values for modulus and flow stress with equivalent viscoelastic relaxation moduli and flow stresses. The time scale for fracture was estimated by using a Dugdale (1960) calculation of the process zone size, which was divided by the crack growth rate. Concurrent with the work of Knauss and Williams and Marshall but indepen- dently, Schapery (1975a, 1975b) demonstrated that indeed viscoelastic creep crack growth could be described using the approach developed for metals by Barenblatt (1962) with the replacement of the elastic modulus with a viscoelastic modulus. He further indicated how to determine the appropriate time at which this viscoelastic modulus should be evaluated using linear viscoelasticity. Finally, by assuming a power law dependence for the viscoelastic constitutive relationship, he was able to obtain a simple analytical relationship between the steady-state crack growth rate and the applied stress intensity. 244 W. BRADLEY ET AL.

These theoretical developments, along with experimental support, have been addressed in recent books by Williams (1987), Kausch (1987), Kinloch and Young (1983), Atkins and Mai (1985), and Anderson (1991). However, there is still a general lack of appreciation for the signiﬁcance of viscoelastic creep crack growth in polymeric materials development and characterization. The variety of relationships found in the literature to relate viscoelastic creep crack growth rate to fundamental material properties can further confuse the person unfamiliar with the ﬁeld. The role of viscoelastic creep crack growth in the newly emerging standards to measure fracture toughness in polymeric materials needs to be addressed. Finally, the role of viscoelastic creep crack growth in slip-stick crack growth and the effect of

adiabatic heating at the tip of a growing crack need to be further clariﬁed, especially G for systems where the critical energy release rate, 1c , increases with increasing temperature. More generally, the effect of crack growth rate and temperature on the fracture process of polymeric materials cannot be understood outside the context of an understanding of viscoelastic creep crack growth. It is to these issues that we will address our attention in this paper. In the next section, we will present an abbreviated and simpliﬁed treatment of Schapery’s theory of viscoelastic fracture mechanics (Schapery, 1975a, 1975b). In Section 3, we will show how the variety of relationships found in the literature based primarily on the work of Williams and co-workers (Marshall and Williams, 1973; Williams, 1972; Williams and Marshall, 1975) can be related to Schapery’s

results, noting especially the difference in assumptions made and how they affect

a_ ' K the exponent, p, in the general expression that .InSection4,wewillexplore the relationship between viscoelastic creep crack growth and unstable crack growth, including slip-stick crack growth, and evaluate when adiabatic conditions at the

crack tp are responsible for unstable crack growth. In Section 5 we will discuss the role of viscoelastic fracture mechanics in fracture toughness characterization of polymeric materials, including a brief discussion of the new American Society for Testing and Materials (ASTM) and the European Society for Integrity of Structures (ESIS) standards on fracture toughness measurements in polymeric materials. We will also consider experimental techniques for conveniently determining crack growth rates as a function of stress intensity in Section 6. In Section 7, we will illustrate the use of viscoleastic fracture mechanics in the prediction of service performance in polymeric materials and indicate the limitations in such applications at present.

2. Schapery’s Theory of Viscoelastic Fracture Mechanics Schapery’s model divides a material with a crack and subjected to loading into two regions (as seen in Figure 1), a process zone at the crack tip which has experienced highly nonlinear viscoelastic (and possibly plastic) deformation and a so-called far ﬁeld region where the deformation may be appropriately characterized as linear viscoelastic, as seen in Figure 1. He then proceeds to show that the stress VISCOELASTIC CREEP CRACK GROWTH 245

Figure 1. Stress around tip of a growing crack in Schapery model (14). distribution in the far ﬁeld, as calculated by Barenblatt (1962) for elastic materials, also describes the stress distribution for viscoelastic materials. He further argues following Barenblatt that the stresses at the tip of the process zone will be ﬁnite

(i.e., all singularities must cancel) if and only if

Z 2 1=2

1=2

K = r =r r;

1 f d (1)

where f is the stress in the process zone as a function of distance, , from the tip of

the crack (see Figure 1) and is the length of the process zone. As in the Dugdale analysis, the size of the process zone in the Barenblatt analysis is determined by

the stresses in the process zone. For the case of uniform stresses in the process zone

= y (which is a key assumption in the Dugdale analysis), where f the Barenblatt

analysis gives the Dugdale result as a special case; namely, !

2 K

= ;

(2)

8 y

K where 1 is the applied stress intensity and y is the yield stress in a non-strain hardening material. Schapery next proves that the displacements at the boundary of the process zone in viscoelastic materials may be determined to a good approximation by using the elastic solution, but replacing the elastic compliance with a viscoelastic creep compliance. He is then able to integrate the work required to fracture a small strip of material as it passes through the process zone (or more appropriately as the crack tip and process zone move past this strip of material) and is gradually stretched to failure using the relationship

246 W. BRADLEY ET AL.

vt; r

r = r;

2 f d (3) r

t; r where v is the displacement at the boundary of the process zone and is the work per unit area of fracture surface created. It should be noted that Equation (3) is

essentially (stress) (displacement) calculated by integrating over the process zone boundary to give work per unit area. The result of the integration of Equation (3), which requires some very clever manipulation to get it into a form that allows use of Equation (1), is

=C tK ;

2 v 1 (4)

t C

where v is the viscoelastic creep compliance, which is a function of the effective ~ time, t. Equation (4) is seen to be the familiar expression

G =

1c (5)

0 E

when it is recognized that

2 1c (6)

0 2

C t E = E= v

and v replaces the equivalent elastic constants, 1(for plane

strain) and E for plane stress.

The use of Equation (4) requires the speciﬁcation of the viscoelastic constitutive ~ relationship and the effective time, t, to be used in the viscoelastic compliance. The

loading of a strip of material as it passes through the process zone occurs gradually _ over a time period that can be estimated as =a, assuming the crack growth rate and the process zone size remain approximately constant over this time interval. However, the use of the creep compliance assumes displacements due to the full load being applied initially. Schapery uses the convolution integral to calculate

an effective time for viscoelastic deformation to occur in the process zone, and

determines that the effective time is proportional to the time given by = ,or

= d

t (7)

when the creep compliance is given by

~ ~

C t=C 1t (8a)

~ ~

t=C +C t :

C 01 (8b) n The proportionality constant, d, is a function of the exponent, , in the creep compliance relationships given in Equation (8), but for a wide range of exponents VISCOELASTIC CREEP CRACK GROWTH 247 it is found to be approximately 1/3. If we consider the special case of a uniform stress in the process zone (which is the Dugdale assumption) and assume the relationship for the viscoelastic creep compliance given in Equation (8a), then we can combine Equations (2), (4–7) and (8a) and solve for crack growth rate as a

function of stress intensity K1 as follows:

1=n

C d

+ =n

1 n 1 1

a = K :

_ 1 (9) G

8 1c If we let the constitutive relationship for the viscoelastic creep compliance assume the more general form of Equation (8b), then again combining Equations (2), (4),

(5), (6), (7) and (8b) gives the result

1=n

d C

+ =n

1 n 1 1

a = K ;

_ 1 (10)

G G

8 1c 10

= K C C where G10 1 0, which is the elastic energy release rate based on 0 alone,

which is the elastic component of the general viscoelastic creep compliance rela- = tionship. Note that for C0 0, Equation (10) simpliﬁes to Equation (9). The simple power-law relationship for steady-state crack growth as a function of stress intensity obtained by Schapery depends on the power law representation of the viscoelastic constitutive relationships assumed by Schapery in Equations (8a) and (8b). There may be some time scales over which the simple power-law relationship is not valid, particularly for polymers with a more narrow relaxation spectra. Over these time scales, the approach of Schapery is still valid but a simple, analytical result may not be possible. This limitation might be especially true for prediction of long-time service failures where the time scale is long compared to the usual measurement time for viscoelastic creep, unless of course, one develops a master curve using time-temperature superposition. The same limitation applies to the results of Williams and Marshall to be described in the next section. To ensure a physical appreciation for the difference in elastic and viscoelastic crack growth, it is instructive to recall the Grifﬁth relationship for a through crack

of length “2a” in a wide plate to the nominal applied stress required to give fracture,

a :

0 E

a (11) a Recall that this relationship was derived by Grifﬁth by assuming that the crack growth would occur when the elastic energy released per unit area of crack growth was greater than the required work of fracture per unit area of fractured surface created. The only difference in a viscoelastic material is that the energy release rate

per unit area of crack growth is time dependent, either partially as in Equation (8b) or 0

entirely, as in Equation (8a). Substituting for the elastic modulus E in Equation (11)

E = C t = G c the appropriate viscoelastic compliance relationship v , letting 2 1,

248 W. BRADLEY ET AL.

= a and recalling that for a through crack in a wide plate, K 1 , Equation (11)

can be rearranged to give G

2 1c

;

K1 (12)

C t v which is in fact identical to Equation (4), as derived by Schapery. Viewed globally, this derivation beginning with the Grifﬁth relationship emphasizes the physically intuitive idea that the primary difference between crack growth in viscoelastic

materials and in an elastic material is that at least some of the elastic energy (that

~ t related to C1 ) is released gradually as a function of time, giving time dependent crack growth rather than instantaneous crack growth. Viewed locally, which is how Knauss and Schapery approach the problem, the work done on the crack tip process zone is time dependent because of the viscoelastic response of the material in the process zone to the applied stress. In their respective analyses, the stresses are shown to be essentially the same as for the elastic case through the correspondence principle.

Equations (9) and (10) based on the work of Schapery give an explicit relation- _

ship between the applied stress intensity, K1, and the resultant crack growth rate, a,

G G

c c y if and only if 1 and y are independent of a.If 1 and are not dependent on

a_ , then the crack growth rate is controlled entirely by the viscoelastic deformation

of the resin in the process zone. However, it is often the case that additional rate G effects are observed due to the rate dependence of the work of fracture, 1c .The

rate dependence here may be associated with the yield (or ﬂow stress) and/or with

G G c

the work of fracture, 1c . For example, at higher rates, the ﬂow stress, and thus, 1

might increase (if the critical crack tip opening displacement c remains constant)

or the material might experience a transition from ductile to brittle fracture, with a

G c concurrent drop in c , and thus, 1 . The effect of rate dependence in the process zone will be developed more fully in the section on unstable crack growth. In the next section, we will compare the relationships for viscoelastic fracture mechanics developed by Schapery with those developed by Williams and Marshall (Marshall and Williams, 1973; Wiliams, 1972; Williams and Marshall, 1975) and popularized in several books (Williams, 1987; Kausch, 1987; Kinloch and Young, 1983; Atkins and Mai, 1985; Anderson, 1991).

3. Williams and Marshall’s Theory of Viscoelastic Fracture Mechanics Williams and Marshall’s theory with the standard relationship for linear elastic fracture mechanics, is equivalent to the Grifﬁth relationship and may be derived from the First Law of Thermodynamics; namely 2

G = G = c c; y

1 1c (13)

E t G

where the work of fracture, 1c , is expressed in terms of the critical crack tip

y opening displacement, c , and the yield strength (or average ﬂow stress for VISCOELASTIC CREEP CRACK GROWTH 249

materials that strain harden). The constant “c” in Equation (13) is the ratio of the principal normal stress at the crack tip required to give shear yielding (where the state of stress may be triaxial tension) and the yield strength one would measure in a

tensile test (where the stress state is uniaxial tension). The value of “ c” is generally found to be between 1.0 (plane stress) and 2.0 (plane strain) and is often referred

to as a constraint factor for the yield stress. For specimens that would craze rather

c c than shear yield, the term y would be replaced by , which is the average stress supported by the material in the craze. Since most of the work reported

in the literature for viscoelastic creep crack growth has been on thermoplastics

that craze, we will use c hereafter. For comparison with the work of Schapery, =

Equtions (9) and (10) were derived for the case of plane stress, with c 1. Thus,

for the assumption of crazing, one needs only replace the yield strength, y ,in

these equations by the craze stress, c . Williams then assumes that “viscoelasticity” enters the fracture mechanics analysis through the viscoelastic behavior of the polymer at the crack tip and potentially through the craze stress level in the process zone. He suggests that these can be modeled by replacing the elastic modulus in Equation (13) by a relaxation modulus, replacing the time independent craze stress by a time dependent craze

stress (or shear yield stress) and assuming that the critical crack tip opening dis-

placement, c , is constant over a wide range of crack growth rates. In mathematical

form, he assumes that

= E t

E 0 (14a)

= t ;

c 0 (14b) where the effective time is considered to be an appropriate time to evaluate the far ﬁeld modulus and also the time scale for crazing or shear yielding in the process zone, as previously discussed. In their earlier papers (Marshall and Williams, 1973; Williams, 1972; Williams and Marshall, 1975) Williams and Marshall assume that the effective time in Equation (12) can be approximated by the plastic zone size

divided by the crack growth rate, or

= =a;_

t (15)

= : which is identical to Schapery’s result (see Equation (7)) if d 1 0. In his book

(Williams, 1987), however, Williams has included d explicitly. Finally, Williams uses the Dugdale model to calculate the plastic zone size (see Equation (2)). Combining Equations (2), (13), (14) and (15), and emphasizing the very important assumption that the crack tip opening displacement is not a function of crack growth

rate, Williams and Marshall (1975) are able to show that

m+n=m+n

21 K

_ :

a = (16)

+ n=m+n

1 2

m=m+n

8 1 2

E c 0 0

250 W. BRADLEY ET AL.

G =

c c

For the special case where the work of fracture, 1c is assumed to be

independent of a, Williams and Marshall’s previous assumption of c constant

leads to the assumption that c must also be constant, which in turn requires that = m 0 in Equations (14) and (16). Equation (16) then reduces to a form quite

similar to Equation (9) from Schapery’s work; namely,

+ =n

21 1 K

: a = _ (17)

1=n 2

G E

8 c

1 0 c

Williams and Marshall (1975) implicitly assume in Equation (15) that the “d”in

d Schapery’s result is unity (see Equation (7)) whereas Schapery calculated =

1=3. They also use a relaxation modulus (Equation (14)) where Schapery uses a creep compliance (Equations (8a) and (8b)). Williams (1987) has shown that the difference in the creep modulus and the inverse of the stress relaxation modulus are small for longer times since

n n

~ ~ ~

t 'Ct = : C t=C +C

v 01 (18)

n t

C t However, for short times where C0 and 1 are of the same order of magnitude, Equations (9) and (17) will both be inaccurate and Equation (10) would be expected to give more satisfactory results, which is supported by the recent results of Pavan (1997).

Another common assumption made in the literature in addressing viscoelastic "

creep crack growth in polymers is that the yield strain, y , is constant. This is

= n equivalent to requiring that m in Equation (14). Effectively, this says that the far ﬁeld relaxation of the modulus has the same time dependence as the time

dependence of the craze or shear yield stress in the process zone. Applying the

= n assumption m in Equation (16) gives the result that

1=n

K E

c 0

a = :

_ (19) n

1=2

d E

8 c 0 0 0

=E = E

Equation (19) may be algebraically rearranged into the form, noting 0 0 y

= n =n

2 1 2 1

a_ = E " K ; c

0 y (20) "

8 y which is in fact equation (6.24) in Atkins and Mai (1985) and equation (5.19)

in Kinloch and Young (1983). However, recent work by Bradley et al. (1998) on m

polyethylene reports values of 0.12 and 0.067 for n and respectively, suggesting

= m that n is not generally a good assumption. A comparison of the exponents in Equations (20) and (9) indicates that the

sensitivity of the crack growth rate, a_ , to the applied stress intensity has been

VISCOELASTIC CREEP CRACK GROWTH 251 _ reduced by more than one half (or the “sensitivity” of K 1 to a has more than

doubled), since

+ =n =n

21 1 1

_ _

K K a 1 goes to a 1 (21a)

or conversely

n+ n

n=2 1

a_ K a_ : K1 goes to 1 (21b) The consequence of a rate dependence of the craze or shear yield stress in the process zone is that it gives an increased resistance to crack growth at higher

rates (in the absence of adiabatic heating), adding to the rate dependence due to a

far ﬁeld viscoelastic behavior and giving a much larger exponent on _ , as seen

= n m = in Equation (21b). Essentially, the assumption that m (rather than 0, as is implicit in Equations (9) and (10)) implies through Equation (14) that the yield strength increases at higher crack growth rates where less time is available

for thermally assisted yielding to occur. This increase in yield strength with crack

growth rate would result in an increase in the c (or ) term in Equations (9) and

G G =

c c c

(10) as well as increasing the work of fracture, 1c ,since 1 . It is clear

G y

from Equations (9) and (10) that increasing 1c and will increase the required

a_ K K1 for a given crack growth rate or reduce at a given applied 1, which accounts for the change in the rate dependence alluded to in Equations (21a) and (21b).

4. Unstable Crack Growth in Viscoelastic Materials In this section we wish to consider the effects of viscoelasticity in the process zone and rate effects on yield strength and critical crack tip opening displacement in the process zone on the stability of crack growth. The considerable effect of specimen geometry and loading conditions (constant load or constant loading rate or constant displacement rate) will not be addressed here since they have been treated extensively elsewhere (Williams, 1987; Kinloch and Young, 1983;

Anderson, 1991). This will allow us to focus our attention on the unique effects that

c process zone viscoelasticity and rate dependence of y and have on crack growth stability. We will ﬁrst consider unstable crack growth for isothermal conditions at the crack tip. Then this treatment will be generalized to include the effect of adiabatic heating at the crack tip on crack growth stability.

4.1. UNSTABLE CRACK GROWTH UNDER ISOTHERMAL CONDITIONS

The general criterion for unstable crack growth in elastic materials is

G G

d 1 d1c

a; a;_ T a; a;_ T :

(22) a da d Equation (22) indicates that unstable crack growth occurs when the change in the driving force for incremental crack growth (left-hand side of Equation (22)) 252 W. BRADLEY ET AL. exceeds (is either more positive or less negative than) the change in the material’s

resistance to crack growth for the same increment of crack growth (right-hand = side of Equation (22)). For the isothermal case (dT 0), the total derivatives in

Equation (22) can be written more explicitly as

@G @G a_ @G @G a_ c

11d1c 1d

> + :

+ (23)

@a_ a @a @a_ a

@a d d

= a The term dG1 d in Equation (23) describes the change in energy release rate with crack growth, and depends on loading conditions (load versus displacement control, for example), specimen geometry, and system stiffness. Displacement control will usually make this term negative, favoring stable crack growth, while a soft testing machine will make it less negative (and thus make crack growth less stable). Load control will make this term positive, favoring unstable crack growth. For our analysis, we will consider the specially designed width-tapered

double cantilevered beam specimen or the double torsion specimen loaded in

= a displacement control, for which dG1 d is zero when a constant displacement rate results in stable crack growth at a constant load (Williams, 1987; Kinloch and Young, 1983; Anderson, 1991). This will allow us to focus on the inﬂuence of viscoelastic material behavior on crack growth stability. However, this analysis can be easily adjusted to take into account other specimen geometries and testing

conditions.

G = a R

The term d 1c d is described by the -curve behavior of the material and is associated with the increasing resistance to crack growth due to the presence of a plastic wake (which in polymers is primarily unrecovered viscoelastic deformation) that is present behind a growing crack. After some amount of crack growth (which

depends on the constitutive properties of the material at a given temperature), the

G = a wake achieves a steady-state value and d 1c d approaches zero.

Finally, for the unstable crack growth condition as deﬁned by Equation (23) we

_ a> have to deﬁne that da=d 0, thus we may divide the remaining two terms by this factor without changing the sign of the inequality, which gives the condition for

unstable crack growth to be

G G

d 1 d 1c :

> (24)

a a_

d _ d

= a_ The term dG1 d in Equation (24) describes the inﬂuence of crack growth rate on the energy release rate, which is determined by the viscoelastic behavior of the polymer. At higher crack growth rates, the time for a viscoelastic deformation

in the process zone is reduced, and thus, this term would normally be negative,

= G a_ which favors stable crack growth. The term d 1c d in Equation (24) describes the dependence of the work of fracture on the crack growth rate, which can include both

changes in the craze stress (or yield stress) and the crack tip opening displacement.

G = a_ G c Thus, d 1c d describes the rate dependence of the work of fracture, 1 ,inthe process zone. At higher crack growth rates, the yield stress will typically increase VISCOELASTIC CREEP CRACK GROWTH 253 while the crack tip opening displacement may remain approximately constant or

decrease dramatically when a change in the fracture process from ductile to brittle

G = a_ behavior is observed. Thus, d 1c d may be either positive or negative for a given material, depending on the crack growth rate and temperature. It is worth noting that Williams (1987), Kinloch and Young (1983), and Atkins

and Mai (1985) incorporate the process zone viscoelasticity and rate dependence

G c of the work of fracture, 1c , into their respective K term, which obscures the physically important distinction of rate dependence in the driving force for crack growth (through viscoelasticity) from rate dependence in the work of fracture through changes in the crack tip ﬂow stress and/or the crack tip strain to failure,

as reﬂected in the critical crack tip opening displacement. Furthermore, the use of G

c as is commonly done is somewhat misleading, implying as it does some unique

work per unit area of fracture surface created. Equations (9) and (10) imply that

K K

a whole range of 1 values, often called c , may be observed at different crack

=G growth rates, even though the work of fracture 2 1c might be quite constant.

Careful examination of Equations (9), (10), (16), and (17) suggests the use of K 1 to

describe the driving force that results from the applied load and which corresponds

G K c to an associated work of fracture of 1c or 2 .Theterm implies some critical

K value at which crack growth begins, rather than reﬂecting the actual situation, K which is crack growth at any value of applied K1, the magnitude of 1 determining only the rate of crack growth and not the existence of crack growth. In order to

distinguish nevertheless the critical moment where unstable crack growth begins,

K G cc the terms cc and might be used (European Group on Fracture, 1988). To address in a more analytical way the case of unstable crack growth under isothermal conditions, we begin with Equation (24). If we consider the case where the far ﬁeld viscoelastic behavior and the viscoelastic behavior in the process zone are described explicitly by Equation (14) and the crack tip opening displacement

is assumed to be either constant or to have an implicit dependence on a_ ,then we may effectively evaluate Equation (24) by differentiating Equation (13), or its

equivalent for our assumed conditions, Equation (16). Solving Equation (16) for _

K1 and differentiating with respect to a gives

+n= m+n

m 2 1

m + n

dK 8

n = m+n

1 0 3m 2 2 1

a_ E

= (25)

a_ E m + n

d c 0 2 1

= a<_ with instability occurring for d K1 d 0. For instability to occur in the absence

of any inﬂuence from specimen geometry or adiabatic heating, considering the

material properties assumed in Equation (14) and noting especially that c

constant, Equation (25) implies that instability occurs when

+ n< :

m 0(26) n

Since “ m”and“ ” are both positive integers, it is clear that unstable crack growth

is not possible under conditions where c constant and no adiabetic heating

254 W. BRADLEY ET AL. K

Figure 2. c of PMMA as a function of crack speed at various temperatures – stable crack growth in Double Torsion tests (Marshall et al., 1974). at the crack tip occurs. Put another way, unstable crack growth in the absence

of specimen geometry/machine stiffness/loading-condition effects and R -curve

G = a = induced stability (i.e., d 1c d 0, in Equation (23)) must be the result of either

adiabatic heating and/or a decrease in the critical crack tip opening displacement,

c , with increasing crack growth rate a. Furthermore, the effect of adiabatic heating and/or a decrease in the critical crack tip opening displacement must be sufficiently great to overcome the stabilizing effect of the process zone viscoelasticity and the tendency under isothermal conditions for the craze stress or flow stress in the process zone to increase at higher crack growth rates. This increase in flow stress can ultimately limit the deformation in the process zone at the crack tip, leading to a reduction in the critical crack tip opening displacement. Furthermore, changes in the fracture process itself are sometimes possible (e.g., more chain scission and less chain pullout). In order to see more clearly we will have to consider the effect of adiabatic heating in the process zone on crack growth stability.

4.2. CRACK GROWTH UNDER ADIABATIC CONDITIONS Since we intend to consider adiabetic conditions at the crack tip, it is useful to incorporate into our instability condition (Equation (24)) an explicit temperature

dependence as follows:

@G T T @G @G @G

c c p

11dp 11d

> + ;

+ (27)

@a_ @T a_ @a_ @T a_ p p d d

where the index p refers to the process zone.

= T

The term dG1 d describes the inﬂuence of an increasing temperature on the G energy release rate (as distinct from the critical energy release rate, 1c ). As one

VISCOELASTIC CREEP CRACK GROWTH 255

Figure 3. Variation of critical crack-opening displacement, tc , with test temperature for a DGEBA epoxy resin cured with 9.8 phr TETA (Gledhill, 1978). would intuitively expect, the energy release rate increases rapidly with increasing temperature through the decrease in the viscoelastic modulus, remembering that

= K =E T; t G1 1 . This has been quantiﬁed by numerous investigations which show that the viscoelastic crack growth rate increases with increasing temperature,

as seen in Figure 2 from work by Marshall (1974). Adiabatic heating during crack

T =

growth is normally localized at the crack tip in the process zone and thus d p da will be positive and can sometimes be substantial. Both Williams (1987) and Atkins and Mai (1985) have discussed the approximate forms of this relationship, indicating that adiabatic heating becomes signiﬁcant at crack growth speeds above 10 2 m/s.

Whether adiabatic heating increases stable crack growth or not depends on how G

1c changes with increasing temperature, with a decrease seen in Equation (27)

T =

favoring instability since d p da is positive. Under what circumstances this is likely to be the case will be discussed next.

The difference in isothermal and adiabatic crack growth is seen by comparison

G p

of Equation (24) to Equation (27), with the term d 1c /dT being the key to the

T =

difference in behavior, with d p da, either approximately zero for crack growth

rates below 10 2 m/s or positive above this crack growth rate. It is helpful to

G =

c c remember that 1c , and thus, the temperature dependence of the craze or shear yield stress and the critical crack tip opening displacement can be considered

256 W. BRADLEY ET AL. T

Figure 4. Toughness versus temperature relative to g for materials A and B, both with a high cross-link density and B with 10% rubber addition (Bradley et al., 1993).

separately. It is well established that the craze or shear yield stress decreases at

= T <

p c y

higher temperatures, or d c d 0. However, a drop in (or )maybe

accompanied by an increase in c (allowing as it does for more crack tip deforma-

G G = T >

c p tion), giving a net increase in 1c (or d 1 d 0), as is typically observed in

epoxies at temperatures from 50 to 150 C below the glass transition temperature T

g (Kinloch and Young, 1983). In fact, direct measurements of the critical crack

tip opening displacement as a function of temperature has been made in an epoxy

@G =@ T >@G =@ T

p p by Gledhill et al. (1978), as seen in Figure 3. When 1c 1,itis

clear from Equation (27) that the effect of adiabatic heating is to stabilize crack

= =@ T < @G G T @G =@ T

c c p growth. Conversely, a negative value for d 1 d (or 1 1p , which is always positive) is seen to destabilize crack growth. In summary, unstable crack growth can occur for at least three reasons. First, high crack growth

rates which would normally require an increased K1 (see Equation (12)) can pro-

duce adiabatic heating and a net decrease in c , and therefore, in the value of

G = G t; a_ = t; a_ t; a_

c c c 1c when constant, 1 (Gledhill et al., 1978). Mar- shall et al. (1974) have modeled the experimental results in Figure 2 to imply a crack tip opening displacement that is surprisingly constant over a wide range of crack growth rates and temperatures for PMMA. However, this may not be true of all polymers, as indicated by results on epoxies (Bradley et al., 1993), as seen in Figure 4. Adiabatic heating, if it softens the polymer without chang-

ing the critical crack tip opening displacement, would decrease the yield or craze

G G c stress, again giving a decrease in 1c with increasing . However, in epoxies 1 has been found to increase with increasing temperature and/or decreasing crack

growth rate, at least up to temperatures within 50 K of T g as previously not-

ed in Figure 4. Thus, adiabatic heating in such a system would serve to further

@G =@ T >@G =@ T

p p stabilize crack growth, if and only if 1c 1. Thus, the effect of temperature and/or crack growth rate on the ﬂow stress and the critical crack tip

VISCOELASTIC CREEP CRACK GROWTH 257

K y

Figure 5. Variation of 1c with cross-head speed, , for a DGEBA epoxy resin cured with

K K K

c ica different stated phr of TETA and tested at 20 C: , ici ; , ; , 1 continuous (after

Kinloch and Young, 1983). G

open displacement can give 1c values which may increase or decrease, depending

on the incremental changes in c and with temperature or crack grow rate, since

G T; a_ = T; a_ T; a_ c

1c . In Figure 4, it is clear that increasing the temperature

G T G

g c to a point increase 1c after which moving closer to gives a decrease in 1 .

A second reason for unstable crack growth that follows from our earlier dis-

cussions is a decrease in c with increasing a. It is worth pointing out that the occurrence of adiabatic heating would normally preclude this kind of ductile to brittle transition. Stick-slip crack growth in epoxies can be explained using Equa- tions (24) and (13) and following the earlier conceptual model of Gledhill et al. (1978). For monotonic and relatively slow loading at a suitable temperature, the yield strength is sufﬁciently low to allow for considerable crack tip blunting and a large crack tip opening displacement. However, once the new crack grows from the blunted tip, further growth is observed effectively with a somewhat higher yield

strength and a smaller crack tip opening displacement (if we are near the ductile G to brittle transition temperature), giving a net decrease in 1c at high crack growth rates, as seen in Figure 5 for an epoxy. Atkins and Mai (1985) have noted that the strain rate at the tip of a growing crack is approximately 10 times the strain rate at the tip of a stationary crack. This may be one of the main factors which accounts for the above described reduction in the critical crack tip open displacement as the transition from initiation of cracking to subsequent growth of the crack occurs. This behavior, however, should only be observed either over a certain range of crack growth rates if the temperature is fixed or, if the crack growth rate is fixed, over a certain temperature range, which would be around the temperature (or crack growth rates) where the material behavior changes from ductile to brittle behavior. Mai and Atkins (1975) have made similar generalizations regarding crack growth instability. A comparison of Equation (10) with Equations (9) and (16) indicates a third factor which may have an influence on when unstable crack growth might occur. 258 W. BRADLEY ET AL.

Figure 6. Schematic indicating conditions for stable and unstable growth in polymers.

When the elastic component of the energy release rate, G 10, becomes a signiﬁcant G fraction of the work of fracture, 1c , then crack growth will tend to become unstable according to Equation (10).

4.3. GRAPHICAL REPRESENTATION OF CRACK GROWTH STABILITY ISSUES

Arguments presented above are illustrated graphically in Figure 6 where a schemat- _ ic representation of log K versus log a is found. Inertial effects are not considered in this diagram, where it is assumed that the behaviors illustrated occur at crack growth rates which are not inhibited by inertial effects Furthermore, this schematic has been

constructed assuming no specimen geometry/machine-stiffness/loading-conditions

= = effects, as per our previous discussion. Instability occurs where d K 1 da 0. It is apparent that a rate dependent ﬂow stress in the process zone that does not effect affect the critical crack tip opening displacement gives more stable crack growth than a rate independent ﬂow stress (see Equations (13) and (27)). When there are

no rate effects in the process zone, the slope of the log stress intensity versus log

n + crack growth rate will have a slope of n=2 1 ; however, when a rate dependent ﬂow stress is assumed in the process zone along with a rate independent critical

crack opening displacement, then the slope increases to a value of 1 =n.

Each curve is seen to asymtotically approach a limiting stress intensity, which

K G c is 1c , which has a corresponding work of fracture 1 . As is clear from Equa- tion (10), as the elastic component of the energy release rate (as distinct from the viscoelastic component) approaches the work of fracture of the polymer at the test temperature and crack growth rate in question, the crack growth rate becomes quite large. The transition from stable to unstable crack growth is associated with approaching this limit. Such behavior might be observed in a constant load or VISCOELASTIC CREEP CRACK GROWTH 259

Figure 7. Actual stress intensity versus crack growth rate in PMMA. monotonically increasing load test of a specimen with a pre-existing crack. Also, note that at any given stress intensity, the crack growth rate will be slower where

the ﬂow stress is assumed to have a rate dependence, which increases the material’s G

resistance to crack growth, 1c , as seen in Equation (13). G

As long as the work of fracture 1c is constant (lower curve in Figure 6) or

increases monotonically with increasing crack growth rate, stable crack growth is G observed (i.e., the slope of the curve remains positive). However, if 1c decreases with increasing crack growth rate (e.g., due to a lowering of the ﬂow stress due to adiabatic heating or due to a decrease in the critical crack tip opening displacement that occurs during a ductile to brittle transition), then unstable, slip-stick crack growth will be observed. This behavior is reﬂected by the dotted lines in Figure 6.

The dotted line in Figure 6 is associated with the reduced K 1 needed to drive

G a_ the crack as 1c decreases with increasing (see Equation (10)). At the point of instability labeled B, the crack growth rate will jump from “B” to “D”, which

is the stable portion of the stress intensity versus crack growth rate associated G with the new lower value of 1c . Since crack growth rate tests are usually run in displacement control, this much more rapid rate of crack growth will give an unloading of the specimen, with a monotonic decrease in the crack growth rate to

crack growth rate “ C ”. Further decrease in the crack growth rate would require an increase in the stress intensity, which cannot occur. Thus, the crack growth rate

jumps back to the original stable branch of the stress intensity versus crack growth a rate curve, to crack growth rate “ _ ” which is so slow that it appears macroscopically to be crack arrest, and is often measured as an arrest stress intensity in slip-stick crack growth, as seen in Figure 5. The schematic in Figure 6 is quite reminescent of actual experimental results published by Kinloch and Young (1983, p. 243) based on the work of four research groups, as seen in Figure 7.

260 W. BRADLEY ET AL.

K y Figure 8.Plot 1ci versus the yield stress , for different formulations of a DGEBA epoxy

resin tested at a variety of rates and temperatures. The regions of different types of crack

N growth are indicated: H, 7–4 phr TETA; , 9–8 phr TETA; , 12–3 phr TETA; , 14–7 phr TETA.

A test machine displacement rate that gives a specimen crack growth rate less than “B” in Figure 6 would not provoke any instability. In fact, only displacement rates that would give specimen crack growth rates in the range between “B” and “D” in Figure 6 would cause instability, slip-stick crack growth . This corresponds to the stable crack growth at high and low yield strengths described by Kinloch and Young(1983, p. 311), with stick-slip crack growth occurring at intermediate values of yield strength, as seen in Figure 8. The high yield strength, stable brittle fracture portion of Figure 8 corresponds to the branch of Figure 6 that passes through point D while the low yield strength, stable ductile region of Figure 8 corresponds to the branch of Figure 6 that passes through point A. The unstable brittle region of Figure 8 corresponds to the behavior between points B and D where the increasing yield strength at higher rates has an associated decrease in the crack tip opening

displacement (or the ﬂow stress due to adiabatic heating), giving a net decrease in G

1c . The range of crack growth rates over which such instabilities are observed is a function of the test temperature, with higher temperatures giving rise to instabilities at higher crack growth rates of “B” in Figure 6. This would be consistent with the observations of unstable crack growth in PMMA occurring at higher crack growth

rates for higher test temperatures, as seen in Figure 2.

a_ The K1 versus behavior for any specimen geometry, loading condition and material properties can be predicted if the various material properties are known as a function of crack growth rate and temperature. While a closed form solution is not usually possible, a ﬁnite difference approach can be used as follows. Each

increment of time will result in an incremental increase in the displacement, or

= + R t;

+ i i 1

VISCOELASTIC CREEP CRACK GROWTH 261

= R t = ;

+ i i 1 (28)

where R is the rate of displacement of the cross head.

Ca ;t a

i i

; P = +

i 1 2 (29)

C ;t C a ;t

i i i i

C P i + + where is the specimen compliance and i 1 is the resultant load at the 1 time

interval or,

P = P +P :

+ i i+ i 1 1 The newly calculated load can be used to calculate the stress intensity on the specimen and this in turn can be used to calculate the viscoelastic creep crack growth rate using Equation 10 or 16, with the appropriate materials constants

determined experimentally, so

K = g P ;a

+ i+ i 1 1 1 (30)

and

a_ = f K :

+ i+ i 1 1 (31)

Finally, the incremental crack growth for this time increment can be calculated as

a = t:

+ i+ i 1 a1 (32) Since these equations are highly nonlinear, especially Equation (30), the result

in Equation (31) should be used in Equation (32) to repeat the calculation with

P a continuing iterations until the values for and are approximately repeated from one iteration to the next. Once this convergence occurs, then the displacement in Equation (28) can be incremented and the process repeated. Both numerical and real system instabilities may arise in this ﬁnite difference model. The numerical instabilities can be minimized by using better approximations in the ﬁnite difference equations using standard approaches and using smaller time steps.

5. Implications of Viscoelastic Creep Crack Growth for Fracture Mechanics Testing Where Linear Elastic Behavior Is Assumed The ASTM and the ESIS have recently developed standards to determine the fracture toughness of polymeric materials which generally mimic the standards developed in 1972 for metals, except for the method used to introduce a suitable sharp crack at the tip of the machined-in notch. What is the relationship between these standards and the above analysis? First, the specimens used in these tests are compact tension or single edge notched bend specimens, with the testing done at a constant crosshead rate, and ideally, in a stiff machine. The system stability and choice of displacement rates 262 W. BRADLEY ET AL.

Figure 9. Schematic of load versus time and crack length as a function of time in a test to K measure 1 c .

will hopefully avoid instability, slip-stick crack growth during the testing, as seen

K K in Figure 6, allowing 1 to reach 1c . Second, a constant displacement rate will give the load and crack length versus time behavior indicated schematically in Figure 9. The shapes of the schematics in Figure 9 are easily predicted using Equation (10). Note that the derivitive of the crack length versus time curve is the crack growth rate predicted by Equation (10) from the load at the same time. A more precise result can obviously be obtained using Equations (28–32), but Equation (10) alone is adequate to get the primary

idea. Low values of the material creep compliance exponent “n” and/or a relatively =C

large ratio of C0 1 from Equations (8a) and (8b) are seen in Equation (10) to

K K ensure that 1 will approach 1c before any signiﬁcant crack growth occurs. However, at higher temperatures where the viscoelastic behavior of the material is more pronounced, the shape of the crack length versus time curve as seen in Figure 9 will give more ambiguous results, limiting the utility of the standard.

Furthermore, instabilities may intervene in the test, causing a measurement of an

K K c apparent 1c value which is quite different from the actual 1 value. On the one hand, such instabilities in service may practically lead to failure, and thus, be the behavior of interest. On the other hand, the measurement of such instabilities is speciﬁc to the compliance characteristics of the specimen/testing machine system, VISCOELASTIC CREEP CRACK GROWTH 263 and thus, is not itself a true property of the material. Furthermore, it may have little relevance to service applications where the system is quite different.

Third, there is a tendency to misinterpret the signiﬁcance of such measurements

K G K c

of 1c or 1 . In metals it is understood that if the applied 1 is maintained below K the material critical 1c , then crack growth will not occur. However, the same is not

necessarily the case for polymeric materials. Since crack growth can and will occur K below the measured “ 1c ”, this result is only useful in determining a material’s ﬂaw tolerance for short time load applications, with little utility for determining ﬂaw tolerance for long time applications such as the pipe examples given in the introduction to this paper.

6. Measuring Time Dependent Crack Growth in Polymers Early efforts to measure the stress intensity as a function of crack growth rate were done using single edge notched specimens or double torsion specimens, both loaded at a constant displacement rate, with test times which were relatively short. However, it has become apparent that service failures often occur after slow crack growth which may occur over a very long period of time under constant load conditions. Polyethylene pipes used for gas distribution systems or polyvinyl chloride pipes used for water distribution or sewage transport are typical examples. Three techniques have been developed in recent years to measure time dependent crack growth under conditions of constant load. The PENT test has been developed by Dr. N. Brown (Lu et al., 1991; Lu and Brown, 1990) has recently been made into an ASTM standard test, D-1473. In this test, a single edge notch specimen is prepared using a sharp razor blade in a specially designed ﬁxture to insure repeatability in the notch tip geometry. A dead weight is then placed on the specimens producing a constant tensile stress in the specimen and the time to specimen failure is noted. Long times to failure correspond to good resistance to time dependent crack growth while short times to failure are associated with poor resistance to crack growth. One can at least qualitatively rank various materials with reference to their resistance to viscoelastic creep crack growth resistance, and using experience based benchmarks, even predict the performance of various polymers. Thus, the test is especially useful for materials development and has some utility for developing engineering data, when used in combination with experienced based benchmarks. A second approach to address the problems of slow crack growth in polymeric pipe has been pioneered by Marshall using a simple fracture mechanics type test that may be empirically calibrated using pipe from service. The result of the efforts of Marshall and others has been the adoption of British Standard 3505 for PVC pipe. In this standard, rings are cut from a piece of extruded pipe, a small section is cut from the ring and directly across from the removed section a sharp notch is placed which is 25% through the pipe wall thickness. The rings are then loaded with various dead weights and the time to failure is recorded. Subsequently, initial 264 W. BRADLEY ET AL. applied stress intensity versus time to failure is plotted semi-logrithmically. The comparison of such results for a given pipe to similar results for both pipe that has performed satisfactorily in service and pipe that has failed in service allows one to give signiﬁcance to the measured results for newly extruded pipe. This approach has proven successful in identifying poorly consolidated pipe that may still appear to be satisfactory in a short time tension test, or even in a fracture toughness tests. This technique could potentially be used with other plastics if suitable benchmarks can be developed. A third approach has been developed by Williams and his student Chung (Chung

and Williams, 1991). In this approach, a standard three point bend specimen with

= : a deep notch (a=W 0 5) is prepared with the tip carefully notched using a razor blade. Again, the specimen is loaded with a dead weight. In this test, the load- line displacement is measured along with the time to failure. Before crack growth begins, the displacement as a function of time is due to viscoelastic deformation in the specimen, which can be used to determine the viscoelastic constitutive relationship for the material being tested. Once crack growth begins, this viscoelastic constitutive relationship in combination with compliance relationships for three point bend specimens with deep cracks (Anderson, 1991) can be used to determine the crack size as a function of time. Differentiation of this crack size as a function of time gives crack growth rates as a function of time and current crack length. The calculation of stress intensity from the dead weight load and current crack length (Anderson, 1991) allows one to determine the crack growth rate as a function of stress intensity from the simple measurement of load-line displacement versus time. Typical results from Chung’s dissertation (1991) are presented in Figure 10. Recent work at Texas A&M University and Phillips Petroleum is presented in Fig- ure 11 (Bradley et al., 1998), both on polyethylene. Chung’s results indicate the importance of constraint in the slope of the stress intensity versus crack growth rate curve, with his results varying from 0.25 for thin, test specimens without side grooves to 0.12 for thick specimens with side grooves. The value of 0.12 for crack growth under full constraint at the crack tip is similar to the viscoelastic exponent,

n, for viscoelastic deformation in polyethylene. This agreement between the power

laws exponent for viscoelastic deformation and for the relationship between K 1 a

and _ is exactly what was presented in Equation (21b) and Figure 6 for viscoelastic

D t D crack growth under conditions of constant c and for 1 0. Bradley et al.’s results in Figure 11 also show a slope that is approximately equal to the viscoelastic

coefﬁcient “ n”, as predicted in Equation (21b) and Figure 6. However, both Chang’s

results and Bradley et al.’s results show a sharp decrease in “ n” at higher crack

growth rates. Where the time to failure in the process gone is sufﬁciently small

D t

(at high crack growth rates) that D0 and 1 become similar in magnitude, the

D t

effective “n” value in Williams assumed constitutive relationship 1 is reduced.

Furthermore, at the higher K and values, the reduced slope in Figure 11 is likely

G G to be the result of 10 approaching 1c in Equation (10), which would also change the slopes in a way that is consistent with Figures 10 and 11. VISCOELASTIC CREEP CRACK GROWTH 265

Figure 10. Crack growth rate at various stress/intensities for several speciment thicknesses,

0 :25

using deeply notched three point bend specimen. Obviously, the deviation from a is an

artifact of shear lips which are larger at larger K values, and thus, have a greater impeding effect on crack growth rate (Chung and Williams, 1991; Chung, 1991).

Figure 11. Calculated rate as a function of stress intensity using deeply notched three point bend specimens. Note the slope is constant at 0.11 until the crack growth rate is very rapid, where it decreases sharply. 266 W. BRADLEY ET AL.

The approach of Williams and Chung (1991), which has not yet been standard- ized, has the advantage over BS3505 and ASTM D-1473 that it separates crack initiation from crack propagation and also measures the relationship of stress intensity to crack growth rate. The other two techniques use specimens which most likely do not give plane-strain constraint and determine only total time to failure, which is speciﬁc to the test geometry and test conditions. The measurement of stress- intensity versus crack growth rate under conditions of plane-strain constraint by Williams and Chung gives fundamental properties of the material that are geometry independent. Furthermore, the test results are not nearly as sensitive to the notching technique, which would effect primarily the initiation time for creep crack growth.

7. Engineering Applications of Viscoelastic Fracture Mechanics In principle, the experimental determination of the stress intensity as a function of crack growth rate and the critical stress intensity at which catastrophic crack growth occurs should allow one to integrate an equation of the form of Equation (21) to determine the time to failure as a function of applied stress and initial flaw size. Unfortunately, the initial flaw size is seldom known and the service loads in the case of pipes are also ill defined, with the cases of interest usually involving rocks or tree roots that occasionally find their way into pipeline beds. In one instance, this approach has been used in a failure analysis to predict accurately the time of introduction of the flaw through pinch clamping (Jones and Bradley, 1993). However, the good agreement found between predicted time from pinch clamping to pipe failure and the actual time interval observed in service is probably fortuitous, since the experimental data used were from the literature and not necessarily for pipe identical to that being studied in the investigation. Nevertheless, this type of information can be used with bench marks from plastics that perform well in service and from plastics that do not perform well in service, for both materials development and for quality control.

8. Summary The study of time dependent crack growth in polymers using a fracture mechanics approach has been reviewed. The time dependence of crack growth in polymers is seen to be the result of the viscoelastic deformation in the process zone, which allows the supply of energy to drive the crack to be provided over time rather than instantaneously, as it is in metals. Additional time dependence in the crack growth process may be due to process zone fracture behavior, where both the craze or yield stress and the critical crack tip opening displacement may be dependent on the crack growth rate. Instability leading to slip-stick crack growth has been seen to be the consequence of an increase in the critical energy release rate with increasing crack growth rate due to adiabatic heating and/or a reduction in the process zone ﬂow stress or a decrease in the crack tip opening displacement due to a ductile VISCOELASTIC CREEP CRACK GROWTH 267 to brittle transition at higher crack growth rates. Finally, various techniques to experimentally characterize the crack growth rate as a function of stress intensity have been critiqued.

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