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The purpose is to quantitatively present in a single equation all the factors that affect the failure time by Slow Crack Growth (SCG) in a semi-crystalline polymer (SCP) under a constant load. The fundamental mechanism of fracture is displayed at the molecular level. The rate of fracturing is determined by the Eyring theory of thermal activation. The resulting equation includes the important molecular properties of therein, the length and density of the tie molecules. The underlying microfracture process is the unfolding of the chains in the crystal under the action of the tie molecules.

When a SCP structure is exposed to a sufficiently low constant tensile stress under plain strain conditions, it will ultimately fail by a brittle fracture process commonly called “slow crack growth” [SCG]. Polyethylene (PE), the most widely used SCP, was first used on a large scale in gas pipes in 1965 manufactured by the DuPont Company. The phenomenon of SCG has been subjected to extensive research throughout the world, because it is very important to know how long a gas pipeline will function before it fails. Much of the published research on PE was produced by Lu and Brown and coworkers at the University of Pennsylvania, as for example in reference [

1) Failure by SCG only occurs when the structure is subjected to a tensile stress less than about 1/2 Y under plain strain conditions where Y is the yield point. When the stress is greater than 1/2 Y, ductile failure is produced by the macroscopic accumulation of plastic strain. Below 1/2 Y, the amount of macroscopic plastic strain is negligible and brittle failure is produced by a crack. The failure in a pressurized pipe exhibits a dramatic difference between the modes of failure above and below 1/2 Y. The failure above 1/2 Y consists of a bulge in the pipe whose dimensions are on the order of the pipe diameter. The failure below 1/2 Y consists of a leak in the form of a slit whose width is a very small fraction of the wall thickness of the pipe.

2) The dependence of the failure time on stress at various temperatures in notched specimens is exhibited in

3) Fracture always initiates at points of stress concentration. It is widely opined that practical structures contain points of stress concentration. The stress concentrators can take many forms such as inclusions, sharp corners, scratches and molding defects. _{I} fracture mode.

The stress concentrator is conventionally described by the stress intensity, K, where K = ga^{1/2}s; g is a geometric factor determined by the ratio of a with respect to the macroscopic dimension of the specimen and s is the global stress. The subsequent analysis shows that c must be included in the description of the stress concentrator order to completely account for all the factors that determine the time to produce failure under a constant load.

4) It is necessary to know the common morphology of all SCP at the molecular level. SCP are linear. Lamella crystals are formed that consists of folded molecules whose thickness may vary from about 40 to 100 nm. The folded chains are parallel to the thickness direction. There is an amorphous boundary about 15 nm thick between the crystals. The microstructure is depicted in

5) When a tensile stress less than 1/2 Y is applied to a structure under plane strain conditions, a deformation zone is produced in the vicinity of the stress concentrators. It consists of coarse fibrils that are oriented in the direction of the applied stress (

6) The crack opening displacement [COD] typically increases with time as shown in

The crack propagates by the successive fracture of the fibrils until the remaining ligament fails by yielding.

It is important to determine the stress in the first fibril because that is where fracture initiates. The geometry of the first fibril is illustrated in _{o}. The cross section area of the fibril is A. The length of the first fibril is the COD. The first fibril evolves from a volume A_{o}c, where c, the sharpness of the stress concentrator, is the thickness of the volume of material from which the first fibril evolves. Conservation of volume gives

A o c = A COD (1)

The validity of Equation (1) is based on the assumption that A is the average cross section area of the fibril and on the facts that the COD is very much greater than c and also that the COD is independent of c.

Stress equilibrium requires

A o Y = A S (2)

where S is the stress in the fibril. Combining Equations (1) and (2)

S = Y COD / c (3)

It is important to know the morphology of the fibril at the molecular level. Based on the extensive research on the morphology of the fibril by Kieth et al. [

S A = n F (4)

where A is the area on which S acts and n is the number of TM. A is also the area from which the TM emerges from the crystal. By definition, the density of TM is ρ = n/A. Combining (3) and (4) gives

F = Y COD / ρ c (5)

In order to better visualize the critical role that F plays in the fracture process, the fundamental morphological unit of the fracture process is exhibited in

The rate of unfolding the chains will be derived from the Eyring theory of thermal activation [

r = V f exp [ − Q / k T ] exp [ F v / k T ] (6)

The value of F from (5) will now be inserted into Equation (6).

r = V f exp [ − Q / k T ] exp [ Y COD v / ρ c k T ] (7)

In order to completely know all the parameters that determine r, the basis for determining the COD must be presented. The value for the COD was derived from a theory independently developed by Dugdale [

hardening after the yielding . Since PE only approximates the mechanical behavior of this material, the Dugdale-Rice formula is an approximation to the observed value of the COD. However, the experimental values of the COD vary in tune with the parameters of the Dugdale-Rice formula that

C O D = K 2 / Y E (8)

K, the stress intensity, is given by

K = g s a 1 / 2 (9)

g is a geometric factor that depends on the geometry of the stress concentrators, s is the applied stress and a is the length of the stress concentrator. E is Youngs modulus.

The experimental values of the COD in PE as measured by Qian et al. [

COD = B s n a m (10)

B is a geometric factor like g. Depending on the resin, n can go from about 3 - 4.5 and m is slightly greater than 1/2n. The experimental value of the COD approximates the Dugdale-Rice prediction. In order to emphasize the importance of the stress intensity, K, the Dugdale-Rice value of the COD is inserted in [

r = V f exp [ − Q / k T ] exp [ K 2 v / E ρ c k T ] (11)

Fracture initiates in the structure when all the TM in a cross section area of the first fibril are pulled out. The time to fracture the first fibril depends on the length average L_{z} of the longest molecules that contain TM. The time also depends on the rate, r_{z} that is controlled by the density of TM, ρ_{z}, whose length is L_{z}. It is straight forward that the time to initiate fracture can be described by

t = L z / r z (12)

Combining (11) and (12), the time to initiate fracture is

t = ( L z / V f ) exp [ Q / k T ] exp [ − K 2 v / E ρ z c k T ] (13)

After the first fibril fractures, the crack propagates by the successive fracture of the following fibrils. The time to fracture in a fibril is still governed by equation (13) with the exception that K increases because the length of the stress concentrator increases every time a fibril fractures. Thus, the curve of COD versus time in

Equation (13) predicts that t is linearly related to the length of the [TM]. Huang and Brown [_{w}. Equation (13) indicates that the ultimate limit on the time to failure depends on the ultimate length of the TM. Experiments with ultra high molecular PE were performed that lasted so long that the time of failure could not be measured under the available laboratory conditions.

The introduction of short chain branches [SCB] greatly increases the failure time in copolymers relative to a homopolymer because the density of TM is directly related to the concentration of the SCB. The reason for this relationship is very simple. Direct measurements show that over 90% of the SCB reside in the amorphous region which is the boundary between adjacent crystals (data from Dow Chemical Co. in private communication). Consequently the probability of forming a TM is directly related to the concentration of SCB.

Huang and Brown [

Equation (13) predicts that the high end of the molecular weight distribution is a significant factor. Scholten and Rikjema [

When the specimen is loaded as shown in ^{3} to 10^{5} gm/mol. They found that the 3 parts whose molecular weight was less than 3000 gm/mol failed immediately upon loading. The other 2 parts with greater molecular weight had failure times on the order of 1000 hr, which is comparable to t for the whole resin.

Huang and Brown [

Industry developed resins with a bimodal molecular weight distribution with no SCB on the low molecular weight side of the distribution. If the low molecular weight molecules contained SCB these SCB would be wasted as soon as the structure is loaded. For some application there is a limit on the concentration of SCB because the yield point decreases as the concentration of SCB is increased. When designing a structure, the engineer often optimizes the performance by making a compromise between the yield strength and failure time by SCG.

The inclusion of the notch sharpness factor, c, in Equation (13) is supported by the experimental observation by Lu et al. [

Lagaron et al. [

When a SCG test is measured in an Igepal environment, there is a significant decrease in the failure time (Paper by Ward et al. [

It is also interesting to know why the activation energy is about 100 kj/mol = 1 ev. The activation energy, Q − Fv, is largely determined by Q, the bonding energy between a folded chain and the crystal. Although the bonding is based on the Van der Waals bond whose energy is about 0.01 ev, Q is about 1 ev because it is an integration of all the bonds that surround a chain.

It is also interesting to note that whether the chains are regularly or irregularly folded the basic theory is not changed. The only difference is that the Q for irregularly folded chains is expected to be less than that for regularly folded chains.

1) The rate of SCG is thermally activated as described by the Eyring theory.

2) Equation (13) contains all the quantities that determine the time for the initiation of fracture by SCG in a semi-crystalline polymer. This is the first time that the process of SCG has been completely described by an equation.

3) The morphological features of the resin that determine the failure time are L_{z}, the length of the TM, whose molecular weight is M_{z}, and the density of the TM, ρ_{z} whose molecular weight is M_{z}. These morphological features are controlled by the conditions under which the resin was polymerized.

4) The fundamental mechanism of SCG is the unfolding of the chains in the crystal under the action of the TM.

5) The time to initiate fracture in the first fibril occupies a major part of the total time for failure by SCG.

6) The density of the TM is proportional to the concentration of SCB because over 90% of the SCB are located in the boundary between the crystals.

Dr. Campbell Laird edited the manuscript.

The author declares no conflicts of interest regarding the publication of this paper.

Brown, N. (2019) Fundamental Mechanism of Slow Crack Growth in Semi-Crystalline Polymers under a Constant Load. Materials Sciences and Applications, 10, 721-731. https://doi.org/10.4236/msa.2019.1011052