Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene

Engineering Fracture Mechanics xxx (2017) xxx–xxx

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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene Jung-Wook Wee, Byoung-Ho Choi ⇑ School of Mechanical Engineering, Korea University, Seoul 136-701, Republic of Korea

a r t i c l e

i n f o

Article history: Received 16 May 2016 Received in revised form 24 October 2016 Accepted 30 January 2017 Available online xxxx Keywords: Crack layer theory Slow crack growth Modeling High density polyethylene Parametric study

a b s t r a c t The crack layer theory, proposed by A. Chudnovsky many years ago, can be a very effective tool to quantitatively model slow crack growth in engineering thermoplastics. However, the large number of input parameters required by this theory has been a technical hurdle for its use in industrial and research applications. Thus, in order to achieve a practical applicability of the crack layer theory, a clear understanding of the effect of each parameter on slow crack growth is quite important. In this study, simulations of slow crack growth in high density polyethylene are performed using the crack layer theory for various key parameters, and the effect of each parameter on crack growth behavior in high density polyethylene is investigated. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction For the lifetime prediction of long-term failure in engineering thermoplastics, which is a main failure mode in structural applications of thermoplastics, a large effort has been devoted to the development of slow crack growth (SCG) models and the linkage between accelerated testing and field failure. Engineering thermoplastics usually show different failure modes depending on the applied load and environmental conditions [1,2]. Under high load, most engineering thermoplastics fail in a short time by ductile failure. In the case of intermediate load levels, it is commonly observed that engineering thermoplastics fail in a quasi-brittle manner with significant SCG, and many unexpected field failures of load-bearing engineering thermoplastics are related to this mode of failure [1,2]. Moreover, reactive chemicals surrounding the engineering thermoplastics also affect the lifetime of the material, leading to relatively long lifetimes with failure modes of environmental stress cracking or stress corrosion cracking depending on the chemical and/or physical damage of the material [3,4]. Therefore, the prediction of the lifetime to failure, in the case of quasi-brittle fracture, and the development of accelerated tests to predict field failures is very important. Simple extrapolations of experimental data without proper justification must be avoided due to transitions of the SCG mechanism and kinetic over time [2,5,6]. In general, empirical models such as Paris’ equation have limited applicability to polymers due to the complexity of crack growth characteristics in engineering thermoplastics. In particular, the discontinuous crack growth behavior under both creep and fatigue loading conditions has been widely observed in brittle fracture of engineering thermoplastics [7–12]. Such crack growth behavior cannot be simulated by conventional models. A more general model such as the crack layer (CL) theory proposed by A. Chudnovsky in [13,14], is required in order to capture the complexity of cracks and the surrounding damage zone interaction resulting in continuous, ⇑ Corresponding author at: School of Mechanical Engineering, Korea University, 1 5-ga, Anam-dong, Sungbuk-gu, Seoul 136-701, Republic of Korea. E-mail address: [email protected] (B.-H. Choi). http://dx.doi.org/10.1016/j.engfracmech.2017.01.028 0013-7944/Ó 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Wee J-W, Choi B-H. Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.01.028

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Nomenclature A, m E0 G CR J PZ 1 , J1 kCR , kPZ KI K Ic lpz, lCR ti tf t X PZ , X CR

c c0 ctr dtot k

r1 rdr

coefficients of the relation between slow crack growth rate and stress intensity factor plane strain elastic modulus Gibbs potential elastic energy release rate at process zone and crack tips kinematic coefficients of crack and process zone growths stress intensity factor critical stress intensity factor length of a process zone and a crack elapsed time for the transformation from fresh material into process zone material time to failure characteristic time for degradation driving forces for a process zone and a crack specific fracture energy initial SFE of fresh material transformation energy per unit volume crack opening displacement at the crack tip natural draw ratio remote stress drawing stress

discontinuous, and mixed mode crack growth [2]. The crack layer theory considers the crack and its surrounding damage zone (commonly called ‘‘process zone”) as a single entity (crack layer) with a few degrees of freedom, and makes use of thermodynamic considerations for modeling the CL evolution. In order to understand the SCG behavior of engineering thermoplastics in the most general manner, a crack and the damage zone around the crack may be considered as a single entity and applied in SCG simulations of engineering thermoplastics [13]. The CL theory can be used to simulate the different SCG modes of engineering thermoplastics, i.e., continuous, discontinuous, or the transition between these two modes [13–17]. According to [13], the CL is a system of a main crack with its surrounding damage zone, also known as the process zone (PZ). In the CL theory, the thermodynamic interaction between the main crack and different surrounding damaged areas is established, considering the energy dissipation required to generate micro-cracks or micro-damages. The applicability of the CL theory has been convincingly demonstrated in various thermoplastics such as high density polyethylene (HDPE), polystyrene, and polycarbonate, which show dissimilar damage structures [13–19]. In other words, the CL theory has the potential to connect the morphological structures with the crack and damage zone growth in a general way. From the fact that the CL simulation provides not only the lifetime or SCG rate but also elucidates the entire crack and damage zone growth process; it has a great potential to be employed in industrial settings, replacing the conventional empirical models. Despite the many scientific benefits of the CL theory, there are several problems to be solved regarding the application of the CL theory in the simulation of the SCG behavior of engineering thermoplastics. In the CL theory, two coupled driving-force equations, which are related to the crack and PZ growth respectively, are derived from nonlinear differential equations. Consequently, the driving forces should be solvable by numerical methods [1,17]. In particular, the diverse input parameters such as the initial specific fracture energy (SFE), transformation energy, characteristic time for material degradation, and so on, make the use of the CL theory quite complicated. Actually, most studies on the CL theory have concentrated on the reproduction of specific test results provided by CL simulations with limited input parameters [18,19]. An understanding of how each parameter affects the overall CL growth process must be established so as to efficiently apply the CL theory to diverse practical cases. In this study, to address the above technical issues, a detailed parametric study with a variety of input parameters of the CL theory for HDPE was performed on single-edge-notched (SEN) specimens in creep conditions. The effect of each input parameter on the CL growth, i.e. the growth of the crack and the PZ, is elaborated. To quantitatively investigate the effect of each parameter of the CL growth, the conventional relationship between the SCG rate and the stress intensity factor is constructed and compared. Based on the constructed relationship, the time to failure of HDPE under discontinuous SCG is predicted for various input parameters. In addition, the detailed backgrounds of the CL theory and a CL simulation algorithm are also introduced.

2. Application of the crack layer theory to high density polyethylene 2.1. Overview of the crack layer theory Detailed observations of the fracture propagation process in HDPE are required in order to understand the SCG behavior and link it to the mechanisms and kinetics of crack growth. In HDPE, the PZ is always present in front of a crack and a strong crack-PZ interaction affecting the SCG should be explicitly addressed. Please cite this article in press as: Wee J-W, Choi B-H. Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.01.028

J.-W. Wee, B.-H. Choi / Engineering Fracture Mechanics xxx (2017) xxx–xxx

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Thermodynamic considerations of a closely coupled system of crack and PZ have been proposed by Chudnovsky [14,15]. Such a thermodynamic system was denoted as a crack layer (CL). In accordance with the general thermodynamics framework, the thermodynamic forces responsible for the CL evolution have been presented in [14] as the derivative of the Gibbs free energy (G) of the system with respect to the geometrical characteristics of the CL, such as the crack and PZ dimensions. It has been widely reported in the literature that in HDPE the PZ appears as a thin wedge shaped zone of transformed (i.e., cold drawn) material in front of the crack separated from the original material by a well-defined boundary [2,2018]. The boundary traction between the PZ and the elastic solid can be assumed as a constant drawing stress (rdr ) based on the observation of the PZ of HDPE. In Fig. 1, schematics of the superposition methodology of crack layer systems of HDPE for a single edgenotched tension (SENT) specimen are shown. As shown in Fig. 1, the PZ length, lPZ , is sufficient so as to characterize the PZ geometry since the PZ width is determined via compatibility conditions and the HDPE natural draw ratio k [18,20]. Therefore, there are only two CL thermodynamic forces in HDPE, one responsible for the crack growth into the PZ and another one controlling the PZ evolution. According to Chudnovsky [13,14], the general expressions for the crack and PZ driving forces in HDPE are represented as:

X CR ¼ 

@G @G ; X PZ ¼  @lCR @lPZ

ð1Þ

Fig. 1. Schematics of the superposition methodology of crack layer systems for single edge-notched tension (SENT) specimen (The boundary traction between PZ and elastic solid is assumed as a constant drawing stress (rdr )).

Fig. 2. Descriptive illustration for the successive development of crack layer.

Please cite this article in press as: Wee J-W, Choi B-H. Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.01.028

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The evaluation of the thermodynamic forces for a CL system with a thin PZ consisting of transformed (oriented) material was conducted in [18,20]. The results can be expressed as: PZ X CR ¼ J CR ¼ J PZ 1  2c; X 1 

ctr

k1

dtot

ð2Þ

PZ where J CR 1 is the elastic energy release rate (ERR) due to unit crack advance into PZ, similarly J 1 is the elastic energy release tr rate due to unit PZ length increment; c and c represent the specific fracture energy of the PZ material and the HDPE transformation energy per unit volume of original HDPE into a highly oriented state of the PZ material. In addition, dtot is the crack opening displacement at the crack tip. It should be noted that the SFE at the crack tip is expected to degrade with time; the degradation can be either mechanical and/or chemical, depending on the environment. For the discontinuous SCG behavior, the initial SFE is higher than the ERR at the crack tip, which means that the value of the crack driving force is negative, and the crack would be in a stationary state, i.e. the growth of the crack would not occur. When the crack driving force becomes positive as a consequence of the material degradation, the crack starts to grow, followed by a crack arrest due to the encounter with fresh material with the initial SFE, which still has a higher value than the ERR. Thus, the stable PZ length can be equivalent to the subsequent crack jump length, and phenomena such as discontinuous crack jumps are observed. The structure of the CL simulation algorithm in this study adopts the driving forces of Eqs. (1) and (2). The expression of the mechanical decay f ðti Þ of the SFE is as follows [15]:

Fig. 3. Brief algorithm of Crack layer (CL) simulation.

Please cite this article in press as: Wee J-W, Choi B-H. Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.01.028

J.-W. Wee, B.-H. Choi / Engineering Fracture Mechanics xxx (2017) xxx–xxx

c 1
 ¼ f ðt i Þ ¼ c0 1 þ tti

5

ð3Þ

where c0 is the initial SFE of fresh material and ti denotes the elapsed time from the moment that the material is transformed into PZ material. The time t is a characteristic time for degradation and controls the rate of degradation. The higher the value of t  , the slower the degradation. Provided that the driving force is positive, the crack or PZ propagates at a speed obtained by utilizing the following linear relationships, Table 1 Summary of input variables for the parametric study. Parameter

Unit

Range of value

E0

MPa MPa MPa mJ/mm2 mJ/mm3 – sec mm2/mJ s pffiffiffiffiffi MPa m

520–760 16–24 6–10 5–25 4–12 2–10 300–700 0.002–0.006 3

rdr r1 c0 ctr k t kPZ and kCR K Ic

σ∞= 6 MPa σ∞= 8 MPa σ∞= 10 MPa

Length (mm)

10 8 6 4 2

5.0x104 1.0x105 1.5x105 2.0x105

0.0

Time (sec)

-3.2

σ ∞= 6 MPa σ ∞= 7 MPa σ ∞= 8 MPa σ ∞= 9 MPa σ ∞= 10 MPa

log (da/dt)

-3.6 -4.0 -4.4 -4.8

6

68

.2 =4 em

p

Slo

-5.2 -5.6 1.3

1.4

1.5

1.6

1.7

1.8

log (KΙ )

Failure time tf (hour)

60 50 40 30 20 10 0

6

7

8

9

10

Applied stress σ∞ (MPa)

Fig. 4. Simulated results for various applied remote stresses (6, 7, 8, 9 and 10 MPa).

Please cite this article in press as: Wee J-W, Choi B-H. Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.01.028

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dlPZ ¼ kPZ  X PZ dt

ð4Þ

dlCR ¼ kCR  X CR dt

ð5Þ

where kCR and kPZ are the kinetic coefficients of crack growth and PZ growth, respectively. At the CL equilibrium, the thermodynamic forces vanish: X CR ¼ 0 and X PZ ¼ 0. Therefore, following the Onsager’s principle, the proportionality of the crack and PZ growth rates with respect to the corresponding non-negative thermodynamic forces can be assumed for a quasiequilibrium CL growth [21]. 2.2. Simulation algorithm of the crack layer theory for HDPE The simulation algorithm of the CL evolution can be implemented based on the loop statement, and several stages constitute an individual loop of calculations. At the i-th loop, lCR ði þ 1Þ and lPZ ði þ 1Þ are first calculated using Eqs. (4) and (5), as follows:

lCR ði þ 1Þ ¼ lCR ðiÞ þ

dlCR ðiÞ  dt dt

ð6Þ

lPZ ði þ 1Þ ¼ lPZ ðiÞ þ

dlPZ ðiÞ  dt dt

ð7Þ

σdr= 16 MPa σdr= 20 MPa σdr= 24 MPa

Length (mm)

10 8 6 4 2

3.0x10 4

0.0

6.0x10 4

9.0x10 4

Time (sec)

-3.2

log (da/dt)

-3.6 -4.0 σ dr= 16 MPa σ dr= 18 MPa σ dr= 20 MPa σ dr= 22 MPa σ dr= 24 MPa

-4.4 -4.8 -5.2 -5.6 1.3

1.4

1.5

1.6

1.7

1.8

log (KΙ ) 35

4.6

Slopem

25 4.4

20

4.3

15 10

4.2 4.1

Slope m Failure time tf

16

18

20

5 22

24

Failure time tf (hour)

30

4.5

0

Drawing stress σdr (MPa)

Fig. 5. Simulated results for various drawing stresses (16, 18, 20, 22 and 24 MPa).

Please cite this article in press as: Wee J-W, Choi B-H. Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.01.028

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where dt is the time increment of the loop simulation. To determine the crack and PZ growth rates, the related driving forces, X CR ðiÞ and X PZ ðiÞ, should be calculated beforehand using Eqs. (1) and (2). Indeed, J CR 1 ðiÞ can be directly obtained through the applied load and specimen geometries including the crack length, lCR ðiÞ. At the initial loop, the SFE at the crack tip is equal to the initial SFE; the subsequent SFE will be calculated in the following stage. The ERR at the PZ tip, J PZ 1 ðiÞ, and the crack opening displacement (COD) at the crack tip, dtot jx¼lCR ðiÞ, are calculated using a superposition method [16]. The contribution of the remote and drawing stresses can be obtained from the Green’s function of the stress intensity factor (SIF) and the COD [21]. If the calculated driving force is negative, the corresponding length is not changed. In the second stage, cði þ 1Þ, which denotes the SFE at the crack tip to be used at the next ‘‘i þ 1”-th loop, can be identified by finding the location of the crack using a discriminant code as shown in Fig. 2; for example, if the calculated lCR ði þ 1Þ at the first stage is determined to be in the range of lPZ ði  2Þ 6 lCR ði þ 1Þ < lPZ ði  1Þ, the corresponding SFE, cði þ 1Þ, can be defined as c0  f ð3dtÞ. In the same way, in the case of lPZ ðiÞ 6 lCR ði þ 1Þ < lPZ ði þ 1Þ, cði þ 1Þ is c0  f ðdtÞ. If lCR ði þ 1Þ ¼ lPZ ði þ 1Þ, the SFE at the crack tip equals the initial SFE and cði þ 1Þ ¼ c0  f ð0Þ ¼ c0 . The calculated cði þ 1Þ can then be used for calculating X CR ði þ 1Þ at the next ‘‘i þ 1”-th loop. In the third stage, two criteria for the failure of the specimen are applied to determine whether the calculation should be terminated. The first criterion is applied when the SIF at the crack tip exceeds the fracture toughness (K Ic ) of HDPE. The second criterion is applied when there is no remaining elastic ligament in the specimen, but the crack opening of the specimen is still considerable. In this stage, provided that one of the failure criterion is satisfied, the loop is terminated and the corresponding time becomes the failure time of the specimen. Otherwise the same stages are re-calculated with i þ 1 replacing i. Finally, after the specimen is determined to have failed, lCR ðtÞ and lPZ ðtÞ can be plotted as functions of time t. Through this stage, we can see the entire CL growth history and failure time as well as each step jump length and time to step jump in

Length (mm)

10

tr

3

tr

3

γ =6 mJ/mm γ =8 mJ/mm

8

tr

3

γ =12 mJ/mm

6 4 2 4

0.0

4

3.0x10

4

6.0x10

9.0x10

Time (sec)

-3.2

log (da/dt)

-3.6 -4.0 -4.4

tr

γ = 4 mJ/mm tr 3 γ = 6 mJ/mm tr 3 γ = 8 mJ/mm 3 tr γ = 10 mJ/mm 3 tr γ = 12 mJ/mm

-4.8 -5.2 -5.6 1.3

1.4

1.5

1.6

3

1.7

1.8

log (K Ι ) 4.6

35

Slope m

25 4.4

20

4.3

15 10

4.2 4.1

Slope m Failure time tf

4

6

5 8

10 tr

12

Failure time tf (hour)

30

4.5

0

3

Transformation energy γ (mJ/mm )

Fig. 6. Simulated results for various transformation energy (5, 10, 15, 20 and 25 MPa).

Please cite this article in press as: Wee J-W, Choi B-H. Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.01.028

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discontinuous crack growth. This gives the average SCG rate for each crack jump; thus, the relationship between the crack growth rate and the corresponding SIF at the crack tip can be obtained. The entire simulation algorithm is schematically expressed as a flow chart in Fig. 3. The SEN specimen used in this simulation has a width of 10 mm and an initial notch length of 2 mm, in plane strain conditions. The input parameters are varied as shown in Table 1. A total of eight key parameters in the CL simulation are varied in a range where reasonable discontinuous crack growth occurs, with 5 equal intervals. E0 is the plane strain elastic modulus, while r1 and rdr stand for the remote and drawing stress, respectively, along the PZ boundary. The fracture toughness K Ic of pffiffiffiffiffi HDPE is assumed to be 3 MPa m for this study. 3. Results and discussion 3.1. Effect of remote stress As shown in Fig. 4(a), both the step jump length (Dl) and the step jump time (Dt) varied with applied stress. Indeed, the step jump length is related to the PZ driving force, X PZ , because a stable PZ is formed when X PZ ¼ 0, and the subsequent crack jump length equals the stable PZ length as elucidated. Thus, the variation of any parameters included in the expression of X PZ , Eq. (2), would change the step jump lengths; so do r1 and rdr through which J PZ 1 ðiÞ and the COD at the crack tip are calculated. The driving force for the crack, X CR , affects the time duration (Dt) of the crack initiation. At the moment that X CR becomes positive, the thermodynamic equilibrium is lost, and the crack starts to propagate through the PZ material to establish a new

λ= 2 λ= 6 λ = 10

Length (mm)

10 8 6 4 2 0.0

5.0x10

4

1.0x10

5

1.5x10

5

Time (sec)

-3.2

log (da/dt)

-3.6 -4.0 -4.4

λ=2 λ=4 λ=6 λ=8 λ = 10

-4.8 -5.2 -5.6 1.3

1.4

1.5

1.6

1.7

1.8

log (K Ι )

35

4.6

Slope m

25 4.4

20

4.3

15 10

4.2 4.1

Slope m Failure time tf

2

4

5 6

8

10

Failure time tf (hour)

30

4.5

0

Drawing ratio λ

Fig. 7. Simulated results for various draw ratio (2, 4, 6, 8 and 10).

Please cite this article in press as: Wee J-W, Choi B-H. Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.01.028

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CR equilibrium state. Because J CR 1 increases with the applied stress, the required degradation time for achieving a positive X , i.e., the PZ lifetime, is reduced. Since K I at the crack tip can be expressed as the applied stress multiplied by a geometric factor, the data of the relationship between slow crack growth rate and stress intensity factor with various values of applied stress can be fitted with a

linear function, as shown in Fig. 4(b). The slope m of this line is equal to the exponent in the expression dldtCR ¼ AðK I Þm , and this value is determined by curve fitting to be about 4, which is in agreement with previous experiments and theoretical analysis [22,23]. This result demonstrates the validity of the simulation. The failure time, t f , decreases with the remote stress in a parabolic form. In particular, the lifetime reduction rate decreases as the applied stress is increased (see Fig. 4(c)). 3.2. Effect of the drawing stress According to the equations for the driving forces (1) and (2), the drawing stress rdr is included only in the expression for X PZ ; therefore, the equilibrium PZ length, which is the same as the crack jump length in the corresponding step, varies with CR rdr . In addition, since rdr is not contained in either JCR 1 or 2c, which are the only terms included in X , the PZ duration time

during the first step (Dt1 ), in which the crack length always has the same value as the initial notch size (l0 ), does not change as a function of rdr . However, beyond the first step, the crack length, which equals the sum of l0 and the PZ length, will be

CR ¼ 0 varies. That is, if different depending on rdr ; J CR 1 is also dependent on rdr . Consequently, the time required to achieve X rdr is increased, the PZ length for the PZ equilibrium is decreased and therefore the crack lengths at subsequent steps are also

Length (mm)

10 8

E' = 520 MPa E' = 640 MPa E' = 760 MPa

6 4 2 0.0

4

4

2.0x10

4.0x10

4

6.0x10

4

8.0x10

Time (sec)

-3.2

log (da/dt)

-3.6 -4.0 -4.4

E' = 520 MPa E' = 580 MPa E' = 640 MPa E' = 700 MPa E' = 760 MPa

-4.8 -5.2 -5.6 1.3

1.4

1.5

1.6

1.7

1.8

log (KΙ )

35

4.6

Slope m

25 4.4

20

4.3

15 10

4.2

Slope m Failure time tf

5

4.1 540

600

660

720

Failure time tf (hour)

30

4.5

0 780

Plane strain elastic modulus E' (MPa)

Fig. 8. Simulated results for various elastic modulus (520, 580, 640, 700 and 760 MPa).

Please cite this article in press as: Wee J-W, Choi B-H. Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.01.028

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decreased (Fig. 5(a)). Similarly, the driving term of X CR , J CR 1 , is also reduced, and Dt at the same step increases with rdr except that the first PZ duration Dt 1 is not changed. The slope m decreases and the lifetime t f increases as rdr increases (Fig. 5 (b) and (c)). In particular, tf increases linearly with rdr .

3.3. Effect of transformation energy per unit volume and natural draw ratio The energy per unit volume, ctr , required for the transformation of the material into PZ material and the natural draw ratio k are included only in the PZ driving term, (see Eq. (1)); this implies that these parameters have an effect on the CL growth that is similar to that of rdr . Just as in the case of rdr , the step lengths differ with ctr and k; however, at the first step, where the crack length is always the same, Dt 1 maintains a constant value regardless of ctr and k (Figs. 6(a) and 7(a)). After the first step, Dt varies with ctr in the same way that it varies with rdr . An increase of k causes a decrease in the amount of material that needs to be transformed to achieve the PZ equilibrium. Thus the amount of energy dissipation caused by damage formation, which serves as an energy barrier against crack growth, is reduced with the decrease in ctr and the increase in k; accordingly, the CL growth becomes rapid, and the SCG rate increases at the same SIF (Figs. 6 (b) and 7(b)). As illustrated in Figs. 6(c) and 7(c), the lifetime tf increases slightly with ctr and asymptotically decreases with k for k > 4. Thus, it may be thought that the material parameters included in the expression of the volumetric quantity of PZ material, ctr

, may not affect the variation of the slope m within the range of ctr and k parameters considered in this study.

Length (mm)

k1

10

γ 0= 5 mJ/mm2

8

γ 0= 25 mJ/mm2

γ 0= 15 mJ/mm2

6 4 2 4

0.0

4

4.0x10

5

8.0x10

1.2x10

Time (sec)

-3.2 γ 0 = 5 mJ/mm2 γ 0 = 10 mJ/mm2

log (da/dt)

-3.6 -4.0 -4.4 -4.8

γ 0= 15 mJ/mm2

γ 0= 20 mJ/mm2

-5.2

γ 0= 25 mJ/mm2

-5.6 1.3

1.4

1.5

1.6

1.7

1.8

log (KΙ ) 35

4.6

Slope m

25 4.4

20

4.3

15 10

4.2 4.1

Slope m Failure time tf

5

10

15

20

25

5

Failure time tf (hour)

30

4.5

0

Initial surface energy γ 0 (mJ/mm ) 2

Fig. 9. Simulated results for various initial surface energy (5, 10, 15, 20 and 25 MPa).

Please cite this article in press as: Wee J-W, Choi B-H. Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.01.028

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3.4. Effect of plane strain elastic modulus The plane strain elastic modulus of HDPE (E0 ) was also changed to assess the effect on CL growth. It is worth noting that although E0 is included in X PZ as well as in the parameters, rdr , ctr , and k, the equilibrium PZ lengths (Dl) do not change with E0 0 0 (see Fig. 8(a)). This result arises from the fact that in the calculation of X PZ , J PZ 1 and dtot have a 1/E term in common. Thus, E

does not make any contribution to the PZ equilibrium, and X PZ ¼ 0. Meanwhile, since the increase of E0 reduces J CR 1 , which is the dominant term in X CR , the time required for the degradation of the PZ material for the positive X CR is greater, resulting in an increase of Dt. The lifetime t f increases linearly and the slope m is nearly constant with varying E0 (Fig. 8(b) and (c)).

3.5. Effect of initial SFE and characteristic time The SFE of fresh material, denoted by c0 , and the characteristic time for material degradation t are included only in X CR . Therefore, the equilibrium PZ length Dl does not vary with c0 or t  (see Figs. 9(a) and 10(a)). For crack growth to occur, X CR PZ should be positive, which is equivalent to saying that 2c is smaller than J PZ 1 . During the crack arrest period, J 1 is constant and  the SFE 2c undergoes a mechanical decay with time that can be expressed according to Eq. (4). As c0 and t increase, the time

required for a positive X CR would be increased; such a situation is displayed in Figs. 9(a) and 10(a), as an increase in Dt and a decrease of the SCG rate in Figs. 9(b) and 10(b). As illustrated in Figs. 9(c) and 10(c), the failure time, tf , also increases linearly with c0 and t  . The slope m, however, shows an opposite tendency with varying c0 or t . The increase in c0 , which implies an

t* = 200 sec t* = 500 sec t* = 800 sec

Length (mm)

10 8 6 4 2 0.0

4.0x10

4

8.0x10

4

1.2x10

5

Time (sec)

(a) Entire crack layer (CL) growth -3.2

log (da/dt)

-3.6 -4.0 -4.4

t* = 200 sec t* = 350 sec t* = 500 sec t* = 650 sec t* = 800 sec

-4.8 -5.2 -5.6 1.3

1.4

1.5

1.6

1.7

1.8

log (KΙ )

(b) Slow crack growth rate 35

4.6

Slope m

25 4.4

20

4.3

15 10

4.2 4.1

Slope m Failure time tf

200

350

500

650

800

5

Failure time tf (hour)

30

4.5

0

Characteristic time t* (sec)

Fig. 10. Simulated results for various characteristic times (200, 350, 500, 650 and 800 s).

Please cite this article in press as: Wee J-W, Choi B-H. Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.01.028

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J.-W. Wee, B.-H. Choi / Engineering Fracture Mechanics xxx (2017) xxx–xxx

overall upward shift of the degradation curve, gives a decrease in the slope m in a consistent way. On the other hand, an increase of t  , which leads to the slower degradation rate with the same initial value of the SFE, clearly increases the slope m. Thus, it could be said that the slope m varies depending on which mechanical degradation parameter is adjusted: i.e., the speed-controlling parameter or the initial value. 3.6. Effect of kinetic coefficients Fig. 11 shows the effect of the kinetic coefficients that control the speed of the crack jump. These coefficients could produce an effect provided that the corresponding driving force is positive, as depicted in Fig. 11(a). As the coefficients are increased, faster stepwise crack jumps take place. Since the stepwise crack jump periods occupy only a small portion of the total lifetime, there is a subtle decrease in tf with these coefficients (Fig. 11(c)). The SCG growth for various kinetic coef-

Fig. 11. Simulated results for various kinematic coefficients (0.002, 0.003, 0.004, 0.005 and 0.006).

Please cite this article in press as: Wee J-W, Choi B-H. Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.01.028

J.-W. Wee, B.-H. Choi / Engineering Fracture Mechanics xxx (2017) xxx–xxx

13

ficients is substantially overlapped in the initial and middle phases, but a small distinction is noticeable near the catastrophic failure stage as shown in Fig. 11(a). The slope m for various kinetic coefficients can be plotted as a single line similar to the case of remote stress variation (Fig. 11(b)). 4. Conclusion A parametric study of crack layer (CL) simulations and a detailed simulation procedure on discontinuous SCG behaviors has been presented in this study. A total of eight key parameters were varied over a reasonable range, and the effect of each parameter was investigated. This study reveals how each parameter changes the entire discontinuous SCG process and how much effect each parameter produces; in particular, the variation of the slope m and time to failure t f with these parameters was evaluated quantitatively. Such a close examination of each parameter allows to evaluate the contribution of each parameter in practical applications of the CL theory. Here is the summary of key findings from this study. The structure of the CL simulation was based on a loop statement. In an individual loop, the i-th loop for example, several stages are executed. After the crack length and process zone (PZ) length, lCR ði þ 1Þ and lPZ ði þ 1Þ, are calculated by using the governing equations, the specific fracture energy (SFE) at the crack tip is determined. Next, the appropriate failure criteria are applied to determine whether the specimen has failed or not. Finally, plotting of both crack length and PZ length is carried out in order to obtain the entire CL growth process. The parameters included in the PZ driving force X PZ (i.e., the remote stress (r1 ), drawing stress along PZ boundary (rdr ), transformation energy (ctr ), and natural draw ratio (k)) all affect both the time to step jump (Dt) and the stepwise jump length (Dl). Because the equilibrium PZ size depends on the equation for X PZ , and each subsequent crack jumps by the PZ CR length, the energy release rate at the crack tip (J CR 1 ), which is a driving term of the crack driving force (X ), also varies. Con0 sequently, Dt and Dlare changed. The plane strain elastic modulus (E ), the characteristic time for degradation (t⁄), and the

initial SFE (c0 ) can affect only the time to step jump. Although E0 is used in the calculation of X PZ , this term cannot affect the equilibrium PZ length, which can be achieved when X PZ ¼ 0. The variables t⁄ and c0 are included only in X CR , the crack driving force. Therefore, the equilibrium PZ size (Dl) is not changed by these parameters. The coefficients kCR and kPZ (i.e., the kinetic coefficients for crack growth and PZ growth, respectively) control the step jump rate. Because the crack arrest durations constitute the major part of the total lifetime (t f ) in such a discontinuous growth, these parameters could hardly affect the entire CL growth. The total lifetime tf increases linearly with rdr , ctr , E0 , c0 , and t⁄, and decreases in a parabolic shape with r1 and k. The slope m in the linear relationship between log(SCG rate) and log(K 1 ) does not vary consistently with ctr , k, and E0 , while m decreases with rdr , c0 , and increases with t⁄. Acknowledgements This work was supported by the Nuclear Research and Development Program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Trade, Industry and Energy (No. 20141510101640). This work is also supported by Korea University. References [1] Willams JG. Fracture mechanics of polymers. Ellis Horwood, Ltd; 1984. [2] Chudnovsky A, Zhou Z, Zhang H, Sehanobish K. Lifetime assessment of engineering thermoplastics. Int J Eng Sci 2012;59:108–39. [3] Choi BH, Chudnovsky A, Paradkar R, Michie W, Zhou Z, Cham PM. Experimental and theoretical investigation of stress corrosion crack (SCC) growth of polyethylene pipes. Polym Degrad Stab 2009;94:859–67. [4] Choi BH, Chudnovsky A. Observation and modeling of stress corrosion cracking in high pressure gas pipe steel. Metall Mater Trans A 2011;42A:383–95. [5] Plummer CJ, Goldberg A, Ghanem A. Micromechanisms of slow crack growth in polyethylene under constant tensile loading. Polymer 2001;42:9551–64. [6] Lu X, Brown N. The ductile-brittle transition in a polyethylene copolymer. J Mater Sci 1990;25:29–34. [7] Shah A, Stepanov EV, Capaccio G, Hiltner A, Baer E. Stepwise fatigue crack propagation in polyethylene resins of different molecular structure. J Polym Sci Part B: Polym Phys 1998;36:2355–69. [8] Parsons M, Stepanov E, Hiltner A, Baer E. Effect of strain rate on stepwise fatigue and creep slow crack growth in high density polyethylene. J Mater Sci 2000;35:1857–66. [9] Parsons M, Stepanov EV, Hiltner A, Baer E. Correlation of fatigue and creep slow crack growth in a medium density polyethylene pipe material. J Mater Sci 2000;35:2659–74. [10] Showaib EA, Moet A, Sehanobish K. Effect of short chain branching on the viscoelastic behavior during fatigue fracture of medium density ethylene copolymers. Polym Eng Sci 1995;35:786–93. [11] Lu X, Qian R, Brown N. Discontinuous crack growth in polyethylene under a constant load. J Mater Sci 1991;26:917–24. [12] Parsons M, Stepanov E, Hiltner A, Baer E. Correlation of stepwise fatigue and creep slow crack growth in high density polyethylene. J Mater Sci 1999;34:3315–26. [13] Chudnovsky A. Crack layer theory, NASA, Report #174634; 1984. [14] Chudnovsky A. Slow crack growth, its modeling and crack layer approach: a review. Int J Eng Sci 2014;83:6–41. [15] Choi BH, Balika W, Chudnovsky A, Pinter G, Lang RW. The use of crack layer theory to predict the lifetime of the fatigue crack growth of high density polyethylene. Polym Eng Sci 2009;49:1421–8. [16] Kim A, Song SH. Time dependent process zone growth in polycarbonate. KSME J 1995;9:421–7. [17] Botsis J. Crack and damage propagation in polystyrene under fatigue loading. Polymer 1988;29:457–62.

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Please cite this article in press as: Wee J-W, Choi B-H. Impact of material and physical parameters of the crack layer theory on slow crack growth behavior of high density polyethylene. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.01.028