Discontinuous slow crack growth modeling of semi-elliptical surface crack in high density polyethylene using crack layer theory

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Discontinuous slow crack growth modeling of semi-elliptical surface crack in high density polyethylene using crack layer theory Jung-Wook Wee a, Alexander Chudnovsky b, Byoung-Ho Choi c,∗ a

Korea University, Seoul, Republic of Korea The University of Illinois at Chicago, Chicago, IL 60607, USA c School of Mechanical Engineering, College of Engineering, Korea University, 5-ga Anam-dong, Sungbuk-ku, Seoul, Republic of Korea b

a r t i c l e

i n f o

Article history: Received 14 May 2018 Revised 2 August 2019 Accepted 15 August 2019 Available online xxx Keywords: Slow crack growth modeling High density polyethylene Crack layer theory Semi-elliptical surface crack

a b s t r a c t Surface flaws in structural materials are most likely to be generated during service time. One of the most general shapes of surface flaws is the semi-elliptical surface crack. Frequently, the slow crack growth (SCG) of a crack has been fitted based on the conventional Paris–Erdogan relationship. However, in the case of engineering plastics, which generally reveal a severe unrecoverable damage zone at the crack tip, the conventional Paris–Erdogan relationship is not appropriate to describe the SCG of a crack. Especially, SCG kinetics of some engineering polymers such as high-density polyethylene (HDPE) affect the SCG characteristics, i.e. non-conventional discontinuous SCG behavior, which cannot be simulated by conventional Paris–Erdogan relationship. It is known that the crack layer (CL) theory, which deals with driving forces of crack and process zone (PZ) together, can be a good mathematical model to simulate such SCG behavior. In this study, the discontinuous SCG of a semi-elliptical surface flaw in HDPE plate under cyclic tensile stress was simulated using the CL theory. Although the CL theory has the advantage of simulating the discontinuous SCG of HDPE accurately, until now the applications have been concentrated on onedimensional slow crack growth. In this paper, the CL model for two-dimensional surface crack growth for axial and bending loading conditions was developed for the first time. The proposed model is validated with actual test results, and the role of some key CL parameters on discontinuous SCG behavior of a semi-elliptical surface flaw is investigated by intensive parametric studies. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Considering the extensive use of high-density polyethylene (HDPE) in structural components such as gas or water distributing pipes, it is critical to assess their lifespan under the prescribed service conditions. It is well known that the pressurized HDPE pipes have various failure modes depending on the circumferential (hoop) stress, i.e. (1) localized ductile failure with ballooning at excessive internal pressure, (2) brittle cracking at intermediate internal pressure, and (3) stress corrosion cracking (SCC) at low internal pressure (Chudnovsky et al., 2012; Frank et al., 2009; Hutarˇ et al., 2011). The representative long-term failure mechanisms of HDPE pipes are related to the quasi-brittle fracture (brittle cracking), and a crack is commonly initiated from near-surface defects and grow to the other wall of the pipe to become a through crack (Barker et al., 1983; Choi et al., 2007; Lang et al., 1997). Accordingly, many studies on characterizing the slow crack growth

∗

Corresponding author. E-mail address: [email protected] (B.-H. Choi).

(SCG) of one-dimensional crack in pipe grade HDPE have been performed. HDPE reveals different SCG modes, i.e. in discontinuous, continuous, and combined manners, depending on the load level and temperature, even in the creep condition (Lu et al., 1991; Parsons et al., 1999). Such SCG kinetics cannot be explained by employing the conventional fracture mechanics parameters exclusively, i.e., the stress intensity factor (SIF), J-integral, or crack opening displacement (COD). Similar to other polymeric materials, HDPE shows a severe and considerable damage zone encompassing the main crack tip (Chudnovsky et al., 1995; Plummer et al., 2001). To solve such uncommon SCG characteristics, theoretical approaches considering the interactions between the main crack and surrounding damages are required. In the crack layer (CL) theory, the main crack and surrounding damage zone, the so-called process zone (PZ), are regarded as a multi-phase system (Chudnovsky, 1984). The PZ is comprised of micro-crazing, shear bands, or combination of these depending on the type of polymers. Then, the physical interactions between the crack and the PZ are explained by the irreversible thermodynamics (Choi et al., 2009; Kadota and Chudnovsky, 1992; Onsager, 1931). It has been demonstrated by previ-

https://doi.org/10.1016/j.ijsolstr.2019.08.016 0020-7683/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: J.-W. Wee, A. Chudnovsky and B.-H. Choi, Discontinuous slow crack growth modeling of semi-elliptical surface crack in high density polyethylene using crack layer theory, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr. 2019.08.016

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ous research that the discontinuous (stepwise) and continuous SCG of HDPE can be accurately regenerated by CL simulation with full scale of SCG kinetics (Wee and Choi, 2016b; Zhang et al., 2014). Despite its potential for broad applicability as the theory reflects various damage morphologies depending on the polymers, CL simulations have been devoted to one-dimensional crack growth in technically designed HDPE configurations, such as single edge notched tension (SENT), compact tension (CT), and stiff constant-K (SCK) specimens (Kadota and Chudnovsky, 1992; Wee and Choi, 2016b; Zhang et al., 2014). Although CL growth models for circular notched bar (CNB) specimens have recently been developed by expanding the CL theory to axisymmetric solids, it is still restricted to one-dimensional crack growth in a polar coordinate system (Wee and Choi, 2016a). Practically, as mentioned above, the actual SCG in HDPE pipes is generally initiated from the surface cracks. The typical surface cracks have a semi-elliptical shape, and they mainly grow on the planes containing the initial semi-elliptical flaw, two-dimensionally (Barker et al., 1983). The modeling of SCG from the semi-elliptical surface crack by using the Paris–Erdogan relationship has been conducted in numerous studies for metallic materials (Connolly and Collins, 1987; Lin and Smith, 1999; McFadyen et al., 1990; Paris and Erdogan, 1963; Wu, 1985). The Paris–Erdogan relationship, however, has no physical background and it is purely empirical. It uses one linear elastic fracture mechanics (LEFM) parameter, SIF, as a driving term, and other two fitting parameters must be obtained by experiments. Thus, its application to a polymeric material, which generally involves a process zone with large amount of irreversible damage in the vicinity of the crack tip, may be fundamentally inappropriate. Moreover, the discontinuous SCG kinetics, which is one of the typical SCG characteristics in HDPE, cannot be addressed in this manner. Therefore, further expansion of the CL theory to the semielliptical surface crack may enable us to establish a theoretical SCG model of the HDPE pipes in practical aspects. In this study, the discontinuous SCG of a semi-elliptical surface crack in HDPE plates under fatigue loading is modeled by expanding the current CL theory. The discontinuous semi-elliptical surface crack is considered to mimic the typical SCG behavior of HDPE, and a detailed simulation algorithm based on the CL theory is developed considering Green’s functions. Extensive parametric studies for key input CL parameters are performed. For validating the newly developed CL model for a semi-elliptical surface crack, actual test results are simulated by the developed CL model. Moreover, the developed CL model is also applied to simulate the discontinuous SCG growth under bending moment.

2. Crack layer theory for semi-elliptical surface cracks 2.1. Kinetic equations of semi-elliptical crack layer growth Let us consider an HDPE plate with a surface semi-elliptical crack under the maximum tensile stress σ ∞ and zero R-ratio (Fig. 1a). W is thickness of plate. The crack layer (CL) is the system composed of the main crack and surrounding damaged zone, which is also called the process zone (PZ), in which the damage morphologies depends on the polymers (Chudnovsky, 1984). The PZ can be separated further into two regions, i.e., active zone (AZ) and wake zone (WZ). In the AZ, the damage density (ρ ) increases with the elapsed time (ρ˙ > 0). In contrast, the damage density in the WZ is consistent with regard to time (ρ˙ = 0) owing to the traction-free condition. As the fatigue cycle runs, the crack and PZ would propagate with elapsed time, and such growths can be considered irreversible, because the recovering processes are physically impossible. In this article, it is assumed that the crack and PZ always have a semi-elliptical shape, with a constant center point as origin in x–y coordinates (Fig. 1b). Thus, provided that the depth and surface lengths of crack and PZ are calculated, the shapes of the two can be defined. To construct the kinetic equations for CL growth through irreversible thermodynamics, we divide the semi-elliptical CL into a number of fanwise segments with angle of θ (Fig. 1b). Onehalf of the semi-elliptical CL facet is divided into n segments. The cross-sectional view of the ith segment is presented in Fig. 1c. According to previous articles, HDPE typically reveals a PZ of narrowwedge shape in front of the main crack tip. The configurational lengths relevant to CL, i.e., the crack length (liCR ) and CL length (Li ) of the ith segment, are indicated in Fig. 1c. By employing the generalized damage parameter P of CL, the rate of total dissipation function of CL (˙ tot ) under isothermal condition is given by (Chudnovsky, 1984)

˙ tot =

2n  i=1

Ti S˙ irr,i = D˙ −

∂ ( + H )P˙ , ∂P

(1)

where the Ti and S˙ irr,i are the temperature and entropy production rate of the ith segment. D˙ is a plastic work rate that contributes to the damage evolutions.  and H represent the potential energy and enthalpy of the damaged solid, respectively. The rate of generalized damage parameter P˙ can also be expressed by the rate of configurational areas associated with the CL, considering the symmetricity on the y-axis in Fig. 1b, as follows (Wee and

Fig. 1. High density polyethylene (HDPE) plate with semi-elliptical surface crack under tensile stress. (a) Geometry of the plate with semi-elliptical surface crack, (b) A x-y plane with configurational parameters, and (c) Section view of ith fanwise segment in (b).

Please cite this article as: J.-W. Wee, A. Chudnovsky and B.-H. Choi, Discontinuous slow crack growth modeling of semi-elliptical surface crack in high density polyethylene using crack layer theory, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr. 2019.08.016

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3

Fig. 2. Notations of configurational areas and lengths. Subscript C and A stand for the surface point and deepest point, respectively.

Choi, 2016a):

P˙ = 2

n 



i=1



∂ P ˙ CR ∂ P ˙ L Ai + A . ∂ ALi i ∂ ACR i

(2)

In Eq. (2), ACR and ALi denote the crack and crack layer area of the i ith segment, respectively. Substituting Eq. (2) into Eq. (1) gives

˙ tot =

n 



Ti S˙ irr,i = D˙ + 2

i=1

n  

XiCR A˙ CR i

+

 L

XiL A˙ i



,

(3)

where the XiCR and XiCR are the thermodynamic forces (TFs) for growth of the corresponding areas; for instance, XiCR =

− ∂CR ( + H ) is a TF for growth of ACR in which other areas are i ∂ Ai

constant. Considering the rate of configurational areas as thermodynamic fluxes, the introduction of the linear irreversible thermodynamics without cross influences yields (Kadota and Chudnovsky, 1992; Onsager, 1931) CR CR A˙ CR i = ki Xi

(4)

for i = 1, 2, …, n. The kCR and kLi in Eq. (4) indicate the kinetic coefi ficient of CL evolution. The abovementioned time rate terms of the configurational areas, A˙ CR and A˙ Li , can be also expressed by the cori responding lengths, l˙CR and L˙ i , respectively, through the time differi

entiation of the configurational areas (see Fig. 1b): CR ˙CR A˙ CR i = l i l i θ

A˙ Li = Li L˙ i θ .

(5)

Substituting Eq. (5) into Eq. (4) gives

l˙iCR = L˙ i =

kCR i

liCR θ kLi

L i θ

κiCR

XiCR =

XiL =

liCR

κ

L i

Li

∂ AA

∂ AA

i=1

A˙ Li = kLi XiL ,

Hereafter, the 1st segment and nth segment in Fig. 1b will be denoted as C (surface position) and A (deepest position), respectively. Consequently, the total four TFs are to be calculated. These equations of TFs for HDPE can be simplified owing to the narrow wedge shape of the PZ with clear boundary. First, the variation of the driving terms with the configurational areas are given by − ∂ = J1CR , − ∂ L = J1PZ,A , − ∂  = J1CR , and − ∂ L = J1PZ,A . The J1CR,A and J1CR,C are CR CR ,A ,A



J1PZ,A = J1PZ,C =



K∞,A + Kdr,A E K∞,C + Kdr,C

2

2

E

XiL ,

(6) kCR

kL

where the constants κiCR = i θ and κiL = iθ . Hence, providing that the TFs are calculated under the given conditions, the rates of CL growth also can be defined by Eq. (6). 2.2. Application to HDPE Because the CL configuration is assumed as a semi-elliptical shape with a constant origin, the deepest point (A) and surface point (C) are sufficient to determine the shape of CL (see Fig. 2).

= =

(Ktot,A )2 E

(Ktot,C )2 E

,

(7)

where the K∞,A denotes the stress intensity factor (SIF) at the deepest position, owing to the remote stress (σ ∞ ), and Kdr,A for the drawing stress (σ dr ). E’ is the plane strain elastic modulus. Next, the enthalpy variations with two crack areas are given by ∂ H =2γ and ∂ H =2γ . The 2γ and 2γ indicate the specific fracC A A C CR CR ∂ AA

∂ AC

ture energy (SFE) at the deepest and surface position, respectively. The enthalpy variations with CL areas are ∂ HL = γh ∂ L ( V ρ dVA ) and ∂ H = γ ∂ ( h ∂ AL VAZ,C ρ dVC ), ∂ ACL C

XiCR

∂ AA

∂ AC

the energy release rate (ERR) at the crack tip along the deepest and surface position, respectively. Similarly, the J1PZ,A and J1PZ,C represent the ERR at the PZ tip, at deepest and surface position, respectively. In the case of HDPE, the clear boundary between the AZ and nondamaged surrounds allows the decomposition of CL into PZ cutoff undamaged surrounds and PZ material, and it can be thought that the constant drawing stress (σ dr ) acts on the AZ boundary, as in Fig. 3 (Chudnovsky et al., 2012; Stojimirovic et al., 1992; Zhang et al., 2014). Under such assumption, the ERR at the PZ tip can be written as

∂ AA

∂ AA

AZ,A

where the γ h is the enthalpy jump owing to

the unit discontinuous area, ρ the discontinuous area per unit volume (damage density), VAZ,A the AZ volume of segment A, and VAZ,C the AZ volume of segment C. It is acceptable that the damage den∂V sity in the AZ of HDPE is almost constant, thus ∂ HL = γ tr AZ,A and L ∂ H =γ tr ∂ VAZ,C ∂ ACL ∂ ACL

, where the

γ tr

∂ AA

∂ AA

= ργ h indicates the density of enthalpy

required for material transformation. The AZ volume of ith seg L ment is given by VAZ,i = CRi wo,i (r ) · r · θ · dr, where the r means li

the distance of the integral element from the origin. As illustrated in Fig. 3, wo,i (r) denotes the width of the original material, and it δ

(r )

is assumed to be equal to tot,i λ−1 , where the δ tot,i (r) represents the total COD along the AZ region owing to the combination of the re-

Please cite this article as: J.-W. Wee, A. Chudnovsky and B.-H. Choi, Discontinuous slow crack growth modeling of semi-elliptical surface crack in high density polyethylene using crack layer theory, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr. 2019.08.016

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Fig. 3. The method of superposition. (a) The entire crack layer (CL) system under tensile stress, (b) PZ cutoff elastic solid, and (c) PZ material.

√ where F(a/W) is a factor in the SIF formula, KI = σ0 π a F (a/W ). σ 0 is a characteristic stress. The G is determined from the selfconsistency requirement of weight function as follows:

G

a W



= S1 ( a ) − 4 · F

a √ W

 √a

a · S2 ( a )

·

S3 ( a )

.

(12)

The factors S1 –S3 are

S1 ( a ) = Fig. 4. The displacement field at the semi-elliptical crack surface due to remote tensile stress. The crack depth and a half length are denoted by a and c, respectively. The thickness of plate is W.

mote stress (σ ∞ ) and drawing stress (σ dr ) acting on the AZ boundary (Zhang et al., 2014). λ is the natural drawing ratio of HDPE. Therefore, the AZ volumes at the deepest and surface positions are given by

VAZ,A = VAZ,C



θ λ−1



θ = λ−1

LA

δtot,A (y )ydy

lACR LC

(8)

Inserting Eqs. (5)–(8) in the TF equations yields the final formulations of TFs for HDPE as

XCCR = J1CR,C − 2γC XCL

=

(Ktot,C )2 E

γ tr ∂ − LC (λ − 1 ) ∂ LC



LC lCCR

x · δtot,C (x )dx

(9)

at the surface position, and

XACR = J1CR,A − 2γA

(Ktot,A )

γ

2

XAL =

E

−



∂

tr

LA ( λ − 1 ) ∂ LA

LC

ur (a, y ) =

√

E 2

4·F

a √ W

·



a 0 a 0

a

  a 2 F

0

lACR

y · δtot,A (y )dy

(10)

√ a· a−y+G

 a  a − y 3/2  ( ) W

·

√ a

,

(11)

W

a da

√

σr (y ) · a − y dy σr (y ) · (a − y )1.5 dy .

(13)

Under the uniform tensile stress, the reference stress σ r = σ ∞ is a constant value. Second, the displacement along the x-direction is assumed as the Griffith-crack opening profile (Mattheck et al., 1983a),

u(x, y = 0 ) = umax

1−

 x 2 c

,

(14)

where the umax is a maximum displacement at the crack mouth, ur (a,y = 0) by Eq. (11). Finally, the COD along each direction is equal to twice the crack face displacements. Mattheck et al. (1983b) demonstrated that such methodology accurately estimates the opening displacement under uniform tensile stress and pure bending. The remaining component required to calculate the TFs is δ dr , which is COD owing to the constant σ dr acting on the AAZ (Fig. 2). In this study, it is obtained by the FEM. The normalized geometric factors affecting the δ dr at the deepest and surface position are

μ1 =



at the deepest position. K∞ at the deepest and surface position are obtainable from (Wang and Lambert, 1995). Kdr,A and Kdr,C can be calculated by the surface integral of the point-wise weight function for a semielliptical surface crack (Jin and Wang, 2013). The corresponding integration area for Kdr is AAZ in Fig. 2. The COD owing to the remote stress (δ ∞ ) was determined by using the method suggested in (Mattheck et al., 1983b; Petroski and Achenbach, 1978). First, the reference displacement field along the y-direction is calculated under the assumption of the plane crack problem (Fig. 4).

σ0

S3 ( a ) =







δtot,C (x )xdx .

lCCR

S2 ( a ) =

√

π 2 σ0

δdr,C (ξ ) · δdr,A (η ) ·

μ2 =

LA LC ,

E 2σdr LC E

2σdr LA

μ3 =

lACR LA ,

lCR

and μ4 = LC . Then, the CODs along C the AZ boundary at the surface and deepest positions owing to the drawing stress can be expressed on the basis of the universal form of the COD profile (Wu and Carlsson, 1991), LA W,

= M1 ξ 0. 5 + M2 ξ 1. 5 + M3 ξ 2. 5 = M4 η 0. 5 + M5 η 1. 5 + M6 η 2. 5 ,

(15)

where the reference coordinates ξ and η are (Fig. 2)

ξ =

LC − x LC − lCCR

η=

LA − y . LA − lACR

(16)

The coefficients in Eq. (15) must be function of geometries, Mk = fk (μ1 ,μ2 ,μ3 ,μ4 ) for integer k = 1–6. The coefficients Mk with

Please cite this article as: J.-W. Wee, A. Chudnovsky and B.-H. Choi, Discontinuous slow crack growth modeling of semi-elliptical surface crack in high density polyethylene using crack layer theory, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr. 2019.08.016

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J.-W. Wee, A. Chudnovsky and B.-H. Choi / International Journal of Solids and Structures xxx (xxxx) xxx Table 1 M1 –M6 for crack opening displacement (COD) due to drawing stress.

μ1

μ2

μ3

μ4

M1

M2

M3

M4

M5

M6

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.8 0.8 0.8

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 1 1 1 1 1 1 1 1 1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 1 1 1 1 1 1 1 1 1 0.5 0.5 0.5

0.6 0.6 0.6 0.8 0.8 0.8 1 1 1 0.6 0.6 0.6 0.8 0.8 0.8 1 1 1 0.6 0.6 0.6 0.8 0.8 0.8 1 1 1 0.6 0.6 0.6 0.8 0.8 0.8 1 1 1 0.6 0.6 0.6 0.8 0.8 0.8 1 1 1 0.6 0.6 0.6 0.8 0.8 0.8 1 1 1 0.6 0.6 0.6 0.8 0.8 0.8 1 1 1 0.6 0.6 0.6 0.8 0.8 0.8 1 1 1 0.6 0.6 0.6

0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1

0.3568 0.2209 0 0.3399 0.2171 0 0.3372 0.2029 0 0.6322 0.3311 0 0.6214 0.3129 0 0.5933 0.3016 0 0.6994 0.3289 0 0.6844 0.3222 0 0.6613 0.3059 0 0.7315 0.3425 0 0.6939 0.3134 0 0.6672 0.293 0 0.4319 0.2593 0 0.4187 0.2394 0 0.3954 0.2246 0 0.7719 0.4318 0 0.7258 0.3896 0 0.6966 0.3609 0 0.863 0.4395 0 0.8128 0.4019 0 0.7825 0.3784 0 0.883 0.4342 0 0.8316 0.3935 0 0.8009 0.3726 0 0.862 0.4518 0

0.2388 0.1988 0 0.2572 0.189 0 0.2778 0.2082 0 0.4099 0.3619 0 0.2867 0.3135 0 0.3685 0.2849 0 0.5972 0.4619 0 0.5106 0.3738 0 0.4896 0.3409 0 0.6385 0.448 0 0.5859 0.3889 0 0.5685 0.3749 0 0.2471 0.1911 0 0.2365 0.1892 0 0.2599 0.1894 0 0.2781 0.2371 0 0.2618 0.1993 0 0.2422 0.1887 0 0.3682 0.3356 0 0.3196 0.263 0 0.2876 0.2204 0 0.4192 0.3661 0 0.3651 0.2984 0 0.3315 0.2547 0 0.3689 0.323 0

−0.3261 −0.2349 0 −0.366 −0.2436 0 −0.3954 −0.2682 0 −0.5883 −0.406 0 −0.5286 −0.389 0 −0.6035 −0.3797 0 −0.7616 −0.4783 0 −0.727 −0.4379 0 −0.7258 −0.4214 0 −0.8023 −0.466 0 −0.7848 −0.438 0 −0.7837 −0.4386 0 −0.3244 −0.2273 0 −0.3446 −0.245 0 −0.3832 −0.2572 0 −0.5038 −0.3319 0 −0.5214 −0.3205 0 −0.5236 −0.3274 0 −0.6228 −0.4172 0 −0.6086 −0.3778 0 −0.5965 −0.3529 0 −0.675 −0.4404 0 −0.6525 −0.4019 0 −0.6361 −0.3758 0 −0.5678 −0.3885 0

0.7689 0.727 0.7334 0.3213 0.309 0.2958 0 0 0 0.7334 0.6872 0.6709 0.3249 0.2929 0.2873 0 0 0 0.6448 0.6066 0.6172 0.3212 0.2873 0.2684 0 0 0 0.618 0.5843 0.5627 0.3049 0.2748 0.2569 0 0 0 0.9349 0.8643 0.8482 0.4145 0.3458 0.3094 0 0 0 0.9063 0.8345 0.7891 0.4437 0.3825 0.3455 0 0 0 0.781 0.7251 0.691 0.3976 0.3554 0.3276 0 0 0 0.7204 0.6755 0.6467 0.3794 0.3381 0.3137 0 0 0 0.9256 0.8504 0.8132

0.4813 0.4702 0.4129 0.3181 0.2942 0.2686 0 0 0 0.4586 0.408 0.3682 0.3521 0.2971 0.2591 0 0 0 0.4808 0.4227 0.3326 0.3358 0.2849 0.2568 0 0 0 0.435 0.3857 0.3551 0.3305 0.2826 0.2553 0 0 0 0.6715 0.5609 0.4679 0.3999 0.3401 0.3093 0 0 0 0.2993 0.2393 0.2113 0.248 0.2025 0.1841 0 0 0 0.2843 0.2385 0.2096 0.246 0.1915 0.1684 0 0 0 0.2694 0.227 0.2005 0.2305 0.1911 0.1685 0 0 0 0.488 0.3783 0.3049

−0.6533 −0.6486 −0.6176 −0.3737 −0.3642 −0.3473 0 0 0 −0.6271 −0.603 −0.5851 −0.3892 −0.3557 −0.3381 0 0 0 −0.6163 −0.5935 −0.551 −0.3736 −0.3476 −0.3326 0 0 0 −0.5769 −0.5631 −0.5524 −0.3639 −0.342 −0.3287 0 0 0 −0.7412 −0.6803 −0.6339 −0.4158 −0.3838 −0.3667 0 0 0 −0.5191 −0.4933 −0.4821 −0.332 −0.3096 −0.3003 0 0 0 −0.4969 −0.4839 −0.4734 −0.326 −0.2993 −0.288 0 0 0 −0.4753 −0.4671 −0.4597 −0.3124 −0.2956 −0.2852 0 0 0 −0.6037 −0.5642 −0.5392

(continued on next page)

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μ1

μ2

μ3

μ4

M1

M2

M3

M4

M5

M6

0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8

0.5 0.5 0.5 0.5 0.5 0.5 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 1 1 1 1 1 1 1 1 1

0.8 0.8 0.8 1 1 1 0.6 0.6 0.6 0.8 0.8 0.8 1 1 1 0.6 0.6 0.6 0.8 0.8 0.8 1 1 1

0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1

0.7995 0.3944 0 0.7615 0.3613 0 0.9909 0.5077 0 0.9164 0.4451 0 0.8773 0.4114 0 0.9871 0.4862 0 0.9178 0.429 0 0.8809 0.4041 0

0.3253 0.27 0 0.2961 0.243 0 0.3244 0.2884 0 0.2785 0.2275 0 0.2427 0.1973 0 0.3714 0.3227 0 0.3176 0.2674 0 0.2834 0.2169 0

−0.5678 −0.3684 0 −0.5633 −0.3606 0 −0.5991 −0.3871 0 −0.5858 −0.356 0 −0.572 −0.3416 0 −0.6482 −0.4105 0 −0.6245 −0.3837 0 −0.6084 −0.3518 0

0.4368 0.3762 0.3491 0 0 0 0.8581 0.7993 0.768 0.4455 0.3917 0.3717 0 0 0 0.7782 0.7303 0.7114 0.4164 0.3698 0.354 0 0 0

0.3351 0.2607 0.2118 0 0 0 0.2443 0.1757 0.1217 0.2025 0.1544 0.1041 0 0 0 0.2225 0.1708 0.1131 0.1896 0.1522 0.1055 0 0 0

−0.3687 −0.3374 −0.3172 0 0 0 −0.4572 −0.4401 −0.4213 −0.2959 −0.2772 −0.2525 0 0 0 −0.438 −0.431 −0.409 −0.2859 −0.2732 −0.2503 0 0 0

Fig. 5. Algorithm of crack layer (CL) simulation of semi-elliptical surface crack growth.

regard to μ1 ∼μ4 are arranged in Table 1. Finally, under the given geometries, physical properties, and loading conditions, the rates of crack and PZ growth can be calculated through Eqs. (9), (10), and (6). 2.3. Development of the SCG algorithm based on the proposed model to HDPE The key equations in the crack layer (CL) simulation are nonlinearly combined themselves. Thus, the entire simulation for the crack and PZ growth must be proceeded by the numerical method. The structure of the algorithm is based on the loop statement. At the jth loop, the main and minor axes of the semi-elliptical crack

and PZ for the next step (j + 1) are calculated using the crack and PZ configuration at the jth step, as follows.

κACR · XACR ( j ) t lACR ( j ) κ L · X L( j) LA ( j + 1 ) = LA ( j ) + A A t LA ( j ) κ CR · X CR ( j ) lCCR ( j + 1 ) = lCCR ( j ) + C CRC t lACR ( j + 1 ) = lACR ( j ) +

lC

LC ( j + 1 ) = LC ( j ) +

κ · XCL ( j ) t , LC ( j ) L C

(17)

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Fig. 6. The variation of crack shape with the elapsed time, at the initial aspect ratio of 1. (a) Crack and CL length on depth direction, (b) Crack and CL length on surface direction, and (c) aspect ratio of crack and CL profiles.

7

Fig. 7. The variation of crack shape with the elapsed time, at the initial aspect ratio of 0.6; (a) Crack and CL length on depth direction, (b) Crack and CL length on surface direction, and (c) Aspect ratio of crack and CL profiles.

where t is a prescribed time increment. The TFs, XACR , XAL , XCCR , and XCL in Eq. (17), are to be calculated through the Eqs. (9) and (10). In the TFs for crack growth, the surface fracture energy (SFE) at the deepest and surface position, 2γ A and 2γ C , should decay with the elapsed time after the transformation of fresh material into active zone (AZ) (Chudnovsky and Shulkin, 1999). The degradation function used in this study is 2γ = 2γ 0 /{1 + (ti /t∗ )}, where the 2γ 0 is SFE of undamaged solid, ti the elapsed time, and t∗ the characteristic time for material degradation. Thus, t∗ controls the rate of SFE decay. Practically, the CL simulation should be stopped when the instability conditions of the given system are satisfied. However, the pointwise weight function adopted for Kdr is valid in 0.2 ≤ LA /LC ≤ 1.0 and 0.2 ≤ LA /W ≤ 0.8 (Jin and Wang, 2013). Thus, in this study, the CL simulation is terminated providing that not only the stress intensity factor (SIF) at the crack tip is larger than the fracture toughness of HDPE, but also one of the above geometrically valid ranges is violated. The overall algorithm is depicted in Fig. 5. 3. Results and discussions 3.1. Parametric study and validation of developed model To confirm the validity of the suggested model, a parametric study was conducted for several important parameters. A sufficiently wide and long HDPE plate with a surface semi-elliptical crack is considered. The plate is loaded by uniform tensile fatigue stress with a R-ratio of zero at room temperature. The input parameters are listed in Table 2. As the half width of the plate is 5 times its thickness, the finite width effect can be negligible. Figs. 6–8 illustrate the SCG kinetics of crack length and CL length at the deepest (Figs. 6–8(a)) and surface (Figs. 6–8(b)) positions under uniform tensile fatigue stress, when the initial aspect ratios are 1.0, 0.6, and 0.25, respectively. The solid and dotted curves represent the crack and CL length, respectively. The typi-

Fig. 8. The variation of crack shape with the elapsed time, at the initial aspect ratio of 0.25; (a) Crack and CL length on depth direction, (b) Crack and CL length on surface direction, and (c) Aspect ratio of crack and CL profiles.

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J.-W. Wee, A. Chudnovsky and B.-H. Choi / International Journal of Solids and Structures xxx (xxxx) xxx Table 2 The input parameters used in the parametric study. Parameter symbol

Description

Unit

Value

W a0

Thickness of HDPE plate Initial crack depth Aspect ratio of initial semi-elliptical crack Applied stress Drawing stress Characteristic time Transformation energy density Plane strain elastic modulus Natural drawing ratio

mm mm – MPa MPa s mJ/mm3 MPa –

10 2 0.25, 0.4, 0.6, 0.8, 1.0 6, 8, 10 15 50, 100, 150 20, 30, 40 1000 5

α0 σ∞ σ dr t∗

γ tr E’

λ

cal SCG mode of HDPE, discontinuous propagation, is manifested in the stepwise jumps of crack and CL length with time. The variations of aspect ratio of the crack and PZ tip with regard to the elapsed time are also given in Figs. 6–8c. At the first arresting step of the semi-circular initial crack (α 0 = 1), the required times for crack to grow at the surface and deepest positions are not the same. The crack in the vicinity of the surface grows first in this case, and the PZ tip at this position also starts to propagate to obtain a new equilibrium configuration, indicating zero TFs. The changed crack and PZ tip at the surface also affect the TFs at the deepest position, and they grow to achieve another equilibrium. Thus, it can be said that the considered CL model is a 4-degree of freedom (DOF) system in which each component affects the other. These complex processes are continued until the prescribed instability condition, which results in disturbed plots, not smooth curves, of the aspect ratio of crack and PZ tip with the elapsed time (Figs. 6–8(c)). The development profiles of semi-elliptical flaws with the different initial aspect ratios at certain elapsed times (t1 –t6 ) are depicted in Fig. 9, where the t1 –t6 are indicated in Figs. 6–8c. At the early stage of SCG, the crack profile aspects are quite different with initial aspect ratio. For example, for the initial aspect ratio of 1, the surface and deepest regions start to propagate at a similar speed, and it is also indicated in the invariant aspect ratio of crack and PZ tip, during the first half (Fig. 6c). In case of the shallow initial crack, i.e., the initial aspect ratio of 0.25, the crack and PZ mainly grow along the crack depth direction (y-direction in Fig. 1b). It results in the consistent growth of aspect ratio with time (Fig. 8c). However, the final profiles of the crack are approximately equal. Such tendencies can be clearly seen in Fig. 10, depicting the variation of aspect ratio of crack and PZ tip with the normalized crack depth (lACR /W ). The different initial aspect ratio at the early stage converges at about 0.7, which can be thought as a stable profile regardless of the initial aspect ratio. These trends have been exhibited by experiments and models in many studies in which the semi-elliptical crack growth is predicted by considering the two points (surface and deepest points) and using the Paris–Erdogan relationship (Connolly and Collins, 1987; Wu, 1985). Further, Lin and Smith (Lin and Smith, 1999) demonstrated such converging trends by 3D FE-step by step method. Therefore, the CL simulation, which is based on the theoretical approach, can reproduce the proper tendencies that have been proven by various methods. Fig. 11 presents the effect of maximum fatigue tensile stress (σ ∞ ) on the CL growth of the initial semi-circular defect. With the applied tensile stress, the crack and PZ growth rates at the deepest Fig. 11a) and surface (Fig. 11b) positions are increased. This results from the increased driving terms (JiCR and JiPZ ) in the TFs equations, Eqs. (9) and (10). The equilibrium AZ sizes to make zero TF, therefore, are expanded. Furthermore, the crack arresting time, which equals to the time requirement for positive TF for crack growth, is reduced. Accordingly, the aspect ratio varies faster with the elapsed time (Fig. 11c). However, the variation of the aspect

Fig. 9. The variation of crack profile with elapsed time under tensile stress; (a) Initial aspect ratio of 1, (b) Initial aspect ratio of 0.6, and (c) Initial aspect ratio of 0.25.

ratio with crack depth shows overlapped curves with the stress (Fig. 11d). The effect of transformation energy density (γ tr ) is shown in Fig. 12. Physically, the γ tr denotes the required energy density for the undamaged material to transform to the highly drawn matter, comprising the AZ of HDPE. Therefore, the increase in γ tr would result in the decreased growth rate of the CL, as Eqs. (9) and (10) indicate. Such physical interpretations clearly appear in Fig. 12a and b. At the first step jump, the TFs for crack growth are invariant, with entered value of γ tr . Thus, the first crack arresting time is not varied. The equilibrium AZ length, however, is absolutely different from the TFs for PZ growth. This discrepancy causes the difference in SCG after the first jump. Similar to the case of maximum tensile stress, the development of the aspect ratio with normalized crack depth is not very different with γ tr . For the last parametric study, the characteristic time for material degradation (t∗ ) was changed. The SFE (2γ ) of the AZ material

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Fig. 10. The aspect ratio changes with the normalized crack depth under tensile stress, at the various initial aspect ratio.

undergoes decrease with elapsed time (ti ). In this situation, the t∗ controls the rate of degradation, i.e., the lower the t∗ , the faster the decrease in SFE. For the stable crack to propagate, the TF for crack (XiCR ) should become positive. In coincidence with the TFs for crack growth, Eqs. (9) and (10), the crack arresting time may be shortened as the t∗ decreases (Fig. 13a and b). It is worth noting that the equilibrium AZ length is not varied, because the TFs for the PZ growth are not affected with regard to t∗ , and the required AZ lengths for equilibrium too. This results in equal step jump lengths regardless of t∗ . The aspect ratio converges faster with the decrease in t∗ (Fig. 13c). The aspect ratio of crack profile with the normalized crack depth shows perfectly overlapped plots in Fig. 13d. It implies that the t∗ serves as a time scale parameter in the life-

9

time of the considered system. The influences of σ ∞ , γ tr , and t∗ on the SCG kinetics at the surface and deepest positions show the same tendency highlighted, in which the SCG of HDPE in a circular notched bar (CNB) specimen was modeled by CL theory (Wee and Choi, 2016a). A comparison between an actual test of discontinuous SCG in an HDPE pipe and the CL simulation was also conducted. The experimental fracture surface was adopted from Barker et al. (1983). The experiment was performed on an HDPE pipe under fatigue internal pressure. The maximum hoop stress was 4.93 MPa with zero R-ratio at 80 °C. The outer diameter and SDR of the pipe was reported as 63 mm and 11, respectively. The pipe can be considered as a thin walled pipe (SDR >10), thereby applying the same simulation modules, the semi-elliptical surface flaw in the HDPE plate. The discontinuous SCG can be accurately estimated by CL simulation (Fig. 14). The striations from the CL simulation are depicted by dotted curves (Fig. 14b). The CL kinetics along the depth and surface directions are also given in Figs. 14a and c. The variation of aspect ratio with elapsed time is shown in Fig. 14d. It demonstrates that the developed CL model is applicable to predict the 2D SCG, such as the semi-elliptical surface crack growth. 3.2. Expansion of the proposed model for the HDPE plate under pure bending moment The validity of the crack layer (CL) growth modeling of a semielliptical surface flaw in an HDPE plate under tensile fatigue is obtained by the parametric study, and it reproduces the actual experiments. The bending fatigue is also one of the prevalent loading conditions on mechanical components. In this section, the SCG of a semi-elliptical surface flaw in an HDPE plate under the pure bending fatigue is simulated by modifying the CL model previously developed (Fig. 15). The maximum bending stress from the remote bending moment, Mb , is denoted by σ b . In the equations of TFs, Eqs. (9) and

Fig. 11. The crack layer growth in the semi-elliptical surface crack with elapsed time at the various maximum tensile stresses; (a) Crack and CL length on the depth direction, (b) Crack and CL length on surface direction, (c) Aspect ratio of crack and CL, and (d) Aspect ratio with normalized crack depth.

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Fig. 12. The crack layer growth in the semi-elliptical surface crack with elapsed time at the various transformation energy densities; (a) Crack and CL length on the depth direction, (b) Crack and CL length on surface direction, (c) Aspect ratio of crack and CL, and (d) Aspect ratio with normalized crack depth.

Fig. 13. The crack layer growth in the semi-elliptical surface crack with elapsed time at the various characteristic time for material degradation; (a) Crack and CL length on the depth direction, (b) Crack and CL length on surface direction, (c) Aspect ratio of crack and CL, and (d) Aspect ratio with normalized crack depth.

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Fig. 14. Comparison of crack growth between the actual test and crack layer simulation; (b) The experimental discontinuous crack growth, (a) Crack and CL growth on depth direction, (c) Crack and CL growth on surface direction, and (d) Aspect ratio of crack and CL profiles. The fracture surface in (b) is adopted from Barker et al. (1983) with permission.

Fig. 15. High density polyethylene (HDPE) plate with semi-elliptical surface crack under bending moment.

(10), the components to be revised are the stress intensity factor (SIF) at the deepest and surface positions (K∞,A and K∞,C ) and COD at these positions (δ ∞,A and δ ∞,C ) owing to the bending moment. The K∞,A and K∞,C were reported in a previous article Newman and Raju, 1981). The δ ∞,A and δ ∞,C can be obtained by (Mattheck et al., 1983b), as in Eqs. (11)–(14). The reference stress σ r (y) is changed to

  2y σr ( y ) = σb 1 − W

(18)

Fig. 16. The variation of crack profile under bending moment; (a) Initial aspect ratio of 1, (b) Initial aspect ratio of 0.6, and (c) Initial aspect ratio of 0.25.

The characteristic stress σ 0 in Eqs. (11) and (13), becomes σ 0 = σ b . The remaining simulation procedures are same as the tensile loading condition. Fig. 16 shows the crack growth profiles under the cyclic bending moment with regard to the various initial aspect ratios, 0.25, 0.6, and 1.0. The maximum bending stress was σ b = 9 MPa. It can

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the key parameters. The discontinuous fracture surface of the actual test of HDPE pipes under cyclic internal pressure was accurately regenerated by the developed model. The simulated results were compared with actual test results to validate the proposed model, and it was shown that the proposed model could predict the discontinuous SCG behavior of thick-walled pipe accurately. (3) Further modification was conducted to apply the proposed model to the plate under pure bending fatigue loading conditions. The newly developed CL model could be applied to predict SCG characteristics under bending fatigue loading condition, and it was expected that the prediction of the lifetime of HDPE pipe with various loading conditions could be accurately predicted even for complicated surface cracks as well as mixed loading conditions. Acknowledgment

Fig. 17. The aspect ratio changes with the normalized crack depth under bending moment, at the various initial aspect ratio.

be seen that the CL growth at the surface position is much faster than that at the deepest position owing to the stress gradient, i.e., the larger SIF at the surface position from the higher membrane stress on the surface of plate (x-z plane in Fig. 15). However, similar to the case of tensile loading, the final profiles are nearly the same. Such simulation results can be more quantitatively observed in Fig. 17, depicting the variations of aspect ratio of crack and PZ with regard to the normalized crack depth. Every case of the initial aspect ratio is asymptotically converged to the dotted line, which is considerably lower than in the tensile case. The simulation results of stable crack profiles under bending moment, shallower than in the case of tensile loading, have been reported in diverse studies (Connolly and Collins, 1987; Lin and Smith, 1999; McFadyen et al., 1990; Wu, 1985). These results, in accordance with the previous studies, indicate that the present CL growth modeling for surface semi-elliptical flaws properly applies the CL theory to the loading conditions of both tensile stress and bending moment. 4. Conclusions One of the typical long-term failure modes of thermoplastic pipes such as HDPE pipes is brittle fracture initiated from surface flaws among three distinctive fracture mechanisms. Fundamental approaches are required to model brittle fracture behaviors owing to the unique SCG characteristics of HDPE, i.e., discontinuous or stepwise SCG. To predict the lifespan of HDPE pipes under service conditions, such SCG behavior from the surface defects must be theoretically modeled. It has been proven that the CL theory can be a good mathematical tool to analyze such SCG behaviors, but most studies are restricted in one dimensional SCG. In this study, the discontinuous SCG kinetics of a surface semi-elliptical flaw under fatigue loading was analyzed and simulated using the CL theory for the first time. Key findings from this study are listed below. (1) The new SCG model for a semi-elliptical surface crack was developed based on the expansion of the CL theory. The governing equations were physically and analytically driven for HDPE, and the detail algorithm to simulate SCG behaviors for a semi-elliptical crack of a HDPE plate was also introduced. (2) The discontinuous SCG, which is a typical SCG mode in HDPE, was simulated and validated. The parametric study with various CL variables revealed reasonable trends with

This work was supported by the Technology Innovation Program Project (No. 10076562) of Korea Evaluation Institute of Industrial Technology (KEIT) funded By the Ministry of Trade, industry & Energy (MI, Korea). This work is also supported by the K-CLOUD research project (No. 2017-Technology-15) funded by Korea Hydro & Nuclear Power Co. LTD. References Barker, M., Bowman, J., Bevis, M., 1983. The performance and causes of failure of polyethylene pipes subjected to constant and fluctuating internal pressure loadings. J. Mater. Sci. 18, 1095–1118. Choi, B.-.H., Chudnovsky, A., Sehanobish, K., 2007. Stress corrosion cracking in plastic pipes: observation and modeling. Int. J. Fract. 145, 81–88. Choi, B.H., Balika, W., Chudnovsky, A., Pinter, G., Lang, R.W., 2009. The use of crack layer theory to predict the lifetime of the fatigue crack growth of high density polyethylene. Polym. Eng. Sci. 49, 1421–1428. Chudnovsky, A., 1984. Crack Layer Theory. NASA report, N174634. Chudnovsky, A., Shulkin, Y., 1999. Application of the crack layer theory to modeling of slow crack growth in polyethylene. Int. J. Fract. 97, 83–102. Chudnovsky, A., Shulkin, Y., Baron, D., Lin, K., 1995. New method of lifetime prediction for brittle fracture of polyethylene. J. Appl. Polym. Sci. 56, 1465–1478. Chudnovsky, A., Zhou, Z., Zhang, H., Sehanobish, K., 2012. Lifetime assessment of engineering thermoplastics. Int. J. Eng. Sci. 59, 108–139. Connolly, M., Collins, R., 1987. The measurement and analysis of semi-elliptical surface fatigue crack growth. Eng. Fract. Mech. 26, 897–911. Frank, A., Freimann, W., Pinter, G., Lang, R.W., 2009. A fracture mechanics concept for the accelerated characterization of creep crack growth in PE-HD pipe grades. Eng. Fract. Mech. 76, 2780–2787. Hutarˇ, P., Ševcˇ ík, M., Náhlík, L., Pinter, G., Frank, A., Mitev, I., 2011. A numerical methodology for lifetime estimation of HDPE pressure pipes. Eng. Fract. Mech. 78, 3049–3058. Jin, Z., Wang, X., 2013. Weight functions for the determination of stress intensity factor and T-stress for semi-elliptical cracks in finite thickness plate. Fatigue Fract. Eng. Mater. Struct. 36, 1051–1066. Kadota, K., Chudnovsky, A., 1992. Constitutive equations of crack layer growth. Polym. Eng. Sci. 32, 1097–1104. Lang, R., Stern, A., Doerner, G., 1997. Applicability and limitations of current lifetime prediction models for thermoplastics pipes under internal pressure. Die Angew. Makromol. Chem. 247, 131–145. Lin, X., Smith, R., 1999. Finite element modelling of fatigue crack growth of surface cracked plates: part II: crack shape change. Eng. Fract. Mech. 63, 523–540. Lu, X., Qian, R., Brown, N., 1991. Discontinuous crack growth in polyethylene under a constant load. J. Mater. Sci. 26, 917–924. Mattheck, C., Morawietz, P., Munz, D., 1983a. Stress intensity factor at the surface and at the deepest point of a semi-elliptical surface crack in plates under stress gradients. Int. J. Fract. 23, 201–212. Mattheck, C., Munz, D., Stamm, H., 1983b. Stress intensity factor for semi-elliptical surface cracks loaded by stress gradients. Eng. Fract. Mech. 18, 633–641. McFadyen, N.B., Bell, R., Vosikovsky, O., 1990. Fatigue crack growth of semi-elliptical surface cracks. Int. J. Fatigue 12, 43–50. Newman, J., Raju, I., 1981. An empirical stress-intensity factor equation for the surface crack. Eng. Fract. Mech. 15, 185–192. Onsager, L., 1931. Reciprocal relations in irreversible processes. I.. Phys. Rev. 37, 405. Paris, P., Erdogan, F., 1963. A critical analysis of crack propagation laws. J. Basic Eng. 85, 528–533. Parsons, M., Stepanov, E., Hiltner, A., Baer, E., 1999. Correlation of stepwise fatigue and creep slow crack growth in high density polyethylene. J. Mater. Sci. 34, 3315–3326. Petroski, H., Achenbach, J., 1978. Computation of the weight function from a stress intensity factor. Eng. Fract. Mech. 10, 257–266.

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Please cite this article as: J.-W. Wee, A. Chudnovsky and B.-H. Choi, Discontinuous slow crack growth modeling of semi-elliptical surface crack in high density polyethylene using crack layer theory, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr. 2019.08.016