Journal of Mechanical Science and Technology 25 (5) (2011) 1201~1206

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0225-2

Crack-tip opening angle-based numerical implementation

for fully plastic crack growth analyses† Sunghwan Jung1,* and Hyungyil Lee2

1Department of Mechanical Engineering, Dankook University, JukJeon, 448-701, Korea 2Department of Mechanical Engineering, Sogang University, Seoul 121-742, Korea

(Manuscript Received October 28, 2010; Revised January 20, 2011; Accepted February 7, 2011)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract This study introduces implementation of a nodal release technique into a FEM/continuum model to enable simulation of fully plastic

crack growth. The nodal release technique is implemented in the user-defined element form on the symmetry line of a deeply single-edge cracked specimen so that the force at the crack-tip node on the symmetry line is made zero after several steps upon the satisfaction of a chosen fracture criterion, and an incremental crack extension is achieved. The fracture criterion adopts the crack-tip opening angle (CTOA) which is determined from the specimen’s loading geometry [1]. For evaluation of the present model, the crack growth simula-tion results from the present FEM model were compared to those from the line-spring model of Lee and Parks [2].

Keywords: Nodal release technique; FEM; Fully plastic crack growth; Single-edge cracked specimen; Crack–tip opening angle (CTOA); Line spring model ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

CTOA has been considered the most suitable local parame-ter to characterize the crack-tip field of stable crack growth. In the past, various mechanisms were studied and proposed to estimate CTOA for stable crack growth [1-7]. Especially for fully plastic crack growth in low strength materials, the slid-ing-off and shear-cracking model suggested by McClintock et al. [1], among others, was found to best serve to estimate CTOA.

In the sliding-off and shear-cracking model, the crack is as-sumed to grow in a zigzag fashion, sliding off by s and crack-ing by c along a shear band before changing direction, as shown in Fig. 1. The geometry gives the form of CTOA in terms of slip line angle θs and s/c as by

sin tantan

2 ( )cos 2( ) 1s s

s

sCTOAc c s c/s

θ θθ

⎛ ⎞ = =⎜ ⎟ + + +⎝ ⎠. (1)

The total strain in the advancing shear band at fracture can be thought of as the sum of strains required to make holes initiate, grow and link to the form of micro-cracks. The width of the model shear band produced during one step of zigzagging is (s+c) sinθs. Then, the geometry gives the corresponding shear

(fracture) strain, γf, as

sf cs

sθ

γ2sin)( +

= . (2)

Substituting (c/s) from Eq. (2) into Eq. (1) gives

1)sin/(2tan

2tan

−=⎟

⎠⎞

⎜⎝⎛

sf

s

θγθCTOA . (3)

The fracture strain depends on the mean normal stress across the slip line at the crack-tip and on material properties. Ac-counting for hole nucleation and growth in a shear band, McClintock et al. [1] suggested a semi-empirical functional form for fracture strain γf , as

)(]/)/11sinh[(

)/11(s

osf B

nAn σ

τσγ +

−−

= . (4)

Here, τo is the yield strength in shear, σs is the mean normal stress and n is the strain hardening exponent [plastic strain proportional to (stress)n]. The dimensionless constant A and function B (σs) are material properties considered to be deter-minable from fully plastic crack growth tests. The first term on the right hand side of Eq. (4) represents a strain for hole growth to linkage by micro-rupture, while the second term represents the strain for hole nucleation. Based on the model of hole growth in a shear band [8], the parameter A is related

† This paper was recommended for publication in revised form by Associate EditorChongdu Cho

*Corresponding author. Tel.: +82 31 8004 3506, Fax.: +82 31 8005 3509 E-mail address: shjung@dankook.ac.kr

© KSME & Springer 2011

1202 S. Jung and H. Lee / Journal of Mechanical Science and Technology 25 (5) (2011) 1201~1206

to the critical hole growth ratio at fracture by A = 2 ln (Rf / Ro), where Ro is the initial hole size and Rf is the hole size at point of micro-localization between two grown holes.

Subsequently, it was proposed that the value of CTOA can be directly predicted from the loading conditions through the sliding-off and shear-cracking model [1]. Further, the line spring model [2] was constructed based on the loading geome-try-based CTOA to study fully plastic crack growth simulation.

In this study, an FEM model is constructed with a nodal re-lease technique implemented for crack growth simulation and CTOA is chosen as the fracture criterion. The crack-tip node of the FEM model is set to be released upon satisfaction of the CTOA criterion. The main body uses the plane strain single-edge cracked specimen composed of elastic/plastic material subject to pure extension (no rotational displacement is im-posed at the loading point).

2. Numerical implementation of crack growth

The CTOA fracture criterion and nodal release algorithm are implemented into the FEM model in the form of user-defined element of a commercial finite element program ABAQUS [9]. Only the upper half of a plane strain single –edge cracked specimen is considered to save the computa-tional effort as shown in Fig. 2. The user-defined elements for nodal release are arranged along the symmetry line of the specimen. The loading condition is imposed on the top edge of the specimen.

First, initial arrangement of the user-defined elements is de-scribed along the symmetry line. Subsequently, CTOA frac-ture criterion and nodal release algorithm coded into the ele-ments are explained. Finally, a comprehensive crack growth algorithm is presented in full detail.

2.1 Symmetric boundary condition/crack path

Two-node spring elements are implanted in the user-defined

element form in order for the nodes of the spring elements to be released for crack growth simulation. The spring elements are vertically oriented and arranged along the symmetry line, which is the predefined crack path. Before the CTOA fracture criterion is satisfied, the spring elements are fixed in the verti-cal direction with high stiffness as shown in Fig. 2. The verti-cal stiffness of the spring element is set to be much greater (ten thousand times or greater) than the uniaxial stiffness per unit thickness of one half of the specimen, which is given by

sspecimen L

EwK =

(5)

where E is the Young’s modulus of the material, w is the width of the specimen and Ls is the length of a half of the specimen. Upon the satisfaction of the CTOA criterion with respect to each crack-tip node, the crack-tip node is released.

2.2 Finite element formulation for the loading geometry-

based CTOA criterion

In the finite element formulation, the CTOA fracture crite-rion is satisfied when the local opening angle (CTOAc) as shown in Fig. 3 reaches the instantaneous CTOA (µ), which is determined by the current loading ratio, µ [1]. The loading ratio µ is calculated from the nodal reactions as follows. Summing all of the nodal reactions along the symmetry line gives the total axial force N:

∑=i

iFN

(6)

where Fi is the nodal reaction at node i. The bending moment taken at the mid-ligament, M, is determined by

( )/ 2i ii

M F x l= −∑

(7)

Fig. 1. Sliding-off and shear cracking model for fully plastic, planestrain quasi-steady crack growth [1]. CTOA and cracked surfaces arepresented by dashed lines and thick solid lines, respectively.

Fig. 2. Half of the single-edge cracked specimen. User-defined spring elements are arranged on the symmetry line in the uncracked part to enable nodal release in conjunction with evolution of CTOA upon the loading condition. The loading point is set at the center on top edge.

S. Jung and H. Lee / Journal of Mechanical Science and Technology 25 (5) (2011) 1201~1206 1203

where xi represents the distance of node i from the current crack-tip node and l represents the current ligament length. Then, the loading ratio µ is given as

NlM

=µ . (8)

Finally, the instantaneous CTOA, CTOA (µ) is determined from the current value of the loading ratio through the relation shown in Fig. 4. It should be noted that the µ− CTOA (µ) relation in Fig. 4 is set with the hole nucleation parameter B(σs) to zero.

2.3 Nodal release

Nodal release of the crack-tip node is initiated when the lo-cal opening angle (CTOAc) with respect to the crack-tip node reaches the instantaneous CTOA (µ). During the release of the crack tip node, the (adhesive) force of the crack tip node is coded to be defined with the far field displacement rather than directly associated with the vertical displacement of the crack tip node. Relating the force of the crack tip node to the vertical displacement of the crack node during release in the computa-tion would introduce negative stiffness, which is thereby ex-cluded here. Instead, for the case where the crack growth in-volves large plastic deformation, the crack tip displacement can be largely predicted with the far field displacement, which is adopted here to define the force during release (the kine-matic relationship between the crack-tip and the far field dis-placement is presented with Eq. (11)). During the release, the nodal force of the spring element at the crack-tip node, Ftip (as shown in Fig. 5) is given as

⎥⎦

⎤⎢⎣

⎡−=

CTOAδδ∆ tf

tiptip fFF /* 1 (9)

where *

tipF is the nodal force at the time when the nodal re-lease initiates, tf /δ∆ and CTOAδ are explained below, and f will be discussed in detail in conjunction with Eq. (12). In Eq.

(9), CTOAδ is given as

( )CTOA otan CTOA / 2caδ µ⎡ ⎤= ∆ ⎣ ⎦

(10)

where µo is the loading ratio as defined in Eq. (8) at the time when the nodal release starts, and ca∆ is the length of finite element at the crack-tip in the undeformed configuration. Also, in Eq. (9) tf /δ∆ is calculated as

ff

ftf

wwaL δ∆

δ∆θ∆

δ∆⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −+=

211 2/

(11)

where the dimensions of a and w refer to Fig. 2, fδ∆ and

fθ∆ are the values of far-field displacement and rotation (at the loading point as illustrated in Fig. 2) at the current time

Fig. 3. FEM formulations for CTOAc with the releasing node andCTOA (µ). The mirror images for the finite elements below the sym-metry line are represented with dashed lines and the spring elementsare not shown.

µ = M/Nl

0.0 0.2 0.4 0.6 0.8 1.0

CTO

A (d

eg)

0

30

60

90

1.2 = A

1.0

0.8

0.6

0.4

0.2

Fig. 4. CTOA (µ) vs. the loading ratio µ with non-hardening plane strain single-edge-cracked specimen for B (σs) = 0 [1].

Fig. 5. Vertical force of the releasing node controlled with the far field displacement as presented in Eq. (9). No shear force is engaged on the symmetry line.

1204 S. Jung and H. Lee / Journal of Mechanical Science and Technology 25 (5) (2011) 1201~1206

subtracted by the values at the time of initiation, and L2 repre-sents the degree of negative relative rotation of the crack flank with respect to the far-field rotation (the kinematic relation of Eq. (11) is found suitable to estimate the local tip displace-ment using the far field displacement in fully plastic region [10]). With the relation of Eq. (11), the force release rate can-not yet be precisely managed to have CTOAc and CTOA (µ) (i.e. δtip and δCTOA) coincide when Ftip reaches 0. Therefore, three cases can be introduced when Ftip reaches 0 and they are:

Case1 : CTOAc < CTOA (µ) Case2 : CTOAc = CTOA (µ) Case3 : CTOAc > CTOA (µ).

Clearly, Case 2 is ideal for the crack growth simulation. For Case 1, the specimen must be reloaded until CTOAc reaches CTOA (µ) for the next crack-tip node to be released. In Case 3, the specimen is overloaded, which must be avoided. To mini-mize reloading of Case 1 and prevent overloading of Case 3, the force release rate with Eq. (9) is further optimized with the correction factor, f . f for the release of each crack tip node is given as

** ff tip

CTOAδδ

= (12)

where f* represents the value of the correction factor set for the preceding crack tip node and δ tip* is the vertical displacement of the preceding crack tip node at the moment when Ftip of the preceding crack tip node reaches 0. f is set to 0.9 for the initial crack tip node and, afterward, adaptively set for the current crack tip node upon the completed release of the preceding crack tip node using Eq. (12).

2.4 Crack growth algorithm

The comprehensive crack growth algorithm including crack initiation is presented in this section. A crack-tip opening dis-placement (CTOD) based on the work of Ref. [11] is imple-mented into the present model for crack initiation; the first crack-tip node is released when the vertical displacement of at the crack-tip reaches a critical value, as shown in Fig. 6. Once cracking initiates, the CTOA criterion is activated and the nodes on the symmetry line are sequentially released as ex-plained in Section 2.3. Fig. 8 shows the mesh configuration near the initial crack-tip node of the plane strain single-edge specimen.

The details of the simulation algorithm of the CTOA-based crack growth following the CTOD-based crack initiation are presented in Fig. 7 and explained as follows. For the crack initiation, the displacement from the symmetry line of Node 0 is taken as half of CTOD as shown in Fig. 6. At the beginning of each time step, the value of the displacement of Node 0 (i.e. CTOD) from the previous time step is read and compared with a given CTODcr. When the value of the displacement of

Fig. 6. FEM formulations for CTODcr and CTOD. The nodes on the symmetry line (i.e. crack path) are numbered. The node numbered 1 represents the initial crack-tip node.

Fig. 7. Numerical algorithm of nodal release for crack initiation and crack growth simulations. Crack growth continues unless specified otherwise.

Fig. 8. Mesh configuration near the initial crack-tip node of the plane strain single edge specimen.

S. Jung and H. Lee / Journal of Mechanical Science and Technology 25 (5) (2011) 1201~1206 1205

Node 0 reaches half of CTODcr, the nodal force of the crack-tip, Node 1, starts to decrease to zero in the manner of Eq. (9). When Node 1 is fully released (it is coded that in case the nodal force is found to be less than zero by Eq. (9), then the force is reset to zero), Node 2 is set to be the crack-tip node and CTOA (µ) with respect to Node 2 is updated with the new ligament length, l, and current values of M and N. After com-pletion of the nodal release, the specimen is reloaded until the CTOA criterion is met with respect to the new crack-tip, Node 2. During reloading, at every time step the CTOAc computed from the previous time step is read and compared with CTOA (µ). Once CTOAc reaches CTOA (µ), the nodal force of Node 2 is set to decrease in the manner of Eq. (9). After completion of the nodal release, reloading ensues if the CTOA criterion is not met. The cycle of nodal release and reloading for each node on the symmetry line is repeated unless specified other-wise and continuing crack growth is achieved.

3. Model evaluation

The present CTOA-based crack growth algorithm is evalu-ated against the line spring model of Ref. [2], which has been developed to simulate fully plastic crack growth. The main body of the present FEM model is meshed with standard ABAQUS 4-node (full integration) plane strain elements along with the spring elements arranged on the symmetry line enabling crack growth (the mesh with 8 node elements is not considered here, since a sufficient number of the present 4 node elements are arranged, achieving the solution accuracy that the 8 node elements could provide). Elastic/perfectly plas-tic material is chosen with tensile yield strain εy set to 2.27x10-3, yield shear strength τy set to 271MPa, Young’s modulus set to 200GPa and Poisson’s ratio set to 0.3. The dimensionless constant parameter A is set to 0.5 and the hole nucleation pa-rameter B (σs) is set to zero. Pure extension (i.e. θ f = 0) load-ing condition is applied. Since θ f = 0 is imposed, tf /δ∆

can be calculated using the simplified form of

ftf δ∆δ∆ =/ (13)

rather than fully invoking Eq. (11). The mesh configuration near the first crack-tip node is shown in Fig. 8, which is re-fined in the region of the first quarter of the ligament and ori-ented in 45° to be aligned with the deformation flow expected from the theoretical solution for pure extension loading condi-tion [12]. The ratios of the sizes of each element (in the region near the initial crack-tip) to the size of the initial ligament is ~ 3 10-3. The relative initial crack depth wa / and the rela-tive length of the specimen, wLs / are 0.5 and 3. The crack growth simulation is conducted under the small geometry change assumption. L2 is undefined since no far field rotation is introduced. The factor, f in Eq. (9) is adaptively set for each respective crack tip node as presented with Eq. (12), limiting the reloading portion to be less than one tenth of nodal release cycle of each crack tip node. Each increment size of the far

field loading displacement fδ was set to ~one fiftieth of CTOAδ allowing ~50 increments to be engaged for the release

of each crack tip node. Crack initiates by CTODcr and grows by CTOA (µ). The relative ratio of CTODcr / lo where lo is the size of the initial ligament, is set to 0.02, which is sufficient to achieve fully plastic deformation before cracking initiates.

For comparison, the line spring model and the present FEM model are set with equal specifications. Both simulations are conducted under the small geometry change assumption. Fig. 9 shows load-deflection curves from the simulations in terms of normalized axial force, N / (2τy lo) vs. loading point dis-placement, δ f / lo. The grey circle indicates crack initiation and the axial force decreases after cracking. Excellent agreement between the two models is clearly demonstrated in Fig. 9. With a coarse mesh configuration with the ratio of size of each element to the size of the initial ligament is equal to 0.035, the load declines faster than with the present refined mesh con-figuration. The loading deflection curve of the FEM model deviates from the prediction of the line spring model by 10%, which is not presented here.

4. Conclusion

FEM formulations of the nodal release technique and the loading geometry-based CTOA criterion were developed for crack growth simulation study. In numerical formulation of the nodal release, the current crack-tip node is set to be re-leased upon the satisfaction of the CTOA fracture criterion and thereby an incremental crack extension is achieved. The nodal force of releasing node is successfully controlled using the far field. The present FEM model was evaluated against the line spring model. Two models were found in excellent agreement for the pure extension loading condition under the assumption of small geometry change.

The subsequent paper will explore the instantaneous local crack-tip field during growth using the present nodal release

δf / lo

0.00 0.05 0.10 0.15

N /

(2τ yl o)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

FEM modelLine spring

Fig. 9. Normalized axial force N / (2τy lo) vs. far field loading point dis-placement δ f / lo of plane strain single-edge-cracked specimen with initial crack depth a / w = 0.5 under pure extension (N > 0, θ f = 0) for A= 0.5 with B (σs) = 0.

1206 S. Jung and H. Lee / Journal of Mechanical Science and Technology 25 (5) (2011) 1201~1206

technique. In particular, large geometry changes produced by the shear band flow of fully plastic deformation during crack growth will be fully investigated and characterized in conjunc-tion with the crack tip field.

Acknowledgment

The present research was supported by the research fund of Dankook University in 2009. The authors thank Prof. David Parks for valuable comments.

References

[1] F. A. McClintock, Y.-J. Kim and D. M. Parks, Criteria for plane strain, fully plastic quasi-steady crack growth, Interna-tional Journal of Fracture, 72 (1995) 197-221.

[2] H. Lee and D. M. Parks, Line spring finite element for fully plastic crack growth-I. Formulation and one dimensional re-sults, International Journal of Solids and Structure, 35 (36) (1998) 5115-5138.

[3] H. Lee and D. M. Parks, Line spring finite element for fully plastic crack growth-II. Surface-cracked plates and pipes, In-ternational Journal of Solids and Structure, 35 (36) (1998) 5139-5158.

[4] J. C. Newman Jr., M. A. James, and U. Zerbst, A review of the CTOA/CTOD fracture criterion, Engineering Fracture Mechanics, 70 (2003) 371-385.

[5] I. Schieder, M. Schodel, W. Brocks, and W. Schonfeld, Crack propagation analyses with CTOA and cohesive model: Comparison and experimental validation, Engineer-ing Fracture Mechanics, 73 (2006) 252-263.

[6] J. Heerens, and M. Schodel, Characterization of stable crack extension in aluminum sheet material using the crack-tip opening angle determined optically and by the δ5 clip gauge technique, Engineering Fracture Mechanics, 76 (2009) 101-113.

[7] T. Chau-Dinh, G. Zi and J. Kim, Predicting residual strength of multi-cracked thin sheet plates based on CTOA or cohe-sive crack model using the extended finite element model, IOP Conference series: Material Science and Technology, 10 (2010) 012063.

[8] F. A. McClintock, Crack growth in full plastic grooved ten-sile specimen, in Physics of Strength and Plasticity: Orowan Anniversary Volume, Argon. A.S., Ed., MIT Press, Cam-bridge (1969) 307-326.

[9] ABAQUS User’s Manual/Standard, Version 6.8, Simulia Co., Providence, RI, USA (2008).

[10] H. Lee and D. M. Parks, Fully plastic analyses of plane strain single-edge-cracked specimens subject to combined tension and bending, International Journal of Fracture, 63 (1993) 329-349.

[11] J. W. Hancock, W. G. Reuter and D. M. Parks, Constraint and toughness parameterized by T, in Constraint Effects in Fracture, ASTM STP 1171, E. M. Hackett, K.-H. Schwalbe and R. H. Dodds, Jr., Eds., American Society for testing and materials, Philadelphia, USA (1993) 21-40.

[12] F. A. McClintock, Plasticity aspects of fracture, Fracture, 3, Academic Press, New York, USA (1971) 47-225.

Sunghwan Jung received a B.S. degree from the University of Iowa, Iowa City, in 1993 and M.S. and Ph.D. degrees from MIT, Cambridge, MA, in 1995 and 2007, respectively, in mechanical engi-neering. From 1996 to 2002, he was a senior research engineer at Samsung Advanced Institute of Technology. From

2006 to 2009, he worked at Atomate Corporation as a senior scientist. He is now an Assistant Professor with Dankook University, Korea. His research interests include fracture me-chanics, micro/nano self-assembly, design and fabrication of MEMS, microfluidics, and carbon nanotube synthesis.