- Email: [email protected]
Crack propagation toward a desired path by controlling the force direction
Engineering Fracture Mechanics 76 (2009) 2554–2559
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Note
Crack propagation toward a desired path by controlling the force direction Baoxing Xu a, Xi Chen a,*, Haim Waisman b a b
Department of Earth and Environmental Engineering, Columbia University, 500 W 120th Street, New York, NY 10027, USA School of Engineering and Applied Sciences, Columbia University, 500 W 120th Street, New York, NY 10027, USA
a r t i c l e
i n f o
Article history: Received 13 August 2009 Received in revised form 13 September 2009 Accepted 15 September 2009 Available online 19 September 2009
a b s t r a c t Controlling the crack propagation along a desired path has important applications in fabrication. We propose a method for extending a crack along a desired trajectory by controlling the direction of an applied external point force. Examples of crack propagation along arc and sinusoidal paths are illustrated, and verified through numerical simulations based on the extended finite element method (XFEM). Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Crack Desired path Force direction XFEM
1. Introduction Cracks are usually regarded as threats to the mechanical integrity of materials and structures, and various strategies were proposed to enhance the fracture toughness so as to reduce or avoid cracking [1,2]. Meanwhile, mechanical self-assembly based on ordered crack morphology has emerged as a competitive approach for micro- and nano-fabrication [3–7], where the crack patterns formed in a specimen (e.g. a thin film on a substrate) may be employed as molds for depositing nanowires or templates for nanochannels. In addition, an important step in a cutting process is to create fracture along a desired contour, such that the specimen can be broken into pieces with a precise shape. In MEMS fabrication the first step is usually the wafer dicing process, with the purpose of making the wafer to break along a prescribed direction [8,9]. The key challenge in all of these examples above is to drive the crack propagating along a desired path. Compliant inclusions [10,11] or voids [12–14] embedded in a brittle matrix can strongly affect the local stress field, and therefore deviate crack path or arrest the crack. The resulting crack path, however, is often a complex function [13] and simple yet precise trajectories (e.g. arc or sinusoidal) are difficult to obtain. By incorporating inclusions with sharp corners (e.g. triangular or square-shaped [6,7]), straight cracks may be initiated from these corners, however desired crack paths are not trivial to achieve. Moreover, the incorporation of inclusions or voids at precise (desired) locations often requires nontrivial effort of material processing (and the inclusions may change the physical properties of the base material), thus not practical for various manufacturing processes, such as cutting and micro-fabrication, where a simpler solution is needed. Inspired by the fact that during a typical cutting process, the force direction is much easier to control, we explore a simple strategy of controlling crack propagation along a desired path by changing only the direction of a single applied point force. The suggested method is essentially an inverse problem, which focuses on how to change the direction of the force with respect to the crack state.
* Corresponding author. E-mail address: [email protected] (X. Chen). 0013-7944/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2009.09.007
B. Xu et al. / Engineering Fracture Mechanics 76 (2009) 2554–2559
2555
2. Model and computational method Consider a model specimen, which is an elastic plate with width of w and height of h in Fig. 1, and its bottom edge is clamped. In this study we focus on the case w = h. The plate contains a very small initial edge crack of length a on the left free boundary, whose distance to the top-left corner is l. A concentrated force F is applied at the top-left corner, and its angle
y
F θ
α
l
h a
w
o
x
Fig. 1. Geometrical parameters and boundary conditions for the propagation of an edge crack in a rectangular/square plate; in this example the desired crack trajectory (dash curve) is an arc. The optimum variation history of the direction of applied force, h, is sought after such that the crack would propagate along the desired trajectory.
(a)
l/h =0.5
25
α = π /6 α = π /4
-5
K I /(El )/(10 )
α = 5 π /12
1/2
2.5
0.25
2.5
0.25
0.5
1
θ
1.5
2
2.5
3 3.5 4
1.5
2
2.5
3 3.5 4
α = π/6 l/h =0.25 l/h =0.75 l/h =0.50
1/2
-5
(b)
K I /(El )/(10 )
0.025
0.025 0.5
1
θ pffi Fig. 2. Variation of the normalized stress intensity factor K I =ðE lÞ with applied force direction h with: (a) l/h = 0.5 and varying a; and (b) a = p/6 with ~ varying l/h. The putative crack extension direction q is along the desired arc path.
2556
B. Xu et al. / Engineering Fracture Mechanics 76 (2009) 2554–2559
(direction) with respect to the x-axis is h. The force magnitude is fixed while its direction is adjustable via h. The specimen is assumed homogeneous and isotropic with Young’s modulus E and Poisson’s ratio m. The brittle material is initially defect-free (except the small initial edge crack), and crack propagation is assumed to obey the maximum circumferential stress principle (to be consistent with numerical simulation below) [15]. If h is fixed, the crack would propagate along a relatively complex parabolic-like path (not shown here) which is also dependent on h. The fundamental question is: can we enforce the crack to propagate along a desired path, such as the arc with radius l illustrated in Fig. 1, by varying the force direction h in a specific way? In the problem illustrated in Fig. 1, suppose the crack has already advanced a segment along the desired path, and the corresponding arc angle of the current crack-tip is a. To solve this type of inverse problem [16] for an optimal force direction h, the following iterative procedure is proposed: First, for a given h, the circumferential (hoop) stress at the crack tip is calculated for all possible values of putative crack advance directions (~ q) using the finite element method (FEM) [17]; the maximum value of hoop stress corresponds to the most likely crack extension direction in the next time step – if such a direction does not fall on the tangent of the desired crack path, the value of h is iterated until the next crack increment falls precisely on the desired trajectory. By advancing the crack along the projected path and repeating the above computations, a unique optimized force history, h(a), is deduced which guarantees the crack to propagate along the desired path. Numerical simulations based on the extended finite element method (XFEM) [18–21] is then used to verify the crack trajectory for a given force history. The nodes at the crack tip are enriched by a crack tip function and have eight additional degrees of freedom (DOFs), and the nodes near the crack trajectory are enriched by the Heaviside step function H(x) with two additional DOFs. The direction of crack propagation hc follows the maximum circumferential stress criterion [15], and can be written as [22]:
(a)
2.4
l/h =0.50 l/h =0.75
2.1 1.8
θ
1.5 1.2
0.9
0.6 0.1
(b)
1
α
(c)
Fig. 3. Crack growth along the desired arc trajectory (with different values of l/h). (a) Variation of the required (optimized) force direction h with crack tip location a; (b) comparison between XFEM simulation (solid red line, based on the deduced force direction history in (a)) and desired (dash blue line) arc crack path for l/h = 0.5; (c) comparison between XFEM simulation (solid red line, based on the deduced force direction history in (a)) and desired (dash blue line) arc crack path for l/h = 0.75. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
B. Xu et al. / Engineering Fracture Mechanics 76 (2009) 2554–2559
2
2557
3
2K II =K I 6 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 hc ¼ 2 arctan 4 1 þ 1 þ 8ðK II =K I Þ2
ð1Þ
The crack growth is represented as a series of small line segments, and the increment is kept the same for all steps. The governing equations of XFEM [16,18–20] are implemented in MATLAB, and verified by comparing with known theoretical and experimental results. 3. Examples: controlling the crack along arc or sinusoidal paths We first illustrate the suggested approach of making the crack to propagate along a desired arc path. For three different ~ crack tip locations pffi (assuming the putative crack extension direction q is along the arc with l/h = 0.5), Fig. 2a shows the varq and h, the optimum iation of K I =ðE lÞ with force direction h. KI is closely related to the maximum hoop stress; by iterating ~ value of h that makes ~ q consistent with the desired trajectory can be determined for a given a. Moreover, FEM calculations show that KI is also a function of the initial crack position l/h, shown in Fig. 2b, indicating that the required optimized force history h(a) is different for different desired crack paths. The deduced optimized force history h(a) is shown in Fig. 3a, for several desired crack paths with different l/h. With crack propagation (i.e. the increase of a), the optimized h decreases nonlinearly; the degradation accelerates with the increase of l/ h due to the effect of boundary constraints. The variation of the optimized h(a) with l/h and a can be fitted by an empirical equation via h = 1.05(l/h) + 7.54(1/a) 1.12(l/h)(1/a) at 0 < l=h 6 0:75 (where the angles are in radians).
(a)
T/ w=0.375, A/ h=0.125 T/ w=0.75, A/ h=0.125
θ
ο
100
10
0.1
(b)
1
α
(c)
Fig. 4. Crack growth along the desired sinusoidal trajectory (y = A sin (Tx)), where A and T are the amplitude and period of the crack path, respectively, and a = x/T; (a) variation of the required (optimized) force direction h with the dimensionless crack tip location a; (b) comparison between XFEM simulation (solid red line, based on the deduced force direction history in (a)) and desired (dash blue line) sinusoidal crack path for T/w = 0.375 and A/h = 0.125; (c) comparison between XFEM simulation (solid red line, based on the deduced force direction history in (a)) and desired (dash blue line) sinusoidal crack path for T/w = 0.75 and A/h = 0.125. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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B. Xu et al. / Engineering Fracture Mechanics 76 (2009) 2554–2559
To validate the findings above, we carry out XFEM simulations with the force direction varied as that in Fig. 3a. The resultant crack path (solid red line) agrees well with the desired one (dash blue line) for both values of l/h examined in Fig. 3b and c. The small deflection at the end of crack path is due to the boundary constraint (when the remaining ligament is quite small), however it does not affect the overall characteristic of crack propagation. Next, we design a sinusoidal crack path, y = A sin (Tx) where A and T are the amplitude and period of the crack path, respectively. The location of the crack tip is defined as a = x/T. Fig. 4a shows the relationship between the optimized h and a determined following the same approach above. Due to the wavy nature of the crack path, the desired load direction varies rapidly like step functions. Fig. 4b and c shows that under such an optimized force direction history h(a), the XFEM simulated crack path agrees well with the desired sinusoidal path. Note that when the crack becomes long, the two crack surfaces may contact with each other in the middle portion of the crack, however experiments show that the crack tip may still remain open in some circumstances [23] and in this study, we neglect the effect of the potential surface contact (in the middle portion of crack) on the stress intensity at crack tip; if the contact region is very close to crack tip, more sophisticated analysis incorporating the crack surface contact effect should be carried out [24,25]. The present analysis is based on the maximum circumferential stress principle [15]; note that there are other fracture criterions available such as the KII = 0 principle and the maximum KI principle [23]. It is found that the optimized force histories h(a) derived from different fracture criterions are quite close. Despite such theoretical agreement, we caution that in real experiments, the crack trajectory does not always coincide with the weakest positions in a solid, and a pre-existing crack may affect the crack initiation/propagation behaviors in brittle materials [26]. Therefore, the predicted crack propagation behavior needs to be verified from future experiments, and depending on specific material behaviors different fracture criterions may be undertaken. 4. Concluding remarks In summary, we propose a new strategy for propagating cracks along desired trajectories by controlling the direction of an applied external force. Numerical examples are conducted on a short edge crack in a square plate where the direction of a concentrated load applied at the corner of a plate is varied in a specific way such that the crack would propagate along arc, sinusoidal, or other desired trajectories. The force history is found via an iterative approach under the principle of maximum circumferential stress, and the result is verified by XFEM simulations. It is envisioned that the proposed method could be extended to different specimen geometry, boundary condition, force/load protocol (e.g. load application point, load type, multiple loads, etc.), as well as to controlling the propagation of multiple cracks simultaneously; corresponding experimental verification will be carried out in future. This study sheds some light on designing crack self-assembly for fabrication and efficient cutting/dicing procedures. Acknowledgement The work is supported by National Science Foundation CMMI-CAREER-0643726. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
Lawn BR. Fracture of brittle solids. Cambridge: Cambridge University Press; 1993. Meyers MA, Mishra A, Benson DJ. Mechanical properties of nanocrystalline materials. Prog Mater Sci 2006;51:427–556. Mani S, Saif TM. Mechanism of controlled crack formation in thin-film dielectrics. Appl Phys Lett 2005;86(20):4669–71. Sardan O, Yalcinkaya AD, Alaca BE. Self-assembly-based batch fabrication of nickel–iron nanowires by electroplating. Nanotechnology 2006;17(9):2227–33. Adelung R, Aktas OC, Franc J, Biswas A, Kunz R, Elbahri M, et al. Strain-controlled growth of nanowires within thin-film cracks. Nat Mater 2004;3(6):375–9. Alaca BE, Sehitoglu H, Saif T. Guided self-assembly of metallic nanowires and channels. Appl Phys Lett 2004;84:4669–71. Sardan O, ErdemAlaca B, Yalcinkaya AD, Bøggild P, Tang PT, Hansen O. Microgrippers: a case study for batch-compatible integration of MEMS with nanostructures. Nanotechnology 2007;18:375501. Chang WJ, Fang TH, Weng CI. Inverse determination of the cutting force on nanoscale processing using atomic force microscopy. Nanotechnology 2004;15:427–30. Venkatakrishnan K, Tan B. Thin silicon wafer dicing with a dual-focused laser beam. J Micromech Microengng 2007;17:2505–15. Xu G, Bower AF, Oritz M. The influence of crack trapping on the toughness of fiber reinforced composites. J Mech Phys Solids 1998;46:1815–33. Evans AG, McMeeking RM. On the toughening of ceramics by strong reinforcements. Acta Metall 1986;34:2435–41. Gao H, Rice JR. A first-order perturbation analysis of crack trapping by arrays of obstacles. J Appl Mech 1989;111:828–36. Liu L, Chen X. Controlled crack arrest in brittle thin films: the effect of embedded voids. Acta Mater 2008;56:6214–23. Salac D, Lu W. Design nanocrack patterns in heterogeneous films. Nanotechnology 2006;17:5185–91. Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear. J Basic Engng 1963;85:519–27. Waisman H, Belytschko T. Parametric enrichment adaptivity by the extended finite element method. Int J Numer Methods Engng 2008;73:1671–92. ABAQUS. ABAQUS 6.7 user’s manual. Pawtucket, Rhode Island: ABAQUS Inc.; 2007. Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Engng 1999;45:601–20. Bordas S, Nguyen VP, Dunant C, Nguyen-Dang H, Guidoum A. An extended finite element library. Int J Numer Methods Engng 2007;71:703–32. Moes N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. Int J Numer Methods Engng 1999;46:131–50. Belytschko T, Gracie R, Ventura G. A review of extended/generalized finite element methods for mechanics. Modell Simulat Mater Sci Engng 2009;17:043001.
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[22] Sukumar N, Prevost JH. Modeling quasi-static crack growth with the extended finite element method. Part I: computer implementation. Int J Solids Struct 2003;40:7513–37. [23] Thouless MD, Evans AG, Ashby MF, Hutchinson JW. The edge cracking and spalling of brittle plates. Acta Metall 1987;35:1333–41. [24] Chen X, Hutchinson JW, He MY, Evans AG. On the propagation and coalescence of delamination cracks in compressed coatings: with applications to thermal barrier coatings. Acta Mater 2003;51:2017–30. [25] Chen X. New stress intensity factor solutions of a periodic array of cracks in residual stress field. Mech Res Commun 2006;33:425–32. [26] Cotterell B, Rice JR. Slightly curved or kinked cracks. Int J Fract 1980;16(2):155–69.
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Note
Crack propagation toward a desired path by controlling the force direction Baoxing Xu a, Xi Chen a,*, Haim Waisman b a b
Department of Earth and Environmental Engineering, Columbia University, 500 W 120th Street, New York, NY 10027, USA School of Engineering and Applied Sciences, Columbia University, 500 W 120th Street, New York, NY 10027, USA
a r t i c l e
i n f o
Article history: Received 13 August 2009 Received in revised form 13 September 2009 Accepted 15 September 2009 Available online 19 September 2009
a b s t r a c t Controlling the crack propagation along a desired path has important applications in fabrication. We propose a method for extending a crack along a desired trajectory by controlling the direction of an applied external point force. Examples of crack propagation along arc and sinusoidal paths are illustrated, and verified through numerical simulations based on the extended finite element method (XFEM). Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Crack Desired path Force direction XFEM
1. Introduction Cracks are usually regarded as threats to the mechanical integrity of materials and structures, and various strategies were proposed to enhance the fracture toughness so as to reduce or avoid cracking [1,2]. Meanwhile, mechanical self-assembly based on ordered crack morphology has emerged as a competitive approach for micro- and nano-fabrication [3–7], where the crack patterns formed in a specimen (e.g. a thin film on a substrate) may be employed as molds for depositing nanowires or templates for nanochannels. In addition, an important step in a cutting process is to create fracture along a desired contour, such that the specimen can be broken into pieces with a precise shape. In MEMS fabrication the first step is usually the wafer dicing process, with the purpose of making the wafer to break along a prescribed direction [8,9]. The key challenge in all of these examples above is to drive the crack propagating along a desired path. Compliant inclusions [10,11] or voids [12–14] embedded in a brittle matrix can strongly affect the local stress field, and therefore deviate crack path or arrest the crack. The resulting crack path, however, is often a complex function [13] and simple yet precise trajectories (e.g. arc or sinusoidal) are difficult to obtain. By incorporating inclusions with sharp corners (e.g. triangular or square-shaped [6,7]), straight cracks may be initiated from these corners, however desired crack paths are not trivial to achieve. Moreover, the incorporation of inclusions or voids at precise (desired) locations often requires nontrivial effort of material processing (and the inclusions may change the physical properties of the base material), thus not practical for various manufacturing processes, such as cutting and micro-fabrication, where a simpler solution is needed. Inspired by the fact that during a typical cutting process, the force direction is much easier to control, we explore a simple strategy of controlling crack propagation along a desired path by changing only the direction of a single applied point force. The suggested method is essentially an inverse problem, which focuses on how to change the direction of the force with respect to the crack state.
* Corresponding author. E-mail address: [email protected] (X. Chen). 0013-7944/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2009.09.007
B. Xu et al. / Engineering Fracture Mechanics 76 (2009) 2554–2559
2555
2. Model and computational method Consider a model specimen, which is an elastic plate with width of w and height of h in Fig. 1, and its bottom edge is clamped. In this study we focus on the case w = h. The plate contains a very small initial edge crack of length a on the left free boundary, whose distance to the top-left corner is l. A concentrated force F is applied at the top-left corner, and its angle
y
F θ
α
l
h a
w
o
x
Fig. 1. Geometrical parameters and boundary conditions for the propagation of an edge crack in a rectangular/square plate; in this example the desired crack trajectory (dash curve) is an arc. The optimum variation history of the direction of applied force, h, is sought after such that the crack would propagate along the desired trajectory.
(a)
l/h =0.5
25
α = π /6 α = π /4
-5
K I /(El )/(10 )
α = 5 π /12
1/2
2.5
0.25
2.5
0.25
0.5
1
θ
1.5
2
2.5
3 3.5 4
1.5
2
2.5
3 3.5 4
α = π/6 l/h =0.25 l/h =0.75 l/h =0.50
1/2
-5
(b)
K I /(El )/(10 )
0.025
0.025 0.5
1
θ pffi Fig. 2. Variation of the normalized stress intensity factor K I =ðE lÞ with applied force direction h with: (a) l/h = 0.5 and varying a; and (b) a = p/6 with ~ varying l/h. The putative crack extension direction q is along the desired arc path.
2556
B. Xu et al. / Engineering Fracture Mechanics 76 (2009) 2554–2559
(direction) with respect to the x-axis is h. The force magnitude is fixed while its direction is adjustable via h. The specimen is assumed homogeneous and isotropic with Young’s modulus E and Poisson’s ratio m. The brittle material is initially defect-free (except the small initial edge crack), and crack propagation is assumed to obey the maximum circumferential stress principle (to be consistent with numerical simulation below) [15]. If h is fixed, the crack would propagate along a relatively complex parabolic-like path (not shown here) which is also dependent on h. The fundamental question is: can we enforce the crack to propagate along a desired path, such as the arc with radius l illustrated in Fig. 1, by varying the force direction h in a specific way? In the problem illustrated in Fig. 1, suppose the crack has already advanced a segment along the desired path, and the corresponding arc angle of the current crack-tip is a. To solve this type of inverse problem [16] for an optimal force direction h, the following iterative procedure is proposed: First, for a given h, the circumferential (hoop) stress at the crack tip is calculated for all possible values of putative crack advance directions (~ q) using the finite element method (FEM) [17]; the maximum value of hoop stress corresponds to the most likely crack extension direction in the next time step – if such a direction does not fall on the tangent of the desired crack path, the value of h is iterated until the next crack increment falls precisely on the desired trajectory. By advancing the crack along the projected path and repeating the above computations, a unique optimized force history, h(a), is deduced which guarantees the crack to propagate along the desired path. Numerical simulations based on the extended finite element method (XFEM) [18–21] is then used to verify the crack trajectory for a given force history. The nodes at the crack tip are enriched by a crack tip function and have eight additional degrees of freedom (DOFs), and the nodes near the crack trajectory are enriched by the Heaviside step function H(x) with two additional DOFs. The direction of crack propagation hc follows the maximum circumferential stress criterion [15], and can be written as [22]:
(a)
2.4
l/h =0.50 l/h =0.75
2.1 1.8
θ
1.5 1.2
0.9
0.6 0.1
(b)
1
α
(c)
Fig. 3. Crack growth along the desired arc trajectory (with different values of l/h). (a) Variation of the required (optimized) force direction h with crack tip location a; (b) comparison between XFEM simulation (solid red line, based on the deduced force direction history in (a)) and desired (dash blue line) arc crack path for l/h = 0.5; (c) comparison between XFEM simulation (solid red line, based on the deduced force direction history in (a)) and desired (dash blue line) arc crack path for l/h = 0.75. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
B. Xu et al. / Engineering Fracture Mechanics 76 (2009) 2554–2559
2
2557
3
2K II =K I 6 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 hc ¼ 2 arctan 4 1 þ 1 þ 8ðK II =K I Þ2
ð1Þ
The crack growth is represented as a series of small line segments, and the increment is kept the same for all steps. The governing equations of XFEM [16,18–20] are implemented in MATLAB, and verified by comparing with known theoretical and experimental results. 3. Examples: controlling the crack along arc or sinusoidal paths We first illustrate the suggested approach of making the crack to propagate along a desired arc path. For three different ~ crack tip locations pffi (assuming the putative crack extension direction q is along the arc with l/h = 0.5), Fig. 2a shows the varq and h, the optimum iation of K I =ðE lÞ with force direction h. KI is closely related to the maximum hoop stress; by iterating ~ value of h that makes ~ q consistent with the desired trajectory can be determined for a given a. Moreover, FEM calculations show that KI is also a function of the initial crack position l/h, shown in Fig. 2b, indicating that the required optimized force history h(a) is different for different desired crack paths. The deduced optimized force history h(a) is shown in Fig. 3a, for several desired crack paths with different l/h. With crack propagation (i.e. the increase of a), the optimized h decreases nonlinearly; the degradation accelerates with the increase of l/ h due to the effect of boundary constraints. The variation of the optimized h(a) with l/h and a can be fitted by an empirical equation via h = 1.05(l/h) + 7.54(1/a) 1.12(l/h)(1/a) at 0 < l=h 6 0:75 (where the angles are in radians).
(a)
T/ w=0.375, A/ h=0.125 T/ w=0.75, A/ h=0.125
θ
ο
100
10
0.1
(b)
1
α
(c)
Fig. 4. Crack growth along the desired sinusoidal trajectory (y = A sin (Tx)), where A and T are the amplitude and period of the crack path, respectively, and a = x/T; (a) variation of the required (optimized) force direction h with the dimensionless crack tip location a; (b) comparison between XFEM simulation (solid red line, based on the deduced force direction history in (a)) and desired (dash blue line) sinusoidal crack path for T/w = 0.375 and A/h = 0.125; (c) comparison between XFEM simulation (solid red line, based on the deduced force direction history in (a)) and desired (dash blue line) sinusoidal crack path for T/w = 0.75 and A/h = 0.125. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
2558
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To validate the findings above, we carry out XFEM simulations with the force direction varied as that in Fig. 3a. The resultant crack path (solid red line) agrees well with the desired one (dash blue line) for both values of l/h examined in Fig. 3b and c. The small deflection at the end of crack path is due to the boundary constraint (when the remaining ligament is quite small), however it does not affect the overall characteristic of crack propagation. Next, we design a sinusoidal crack path, y = A sin (Tx) where A and T are the amplitude and period of the crack path, respectively. The location of the crack tip is defined as a = x/T. Fig. 4a shows the relationship between the optimized h and a determined following the same approach above. Due to the wavy nature of the crack path, the desired load direction varies rapidly like step functions. Fig. 4b and c shows that under such an optimized force direction history h(a), the XFEM simulated crack path agrees well with the desired sinusoidal path. Note that when the crack becomes long, the two crack surfaces may contact with each other in the middle portion of the crack, however experiments show that the crack tip may still remain open in some circumstances [23] and in this study, we neglect the effect of the potential surface contact (in the middle portion of crack) on the stress intensity at crack tip; if the contact region is very close to crack tip, more sophisticated analysis incorporating the crack surface contact effect should be carried out [24,25]. The present analysis is based on the maximum circumferential stress principle [15]; note that there are other fracture criterions available such as the KII = 0 principle and the maximum KI principle [23]. It is found that the optimized force histories h(a) derived from different fracture criterions are quite close. Despite such theoretical agreement, we caution that in real experiments, the crack trajectory does not always coincide with the weakest positions in a solid, and a pre-existing crack may affect the crack initiation/propagation behaviors in brittle materials [26]. Therefore, the predicted crack propagation behavior needs to be verified from future experiments, and depending on specific material behaviors different fracture criterions may be undertaken. 4. Concluding remarks In summary, we propose a new strategy for propagating cracks along desired trajectories by controlling the direction of an applied external force. Numerical examples are conducted on a short edge crack in a square plate where the direction of a concentrated load applied at the corner of a plate is varied in a specific way such that the crack would propagate along arc, sinusoidal, or other desired trajectories. The force history is found via an iterative approach under the principle of maximum circumferential stress, and the result is verified by XFEM simulations. It is envisioned that the proposed method could be extended to different specimen geometry, boundary condition, force/load protocol (e.g. load application point, load type, multiple loads, etc.), as well as to controlling the propagation of multiple cracks simultaneously; corresponding experimental verification will be carried out in future. This study sheds some light on designing crack self-assembly for fabrication and efficient cutting/dicing procedures. Acknowledgement The work is supported by National Science Foundation CMMI-CAREER-0643726. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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