Research articleFluid-driven cracks in an elastic matrix in the toughness-dominated limit
Abstract
The dynamics of fluid-driven cracks in an elastic matrix is studied experimentally. We report the crack radius R(t) as a function of time, as well as the crack shapes w(r,t) as a function of space and time. A dimensionless parameter, the pressure ratio Δpf/Δpv, is identified to gauge the relative importance between the toughness (Δpf) and viscous (Δpv) effects. In our previous paper (Lai et al. 2015 Proc. R. Soc. A 471, 20150255. (doi:10.1098/rspa.2015.0255)), we investigated the viscous limit experimentally when the toughness-related stresses are negligible for the crack propagation. In this paper, the experimental parameters, i.e. Young’s modulus E of the gelatin, viscosity μ of the fracturing liquid and the injection flow rate Q, were chosen so that the viscous effects in the flow are negligible compared with the toughness effects, i.e. Δpf/Δpv≫1. In this limit, the crack dynamics can be described by the toughness-dominated scaling laws, which give the crack radius R(t)∝t2/5 and the half maximum crack thickness W(t)∝t1/5. The experimental results are in good agreement with the predictions of the toughness scaling laws: the experimental data for crack radius R(t) for a wide range of parameters (E,μ,Q) collapse after being rescaled by the toughness scaling laws, and the rescaled crack shapes w(r,t) also collapse to a dimensionless shape, which demonstrates the self-similarity of the crack shape. The appropriate choice of the viscous or toughness scaling laws is important to accurately describe the crack dynamics.
This article is part of the themed issue ‘Energy and the subsurface’.
1. Introduction
The success of hydraulic fracturing for shale gas recovery, i.e. the injection of pressurized water to fracture a shale reservoir and access the trapped natural gas, has motivated much research interest in understanding the underlying mechanisms. Several competing forces involved in the crack dynamics have been identified, such as elastic, viscous and toughness-related stresses [1–3]. The fracturing process of liquid intrusion into elastic solids has also been used to study magma transport, in which the cracks are mainly driven by buoyancy due to the density difference between the injected fluid and rock [4–6].
Hydraulic fractures have been extensively studied theoretically and numerically [2,3,7–10], but experimental observations of the fracture geometry are relatively limited. A summary of the previous experimental work is presented in table 1. Qualitative observations of hydraulic fractures in laboratory experiments were reported by Hubbert & Willis [11]. Experiments were also conducted to study the propagation of fractures, either in a cement block, or along the interface between an acrylic block and a rubber block, as summarized in [18,19]. In some experimental studies on magma-filled cracks, also referred to as buoyancy-driven cracks, gelatin has been used to visualize the fracturing process in an elastic material due to its transparent, elastic and brittle properties [12–14]. Fewer quantitative experiments on the detailed fracture geometry have been reported for hydraulic fractures. Shape measurements of penny-shaped crack tips were performed at a particular time by Bunger [20]. We focus on the experimental developments to obtain the time-dependent information for the fluid-driven crack radius and crack shape in a gelatin matrix, and investigate the dynamics of the crack propagation process in this paper.
| authors | physical mechanisms | L(t) | w(r,t*) | w(r*,t) | w(r,t) | motivation | |
|---|---|---|---|---|---|---|---|
| Hubbert & Willis [11] |  |
— | — | — | — | H. F. | |
| Takada [12] |  ,  |
 |
 |
— | — | — | M. T. |
| Lister & Kerr [5] |  ,  |
 ,  |
 |
— | — | — | M. T. |
| Menand & Tait [13] |  ,  |
 |
 |
— | — | — | M. T. |
| Kavanagh et al. [14] |  ,  |
 ,  |
 |
— | — | — | M. T. |
| Bunger & Detournay [15] |  |
 ,  |
— |  |
— | — | H. F. |
| Jeffrey & Bunger [16] |  |
 |
 |
 |
 |
— | H. F. |
| Lai et al. [17] |  |
 |
 |
 |
 |
 |
H. F. |
| this study |  |
 |
 |
 |
 |
 |
H. F. |
Two possible scaling laws can be obtained considering different force balances. When a crack in an elastic solid is created due to the intrusion of a fluid, the fluid pressure balances the elastic stresses at the crack surface, which serves as the driving force for the propagation of hydraulic fractures. The resisting forces during crack propagation include the viscous forces of the flow in the crack, and the forces required to open new crack surfaces in the elastic solid [21]. The viscous-dominated scaling is obtained from the balance between the elastic stresses and the viscous stresses [2], whereas a balance between elastic- and toughness-related stresses gives the toughness-dominated scaling [8,21].
In our previous work, we reported experimental measurement of the time evolution of the crack profile shapes in the viscous-dominated regime [17]. In this paper, we provide the time evolution of crack profile shapes in the toughness-dominated regime, which further allows us to compare the experimental results in the viscous and toughness-dominated regimes. We also identify a dimensionless parameter, the pressure ratio Δpf/Δpv, to gauge the relative importance between the toughness and viscous stresses. The importance of the pressure ratio is to determine the proper scaling laws to predict the crack dynamics. For example, the crack size estimated by the toughness scaling law will be an order of magnitude greater than that estimated by the viscous scaling law, if the pressure ratio Δpf/Δpv≈0.03, as discussed in an example in §5.
The paper is structured as follows: §2 provides a summary of both the viscous and toughness scaling arguments, and the derivation of a dimensionless parameter, the pressure ratio Δpf/Δpv that gauges the viscous and toughness effects. Section 3 explains the experimental method designed to study fluid-driven cracks in the toughness-dominated regime. The experimental results are presented in both dimensional and dimensionless forms in §4. A comparison of experiments in both the viscous- and toughness-dominated regimes, and the significance of identifying the proper regime of a fluid-driven propagating crack are discussed in §5. Concluding remarks are provided in §6.
2. Scaling arguments for fluid-driven cracks
We summarize in this section the scaling arguments for fluid-driven cracks in two asymptotic regimes: viscous and toughness regimes [2,3,8]. We identify a dimensionless parameter, the pressure ratio, to determine when the toughness effects are more significant than the viscous effects or vice versa. For the mechanics of crack opening we denote R,W and P, respectively, as the crack radius, the crack maximum half thickness and the scale of fluid pressure in the crack, which balances the elastic stress applied on the crack surface. Also, E is Young’s modulus, σ is Poisson’s ratio and 2γs is the energy per unit area required to open new crack surfaces in the elastic solid.
(a) Toughness and viscous scaling arguments
When a crack is opened in an elastic medium due to the intrusion of fluid, the magnitude of the fluid pressure, which balances the elastic stress (equation (.1)) exerted on the crack surface, is
is the mode I stress intensity factor for a penny-shaped crack (equation (.2)) and 
is the fracture toughness [13,23], which is a material property that represents the resistance of a material to the opening of new crack surfaces. Thus, the typical fluid pressure P, or the elastic stress, for a crack radius R in a material dominated by the toughness of opening the crack is
For the fluid flow in the crack, lubrication theory can be applied due to the narrow geometry of the crack (W≪R), as inertial effects are expected to be small. Viscous stresses cause a radial pressure gradient in the flow. In this viscous-dominated flow limit, the change of crack thickness W with time can be shown [2] to scale as
In the viscous-dominated regime, the elastic stresses on the crack (equation (2.1)) balance the fluid pressure in the viscous flow (equation (2.3)), and the toughness effects of the elastic material are assumed negligible. By contrast, in the toughness-dominated regime, there is a balance between the fluid pressure, or the elastic stresses, set by the elastic strain (equation (2.1)) and the fracture toughness (equation (2.2)), while the viscous effects are assumed negligible. Thus, the scaling laws for the time dependence of R,W and P in the viscous and toughness regimes can be obtained using equations (2.1), (2.3), (2.4) and (2.1), (2.2), (2.4), respectively, as summarized in table 2.
| regime | viscous regime | toughness regime |
|---|---|---|
| crack radius |  |
 |
| crack thickness |  |
 |
| fluid pressure |  |
 |
(b) Pressure ratio
To examine the viscous effects, which are neglected in the toughness regime, we compare the typical toughness-related stress 
(equation (2.2) without the respective coefficients) to the typical viscous stress Δpv≈μQ/W3 (equations (2.3) and (2.4) without the constant coefficients). A dimensionless parameter, the pressure ratio Δpf/Δpv, can be identified to gauge the relative importance of the toughness and viscous effects [8,17],
By contrast, if we substitute the viscous-dominated scaling laws for R and W into equation (2.5), we obtain the pressure ratio
3. Experimental design for the toughness-dominated regime
To conduct experiments on fluid-driven cracks in an elastic medium in the toughness-dominated regime, we tune the parameters E,Q and μ so that [Δpf/Δpv]t≫1, as shown in table 3. The goal of the experiments is to visualize the time evolution of the shape of the crack w(r,t) and the propagation of the crack radius R(t). We have previously reported extensive experimental results in the viscous-dominated regime [17]. Compared with the experimental set-up designed for the viscous-dominated regime, for the toughness-dominated experiments reported here, we added a long tube, which was connected to a syringe to precisely calibrate the injection flow rate of the fracturing liquid. The details of the method are described below.
| fracturing liquid | viscosity μ (mPa s) | Young’s modulus E (kPa) | injection rate Q (ml min−1) | pressure ratio [Δpf/Δpv]t |
|---|---|---|---|---|
| water | 1 | 74±7.4 | 1±0.07 | 1.8×104 |
| water | 1 | 24±2.4 | 1±0.07 | 4.3×104 |
| water | 1 | 16±1.6 | 1±0.07 | 6.4×104 |
| water | 1 | 55±5.5 | 1±0.07 | 2.3×104 |
| water | 1 | 120±12 | 3±0.21 | 4.1×103 |
| mineral oil | 11.5±0.2 | 21±2.1 | 0.5±0.04 | 8.9×103 |
| mineral oil | 26.9±0.1 | 21±2.1 | 0.1±0.007 | 1.9×104 |
(a) Method
A transparent block of solid gelatin was prepared as the elastic matrix to be fractured. This material has been used in several fluid-driven fracture experiments [12–14,24]. Young’s modulus of the matrix can be changed by varying the concentration of the gelatin powder (Gelatin type A, ThermoFisher Scientific, USA), and Young’s modulus was measured using the indentation method [17,25]. A long tube was connected to a syringe and a needle, which was placed inside the gelatin (figure 1a). The syringe and part of the tube were filled with air, while the rest of the tube and the needle were filled with the fracturing liquid (water or mineral oil). At the beginning of the injection process, the air–liquid interface was stationary, and the air was compressed. When a threshold air pressure was reached, the air–liquid interface started to move (shown by the red arrow in figure 1a) and reached the speed calculated from the injection rate Q set by the syringe pump (Harvard apparatus, PhD Ultra) within a 7% error. A USB IDS uEye camera (1280×1024 pixels, frame rate=10 fps) was placed on top of the block and a Nikon D5100 camera (640×424 pixels, frame rate=29 fps) was placed on the side to record the three-dimensional shape of the crack. The camera configuration is shown in figure 1a and typical side and top views of a growing crack are shown in figure 1b,c.
Figure 1. (a) Schematic of the experimental set-up. A needle connected to a long tube is inserted into a block of elastic gelatin. The long tube and the needle are filled with the fracturing liquid. An air–liquid interface in the long tube is used to calibrate the flow rate of the liquid. A syringe filled with air first compresses the air column, then pushes the liquid forward to create a penny-shaped crack inside the gelatin block. The direction of liquid flow is marked by the arrow above the air–liquid interface. A blue light panel, facing the camera, was placed behind the crack for visualization purposes. (b) The side view and (c) top view of a penny-shaped crack. t=0 is defined as the time when the fracturing liquid emerges from the needle tip. The flow rate at which the fracturing liquid is injected is 
, the fracturing liquid is water (mixed with fluorescein and red dye) and Young’s modulus of the gelatin matrix is E=77±7.7 kPa. Rin represents inner radius.
When the liquid enters the solid matrix, it creates a penny-shaped crack. We define time zero as the moment when the liquid, dyed red for visualization as shown in figure 1, appears at the tip of the needle. The error from this determination is ±0.5 s. In table 3, we summarize the typical values of μ,E and Q used in our experiments. The viscosity μ varied by a factor of 27, E by a factor of 8 and Q by a factor of 30. We used γs≈1 J m−2 measured by Menand & Tait [13]. In the calculation of the pressure ratio (equation (2.6)), t is chosen to be the time half-way through the experiments. The pressure ratio [Δpf/Δpv]t≈103−104 in our experiments, which confirms that the experiments are in the toughness-dominated regime.
In all our experiments, the buoyancy effects from the density difference between the fracturing liquid and the elastic solid are negligible. For example, in a typical experiment, the crack radius is R≈2 cm, the density difference is Δρ≈900 kg m−3 and the gelatin Young’s modulus is E≈20 kPa. Consequently, we can estimate the typical buoyancy-related stress to be ΔρgR≈180 Pa, and the toughness-related stress is 
. Thus, the buoyancy effects are negligible in the experiments. In addition, the viscous drag from flow along the needle and the long tube is also negligible compared with the toughness-related stresses. For example, given the inner radius (Rin) and the length (L) of the needle and tube (figure 1), μ≈26.9 mPa s and 
, the viscous stresses from the flow along the needle and the tube is estimated to be μQL/(Rin)4≈88 Pa, which is negligible compared with the typical toughness-related stresses Δpf≈103 Pa. An experimental verification of the lack of significant needle-size effects is also provided in appendix C.
To estimate the plastic effects, we calculate the size of the process zone rp at the crack tip where the inelastic deformation occurs. For a gelatin of a typical yield stress τy≈104−105 Pa [26], assuming K=KIC, the process zone 
–0.4 mm, which is much smaller than the representative crack radius R≈20 mm in our experiments. Thus, the plastic effects are negligible for the experiments of the current paper.
(b) Measurement of crack shape
The advantage of using transparent gelatin is that we can visualize the shape of the crack. We used Bunger’s [20] imaging technique, measuring the change of light intensity as light passes through the liquid-filled crack, to obtain the time-dependent crack thickness w(r,t) [17]. This technique has been used previously to obtain the crack thickness for a penny-shaped crack [15,20] and for a saucer-shaped crack near a free surface [27].
We set a uniform blue light background of light intensity I0 behind the crack, dye the fracturing liquid with fluorescein (15 μl fluorescent yellow 131 SC, Keystone +10 g fracturing liquid) and measure the intensity of light I after it passes through the liquid-filled crack. We calibrate the change of the blue light intensity (I/I0), as it travels through the fluorescein-filled liquid, as a function of liquid thickness d. Thus, we are able to convert the information of the measured blue light intensity I to the crack thickness 2w (for technical details, see [17]). A typical result is shown in figure 2.
Figure 2. The thickness distribution (2w) of the entire crack measured at t= 116 s. Experimental parameters: μ=11.5±0.2 Pa s, E=77±7.7 kPa and 
. We note that the signal at the centre of the crack was blocked by the needle.
4. Results and observations
Once the liquid enters the solid matrix, it creates a penny-shaped crack (see figure 1b, c for the side and top views, respectively). The thickness of a crack 2w at any given time can be obtained using the imaging technique described in §3b, as shown in figure 2. We will show in this section the time dependence of both the crack radius R(t) and the crack shape 2w(r,t), and the dimensionless data rescaled by the toughness scaling laws (table 2).
(a) Dimensional results
We measured the radius of the crack as a function of time R(t) for various liquid viscosities μ, Young’s moduli E and injection flow rates Q. The results are shown in figure 3a. We chose carefully the experimental parameters so that 4×103≤[Δpf/Δpv]t≤6×104 (table 3); thus, the viscous effects are negligible. The results show that the crack radius R exhibits a power-law dependence on time, as shown in the log–log plot (figure 3a).
Figure 3. (a) Crack radius R(t) as a function of time for various values of μ,E and Q. (b) The time evolution of the crack profile shapes w(r,t). Each curve is the crack profile measured at a fixed cross section. The time difference Δt between two adjacent curves is 10 s. The flat curve at the bottom corresponds to t=0. Note that r=0 is defined at the centre of the crack, where the light is blocked by the needle and no data were recorded. Experimental parameters: μ=11.5±0.2 Pa s, E=77±7.7 kPa and 
.
Also, for a given experiment we report the crack half thickness profile w(r,t) at a cross section through the centre of the crack, as shown in figure 3b. The red flat curve represents the crack profile at t=0 and the crack profiles are measured every 10 s. For all curves there is a region at the centre of width 6 mm, where the light signal is blocked by the needle.
(b) Dimensionless results
To compare the results R(t) and w(r,t) with the toughness scaling laws, we rescaled the raw experimental data in figure 3a,b by the toughness scaling laws provided in table 2. When the data are rescaled using the scaling laws in table 2, good collapse of the rescaled crack radius was obtained, as shown in figure 4a. We fit each rescaled experimental curves with a power-law function, and obtain an average pre-factor k=1.14±0.35 and exponent α=0.46±0.06, which shows a good agreement with the toughness scaling law.
Figure 4. (a) The rescaled crack radius using the toughness scaling arguments in table 2. The collapse of experimental data supports the toughness scaling laws. We fit each rescaled curve at late times with a power-law form (see the equation shown on the plot) and obtain an average pre-factor k=1.14±0.35 and exponent α=0.46±0.06. The power-law curve with an average k and α is shown by the dark line. (b) The dimensionless crack profiles, in which r is rescaled by R and w is rescaled by W. The black curve represents the best elliptical fit, equation (4.1), where A≈0.61. The asymptotic solution in equation (.14) gives A≈0.47 and the difference in A is likely due to the uncertainties in measuring E,Q,w,r,t and the values of γs, σ used in this paper. The convergence of the dimensionless crack shapes to a universal shape demonstrates the self-similarity of the crack shapes.
We next rescale the time-dependent crack profile shapes (figure 3b). Since the dependence of both the crack radius R and maximum half thickness W on time is available (table 2), we can rescale r and w(r,t) using the scaling laws for R and W. Here, instead of γs≈1 J m−2, we use γs≈5.15 J m−2 determined by fitting the crack radius R with the theoretical prediction in equation (.12). We obtained convergence of the dimensionless profile shapes to a universal dimensionless crack shape after around 30 s, as shown in figure 4b. The collapse of the crack shapes suggests that the profile shapes are self-similar.
We also fit the dimensionless crack shapes with an elliptical functional form (to be consistent with the theoretical prediction, equation (.14)):
(c) Crack roughness
We note that for water as the fracturing liquid, the crack surface appeared smooth, while with mineral oils as the fracturing liquid, the crack surface appeared rough. The detailed structure of a rough crack surface with mineral oils as the fracturing liquid can be clearly observed, as shown in figure 5a, as we change the direction of illumination (the direction of the excitation blue light is into the page) and increase the dye concentration (200 μl fluorescent yellow 131 SC, Keystone +10 g mineral oil). In comparison, figure 5b shows a smooth crack surface with water as the fracturing liquid.
Figure 5. (a) The roughness of a crack surface is visualized using a higher concentration of fluorescein (200 μl fluorescent yellow 131 SC, Keystone +10 g mineral oil) in the fracturing liquid than the concentration used for the crack shape measurement. The crack surface appears to be rough when we use mineral oils as the fracturing liquid. The fracturing liquid in this figure is mineral oil of viscosity μ=11.5 mPa s. The direction of the excitation blue light is into the page. (b) The crack surface appears to be smooth when the injected liquid is water (dyed red).
5. Discussion
In §4, we have shown the self-similarity of the time evolving crack shapes. In addition, there is good agreement between the experimental results and the predictions of the scaling laws in the toughness-dominated limit. Similar procedures had been developed by Lai et al. [17] for the experiments and scaling laws in the viscous-dominated regime. Now we compare the toughness data [Δpf/Δpv]t≫1, with the viscous data [Δpf/Δpv]v<1.
We plotted the time-dependent crack radius R(t) from experiments in both the viscous and toughness regimes, both rescaled by the toughness scaling laws, as shown in figure 6. We observe that the functional forms of the viscous and toughness scaling laws are very similar in terms of time dependence (t4/9 and t2/5, respectively), and the dependence on μ,E and Q shows modest differences (table 2). Given these similarities in the functional forms, within the parameter limits of our experimental system the toughness scaling law seems to collapse the experimental data in both the viscous and toughness regimes.
Figure 6. A comparison of the experimental data between the toughness and the viscous regimes, both rescaled by the toughness scaling laws. The data in the viscous regime, where 0.62<[Δpf/Δpv]v<0.97, were adapted from reference [17] and shown with the empty symbols. The data in the toughness regime, where 4.1×103<[Δpf/Δpv]t<6.4×104, are shown with solid symbols. We note that the functional form of the viscous and toughness scaling laws (table 2) are very similar, thus the toughness scaling laws also seem to collapse the data in the viscous regime. However, there is a gap in the pre-factor from fitting the toughness and viscous datasets. The toughness data suggest an average pre-factor k=1.14±0.35 (solid line), while the viscous data suggest an average pre-factor k=0.7±0.14 (dashed line).
Another important issue is that the pre-factors in the power-law forms are different between our experimental data in the viscous and toughness regimes. In particular, for experimental data in the toughness limit, the pre-factor is k=1.14±0.35, as shown by the solid line in figure 6. By contrast, for experimental data in the viscous regime, the pre-factor is k=0.7±0.14, as shown by the dashed line in figure 6.
However, an experiment in the viscous limit with an extremely small pressure ratio will do less well in collapsing the viscous data in figure 6, when rescaled by the toughness scaling law. This feature is because the crack radius predicted by the viscous and toughness scaling laws, denoted by Rv and Rt, respectively, vary with the pressure ratio according to
6. Conclusion
A laboratory experiment was developed to study fluid-driven cracks inspired by the process of hydraulic fracturing. Two possible sets of scaling arguments, the viscous and toughness scalings, were discussed, considering whether the resistance of crack propagation is from the viscous stresses or the toughness-related stresses. In our previous work [17], we chose gelatin as an analogue for rocks and investigated the dynamics of fluid-driven cracks in the viscous-dominated regime. In this paper, we redesigned the experiments to study the crack dynamics in the toughness-dominated regime. Time-dependent information of the crack radius R(t) and crack shape w(r,t) were obtained in our experiments, and the dimensional data were rescaled based on the toughness scaling laws. The collapse of dimensionless data for a wide range of experimental parameters (E,Q,μ) demonstrates the success of the toughness scaling arguments.
In particular, the pressure ratio (equation (2.5)), a dimensionless parameter defined as the ratio of toughness-related stresses to viscous stresses, is identified to be a guideline for the choice of the appropriate scaling laws to describe the crack propagation. We note that the difference between the predictions for the crack size in the viscous- and toughness-dominated regimes can be significant when the pressure ratio departs from unity.
Our experimental system provides an idealized model to study fluid-driven fracturing operations such as hydraulic fracturing. Alternative experimental models, considering the heterogeneity of the rocks, the leak-off of liquid from the cracks, the effects of proppants in the fluid flow and the porelastic effects of the solid matrix, could be potentially explored from proper modifications of the experimental system. We hope to report some of these studies in future work.
Data accessibility
The data are available at http://arks.princeton.edu/ark:/88435/dsp01pv63g266t.
Authors' contributions
All authors contributed to all aspects of this work.
Competing interests
We declare that we have no competing interests.
Funding
We thank the National Science Foundation for funding via grant no. CBET-1509347 and the Andlinger Center for Energy and the Environment at Princeton University for partial support of this research. Also, Z.Z. thanks the support from the Princeton Environmental Institute through the CMI Young Investigator Award.
Acknowledgements
We thank Jesse Ault, Matthew Edwards, Jie Feng, Hyoungsoo Kim, Janine Nunes, Jean H. Prévost, Barrie S. H. Royce, Allan Rubin, Robert H. Socolow, Sankaran Sundaresan and Jason S. Wexler for helpful discussions.
Appendix A. Governing equations in the toughness-dominated regime
Let us consider an axisymmetric penny-shaped crack in an infinite elastic medium. We define r=0 at the centre of the crack and r=R(t) at the crack radius. The z-axis is aligned with the axial direction. We assume the shear traction on the crack surface is small compared with the normal traction. The elastic stress σzz=−p(r,t) is related to the shape of the crack by [2]
In the realm of linear elastic fracture mechanics, the stress singularity at the crack tip can be characterized by the stress intensity factor. The mode I stress intensity factor for a penny-shaped crack is Rice [21]:
Lastly, the fluid flow inside the crack is incompressible, and thus the crack volume is equal to the volume of fracturing liquid supplied at flow rate Q,
The scaling argument based on equations (.1)–(.4) can be obtained. This scaling argument (toughness limit) is different from the scaling argument in Lai et al. [17], which is in the viscous limit. Both the viscous and toughness scaling arguments are summarized in table 2.
Appendix B. Crack shape in the toughness-dominated regime
In this section, we provide an approximate solution for the crack shape and front location in the toughness regime. We first note that the equation for the pressure distribution (.1) can also be rewritten as a double integral equation (e.g. [8,28])
In the toughness regime, given that the pressure ps(t) is only time dependent, the crack propagation criterion (.3) and the stress intensity factor (.2) can be rewritten to provide an explicit expression for pressure ps(t) as a function of crack radius R(t):
We now combine (.6) and (.7) to obtain an expression for the crack half thickness w(r,t) with only R(t) to be determined:
Then, we can substitute the profile shape (.8) into the global mass constraint (.4), and we obtain the time evolution of the crack radius R(t):
From the scaling arguments in the toughness regime presented in table 2, we can define the characteristic scales for the pressure Pc, crack thickness Wc and radius Rc
Finally, we can now compare the scaling results and the detailed calculations of the asymptotic solutions, which suggests that
Appendix C. Examination of the effect of the needle size
The size effects of the injection needle are not considered in the scaling laws (table 2). We experimentally investigated the effect of the needle size. For the similar parameter regimes described in §4a, a set of experiments with different needle sizes was conducted, and very good data collapse (figure 7) was observed based on the toughness scaling laws (table 2). Thus, the needle size does not appear significant in our experiments and does not influence the scaling laws.
Figure 7. The effects of needle size on the dimensionless crack radius versus time. Rin represents the inner radius of the needle. The collapse of experimental data indicates that the effects of the needle size are not important in our experiments. The dark line is the same line as shown in figure 4a with k≈1.14 and α≈0.46.
