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Soft Matter Accepted Manuscript
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DOI: 10.1039/C5SM00230C
Puncture Mechanics of Soft Solids† Sami Fakhouri,a Shelby B. Hutchens,b‡ and Alfred J. Crosby,∗a Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX First published on the web Xth XXXXXXXXXX 200X
Gels and other soft elastic networks are a ubiquitous and important class of materials whose unique properties enable special behavior, but generally elude characterization due to the inherent difficulty in manipulating them. An example of such behavior is the stability of gels to large local deformations on their surface. This paper analyzes puncture of model soft materials with particular focus on the force response to deep indentation and the critical load for material failure. Large-strain behavior during deep indentation is described with a neo-hookean contact model. A fracture process zone model is applied to the critical load for puncture. It is found that the indenter geometry influences the size of the fracture process zone, resulting in two distinct failure regimes: stress-limited and energy-limited. The methods outlined in this paper provide a simple means for measuring Young’s modulus, E, as well as the material’s maximum cohesive stress, σ0 , fracture energy, Γ0 , and the intrinsic length scale linking the two, l0 , all without requiring specialized sample preparation.
1
Introduction
Penetration of soft materials by acicular objects bears relevance to many fields including bite mechanics, design of protective equipment, percutaneous drug delivery, and surgical instrumentation and technique; however, understanding of the resisting forces to such punctures has remained largely empirical to date. 1–11 Figure 1 demonstrates the astounding ability of soft materials to resist puncture. A glass bead at the tip of a needle indents a hydrogel to great depth without breaking the surface. The resulting stresses are so severe that they are able to be visualized under cross-polarized light despite the fact that the gel consists of only 4 percent by weight polymer. In fact, we find that indentation depths and nominal stresses associated with puncture can reach two orders of magnitude larger than the indenter radius and material modulus, respectively. The source of difficulty in predicting puncture failure of soft materials is two-fold. The first arises from the relative strength of highly elastic networks in comparison to their modulus. In such highly compliant systems failure is preceded by finite strains exceeding the limits of linear elasticity. 12–15 Indeed, the surfaces of soft materials have been shown to be stable to large local deformations 16–19 and as such, are difficult to cut or tear by simple normal force. 20–22 These being the precise conditions for puncture gives rise to the second challenge: the application of load directly to the point of failure in the absence of an initial defect defies treatment by Linear Elastic Fracture Mechanics (LEFM). 23 That is, the conditions for failure must be determined without making assumptions regarding the crack geometry or the path of propagation. Cohesive zone models (CZM) have been employed to ad-
Fig. 1 Deformation in a 4w% Poly(acrylamide) hydrogel contacted by a spherically-tipped indenter, pre-puncture.
dress both the aforementioned challenges; namely, to approximate stress states of nonlinear materials at failure, and to predict failure of materials with blunted or no defects. 24 Barenblatt defined a region, the cohesive zone, within which the attractive potentials between molecules on crack faces provide cohesive forces resisting crack opening; the summation of which constitute a “modulus of cohesion” indicating a critical stress at the crack tip above which the crack will propagate. 25 Hillerborg later expanded this approach to include not only crack propagation, but also the nucleation of a crack at any arbitrary location where such a stress is achieved. 26 1–9 | 1
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DOI: 10.1039/b000000x
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DOI: 10.1039/C5SM00230C
Gc = σ 0 ∗ l
(1)
where Gc is the critical strain energy release rate, σ0 is the maximum cohesive stress of the material and l is the representative size of the FPZ. 14 It follows, then, that predicting fracture behavior requires knowledge a priori of at least two of the three variables of Equation 1. In most analyses, l and σ0 are taken to be a material constants, thus creating an equivalence between stress and energy descriptions of failure. Lake and Thomas took l to be the end-to-end distance between crosslinks in order to account for the discrepancy in the observed fracture energies and intrinsic bond strengths of elastic networks. 13 Similarly, Hui and Jagota developed criteria for the elastic blunting of cracks in order to account for the strength of elastic networks where l becomes the opening displacement of the stretched crack. 14 However, the dependence of the fracture energy on the size of the FPZ has yet to be demonstrated by controlling the latter as the independent variable. This paper analyzes the mechanical response of soft materials to deep indentation by high aspect ratio indenters, with particular interest in the critical load for puncture, Pc , the load at which the needle breaks through the surface of the material. The relationship between applied displacement and resulting force is quantified and understood for small and large strains for two indenter tip geometries, spherical and flat. This analysis, combined with the application of FPZ, leads to understanding of Pc in terms of materials properties and indenter geometry. We find that the average puncture stress, which can be several orders of magnitude greater than the material’s elastic modulus, is defined by a mechanism that transitions from an energy-based criterion to a stress-based criterion as a function of indenter length scale. The findings have profound implications for the characterization of soft materials, and even more importantly, the design criteria for applications involving soft materials at a wide range of length scales. 2|
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Fig. 2 (a) Schematic of puncture apparatus and representative program for indenter displacement. Zoom-in establishes key definitions: radius, R, depth, d, and load, P. (b) Representative force profile for A25 B116 A25 gel indicating critical load for puncture, Pc , and critical depth for puncture, dc .
2 2.1
Experimental Materials
Acrylic triblock copolymer gels were chosen as primary model soft materials due to their elasticity at large strain and thermoreversible gel transition. 28,29 ABA copolymers of poly(methyl methacrylate) end blocks (PMMA) and poly(nbutyl acrylate) mid block (PnBA) with molecular weights kg kg of 25 mol and 116 mol , respectively (A25 B116 A25 ) were acquired from Kuraray Co. and used as received. Gels were formed by dissolving polymer in 2-ethylhexanol at 150◦ C and then cooling to room temperature, 25◦ C; the same thermal treatment was applied in between puncture experiments to erase damage history. Poly(acrylamide) hydrogels were prepared from 30:1 acrylamide/bisacrylamide stock solution from Bio-Rad Laboratories, Inc. with ammonium persulfate initiator and tetramethylethylenediamine (TEMED) catalyst. Poly(dimethyl siloxane) elastomer (PDMS) was preR pared from a Sylgard 184 kit obtained from KR Anderson Co. A 40:1 ratio of prepolymer to curing agent was used. Poly(urethane) elastomer was prepared from BJB Enterprises PC-12 A/B elastomer kit as directed. Puncture experiments were performed in A25 B116 A25 gels unless otherwise specified.
† Electronic Supplementary Information (ESI) available. See DOI: 10.1039/b000000x/ a Polymer Science and Engineering Department, University of Massachusetts, 120 Governors Drive, Amherst, MA, USA 01003. Fax: +1 413 545 0082; Tel: +1 423 577 1313; E-mail: [email protected]
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The CZM approach to fracture initiation, i.e. critical stress, has lead to the development of Fracture Process Zone (FPZ) models, which provide a physical description of energy dissipation in systems experiencing nonlinear processes during failure. The FPZ represents the volume of material wherein under steady state crack propagation the dissipative work is constant. 27 In this way, the critical stress approach of CZM is related to the critical energy approach of Griffith 23 by the size of the FPZ as
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Fig. 3 (a) Representative force profiles for insertion with an R = 283µm indenter in 4 unique materials: 20 w% A25 B116 A25 gel, poly(dimethyl siloxane) elastomer with 40:1 ratio of prepolymer to curing agent, OO-20 durometer poly(urethane) elastomer, and 4 w% poly(acrylamide) hydrogel. On the y axis load, P, is normalized by load at puncture, Pc , and on the x axis depth, d, is normalized by depth at puncture, dc . (b) Representative force profiles for deep indentation by two indenter shapes: spherical and flat-punch. Inset provides the same data on linear axes to emphasize second power scaling at large depth. (c) Uniaxial compression data for A25 B116 A25 gel fit with Equation 3a. (d) Schematic of strain approximation for deep indentation by flat punch indenter. A cylindrical volume of gel under the indenter is considered to have (i) initial radius R equal to that of the indenter and (ii) stretched radius R + kd, a multiple of the indentation depth.
2.2
Methods
2.2.1 Puncture. Puncture experiments were carried out on a custom apparatus consisting of a Burleigh Instruments EXFO IW-820 Nanopositioner and Inchworm 8200 controller. A custom load cell was used consisting of aluminum cantilevers of dimensions 44.45 x 15.875 x 6.35 mm and 44.45 x 15.875 x 6.35 mm, corresponding to stiffnesses of 0.001 and 0.01 mmN −1 , respectively, coupled to a PISeca D-510.020 single-electrode capacitive sensor and PISeca E-852.10 signal conditioner. Indenters were fixed to the nanopositioner in-line with the load cell and translated at a rate of 250 µms−1 , unless otherwise specified. Experiments involve translation of the indenter into the gel such that the indenter breaks through the surface of the gel before reversal (Fig. 2(a,b)). A representative force trace of this complete cycle is shown in Figure 2(b); however, the work described here is focused only on the portion of the force response leading up to and including the point of puncture, which is marked by a sharp peak formed by a rapid drop-off in load coinciding with penetration of the needle through the surface of the gel. The critical load for puncture, Pc , is taken as the maximum force associated with the peak in the loading curve, and the critical depth, dc , is that corresponding to the critical load. Data points correspond to mean values of between 5 and 10 punctures and error bars represent one standard deviation above and below the mean. Fits to these data utilize standard deviations in instrumental weighting of least squares parameters. Flat punch indenters used were blunt-tipped needles obtained from Hamilton Company and back-filled with epoxy to
avoid coring. These needles come from the manufacturer with a flat tip profile and filleted edge. Spherically-tipped indenters were fabricated by gluing spherical beads to the ends of hollow blunt-tipped needles. Where indenter diameters were too small for this method, indenters were fabricated by pulling capillary tubes in a Narishige PC-10 and subsequently annealing the tips in a microforge. Spherical profiles resulted from surface tension and were confirmed by microscopy (ESI Fig. 1). Being that boundary conditions are critical for understanding the deformation mechanisms of highly strained materials, puncture experiments were conducted in such a manner as to simulate contact with an infinite elastic half-space. That is, gels were contained in vessels sufficiently large that no surface deformation was able to be visually detected at the walls of the container at any time during puncture, and with a depth at least 10 times the depth of puncture. In addition, as the gels used have significant liquid content (between 60 and 90 percent), the interface between the indenter and the gel was not modified in any way, but is still considered to be frictionless for the purposes of analysis. We consider the universality of the force profiles observed across multiple materials and formulations to be indicative of of the consistency of these boundary conditions in our experiments. 2.2.2 Rheology. Triblock copolymer gels (A25 B116 A25 ) were prepared at various weight percent polymer in order to achieve a range of materials properties. The linear relationship between weight percent polymer and shear storage modulus, G0 , obtained by shear rheology is demonstrated in ESI Figure 2. Rheological measurements were made on an AR 2000 rheometer using 25 mm steel ETC parallel plates. Triblock 1–9 | 3
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DOI: 10.1039/C5SM00230C
2.2.3 Mechanical Testing. Compression and tension R mechanical testing appatests were performed on an Instron ratus fitted with a 50 N load cell. For compression, triblock gel cylinders of 16 mm height and 20 mm diameter were placed between metal plates lubricated with silicon oil to reduce friction. Samples were compressed at a strain rate of 0.001 s−1 until failure. Tensile tests were intended to serve as complements to puncture tests; therefore, strain rates were matched accordingly. Secant strain rates for puncture were calculated from the stress at puncture, σc , the Young’s modulus, E, determined from the loading curve, and the time-to-puncture, t, as ε˙ = σEtc . Tensile rupture tests were performed on “dog-bone” samples of ASTM D412-A dimensions. Fracture energies were taken as the area under the load-displacement curve from the start of the test to the point of failure normalized by the initial cross-sectional area of the dog bone gauge. Notch tests to calculate strain energy release rates were conducted on samples of dimensions 60 · 50 · 1 mm3 with a volume 10 · 50 · 1 mm3 exposed between grips following the procedure outlined by Rivlin. 12 Notches were cut at the mid-point with a razor blade.
3 3.1
Results and Discussion Loading
Deep indentation experiments in four unique materials demonstrate the universality of the force profile pre-puncture (Figure 3(a)). Such a profile is qualitatively consistent with those observed in a variety of synthetic and biological materials. 3,5,6,30 Specifically, the force response pre-puncture indicates an increasing stiffness with depth. Figure 3(b) shows force profiles in A25 B116 A25 gels for indenters with two distinct tip geometries: flat, and spherical. At small depth, both indenters follow the force-displacement behavior predicted by for small-strian contact; 31 that is, 8 P ∼ ERd 3
(2a)
3 1 16 (2b) ER 2 d 2 9 for flat (a) and spherical (b), respectively, where P is the load, E is Young’s modulus of the indented material and R is the radius of the indenter. At large depth, however, both of these models break down, and the force responses of both indenters transition to a second order dependence on depth. This
P∼
4|
1–9
Fig. 4 Empirical determination of the coefficient k0 of Equation 5 for flat-punch indenters in A25 B116 A25 gels of various modulus.
can be attributed to the expected behavior for compression of materials described by a Neo-Hookean consititutive relationship, as demonstrated previously for these ABA gels. 29 To confirm this, uniaxial compression experiments were performed on bulk A25 B116 A25 gels. Under these conditions, the nominal engineering and true stresses in an incompressible neoHookean material, σz, eng , and σz, true , respectively, obey the following expressions: 32 E 1 2 σz, eng = λz − (3a) 3 λz E 1 σz, true = λz − 2 (3b) 3 λz In this case the stretch ratio, λz , is defined as the ratio of the deformed height to the undeformed height, L/L0 . The compression data were fit with Equation 3a using the Young’s modulus, E, as an adjustable parameter (Fig. 3(c)). By redefining the stretch ratio, this neo-Hookean model can be used to describe deep indentation experiments. Consider a volume of unstretched gel underneath a flat punch indenter with radius equal to that of the indenter(Fig. 3(d.i)). Upon compression, the gel is forced out along the side of the indenter such that its radial dimension is increased by a constant multiple of the indentation depth, kd (Fig. 3(d.ii)). Thus, the 2 R stretch ratios can be defined as λr = R+kd R , λz = R+kd , and Equation 3b can then be written as " 4 # E R kd + R 2 − (4) σz = 3 kd + R R Equation 4 can be expanded and multiplied by R2 to give an expression for the load, P, as a function of the depth, d, which after eliminating higher order terms predicts the load at large depth to follow P = −σz R2 = k0 Ed 2 + k00 ERd
(5)
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solutions were brought to 150◦ C, injected between plates preset at 80◦ C and then equilibrated and held at 25◦ C for testing. Frequency sweeps from 1 to 10Hz were performed at 1 Pa and the shear storage modulus, G0 , was taken as the average over the frequency range recorded (ESI Fig. 2).
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Fig. 5 Critical load for puncture, Pc ; critical nominal stress at puncture, σc , normalized by shear modulus, G0 ; and critical depth at puncture, dc , versus radius, R, for flat punch indenters in A25 B116 A25 gels. (a,b) Flat punch indenters demonstrate stress-limited failure occurring at a critical nominal stress σc = RPc2 indicative of the maximum cohesive stress of the gel, which can surpass the modulus by two orders of magnitude. (c) The critical depth, dc , is found to be in excess of ten times the radius, R. (d,e) Spherical tip indenters exhibit a transition between energy-limited and stress-limited failure at a critical transition radius, RT . (f) The critical depth, dc is found to be proportional to the square root of the radius, R, in the energy-limited regime as predicted by equations 5 and 8. The critical depth, dc , is found to be loosely approximated by dc = 3.5R as predicted by equations 5 and 7 for the stress-limited regime.
where k0 and k00 are constants related to k. The linear term of Equation 5 captures small-strain contact of a flat punch at small depth (Equation 2a). Setting k00 = 38 accordingly, indentation force profiles for 30 combinations of flat punch indenter radius and A25 B116 A25 modulus were fit with Equation 5 using k0 E as a fitting parameter. The resulting values of k0 E are plotted against storage modulus obtained from shear rheology in Figure 4, the good linear fit confirming the accuracy of the model. The storage modulus, E 0 , is calculated from the shear modulus, G0 , assuming incompressibility, i.e. ν = .5 and E 0 = 2G0 (1 + ν). The slope of Figure 4 gives an empirical value k0 = .26 ± .01. Figure 3 of ESI demonstrates the efficacy of Equation 5 in describing the loading behavior of spherical indenters. These results indicate that spherical indenters behave similarly to flat punch indenters for d > R. In addition, the second-power term of Equation 5, which dominates at large depth, contains no reference to the radius of the
indenter. This absence suggests that the relationship between the load on the indenter and the indentation depth becomes insensitive to the specific geometry of the indenter at large depth. With this model, we have demonstrated a large-strain contact mechanical approach to modulus measurement of soft materials that offers increased sensitivity and simplicity compared to traditional contact mechanics measurements which require precise alignment of the indenter, knowledge of the area of contact, and are limited to small strains. 33 3.2
Spherical Tip Indenter Puncture
3.2.1 Observations. The critical load for puncture, Pc , for spherical tip indenters is found to be linearly proportional to R below a critical size, RT , and proportional to R2 above RT (Fig. 5(d)). Normalizing the critical load by the cross-sectional area of the indenter defines a critical nominal stress at puncture, 1–9 | 5
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Fig. 6 (a) Schematic of fracture process zone surrounding fracture plane and cohesive forces resisting crack opening. (b) Representative plot of cohesive forces, F, versus crack opening displacement, x, showing critical force for crack nucleation, Fc and displacement at complete separation, xc . (c) Schematic of fracture process zone under (i) flat punch and (ii) spherical tip indenters. Shaded areas depict highly stressed material, while dashed lines indicate the size of the fracture process zone, l. Together the average stress in the fracture process zone, σ , and the size of the fracture process zone, l, determine the available energy for fracture nucleation.
σc . Accordingly, σc is found to decrease with R, eventually plateauing for R > RT (Fig. 5(e)). The critical depth for puncture, dc also exhibits a transition at RT (Fig. 5(f)). Below RT , the dependence of dc on R can be approximated by substituting Pc ∼ R into Equation 5 and solving for dc . As puncture occurs at depths considerably larger than the radius, the second term of Equation 5 can be eliminated for simplicity, 1 yielding dc ∼ R 2 . This result is found to fit the experimental data in Figure 5(f) well for small R. Above RT , substituting Pc = σc R2 into Equation 5 predicts dc = 3.5R. This result is found to loosely approximate the data in Figure 5(f) for large R. 3.2.2 Analysis. We propose that puncture results from nucleation and growth of a crack in a region of highly stressed material under the indenter tip. The stress-concentrating ability of the system, i.e. the distribution of stress under the indenter, is dictated by the geometry of the tip. In the same way, the geometry dictates the volume of material whose strain energy is able to participate in crack formation. These effects and their implications on material failure are discussed in the context of a fracture process zone (FPZ) model below. Consider two half-spaces which meet along a fracture plane as in Figure 6(a). For a crack to form, two conditions must be met: 1) the stress across the fracture plane must be sufficiently high to break chemical bonds, and 2) there must be sufficient energy in the system to perform the work of separating the newly created crack faces. These conditions are depicted graphically in figure 6(b), where the attractive force acting on the faces, F, is plotted as a function of the crack opening displacement, x. At a critical force, Fc , the crack begins to grow and the force resisting crack opening is a monotonically decreasing function of the opening displacement, x, 6|
1–9
which eventually decays to zero at large displacement. 25 Here, we use a linear approximation for simplicity. In this way, the work of separation can be easily defined geometrically as half the product of the critical force, Fc , and the critical displacement, xc , where the force is zero. The energy to perform this work, U, must be supplied by strain energy in the surrounding material. This energy is contained in a finite volume of highly strained material, the FPZ, within the vicinity of the crack plane. A characteristic FPZ size, l0 , can be calculated by taking the critical energy per area, UA = Γ0 (the fracture energy), divided by the critical force per area, FAc = σ0 (the maximum cohesive stress). For illustration, l0 = xc in our example. A FPZ of size l0 will simultaneously satisfy the critical stress and critical energy requirements: σ0 l0 = Γ0
(6)
Equation 6 is a modified form of Equation 1, where the fracture nucleation energy, Γ0 , takes the place of the strain energy release rate Gc . The characteristic FPZ size, l0 , is a material constant that represents a transition between stress-limited and energylimited failure modes. If the size of the FPZ l ≥ l0 , the material fails when the stress reaches σ0 , as by definition the energy criterion will have already been met. This case is therefore stress-limited. Conversely, if stress is concentrated in a FPZ of size l RT . These results are also found to be consistent with our contact model by solving Equation 5 at the point of puncture, i.e., σc ∼ 145G0 from Figure 5(b). It is interesting to note that these data also indicate that dc for these indenters is independent of the material modulus, E, as shown in ESI Figure 4. 3.3.2 Analysis. The stress fields surrounding spherical and flat geometries during indentation have not been studied in this work, and therefore, the specifics of stress concentration at the point of failure cannot be directly addressed. However, 1–9 | 7
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the experimental data suggest that the FPZ size, l, is larger for flat indenters than spherical. From this, we will employ the assumption that the FPZ size is in fact equal to the radius, i.e., that the stress field under the flat punch indenter is uniform, in order to make interesting inferences about failure in these systems. Assuming that for flat punch indenters α, and consequently β , are both approximately equal to 1 (Fig. 6(c)), σc is then approximately equal to the maximum cohesive stress of the gel, σ0 . From this, we can calculate the characteristic FPZ size, l0 , for a 25w% gel by employing Equation 6. Using σ0 ∼ 1.1 ± .18 MPa from Figure 5(b) and Γ0 = 142 ± 40N/m from tensile testing (Fig. 7(b)), we can estimate l0 = 130 ± 42µm, which is similar in magnitude to the smallest radii flat punch indenters used. While this analysis is convenient and selfconsistent, a more detailed treatment of the stress field during indentation is needed, and the precise relationship between indenter geometry and FPZ size remains an open question.
4
Conclusions
Puncture experiments were conducted in model soft materials across a range of material moduli, fracture energies, and indenter shapes and sizes. Experiments indicate uniform force profiles for deep indentation; namely that the force scales with the square of the depth and with little influence of indenter geometry. This behavior is well described by defining the strain as approximately the ratio of the depth to the radius and assigning a neo-Hookean constitutive relation. The point of puncture is found to be determined by fracture nucleation within a fracture process zone (FPZ) under the indenter. Influencing the size of the FPZ by changing the shape and size of the indenter allows for observation of two distinct failure regimes arising from the necessity for two criteria to be met or failure: a critical stress, and a critical energy. A stresslimited regime occurs when the FPZ is large enough to store the necessary energy for failure at small strain. When the FPZ is small, very large stresses, exceeding the modulus by two orders of magnitude, are necessary to reach the critical energy, which becomes the limiting criterion. The ability to probe each of these regimes independently allows for measurement of the cohesive stress and fracture nucleation energy simply by pushing on a soft material with a high aspect ratio indenter. In addition, modulus measurements can be made from the loading curve with increased sensitivity and simplicity relative to small-strain contact mechanics.
References 1 O. Shergold and N. Fleck, Journal of biomechanical engineering, 2005, 127, 838–48.
8|
1–9
2 L. B. Whitenack and P. J. Motta, Biological Journal of the Linnean Society, 2010, 100, 271–286. 3 N. Abolhassani, R. Patel and M. Moallem, Medical engineering & physics, 2007, 29, 413–31. 4 H. Kataoka, T. Washio and K. Chinzei, Medical Image Computing and Computer-Assisted Intervention, 2002, 216–223. 5 A. M. Okamura, C. Simone and M. D. O’Leary, IEEE transactions on bio-medical engineering, 2004, 51, 1707–16. 6 C. Simone and A. M. Okamura, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292), 2002, 2, 2085–2091. 7 D. J. van Gerwen, J. Dankelman and J. J. van den Dobbelsteen, Medical engineering & physics, 2012, 34, 665–80. 8 D. Richter, F.-R. Matuschka, A. Spielman and L. Mahadevan, Proceedings of the Royal Society B, 2013, 280, 1–6. 9 W. K. Cho, J. a. Ankrum, D. Guo, S. a. Chester, S. Y. Yang, a. Kashyap, G. a. Campbell, R. J. Wood, R. K. Rijal, R. Karnik, R. Langer and J. M. Karp, Proceedings of the National Academy of Sciences, 2012, 109, 21289–21294. 10 S. Das and A. Ghatak, Journal of Materials Science, 2010, 46, 2895– 2904. 11 S. Das, S. Laha and A. Ghatak, Soft matter, 2014, 10, 6059–67. 12 R. S. Rivlin and A. G. Thomas, Journal of Polymer Science, 1953, 10, 291–318. 13 G. Lake and A. Thomas, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1967, 300, 108–119. 14 C.-Y. Hui, J. A., S. J. Bennison and J. D. Londono, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2003, 459, 1489–1516. 15 M. Boyce and E. Arruda, Rubber chemistry and technology, 2000, 73, 504–523. 16 S. Cai, D. Chen, Z. Suo and R. C. Hayward, Soft Matter, 2012, 8, 1301. 17 D. Chen, L. Jin, Z. Suo and R. C. Hayward, Materials Horizons, 2014, 1, 207. 18 A. Chakrabarti and M. K. Chaudhury, Langmuir : the ACS journal of surfaces and colloids, 2013, 29, 6926–35. 19 M. Chaudhury and A. Chakrabarti, Langmuir, 2013, 29, 15543–15550. 20 E. Reyssat, T. Tallinen, M. Le Merrer and L. Mahadevan, Physical Review Letters, 2012, 109, 244301. 21 P. S. L. Anderson and E. J. Rayfield, Journal of the Royal Society, Interface, 2012, 9, 1846–1855. 22 P. S. L. Anderson, The Journal of experimental biology, 2009, 212, 3627– 32. 23 B. Lawn and T. Wilshaw, Fracture of Brittle Solids, Cambridge Univ Press, 1975. 24 M. Elices, G. Guinea, J. G´omez and J. Planas, Engineering Fracture Mechanics, 2002, 69, 137–163. 25 G. Barenblatt, Advances in applied mechanics, 1962, 1–109. 26 A. Hillerborg, M. Modeer and P.-E. Petersson, Cement and Concrete Research, 1976, 6, 773–782. 27 B. Cotterell, Engineering fracture mechanics, 2002, 69, 533–553. 28 M. E. Seitz, D. Martina, T. Baumberger, V. R. Krishnan, C.-Y. Hui and K. R. Shull, Soft Matter, 2009, 5, 447. 29 M. E. Seitz, W. R. Burghardt, K. T. Faber and K. R. Shull, Macromolecules, 2007, 1218–1226. 30 C. Gokgol, C. Basdogan and D. Canadinc, Medical engineering & physics, 2012, 34, 882–91. 31 A. C. Fischer-Cripps, Introduction to Contact Mechanics, Springer Science, 2nd edn, 2007. 32 R. Rivlin, Philosophical Transactions of the Royal Society of . . . , 1948, 240, 459–490. 33 K. R. Shull, Materials Science and Engineering: R: Reports, 2002, 36,
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Soft Matter View Article Online
DOI: 10.1039/C5SM00230C
Puncture Mechanics of Soft Solids† Sami Fakhouri,a Shelby B. Hutchens,b‡ and Alfred J. Crosby,∗a Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX First published on the web Xth XXXXXXXXXX 200X
Gels and other soft elastic networks are a ubiquitous and important class of materials whose unique properties enable special behavior, but generally elude characterization due to the inherent difficulty in manipulating them. An example of such behavior is the stability of gels to large local deformations on their surface. This paper analyzes puncture of model soft materials with particular focus on the force response to deep indentation and the critical load for material failure. Large-strain behavior during deep indentation is described with a neo-hookean contact model. A fracture process zone model is applied to the critical load for puncture. It is found that the indenter geometry influences the size of the fracture process zone, resulting in two distinct failure regimes: stress-limited and energy-limited. The methods outlined in this paper provide a simple means for measuring Young’s modulus, E, as well as the material’s maximum cohesive stress, σ0 , fracture energy, Γ0 , and the intrinsic length scale linking the two, l0 , all without requiring specialized sample preparation.
1
Introduction
Penetration of soft materials by acicular objects bears relevance to many fields including bite mechanics, design of protective equipment, percutaneous drug delivery, and surgical instrumentation and technique; however, understanding of the resisting forces to such punctures has remained largely empirical to date. 1–11 Figure 1 demonstrates the astounding ability of soft materials to resist puncture. A glass bead at the tip of a needle indents a hydrogel to great depth without breaking the surface. The resulting stresses are so severe that they are able to be visualized under cross-polarized light despite the fact that the gel consists of only 4 percent by weight polymer. In fact, we find that indentation depths and nominal stresses associated with puncture can reach two orders of magnitude larger than the indenter radius and material modulus, respectively. The source of difficulty in predicting puncture failure of soft materials is two-fold. The first arises from the relative strength of highly elastic networks in comparison to their modulus. In such highly compliant systems failure is preceded by finite strains exceeding the limits of linear elasticity. 12–15 Indeed, the surfaces of soft materials have been shown to be stable to large local deformations 16–19 and as such, are difficult to cut or tear by simple normal force. 20–22 These being the precise conditions for puncture gives rise to the second challenge: the application of load directly to the point of failure in the absence of an initial defect defies treatment by Linear Elastic Fracture Mechanics (LEFM). 23 That is, the conditions for failure must be determined without making assumptions regarding the crack geometry or the path of propagation. Cohesive zone models (CZM) have been employed to ad-
Fig. 1 Deformation in a 4w% Poly(acrylamide) hydrogel contacted by a spherically-tipped indenter, pre-puncture.
dress both the aforementioned challenges; namely, to approximate stress states of nonlinear materials at failure, and to predict failure of materials with blunted or no defects. 24 Barenblatt defined a region, the cohesive zone, within which the attractive potentials between molecules on crack faces provide cohesive forces resisting crack opening; the summation of which constitute a “modulus of cohesion” indicating a critical stress at the crack tip above which the crack will propagate. 25 Hillerborg later expanded this approach to include not only crack propagation, but also the nucleation of a crack at any arbitrary location where such a stress is achieved. 26 1–9 | 1
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DOI: 10.1039/b000000x
Soft Matter
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DOI: 10.1039/C5SM00230C
Gc = σ 0 ∗ l
(1)
where Gc is the critical strain energy release rate, σ0 is the maximum cohesive stress of the material and l is the representative size of the FPZ. 14 It follows, then, that predicting fracture behavior requires knowledge a priori of at least two of the three variables of Equation 1. In most analyses, l and σ0 are taken to be a material constants, thus creating an equivalence between stress and energy descriptions of failure. Lake and Thomas took l to be the end-to-end distance between crosslinks in order to account for the discrepancy in the observed fracture energies and intrinsic bond strengths of elastic networks. 13 Similarly, Hui and Jagota developed criteria for the elastic blunting of cracks in order to account for the strength of elastic networks where l becomes the opening displacement of the stretched crack. 14 However, the dependence of the fracture energy on the size of the FPZ has yet to be demonstrated by controlling the latter as the independent variable. This paper analyzes the mechanical response of soft materials to deep indentation by high aspect ratio indenters, with particular interest in the critical load for puncture, Pc , the load at which the needle breaks through the surface of the material. The relationship between applied displacement and resulting force is quantified and understood for small and large strains for two indenter tip geometries, spherical and flat. This analysis, combined with the application of FPZ, leads to understanding of Pc in terms of materials properties and indenter geometry. We find that the average puncture stress, which can be several orders of magnitude greater than the material’s elastic modulus, is defined by a mechanism that transitions from an energy-based criterion to a stress-based criterion as a function of indenter length scale. The findings have profound implications for the characterization of soft materials, and even more importantly, the design criteria for applications involving soft materials at a wide range of length scales. 2|
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Fig. 2 (a) Schematic of puncture apparatus and representative program for indenter displacement. Zoom-in establishes key definitions: radius, R, depth, d, and load, P. (b) Representative force profile for A25 B116 A25 gel indicating critical load for puncture, Pc , and critical depth for puncture, dc .
2 2.1
Experimental Materials
Acrylic triblock copolymer gels were chosen as primary model soft materials due to their elasticity at large strain and thermoreversible gel transition. 28,29 ABA copolymers of poly(methyl methacrylate) end blocks (PMMA) and poly(nbutyl acrylate) mid block (PnBA) with molecular weights kg kg of 25 mol and 116 mol , respectively (A25 B116 A25 ) were acquired from Kuraray Co. and used as received. Gels were formed by dissolving polymer in 2-ethylhexanol at 150◦ C and then cooling to room temperature, 25◦ C; the same thermal treatment was applied in between puncture experiments to erase damage history. Poly(acrylamide) hydrogels were prepared from 30:1 acrylamide/bisacrylamide stock solution from Bio-Rad Laboratories, Inc. with ammonium persulfate initiator and tetramethylethylenediamine (TEMED) catalyst. Poly(dimethyl siloxane) elastomer (PDMS) was preR pared from a Sylgard 184 kit obtained from KR Anderson Co. A 40:1 ratio of prepolymer to curing agent was used. Poly(urethane) elastomer was prepared from BJB Enterprises PC-12 A/B elastomer kit as directed. Puncture experiments were performed in A25 B116 A25 gels unless otherwise specified.
† Electronic Supplementary Information (ESI) available. See DOI: 10.1039/b000000x/ a Polymer Science and Engineering Department, University of Massachusetts, 120 Governors Drive, Amherst, MA, USA 01003. Fax: +1 413 545 0082; Tel: +1 423 577 1313; E-mail: [email protected]
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The CZM approach to fracture initiation, i.e. critical stress, has lead to the development of Fracture Process Zone (FPZ) models, which provide a physical description of energy dissipation in systems experiencing nonlinear processes during failure. The FPZ represents the volume of material wherein under steady state crack propagation the dissipative work is constant. 27 In this way, the critical stress approach of CZM is related to the critical energy approach of Griffith 23 by the size of the FPZ as
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Fig. 3 (a) Representative force profiles for insertion with an R = 283µm indenter in 4 unique materials: 20 w% A25 B116 A25 gel, poly(dimethyl siloxane) elastomer with 40:1 ratio of prepolymer to curing agent, OO-20 durometer poly(urethane) elastomer, and 4 w% poly(acrylamide) hydrogel. On the y axis load, P, is normalized by load at puncture, Pc , and on the x axis depth, d, is normalized by depth at puncture, dc . (b) Representative force profiles for deep indentation by two indenter shapes: spherical and flat-punch. Inset provides the same data on linear axes to emphasize second power scaling at large depth. (c) Uniaxial compression data for A25 B116 A25 gel fit with Equation 3a. (d) Schematic of strain approximation for deep indentation by flat punch indenter. A cylindrical volume of gel under the indenter is considered to have (i) initial radius R equal to that of the indenter and (ii) stretched radius R + kd, a multiple of the indentation depth.
2.2
Methods
2.2.1 Puncture. Puncture experiments were carried out on a custom apparatus consisting of a Burleigh Instruments EXFO IW-820 Nanopositioner and Inchworm 8200 controller. A custom load cell was used consisting of aluminum cantilevers of dimensions 44.45 x 15.875 x 6.35 mm and 44.45 x 15.875 x 6.35 mm, corresponding to stiffnesses of 0.001 and 0.01 mmN −1 , respectively, coupled to a PISeca D-510.020 single-electrode capacitive sensor and PISeca E-852.10 signal conditioner. Indenters were fixed to the nanopositioner in-line with the load cell and translated at a rate of 250 µms−1 , unless otherwise specified. Experiments involve translation of the indenter into the gel such that the indenter breaks through the surface of the gel before reversal (Fig. 2(a,b)). A representative force trace of this complete cycle is shown in Figure 2(b); however, the work described here is focused only on the portion of the force response leading up to and including the point of puncture, which is marked by a sharp peak formed by a rapid drop-off in load coinciding with penetration of the needle through the surface of the gel. The critical load for puncture, Pc , is taken as the maximum force associated with the peak in the loading curve, and the critical depth, dc , is that corresponding to the critical load. Data points correspond to mean values of between 5 and 10 punctures and error bars represent one standard deviation above and below the mean. Fits to these data utilize standard deviations in instrumental weighting of least squares parameters. Flat punch indenters used were blunt-tipped needles obtained from Hamilton Company and back-filled with epoxy to
avoid coring. These needles come from the manufacturer with a flat tip profile and filleted edge. Spherically-tipped indenters were fabricated by gluing spherical beads to the ends of hollow blunt-tipped needles. Where indenter diameters were too small for this method, indenters were fabricated by pulling capillary tubes in a Narishige PC-10 and subsequently annealing the tips in a microforge. Spherical profiles resulted from surface tension and were confirmed by microscopy (ESI Fig. 1). Being that boundary conditions are critical for understanding the deformation mechanisms of highly strained materials, puncture experiments were conducted in such a manner as to simulate contact with an infinite elastic half-space. That is, gels were contained in vessels sufficiently large that no surface deformation was able to be visually detected at the walls of the container at any time during puncture, and with a depth at least 10 times the depth of puncture. In addition, as the gels used have significant liquid content (between 60 and 90 percent), the interface between the indenter and the gel was not modified in any way, but is still considered to be frictionless for the purposes of analysis. We consider the universality of the force profiles observed across multiple materials and formulations to be indicative of of the consistency of these boundary conditions in our experiments. 2.2.2 Rheology. Triblock copolymer gels (A25 B116 A25 ) were prepared at various weight percent polymer in order to achieve a range of materials properties. The linear relationship between weight percent polymer and shear storage modulus, G0 , obtained by shear rheology is demonstrated in ESI Figure 2. Rheological measurements were made on an AR 2000 rheometer using 25 mm steel ETC parallel plates. Triblock 1–9 | 3
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DOI: 10.1039/C5SM00230C
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DOI: 10.1039/C5SM00230C
2.2.3 Mechanical Testing. Compression and tension R mechanical testing appatests were performed on an Instron ratus fitted with a 50 N load cell. For compression, triblock gel cylinders of 16 mm height and 20 mm diameter were placed between metal plates lubricated with silicon oil to reduce friction. Samples were compressed at a strain rate of 0.001 s−1 until failure. Tensile tests were intended to serve as complements to puncture tests; therefore, strain rates were matched accordingly. Secant strain rates for puncture were calculated from the stress at puncture, σc , the Young’s modulus, E, determined from the loading curve, and the time-to-puncture, t, as ε˙ = σEtc . Tensile rupture tests were performed on “dog-bone” samples of ASTM D412-A dimensions. Fracture energies were taken as the area under the load-displacement curve from the start of the test to the point of failure normalized by the initial cross-sectional area of the dog bone gauge. Notch tests to calculate strain energy release rates were conducted on samples of dimensions 60 · 50 · 1 mm3 with a volume 10 · 50 · 1 mm3 exposed between grips following the procedure outlined by Rivlin. 12 Notches were cut at the mid-point with a razor blade.
3 3.1
Results and Discussion Loading
Deep indentation experiments in four unique materials demonstrate the universality of the force profile pre-puncture (Figure 3(a)). Such a profile is qualitatively consistent with those observed in a variety of synthetic and biological materials. 3,5,6,30 Specifically, the force response pre-puncture indicates an increasing stiffness with depth. Figure 3(b) shows force profiles in A25 B116 A25 gels for indenters with two distinct tip geometries: flat, and spherical. At small depth, both indenters follow the force-displacement behavior predicted by for small-strian contact; 31 that is, 8 P ∼ ERd 3
(2a)
3 1 16 (2b) ER 2 d 2 9 for flat (a) and spherical (b), respectively, where P is the load, E is Young’s modulus of the indented material and R is the radius of the indenter. At large depth, however, both of these models break down, and the force responses of both indenters transition to a second order dependence on depth. This
P∼
4|
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Fig. 4 Empirical determination of the coefficient k0 of Equation 5 for flat-punch indenters in A25 B116 A25 gels of various modulus.
can be attributed to the expected behavior for compression of materials described by a Neo-Hookean consititutive relationship, as demonstrated previously for these ABA gels. 29 To confirm this, uniaxial compression experiments were performed on bulk A25 B116 A25 gels. Under these conditions, the nominal engineering and true stresses in an incompressible neoHookean material, σz, eng , and σz, true , respectively, obey the following expressions: 32 E 1 2 σz, eng = λz − (3a) 3 λz E 1 σz, true = λz − 2 (3b) 3 λz In this case the stretch ratio, λz , is defined as the ratio of the deformed height to the undeformed height, L/L0 . The compression data were fit with Equation 3a using the Young’s modulus, E, as an adjustable parameter (Fig. 3(c)). By redefining the stretch ratio, this neo-Hookean model can be used to describe deep indentation experiments. Consider a volume of unstretched gel underneath a flat punch indenter with radius equal to that of the indenter(Fig. 3(d.i)). Upon compression, the gel is forced out along the side of the indenter such that its radial dimension is increased by a constant multiple of the indentation depth, kd (Fig. 3(d.ii)). Thus, the 2 R stretch ratios can be defined as λr = R+kd R , λz = R+kd , and Equation 3b can then be written as " 4 # E R kd + R 2 − (4) σz = 3 kd + R R Equation 4 can be expanded and multiplied by R2 to give an expression for the load, P, as a function of the depth, d, which after eliminating higher order terms predicts the load at large depth to follow P = −σz R2 = k0 Ed 2 + k00 ERd
(5)
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solutions were brought to 150◦ C, injected between plates preset at 80◦ C and then equilibrated and held at 25◦ C for testing. Frequency sweeps from 1 to 10Hz were performed at 1 Pa and the shear storage modulus, G0 , was taken as the average over the frequency range recorded (ESI Fig. 2).
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Fig. 5 Critical load for puncture, Pc ; critical nominal stress at puncture, σc , normalized by shear modulus, G0 ; and critical depth at puncture, dc , versus radius, R, for flat punch indenters in A25 B116 A25 gels. (a,b) Flat punch indenters demonstrate stress-limited failure occurring at a critical nominal stress σc = RPc2 indicative of the maximum cohesive stress of the gel, which can surpass the modulus by two orders of magnitude. (c) The critical depth, dc , is found to be in excess of ten times the radius, R. (d,e) Spherical tip indenters exhibit a transition between energy-limited and stress-limited failure at a critical transition radius, RT . (f) The critical depth, dc is found to be proportional to the square root of the radius, R, in the energy-limited regime as predicted by equations 5 and 8. The critical depth, dc , is found to be loosely approximated by dc = 3.5R as predicted by equations 5 and 7 for the stress-limited regime.
where k0 and k00 are constants related to k. The linear term of Equation 5 captures small-strain contact of a flat punch at small depth (Equation 2a). Setting k00 = 38 accordingly, indentation force profiles for 30 combinations of flat punch indenter radius and A25 B116 A25 modulus were fit with Equation 5 using k0 E as a fitting parameter. The resulting values of k0 E are plotted against storage modulus obtained from shear rheology in Figure 4, the good linear fit confirming the accuracy of the model. The storage modulus, E 0 , is calculated from the shear modulus, G0 , assuming incompressibility, i.e. ν = .5 and E 0 = 2G0 (1 + ν). The slope of Figure 4 gives an empirical value k0 = .26 ± .01. Figure 3 of ESI demonstrates the efficacy of Equation 5 in describing the loading behavior of spherical indenters. These results indicate that spherical indenters behave similarly to flat punch indenters for d > R. In addition, the second-power term of Equation 5, which dominates at large depth, contains no reference to the radius of the
indenter. This absence suggests that the relationship between the load on the indenter and the indentation depth becomes insensitive to the specific geometry of the indenter at large depth. With this model, we have demonstrated a large-strain contact mechanical approach to modulus measurement of soft materials that offers increased sensitivity and simplicity compared to traditional contact mechanics measurements which require precise alignment of the indenter, knowledge of the area of contact, and are limited to small strains. 33 3.2
Spherical Tip Indenter Puncture
3.2.1 Observations. The critical load for puncture, Pc , for spherical tip indenters is found to be linearly proportional to R below a critical size, RT , and proportional to R2 above RT (Fig. 5(d)). Normalizing the critical load by the cross-sectional area of the indenter defines a critical nominal stress at puncture, 1–9 | 5
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Fig. 6 (a) Schematic of fracture process zone surrounding fracture plane and cohesive forces resisting crack opening. (b) Representative plot of cohesive forces, F, versus crack opening displacement, x, showing critical force for crack nucleation, Fc and displacement at complete separation, xc . (c) Schematic of fracture process zone under (i) flat punch and (ii) spherical tip indenters. Shaded areas depict highly stressed material, while dashed lines indicate the size of the fracture process zone, l. Together the average stress in the fracture process zone, σ , and the size of the fracture process zone, l, determine the available energy for fracture nucleation.
σc . Accordingly, σc is found to decrease with R, eventually plateauing for R > RT (Fig. 5(e)). The critical depth for puncture, dc also exhibits a transition at RT (Fig. 5(f)). Below RT , the dependence of dc on R can be approximated by substituting Pc ∼ R into Equation 5 and solving for dc . As puncture occurs at depths considerably larger than the radius, the second term of Equation 5 can be eliminated for simplicity, 1 yielding dc ∼ R 2 . This result is found to fit the experimental data in Figure 5(f) well for small R. Above RT , substituting Pc = σc R2 into Equation 5 predicts dc = 3.5R. This result is found to loosely approximate the data in Figure 5(f) for large R. 3.2.2 Analysis. We propose that puncture results from nucleation and growth of a crack in a region of highly stressed material under the indenter tip. The stress-concentrating ability of the system, i.e. the distribution of stress under the indenter, is dictated by the geometry of the tip. In the same way, the geometry dictates the volume of material whose strain energy is able to participate in crack formation. These effects and their implications on material failure are discussed in the context of a fracture process zone (FPZ) model below. Consider two half-spaces which meet along a fracture plane as in Figure 6(a). For a crack to form, two conditions must be met: 1) the stress across the fracture plane must be sufficiently high to break chemical bonds, and 2) there must be sufficient energy in the system to perform the work of separating the newly created crack faces. These conditions are depicted graphically in figure 6(b), where the attractive force acting on the faces, F, is plotted as a function of the crack opening displacement, x. At a critical force, Fc , the crack begins to grow and the force resisting crack opening is a monotonically decreasing function of the opening displacement, x, 6|
1–9
which eventually decays to zero at large displacement. 25 Here, we use a linear approximation for simplicity. In this way, the work of separation can be easily defined geometrically as half the product of the critical force, Fc , and the critical displacement, xc , where the force is zero. The energy to perform this work, U, must be supplied by strain energy in the surrounding material. This energy is contained in a finite volume of highly strained material, the FPZ, within the vicinity of the crack plane. A characteristic FPZ size, l0 , can be calculated by taking the critical energy per area, UA = Γ0 (the fracture energy), divided by the critical force per area, FAc = σ0 (the maximum cohesive stress). For illustration, l0 = xc in our example. A FPZ of size l0 will simultaneously satisfy the critical stress and critical energy requirements: σ0 l0 = Γ0
(6)
Equation 6 is a modified form of Equation 1, where the fracture nucleation energy, Γ0 , takes the place of the strain energy release rate Gc . The characteristic FPZ size, l0 , is a material constant that represents a transition between stress-limited and energylimited failure modes. If the size of the FPZ l ≥ l0 , the material fails when the stress reaches σ0 , as by definition the energy criterion will have already been met. This case is therefore stress-limited. Conversely, if stress is concentrated in a FPZ of size l RT . These results are also found to be consistent with our contact model by solving Equation 5 at the point of puncture, i.e., σc ∼ 145G0 from Figure 5(b). It is interesting to note that these data also indicate that dc for these indenters is independent of the material modulus, E, as shown in ESI Figure 4. 3.3.2 Analysis. The stress fields surrounding spherical and flat geometries during indentation have not been studied in this work, and therefore, the specifics of stress concentration at the point of failure cannot be directly addressed. However, 1–9 | 7
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DOI: 10.1039/C5SM00230C
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the experimental data suggest that the FPZ size, l, is larger for flat indenters than spherical. From this, we will employ the assumption that the FPZ size is in fact equal to the radius, i.e., that the stress field under the flat punch indenter is uniform, in order to make interesting inferences about failure in these systems. Assuming that for flat punch indenters α, and consequently β , are both approximately equal to 1 (Fig. 6(c)), σc is then approximately equal to the maximum cohesive stress of the gel, σ0 . From this, we can calculate the characteristic FPZ size, l0 , for a 25w% gel by employing Equation 6. Using σ0 ∼ 1.1 ± .18 MPa from Figure 5(b) and Γ0 = 142 ± 40N/m from tensile testing (Fig. 7(b)), we can estimate l0 = 130 ± 42µm, which is similar in magnitude to the smallest radii flat punch indenters used. While this analysis is convenient and selfconsistent, a more detailed treatment of the stress field during indentation is needed, and the precise relationship between indenter geometry and FPZ size remains an open question.
4
Conclusions
Puncture experiments were conducted in model soft materials across a range of material moduli, fracture energies, and indenter shapes and sizes. Experiments indicate uniform force profiles for deep indentation; namely that the force scales with the square of the depth and with little influence of indenter geometry. This behavior is well described by defining the strain as approximately the ratio of the depth to the radius and assigning a neo-Hookean constitutive relation. The point of puncture is found to be determined by fracture nucleation within a fracture process zone (FPZ) under the indenter. Influencing the size of the FPZ by changing the shape and size of the indenter allows for observation of two distinct failure regimes arising from the necessity for two criteria to be met or failure: a critical stress, and a critical energy. A stresslimited regime occurs when the FPZ is large enough to store the necessary energy for failure at small strain. When the FPZ is small, very large stresses, exceeding the modulus by two orders of magnitude, are necessary to reach the critical energy, which becomes the limiting criterion. The ability to probe each of these regimes independently allows for measurement of the cohesive stress and fracture nucleation energy simply by pushing on a soft material with a high aspect ratio indenter. In addition, modulus measurements can be made from the loading curve with increased sensitivity and simplicity relative to small-strain contact mechanics.
References 1 O. Shergold and N. Fleck, Journal of biomechanical engineering, 2005, 127, 838–48.
8|
1–9
2 L. B. Whitenack and P. J. Motta, Biological Journal of the Linnean Society, 2010, 100, 271–286. 3 N. Abolhassani, R. Patel and M. Moallem, Medical engineering & physics, 2007, 29, 413–31. 4 H. Kataoka, T. Washio and K. Chinzei, Medical Image Computing and Computer-Assisted Intervention, 2002, 216–223. 5 A. M. Okamura, C. Simone and M. D. O’Leary, IEEE transactions on bio-medical engineering, 2004, 51, 1707–16. 6 C. Simone and A. M. Okamura, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292), 2002, 2, 2085–2091. 7 D. J. van Gerwen, J. Dankelman and J. J. van den Dobbelsteen, Medical engineering & physics, 2012, 34, 665–80. 8 D. Richter, F.-R. Matuschka, A. Spielman and L. Mahadevan, Proceedings of the Royal Society B, 2013, 280, 1–6. 9 W. K. Cho, J. a. Ankrum, D. Guo, S. a. Chester, S. Y. Yang, a. Kashyap, G. a. Campbell, R. J. Wood, R. K. Rijal, R. Karnik, R. Langer and J. M. Karp, Proceedings of the National Academy of Sciences, 2012, 109, 21289–21294. 10 S. Das and A. Ghatak, Journal of Materials Science, 2010, 46, 2895– 2904. 11 S. Das, S. Laha and A. Ghatak, Soft matter, 2014, 10, 6059–67. 12 R. S. Rivlin and A. G. Thomas, Journal of Polymer Science, 1953, 10, 291–318. 13 G. Lake and A. Thomas, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1967, 300, 108–119. 14 C.-Y. Hui, J. A., S. J. Bennison and J. D. Londono, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2003, 459, 1489–1516. 15 M. Boyce and E. Arruda, Rubber chemistry and technology, 2000, 73, 504–523. 16 S. Cai, D. Chen, Z. Suo and R. C. Hayward, Soft Matter, 2012, 8, 1301. 17 D. Chen, L. Jin, Z. Suo and R. C. Hayward, Materials Horizons, 2014, 1, 207. 18 A. Chakrabarti and M. K. Chaudhury, Langmuir : the ACS journal of surfaces and colloids, 2013, 29, 6926–35. 19 M. Chaudhury and A. Chakrabarti, Langmuir, 2013, 29, 15543–15550. 20 E. Reyssat, T. Tallinen, M. Le Merrer and L. Mahadevan, Physical Review Letters, 2012, 109, 244301. 21 P. S. L. Anderson and E. J. Rayfield, Journal of the Royal Society, Interface, 2012, 9, 1846–1855. 22 P. S. L. Anderson, The Journal of experimental biology, 2009, 212, 3627– 32. 23 B. Lawn and T. Wilshaw, Fracture of Brittle Solids, Cambridge Univ Press, 1975. 24 M. Elices, G. Guinea, J. G´omez and J. Planas, Engineering Fracture Mechanics, 2002, 69, 137–163. 25 G. Barenblatt, Advances in applied mechanics, 1962, 1–109. 26 A. Hillerborg, M. Modeer and P.-E. Petersson, Cement and Concrete Research, 1976, 6, 773–782. 27 B. Cotterell, Engineering fracture mechanics, 2002, 69, 533–553. 28 M. E. Seitz, D. Martina, T. Baumberger, V. R. Krishnan, C.-Y. Hui and K. R. Shull, Soft Matter, 2009, 5, 447. 29 M. E. Seitz, W. R. Burghardt, K. T. Faber and K. R. Shull, Macromolecules, 2007, 1218–1226. 30 C. Gokgol, C. Basdogan and D. Canadinc, Medical engineering & physics, 2012, 34, 882–91. 31 A. C. Fischer-Cripps, Introduction to Contact Mechanics, Springer Science, 2nd edn, 2007. 32 R. Rivlin, Philosophical Transactions of the Royal Society of . . . , 1948, 240, 459–490. 33 K. R. Shull, Materials Science and Engineering: R: Reports, 2002, 36,
Soft Matter Accepted Manuscript
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DOI: 10.1039/C5SM00230C
Soft Matter Accepted Manuscript
Published on 11 May 2015. Downloaded by Carleton University on 13/05/2015 09:34:40.
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