The mechanical behavior of soft solids is critical for evaluating their suitability for use as tissue scaffolds, [1][2][3] drug delivery systems, 4,5 and 3D-printing inks. 6,7 Designing materials for these applications requires characterization of the elastic and failure responses to predict both how resistant a material will be to deformation and the extent of deformation at which it will fail. The elastic and failure responses of soft solids are typically quantified by a core set of conventional mechanical characterization techniques, such as uniaxial extension and notch tests. 8,9 While conventional mechanical characterization techniques are preferred methods for evaluating materials, a growing body of literature suggests that there are instances in which a nonconventional measurement, such as deep indentation and puncture, is a more appropriate choice. For example, such instances arise when samples have a limited volume, 10 extreme reaction kinetics, 11 or brittle behavior. 12 Beyond challenging samples, conventional techniques often fall short in their ability to access high-throughput screening due to difficulty in automating the loading and unloading of samples from instrument clamps. Contact-based methods, such as indentation, currently occupy this space as a high-throughput testing method for soft solids; however, protocols often operate at low indentation depths (i.e., small strains) where analysis is limited to estimates of stiffness and adhesive strength. 13,14 By contrast, in instances where the large-strain or failure response is a desired screening parameter, 15 puncture tests are a promising method for the high-throughput characterization of the nonlinear behavior of soft materials.

Recent developments in the deep indentation and puncture of soft solids suggests that puncture tests are information-rich, where one test can be used to characterize the linear and nonlinear elastic, rupture, fracture, and frictional behavior of a material. 10,[16][17][18][19][20][21][22] Puncture tests consist of inserting a needle into a material at a controlled displacement rate, while monitoring the force. After a prescribed ''turnaround'' displacement beyond the point of puncture, the displacement rate is reversed and the needle is retracted from the material. A typical force-displacement curve for this test (Fig. 1) shows the four stages of deep indentation and puncture, 23,24 which broadly fall into two regimes-the elastic loading and puncture regime, and the post-puncture regime. The first stage (i) includes the linear and nonlinear elastic loading of

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the material. The material response is described by Hertzian contact mechanics at low indentation depths while the nonlinear elastic response at large depths has been found to scale with Young's modulus E and the square of the displacement d 2 . 16 This stage terminates at the critical force for puncture F c , which is followed by an abrupt drop in the force that signals the transition to the post-puncture regime. The second stage (ii) is the ''embedding stage'' where the force response is dominated by friction at the walls and a steady-state crack propagation force at the needle tip. This stage terminates at the turnaround displacement of the needle. The third stage (iii) consists of retraction to reverse the shear direction at the walls and decompress the material below the needle tip. This stage terminates once the needle tip peels from the bottom of the crack generated during the embedding stage. 23 The final stage is the shear retraction stage (iv) where the forces are determined solely by the sliding friction at the needle wall. The elastic and failure responses of soft solids can be quantified by analyzing the behavior in both the elastic loading and puncture as well as the post-puncture regimes during puncture tests.

Despite the promise that puncture tests show for characterizing soft solids, particularly in a high-throughput manner, current experimental protocols do not provide quantitative advice for setting sample dimensions relative to the indenter dimensions. As a result, the volume of sample needed to perform a puncture test is ambiguous. This ambiguity allows for significant differences in both measurement and interpretation between users due to contributions in the measured force due to finite size effects -i.e., the sample not being an infinite halfspace and thus being sensitive to the boundary conditions at the edges. Two major roadblocks exist for developing quantitative guidelines for setting puncture test geometries. First, numerical modeling of deep indentation and puncture is difficult due to the fine resolution of the mesh elements required to resolve large distortions that occur near the indenter tip. Treating these distortions is a substantial computational undertaking and is left to future work. The second roadblock for developing such guidelines is the lack of experiments that systematically vary the relative sample dimensions to serve as a benchmark against which future modeling can be compared.

Here, we aim to address this lack of available data by experimentally documenting how finite sample size effects influence the observed elastic and failure response of a wellcharacterized material during deep indentation and puncture. This is accomplished by performing deep indentation and puncture measurements on a well-characterized silicone elastomer with varied sample heights and radii relative to the indenter needle. First the materials and methods employed in this work, including those for the core puncture experiments and the additional material characterization, are presented. Then the results from the elastic loading and puncture regime as well as the post-puncture regime are presented and discussed. Finally, the limitations of this data are presented alongside a discussion of the opportunities that this work enables. These findings enable the development of quantitative guidelines for setting sample dimensions (and by extension, volume) during puncture tests and will allow puncture tests to be exploited as a high-throughput screening method.

Measurements were performed on a silicone blend elastomer. 25 Samples were formed by mixing a commercially available polydimethylsiloxane (PDMS) elastomer kit, Sylgard 184, at a 30 : 1 prepolymer : curing agent ratio. This pre-cured mixture was then diluted with 100 cSt trimethylsiloxy terminated linear polydimethylsiloxane chains to a weight fraction of 0.75. The added PDMS chains (approximately 6000 g mol À1 ) act as a nonvolatile, unreactive solvent and are too short to form entanglements. Samples were mixed and then degassed under vacuum before being poured into vials and 4 inch square polystyrene Petri dishes to form samples for puncture and other additional characterization tests. All samples were cured at 70 1C for 21 hours and then immediately removed and left to cool at room temperature (22 1C) for several hours before testing.

Silicone blend samples and needles were subjected to a series of characterization tests beyond the puncture measurements that form the core experiments of this work. The intent of this additional characterization was to quantify the properties that are most relevant to future modeling of deep indentation and puncture. These additional characterization methods include small strain indentation, uniaxial extension, uniaxial compression, tear tests, profilometry of the needle tip, rheometry, and sliding friction measurements. Radially confined compression has also been performed in a previous publication. This journal is © The Royal Society of Chemistry 2024

The methods and instrumentation for each of the characterization methods used in this work are documented in the ESI. † A summary of the properties provided by these measurements is contained in Table 1.

Deep indentation and puncture were performed on a TA.XTPlus Connect Texture Analyzer with a 50 N load cell. A 22 gauge blunt-tipped steel needle (inner and outer radius of 0.207 mm and 0.359 mm, respectively) was sourced from the Hamilton Company and mounted to the instrument using the barrel of a 3 mL luer lock tipped syringe mounted into a drill chuck fixture that connects directly to the load cell. Puncture measurements were performed in two vials with different internal dimensions. The larger vial was a 20 mL scintillation vial and had an internal radius of R = 13 mm while the smaller vial had an internal radius of R = 7.1 mm. Samples were prepared in each vial with different amounts of material to alter the sample height H. Ten different heights (3-15 mL of sample) were tested in the R = 13 mm vials and six different heights (1-5 mL of sample) were tested in the R = 7.1 mm vials. The vials of samples tested in the variable height measurements were not clamped and thus the data obtained during the retraction stage of the test are only useful up to the point where the frictional forces become equal to the weight of the sample. The majority of samples were tested with a displacement rate of 1 mm s À1 ; however, samples with dimensions (R = 13 mm, H= 28.3 mm) and (R = 7.1 mm, H = 31.4 mm) were also tested with displacement rates of 0.1 and 10 mm s À1 . Four measurements with clamped vials were performed at each displacement rate. The starting height of the needle tip was recorded before measurements were performed and samples were determined not to puncture if the indenter tip reached within 100 mm of the base of the vial.

Previous work during the elastic loading regime showed that, in the absence of any near-wall effects, the relationship between F and d is well-approximated by, 16

where a is the indenter radius and both k 0 and k 00 are empirically measured constants. This established relationship assumes that the sample is sufficiently large that the boundary conditions at the edge of the sample (i.e., the cylindrical vial walls) do not influence the indentation measurement. k 0 is typically viewed as being material-specific and reported values of this property typically span 0.25-0.5, 11,[16][17][18] though one work has observed values on the order of 1. 10 This relationship suggests that a plot of F against Ed 2 should be linear at large d where k 0 Ed 2 c k 00 Ead in the absence of near-wall effects to the force, as k 0 should be constant for a given material.

Plots showing representative curves of F against Ed 2 for the vials where (a) R = 13 mm and (b) R = 7.1 mm with varied H are contained in Fig. 2. Note that all runs from both of these plots have been individually plotted in the ESI † for additional clarity.

Since a = 0.359 mm, R a ¼ 36 and R a ¼ 20 for the large and small vial, respectively. In the large vial samples, an apparent linear relationship is observed at large H while a nonlinear relationship where the sample appears to stiffen is observed at smaller H. The transition between these two behaviors appears to occur when H = 11.647 mm H a ¼ 32

indicating that finite size effects can influence measurements when the sample height is significantly larger than the indenter dimensions. Since the underlying constitutive response of the material is unlikely to change across the samples, this observed stiffening likely indicates that the measurement is being influenced by traction at the sample-vial interface. These tractions should generate some hydrostatic stress that would stiffen the apparent material response consistent with the behavior observed in Fig. 2a. In contrast to the large vials, the material response of the small vial samples (R = 7.1 mm, R a ¼ 20)

does not appear to be significantly influenced at different sample heights. There is some nonlinearity in the two shortest samples; however, it is less pronounced than in the large vials. This appears to be counter-intuitive as one would expect radial confinement to also cause a stiffening effect. To further examine this behavior, it is useful to consider the actual values of k 0 . A plot of the apparent k 0 values for samples calculated as F c /(Ed c

2 ) (setting k 00 = 0), where F c and d c are respectively the critical puncture force and displacement, against sample height is contained in Fig. 2c. We emphasize that these are apparent values -and not intrinsic material properties -as they are calculated in a manner that ignores finite size effects. In the large vial samples (R = 13 mm), k 0 has a value of approximately 0.4 at large H with a steep increase in value to a maximum of approximately 1 at small H. This trend is consistent with the full F vs. Ed 2 curve (Fig. 2a), indicating that deviations from the apparent k 0 value at large H may be a reliable indicator of the onset of finite size effects. In the small vial samples (R = 7.1 mm), the apparent k 0 value is largely insensitive to H and is approximately 0.75. This trend is consistent with the previous observation from the full F vs. Ed 2 curve, which lacks a pronounced nonlinear curvature at smaller H. Importantly, the apparent k 0 values for both sample vial radii do not agree at large H, with those for the small vials being almost a factor of 2 higher. This suggests that the influence of a finite sample height on the observed stiffness can be neglected if the indentation response is already dominated by the effects of the finite sample radius. While there is a straightforward trend showing that the stiffening driven by a finite sample height is negligible when d c 4 H, estimating the effective halfspace value of k 0 at large sample radius requires further analysis. To aid in this analysis, an additional data point was taken at large sample height (H = 30 mm) in 100 mL beakers that had an internal radius of 23.5 mm. A plot of the apparent k 0 values for the largest sample heights against 1 R is shown in Fig. 2d. Note that the inset of k 0 plotted against R shows an inverse dependence with R. When plotted against 1 R it is apparent that k 0 has a nonlinear relationship with 1 R . This data was fit to a power law fit and it was found that k 0 $ 1 R 2:285AE0:487

. Additionally extrapolation found that the predicted halfspace value (intercept in Fig. 2d) was 0.307 AE 0.033 which is in the lower range of values often observed during puncture tests. 11,[16][17][18] While general recommendations for managing finite size effects will require numerical modeling, this data shows that measurements of apparent k 0 values can be used to quantify their influence. Finite height effects are negligible when d c 4 H and finite radial size effects This journal is © The Royal Society of Chemistry 2024 can be quantified by extrapolating to 0 on the apparent k 0 vs. 1 R plot. Performing measurements to quantify this behavior requires a minimum of 3 samples; however, it can require more if the initial trials show that d c E H.

Puncture of the sample occurs when a crack nucleates below the indenter and releases some of the strain energy stored in the material. The release of this energy leads to a characteristic drop in the force-displacement curve, as shown in Fig. 1. The critical puncture displacement d c and puncture force F c at which the crack nucleates are often used as characteristic measures of the fracture response of a material. 10,11,16 d c and F c are measured without notching the sample and are determined by a rupture process where the fracture energy is convoluted with the crack nucleation energy, as discussed previously. 16 In this work, and consistent with prior work, 18,20 we make the simplifying assumption that fracture energy scales with the rupture energy and thus use d c and F c to quantify the fracture response.

Plots of (a) d c and (b) F c at different sample heights are contained in Fig. 3. At sufficiently large values of H, d c appears to be relatively constant in both vial sizes with approximate values of 10 mm and 7.5 mm in the large and small vials, respectively. The lower observed values of d c in the small radius vials indicate the influence of a finite radial sample size. The limit where d c = H is marked in Fig. 3a by the solid line. For the larger vials, the samples appear to approach this limit when H E 12 mm, which is consistent with the onset of stiffening observed previously in the F vs. Ed 2 curves. This further suggests that the stiffening observed in the large vials occurs due to the indenter tip approaching the interface at the base of the vial. In the small vials, the samples appear to approach this limit when H E 10 mm, suggesting that stiffening driven by a finite sample height should not be apparent in these samples. This is consistent with the observations in Fig. 2 that suggested the small samples are mainly sensitive to the interface at the wall of the vial. Unlike d c , F c does not appear to have a strong dependence on the sample dimensions. As shown in Fig. 3b, F c has significant scatter and an approximate value of 3 N in both vial sizes at all sample heights.

Similar trends and values of F c are observed in both vial sizes regardless of the indenter velocity (i.e., the strain rate) (Fig. 3c). Both samples were measured at large values of H where finite sample height effects should not influence the measurement. In the large vial, F c decreases as the indenter velocity increases from 0.1 to 1 mm s À1 and then slightly increases when the indenter velocity increases to 10 mm s À1 . In the small vial, F c decreases and becomes constant within the spread of the measurement at 1 and 10 mm s À1 . Neither of these trends are consistent with that observed in the tear tests, performed in the ESI, † where G c was found to monotonically increase with crack velocity. This suggests that, while previous measurements have observed F c monotonically increases with indenter velocity as G c increases, 18 this may not be the case for all materials, geometries, and indenter velocities.

Together, the results from the elastic loading and puncture regimes suggest that the small and large vials represent two different limiting cases where finite size effects matter. In the large R a ¼ 36

vial, stiffening is driven by mechanical interactions at the interface between the sample and the base of the vial. This is consistent with the observation that, as d c -H, the measurements display both a pronounced nonlinearity in the F vs. Ed 2 curves and increasing apparent k 0 values. On the other hand, the stiff behavior of the small vial R a ¼ 20

appears to be driven primarily by mechanical interactions at the interface between the sample and the vial walls. This is consistent with the observation that d c does not impinge upon the limit where d c = H and displays both a lack of pronounced nonlinearity in the F vs. Ed 2 curves as well as constant apparent k 0 values that are higher than those in the tall sample limit of the large vials.

The consistently increased apparent stiffness in the small vials is likely due to the indenter being equidistant from the vial walls as the sample height is altered. This means that the large vial represents a case where the finite sample height primarily alters the apparent response whereas the small vial represents a case where the finite sample radius primarily alters the apparent response.

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3 Post-puncture regime Analysis of deep indentation and puncture in the post-puncture regime is an active area of investigation. [22][23][24][27][28][29] The current understanding of crack morphology during deep indentation and puncture is that it relates to the indenter tip geometry, materials properties, distribution of materials defects, and large strain stiffening behavior. 21,30 Currently this link is qualitative and prohibits researchers from predicting the crack morphology without making measurements. The analysis used here assumes that the crack that nucleates upon puncture can be represented as an idealized planar crack and further assumes that the crack has sufficient room to reach a steady state of propagation before the turnaround displacement is reached. While these assumptions are appropriate for many samples, there are more complex crack morphologies and experimental scenarios where this analysis would not apply. 11,21,31,32 Indeed, as seen in Fig. 4a, samples that impinge upon the d c = H limit puncture close to the turnaround displacement and fail to reach steady state propagation before the indenter is retracted from the sample. Note that all runs from this plot have been individually plotted in the ESI † for additional clarity. As a result, we focus further analysis on the experiments performed using samples with large heights and only probe the influence of a finite radial size. After puncture, the force resisting further insertion can be separated into two contributions, 22,33,34

that represent the force at the tip needed to propagate the crack F prop and the sliding friction that develops at the walls of the indenter F slide . Assuming that the crack propagates in a self-similar manner, F prop should not vary as the displacement increases. Assuming a penny-shaped crack geometry at the indenter tip gives

ffiffiffiffiffiffiffiffi ffi EG c 3 r when one assumes a linear elastic response for the fracture propagation. 35 F slide can be determined by multiplying the critical sliding stress t c by the area A of contact at the needle walls. 22 A is estimated as the circumference of the needle 2pa multiplied by the contact length l which should scale directly with d. Thus the total force can be expressed as This journal is © The Royal Society of Chemistry 2024

Here, both terms that determine F depend on velocity through the dependence of E, G c , and t c on velocity.

Notably, F prop does not depend on d while F slide does through l. This means that the slope on the force-displacement curve @F @d provides characteristic information about the sliding friction. It also suggests that F prop can be quantified by extrapolating to l = 0 where F slide = 0, which should occur when d = 0.

This possible treatment of the data is illustrated in Fig. 4b. @F @d is calculated from the embedding stage and F prop is calculated by extrapolating this line to d = 0.

A plot of the resulting values of @F @d and the friction coefficient, measured as part of the additional materials characterization presented in the ESI, † against the sliding velocity is shown in Fig. 4c. These data show that both @F @d and the friction coefficient increase as the sliding velocity increases. This suggests that the frictional stresses increase due to an increase in the friction coefficient. The critical sliding stress is smaller in the small vials at the 0.1 and 1 mm s À1 sliding velocities while no difference is observed at 10 mm s À1 . This suggests that finite radial size effects become less important as the sliding velocity increases.

A plot of F prop against G c , measured via tear tests discussed in the ESI, † is contained in Fig. 4d. For all sliding velocities, F prop is observed to be larger in the smaller vials. Notably, F prop decreases to the point that it eventually becomes negative as G c increases. These negative values are nonphysical as F prop would then work to close the crack instead of open it so that the crack can propagate. The decreasing trend and negative values of F prop at higher sliding velocities possibly indicates a diminishing and eventual loss of the ability to extract F prop as the frictional component becomes too large to sense the crack tip propagation stress. Further, the observation that F prop varies between vials suggests that the relative magnitudes of F prop and F slide are sensitive to boundary conditions. Taking the lowest sliding velocities, where the frictional stresses are lowest, the predicted magnitude of F prop can be calculated by using independently measured reference values (Table 1) in the first term in eqn (3). This gives a predicted value of 82 mN, which is an order of magnitude lower than the approximate value of 500 mN found through the force-intercept method. Given that G c B F prop 2 , estimating G c from the experimentally observed F prop would overestimate G c by two orders of magnitude. This discrepancy suggests that either the method of extracting F prop is flawed or the theoretical form of F prop , which assumes that linear elastic fracture mechanics are valid, is incorrect.

Recently, it has been suggested that the slope in the shear retraction stage should be the negative value of that in the embedding stage. 22 Examining the force-displacement curve in Fig. 4a, it is apparent that, while the curve is fairly linear in the embedding stage, it is curved and not as smooth in the shear retraction stage. This is caused by the jagged nature of the crack that forms during puncture, and an example image of this can be seen in Fig. 1a (enlarged version in ESI †). This jaggedness results in a non-uniform contact pressure and causes jumps in the force-displacement curve due to stick-slip events that occur as the tip slides across these features. As a result, the comparison of these slopes is not performed in this work, though previous measurements have found that these slopes are roughly equal and opposite in sign. 36 Together, these results suggest that finite radial size effects influence measurements in the post-puncture regime by increasing the apparent values of F prop and decreasing the apparent values of F slide in smaller vials. It was also found that characterizing sliding friction from F slide is very promising while extracting G c from F prop merits additional consideration. Further it was found that examining the post-puncture mechanics was only useful in the case where d c { H so that a steady state crack propagation process could develop.

This work examined the influence of finite size effects on the apparent material response during deep indentation experiments. Finite size effects were found to be important in the elastic loading and puncture as well as the post-puncture regimes. For the materials studied here, confinement effects in the loading direction are apparent when H d c $ o 1, whereas radial confinement effects are detectable up to at least R a $ 65:5. While the measurements presented show that deep indentation and puncture is sensitive to the influence of boundary conditions at the sample edges, there are some limitations to these findings. First, while an experimental pathway for quantifying the effect of finite size was demonstrated, general guidelines for managing them were not developed.

Creating such guidelines will require numerical modeling of deep indentation and puncture which is challenging given the large distortions that occur near the indenter tip and is left to future work. Second, these results highlight the current gap in understanding of the mechanical response in the postpuncture regime. Additional measurements in a system that has a lower or tunable friction coefficient to further examine the prospect of extracting a fracture energy is of interest in the future. Together these measurements are a vital step towards implementation of deep indentation and puncture as a highthroughput screening method.