Hyperelastic characterization via deep indentation

Mattia Bacca; Mohammad Shojaeifard

REVIEW 4 major objections 5 minor 11 references

Hyperelastic characterization via deep indentation

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Deep indentation of soft solids yields a parabolic force-depth law from which both the elastic modulus and the strain-stiffening exponent can be read from a single measurement.

desk verdict A useful, honest empirical calibration for extracting Ogden parameters from deep indentation, but the friction caveat is real and needs to be addressed before the low-alpha claims hold. read the letter →

arxiv 2506.05371 v2 pith:6BJOP7MJ submitted 2025-05-28 cond-mat.soft cond-mat.mtrl-sci

Mohammad Shojaeifard , Mattia Bacca This is my paper

classification cond-mat.softcond-mat.mtrl-sci

keywords deepindentationhyperelasticityOgdenmodelsoftmaterialscontactmechanicsstrainstiffeningin-situmechanicaltestingfiniteelementanalysis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that deep indentation—pushing a small spherical-tipped cylinder far into an incompressible soft solid—carries enough information to determine both parameters of a one-term Ogden hyperelastic model. In the deep regime, where indentation depth $D$ exceeds ten times the indenter radius $R$, the force-depth curve becomes parabolic, $F = \beta E D^2$, with the radius dropping out and the coefficient $\beta$ tied to the strain-stiffening exponent $\alpha$ by $\beta \approx 0.15\,\alpha^{0.85}$ for $\alpha>0$. If true, this replaces dumbbell-shaped uniaxial specimens with a quick, in-situ, sample-free measurement that works on tissues, gels, and elastomers. The paper supports the claim with finite-element simulations and with matching uniaxial and indentation experiments on three silicone elastomers and porcine skin, extracting $E$ and $\alpha$ from both methods with discrepancies mostly under 11%.

What carries the argument

The load-bearing object is the parabolic deep-indentation regime, $F\approx\beta E D^2$ for $D>10R$, which emerges when penetration depth greatly exceeds the probe radius so that geometry, not probe size, sets the resistance. The coefficient $\beta$ is the measurable that carries the strain-stiffening information: through the empirical maps $\beta\approx0.15\,\alpha^{0.85}$ and $\beta\approx0.33|\alpha|^{0.85}$, a single deep curve constrains $\alpha$, while the Hertzian intercept $F=(16/9)E R^{0.5}D^{1.5}$ fixes $E$ independently. An exponential blending formula combines the two power laws into a single expression that fits full indentation curves across both regimes.

What would settle it

Perform deep indentation on the same low-$\alpha$ material (for instance Ecoflex 30) under three deliberately different interfacial conditions—dry steel, lubricated steel, and roughened steel—and compare the extracted $\beta$ values; if $\beta$ shifts by more than the reported extraction error, the frictionless calibration is insufficient and extracted $\alpha$ values for low-$\alpha$ materials need a friction correction.

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Extended reading notes

Core claim

For a one-term incompressible Ogden solid, the indentation force no longer follows the shallow Hertzian scaling $F=(16/9)E R^{0.5}D^{1.5}$ once $D>10R$; instead it becomes $F\approx\beta E D^2$, independent of $R$. The paper identifies three regimes—Hertzian, intermediate, and parabolic—and fits the dimensionless coefficient $\beta$ to the Ogden exponent $\alpha$, obtaining $\beta\approx0.15\,\alpha^{0.85}$ for $\alpha>0$ and $\beta\approx0.33|\alpha|^{0.85}$ for $\alpha<0$. Coulomb friction raises $\beta$ and can mask strain stiffening for small $\alpha$, but the effect becomes negligible for $\alpha>3$. Experiments on Ecoflex 00-10, Ecoflex 00-30, Mold Star 16 Fast, and porcine skin give $\alpha$ values from indentation within about 2–10% of uniaxial values for the elastomers, and a 1.9% difference for pig skin's $\alpha$ (with a 22.8% difference in $E$, attributed to anisotropy).

Load-bearing premise

The load-bearing premise is that friction at the steel-elastomer contact can be ignored during calibration, yet the actual Coulomb friction coefficient is never measured, and the paper's own simulations show friction inflates $\beta$ most for low-$\alpha$ materials such as Ecoflex 30.

Editorial extensions

If this is right

A single deep indentation curve yields both $E$ (from the Hertzian intercept) and $\alpha$ (from the parabolic $\beta$), removing the need for dumbbell-shaped specimens and clamp alignment.
Because the indenter radius drops out in the deep regime, smaller probes reach $D>10R$ at modest depths, making the method practical in-situ, in-vivo, and on samples with unknown thickness.
For materials with $\alpha>3$, Coulomb friction barely changes $\beta$, so dry-contact indentation gives reliable parameters without lubrication or friction measurement.
Combining deep indentation with uniaxial tension resolves the sign ambiguity of $\alpha$ (for example $\alpha=10$ versus $\alpha=-20$), where each method alone cannot distinguish the two.
Materials with negative $\alpha$ (such as brain tissue) show a roughly two-fold larger $\beta$, entering the parabolic regime at shallower depths, so deep indentation should work especially well on them.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the $F\propto D^2$ scaling is not tied to the one-term Ogden form; it notes the power laws persist for other hyperelastic models, so the same deep-indentation protocol could be recalibrated for Arruda-Boyce or compressible Ogden parameters, with multiple probe geometries providing the extra independent trends.
An unmeasured friction coefficient is the main hidden variable for low-$\alpha$ materials: measuring the interface friction directly, or testing lubricated versus dry contacts, would extend the method's accuracy to neo-Hookean-like materials where $\alpha$ is near 2.
For anisotropic tissues, the 22% error in $E$ for pig skin hints that combining indentation along multiple directions—or using a microneedle array—could reconstruct directional stiffness rather than treating the material as isotropic.
A direct test of generality: run deep indentation on a hydrogel family with controlled crosslink density, compare $\alpha$ from $\beta$ against uniaxial measurements, and check whether $\beta\approx0.15\alpha^{0.85}$ holds outside silicones.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The genuinely new thing here is the empirical calibration beta ≈ 0.15 alpha^0.85 (and the negative-alpha counterpart), plus the experimental demonstration that deep indentation recovers E and alpha within about 11% for three elastomers. The parabolic F ~ E D^2 regime itself is not new—Li and Masen (2024) and Mu et al. (2025) already describe it—but the paper's specific mapping from the parabolic coefficient to the Ogden exponent, validated independently against uniaxial tests, is a legitimate incremental contribution. The FEA work is careful: quasi-static checks, mesh convergence, and friction effects shown explicitly. The uniaxial experiments are independent of the indentation fit, so the 11% agreement for Moldstar (alpha = 3.1) and the 1.9% error for pig skin (alpha = 10) give real support to the method in the alpha > 3 regime. The soft spots are in proportion. The stress-test note is right: the friction coefficient at the steel–elastomer interface is never measured, and two of the three elastomers have indentation-extracted alpha below 3 (Ecoflex 10: 2.8, Ecoflex 30: 2.4), where the paper's own FEA says Coulomb friction inflates beta. The observed alpha errors are +8.4% and +10.1%, exactly the sign a frictional contribution would produce. The paper acknowledges this but does not bound it, so the claimed accuracy for low-alpha materials is not fully supported. That is a real caveat, but it does not sink the paper; it means the method is most reliable for alpha > 3, which the authors themselves state. Second, the beta–alpha relation is an empirical fit to FEA at five positive alpha values; deriving alpha from indentation is interpolation on that fitted curve, so there is partial circularity. The independent uniaxial validation breaks most of that circularity, but reporting uncertainty propagation for alpha and beta would make the claim cleaner. Third, the neo-Hookean alpha = 2 point never reaches the parabolic regime within the explored depth, so the method is not universal, and the authors should state that more prominently. Eq. (9) is ad hoc, but it matches the data and is not load-bearing. Bottom line: the paper deserves a serious referee. I would send it out with a request to measure or bound friction for low-alpha materials, add uncertainty bars, and clearly delineate what is new versus Li and Masen and Mu et al. The central method holds up for the high-alpha regime, and the experimental validation is a real asset. Worth a reading group discussion.

Referee Report

4 major / 5 minor

Summary. This paper proposes a deep-indentation method for characterizing soft incompressible materials described by a one-term Ogden model. Through axisymmetric FEA, the authors identify a Hertzian regime (D < 0.1R) and a parabolic regime (D > 10R) in which F = β E D^2, with β correlated empirically to the Ogden exponent α. They propose extracting E from the Hertzian intercept and α from β via the fitted relations β ≈ 0.15 α^0.85 (α > 0) and β ≈ 0.33 |α|^0.85 (α < 0). The method is validated by comparing uniaxial tension and deep indentation on Ecoflex 00-10, Ecoflex 00-30, Mold Star 16 Fast, and porcine skin, with reported parameter errors up to 11% for elastomers and 22.8% for skin.

Significance. If the central claim holds, the method offers a practical, in-situ alternative to dogbone uniaxial testing for two-parameter Ogden characterization of soft materials. The paper's strengths include a parametric FEA campaign spanning α and Coulomb friction, explicit independent comparison between uniaxial and indentation extraction, and a clear presentation of the two power-law regimes. The experimental agreement on elastomers is encouraging, and the authors are transparent about several limitations (anisotropy, model simplicity, friction sensitivity). However, the quantitative value of the method rests on the empirical β(α) calibration and on the unmeasured friction coefficient for low-α materials, both of which need strengthening before the claimed accuracy is fully supported.

major comments (4)

[Experimental Method (Deep Indentation) and Discussion] The Coulomb friction coefficient f at the steel-elastomer interface is never measured, yet the FEA results in Figure 3 show that friction inflates β for low α and that its effect is negligible only for α > 3. Two of the three validation elastomers have uniaxial α below this threshold (Ecoflex 10: α = 2.8; Ecoflex 30: α = 2.4), and the indentation-extracted α exceeds the uniaxial value by 8.4% and 10.1%, respectively—the sign expected from a frictional contribution. Without a measured or bounded f, Eq. (8a) is not quantitatively validated for the very materials used to support the method, and α is not identifiable from a single frictionless-calibration indentation curve.
[Deep Indentation via Spherically-tipped Cylinders, Eqs. (8a)-(8b) and Figure 3-right] The β(α) mapping is an empirical fit to FEA results at only five α values (2, 3, 5, 9, −9, −20), and the paper reports no fit residuals, confidence intervals, or independent verification of the functional form β ≈ 0.15 α^0.85. Because this relation is the sole route from a measured β to an extracted α, its interpolation uncertainty propagates directly into every reported α value. The authors should provide the fit statistics, test additional α values, or supply a mechanical derivation of the scaling.
[Deep Indentation via Spherically-tipped Cylinders, paragraph on α = 2] The treatment of α = 2 is internally inconsistent. The text states that for α = 2, corresponding to neo-Hookean behavior, “the indentation response does not exhibit a parabolic behavior within the explored depth range” and that the transition depth may be indefinitely large. Yet Figure 3-right and Eq. (8a) appear to include α = 2 in the positive-α fit. If α = 2 is excluded, the positive branch of the fit rests on only three points (3, 5, 9); if it is included, β is undefined for that point. The authors must clarify which data points enter the fit and how β was assigned at α = 2.
[Discussion and Conclusions and Table 1] The experimental validation reports only point estimates of E and α from the uniaxial and indentation fits, without standard errors, confidence intervals, or goodness-of-fit measures for the fits to Eqs. (8a) and (9). The claim of “good agreement” is therefore not quantitatively bounded, and the situation is especially concerning for pig skin, where the E error is 22.8% and the isotropic model is acknowledged to be inadequate. Please report fit uncertainties and, if feasible, a sensitivity analysis of extracted α with respect to f.

minor comments (5)

[Uniaxial Tension] There are typos in the text: “puling stress” and “puling force” should be “pulling stress” and “pulling force.”
[Equation (9)] The empirical blending form in Eq. (9), with the exponential decay term, is introduced without a rationale for its functional form; a sentence explaining the choice would improve reproducibility.
[Experimental Method (Deep Indentation)] Indentation tests were performed on only two samples per elastomer and results were averaged; a statement of the resulting uncertainty or a repeat count comparable to the uniaxial tests (n = 3) would strengthen the validation.
[Experimental Method (Deep Indentation)] The porcine skin samples were built by stacking 14 layers bonded with cyanoacrylate; the potential effect of the adhesive layers on the measured force-depth response is not discussed.
[Abstract and Deep Indentation section] The abstract states the parabolic regime occurs for D > 10R, but the paper itself shows this threshold does not apply to α = 2; the threshold should be stated as valid for sufficiently large α or qualified explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the beta-alpha calibration is a forward-FEA surrogate independently benchmarked against uniaxial tests.

full rationale

The derivation chain is self-contained and externally benchmarked. The central calibration, Eq. (8a) beta approx 0.15 alpha^0.85 for alpha > 0, is obtained by fitting forward finite-element simulations of a prescribed one-term incompressible Ogden solid (alpha = 2, 3, 5, 9, -9, -20); it is not fitted to the experimental indentation curves that are later used for validation. The deep-indentation alpha is then inferred from the measured beta using this independently generated surrogate, while the uniaxial alpha comes from separate dogbone tests fitted to Eq. (4). Agreement between the two protocols (Table 1) is therefore a genuine external check rather than a consistency loop. The self-citations (Fregonese and Bacca 2021, 2022) are used only to motivate practical feasibility, such as deep puncture thresholds and the relevance of friction, and are not load-bearing for the beta-alpha mapping; moreover, the experiments independently reach D/R approximately 14-16 without puncture. The unmeasured friction coefficient is a potential identifiability and accuracy limitation for low-alpha materials, but it is a correctness risk, not a circularity: the paper openly reports that friction inflates beta for small alpha and does not use a friction-fitted calibration to manufacture agreement. No step reduces, by construction, to its own inputs.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new physical entities, particles, forces, or conserved quantities are introduced; the only new constructs are empirical fitted coefficients beta(alpha) and an interpolation formula, which are accounted for in free_parameters.

free parameters (5)

beta_alpha_prefactor_positive = 0.15
Fitted to FEA-computed beta for alpha = 3, 5, 9 (Eq. 8a); used to convert experimental beta into alpha.
beta_alpha_exponent_positive = 0.85
Fitted exponent in Eq. 8a from the same FEA points.
beta_alpha_prefactor_negative = 0.33
Fitted to FEA-computed beta for alpha = -9, -20 (Eq. 8b).
beta_alpha_exponent_negative = 0.85
Fitted exponent in Eq. 8b.
regime_thresholds = D/R = 0.1 and D/R = 10
Chosen by inspection of log-log FEA curves to define Hertzian and parabolic regimes; thresholds are not derived.

assumptions (5)

domain assumption The tested materials are incompressible and described by a one-term Ogden strain energy with only E and alpha.
Used throughout (Eq. 1); the authors acknowledge it limits capture of volumetric compressibility and multi-term behavior (Discussion).
domain assumption The finite-element model with sample radius and height 100R and a rigid indenter accurately represents the experimental half-space condition.
SI mesh and boundary conditions; experiments use B/R = 80 and H/R = 90, so edge effects are assumed negligible; stacked pig-skin layers bonded with cyanoacrylate add interface uncertainty.
domain assumption Coulomb friction with a single coefficient f describes the steel-elastomer interface, and f is either known or negligible in experiments.
FEA varies f = 0, 0.1, 1, but no friction measurement is reported for the actual experiments (Discussion).
standard math Hertz contact theory F = (16/9)E R^0.5 D^1.5 applies at D < 0.1R and is independent of alpha and friction.
Used as Eq. 6 to extract E; based on classical linear elastic contact.
domain assumption Quasi-static conditions are achieved when the kinetic-to-internal energy ratio is below 2% in FEA and at 0.1 mm/s in experiments.
SI and Experimental Method; rate effects are assumed absent.

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Cite this review

Pith. "Pith review of Hyperelastic characterization via deep indentation." pith.science (2026). https://pith.science/paper/6BJOP7MJ

@misc{pith2026250605371,
  author       = {Pith},
  title        = {Pith review of: Hyperelastic characterization via deep indentation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6BJOP7MJ}},
  note         = {Machine review of arXiv:2506.05371}
}

abstract

Hyperelastic material characterization is crucial for understanding the behavior of soft materials -- such as tissues, rubbers, hydrogels, and polymers -- under quasi-static loading before failure. Traditional methods typically rely on uniaxial tensile tests, which require the cumbersome preparation of dumbbell-shaped samples for clamping in a uniaxial testing machine. In contrast, indentation-based methods, which can be conducted \textit{in-situ} without sample preparation, have been underexplored. To characterize the hyperelastic behavior of soft materials, deep indentation is required, where the material response extends beyond linear elasticity. In this study, we perform finite element analysis to link the force ($F$) versus indentation depth ($D$) curve with the hyperelastic behavior of a soft incompressible material, using a one-term Ogden model for simplicity. We identify three indentation regimes based on the ratio between indentation depth and the radius ($R$) of the spherical-tipped cylindrical indenter: the Hertzian regime ($D < 0.1R$), where $F = \frac{16}{9} E R^{0.5} D^{1.5}$; the parabolic regime ($D > 10R$), where $F = E D^2 \beta$ and the indenter radius becomes irrelevant; and an intermediate regime ($0.1R < D < 10R$) bridging the two extremes. We find that the Ogden strain-stiffening coefficient ($\alpha$) increases the parabolic indentation coefficient ($\beta$), allowing for the estimation of $\alpha$ from $\beta$. Furthermore, we observe that Coulomb friction increases $\beta$, potentially masking the effect of strain-stiffening for small $\alpha$. However, for $\alpha > 3$, friction has a negligible effect. Finally, our results show good agreement with experimental data, demonstrating that deep indentation can be an effective method for extracting hyperelastic properties from soft materials through \textit{in-situ} testing.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Traditional methods typically rely on uniaxial tensile tests, which require the cumbersome preparation of dumbbell-shaped samples for clamping in a uniaxial testing machine

1 Hyperelastic characterization via deep indentation Mohammad Shojaeifard1, Mattia Bacca1* 1Mechanical Engineering Department, Institute of Applied Mathematics School of Biomedical Engineering University of British Columbia, Vancouver BC V6T 1Z4, Canada *Corresponding author: mbacca@mech.ubc.ca Abstract Hyperelastic material characterization is crucial fo...

work page 2009
[2]

to reveal the material’s anisotropy. Force was measured using the load cell of an Instron® universal testing machine, while stretch 𝜆 was obtained optically by tracking surface deformation from 4K-resolution video recordings (100 Hz). Horizontal fiducial lines drawn on the specimen surface were tracked frame-by-frame (SI: Figure S2) using Tracker video an...

work page 2024
[3]

These geometric considerations activate the nonlinear behavior of the material but introduce complexity into the measurement

typically focus on shallow indentation depths and require knowledge of the substrate’s thickness and/or curvature. These geometric considerations activate the nonlinear behavior of the material but introduce complexity into the measurement. In this study, we present an alternative method based on deep indentation using small cylindrical probes with spheri...

work page 2022
[4]

Elastomer samples were cast into 3D-printed molds to the required dimensions

of Ecoflex® 00-10, Ecoflex® 00-30, Mold Star™ 16 Fast (Smooth-On, Inc.), and fresh porcine skin, the same materials tested in uniaxial tension. Elastomer samples were cast into 3D-printed molds to the required dimensions. Porcine skin specimens were prepared by trimming subcutaneous fat to a thickness of ~ 2 𝑚𝑚, cutting circular disks (radius 𝐵=40 𝑚𝑚), an...

work page 1993
[7]

The indentation speed was selected to maintain quasi-static conditions, keeping the kinetic-to-internal energy ratio below 2%

Simulations were conducted for various strain-stiffening parameters 𝛼=2, 3, 5, 9, −9, −20. The indentation speed was selected to maintain quasi-static conditions, keeping the kinetic-to-internal energy ratio below 2%. The reaction force was extracted and plotted versus indentation depth (Figure 3). At small indentation depths (𝐷≪𝑅), the indentation force ...

work page 2025
[9]

This skin may affect both the Hertzian and parabolic responses, meaning deep indentation may reflect surface rather than bulk properties

and sometimes present in synthetic materials due to surface oxidation or environmental exposure. This skin may affect both the Hertzian and parabolic responses, meaning deep indentation may reflect surface rather than bulk properties. However, if the skin is thin compared to the indentation depth, its influence is likely minimized, allowing access to more...

work page 2025
[11]

Journal of the Royal Society Interface, 21(219), p.20240311

Being thin-skinned can still reduce damage from dynamic puncture. Journal of the Royal Society Interface, 21(219), p.20240311. 13 SI - Hyperelastic characterization via deep indentation Mohammad Shojaeifard1, Mattia Bacca1* 1Mechanical Engineering Department, Institute of Applied Mathematics School of Biomedical Engineering University of British Columbia,...

work page 2024
[1972]

!−3C (2) The nominal (engineering) puling stress 𝑆=𝐹/𝐴!, with 𝐹 puling force and 𝐴! initial cross-section area, is 𝑆=𝜕𝜓/𝜕𝜆, giving 𝑆=%&'(A𝜆(*$−𝜆*

for this study, where the strain energy density (SED) is 𝜓=%&'(!4𝜆̅$(+𝜆̅%(+𝜆̅'(−39 (1) Here, 𝐸 is the (zero-strain) Young’s modulus, 𝛼 is the strain-stiffening coefficient (Ogden, 1972), and 𝜆̅)=𝜆)𝐽*$/' represents the deviatoric component of the principal stretch 𝜆), with 𝐽 the swelling ratio. Note that 𝜆)=𝑑𝑙)/𝑑𝐿), where 𝑑𝑙) and 𝑑𝐿) are the current and re...

work page 1972

Show all 11 references

[2020]

However, this ambiguity is less problematic in deep indentation, thus highlighting the robustness of our proposed method

exhibit negative values for 𝛼. However, this ambiguity is less problematic in deep indentation, thus highlighting the robustness of our proposed method. We also discuss the limitations of using a small set of hyperelastic parameters, which restricts our ability to capture volu...

2023
[2024]

It is crucial to go beyond linear elastic regimes when characterizing hyperelastic materials to capture the full extent of their behavior

offer greater flexibility, as they can be performed in situ and in vivo and are non-destructive, eliminating the need for sample preparation. It is crucial to go beyond linear elastic regimes when characterizing hyperelastic materials to capture the full extent of their behavi...

2019
[2025]

arXiv preprint arXiv:2506.11461

Universal Scaling Laws for Deep Indentation Beyond the Hertzian Regime. arXiv preprint arXiv:2506.11461. Ogden, R.W.,

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Traditional methods typically rely on uniaxial tensile tests, which require the cumbersome preparation of dumbbell-shaped samples for clamping in a uniaxial testing machine

1 Hyperelastic characterization via deep indentation Mohammad Shojaeifard1, Mattia Bacca1* 1Mechanical Engineering Department, Institute of Applied Mathematics School of Biomedical Engineering University of British Columbia, Vancouver BC V6T 1Z4, Canada *Corresponding author: mbacca@mech.ubc.ca Abstract Hyperelastic material characterization is crucial fo...

work page 2009

[2] [2]

to reveal the material’s anisotropy. Force was measured using the load cell of an Instron® universal testing machine, while stretch 𝜆 was obtained optically by tracking surface deformation from 4K-resolution video recordings (100 Hz). Horizontal fiducial lines drawn on the specimen surface were tracked frame-by-frame (SI: Figure S2) using Tracker video an...

work page 2024

[3] [3]

These geometric considerations activate the nonlinear behavior of the material but introduce complexity into the measurement

typically focus on shallow indentation depths and require knowledge of the substrate’s thickness and/or curvature. These geometric considerations activate the nonlinear behavior of the material but introduce complexity into the measurement. In this study, we present an alternative method based on deep indentation using small cylindrical probes with spheri...

work page 2022

[4] [4]

Elastomer samples were cast into 3D-printed molds to the required dimensions

of Ecoflex® 00-10, Ecoflex® 00-30, Mold Star™ 16 Fast (Smooth-On, Inc.), and fresh porcine skin, the same materials tested in uniaxial tension. Elastomer samples were cast into 3D-printed molds to the required dimensions. Porcine skin specimens were prepared by trimming subcutaneous fat to a thickness of ~ 2 𝑚𝑚, cutting circular disks (radius 𝐵=40 𝑚𝑚), an...

work page 1993

[5] [7]

The indentation speed was selected to maintain quasi-static conditions, keeping the kinetic-to-internal energy ratio below 2%

Simulations were conducted for various strain-stiffening parameters 𝛼=2, 3, 5, 9, −9, −20. The indentation speed was selected to maintain quasi-static conditions, keeping the kinetic-to-internal energy ratio below 2%. The reaction force was extracted and plotted versus indentation depth (Figure 3). At small indentation depths (𝐷≪𝑅), the indentation force ...

work page 2025

[6] [9]

This skin may affect both the Hertzian and parabolic responses, meaning deep indentation may reflect surface rather than bulk properties

and sometimes present in synthetic materials due to surface oxidation or environmental exposure. This skin may affect both the Hertzian and parabolic responses, meaning deep indentation may reflect surface rather than bulk properties. However, if the skin is thin compared to the indentation depth, its influence is likely minimized, allowing access to more...

work page 2025

[7] [11]

Journal of the Royal Society Interface, 21(219), p.20240311

Being thin-skinned can still reduce damage from dynamic puncture. Journal of the Royal Society Interface, 21(219), p.20240311. 13 SI - Hyperelastic characterization via deep indentation Mohammad Shojaeifard1, Mattia Bacca1* 1Mechanical Engineering Department, Institute of Applied Mathematics School of Biomedical Engineering University of British Columbia,...

work page 2024

[8] [1972]

!−3C (2) The nominal (engineering) puling stress 𝑆=𝐹/𝐴!, with 𝐹 puling force and 𝐴! initial cross-section area, is 𝑆=𝜕𝜓/𝜕𝜆, giving 𝑆=%&'(A𝜆(*$−𝜆*

for this study, where the strain energy density (SED) is 𝜓=%&'(!4𝜆̅$(+𝜆̅%(+𝜆̅'(−39 (1) Here, 𝐸 is the (zero-strain) Young’s modulus, 𝛼 is the strain-stiffening coefficient (Ogden, 1972), and 𝜆̅)=𝜆)𝐽*$/' represents the deviatoric component of the principal stretch 𝜆), with 𝐽 the swelling ratio. Note that 𝜆)=𝑑𝑙)/𝑑𝐿), where 𝑑𝑙) and 𝑑𝐿) are the current and re...

work page 1972