Biaxial characterization of soft elastomers: experiments and data-adaptive configurational forces for fracture

Ali Esmaeili; Miguel Angel Moreno-Mateos; Mokarram Hossain; Paul Steinmann; Simon Wiesheier

REVIEW 4 major objections 6 minor 1 cited by

Biaxial characterization of soft elastomers: experiments and data-adaptive configurational forces for fracture

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that the magnitude of the total crack-tip configurational force at crack onset gives a finite-strain fracture-toughness criterion for soft elastomers under equi-biaxial loading.

desk verdict Solid biaxial dataset and a coherent pipeline, but the headline fracture criterion is calibrated to the same onset data and is not yet a prediction. read the letter →

arxiv 2505.20244 v1 pith:3WUVJKT4 submitted 2025-05-26 cond-mat.soft cond-mat.mtrl-sci

Miguel Angel Moreno-Mateos , Simon Wiesheier , Ali Esmaeili , Mokarram Hossain , Paul Steinmann This is my paper · ORCID

classification cond-mat.softcond-mat.mtrl-sci MSC 74B2074R1074S05

keywords configurationalforcesfracturetoughnesssoftelastomersequi-biaxialloadinghyperelasticityJ-integraldata-drivenconstitutivemodelingdigitalimagecorrelation

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 fracture onset in soft elastomers under equi-biaxial stretching can be predicted by a single computed number: the total configurational force at the crack tip. The authors run equi-biaxial experiments to rupture on five elastomers (Sylgard, three Elastosil blends, and VHB tape) and observe a wide behavioral spectrum, from brittle Sylgard to VHB with crack-tip strains above 150%. They combine these experiments with a data-adaptive B-spline hyperelastic model and a post-processing Configurational Force Method that estimates the J-integral without contour integration. At the experimentally observed crack onset, the critical configurational force values (0.0386 to 0.4228 N/m) rank the materials consistently with their measured total work of fracture. If the criterion holds, the critical configurational force offers a computationally efficient, finite-strain fracture-toughness parameter for soft solids.

What carries the argument

The framework rests on two components. First, a data-adaptive hyperelastic strain energy density is represented by additive cubic B-spline interpolation over the isochoric invariants \bar I_1 and \bar I_2, with interpolation points and values identified by nonlinear least-squares optimization against reaction forces and DIC displacement fields from the biaxial tests; the optimizer consistently set the \bar I_2 contribution to zero, leaving \bar I_1-only energy functions. Second, the Configurational Force Method is implemented as a post-processing algorithm: after solving the forward boundary-value problem with the identified energy, the Eshelby stress tensor is computed and integrated against the FE basis-function gradients to produce nodal configurational forces, and the nodal forces inside a 0.1 mm radius cylinder around the crack tip are summed to give the total crack-tip configurational force, which the paper equates with the J-integral.

What would settle it

Refine the finite-element mesh near the crack tip and vary the summation radius around 0.1 mm while recomputing the total configurational force at crack onset; if the reported critical values (0.0386 to 0.4228 N/m) change substantially, the proposed fracture criterion is not reliable. An independent check would evaluate the contour J-integral on a Pacman-shaped domain in the same virtual experiments and test whether it equals the summed nodal configurational force.

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Extended reading notes

Core claim

The central claim is that the magnitude of the total crack-tip configurational force at crack onset serves as a measure of the material's fracture toughness, leading to a configurational-force-based fracture criterion: fracture onset occurs when the total configurational force reaches a critical value, F_CNF = F_CNF,C. This force is computed by summing nodal configurational forces derived from the Eshelby stress inside a small cylindrical region of radius 0.1 mm around the crack tip, and is presented as a computationally efficient estimate of the J-integral under finite strains and biaxial loading. The authors report F_CNF,C values of 0.0386 N/m for Elastosil 2:1, 0.0994 N/m for Elastosil 8:5, 0.2149 N/m for Elastosil 1:1, 0.1286 N/m for Sylgard, and 0.4228 N/m for VHB. They note that Sylgard's intermediate critical force combined with its low total work of fracture indicates brittle behavior, while VHB's high values mark a highly deformable fracture response. For the sideways-fracturing Elastosil 1:1, the method yields a forward-oriented configurational force, which the authors attribute to the isotropic hyperelastic model lacking anisotropic fracture resistance.

Load-bearing premise

The numerical estimate of the total configurational force equals the true J-integral for the chosen mesh and the 0.1 mm summation radius, an equality inherited from prior work rather than demonstrated by a convergence study in this paper.

Editorial extensions

If this is right

Fracture onset under biaxial loading could be predicted without contour integration or crack-growth simulation by stopping at the loading stage where the total configurational force reaches the material's critical value.
The reported critical values provide a finite-strain toughness ranking across a stiffness range of roughly 5 to 350 kPa, with VHB about eleven times tougher than Elastosil 2:1 by this measure.
The data-adaptive optimization identifies \bar I_1-only energy functions as sufficient for the biaxial constitutive response of these elastomers, supporting simpler calibrations for this class of materials.
The same virtual testbed could be reused for non-equi-biaxial loading paths and, with added anisotropic fracture resistance, for post-onset crack propagation predictions.

Reading between the lines

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

Because the paper states that nodal configurational forces depend on the FE mesh yet reports no mesh-convergence study, the specific F_CNF,C values should be treated as provisional until a denser-mesh verification stabilizes them.
A natural next experiment would be to apply the same B-spline calibration to uniaxial or shear data and test whether the resulting \bar I_1-only energy reproduces the biaxial force-displacement curves, which would probe the transferability of the data-adaptive model across loading modes.
The reported consistency between F_CNF,C and total work of fracture suggests that the method could rank fracture toughness in other soft solids, but the directional limitation seen in Elastosil 1:1 implies that the magnitude, not the crack-growth direction, is the trustworthy output for isotropic models.
Comparing F_CNF,C against an independent contour J-integral evaluated on the same finite-element solution would directly test whether the 0.1 mm summation radius is the load-bearing numerical choice.

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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

Short version: read this for the experiments, not for the claimed fracture criterion. The equi-biaxial dataset on five elastomers is a real asset, and the data-adaptive hyperelastic calibration looks careful. But the paper says fracture onset is 'predicted' when the total configurational force reaches FCNF,C, and then reads FCNF,C off the simulation at the experimentally observed onset displacement. That is a fit, not a prediction. No independent geometry, crack length, or loading case is used to test the criterion, and no comparison with an established toughness measurement is offered.

What is actually new: original force-displacement curves, full-field DIC strain maps, and crack-onset values for five soft elastomers under equibiaxial loading, with the raw data deposited on Zenodo. That alone is a useful contribution. The combination of B-spline hyperelastic energies (their own prior work) with the configurational force method as a J-estimator is a legitimate integration, and the calibration produces force and displacement errors that look small across four repetitions. They are also transparent about the sideways-fracture case: they admit the method predicts the wrong crack direction for Elastosil 1:1 and explain why.

The main soft spot is the circularity, and it is load-bearing. The constitutive model is calibrated on these same tests, and the critical configurational force is evaluated at the measured onset displacement. So the reported ranking of materials by FCNF,C is a description of the calibration data, not an independent inference. I'd add two smaller issues: there is no mesh or summing-radius convergence study for the configurational force, even though the paper notes nodal forces are mesh-dependent; and the FCNF,C values are reported as points without uncertainty, while Table 1 shows large run-to-run scatter in onset displacement (VHB: 48.8 ± 13.7 mm). The code is also not released.

These are fixable. An independent validation—e.g., a different specimen geometry or a single-edge notch test—would give the criterion real predictive teeth. A mesh/radius study and propagating the onset scatter into FCNF,C would make the reported numbers credible.

Useful for: experimentalists working on soft fracture who want multiaxial data; computational mechanicians who want a worked example of data-adaptive constitutive fitting with FEMU. It deserves serious peer review, but the review should be conditional and push hard on the predictive test.

Referee Report

4 major / 6 minor

Summary. The manuscript combines original equi-biaxial fracture experiments on five soft elastomers (Elastosil P7670 at three mixing ratios, Sylgard 184, and VHB 4905) with a data-adaptive hyperelastic constitutive framework and a post-processing Configurational Force Method. The constitutive model represents the isochoric strain energy as B-spline functions of the invariants, calibrated by finite element model updating against DIC displacement fields and reaction forces. The authors then compute the total crack-tip configurational force as an estimator of the J-integral and report critical values at crack onset, proposing a fracture criterion based on a critical configurational force FCNF,C. The paper also provides a public dataset on Zenodo.

Significance. If the quantitative claims are robust, the paper would offer a useful finite-strain alternative to contour-integral J evaluation for soft elastomers under multiaxial loading, and the B-spline data-adaptive constitutive framework is a flexible contribution. The experimental dataset, including full-field DIC data and force-displacement curves for five materials, is a valuable community resource. The authors are transparent about limitations, including the ad hoc I1-only reduction, the directional shortcoming for Elastosil 1:1, and a discrepancy with an earlier value in [52]. However, the central fracture-criterion claim is not independently validated, and the numerical convergence of the reported FCNF,C values is not established.

major comments (4)

[§4.2, Table 1] The proposed fracture criterion in §4.2 is not demonstrated as predictive. The critical values FCNF,C are computed by running each forward simulation up to the experimentally measured average displacement at crack onset (Table 1) and reading off the configurational force at that loading stage. Therefore, FCNF,C is a calibration output on the same experiments used to fit the constitutive model, not a prediction of onset. No test applies FCNF,C to a different geometry, crack length, or loading condition, and no independent toughness comparison is provided. I recommend either an out-of-sample predictive test or a revised framing that presents FCNF,C as a descriptive characterization rather than a fracture criterion.
[§4.1, §4.2] The numerical robustness of FCNF,C is not established. The paper states in §4.2 that nodal configurational forces depend on the FE mesh, yet no mesh-convergence study is reported. Likewise, the choice of a 0.1 mm cylinder radius for summing spurious configurational forces (§4.1) is not justified by a radius-sensitivity study. Since the reported critical values and the material ranking depend on these two numerical choices, a convergence and sensitivity analysis (e.g., two additional mesh densities and radii) is needed before the quantitative values can be relied upon.
[Table 1, §4.2] Uncertainty in the experimentally measured onset displacement is not propagated to FCNF,C. For example, VHB has onset displacement 48.8 ± 13.7 mm and Sylgard 5.6 ± 0.4 mm, but FCNF,C is reported as a single number computed at the mean onset displacement. Because the configurational force is a nonlinear function of displacement, the value at the mean need not equal the mean value, and the ordering of materials (e.g., Sylgard 0.1286 N/m vs Elastosil 1:1 0.2149 N/m) may not be robust. The authors should compute FCNF,C for each experimental repetition and report the distribution, or provide a sensitivity analysis.
[§3.3, §3.4] The data-adaptive constitutive functions are validated only in-sample: the force and displacement errors in Figures 7–11 are on the same biaxial experiments used for calibration. Since the accuracy of the configurational forces depends directly on the constitutive model, an independent check (e.g., prediction of a uniaxial or different biaxial test, or comparison with an independent modulus or toughness measurement) would substantially strengthen the claim. The paper itself acknowledges in §3.4 that the I1-only reduction is ad hoc, which further motivates such a check.

minor comments (6)

[§3 heading] The heading contains a typo: “postpocessing” should be “post-processing.”
[§5.1] The phrase “fracture fracture” appears in the Discussion; one of the two words should be removed.
[§3.3] The word “compreshensive” is misspelled; it should be “comprehensive.”
[§4.2] The term “configurational-forced-based” should be “configurational-force-based.”
[Figure 12] The caption lists panels (a), (b.1), and (b.2), but the text refers to panel (c) for the critical-force bar plot; the panel labels should be made consistent.
[Eqs. (10)–(11)] The objective function is split across two numbered equations with dangling terms; it would be clearer to present it as a single equation or to explicitly reference the undisplayed remainder.

Circularity Check

1 steps flagged · score 6.0 of 10

The fracture criterion is calibrated at the experimentally measured onset displacement, so FCNF = FCNF,C 'predicts' onset by reproducing its own input; no independent validation is provided.

fitted input called prediction [Section 4.2 (Results), Figure 12 caption, 'Total configurational force at crack onset yields a fracture criterion' paragraph]
"The analysis is carried out up to the point at which fracture propagation begins, as determined experimentally from the average displacement at crack onset (see Table 1). ... The loading stage selected in the virtual experiment (computation) is exactly the one corresponding to the displacement at crack onset (average values in Table 1). ... This leads to what we term a configurational-forced-based fracture criterion, in which fracture onset (not complete rupture) is predicted when the magnitude of the total configurational force reaches a critical value, FCNF = FCNF,C."

FCNF,C is not predicted or measured independently: it is read off from the simulation at the experimentally known onset displacement, and the paper states 'These values correspond to the final points of the force-displacement curves in Figure 12.b.' The criterion FCNF = FCNF,C therefore asserts that onset occurs when the simulation reaches the value it was forced to attain at the measured onset, so the 'prediction' reproduces its calibration input by construction for every material. The constitutive energy functions used in the same model were fitted to the same biaxial experiments, so ranking FCNF,C against the measured work of fracture is an internal-consistency check within one dataset, not an independent test.

full rationale

The single clear circular step is the fracture criterion: Section 4.2 defines FCNF,C as the total crack-tip configurational force evaluated at the experimentally measured average onset displacement (Table 1), then describes onset 'is predicted' when FCNF = FCNF,C. Because the critical value is extracted exactly where onset was already known, the rule's output duplicates its input for all five materials; this is a fitted input called a prediction. The constitutive modeling itself is not circular: the B-spline energy functions are calibrated by FEMU against pre-crack reaction forces and DIC displacement fields, with fit quality checked (Figures 7-11), and the configurational force genuinely depends on those fitted functions (footnote 13 acknowledges the VHB value changed with the improved model, indicating real dependence). The J-integral equivalence is imported from the authors' own prior work [52], but the same equivalence is established in the wider configurational-mechanics literature (refs. [53, 55]), so that self-citation is corroborated rather than load-bearing in a way that would raise the score. The missing mesh-convergence study and the fixed 0.1 mm summation radius ('Nodal configurational forces are ... a consequence of the FE spatial discretization') are correctness risks rather than circular steps, and the acknowledged failure to predict the sideways fracture direction for Elastosil 1:1 is an honest falsification of the direction claim, not a tautology. The underlying configurational-force framework retains independent mathematical content, so the circularity is partial: the central 'predictive' criterion reduces by construction to the measured onset it claims to forecast, with no independent validation on other geometries, crack lengths, or loading conditions, warranting score 6.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The central new numerical content is a set of fitted B-spline control values and domain boundaries; these are legitimate constitutive parameters but they are fitted to the same data used to evaluate the fracture criterion. No new physical entities are introduced.

free parameters (5)

B-spline interpolation values w(1)_i (I1-only strain energy function) = Not tabulated; shown as curves in Figs. 7-11a
These are the principal unknowns in the FEMU calibration (Eqs. 10-13), fitted to reaction forces and DIC displacement fields for each of the five materials.
Last interpolation point I1^(n1) per material (domain boundary) = Varies by material: 5.76 (Elastosil 2:1), 5.06 (8:5), 4.35 (1:1), 3.07 (Sylgard), 9.77 (VHB)
The interpolation domain is also optimized, with a KDE-based regularization pushing it toward the 95th percentile of sampled invariants (Eq. 11, Section B).
Cylinder radius for summing spurious configurational forces = 0.1 mm
Chosen in Section 4.1 for all materials; no mesh or radius convergence study is provided, and nodal configurational forces are stated to be mesh-dependent.
Objective function weights omega_f and omega_u = 1/max(f_exp) and 1/max(u_exp)
Normalization scheme in Eq. (12) to combine force and displacement residuals.
ROI dimensions for sampling DIC displacements = 20 mm x 20 mm, except 17 mm x 17 mm for Elastosil 8:5
Manual choice of calibration region around the crack; its influence on the identified energy functions is not studied.

assumptions (3)

domain assumption The elastomers are incompressible and isotropic hyperelastic; volumetric behavior is enforced by a Lagrange multiplier p (Eqs. 4-6).
Used to set up the forward boundary value problem (Section 3.1). Sideways fracture of Elastosil 1:1 suggests deformation-induced anisotropy that the isotropic model cannot capture.
standard math The configurational force at the crack tip equals the J-integral under finite strains.
Assumed from prior work [52] and used to interpret F_CNF as J (Sections 1.1 and 4.1). This equivalence is not re-derived here.
domain assumption The DIC displacement fields and machine reaction forces are accurate and the FE model reproduces the experimental boundary conditions with a 45-degree notch.
All calibration and fracture metrics rest on this data-fidelity assumption (Section 3.3).

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Pith. "Pith review of Biaxial characterization of soft elastomers: experiments and data-adaptive configurational forces for fracture." pith.science (2026). https://pith.science/paper/3WUVJKT4

@misc{pith2026250520244,
  author       = {Pith},
  title        = {Pith review of: Biaxial characterization of soft elastomers: experiments and data-adaptive configurational forces for fracture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3WUVJKT4}},
  note         = {Machine review of arXiv:2505.20244}
}

read the original abstract

Understanding the fracture mechanics of soft solids remains a fundamental challenge due to their complex, nonlinear responses under large deformations. While multiaxial loading is key to probing their mechanical behavior, the role of such loading in fracture processes is still poorly understood. Here, we present a combined experimental-computational framework to investigate fracture in soft elastomers under equi-biaxial loading. We report original equi-biaxial quasi-static experiments on five elastomeric materials, revealing a spectrum of material and fracture behavior, from brittle-like to highly deformable response with crack tip strains exceeding 150 %. Motivated by these observations, we develop a hybrid computational testbed that mirrors the experimental setup and enables virtual biaxial tests. Central to this framework are two components: a data-adaptive formulation of hyperelastic energy functions that flexibly captures material behavior, and a post-processing implementation of the Configurational Force Method, providing a computationally efficient estimate of the J-integral at the crack tip. Our data-adaptive framework for hyperelastic energy functions proves versatility to capture with high accuracy the hyperelastic behavior observed in the biaxial experiments. This is important because accurately capturing the constitutive behaviour of soft solids is key for a reliable application of the Configurational Force Method to soft solids. In the limit of crack onset, a critical value of the crack tip configurational force allows for a criterion of fracture toughness. Together, our experimental, theoretical, and computational contributions offer a new paradigm for characterizing and designing soft materials with tailored fracture properties.

Figures

Figures reproduced from arXiv: 2505.20244 by the authors.

**Figure 1.** Experimental setup for biaxial stretch tests. (a) Close-up view of the grips used in the experimental testbed. Each grip consists of three independent clamps designed to hold rectangular specimens. A sliding grip system allows the gripping fingers to translate perpendicular to the loading direction, i.e., in the direction of the edge of the sample. To that end, the gripping fingers can freely move in this direction.… view at source ↗

**Figure 2.** Experimental results for the biaxial experiments. The two independent axes move identically at a displacement rate of 0.85 mm s−1 , which renders a strain rate of 0.01 s−1 . (a.1-5) Force displacement results for biaxial experiments and for each of the axes of the machine (blue and orange colored-lines). The results for each experimental repetition are marked with ⃝1 , ⃝2 , ⃝3 , and ⃝4 so that they can be easily ide… view at source ↗

**Figure 3.** Experimental total work of fracture and strain at failure for the biaxial experiments. (a) Work of fracture for each material calculated as the total work performed by both axes during the deformation of the sample until full propagation of the crack divided by the initial volume of the sample. The volume of the samples is 85 mm × 85 mm × 2 mm (thickness) for all samples except for VHB. For VHB, thickness is 0.5 mm.… view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: Strain fields at crack onset. The fields are engineering strain fields computed from the Lagrange strain tensor. The fields correspond to the loading stages described in [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: Schematics of the data-adaptive hyperelastic constitutive modeling framework. The invariants I¯1, I¯2 are discretized with interpolation points (black circles) and the corresponding interpolation values (blue circles) represent unknown material parameters. Interpolatio…

**Figure 6.** Figure 6: Region of interest used to sample the experimental displacement fields to the FE mesh and calibrate data-adaptive strain energy functions. A square region of interest (ROI) with dimensions 20 mm x 20 mm in the vicinity of the crack contour is defined. The ROI is chosen…

**Figure 7.** Figure 7: Data-adaptive strain energy function for Elastosil 2:1 obtained from biaxial experimental data: reaction forces and full-field DIC measurements. (a) Data-adaptive hyperelastic strain energy density as a function of the first invariant, I¯1. The energy function is recon…

**Figure 8.** Figure 8: Data-adaptive strain energy function for Elastosil 8:5 obtained from biaxial experimental data: reaction forces and full-field DIC measurements. (a) Data-adaptive hyperelastic strain energy density as a function of the first invariant, I¯1. The energy function is recon…

**Figure 9.** Figure 9: Data-adaptive strain energy function for Elastosil 1:1 obtained from biaxial experimental data: reaction forces and full-field DIC measurements. (a) Data-adaptive hyperelastic strain energy density as a function of the first invariant, I¯1. The energy function is recon…

**Figure 10.** Figure 10: Data-adaptive strain energy function for Sylgard obtained from biaxial experimental data: reaction forces and full-field DIC measurements. (a) Data-adaptive hyperelastic strain energy density as a function of the first invariant, I¯1. The energy function is reconstruc…

**Figure 11.** Figure 11: Data-adaptive strain energy function for VHB obtained from biaxial experimental data: reaction forces and full-field DIC measurements. (a) Data-adaptive hyperelastic strain energy density as a function of the first invariant, I¯1. The energy function is reconstructed …

**Figure 12.** Figure 12: Results for data-adaptive configurational forces ante crack onset and fracture criterion based on their critical values. (a) Nodal configurational forces at the crack tip and spurious configurational forces at the crack tip vicinity are depicted for Elastosil 2:1, Ela…

**Figure 13.** Figure 13: Strain fields at onset of crack growth. The fields are engineering strain fields computed from the Lagrange strain tensor. The fields correspond to the loading stages described in [PITH_FULL_IMAGE:figures/full_fig_p035_13.png]

**Figure 14.** Figure 14: Strain fields at onset of crack growth. The fields are engineering strain fields computed from the Lagrange strain tensor. The fields correspond to the loading stages described in [PITH_FULL_IMAGE:figures/full_fig_p036_14.png]

**Figure 15.** Figure 15: Strain fields at onset of crack growth. The fields are engineering strain fields computed from the Lagrange strain tensor. The fields correspond to the loading stages described in [PITH_FULL_IMAGE:figures/full_fig_p037_15.png]

**Figure 16.** Figure 16: Kernel Density Estimation (KDE) employed to obtain a differentiable representation of the probability density function (PDF) for the sampled invariants I¯1, I¯2. At each optimization iteration, the sampled invariants (¯I1, ¯I2) at the quadrature points of the FE-mesh …

**Figure 17.** Figure 17: Sampled invariants predicted by the FE forward simulation using the identified data-adaptive hyperelastic strain energy functions. The vertical line marks the 95th-percentile of the invariants. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_17.png]

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Configurational forces explain echelon cracks in soft materials
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Configurational force computations on idealized crack geometries show that facet tilting, local binormal shear, and facet spacing govern echelon crack growth and coalescence in soft materials.

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