On modeling fracture of soft polymers

Soft polymers are ubiquitous materials in nature and as engineering materials with properties varying from rate-independent to rate-dependent. Current fracture toughness measures are non-unique for rate-dependent soft materials for varying loading profiles and specimen geometries. Works on modeling fracture in rate-dependent soft polymers are limited to specific pre-cracked geometries. There is no generally agreed-upon model for the fracture of soft polymers. We propose and show that a critical value of stress work can be used as a measure of fracture resistance for a certain class of soft polymers. We develop a damage model to predict fracture in soft polymers. In the model, the energetic part of the critical stress work is proposed as a damage initiation criterion that has the ability to capture failure surfaces. The damage growth is modeled through a generalized gradient-damage framework. The fracture model is validated for both elastomers and viscous soft polymers by comparing model predictions against experimental results for different materials (ethylene propylene diene monomer - EPDM, EPS25 vitrimer, styrene butadiene rubber - SBR, natural rubber - NR, and polyborosiloxane - PBS), a variety of specimen geometries, and loading conditions. The model can predict key physical phenomena such as brittle and ductile responses and different fracture profiles. The microstructural quantities, such as subchain dissociation energy during the fracture of polymers, can be predicted from the macroscopic model parameters.

💡 Research Summary

The paper addresses the long‑standing challenge of describing fracture in soft polymers—materials that span from highly elastic elastomers to highly viscous, rate‑dependent polymers—by introducing a unified physical measure and a comprehensive computational model.
Critical work of stress (Wcr). The authors define a single scalar quantity, the critical work of stress, as the total mechanical work per unit volume required for complete failure of a material point under a given loading rate. Unlike conventional fracture toughness (energy release rate) or stress intensity factors, Wcr incorporates elastic storage, viscous dissipation, and the energy associated with breaking and reforming dynamic cross‑links. In practice, Wcr can be obtained from the area under a stress‑strain curve in a homogeneous uniaxial test, and it can be evaluated for arbitrary deformation states through the virtual‑power framework. For dynamically cross‑linked polymers, Wcr directly reflects sub‑chain dissociation energy and the rate‑dependent visco‑elastic response.
Generalized gradient‑damage framework. Building on the method of virtual power, the free‑energy density is split into an entropic (viscous) part and a network‑resistance part that captures cross‑link kinetics. Damage initiation is governed by the energetic contribution ψ+cr, which becomes active when the local Wcr exceeds a material‑specific threshold. Damage evolution follows a higher‑order partial differential equation containing a Laplacian term (∇²d), allowing damage to diffuse spatially and to reproduce realistic blunting of notches before crack propagation. This gradient‑damage approach overcomes limitations of variational phase‑field methods that cannot capture experimentally observed failure surfaces.
Experimental validation. The model is calibrated and validated against a broad data set comprising five polymers (EPDM, SBR, NR, EPS25 vitrimer, and polyborosiloxane PBS), multiple specimen geometries (single notch, double notch, plate, cylinder), and a range of loading rates (6 mm s⁻¹ to 60 mm s⁻¹). For the three elastomers, the model predicts the transition from ductile, high‑elongation failure at low rates to brittle, low‑elongation failure at high rates. For the vitrimer and PBS, it captures rate‑dependent phenomena such as extensive notch blunting, a “trumpet‑shaped” secondary notch ahead of the primary crack, and the abrupt loss of extensibility at high strain rates. Quantitative agreement is also shown between the macroscopic Wcr values and independently measured sub‑chain dissociation energies, confirming that the macroscopic measure faithfully encodes microscopic fracture mechanisms.
Contributions and limitations. The work contributes (i) a geometry‑ and rate‑independent fracture resistance metric, (ii) a thermodynamically consistent damage model that includes both material strength and cross‑link dynamics, and (iii) a computational tool capable of predicting crack nucleation, propagation, and coalescence in arbitrary, complex geometries. Limitations include the need for extensive experimental data to identify model parameters and potential numerical stability issues under extreme loading or multi‑crack scenarios, which the authors acknowledge as avenues for future research.
Overall, the study provides a robust, physics‑based framework for predicting fracture in a wide class of soft polymers, offering valuable guidance for the design of biomedical devices, impact‑mitigation systems, soft actuators, and self‑healing materials.