Modeling cavitation and fibrillation in elastomers and adhesives. Part I: Cohesive instability

Technical Deep Dive

Overview

This paper presents a gradient-enhanced continuum framework to model the cohesive instability that can trigger a phase transition from a dense to a rare phase, leading to cavitation and fibrillation in soft elastomers and adhesives. The key aspects are:

• Modeling elastomers as crosslinked van der Waals fluids, capturing intermolecular cohesion and excluded volume effects that naturally lead to an unstable response
• Introducing a nonlocal volume ratio field and a thermodynamically consistent derivation to enable a C^0 finite element formulation
• Capturing the unstable onset of cavitation in constrained biaxial tension tests, closely following analytically predicted thresholds
• Examining the effects of mesh refinement, viscous dissipation, and the material length scale controlling interfacial excess energy
• Demonstrating the ability to capture spontaneous multi-cavity nucleation in a "pure shear" test setup, consistent with experimental observations
• Observing the dependence of cavitation patterns on the specimen aspect ratio, matching experimental trends

Methodology

The key elements of the proposed framework are:

Modeling Elastomers as Crosslinked van der Waals Fluids

• The total free energy density combines a compressible Neo-Hookean contribution for network elasticity and a van der Waals fluid contribution to capture cohesive interactions and excluded volume effects.
• This non-polyconvex free energy allows for cohesive instabilities and a phase transition from a dense to a rare phase, without requiring pre-existing defects.

Gradient-Enhanced Continuum Formulation

• A nonlocal volume ratio field bar{J} is introduced to enable a C^0 finite element formulation.
• A thermodynamically consistent derivation is presented, including viscous dissipation effects associated with the phase transition.
• The model is implemented using the open-source FEniCS library, employing a mixed finite element approach.

Parametric Studies

1. Constrained Biaxial Tension:
   • Examined using both Neo-Hookean and monodisperse network models.
   • Investigated mesh sensitivity and the effect of viscosity of the phase transition.
   • Proposed a relaxation function to maintain a constant cavity radius in the reference configuration.
   • Studied the influence of polymer chain length on the cavitation response.
2. "Pure Shear" Test:
   • Captured spontaneous multi-cavity nucleation without pre-existing defects.
   • Examined the effect of mesh density and nonlocal length scale.
   • Observed the dependence of cavitation patterns on the specimen aspect ratio.

Results

Constrained Biaxial Tension

• Both Neo-Hookean and monodisperse network models captured the characteristic traction peak followed by an unstable drop, corresponding to cavity nucleation.
• The onset of the instability closely matched the analytically predicted threshold.
• Increasing mesh refinement and decreasing viscosity of the phase transition brought the numerical results closer to the analytical prediction.
• The monodisperse network model showed a weak dependence of the cavitation response on the polymer chain length (N).

"Pure Shear" Test

• The model spontaneously captured a cascade of cavity nucleation events, starting from the edges and propagating inward.
• The location and pattern of the cavities were largely independent of the nonlocal length scale, but the scale influenced the cavity size and merging behavior.
• Thinner specimens formed a dense array of small cavities, while thicker specimens led to fewer, larger cavities - consistent with experimental observations.

Interpretation

• The proposed framework provides a physically grounded approach to modeling cavitation in soft elastomers and adhesives, based on a cohesive instability and phase transition perspective, without requiring pre-existing defects.
• The ability to capture multi-cavity nucleation and the dependence on aspect ratio aligns with experimental observations, demonstrating the predictive capability of the model.
• The inclusion of viscous effects and the material length scale controlling interfacial excess energy are important for accurately capturing the cavitation process.
• While the current formulation focuses on cavity nucleation and growth, the authors note that future work will incorporate chain damage and viscoelastic effects to simulate the full cavitation-to-failure cascade.

Limitations and Uncertainties

• The current model does not yet incorporate damage mechanisms, such as chain scission, which are expected to dominate at high deformations within the cavitated regions.
• The relationship between the material length scale and the size of the zone that undergoes the phase transition remains an open question.
• The model has been validated against limited experimental data, and further comparison with a broader range of experimental observations would be beneficial.

Next Steps

• Extend the framework to include damage mechanisms, such as chain scission, to capture the complete cavitation-to-failure cascade.
• Investigate the connection between the material length scale and the size of the cavitated region, potentially linking it to network imperfections and polydispersity.
• Perform more extensive validation against a broader range of experimental data, including different material systems and loading conditions.
• Explore the implementation of the model in a computationally efficient manner to enable its use in practical engineering applications.