A research group from the University of Osaka, ZEN University, and the University of Tokyo has mathematically uncovered the mechanism that causes crack tips to sharpen during the rapid fracture of rubber.

The bursting of rubber balloons or tire blowouts is caused by rapid fracture, a phenomenon in which a small crack propagates instantaneously. During this process, the crack tip sharpens, accelerating the fracture. However, the reason behind this sharpening had long remained unexplained. Traditionally, it was believed to result from the material’s complex nonlinear effects^*1.

The research group—comprising Hokuto Nagatakiya, a doctoral student; Shunsuke Kobayashi, Assistant Professor; and Ryuichi Tarumi, Professor at the University of Osaka; along with Naoyuki Sakumichi, Associate Professor at ZEN University and Project Associate Professor at the University of Tokyo—has mathematically solved the problem of crack propagation. They derived equations that describe both the shape of the crack and the overall deformation of the material. This breakthrough demonstrates that crack tip sharpening in polymer materials^*2 such as rubber arises solely from their fundamental property of viscoelasticity^*3. Furthermore, the team mathematically proved the viscoelastic trumpet theory^*4 —proposed nearly 30 years ago by Nobel Laureate in Physics, Pierre-Gilles de Gennes^*5—based on the fundamental equations of continuum mechanics.

These findings lay a theoretical foundation for controlling fractures in a wide range of viscoelastic materials—from tires to medical products—contributing to improved durability, accident prevention, and reduced environmental impact through longer product lifespans.

This research was supported by the following programs from JST the Strategic Basic Research Programs: ERATO under the Sakai Real and Abstract Gel Project; PRESTO through the research project on the Multi-Scale Mechanics of Material Manifold; and the FOREST Program, through the research project on the establishment of universal thermodynamics, dynamics, and fracture mechanics of polymer gels. Additional support was provided by the Japan Society for the Promotion of Science (JSPS) through a Grant-in-Aid for Scientific Research (B).


(*1) Nonlinear effects
These effects arise when the relationship between the force applied to a material and the resulting deformation deviates from proportionality. Under small deformations, doubling the force typically results in a doubling of the deformation. However, under large deformations, this proportional relationship no longer holds. Such nonlinear behavior is particularly pronounced in soft and highly stretchable polymer materials, including rubber, gels, and plastics.

(*2) Polymer materials (macromolecular materials)
Polymers are macromolecules formed by the connection of many small molecules (monomers) into chain-like or network structures. Materials composed of such macromolecules are referred to as polymer materials. Depending on the types of molecular chains and bonding configurations, they exhibit a wide range of properties, including those of plastics, rubber, gels, and fibers. Rubber consists of long molecular chains chemically cross-linked into a network structure, giving it the ability to stretch significantly and return to its original shape. Gels have a similar network structure but contain liquids such as water. Polymer materials are lighter and easier to mold than metals, but they tend to tear when made thin and become brittle when hardened. These challenges highlight the need for developing tougher polymer materials.

(*3) Viscoelasticity
Viscoelasticity is a property that combines elasticity (spring-like behavior) and viscosity (sticky, honey-like behavior). In polymer materials, the mechanical response varies depending on the rate of deformation. When deformed slowly, they behave softly (rubber-like state); when deformed rapidly, they become rigid (glass-like state). At intermediate deformation rates, energy dissipation due to viscosity occurs. This property causes the fracture behavior of the same material to change dramatically depending on the crack propagation speed. When deformation is proportional to the applied force, the behavior is called linear elasticity; when flow is proportional to force, it is called linear viscosity. A material exhibiting both is said to have linear viscoelasticity. This study demonstrated that crack tip sharpening occurs solely due to linear viscoelasticity.

(*4) Viscoelastic trumpet theory
A theory proposed by Pierre-Gilles de Gennes in 1996 to describe the dynamic fracture of viscoelastic materials. It predicts that the region surrounding a crack consists of three distinct zones, with deformation in each zone following a power-law relationship. The theory is named after the trumpet-like shape of the crack. Although it was originally derived from scaling theory and energy balance, its connection to continuum mechanics remained unclear. This study is the first to derive the theory from the fundamental equations of continuum mechanics, thereby proving its theoretical validity after nearly 30 years.

(*5) Pierre-Gilles de Gennes
A French theoretical physicist (1932–2007) who was awarded the Nobel Prize in Physics in 1991. He pioneered a new field in physics by applying theoretical methods originally developed for magnetic and superconducting materials to soft matter such as polymers and liquid crystals. Through his use of scaling theory to extract the essence of complex phenomena, he came to be known as the “modern Newton.”

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