Fracture and healing of elastomers: a phase-transition theory and numerical implementation

Recent experiments, analogous to the classical experiments by Gent and collaborators but carried out at higher spatiotemporal resolution (of 1 micron in space and 60 ms in time), have provided a complete qualitative picture of the nucleation and the ensuing growth and interaction of internal cavities/cracks in elastomers subjected to externally applied quasi-static mechanical loads. In this talk, I will begin by presenting a continuum field theory seemingly capable to explain, describe, and predict all of the classical and recent experimental observations: from the nucleation of cavities/cracks, to their growth to micro-cracks, to their continued growth to macro-cracks, to the remarkable healing of some of the cracks. The theory rests on two central ideas. The first one is to view elastomers as solids capable to undergo finite deformations and capable also to phase transition to another solid of vanishingly small stiffness, whereas the forward phase transition serves to characterize the nucleation and propagation of fracture, the reverse phase transition characterizes the healing. The second central idea is to take the phase transition to be driven by the competition between a combination of strain energy and stress concentration in the bulk and surface energy on the created/healed new surfaces in the elastomer. In the second part of the talk, I will present a numerical implementation of the theory capable of efficiently dealing with large deformations, the typical near incompressibility of elastomers, and the large changes in the deformation field that can ensue locally in space and time from the nucleation of fracture. I will close by confronting its predictions with a number of recent experiments.Publicado en: Mecánica Computacional vol. XXXV, no. 1.Facultad de Ingenierí

Dielectric elastomer composites: the critical role of interphasial phenomena

In this discussion, I will present new theoretical results in conjunction with experiments that reveal the critical role that interphasial phenomena play on the macroscopic electromechanical properties of soft dielectric composites

Some Remarks on the Effect of Interphases on the Mechanical Response and Stability of Fiber-Reinforced Elastomers

In filled elastomers, the mechanical behavior of the material surrounding the fillers — termed interphasial material — can be significantly different (softer or stiffer) from the bulk behavior of the elastomeric matrix. In this paper, motivated by recent experiments, we study the effect that such interphases can have on the mechanical response and stability of fiber-reinforced elastomers at large deformations. We work out in particular analytical solutions for the overall response and onset of microscopic and macroscopic instabilities in axially stretched 2D fiber-reinforced non-linear elastic solids. These solutions generalize the classical results of Rosen (1965) and Triantafyllidis and Maker (1985) for materials without interphases. It is found that while the presence of interphases does not significantly affect the overall axial response of fiber-reinforced materials, it can have a drastic effect on their stability.Engineering and Applied Science

The trousers fracture test for viscoelastic elastomers

Shrimali and Lopez-Pamies (2023) have recently shown that the Griffith criticality condition that governs crack growth in viscoelastic elastomers can be reduced to a fundamental form that involves exclusively the intrinsic fracture energy $G_c$ of the elastomer and, in so doing, they have brought resolution to the complete description of the historically elusive notion of critical tearing energy $T_c$. The purpose of this paper -- which can be viewed as the third installment of the series started by Shrimali and Lopez-Pamies (2023) -- is to make use of this fundamental form to explain one of the most popular fracture tests for probing the growth of cracks in viscoelastic elastomers, the trousers test

Fiber-reinforced hyperelastic solids: a realizable homogenization constitutive theory

A new homogenization theory to model the mechanical response of hyperelastic solids reinforced by a random distribution of aligned cylindrical fibers is proposed. The central idea is to devise a special class of microstructures—by means of an iterated homogenization procedure in finite elasticity together with an exact dilute result for sequential laminates—that allows to compute exactly the macroscopic response of the resulting fiber-reinforced materials. The proposed framework incorporates direct microstructural information up to the two-point correlation functions, and requires the solution to a Hamilton–Jacobi equation with the fiber concentration and the macroscopic deformation gradient playing the role of “time” and “spatial” variables, respectively. In addition to providing constitutive models for the macroscopic response of fiber-reinforced materials, the proposed theory also gives information about the local fields in the matrix and fibers, which can be used to study the evolution of microstructure and the development of instabilities. As a first application of the theory, closed-form results for the case of Neo-Hookean solids reinforced by a transversely isotropic distribution of anisotropic fibers are worked out. These include a novel explicit criterion for the onset of instabilities under general finite-strain loading conditions.Facultad de Ingenierí

Homogenization of elastomers filled with liquid inclusions: The small-deformation limit

This paper presents the derivation of the homogenized equations that describe the macroscopic mechanical response of elastomers filled with liquid inclusions in the setting of small quasistatic deformations. The derivation is carried out for materials with periodic microstructure by means of a two-scale asymptotic analysis. The focus is on the non-dissipative case when the elastomer is an elastic solid, the liquid making up the inclusions is an elastic fluid, the interfaces separating the solid elastomer from the liquid inclusions are elastic interfaces featuring an initial surface tension, and the inclusions are initially $n$-spherical ($n=2,3$) in shape. Remarkably, in spite of the presence of local residual stresses within the inclusions due to an initial surface tension at the interfaces, the macroscopic response of such filled elastomers turns out to be that of a linear elastic solid that is free of residual stresses and hence one that is simply characterized by an effective modulus of elasticity $\bar{\textbf{L}}$. What is more, in spite of the fact that the local moduli of elasticity in the bulk and the interfaces do not possess minor symmetries (due to the presence of residual stresses and the initial surface tension at the interfaces), the resulting effective modulus of elasticity $\bar{\textbf{L}}$ does possess the standard minor symmetries of a conventional linear elastic solid, that is, $\bar{L}_{ijkl}=\bar{L}_{jikl}=\bar{L}_{ijlk}$. As a first application, numerical results are worked out and analyzed for the effective modulus of elasticity of isotropic suspensions of incompressible liquid $2$-spherical inclusions of monodisperse size embedded in an isotropic incompressible elastomer

On microstructure evolution in fiber-reinforced elastomers and implications for their mechanical response and stability

Lopez-Pamies and Idiart (2010, "Fiber-Reinforced Hyperelastic Solids: A Realizable Homogenization Constitutive Theory," J. Eng. Math., 68(1), pp. 57-83) have recently put forward a homogenization theory with the capability to generate exact results not only for the macroscopic response and stability but also for the evolution of the microstructure in fiber-reinforced hyperelastic solids subjected to finite deformations. In this paper, we make use of this new theory to construct exact, closed-form solutions for the change in size, shape, and orientation undergone by the underlying fibers in a model class of fiber-reinforced hyperelastic solids along arbitrary 3D loading conditions. Making use of these results, we then establish connections between the evolution of the microstructure and the overall stress-strain relation and macroscopic stability in fiber-reinforced elastomers. In particular, we show that the rotation of the fibers may lead to the softening of the overall stiffness of fiber-reinforced elastomers under certain loading conditions. Furthermore, we show that this geometric mechanism is intimately related to the development of long-wavelength instabilities. These findings are discussed in light of comparisons with recent results for related material systems.Fil: Lopez Pamies, Oscar. State University of New York; Estados UnidosFil: Idiart, Martín Ignacio. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina. Universidad Nacional de La Plata. Facultad de Ingeniería. Departamento de Aeronáutica; ArgentinaFil: Li, Zhiyun. State University of New York; Estados Unido

Cavitation in rubber: an elastic instability or a fracture phenomenon?

It is by now well established that loading conditions with sufficiently large triaxialities can induce the sudden -appearance of internal cavities within elastomeric (and other soft) solids. The occurrence of such a phenomenon, commonly referred to as cavitation, can be attributed to the growth of pre-existing defects into finite sizes. In the first part of this discussion, I will present a new theory within the context of nonlinear elasticity to study the phenomenon of cavitation in rubber that contrary to earlier approaches: (i) allows to consider general 3D loading conditions with arbitrary triaxiality; (ii) applies to general classes of nonlinear elastic solids; and (iii) incorporates direct information on the initial shape, spatial distribution, and mechanical properties of the underlying defects at which cavitation can initiate. The basic idea is to first cast cavitation in elastomeric solids as the homogenization problem of nonlinear elastic materials containing random distributions of zero-volume cavities, or defects. Then, by means of a novel iterated homogenization procedure, exact solutions are constructed for such a problem. These include solutions for the change in size of the underlying cavities as a function of the applied loading conditions, from which the onset of cavitation – corresponding to the event when the initially infinitesimal cavities suddenly grow into finite sizes – can be readily determined. In the second part of the discussion, I will confront the theory with a variety of cavitation experiments with the objective of establishing whether the phenomenon of cavitation is an elastic instability (and hence depends only on the elastic properties of the rubber), or, on the other hand, a fracture process (and hence depends on the fracture properties of the rubber). REFERENCES [1] Lefèvre, V., Ravi-Chandar, K., Lopez-Pamies, O. Cavitation in rubber: An elastic instability or a fracture phenomenon? International Journal of Fracture. 2014. Submitted. [2] Lopez-Pamies, O., Nakamura T., Idiart, M.I. Cavitation in elastomeric solids: I – A defect growth theory. Journal of the Mechanics and Physics of Solids. 2011, 59, 1464–1487. [3] Lopez-Pamies, O., Nakamura T., Idiart, M.I. Cavitation in elastomeric solids: II – Onset-of-cavitation surfaces for Neo-Hookean materials. Journal of the Mechanics and Physics of Solids. 2011, 59, 1488–1505