Giuseppe Saccomandi

Extreme softness of brain matter in simple shear

Abstract We show that porcine brain matter can be modelled accurately as a very soft rubber-like material using the Mooney–Rivlin strain energy function, up to strains as high as 60%. This result followed from simple shear experiments performed on small rectangular fresh samples (2.5xa0cm3 and 1.1xa0cm3) at quasi-static strain rates. They revealed a linear shear stress–shear strain relationship ( R 2 > 0.97 ) , characteristic of Mooney–Rivlin materials at large strains. We found that porcine brain matter is about 30 times less resistant to shear forces than a silicone gel. We also verified experimentally that brain matter exhibits the positive Poynting effect of non-linear elasticity, and numerically that the stress and strain fields remain mostly homogeneous throughout the thickness of the samples in simple shear.

Methodical fitting for mathematical models of rubber-like materials

A great variety of models can describe the nonlinear response of rubber to uniaxial tension. Yet an in-depth understanding of the successive stages of large extension is still lacking. We show that the response can be broken down in three steps, which we delineate by relying on a simple formatting of the data, the so-called Mooney plot transform. First, the small-to-moderate regime, where the polymeric chains unfold easily and the Mooney plot is almost linear. Second, the strain-hardening regime, where blobs of bundled chains unfold to stiffen the response in correspondence to the ‘upturn’ of the Mooney plot. Third, the limiting-chain regime, with a sharp stiffening occurring as the chains extend towards their limit. We provide strain-energy functions with terms accounting for each stage that (i) give an accurate local and then global fitting of the data; (ii) are consistent with weak nonlinear elasticity theory and (iii) can be interpreted in the framework of statistical mechanics. We apply our method to Treloars classical experimental data and also to some more recent data. Our method not only provides models that describe the experimental data with a very low quantitative relative error, but also shows that the theory of nonlinear elasticity is much more robust that seemed at first sight.

Strain energy function for isotropic non-linear elastic incompressible solids with linear finite strain response in shear and torsion

RM gratefully acknowledges the funding of his PhD by a scholarship from the Irish Research Council. The research of GS is partially funded by GNFM of Istituto Nazionale di Alta Matematica.

Nonlinear elasticity of auxetic open cell foams modeled as continuum solids

Structure evolution during deformation of isotropic auxetic foams is investigated by simple compression experiments. It is shown that the main feature observed in the experimental data can be accurately described by an isotropic hyperelastic model with an Ogden-type strain energy function. The model can be easily implemented in a finite element code and, hence, can be used to simulate foams undergoing complex structural deformations.

Ermakov-Modulated Nonlinear Schrödinger Models. Integrable Reduction

Nonlinear Schrödinger equations with spatial modulation associated with integrable Hamiltonian systems of Ermakov-Ray-Reid type are introduced. An algorithmic procedure is presented which exploits invariants of motion to construct exact wave packet representations with potential applications in a wide range of physical contexts such as, ‘inter alia’, the analysis of Bloch wave and matter wave solitonic propagation and pulse transmission in Airy modulated NLS models. A particular Ermakov reduction for Mooney-Rivlin materials is set in the broader context of transverse wave propagation in a class of higher-order hyperelastic models of incompressible solids.

A note about waves in dissipative and dispersive solids

We study shear waves propagating in a special viscoelastic model proposed first by Fosdick and Yu in 1996. We deduce an asymptotic approximatio nw hich reduces the full balance equations to a system of evolution equation sw hich are av ectorial generalization of the Modified KDV-Burger equation. In such a way we show that the model takes into account not only dissipative e!ects but also dispersive e!ects.

Old Problems Revisited from New Perspectives in Implicit Theories of Fluids

Three of the most studied problems in fluid dynamics are revisited within implicit theories of fluids. Specifically, the onset of convection, the determination of laminar flows and the motion of a fluid down an inclined plane are studied under the assumption that the Cauchy stress tensor and the rate-of-strain tensor are related through implicit constitutive equations. Particular attention is paid to fluids whose viscosities are pressure-dependent.