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Giuseppe Saccomandi

Publications and source records attributed to Giuseppe Saccomandi.

At least 37 records · Page 2Linked to original sources
arXiv2013

Some results on finite amplitude elastic waves propagating in rotating media

Two questions related to elastic motions are raised and addressed. First: in which theoretical framework can the equations of motion be written for an elastic half-space put into uniform rotation? It is seen that nonlinear finite elasticity provides such a framework for incompressible solids. Second: how can finite amplitude exact solutions be generated? It is seen that for some finite amplitude transverse waves in rotating incompressible elastic solids with general shear response, the solutions are obtained by reduction of the equations of motion to a system of ordinary differential equations equivalent to the system governing the central motion problem of classical mechanics. In the special case of circularly-polarized harmonic progressive waves, the dispersion equation is solved in closed form for a variety of shear responses, including nonlinear models for rubberlike and soft biological tissues. A fruitful analogy with the motion of a nonlinear string is pointed out.

cond-mat.soft↗
arXiv2013

Finite amplitude inhomogeneous waves in Mooney-Rivlin viscoelastic solids

New exact solutions are exhibited within the framework of finite viscoelasticity. More precisely, the solutions correspond to finite-amplitude, transverse, linearly-polarized, inhomogeneous motions superposed upon a finite homogeneous static deformation. The viscoelastic body is composed of a Mooney-Rivlin viscoelastic solid, whose constitutive equation consists in the sum of an elastic part (Mooney-Rivlin hyperelastic model) and a viscous part (Newtonian viscous fluid model). The analysis shows that the results are similar to those obtained for the purely elastic case; inter alia, the normals to the planes of constant phase and to the planes of constant amplitude must be orthogonal and conjugate with respect to the B-ellipsoid, where B is the left Cauchy-Green strain tensor associated with the initial large static deformation. However, when the constitutive equation is specialized either to the case of a neo-Hookean viscoelastic solid or to the case of a Newtonian viscous fluid, a greater variety of solutions arises, with no counterpart in the purely elastic case. These solutions include travelling inhomogeneous finite-amplitude damped waves and standing damped waves.

cond-mat.soft↗
arXiv2013

Finite amplitude elastic waves propagating in compressible solids

The paper studies the interaction of a longitudinal wave with transverse waves in general isotropic and unconstrained hyperelastic materials, including the possibility of dissipation. The dissipative term chosen is similar to the classical stress tensor describing a Stokesian fluid and is commonly used in nonlinear acoustics. The aim of this research is to derive the corresponding general equations of motion, valid for any possible form of the strain energy function and to investigate the possibility of obtaining some general and exact solutions to these equations by reducing them to a set of ordinary differential equations. Then the reductions can lead to some exact closed-form solutions for special classes of materials (here the examples of the Hadamard, Blatz-Ko, and power-law strain energy densities are considered, as well as fourth-order elasticity). The solutions derived are in a time/space separable form and may be interpreted as generalized oscillatory shearing motions and generalized sinusoidal standing waves. By means of standard methods of dynamical systems theory, some peculiar properties of waves propagating in compressible materials are uncovered, such as for example, the emergence of destabilizing effects. These latter features exist for highly nonlinear strain energy functions such as the relatively simple power-law strain energy, but they cannot exist in the framework of fourth-order elasticity.

cond-mat.soft↗
arXiv2013

An energetic model for macromolecules unraveling

We propose a simple approach, based on the minimization of the total (entropic plus unfolding) energy of a two-state system, describing the stretch-induced unfolding of macromolecules (proteins, silks, nanopolymers, DNA/RNA). The model is fully analytical and enlightens the role of the different energetic components regulating the unfolding evolution. As an explicit example of application we compare the analytical results with the titin Atomic Force Microscopy experiments showing the ability of the model to quantitatively reproduce the mechanical behavior of macromolecules unfolding.

cond-mat.soft↗
arXiv2013

Nonlinear transverse waves in deformed dispersive solids

We present a phenomenological approach to dispersion in nonlinear elasticity. A simple, thermomechanically sound, constitutive model is proposed to describe the (non-dissipative) properties of a hyperelastic dispersive solid, without recourse to a microstructure or a special geometry. As a result, nonlinear and dispersive waves can travel in the bulk of such solids, and special waves emerge, some classic (periodic waves or pulse solitary waves of infinite extend), some exotic (kink or pulse waves of compact support). We show that for incompressible dispersive power-law solids and forth-order elasticity solids, solitary waves can however only exist in the case of linear transverse polarization. We also study the influence of pre-stretch and hardening. We provide links with other (quasi-continuum, asymptotic) theories; in particular, an appropriate asymptotic multiscale expansion specializes our exact equations of motion to the vectorial MKdV equation, for any hyperelastic material.

cond-mat.soft↗
arXiv2013

Proper formulation of viscous dissipation for nonlinear waves in solids

In order to model nonlinear viscous dissipative motions in solids, acoustical physicists usually add terms linear in dot{E}, the material time derivative of the Lagrangian strain tensor E, to the elastic stress tensor sigma derived from the expansion to the third- (sometimes fourth-) order of the strain energy density e=e(trace(E), trace(E^2), trace(E^3)). Here, it is shown that this practice, which has been widely used in the past three decades or so, is physically wrong for at least two reasons, and that it should be corrected. One reason is that the elastic stress tensor sigma is not symmetric while dot{E) is symmetric, so that motions for which sigma + sigma^T <> 0 will give rise to elastic stresses which have no viscous pendant. Another reason is that dot{E} is frame-invariant, while sigma is not, so that an observer transformation would alter the elastic part of the total stress differently than it would alter the dissipative part, thereby violating the fundamental principle of material frame-indifference. These problems can have serious consequences for nonlinear shear wave propagation in soft solids, as seen here with an example of a kink in almost incompressible soft solids.

cond-mat.soft↗
arXiv2013

Inhomogeneous shear of orthotropic incompressible non-linearly elastic solids: singular solutions and biomechanical interpretation

We present a detailed study of rectilinear shear deformation in the framework of orthotropic nonlinear elasticity, under Dirichlet and mixed-boundary conditions. We take a slab made of a soft matrix, reinforced with two families of extensible fibers. We consider the case where the shear occurs along the bissectrix of the angle between the two privileged directions aligned with the fibers. We show that if the two families of parallel fibers are mechanically equivalent, then only smooth solutions are possible, whereas if the mechanical differences among the two families of fibers is pronounced, then strain singularities may develop. We determine the precise conditions for the existence of such singular solutions for the standard reinforcing orthotropic model. We then extend our findings to some orthotropic models of interest in biomechanical applications, and we discuss the possible relevance of the singular solutions to biomechanics.

cond-mat.soft↗
arXiv2013

Compact travelling waves in viscoelastic solids

We introduce a model for nonlinear viscoelastic solids where traveling shear waves with compact support are possible. We obtain an exact compact solution. We also derive a new Burger's type evolution equation associated with the introduced constitutive equation.

nlin.PS↗
arXiv2013

Simple shear is not so simple

For homogeneous, isotropic, nonlinearly elastic materials, the form of the homogeneous deformation consistent with the application of a Cauchy shear stress is derived here for both compressible and incompressible materials. It is shown that this deformation is not simple shear, in contrast to the situation in linear elasticity. Instead, it consists of a triaxial stretch superposed on a classical simple shear deformation, for which the amount of shear cannot be greater than 1. In other words, the faces of a cubic block cannot be slanted by an angle greater than 45deg by the application of a pure shear stress alone. The results are illustrated for those materials for which the strain energy function does not depend on the principal second invariant of strain. For the case of a block deformed into a parallelepiped, the tractions on the inclined faces necessary to maintain the derived deformation are calculated.

cond-mat.soft↗
arXiv2013

Transverse Waves in Nonlinearly Elastic Solids and the Milne-Pinney (or Ermakov) Equation

We establish a connection between the general equations of nonlinear elastodynamics and the nonlinear ordinary differential equation of Pinney [Proc. Amer. Math. Soc. 1 (1950) 681]. As a starting point, we use the exact travelling wave solutions of nonlinear elasticity discovered by Carroll [Acta Mechanica 3 (1967) 167]. The connection provides a method for finding new exact and approximate dynamic solutions for neo-Hookean and Mooney-Rivlin solids, and for the general third- and fourth-order elasticity models of incompressible solids.

cond-mat.soft↗
arXiv2013

Scalar evolution equations for shear waves in incompressible solids: A simple derivation of the Z, ZK, KZK, and KP equations

We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent, and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy-Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then we introduce dispersive and dissipative terms to deduce the scalar Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and Khokhlov-Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid mechanics.

cond-mat.soft↗
arXiv2013

On the Rectilinear Shear of Compressible and Incompressible Elastic Slabs

We review some pseudo-planar deformations for the equations of incompressible isotropic nonlinear elasticity first introduced in 1985 by Rajagopal and Wineman. We extend this class of deformations to compressible isotropic and transverse isotropic materials, and also consider the influence of gravity. We consider some new approximate solutions and we discuss the possible relevance of such solutions to the understanding of the complex structure of the fields equations of nonlinear elasticity, using weakly nonlinear theories.

nlin.PS↗
arXiv2012

Automated Estimation of Collagen Fibre Dispersion in the Dermis and its Contribution to the Anisotropic Behaviour of Skin

Collagen fibres play an important role in the mechanical behaviour of many soft tissues. Modelling of such tissues now often incorporates a collagen fibre distribution. However, the availability of accurate structural data has so far lagged behind the progress of anisotropic constitutive modelling. Here, an automated process is developed to identify the orientation of collagen fibres using inexpensive and relatively simple techniques. The method uses established histological techniques and an algorithm implemented in the MATLAB image processing toolbox. It takes an average of 15 s to evaluate one image, compared to several hours if assessed visually. The technique was applied to histological sections of human skin with different Langer line orientations and a definite correlation between the orientation of Langer lines and the preferred orientation of collagen fibres in the dermis was observed. The structural parameters of the Gasser-Ogden-Holzapfel (GOH) model were all successfully evaluated. It is expected that the results of this study will assist those wishing to model skin, and that the algorithm described will be of benefit to those who wish to evaluate the collagen dispersion of other soft tissues.

physics.bio-ph↗
arXiv2011

On the Nonlinear Theory of Viscoelasticity of Differential Type

We consider nonlinear viscoelastic materials of differential type and for some special models we derive exact solutions of initial boundary value problems. These exact solutions are used to investigate the reasons of non-existence of global solutions for such equations.

physics.class-ph↗
arXiv2008

Small amplitude waves and stability for a pre-stressed viscoelastic solid

We study the propagation of small amplitude waves superimposed on a large static deformation in a nonlinear viscoelastic material of differential type. We use bulk waves and surface waves to address the questions of dissipation and of material and geometric stability. In particular, the analysis provides bounds on the constitutive parameters and on the pre-deformation that ensure linearized stability in the neighbourhood of a large pre-stretch. This type of result is relevant to the imaging of biological soft tissues using acoustical techniques, where pre-deformation is known to increase contrast and reduce de-correlation noise.

physics.bio-ph↗
arXiv2008

Phenomenological modeling of DNA overstretching

A phenomenological model based on the three-dimensional theory of nonlinear elasticity is developed to describe the phenomenon of overstretching in the force-extension curve for dsDNA. By using the concept of a material with multiple reference configurations a single formula is obtained to fit the force-extension curve.

q-bio.BM↗
arXiv2007

The stress field in a pulled cork and some subtle points in the semi-inverse method of nonlinear elasticity

In an attempt to describe cork-pulling, we model a cork as an incompressible rubber-like material and consider that it is subject to a helical shear deformation superimposed onto a shrink fit and a simple torsion. It turns out that this deformation field provides an insight into the possible appearance of secondary deformation fields for special classes of materials. We also find that these latent deformation fields are woken up by normal stress differences. We present some explicit examples based on the neo-Hookean, the generalized neo-Hookean and the Mooney-Rivlin forms of the strain-energy density. Using the simple exact solution found in the neo-Hookean case, we conjecture that it is advantageous to accompany the usual vertical axial force by a twisting moment, in order to extrude a cork from the neck of a bottle efficiently. Then we analyse departures from the neo-Hookean behaviour by exact and asymptotic analyses. In that process, we are able to give an elegant and analytic example of secondary (or latent) deformations in the framework of nonlinear elasticity.

physics.class-ph↗