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An introduction to nonlinear Viscoelasticity of filled Rubber. A continuum mechanics approach
No full textThe use of rubber in industry has grown rapidly and constantly since the discovery of vulcanization in 1839: tires, belts, seals, gaskets, mounts are but a few of the myriad products made of rubber nowadays. The neat elastomer, natural or synthetic, is employed after the addition of reinforcing fillers to improve thermal stability and mechanical properties. The resulting compound can be stretched to about ten times its original length and possesses both solid-like and fluid-like characteristics, well described by nonlinear viscoelastic models.The purpose of this book is to provide an up-to-date and comprehensive overview of all aspects concerning the mechanical characterization of filled rubber: from experiments to modelling, from parameters identification to numerical simulations. Throughout the book, the modelling aspects are approached from a continuum mechanics perspective: by inferring the behavior of the material under different loading conditions, macroscopic constitutive relations between stress and strain are derived without reference to the internal structure.In detail, the aspects addressed cover: - The basic phenomenology of rubber under static and dynamic loading conditions.- A brief introduction to main aspects of nonlinear elasticity.- A detailed overview of the different nonlinear viscoelastic approaches used in the literature.- A thorough description of nonlinear identification techniques to estimate model coefficients from the experimental data.- An overview of Finite Element codes used to simulate the large strain behavior of filled rubber
Torque-induced reorientation in active fibre-reinforced materials
No full textWe introduce a continuum model for a fibre reinforced material in which the reference orientation of the fibre may evolve with time, under the influence of external stimuli. The model is formulated in the framework of large strain hyperelasticity and the kinematics of the continuum is described by both a position vector and by a remodelling tensor which, in the present context, is an orthogonal tensor representing the fibre reorientation process. By imposing suitable thermodynamical restrictions on the constitutive equation, we obtain an evolution equation of the remodelling tensor governed by the Eshelby torque, whose stationary solutions are studied in absence of any external source terms. It is shown that the fibres reorient themselves in a configuration that minimises the elastic energy and get aligned along a direction that may or may not be of principal strain. The explicit analysis of the Hessian of the strain energy density allows us to discriminate among the stationary solutions, which ones are stable. Examples are given for passive reorientation processes driven by applied strains or external boundary tractions. Applications of the proposed theory to biological tissues, nematic or magneto-electro active elastomers are foreseen
Experimental Testing and Nonlinear Viscoelastic Modeling of Filled Rubber
Full text linkOwing to its unique physical properties, rubber plays a keyrole in countless industrial applications. Tires, vibration absorbers and shoe soles are only but a few of the myriad uses of natural and synthetic rubber in an industry which in 2009 had an estimated market value of 2 billion euro.
Despite a peculiar internal structure, the macroscopic behavior of filled-rubber is reminiscent of several biological soft tissues. While rubber is internally constituted by flexible long chain molecules that intertwine with each other, a similar role is played, in soft-tissues, by
collagen fiber bundles. As a consequence, both classes of materials are able to sustain large strains and exhibit the characteristics of a viscous fluid and an elastic solid.
In industry, the requirement to model complex geometrical structures made of materials
exhibiting a nonlinear constitutive behavior is a compelling reason to use Finite Element
Analysis (FEA) software. The predictive capabilities of these numerical tools strongly rely upon the capabilities of the underlying model to describe the material’s rheological properties.
The possibility of simulating accurately the material behavior over the entire working range avoids the use of excessive number of prototypes, thereby reducing the need for expensive and difficult experimental tests; consequently, development costs can be drastically reduced.
The theory of viscoelasticity is crucial in describing materials, such as filled rubber, which exhibit time dependent stress-strain behavior. In many engineering applications, such as the estimate of the rolling resistance of tires and hysteretic losses in soft biological tissues, the energy dissipation is a primary feature to be predicted. In addition, in the usual
operative range, tires, shock absorbers and other rubber components bear finite dynamic
deformations. Therefore, a reliable constitutive equation must be assessed within the theory of nonlinear viscoelasticity.
A review of the literature revealed significantly more well-established studies dealing with hyperelastic constitutive models, than those dealing with finite viscoelasticity.
Over the years, many hyperelastic models able to describe all the relevant aspects of
the quasi-static response have been introduced. Furthermore, the American norms (ASTM D412, ASTM D575, ASTM D945, ASTM D6147, ASTM D1456) establish all the experimental
techniques to identify the material constitutive parameters. In this context, many authors have recently addressed the problem of finite amplitude wave propagation or focused their interest upon particular boundary value problems.
On the other hand, there is a lack of well-established nonlinear viscoelastic models capable of describing all the relevant effects in the material response. Moreover, a standardization similar to that concerning the static norms is yet to be achieved. The usual methodology provides for small harmonic deformations superimposed on a large static displacement. However, such a prescription does not allow the capture of many of the relevant nonlinear phenomena. In the
literature, experimental evidence concerning finite dynamic deformations is rarely reported
A comparison of nonlinear viscoelastic models for filled-rubber: analytical formulation, experimental modeling and identification
No full textA theory of magneto-elastic nanorods obtained through rigorous dimension reduction
Full text linkStarting from a two-dimensional theory of magneto-elasticity for fiber-reinforced magnetic elastomers we carry out a rigorous dimension reduction to derive a rod model that describes a thin magneto-elastic strip undergoing planar deformations. The main features of the theory are the following: a magneto-elastic interaction energy that manifests itself through a distributed torque; a penalization term that prevent local interpenetration of matter; a regularization that depends on the second gradient of the deformation and models microstructure-induced size effects. As an application, we study a problem involving magnetically-induced buckling and we show that the intensity of the field at the onset of the instability increases if the length of the rod is decreased. Finally, we assess the accuracy of the deduced model by performing numerical simulations where we compare the two-dimensional and the one-dimensional theory in some special cases and we observe excellent agreement
Identification of the viscoelastic properties of soft materials at low frequency: Performance, ill-conditioning and extrapolation capabilities of fractional and exponential models
No full textWe report about the experimental identification of viscoelastic constitutive models for frequencies ranging within 0-10. Hz. Dynamic moduli data are fitted for. several materials of interest to medical applications: liver tissue (Chatelin et al., 2011), bioadhesive gel (Andrews et al., 2005), spleen tissue (Nicolle et al., 2012) and synthetic elastomer (Osanaiye, 1996). These materials actually represent a rather wide class of soft viscoelastic materials which are usually subjected to low frequencies deformations.We also provide prescriptions for the correct extrapolation of the material behavior at higher frequencies. Indeed, while experimental tests are more easily carried out at low frequency, the identified viscoelastic models are often used outside the frequency range of the actual test.We consider two different classes of models according to their relaxation function: Debye models, whose kernel decays exponentially fast, and fractional models, including Cole-Cole, Davidson-Cole, Nutting and Havriliak-Negami, characterized by a slower decay rate of the material memory. Candidate constitutive models are hence rated according to the accurateness of the identification and to their robustness to extrapolation. It is shown that all kernels whose decay rate is too fast lead to a poor fitting and high errors when the material behavior is extrapolated to broader frequency ranges. © 2014
The use of modal curvatures for damage localization in beam-type structures
No full textThe localization of stiffness variation in damaged beams through modal curvatures, i.e., second derivative of mode shapes, is studied by exploiting a perturbative solution of the Euler–Bernoulli equation. It is shown that for low order modes the difference between undamaged and damaged modal curvatures has only one distinct peak if the damage is localized in a narrow region. This phenomenon is independent of the presence of experimental noise and of the technique used to reconstruct the curvature mode shapes from the displacement mode shapes. Broader damages cause the modal curvature difference to have several peaks outside the damage region that could result in a false damage localization. The same effect is present at higher modes for both narrow and broad damages. As a result, modal curvatures can be effectively used to localize structural damages only once they have been properly filtered. Here the perturbative solution is used to introduce an effective damage measure able to localize correctly narrow and broad damages and also single and multiple damages cases
Orientation effects in short fibre-reinforced elastomers
No full textThe large strain behaviour of a short fibre-reinforced composite is studied through numerical simulations. The reinforcing fibres yield the macroscopic response transversely isotropic which is indeed the case of many reinforcements currently used in composites: short carbon fibres, cellulose whiskers, carbon nanotubes. As a result of the analysis, it is shown that the reorientation of the fibres that takes place at large strain has a significant effect on the overall material response by changing the axis of isotropy. This behaviour can be adequately described by using a transversely isotropic model whose strain energy function depends on three invariants: two isotropic and one representing the stretch along the direction of the fibres. To assess its capabilities, the model is compared to the results of experiments carried out by the authors on nickel-coated chopped carbon fibres in a vulcanised natural rubber matrix for which the fibre orientation is achieved by controlling an external magnetic field prior to curing. Possible applications include micro-sized propulsion devices and actuators. Copyright © 2014 by ASME
Multiscale modelling nano-platelet reinforced composites at large strain
No full textWe study the behaviour of an incompressible particle-reinforced neo-Hookean (IPRNC) material when subjected to large plain strain deformation. The peculiarity of the model consists in the rectangular shape of the particle which yields the macroscopic response of the composites non isotropic. This is indeed the case for many reinforcements currently used in composites at all length scales: short-fibres, clays, graphene. The consequence of the anisotropic reinforcement in this model at short strain is evident in the stiffness that is observed to depend strongly on the platelet orientation; a transverse stiffening effect when the platelet is oriented perpendicular to the loading direction proves to be almost as significant as the longitudinal stiffness contribution usually considered for anisotropic reinforcements. The large strain effects of orientation are also significant and an understanding of them is relevant to a number of applications that can take advantage of the large strain non-linear response
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