On The Concepts of State and Free Energy in Linear Viscoelasticity

Piero, Gianpietro Del; Deseri, Luca

doi:10.1007/s002050050035

Cited by 123 publications

(65 citation statements)

References 21 publications

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“…(35)), requires the introduction of a non-equilibrium entropy, functionally dependent on the internal energy density u and on the thermal energy flux q in the form: s ¼ s u; q ð Þ. Under this assumption the local entropy flux shows the functional dependence J ðsÞ l ¼ J ðsÞ l u; q ð Þ¼uðuÞq [28]. In this setting, the non-equilibrium entropy rate reads:…”

Section: The Clausius-planck Relation Of Eit With Fractional-order Hementioning

confidence: 99%

Fractional-order theory of heat transport in rigid bodies

Zingales¹

2014

Communications in Nonlinear Science and Numerical Simulation

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a b s t r a c tThe non-local model of heat transfer, used to describe the deviations of the temperature field from the well-known prediction of Fourier/Cattaneo models experienced in complex media, is framed in the context of fractional-order calculus. It has been assumed (Borino et al., 2011 [53], Mongioví and Zingales, 2013 [54]) that thermal energy transport is due to two phenomena: (i) A short-range heat flux ruled by a local transport equation; (ii) A long-range thermal energy transfer proportional to a distance-decaying function, to the relative temperature and to the product of the interacting masses. The distance-decaying function is assumed in the functional class of the power-law decay of the distance yielding a novel temperature equation in terms of a-order Marchaud fractional-order derivative ð0 6 a 6 1Þ. Thermodynamical consistency of the model is provided in the context of Clausius-Plank inequality. The effects induced by the boundary conditions on the temperature field are investigated for diffusive as well as ballistic local heat flux. Deviations of the temperature field from the linear distributions in the neighborhood of the thermostated zones of small-scale conductors are qualitatively predicted by the used fractional-order heat transport model, as shown by means of molecular dynamics simulations.

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Section: The Clausius-planck Relation Of Eit With Fractional-order Hementioning

confidence: 99%

Fractional-order theory of heat transport in rigid bodies

Zingales¹

2014

Communications in Nonlinear Science and Numerical Simulation

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“…When memory effects are accounted for, the notion of state requires careful definition because different equivalence relations between pairs of histories exist so that the notion of state appears to be rather natural in terms of equivalence classes. Such a question has been tackled variously (see [22,23,25]) on the basis of the abstract approach to thermodynamics proposed in [29] (see also [41][42][43]). …”

Section: History-dependent Measures Of Interaction and Equivalence Ofmentioning

confidence: 99%

Complex bodies with memory: Linearized setting

Mariano

Paoletti

2008

Math Methods in App Sciences

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SUMMARYThe mechanics of complex bodies with memory effects is discussed in linearized setting. The attention is focused on the characterization of free energies in terms of minimum work and maximum recoverable work in the bulk and along a discontinuity surface endowed with its own surface energy, a surface within the body. To this aim, use is made of techniques proposed by Del Piero. Consequences of the Clausius-Duhem inequality are investigated for complex bodies with instantaneous linear elastic response.

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“…IV in [15] for an analog derivation). The observation of (34,35) shows that assuming ↵ = d 1, the solution for " ! 0 yields a displacement-force rheological relation as:…”

Section: Power-law Hereditariness Of Fractal Models Of Bonesmentioning

confidence: 99%

Power‐law hereditariness of hierarchical fractal bones

Deseri

Paola

Zingales

et al. 2013

Numer Methods Biomed Eng

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SUMMARYIn this paper the authors introduce a hierarchic fractal model to describe bone hereditariness. Indeed, experimental data of stress relaxation or creep functions obtained by compressive/tensile tests have been proved to be fit by power-law with real exponent 0   1. The rheological behavior of the material has therefore been obtained, using the Boltzmann-Volterra superposition principle, in terms of real order integrals and derivatives (fractional-order calculus). It is shown that the power-laws describing creep/relaxation of bone tissue may be obtained introducing a fractal description of bone cross-section and the Hausdorff dimension of the fractal geometry is then related to the exponent of the power-law. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 INTRODUCTIONMathematical models of material behavior are fundamental for optimization and for reliable design of engineered devices. Furthermore, the key issue is to be able to track down the multiscale behavior from the nano-to the macrolevel, with particular regard to biological and bioinspired materials.Indeed, the mechanical interactions among biomedical devices and biological tissues play a keyrole for the optimization of physiological functionality of such devices owing to the reduction of immunologic response. In this regard, it is clear that the detailed knowledge of the features of biological tissues at the different scales and their interactions with the devices is a crucial step to optimize the mechanical and physical response of the compound.Physical parameters of biological tissues usually investigated in scientific literature involve stiffness, strength, toughness, permeability, porosity, thermal conductivity among others [1,2,3].Besides these important features, mineralized bone tissues must provide load carrying capabilities and they exhibit a marked time-dependent behavior under applied loads. In this context, the term hereditariness is usually used in the sense that the actual response of bone material in terms of stress/displacement depends on previously applied stress/strain. This feature is macroscopically detectable by stress relaxation and creep observed in classical traction/compression mechanical tests. During a relaxation test, the imposed strain is held constant and a measure of the stress is monitored showing that it is a decreasing function of time; similarly, in a creep test an imposed constant stress is applied and a continuous monitoring of the strain is considered showing that it is an increasing function of time. Both these tests highlight the hereditariness feature of such material; the past undergone stress or strain history influence the future response of the specimen. A similar time-dependent behavior also arises in mineralized tissues as ligaments and tendons. Indeed the high stiffness (but highly brittleness) of the hydroxyapatite cryst...

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On The Concepts of State and Free Energy in Linear Viscoelasticity

Cited by 123 publications

References 21 publications

Fractional-order theory of heat transport in rigid bodies

Fractional-order theory of heat transport in rigid bodies

Complex bodies with memory: Linearized setting

Power‐law hereditariness of hierarchical fractal bones

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