# Energy dissipation for hereditary and energy conservation for non-local fractional wave equations

@article{Zorica2020EnergyDF, title={Energy dissipation for hereditary and energy conservation for non-local fractional wave equations}, author={Du{\vs}an Zorica and Ljubica Oparnica}, journal={Philosophical Transactions of the Royal Society A}, year={2020}, volume={378} }

Using the method of a priori energy estimates, energy dissipation is proved for the class of hereditary fractional wave equations, obtained through the system of equations consisting of equation of motion, strain and fractional order constitutive models, that include the distributed-order constitutive law in which the integration is performed from zero to one generalizing all linear constitutive models of fractional and integer orders, as well as for the thermodynamically consistent fractional…

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## References

SHOWING 1-10 OF 46 REFERENCES

On the fractional generalization of Eringenʼs nonlocal elasticity for wave propagation

- Mathematics
- 2013

Fractional Burgers wave equation

- MathematicsActa Mechanica
- 2019

Thermodynamically consistent fractional Burgers constitutive models for viscoelastic media, divided into two classes according to model behavior in stress relaxation and creep tests near the initial…

Thermodynamical Restrictions and Wave Propagation for a Class of Fractional Order Viscoelastic Rods

- Mathematics
- 2011

We discuss thermodynamical restrictions for a linear constitutive equation containing fractional derivatives of stress and strain of different orders. Such an equation generalizes several known…

Generalized wave equation in nonlocal elasticity

- Mathematics
- 2009

We study the motion of a one-dimensional continuum whose deformation is described by a strain measure of nonlocal type. In particular, we use the Caputo fractional derivatives and a linear relation…

Waves in viscoelastic media described by a linear fractional model

- Mathematics
- 2011

Recently, the classical wave equation has been generalized for the case of viscoelastic media described by the fractional Zener model (cf. [S. Konjik, Lj. Oparnica, and D. Zorica, Waves in fractional…

Distributed-order fractional constitutive stress–strain relation in wave propagation modeling

- MathematicsZeitschrift für angewandte Mathematik und Physik
- 2019

Distributed-order fractional model of viscoelastic body is used to describe wave propagation in an infinite media. Existence and uniqueness of fundamental solution to the generalized Cauchy problem…

A 3-dimensional singular kernel problem in viscoelasticity: an existence result

- Mathematics
- 2018

Materials with memory, namely those materials whose mechanical and/or thermodynamical behaviour depends on time not only via the present time, but also through its past history, are considered.…

On weak regularity requirements of the relaxation modulus in viscoelasticity

- MathematicsCommunications in Applied and Industrial Mathematics
- 2019

Abstract The existence and uniqueness of solution to a one-dimensional hyperbolic integro-differential problem arising in viscoelasticity is here considered. The kernel, in the linear viscoelasticity…

Formulation of thermodynamically consistent fractional Burgers models

- MathematicsActa Mechanica
- 2018

The approach of viscoelastic body constitutive equation fractionalization using rheological representation of the classical Burgers model and considering the Scott–Blair (fractional) elements instead…