Ljubica Oparnica

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

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 models. The restrictions on coefficients are derived by using entropy inequality for isothermal processes. In addition, we study waves in a rod of finite length modelled by a linear fractional constitutive equation. In particular, we examine stress relaxation and creep and compare results with the quasistatic analysis.

Waves in viscoelastic media described by a linear fractional model

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 Zener type viscoelastic media, J. Math. Anal. Appl. (2009), doi:10.1016/j.jma.2009.10.043]). In this article, we use a more general fractional model for a viscoelastic body to describe the wave equation for viscoelastic infinite media, and prove existence and uniqueness of distributional solutions to the corresponding generalized Cauchy problem.

Distributional framework for solving fractional differential equations

We analyse the solvability of a special form of distributed order fractional differential equations within 𝒮′+, the space of tempered distributions supported by [0, ∞).

Distributional solution concepts for the Euler–Bernoulli beam equation with discontinuous coefficients

We study existence and uniqueness of distributional solutions w to the ordinary differential equation with discontinuous coefficients and right-hand side. For example, if a and w are non-smooth the product a · w″ has no obvious meaning. When interpreted on the most general level of the hierarchy of distributional products discussed by Oberguggenberger, M. [1992, Multiplication of distributions and applications to partial differential equations (Harlow: Longman Scientific & Technical)], it turns out that existence of a solution w forces it to be at least continuously differentiable. Curiously, the choice of the distributional product concept is thus incompatible with the possibility of having a discontinuous displacement function as a solution. We also give conditions for unique solvability. §Supported by Ministry of Science of Serbia, project 144016, and the Austrian Science Fund (FWF) START program Y237 on ‘Nonlinear distributional geometry’.

Space–time fractional Zener wave equation

The space–time fractional Zener wave equation, describing viscoelastic materials obeying the time-fractional Zener model and the space-fractional strain measure, is derived and analysed. This model includes waves with finite speed, as well as non-propagating disturbances. The existence and the uniqueness of the solution to the generalized Cauchy problem are proved. Special cases are investigated and numerical examples are presented.

Generalized Hamilton's Principle with Fractional Derivatives

We generalize Hamiltons principle with fractional derivatives in the Lagrangian L(t, y(t), 0Dαty(t), α) so that the function y and the order of the fractional derivative α are varied in the minimization procedure. We derive stationarity conditions and discuss them through several examples.

Microlocal analysis of fractional wave equations

We determine the wave front sets of solutions to two special cases of the Cauchy problem for the space-time fractional Zener wave equation, one being fractional in space, the other being fractional in time. For the case of the space fractional wave equation, we show that no spatial propagation of singularities occurs. For the time fractional Zener wave equation, we show an analogue of non-characteristic regularity.

Solvability and microlocal analysis of the fractional Eringen wave equation

We discuss unique existence and microlocal regularity properties of Sobolev space solutions to the fractional Eringen wave equation, initially given in the form of a system of equations, in which the classical non-local Eringen constitutive equation is generalized by employing space fractional derivatives. Numerical examples illustrate the shape of solutions as a function of the order of the space fractional derivative.

GENERALIZED SOLUTIONS FOR THE EULER-BERNOULLI MODEL WITH ZENER VISCOELASTIC FOUNDATIONS AND DISTRIBUTIONAL FORCES

We study the initial-boundary value problem for an Euler–Bernoulli beam model with discontinuous bending stiffness laying on a viscoelastic foundation and subjected to an axial force and an external load both of Dirac-type. The corresponding model equation is a fourth-order partial differential equation and involves discontinuous and distributional coefficients as well as a distributional right-hand side. Moreover the viscoelastic foundation is of Zener-type and described by a fractional differential equation with respect to time. We show how functional analytic methods for abstract variational problems can be applied in combination with regularization techniques to prove existence and uniqueness of generalized solutions.