Distributed-order fractional wave equation on a finite domain. Stress relaxation in a rod

We study waves in a rod of finite length with a viscoelastic constitutive equation of fractional distributed-order type for the special choice of weight functions. Prescribing boundary conditions on displacement, we obtain case corresponding to stress relaxation. In solving system of differential and integro-differential equations we use the Laplace transformation in the time domain

On a initial value problem arising in mechanics

We study initial value problem for a system consisting of an integer order and distributed-order fractional differential equation describing forced oscillations of a body attached to a free end of a light viscoelastic rod. Explicit form of a solution for a class of linear viscoelastic solids is given in terms of a convolution integral. Restrictions on storage and loss moduli following from the Second Law of Thermodynamics play the crucial role in establishing the form of the solution. Some previous results are shown to be special cases of the present analysis

Stability and optimal shape of Pflüger micro/nano beam

AbstractThis paper deals with optimal shapes against buckling of an elastic nonlocal small-scale Pflüger beams with Eringen’s model for constitutive bending curvature relationship. By use of the Pontryagin’s maximum principle the optimality condition in form of a depressed quartic equation is obtained. The shape of the lightest (having the smallest volume) simply supported beam that will support given uniformly distributed follower type of load and axial compressive force of constant intensity without buckling, is determined numerically. A special attention is paid to the influence of the characteristic small length scale parameter of the nonlocal constitutive law to both critical load and optimal shape of the analyzed beams. For the case when distributed follower type of load is zero, our results reduce to those obtained recently for compressed nonlocal beam. Also the post buckling shape of the optimally shaped rod is studied numerically

On the fractional generalization of Eringen's nonlocal elasticity for wave propagation

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Shape optimization against buckling of micro- and nano- rods

In this paper, we analyze elastic buckling of micro- and nano-rods based on Eringen's nonlocal elasticity theory. By using the Pontryagin's maximum principle, we determine optimality condition for a rod simply supported at both ends and loaded with axial compressive force only. Thus, the problem that we treat represents a generalization of the classical Clausen problem formulated for Bernoulli–Euler rod theory. Several concrete examples are treated in details, and the increase in buckling load capacity is determined. In solving the problem numerically, we used a first integral of the resulting system of equations, which helped us to monitor error of the numerical procedure. Our results show that nonlocal effects decrease the buckling load of optimally shaped rod. However, nonlocal theory leads to the optimal rod with the cross-sectional area at the rod ends different from zero. This is important property since zero value of the cross-section at the ends, which optimally shaped rod according to Bernoulli–Euler rod theory has, is unacceptable in applications