Here, we study the internal variable approach to viscoelasticity. First, we generalize the classical approach by introducing a fractional derivative into the equation for time evolution of the internal variables. Next, we derive restrictions on the coefficients that follow from the dissipation inequality (entropy inequality under isothermal conditions). In the example of wave propagation, we show that the restrictions that follow from entropy inequality are sufficient to guarantee the existence of the solution. We present a numerical solution to the wave equation for several values of the parameters.
We study optimal shape of an inverted elastic column with concentrated force at the end and in the gravitational field. We generalize earlier results on this problem in two directions. First we prove a theorem on the bifurcation of nonlinear equilibrium equations for arbitrary cross-section column. Secondly we determine the cross-sectional area for the compressed column in the optimal way. Variational principle is constructed for the equations determining the optimal shape and two new first integrals are constructed that are used to check numerical integration. Next, we apply the Noether’s theorem and determine transformation groups that leave variational principle Gauge invariant. The classical Lagrange problem follows as a special case. Several numerical examples are presented.
We analyze the classical problem of finding the shape of the column that optimizes certain criteria. The new formulation proposed here may be stated as: given the critical buckling load 𝐹 of the column and the length 𝐿, find crosssectional area 𝐴, such that the volume 𝑊 of the column attains minimal value. This is a classical Clausen problem. However, in this work we shall use the generalized constitutive equations of the column that allows for shear deformation and axis compressibility. This, as well as the novel use of the first integral, are the main novelties of our work. We will formulate a nonlinear boundary value problem for post critical deformation of optimally shaped rod. Finally we show that optimally shaped rod exhibits pitchfork supercritical bifurcation at critical buckling load.
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