Dynamic buckling of linear viscoelastic rods
Murphy, Kieran (1996) Dynamic buckling of linear viscoelastic rods. PhD thesis, Dublin City University.
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This thesis examines the planar bending of a viscoelastic rod subject to a uniaxial load A(i). The rod is assumed to be inextensible, and the torsion and shear of the sections are ignored. The bending moment across a section is assumed to depend on the curvature through a linear hereditary law of Boltzmann type. The rod is composed of a solid material, so the creep function remains bounded for all time. Thus a viscoelastic solid rod in simple extension eventually approaches an equilibrium state. This is equivalent to G'(oo) := lim^ooG(t)>0, where G(t) is the material specific relaxation function. The exact nonlinear dynamic problem can be linearised about the straight equilibrium position to yield an integro-differential equation. It is this linear problem which is investigated here. The initial history of the deflection is allowed to be nontrivial. Usually this initial history is prescribed, but we also consider the problem without this assumption.
For constant loads, Laplace transform techniques can be employed to show that solutions decay if A < XiG(oo) /G(0 ), and grow exponentially if A > XiG(oo) /G(0) , where Ai > 0 is the Euler critical load calculated using the instantaneous elasticity G(0). For th e standard viscoelastic material, we derive necessary and sufficient conditions on the material parameters which ensure that the solution is oscillatory.
For time-varying loads, the evolution equation for the initial history problem generates a semigroup, and has a unique solution which depends continuously on the initial data. This is in contrast to the corresponding results in the qua si-sta tic theory. The Volterra-Graffi energy is used to construct a suitable Lyapunov function, which can be used to demonstrate that the zero solution is stable for a large class of loads satisfying 0<A(t)< A1Gr(oo)/Gr(0).
Multiple scale methods are also used to determine various approximate solutions. For a standard viscoelastic material with long relaxation time, the elastic and creep effects occur on different time scales. If A > AiG(oo)/G(0), an approximate solution is determined and is used to investigate the effect of the different types of initial disturbance on the growth rate of the solution. Also if a standard viscoelastic material is subject to a periodic load A(^), an approximate stability region in parameter space is found when the parametric excitation is near the principal resonance.
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