Abstract
We study the oscillations of a spherical shell of a rate-type viscoelastic solid subject to a pressure difference between the inner and the outer surface. The stable equilibrium configurations, in the class of spherically symmetric deformations, correspond to the minima of the elastic energy function. Numerical simulations indicate that the way in which the equilibrium state is reached, strongly depends on the material parameters.
Highlights
Instabilities of inflated spherical membranes are a stimulating problem in continuum mechanics
The author started with a general elastic strain-energy density W (F) where F is the gradient of the deformation and, adding to W an extra history-type term, i.e., a term which depends via a suitable integral on the history of F up to time t, she introduced a free energy functional to describe the mechanical behavior of the material
The paper is organized as follows: in Sect. 2, we illustrate the constitutive model and, in Sects. 3 and 4, we investigate the radial deformations of a spherical shell
Summary
Instabilities of inflated spherical membranes are a stimulating problem in continuum mechanics (see, e.g., Chap. 7 in [1]). A model similar to the one proposed in [3] has been considered by Fosdick and Yu [5] In this case, the authors studied the stability of the radial oscillations of a sphere via a suitable Lyapunov functional. There has been a renewed interest on this problem [7] studied using the celebrated quasi-linear viscoelastic model proposed by Pipkin and Rogers [8]. Within the framework of traveling waves, we do know that global solutions may not exist for the quasi-linear viscoelastic models (see, e.g., [9]). In the three-dimensional generalization, one major problem is the choice of the objective derivative required to describe stress relaxation. The differences associated with the possible choices of the objective derivative have been pointed out in [12] and in [13]
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