## Dynamic boundary value problems for transversely isotropic cylinders and spheres in finite elasticity

##### Date

2007-02-21T13:02:25Z

##### Authors

Maluleke, Gaza Hand-sup

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##### Abstract

A derivation is given of the constitutive equation for an incompressible transversely
isotropic hyperelastic material in which the direction of the anisotropic director is
unspecified. The field equations for a transversely isotropic incompressible hyperelastic
material are obtained.
Nonlinear radial oscillations in transversely isotropic incompressible cylindrical
tubes are investigated. A second order nonlinear ordinary differential equation,
expressed in terms of the strain-energy function, is derived. It has the same form
as for radial oscillations in an isotropic tube. A generalised Mooney-Rivlin strainenergy
function is used.
Radial oscillations with a time dependent net applied surface pressure are first
considered. For a radial transversely isotropic thin-walled tube the differential equation
has a Lie point symmetry for a special form of the strain-energy function and
a special time dependent applied surface pressure. The Lie point symmetry is used
to transform the equation to an autonomous differential equation which is reduced
to an Abel equation of the second kind. A similar analysis is done for radial oscillations
in a tangential transversely isotropic tube but computer graphs show that the
solution is unstable. Radial oscillations in a longitudinal transversely isotropic tube
and an isotropic tube are the same. The Ermakov-Pinney equation is derived.
Radial oscillations in thick-walled and thin-walled cylindrical tubes with the
Heaviside step loading boundary condition are next investigated. For radial, tangential
and longitudinal transversely isotropic tubes a first integral is derived and effective
potentials are defined. Using the effective potentials, conditions for bounded
oscillations and the end points of the oscillations are obtained. Upper and lower
bounds on the period are derived. Anisotropy reduces the amplitude of the oscillation
making the tube stiffer and reduces the period.
Thirdly, free radial oscillations in a thin-walled cylindrical tube are investigated.
Knowles(1960) has shown that for free radial oscillations in an isotropic tube, ab = 1
where a and b are the minimum and maximum values of the radial coordinate. It
is shown that if the initial velocity v0 vanishes or if v0 6= 1 but second order terms
in the anisotropy are neglected then for free radial oscillations, ab > 1 in a radial
transversely isotropic tube and ab < 1 in a tangential transversely isotropic tube.
Radial oscillations in transversely isotropic incompressible spherical shells are
investigated. Only radial transversely isotropic shells are considered because it is
found that the Cauchy stress tensor is not bounded everywhere in tangential and longitudinal
transversely isotropic shells. For a thin-walled radial transversely isotropic
spherical shell with generalised Mooney-Rivlin strain-energy function the differential
equation for radial oscillations has no Lie point symmetries if the net applied
surface pressure is time dependent.
The inflation of a thin-walled radial transversely isotropic spherical shell of generalised
Mooney-Rivlin material is considered. It is assumed that the inflation proceeds
sufficiently slowly that the inertia term in the equation for radial oscillations
can be neglected. The conditions for snap buckling to occur, in which the pressure
decreases before steadily increasing again, are investigated. The maximum value of
the parameter for snap buckling to occur is increased by the anisotropy.

##### Description

Student Number : 9202983Y -
PhD thesis -
School of Computational and Applied Mathematics -
Faculty of Science

##### Keywords

elasticity, differential equation, isotropic anisotropy, cylinder, sphere, lie point symmetries