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