In this study, we propose the analysis of a tubular structure undergoing expansion along the axis of the cylindrical material under internal pressure. Both kinematic and dynamic aspects are examined, leading to the derivation of an exact analytical solution using a system of partial differential equations. Simulation results demonstrate that the solution exhibits sinusoidal behavior in all cases. Minor variations result in incremental or decremental intervals, while significant changes in radius cause simultaneous increase and decrease intervals with trigonometric patterns. Additionally, we observe that the second component significantly influences the overall solution behavior compared to the first component.
Problems related to the elasticity, studied based on the models represented by the elliptic equations are characterized by their invariable in time. For an isotropic compressible medium, the loss of ellipticity criterion depends on the determinant of the Hessian matrix of the energy potential and therefore invariant tensor Cauchy Green. Necessary and sufficient conditions on the ellipticity or Legendre -Hadamard condition are given from two energy functions of polynomial type from the isotropic compressible case.
We consider a nonlinear hyperelastic tube subjected to a deformation radial. We study then the phenomena of asymptotic stability of the tube. We use techniques for obtaining approximations to periodic time solutions of nonlinear second-order differential equations subject to a harmonic forcing term, and to limit cycles of autonomous equations. These approximations take the form of an expansion in integer powers of a small parameter, having coefficients that are functions of time.
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