Axisymmetric Deformation of a Transversely Isotropic Cylindrical Body: A Hamiltonian State-Space Approach

Abstract

A state-space approach for exact analysis of axisymmetric deformation and stress distribution in a circular cylindrical body of transversely isotropic material is developed. By means of Hamiltonian variational formulation via Legendre's transformation, the basic equations in cylindrical coordinates are formulated into a state-space framework in which the unknown state vector comprises the displacements and associated stress components as the dual variables and the system matrix possesses the symplectic characteristics of a Hamiltonian system. Upon delineating the symplecticity of the formulation, a viable solution approach using eigenfunction expansion is developed. For illustration, an exact analysis of a finite thick-walled circular cylinder under internal and external pressures is presented, with emphasis on the end effects.

A Hamiltonian State Space Approach for 3D Analysis of Circular Cantilevers

“…Table 1 shows the first 25 non-dimensional eigenvalues μ i a of the vertical bending mode, in which the first eigenvalues are zero and repeated with multiplicity 4, resulting in z 3 -dependent displacements and linearly z-dependent stresses. As discussed in [12,13], a simple measure to estimate the extent of the stress disturbance due to the end effect is the characteristic decay length:…”

“…The zero placements in the matrix reveal the decoupling that is crucial to the earlier papers [12][13][14].…”

“…This is a distinct characteristic of the Hamiltonian matrix and is critical in deriving the symplectic orthogonality of the eigensystem. Development of the orthogonality relations for the eigensystem is essentially similar to that in the previous paper [13]. Here it is necessary to consider the case of zero eigenvalue (μ = 0) which is a repeated eigenvalue of even multiplicity.…”

“…In a previous paper [13] we have formulated the axisymmetric problem of a cylindrical body in the Hamiltonian state space. Here it is generalized to 3D by considering θ -dependence.…”

“…Recently, we have obtained some exact solutions for torsion and axisymmetric deformation of elastic circular cylinders [12][13][14], which satisfy exactly the zero displacement condition at the fixed end. To our knowledge, it is difficult, if not impossible, to obtain these exact solutions following the classical approach of elasticity.…”