Critical velocities and displacements of anisotropic tubes under a moving pressure

Abstract

Critical velocities and middle-surface displacements of anisotropic axisymmetric cylindrical shells (tubes) under a uniform internal pressure moving at a constant velocity are derived in closed-form expressions by using the Love–Kirchhoff thin shell theory incorporating the rotary inertia and material anisotropy. The formulation is based on the general three-dimensional constitutive relations for orthotropic elastic materials and provides a unified treatment of orthotropic, transversely isotropic, cubic and isotropic tubes, which can represent various composite and metallic tubes. Closed-form formulas are first obtained for the general case with both the rotary inertia and radial stress effects, which are then reduced to the special cases without the rotary inertia effect and/or radial stress effect. It is shown that when the rotary inertia effect is suppressed and the radial normal stress is neglected, the newly derived formulas for the critical velocities of orthotropic and isotropic tubes recover the two existing ones for thin tubes as special cases. An example for an isotropic tube is provided to illustrate the new formulas, which give the values of the critical velocity and dynamic amplification factor that agree well with those obtained experimentally and computationally by others.

Critical velocities of a two-layer composite tube under a moving internal pressure

“…where u x , u θ and u z are, respectively, the x-, θ -and z-components of the displacement vector u of a point (x, θ , z) in the shell at time t, and u and w are, respectively, the x-and z-displacement components of the corresponding point (x, θ , 0) on the shell middle surface at time t. Note that u θ is identically zero and there is no dependence on θ for all kinematic and kinetic quantities in such an axisymmetric problem (e.g., [11,14]), which differ from those in general deformations of circular cylindrical shells (e.g., [37,40]). From Eq.…”

“…From Eq. ( 1), the components of the infinitesimal strain tensor in the axisymmetric circular cylindrical thin shell can be obtained as (e.g., [11]), with r ≈ R,…”

“…It has been known that dynamic strains in a homogeneous tube under a moving internal pressure can exceed three times of the corresponding static values (e.g., [11,[30][31][32]). This occurs when the velocity of the moving pressure is very close to the lowest critical velocity of the tube.…”

“…Efforts have been made to determine critical velocities of single-layer, homogeneous tubes of materials with various types of symmetry, including isotropic, cubic, transversely isotropic and orthotropic (e.g., [10,11,15,27,30,35,36]). However, for composite tubes containing two or more cylindrical layers of dissimilar materials (e.g., [20,21,29]), very few studies have been conducted to find their critical velocities.…”

“…In Sect. 2, the model for axisymmetric orthotropic cylindrical Love-Kirchhoff thin shells with the rotary inertia effect formulated in Gao and Littlefield [11] is briefly reviewed. In Sect.…”