The finite deformation of a pressurized circular tube for a class of compressible materials
The body under consideration is a hollow circular cylinder (e.g., a tube) subjected to uniform internal or external pressure or both at a time and consists of a homogeneous, isotropic, compressible elastic material of a special type, called a harmonic material, introduced by \textit{F. John} [Comm. Pure Appl. Math. 13, 239-296 (1960; Zbl 0094.370)]. For materials of this type the strain energy density function in plane strain is given by \(W=2\mu [H(\lambda_ 1+\lambda_ 2)-\lambda_ 1\lambda_ 2]\), where \(\lambda_ 1,\lambda_ 2\) denote the principal stretches (both positive), \(\mu\) a positive constant that can be identified with the shear modulus of the linear theory of elasticity, and H a constitutive function satisfying certain restrictions in order to ensure a physically reasonable response. Solutions for the deformation and stress fields are obtained in explicit closed form. The true stress distribution in spatial description proves to be essentially independent of the material properties. In the case of internal pressure alone, there is a critical value of the applied pressure at which the largest hoop stress becomes unbounded. This stress always occurs at the inner wall. In the case of external pressure alone, a critical value exists for which closing of the body's cavity is predicted. For nearly solid cylinders or, equivalently, for a cavity in an unbounded body, explicit results are provided in terms of the corresponding stress concentration factor. A classical linearized stability analysis is carried out with allowance for non-axisymmetric states of plane strain. Internal pressurization leads to no instability. Consequently, the occurrence of the infinite hoop stress is not preceded by an instability. On the other hand, external pressurization effects buckling before the cavity closes. The smallest buckling load is found to be much smaller than 2\(\mu\), which is predicted to be the pressure at which the cavity would close. The occurrence of an unbounded stress at a finite value of the load is due to the so-called harmonic materials, which are postulated on account of the analytical convenience in solving nonlinear partial differential equations that govern finite elasticity, but obviously exhibit a pathological behavior under compression. However, the stress singularity is notably induced by a constitutive effect rather than by a discontinuity in loading or deformational geometry.
homogeneous, isotropic, compressible elastic material
closing of the body's cavity is predicted
non- axisymmetric states of plane strain
scientific article; zbMATH DE number 3858537