Buckling of Thin-Walled Cylinders from Three Dimensional Nonlinear Elasticity

Abstract

The famous bifurcation analysis performed by Flügge on compressed thin-walled cylinders is based on a series of simplifying assumptions, which allow to obtain the bifurcation landscape, together with explicit expressions for limit behaviours: surface instability, wrinkling, and Euler rod buckling. The most severe assumption introduced by Flügge is the use of an incremental constitutive equation, which does not follow from any nonlinear hyperelastic constitutive law. This is a strong limitation for the applicability of the theory, which becomes questionable when is utilized for a material characterized by a different constitutive equation, such as for instance a Mooney-Rivlin material. We re-derive the entire Flügge’s formulation, thus obtaining a framework where any constitutive equation fits. The use of two different nonlinear hyperelastic constitutive equations, referred to compressible materials, leads to incremental equations, which reduce to those derived by Flügge under suitable simplifications. His results are confirmed, together with all the limit equations, now rigorously obtained, and his theory is extended. This extension of the theory of buckling of thin shells allows for computationally efficient determination of bifurcation landscapes for nonlinear constitutive laws, which may for instance be used to model biomechanics of arteries, or soft pneumatic robot arms.

Necking of thin-walled cylinders via bifurcation of incompressible nonlinear elastic solids

“…where r, y, z are to be substituted in place of the subscript Á, while K Áy and K Áz represent the Áy and the Áz components of the Kirchhoff stress. Performing the integration of the incremental equilibrium equations across the thickness of the shell and the subsequent integration by parts with enforcement of the boundary conditions, 24 the incremental equilibrium can be expressed as…”

“…The current configuration of the cylindrical shell is characterized by its length l , external and internal radii r e and r i , respectively, and thus thickness t = r e − r i , mid-radius a = ( r e + r i )/2, the latter defining the mid-surface, as well as the so-called ‘reduced radius’, r̄ = r − a , so that − t /2 ≤ r̄ ≤ t /2. The incremental kinematics of the thin-walled cylindrical shell is assumed in the form 24 v ( r̄ , θ , z ) = v̄ ( θ , z ) + r̄ [ n̄ ( θ , z ) − e r ],where v̄ ( θ , z ) denotes the incremental displacement of the points along the current mid-surface of the shell, whose unit normal is n̄ ≈ e r + ( v̄ θ − v̄ r , θ )/ a e θ − v̄ r , z e z .The incremental displacements can therefore be detailed componentwise asOn the basis of the linearized kinematics in eqn (3.3), the components of gradient of incremental displacement L = grad v areso that the non-trivial components of the Eulerian incremental strain tensor D = ( L + L T )/2 turn out to beWhen substituted into eqn (2.10), eqn (3.5) allow expressing the incremental stress for both the constitutive laws considered here as functions of the constitutive parameters through eqn (2.3) and (2.6), respectively, the axial pre-stretch λ z , and the components of the velocity along the mid-surface of the shell.…”

“…According to the usual shell theory, a set of generalized stresses is introduced, representing the resultant forces and moments per unit length along the mid-surface of the shell, 24 as in Table 1. The relevant incremental stresses, based on the Oldroyd increment, are defined aswhere r , θ , z are to be substituted in place of the subscript ·, while K · θ and K · z represent the · θ and the · z components of the Kirchhoff stress.…”

“…The relevant incremental stresses, based on the Oldroyd increment, are defined aswhere r , θ , z are to be substituted in place of the subscript ·, while K · θ and K · z represent the · θ and the · z components of the Kirchhoff stress. Performing the integration of the incremental equilibrium equations across the thickness of the shell and the subsequent integration by parts with enforcement of the boundary conditions, 24 the incremental equilibrium can be expressed aswhere P = T zz t is the pre-stress load per unit length along the mid-circular surface.…”

“…According to the usual shell theory, a set of generalized stresses is introduced, representing the resultant forces and moments per unit length along the mid-surface of the shell, 24 as in Table 1. The relevant incremental stresses, based on the Oldroyd increment, are defined as…”