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# Does a radially expanding cylinder bend?

The Koiter-Sanders-Budiansky bending strain measure and a nonlinear generalization

We know from strength of materials that non-uniform stretching of fibers along the cross section of a beam produces bending moments. But does this situation necessarily correspond to a 'bending' deformation? For that matter, what do we exactly mean kinematically when we talk about a bending deformation?

To make the question more concrete, consider a cylinder that expands uniformly along all radial rays. Does this deformation of the cylinder correspond to bending? I think it is fair to say that most would say that this is purely a stretching deformation with no bending. But then, what is precisely a bending deformation?

The most classical definition relates the change in the second fundamental form as a bending strain. By this is meant the following. Calculate the gradient of the unit normal fields (i.e. curvature tensors) on the deformed and undeformed shells. The difference of these curvatures, suitably adjusted for the fact that at each material point they are tensors on different tangent spaces that can be oriented very differently, is assigned to be the bending strain. But you see, a stretching of the shell changes the curvature and therefore the radially expanding cylinder would be described as undegoing a bending deformation, according to this classical measure.

So, something isn't quite right here. Starting with KOITER and then followed by SANDERS and BUDIANSKY, a bending strain measure was introduced for linear kinematics that does not have this shortcoming, up to the accuracy of the linear theory. Koiter and Budiansky later proposed nonlinear strain measures that predict vanishing bending strain in biaxial stretching of cylinders, up to second order in the radial and axial displacements.

In my opinion, the Koiter, Sanders, Budainsky developments were pioneering works towards clarifying what one might mean kinematically by bending. However, a clear physical definition leading to an exact mathematical statement of what constitutes bending deformation of a shell was lacking. For one thing, if such a measure was available then it would associate, without approximation, no bending strain to a biaxial stretching of a cylinder.

In the attached paper, I tried to address this question of physical definition and corresponding mathematical form in the nonlinear setting with its connections to the linear KSB measure. The seemingly innocuous question became surprisingly subtle - at least for me - with things like the drill rotation spoiling the show to exact success. The physical definition naturally involves things like the Polar Decomposition of the deformation gradient on a manifold with almost the whole story revolving around being careful about the domain and range of the stretch and rotation tensors.

For those of you who read it, I hope you enjoy it as much as I did working on the problem.

Attachment | Size |
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nonlinear_KSB.pdf | 757.66 KB |

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