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# The Twist-Fit Problem: Finite Torsional and Shear Eigenstrains in Nonlinear Elastic Solids

Eigenstrains in nonlinear elastic solids are created through defects, growth, or other anelastic effects. These eigenstrains are known to be important as they can generate residual stresses and alter the overall response of the solid. Here, we study the residual stress fields generated by finite torsional or shear eigenstrains. This problem is addressed by considering a cylindrical bar made of an incompressible isotropic solid with an axisymmetric distribution of shear eigenstrains. As particular examples, we consider a cylindrical inhomogeneity and a double inhomogeneity with finite shear eigenstrains and study the effect of torsional shear eigenstrains on the axial and torsional stiffnesses of the circular cylindrical bar.

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## Comments

## eigenstrains

Dear Yavari,

Are eigenstrains the strains for each mode calculated with the eigenvalues and then you need to generate the calculate the real strains with a superposition process in dynamics.

## Re: Eigenstrains

Dear Mohammed:

The terms “eigenstrain” and “eigenvalue” are not related. As far as I know “eigen” has a German origin. Eigenstrain refers to any strain that is not caused by stress. Perhaps the easiest example would be eigenstrains due to temperature changes. A change in temperature results in strain that may or may not induce stresses. For example, if you change the temperature of a homogeneous body uniformly and if the boundaries are free to move there won’t be any stresses. An eigenstrain distribution can have many difference sources, e.g. temperature changes, bulk growth, plasticity, etc.

Regards,

Arash

## Paper of interest

Dear Dr. Yavari,

I was very interested to read this paper. Will try to cite it in my current work which is somehow related. The following paper might be of interest for you.

http://www.sciencedirect.com/science/article/pii/S0020768313005155

Regards,

Tais

## Re: Paper of interest

Dear Tais:

Thanks very much for sending the link to your paper. It looks very interesting and I’ll read it with great interest in the next few days. I am familiar with Prof. Zubov’s work on nonlinear mechanics of defects and have cited his book in the past.

Regards,

Arash

## Remarks

Hello Dr Yavari,

What is different between classical mechanical strains generating stresses and those which you have defined?

in my comment you must read "eigenvectors" instead of "eigenvalues" sorry for this misplaced word.

in you paper is the word incompressible the meaning of only a tensile case or is it an undeformable case meaning a rigid body without deformation.

## Re: Remarks

Dear Mohammed:

What you call “mechanical strain” is sometimes referred to as “elastic strain”. A bar under uniaxial tension elongates. Strain corresponding to this deformation is an elastic strain assuming that after unloading the bar goes back to its original configuration. A solid under external loads deforms. However, not any deformation is a result of mechanical loads. As I said earlier, the easiest example is deformation due to temperature changes. Any measure of strain is defined locally and so is eigenstrain. Consider a stress-free solid. Suppose some eigenstrain distribution occurs, for example, due to bulk growth or temperature changes. Imagine that you partition this body into a large number of small pieces. Pick one such small piece and isolate it from the rest of the body. Obviously, this piece being part of a larger body experiences boundary tractions on its boundary. In other words on its boundary there are forces coming from the rest of the body. Let this small piece relax, i.e. remove the boundary tractions. If there are no eigenstrains present in the body any such small piece will go back to its initial configuration in the stress-free body. In general, this small piece will have a relaxed configuration different from its original shape and this difference is due to eigenstrains.

I believe the term “eigenstrain” was introduced by T. Mura. Analysis of eigenstrains in solids has overwhelmingly been restricted to linear elasticity. Here, I’ve been looking at finite shear eigenstrains (in collaboration with my friend Alain) in the framework of nonlinear elasticity.

Incompressibility does not mean a rigid body without deformation. Incompressibility is simply an internal constraint. An incompressible solid can deform but without volume changes anywhere.

Regards,

Arash