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# Inverse eigenstrain analysis based on residual strains in the case of small strain geometric nonlinearity

Hi,

I was wondering if someone knows literature related to "**Inverse eigenstrain analysis** based on residual strains in the case of **small strain geometric nonlinearity**"? (elongated bodies)

In the case of lineralized elasticity I guess one could postulate an eigenstrain distribution as the sum of a finite set of basic eigenstrain distribution and minimize the difference between the predicted and the actual residual strain distribution (retrieved from a synchroton mapping for example).

As done in the paper:

*Alexander M. Korsunsky, Gabriel M. Regino, David Nowell, Variational eigenstrain analysis of residual stresses in a welded plate, International Journal of Solids and Structures, Volume 44, Issue 13, 15 June 2007, Pages 4574-4591, ISSN 0020-7683, DOI: 10.1016/j.ijsolstr.2006.11.037.*

But, I'm afraid this procedure (as well as FFT) is not suited in case of nonlinear geometric effects as one can not add solutions. I'm not even sure if the problem is not ill-posed, as the solution might not be unique. In the context of an isotropic elastic media, can we define an energetically optimized eigenstrain distribution that would result in a residual strain distribution mapping a given objective?

I'd be pleased to read your comments

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## Comments

## Re: inverse eigenstrain analysis/geometric nonlinearity

I have not read the paper you mention. However, I am interested in questions of predicting residual stress in the presence of material and geometric nonlinearity and with my erstwhile student, Saurabh Puri, have complete tools for doing so when the incompatibility of the elastic distortion is given. In the general case, uniqueness is not expected as you suspect.

For your case, could you make what is given and what needs to be found a little more precise.

## Inverse problem

Hi,

thank you Amit for your reply.

Im interested in an elongated isotropic homogeneous elastic media undergoing large transformations (but still small deformation) under distributed eigenstrain (as created by spatially varying expansion coeffecients under a constant temperature field for example) without any other external loads applied. Knowing the initial stress/strain-free geometry and the resulting final shape, thus the residual strain field, Im asking what are the possible eigenstrain distributions and how can we determine them. As Im looking at nonlinear geometric effects (large transformations/rotations), what strategies are there to guess possible eigenstrain distributions?

I was thinking of taking analytical known solutions (as a distribution of ellipsoidal inclusion cf. Eshelby) as a fundamental eigenstrain basis in order to approximate possible solutions.

Thank you for your comments

## solution to residual stress nonlinear elastic inverse problem

Dear Sebastien,

I got a little time last night and wrote out my thoughts. In the way you pose the problem, the solutions are definitely non-unique regardless of the nature of the elastic stress-strain relation. I think I have managed to characterize a large class of residual stress fields in full-blown nonlinear elasticity (not necessarily isotropic with unrestricted material and geometric nonlinearity).

If some more constraints are available to be imposed on the elastic distortion field (e.g. its curl is known), then one can expect uniqueness under certain situations.

Now, I don't know how to attach notes to comments. So I'll open a new post:

http://imechanica.org/node/9523