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# Characterization of residual stress fields in nonlinear elasticity; a question posed by Sebastien Turcaud

In the post

node/9509

Sebastien Turcaud asks the question (my interpretation) of the characterization of all possible residual elastic distortion fields on a given configuration (interpreted as the current configuration). If one in addition introduces a reference configuration then the deformation gradient w.r.t. this reference is known and depending upon how one defines 'eigendeformation' in nonlinear elasticity, corresponding eigendformation fields to the residual elastic distortion fields can be determined. Such eigendeformation fields can contain fields arising from plastic deformation, non-uniform thermal expansion etc.

I give a characterization of possible solutions in a large class. I don't have the time to check if I have characterized the entire class of possible residual stress fields (with some appropriate smoothness etc., etc.) but certainly the allowable domains in the characterization need not be simply-connected (i.e. can contain 'through' holes). To check this issue would require a look at Mort Gurtin's 1963 paper on generalized Beltrami stress functions.

A practical (obvious) implication is the inadequacy of DIC measurements in nailing down the residual stress field in a deformed body.

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## Comments

## DIC?

Amit - does DIC mean Digital Image Correlation? -Nachiket

## RE: DIC?

Nachiket - Yes, you are correct. - Amit

## Answer

Thank you for your script Amit.

Tell me if I'm wrong and I'd be pleased to read your comments.

In my understanding, the effect of an eigenstrain distribution applied to a reference configuration R can be decomposed into two parts:

ε* = εi + εn

where ε* is the total eigenstrain (inhomogeneous in the general case),

εi the stress-free impotent eigenstrain (see Reissner, Mura),

εn the deformation-free nilpotent eigenstrain (see Irschik).

Is the case of an homogeneous total eigenstrain, the nilpotent eigenstrain vanishes and the body remains stress-free and its total deformation strain equals the impotent eigenstrain. However, is the general case of an inhomogeneous eigenstrain field (as resulting from inhomogeneities, inclusions, graded properties...) the incompatibility of the eigenstrain field results in an elastic response as illustrated in Eshelby's "cutting and welding exercices".

εtotal = ε* + εel

where εtotal is the total deformation strain,

ε* is the total eigenstrain,

εel is the elastic response of the medium.

In this context, your "elastic distortion fields" based on generalized Beltrami stress correspond to the elastic response to the imposed eigenstrains. For clarity, lets introduce an intermediate virtual configuration E corresponding to the effect of eigenstrains without the elastic response. The deformation gradient between R and E is noted F* as in your script. Then, Fc corresponds to the deformation gradient between E and C. Noting F the deformation gradient between R and C, we have:

F = F*Fc

Thus leading to F* = F (Fc)-1 which differs slightly from your expressions (?).

I looked at Gurtin's paper about generalized Beltrami stresses and it seems that the standard Beltrami representation you used "can at most represent [...] totally self-equilibrated [solutions]". In the case of periphractic regions "there exist self-equilibrated fields [...] which are not totally self-equilibrated". The generalized Beltrami representation adds a biharmonic vector field to the 2-order tensor field:

σ*km = εkip εmjq φij,pq + ∇2(φk,m + φm,k)- φj,jkm

where, σ*km = σ*mk and ∇4φi = 0.

With this precision, it should enable to characterize all residual elastic stress fields in the current configuration. Given the transformation gradient between the reference and current configuration F, a unique eigenstrain field corresponds to a givent residual stress or distortion field. In the trivial case where FC is equal to identity, F = F* and the body remains stress-free under eigenstrain acutation. In the case where the desired or observed transformation characterized by F can not be processed as an eigenstrain due to manufacturing/process limitations (the range of faisible eigenstrain distribution is limited), the nearest faisible eigenstrain distribution is choosen and the residual elastic distortion is directly given by:

Fc = F (F*)-1

## Re: residual stress in nonlinear elasticity

Thanks for bringing your response to my attention.

Answers to your comments:

1) With the standard rules and notation for the action of a second order tensor on a vector, in your notation the expression should be F = Fc F* and not what you write above. Roughly, a vector in the reference configuration gets operated by F* first and then by Fc.

2) If you look at step 3 of my notes, you will see that I do not use only the standard Beltrami representation. The vector field phi above that you copy from Gurtin is like the field v in my notes and I show there why the characterization I give can generally deal with at least a large class of mulitply connected domains. My v field is not required to be biharmonic and because I knew of Gurtin's result and did not have the time to nail down the precise reason why this difference arises, I had mentioned that reference. This is also why I claimed that I had only characterized a large class and not the entire class.

It would be good to understand the precise differences between using Gurtin's representation and using my argument for the problem we are discussing. For one thing, in using Gurtin, phi and psi can be arbitrarily chosen (up to smoothness etc.) with psi biharmonic. In my argument v crucially depends on Xi, but I can do regions with holes. I would have done this study myself, but do not have the time. Maybe you can read Gurtin in detail and my notes and enlighten us all?

3) The question you had asked me in the clarification under http://www.imechanica.org/node/9509 is

"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?"

This is the question I answered in my notes.