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# Does a single crystal have a net rotation after you load and then unload it ?

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To detail out,

My understanding of multiplicative decomposition of deformation gradient F =FeFp (see the attached figure) is

Fp indicates the shape change without inducing any stress

Fe leads to elastic stretching and rotation

Say, If I load a single crystal and unload it, we will have a residual plastic strain present. As, we removed the load, the effect of Fe would have been gone. So all the elastic effects vanish.

Although, I understand that mathematically we can do a polar decomposition of Fp=RU, which has a rotation R.

According to the very definition of Fp which just indicates just the slip preserving the slip direction and slip normal, Would there be any rotation after we unload ?

Then what does R in the polar decomposition refer to ?

If possible, kindly make me understand with figures !

Thanks,

Prithivi

## Re: decomposition of deformation

Hey Prithivi

It is a very interesting question. We had similar confusion and please find my response below

Say, If I load a single crystal and unload it, we will have a residual plastic strain present. As, we removed the load, the effect of Fe would have been gone. So all the elastic effects vanish.

This is untrue that elastic deformation will be gone after removing load as it will exist as forms of elastic stress (residual stress), lattice rotation (GND density) within your crystalsAccording to the very definition of Fp which just indicates just the slip preserving the slip direction and slip normal, Would there be any rotation after we unload ?

The plastic rotation (continum rotation) differs from lattice rotation which assoicated with shape change and will not be gone after unloading.Then what does R in the polar decomposition refer to ?

Please see the previous comment.We have carried out a comprehensive study using HR-DIC and HR-EBSD to study the deformation compatibility in single crystal Ni. Hope you will find it useful.http://rspa.royalsocietypublishing.org/content/472/2185/20150690.abstractJun

## Few more clarifications !

Thanks a lot Jun for taking time to reply and your views were very useful for my understanding. I closely follow the works of Prof Dunne and Prof Britton, I have read that paper right when it got published !

I have few more clarifications, May I request your views on the following ?

I shall expalin what I understood -

1) As we load the crystal till some point beyond the yield

(a) "elastic " rotation of the crystal occurs, which changes the euler angles of the grains . From a modeling stand point, this is obtained from the polar decomposition of Fe ( = ReUe)

(b) As you mentioned there will be a continuum rotation, which I understand as the shearing angle . This is obtained from Fp = RpUp as per your point.

-- In general, my understanding is that the rotation matrix obtained from the polar decomposition of a deformation gradient (F=UR) rotates every vector (after stretch)

--So, In context of Fp =RpUp, If both slip normal and slip direction rotate , this would contradict the defition of isoclinic intermediate configuration where in there is no rotation of any vector from reference to intermediate

(c) There will be dislocations generated during loading and there will be internal stress fields due to that. But with respect to crystal plasticity modeling, I dont think the internal stresses due to dislocations are captured - Your views ?

2) As we unload the crystal

(a) From a modeling standpoint , all the Re would be reversed !

(b) There would probably be continuum rotation left

(c) There would be some annihilation of dislocations but still there would be RESIDUAL dislocations.

-- There will be ofcourse internal elastic stresses ( physically)

-- You have mentioned that due to GND, there would be a lattice ROTATION. As per my understanding, there would be a lattice CURVATURE associated with GND but not rotation - Your views ?

Thanks,

Prithivi

I have few more clarifications, May I request your views on the following ?

Below is my understanding of the deformation mechanisms based on our EBSD, DIC, HR-EBSD and CPFE work so far, please feel free to disagree with my view.

I shall expalin what I understood -

1) As we load the crystal till some point beyond the yield

(a) "elastic " rotation of the crystal occurs, which changes the euler angles of the grains . From a modeling stand point, this is obtained from the polar decomposition of Fe ( = ReUe)

Correct.

(b) As you mentioned there will be a continuum rotation, which I understand as the shearing angle . This is obtained from Fp = RpUp as per your point.

Correct

-- In general, my understanding is that the rotation matrix obtained from the polar decomposition of a deformation gradient (F=UR) rotates every vector (after stretch)

--So, In context of Fp = RpUp, If both slip normal and slip direction rotate , this would contradict the defition of isoclinic intermediate configuration where in there is no rotation of any vector from reference to intermediate

The shear angle does not contribute to the crystal orientation rotation such that no slip normal and slip direction change in the plastic part of the decomposition.

(c) There will be dislocations generated during loading and there will be internal stress fields due to that. But with respect to crystal plasticity modeling, I dont think the internal stresses due to dislocations are captured - Your views ?

Continuum-based CPFE are not expecting to capture the dislocations and its associated stress field. If you would like to explore it, discrete dislocation dynamics would be the tool to use. However inhomogeneous plastic strain resulted residual stress should be captured by CPFE.

2) As we unload the crystal

(a) From a modeling standpoint , all the Re would be reversed !

Not if you incorporate geometrically necessary dislocation density which is the physical reason for lattice rotation (Dunne and Cocks), although the initial motivation is to capture the size effect e.g. strain gradients. (Ashby’s paper in 1956 Phil. Mag.)

(b) There would probably be continuum rotation left

Yes

(c) There would be some annihilation of dislocations but still there would be RESIDUAL dislocations.

Yes, some dislocations are stored in crystals after unloading and responsible for dislocation forest hardening.

-- There will be of course internal elastic stresses (physically)

No necessarily as in an unconstrained case e.g. single crystal uniaxial tensile test. Just image it is a stack of cards. After cards slipped, there will not residual stress needed to support the final shape.

-- You have mentioned that due to GND, there would be a lattice ROTATION. As per my understanding, there would be a lattice CURVATURE associated with GND but not rotation - Your views ?

You are right and such lattice curvature results in crystal lattice rotation. Curvature effectively cause lattice to rotate. Curvature occurs in 3D rather than 2D.

Thanks,

Hope these help with your current research. Of course there are other world leading research group with impressive work to explain such deformation

## Re: F=F^eF^p

Just a couple of comments:

Unloading a single crystal (locally) via Fe^-1 does not imply that 'all lattice rotations vanish', i.e. R^e = I. In the case of a heterogeneously deformed single crystal, as Jun has pointed out, there will be GNDs, residual stresses, and lattice reorientation. However, even in the case of a uniformly deformed and unloaded single crystal, there will generally be lattice reorientation even in the absence of residual stresses. In this case, U^e = I but R^e /= I. There will be zero lattice reorientation only when there is symmetric slip system activation.

## Thanks !

Thanks for your comments Jason !

## Lattice curvature and Lattice rotation

Thanks Jun, that was really useful!

Could you give me references to get a good understanding of lattice curvature and lattice rotation ?

## Sorry Permalink, I am not

Sorry Permalink, I am not aware of any paper/book chapter explaining this part. You can take the idea that lattice curvature (GND resulted) causes lattice rotation. GNDs are generated due to plastic strain graident (e.g. heterogenous deformation as Jason mentioned).