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# MODELING OF FGM IN ABAQUS

hello every one,

i am Ravitej, doing my Mtech at IIT kanpur, India. Now, i am working on FGM( funtionally Graded Materails). Can i model the FGM as a layered specimen with the youngs modulus varying according to layers. is there any special elements in abaqus, so that we can assign property variation along the thickness.

Problem that i had faced with the layerd model is

1) stress is not continous at the interface of layers, then how can we believe that the solution obtained is correct

2) what could be the youngs modulus at the interface, will it be average of two adjacent layers

please guide me in this aspect to model the FGM correctly..

cheers

Ravitej

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## Comments

## Hi, Ravitej, You

Hi, Ravitej,

You can try user subroutine UMAT, where you can define material properties according to the coordinates of the integration points. In this case, you don't need to divide FGM region into layers. Hope it helps.

Jianlong Xu

Center for Advanced Composites

University of Florida

## Stress is not continuous.

R. Chennamsetti, R&D Engineers, India

In a layered media like composites (Is FGM also like this?) formed by stacking the layers, stress wouldn't be continuous, but, displacement is continuous.

You may explain the gemoetry of your model and direction of variation of material properties for detailed and clear discussion.

- Ramdas

## Strain is not continuous

For recent experiments, it was found that local, small scale strain field is not uniform/continuous. The following paper may help.

Xiaodong Li, Weijie Xu, Michael A. Sutton and Michael Mello, "In-situ Nanoscale In-plane Deformation Studies of Ultrathin Polymeric Films during Tensile Deformation Using Atomic Force Microscopy and Digital Image Correlation Techniques, " IEEE Transactions on Nanotechnology, 6 (2007) 4-12.

## Stress is discontinuous => Strain also...

R. Chennamsetti, Scientist, R&D Engineers, India

When stress is discontinuous, then, obviously strain is also discontinuous. I am referring this from MACROSCOPIC point of view. I don't in nanoscales...

## Stress is discontinuous => Strain also...

I have never been aware of this concept, could you please explain why discontinuity in stress automatically means the same situation for strain? Thank you.

## discontinuity of stress -> discontinuity of strain

For simplicity, consider 1D specimen. Plot stress. A C0 discontinuity in the stress field would mean a sudden vertical jump or an approach (at least a one sided approach) to infinitely large stress value at some point somewhere within the domain.

First, consider the case of the sudden vertical jump of a finite difference in stress levels. To have C0 continuity in the corresponding strain field, you would have to have a precisely compensating sudden jump in the strain-versus-stress [sic] curve. If this condition is not satisfied, a discontinuity of stress would necessarily imply a discontinuity in strain.

Notice that the above conclusion would hold regardless of whether the strain-versus-stress curve was linear or not (and even, for that matter, whether it was, overall, unique-valued or not. Unique-valued-ness also does not matter. What matters is the local behavior of the strain-stress curve---it ought to remain *continuous* in the neighbourhood of those stress values near which there is an asserted discontinuity in the specimen.)

Similarly, as far as the case of the infinitely large values of stresses go, the strain-vs-stress curve would have to carry a compensating approach to infinity.

It so happens that the actual materials show far simpler forms of constitutive relations---they do not possess compensating jumps of either kind. Hence, the conclusion that discontinuity in one leads to another. (Observe that the compensating jumps would have to occur at different levels of strains or stresses to suit the specifics of a given concrete problem. This, by itself, ought to indicate the conceptual unsoundness of the idea that there can be discontinuity in one but not in the other.)

The reasoning remains valid even if you take a matrix of 81 functions (not just constants) as representing the various stress-strain relations.

So, IMO, Ramdas' latter correction (appearing below) seems to be uncalled for.

-----

Of course, the above reasoning does not apply to composites. For example, you could take two cylinders of two different materials and different cross sectional areas, join them axially, and load them, say, in simple compression. Thus, you could artificially arrange to have a jump in stresses but not in strains at the section where the two different materials are joined. But note, this is a rather contrived example, and it does not really represent the case of the FGMs very well.

The reasoning above does apply to FGMs if the variation in the material properties is assumed to be smooth.

## From Hooke's law

This is using Hooke's law. Take a simple 1D problem...

Stress is proportional to stress (within proportional limit).

Stress = E * strain.

The variation of strain is also same as stress. 'E' (Modulus of elasticity) is a scaling factor.

## what if you have different E's?

Unless we are assuming a certain material behavior (such as uniformity for example), why should a discontinuity/continuity in either stress or strain necessarily mean the same in the other one? By appropriately varying material coefficients, can't we have a continuous strain out of a discontinous stress field? This may not be a practical thing to do if the discontinuity occurs at several points, but it is still theoretically possible.

## Different E's wouldn't be enough

It wouldn't be enough to have different values of E's---the constitutive law itself would also have to carry a precisely compensating jump in itself.... Please see my recent reply above.

## Right....

Yes, we can have. When I said in one of my posts

"When stress is discontinuous, then, obviously strain is also discontinuous"I was referring only "Stress = E * strain"But, as you pointed out we can have one field (stress/strain) continuous other one is dicontinuous (strain/stress) by adding appropriate coefficients/constants.

In my first post (about composite material, I assumed linearly elastic material),

the stress/strain are discontinuous across the laminate thickness.## If you have a laminate,

If you have a laminate, wouldn't the stress be constant? Think of springs in series. The strains would be discontinous, becasue of discontinuity in the Es.

## Plies in parallel in a laminate...

R. Chennamsetti, Scientist, R&D Engineers, India

Here, you are stacking laminae (this is like parallel springs), which have different orientations, to get a laminate. When you apply some load, say in bendig, each ply (lamna) experiences different stress.

In a simple isotropic beam beding, we get a continuous stress distribution across the thickness. But, in a composite laminates, the stress will be discontinuous at the ply interfaces. You may refer any book on Mechanics of composites or the following.

R. F. Gibbson, Principles of Composite Material Mechanics, McGraw Hill publications (1994), Page: 195.

## Continuation to above post....

R. Chennamsetti, Scientist, R&D Engineers, India

In continution to my above post and Markos point on continuity/discontinuity I forgot to mention the following....

In a composite laminate, plies of various orientations stacked one above the other, stress variation across the thickness is dicontinuous, but, the strain is continuous.

## For laminated structures,

For laminated structures, the transverse normal and transverse shear stress components must be continuous across the layer surface to satisfy equilibrium. Only the in-plane stess components are not continuous. The displacements are continuous. Strains, let it be transverse or in-plane, they could be not contiuous.

Stress continuity implies strain continuity if all the stress components are continuous.and the material constants are continous functions of position.

## example of modeling of FGM in ABAQUS

i am yahya bayat, . Now, i am working on FGM( funtionally Graded Materails).

i weanted a example of modeling a sphere of FGM IN ABAQUS.

please guide me in this aspect to model the FGM correctly..

## modeling of fgm in ABAQUS

Hi everybody,

I'm looking for some resources (textbooks,webpages,papers) to give me some insights into modeling of FGM in ABAQUS ,stress analysis of functionally graded materials. i appreciate any suggestions.

Thanks

## FGM Refernece

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA434403&Location=U2&doc=GetTRDoc.pdf