# Can "extremal" strain tensors be in the interior of the body

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Chevallier
I am new to elastic theory. I have a question about elasticity. We assume we have a body with no internal forces. Surface forces are applied on the border. Can we leave the elastic domain (reach the yield surface) in an interior point without leaving the elastic domain on the boundary?
If no, is there a reference in the literature which shows that from the equations?

Thank you.

Mentor
I have no idea what you are asking. Are you asking if the strain tensor is continuous across a boundary between two materials

Chevallier
Thanks for asking. I'll try to be more precise:
-Let Omega be a homogeneous and isotropic body.
-This body occupies a part of the space R^3, its border is a surface dOmega.
-We apply forces on dOmega. We don't consider other forces.
At a point in the body when local forces are small, the local deformation is elastic. When the forces get too big, the local deformation leaves the elastic domain.

The question is:
Can we find external forces such that we leave the elastic domain at a point in the interior of the body, without leaving the elastic domain on any point of the bordrer dOmega?

Mentor
I still don't follow what you are asking, but here's my interpretation:

Is it possible to impose a load distribution on the surface of an object such that the elastic limit is exceeded on all parts of the surface while, in the interior of the object, there is a region in which the elastic limit is not exceeded?

Does that come anywhere close to what you are asking?

Gold Member
Possibly question is : Can there be a situation where externally applied forces create stresses below yield in the surface layers of an object but above yield in the interior of the object .

Chevallier
Chestermiller, it's almost that, just the other way around, as Nidum said. Is it possible to exceed the elastic limit at an interior point, while everywhere on the surface the elastic limite is not exeeded?

Gold Member
Chevallier
I would be a bit surprised if it was possible. If it's not, i am interested in how to show that from the equations.

Mentor
I don't think it's possible, but I don't know how to prove it mathematically. What is the motivation for asking this question?

Chevallier
i would like to optimize over elastic deformation of a body (finding the best deformation in some sens), under the constraint that we never reach the elastic limit. In the optimization procedure, i am wondering if i should go over all points of the body and check if the constraint is verified, or if it is sufficient to check on the surface. This would reduce the computational complexity of the optimization.

Gold Member
In the optimization procedure, i am wondering if i should go over all points of the body and check if the constraint is verified, or if it is sufficient to check on the surface. This would reduce the computational complexity of the optimization.

Just checking the surface stress levels could be very unreliable when applied to complex shaped objects and especially to objects with holes in them .

Think about the simple example of a rectangular plate with a hole in it . With suitable loading this plate could have stresses below the elastic limit generally but above the elastic limit around the hole . Do you think that your proposed method would work in this case ?

You can always test any optimisation procedure using analytic methods on objects with simple geometry and FEA on objects of more complexity .