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# Stress Update at every Gauss Point ???

## Stress Update at every Gauss Point ???

(OP)
Please could someone assist me with the following :

(For this, assume the Von Mises failure criterion is used. Also assume that the material is linear elastic with zero hardening (perfect plasticity) and isotropic. Assume the problem is 2D, with no temperature effects. The externally applied load is applied in load increments. )

Its with regard to the stress update (return mapping of elastic trial stresses to the yield surface) at every gauss point within every element. (This is how numerous books say it...)

My understanding is that you first carry out an elastic analysis to determine the overall displacements. Thereafter you compute the strains and elastic trial stresses. This gives you a cauchy tensor ( or voigt vector ) of stress components for the entire element. You dont get a cauchy tensor for each gauss point.

Thereafter you compute the yield criterion to determine if the element response is elastic or plastic.
If the response is plastic a stress update procedure(J2 radial return method) can be used to return the stresses to the yield surface.

I understand that gaussian quadrature is used to approximate the exact solution of the components in the elemental stiffness matrix as well as the components of the elemental internal force vector.

So with the above said, how is it possible that every gauss point is assigned its own cauchy tensor? When the cauchy tensor is applicable to the entire element.

Or should we split the updated cauchy tensor according to the weights of the Gaussian quadrature procedure - and assign these tensors to the gauss points?

### RE: Stress Update at every Gauss Point ???

Why do you only have one Cauchy stress tensor for the whole element? I would expect to do it at each Gauss point. It will be different across the element, so you'd lose information by summarizing it with a single tensor.

In case you do need a single tensor for the whole element. Then perhaps use the mean? That is, the integral of the Gauss point values divided by the element volume.

Weighting values by the Gauss weights is part of the Gaussian integration procedure. So if you're integrating the Cauchy stress tensor, then yes, you'd need to weight the values at the Gaussian points. If you're doing something else with it, like comparing against the yield criterion, then no, I don't think it's correct to weight the values.

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