# How to interpolate stress at unknown points from the stress values available based on geometrical position for constant load?

I am working on a combined contact, bending, and torsion problem. I have data on geometrical points and their instantaneous stress components. However, based on the available data, I have to interpolate the stress values at points where the stress values are not available, i.e., I would like to interpolate the stress values at points where the stresses are not calculated by the computational model. Any inputs on the interpolation functions are highly appreciated.

I did try interpolation function and scattered data function available in MATLAB. I am getting an error as the points are not monotonically increasing and the points are not under a convex hull. I did use mldivide in MATLAB to obtain the fit based on the available data. However the accuracy of the interpolated stress is a question. I am still contemplating whether the use of Airy's stress equation and Prandtl stress equation will be appropriate to this condition. Any inputs on interpolation functions or directions on how to arrive at interpolation function will be highly appreciated.

The mesh is made up of mid noded quadrilateral element. It is 8 noded higher order element. As I mentioned earlier the computational model calculates only instantaneous stress values based on which point is in contact. For other nodes which are not in contact/or will be in contact in future there is no stress value. The idea is to obtain realistic stress histories of all the contact points.

• I have two questions. (1) Do you have a mesh or how are your points placed? (2) How the Airy's and Prandtl's stress functions are related to your problem? Jun 7 at 16:01
• 1A) I have a finite element mesh and to be precise they are 10 noded quadrilateral element. However the point where the two bodies are in contact in current time step / will be in contact in future time step is determined by the contact algorithm. So the point here is, it is not always the neighboring element which will come in contact in the next time step. Jun 8 at 5:55
• 2A) Since rather using a curve fit based on current contact points and the stress values (which I am currently doing), I thought it would be more logical to use Airy's stress function for the stress values and prandtl stress function for the shear values. Let me know if you have any comments on both the questions. Jun 8 at 5:58
• (1) I don't know quadrilaterals of 10 nodes, but that's not a problem. If you already have the stress values on the nodes you could use a visualization software such as ParaView. If you have the values on your Gauss points you could visualize them inside each element or do some averaging yo the nodes. I don't think that what part is in contact is important, though. Jun 8 at 15:09
• (2) Sorry, I don't follow you. Jun 8 at 15:10