# Using Strain Tensors to Recover Stress Tensors

### Disclaimer:

I am very new to the field of mechanics, my background is in electrical engineering. I apologize in advance if anything is horribly wrong.
My references are:
"First Course in FEM, 4th Edition" by Daryl L Logan
"Theory of Elasticity, 3rd Edition" by S. Timoshenko and J.N. Goodier

### Problem Statement

Let's say I have some three dimensional body which undergoes a deformation due to an internal stress profile. Call the undeformed state $$X$$ and the deformed state $$X'$$.

Assume that the undeformed state $$X$$ has some internal stress profile and this stress profile is not in equilibrium. Call this initial stress profile $$S$$. As time proceeds, the body relaxes until equilibrium is achieved. The deformed body in state $$X'$$ ends up with some resulting stress profile $$S'$$. A stress profile is constituted by a stress tensor at every point within the body.

Let's say that I have displacement equations describing the deformation. Concretely, I have three equations $$x' = d_1(x, y, z)$$, $$y' = d_2(x, y, z)$$, and $$z' = d_3(x, y, z)$$. These fully describe how a point $$(x, y, z)$$ moves to $$(x', y', z')$$ under the deformation.

My primary goal is to use the displacement functions to estimate the stress tensor at every point in $$X$$ which resulted in the observed deformation.

Here is my current approach:

1. Assuming small strain, we can relate the displacement equations to the entries in the strain tensors at every point. In particular, we know that $$\epsilon_x (x, y, z) = \frac{\partial d_1}{\partial x} (x, y, z)$$, $$\epsilon_y (x, y, z) = \frac{\partial d_2}{\partial y} (x, y, z)$$, etc.

2. Assuming linear elastic behavior, the stress and strain tensors at any point are linearly related. Thus we can say that $$\sigma = D \epsilon$$ for some matrix $$D$$. We can then use the strain tensors at every point, generated via the technique in step 1, to compute the stress tensors at every point which resulted in the observed deformation.

### Questions

I'm not sure that this technique will produce a reasonable estimate of the stress tensors which caused the deformation. Step 2 relies on linear elasticity, but the deformation from $$X$$ to $$X'$$ is plastic. Do people think that this approach has any hope of estimating the stress profile of $$X$$? Might it only produce reasonable results if the deformation is small?

If we knew the strain tensor, $$\epsilon$$, at every point in a body which underwent a deformation, is there anything wrong with using $$\sigma = D \epsilon$$ to recover the stress tensor, $$\sigma$$, which caused the observed deformation?

Are there existing techniques to recover the stress state which resulted in an observed deformation? I've looked around and haven't found anything useful.

Edits: Cleaned up the assumptions necessary for the 2 step process.

• "but the deformation from X to X′ is clearly plastic" Why? What's the actual problem you're trying to solve? May 10, 2023 at 23:03
• The question I am trying to answer is: Given a deformation of a body which occurred due to its internal stress state, can we measure the deformation and estimate the stress state which caused the deformation? I assume that the deformation from X to X' is plastic because the deformation is permanent. Maybe my use of the term 'plastic' is incorrect.
– yl0l
May 11, 2023 at 0:05
• The term is correct; I want to know what motivates the assumption that the deformation is permanent. Are you just trying to consider the most general case, comprising both elastic and plastic deformation? May 11, 2023 at 0:30
• Ok, let me give you more context. Assume we have a metal workpiece, with some initial residual stress profile, which gets cut in a CNC machine. After the cut happens, the metal workpiece deforms (because of its residual stress profile). I'm trying to devise a numerical technique which can estimate the initial residual stress profile based on the deformation which occurred. I hope to devise a technique which is as general as possible.
– yl0l
May 11, 2023 at 0:48
• Are you asking about the inverse elasticity problem, i.e., finding the set of loads, stresses and local material properties that produce a given set of displacements? Example; example. May 11, 2023 at 2:10