# How to find out the strain distribution of a block in a simple compression test?

I am performing a displacement-controlled compression test on a square block, where the top of the block is attached to a fixed plate, and the bottom of the block is attached to a movable plate that moves up.

1. How will the strain distribution look like in a block (is there any analytical expression for it)? Will it be uniform throughout, or will it be max/min at the center?
2. Since it is displacement-controlled, will the strains be same irrespective of the material (e.g., steel vs rubber)? P.S. I know the stress-strain curve will look different - I want to know if the the strain distribution would look same for different materials at a particular displacement.

## 1 Answer

Strain is displacement normalized by the original distance, so yes, the same displacement applied uniformly to the same original dimension will produce the same strain for that axis for any material.

The qualifiers are important: all bets are off if the sample buckles or otherwise fails, for example, and different materials have different Poisson ratios, meaning that they strain to different degrees laterally in response to a given strain along a certain axis.

There’s a simple answer and a complex answer to the strain state you describe. If the platens are frictionless—meaning that the ends of the sample are free to expand—then a single uniaxial strain exists throughout the sample upon compression. Very straightforward. But this usually isn’t the case; the platens usually constrain the ends through friction, and so the sample swells from Poisson effects at the middle but not the ends:

This creates a complex stress and strain state that would probably have to be analyzed numerically, through finite element analysis. You could still use a single uniform strain value as an approximation, though.

• Very good point about the friction at the platens. In fact, my ultimate goal is to model the compression test on an elastomer using FEM and validate it experimentally using DIC. I am interested in how the strain varies along the Z (vertical) direction. Assuming the platens to be frictionless (I can achieve this by lubricant it), will the strains be uniform throughout? My FEM simulation shows a minimum at the center and maximum at the platens, but I'm not sure if this is correct. Even if it is, I'm unable to get a physical/mathematical intuition for it. Commented Jun 24, 2023 at 4:18
• How much did the FEM values differ? What were the exact conditions of the simulation? If you apply a uniaxial displacement to a free object, you'll get a single uniform strain, so there must be additional unspecified details about the results being described. Commented Jun 25, 2023 at 19:00