## Introduction to Simple Shear

In this article I will analyze a cube that is deformed in simple shear, as shown in the figure above. The cube has a side length of 10 mm, and its top surface is sheared to the right a distance of 10 mm.

**What do you think the shear strain is in the block after it has been deformed?**

## Finite Element Model

To determine the shear strain in the block I will run a large-strain finite element simulation. The following boundary conditions are applied:

- Back side: U3=0
- Bottom side: U1=U2=0
- Top side: U2=0, U1=linearly increasing displacement to 10 mm.

The FE simulation is performed with Abaqus using one C3D8H element.

After the simulation has been run, Abaqus reports that the nominal shear strain is not 1.0, but instead 0.894.

Abaqus also reports that the true (logarithmic) shear strain is 0.8608.

To better understand the strains, I also plotted the shear angle gamma and the two strain quantities provided by Abaqus, as a function of gamma.

Clearly all 3 quantities (gamma, NE12, LE12) are the same for very small strains, **but they are not the same for finite strains!**

At finite strain you should use nominal (engineering) or true strain, not the gamma quantity. In this case, if we plug in gamma=1 in the equation in the figure we get NE12=0.894. This is indeed the value that Abaqus reports.

The following Julia code shows how you can calculate the true shear strain from the applied deformation gradient:

` ````
```julia> F = [1 1 0; 0 1 0; 0 0 1]
3×3 Array{Int64,2}:
1 1 0
0 1 0
0 0 1
julia> strain = log(sqrt(F * transpose(F)))
3×3 Array{Float64,2}:
0.215204 0.430409 0.0
0.430409 -0.215204 0.0
0.0 0.0 0.0
julia> strain12 = 2 * strain[1,2]
0.8608178819280081

## Out of Plane Deformations

If we study the Abaqus results a bit more we see that there is an out-of-plane strain. This out of plane strain occurs simply because we allow it to. If we don’t want any out-of-plane deformation then we need to set U3=0 on the front surface.

**In other words, there are two different ways to apply a simple shear deformation: (1) free deformation in the out-of plane direction, or (2) no out-of-plane deformation.**

To correctly analyze simple shear data one therefore needs to know how the simple shear experiments were performed. My preferred experimental method is to use double lap shear tests. In these tests the specimen is constrained from moving the out-of-plane direction.

MCalibration allows you to specify if your simple shear test was performed using free out-of-plane deformations (loading mode “Simple Shear”), or if the the out-of-plane deformations were fixed (loading mode “Simple Shear Fixed”).

## Edge Conditions

One other serious complication that occurs in simple shear is that the stress and strain fields are not homogeneous. The contours of Mises stress for a multi-element FE model show that the max Mises stress is 10X larger than the min Mises stress. That is not so good!

**One way to reduce the influence of the edge conditions is to make the shear specimen wider than it is tall.** In this case the stress and strain fields become more homogeneous. As is shown in the following images, using a shear specimen that is 5X wider than its height reduces the error in the experimentally determined stress from 17% to 3%. That is a significant improvment.

## Summary

Simple shear testing is a convenient way to test a material in a loading mode that is not uniaxial. There are a 5 common problems that can occur when analyzing simple shear test data:

- The engineering shear strain is only approximately equal to gamma when gamma is much less than 1.
- There are 2 types of simple shear in MCalibration: “Simple Shear” and “Simple Shear Fixed”. Most of the time you should use “Simple Shear Fixed”.
- Do not import shear strain as gamma into MCalibration. Always convert it to either engineering or true strain.
- A simple shear experiment is not “clean”. The edges of the test specimen will not be deformed in simple shear. To get more accurate shear results use shear specimens that are 5X wider than their height.
- If you cannot use wide shear specimens then you may need to use an inverse material model calibration approach. That is easy to do using MCalibration.