As an engineer initially studied concrete structures, I was confused for a long time as to why in geotechnics a ‘deviatoric stress’ is a shear stress, and vice versa. It was after many years into my professional career that I finally gained some understanding on this, and I am sharing it in this blog. No equation, no matrix, just common sense.
Firstly, to set the scene, deviatoric stress is the difference between the major principal stress and the minor principal stress. Principal stresses are the normal stresses of an element at the only angle you can find where there is zero shear stress developed. Look up on Google if you are not sure.
Shear is the sliding movement between two adjoining parts of a material. This post explains shear in soil.
Below provides a few angles of perspectives to the explanation of this issue. They are seemingly separate but actually very much related. Hopefully by the end of this blog I have sufficiently explained why deviatoric stress and shear stress are just two different angles of looking at the same thing.
Fluids vs. solids
Fluids (e.g. water) has no shear capacity. Its particles can move freely relative to each other with almost zero friction. The triaxial stress condition for any fluid under any loading conditions should be isotropic (or hydrostatic), i.e. same stress in all directions. If you pour fluids into any container or surface, it will self-level. Therefore, the tri-axial stress condition of a fluid is always hydrostatic.
It is only when a material’s mechanical properties move from fluids to solids, it starts to gain shear capacity by generating friction between its particles, and it is only then the material starts to gain the tendency to stay in shape when non-uniformly loaded, and be able to sustain deviatoric stress (non-hydrostatic stress). If you pour solids onto the ground, it will not self-level, but aggregate into a slope. The more ‘frictional’ the material is, the steeper the slope is, up to 45 degrees maximum. When there is bonding between the particles of a solid, upon reaching certain level, the material can form even steeper slope, and finally reach a state where it can ‘self-stand’ with a vertical cliff.
Here we can see shear capacity and deviatoric stress go hand-in-hand..
The triaxial test
The triaxial test is probably the best demonstration of deviatoric stress condition. You may notice the typical failure plain of a triaxial test sample is like below:
Essentially, the sample fails due to the relative sliding of its two adjoining pieces. This is a typical phenomenon of shear failure, for materials based on frictions rather than bonding, with soil being an excellent example. When loaded, the sample will automatically find its lowest possible angle of slope as its failure plane, as it takes the least effort to break at this angle, because there is a smallest area of interface friction to be overcome.
By breaking down the stresses using trigonometry, the diagram below shows how a ‘failure wedge’ can be found for a given material, when the deviatoric stress is sufficiently large. It shows how deviatoric stress generates shear stress across a failure plane that the sample automatically finds by itself. The angle of slope of the ‘failure wedge’ is steeper for a material with more friction between its particles.
What is shear really for a concrete beam
Having understood the above, let’s now look at shear in a concrete beam from a whole new angle. For a uniformly loaded simply supported concrete beam, the ‘stress flow’, which means the directions of its principal stresses, is like below. Green or broken line means compression and red or solid line means tension. At its very core, the so-called ‘shear’ failure of a beam is essentially a failure of deviatoric stress conditions, with one direction in tension and the other in compression. Concrete has bonding capacity and reinforced with steel, so unlike soil it does not solely rely on friction and can sustain this kind of tension-dominated deviatoric stress to a large extent.
