shear stress and shear strain
So far the stress has been normal stress, pulling or pressing straight on a face. What happens when the force runs parallel to the face, along it? It does not pull. It slides. Scissors cutting paper, and a bolt snapping between two plates, are both this. The stress then is the shear stress, and the skew that follows is the shear strain. The surprise is that the tension story repeats once more, as it stands. Put shear stress where stress was, shear strain where strain was, and put G where E was. τ = G·γ, and the shape is the same. A sibling that differs only in name and direction.
First, get shear in the hand. Think of a stack of cards or a thick book. Push the top sideways and it does not lengthen. Each sheet slides a little against the one below, so the tidy stack becomes a slanted parallelogram. Push the top in the picture and the layers step aside and the whole stack leans. That is the nature of shear. A force parallel to a face slides the material along that face. If tension pulls the two ends apart, shear pushes the layers sideways until they sit askew. The step between layers is shear.
Now turn that shear force into a stress. It is the same move as in A1. Divide the force V that pushes along the face by the area A on which it acts, and that is the shear stress. τ = VA. When one pin is pulled up and down out of line, it tries to cut along the middle face. That face is the shear plane, and the stress there is the shear stress. Move V and the shear plane heats toward a deeper color. The formula is the same as the normal stress σ = PA. Only one thing differs, whether the force is perpendicular to the face or parallel to it. Perpendicular is normal stress, parallel is shear stress, and the unit of both is MPa.
When shear stress acts, the material skews. The shear strain seen in A2 arrives properly here. Give a square element a shear and the tidy right angle collapses, so the element leans into a parallelogram. That collapsed angle is the shear strain. If normal strain was a ratio of lengths, shear strain is a change of angle. It is a deformation you cannot measure as a length, so you measure it as an angle. A larger stress means a larger angle. Stress and deformation are both here, so the next step is to join them. Read it as an angle or the size of the skew stays hidden. Length alone cannot show this deformation.
Hooke's law, which in tension was σ = E·ε, holds for shear in the same way. τ = G·γ. Move the shear strain and, in the elastic range, the shear stress is proportional to it, and the slope of that line is the shear modulus G. G is how hard the material resists sliding, the shear stiffness. If E was the stiffness against lengthening, G is the stiffness against skewing. They are not strangers. Inside one material, E, G, and the Poisson ratio of the next lesson are tied by one formula. Usually G is about 40% of E. For steel, E is about 200 GPa and G is about 80 GPa. Two stiffnesses of one material.
The place shear shows up most often is a bolt or a pin. Join two plates with a bolt and pull from both sides, and the bolt tries to cut clean along the face where the plates meet. That is shear. Compare single shear and double shear with the buttons. Single shear has one face being cut, so τ = VA acts as it stands. Double shear wraps the middle plate from both sides, so two faces are cut. The same force V is then shared by two faces, τ = V(2A), and the stress drops by half. Important joints are therefore designed as double shear on purpose. When you choose a pin or a rivet, the central check is whether the shear stress stays within the shear strength the material allows. The allowed shear strength is the limit.