what is strain
Pull a bar and it lengthens a little. Call that added length ΔL. But if you only say it grew by 1mm, is that a lot? On a 10cm ruler it is enormous, and on a 100m bridge it is almost nothing. The added length means something only when you set it against the original length. That ratio, ΔL divided by L, is the strain. You throw the length away and keep only how much it grew compared with its own length. Strain therefore has no unit, and it is usually very small. A sliding deformation is measured by an angle instead of a length ratio. Pair this strain with stress and the formulas of the next lessons open.
Look first at the added length itself. Move P and pull the bar, and the bar grows past its original end. The dashed line is the original length L, and what sticks out beyond it is ΔL, the elongation. A larger pull makes ΔL larger too. For now, notice that this ΔL is an absolute length. It is a real distance you can measure in millimeters. This one number cannot tell you whether the bar stretched hard or only a little. The standard to compare against is missing. Without a standard, the size cannot be named.
The missing standard is the original length. Divide ΔL by L and you get how much it grew compared with its own length, and that is the strain ε = ΔLL. Move ΔL and ε changes together as a fraction, as a percent, and as microstrain. They are all the same number. Length divided by length has no unit, which is to say it is dimensionless. Strain therefore says fairly how much a material grew, regardless of the size of the ruler or the bar. Whether it is 1m long or 1km long, the same ε means it grew by the same ratio. The ratio is what makes the comparison fair.
Pin the difference down with two bars. Give the short bar A and the longer bar B the same strain. Move that value and both grow by the same ratio, but the added length ΔL differs. The long B grows more. Because ΔL = ε · L, a longer bar has a larger absolute growth even at the same ratio. Looking only at ΔL fools you. Whether a bridge that grew 30cm or a bolt that grew 3mm is more dangerous depends on the length. Strain removes that trap. How close a material is to its limit is told by the strain, not by ΔL. The ratio is what takes the trap away.
Stretching is not the only deformation. There is also the shear seen in A1, a sliding along the face. You cannot measure that as a length ratio. It is not a lengthening. It is a skewing. Shear strain is therefore measured as an angle. Move γ and push a square element, and the right angle that was square skews away from 90 degrees, and that skewed angle is the shear strain γ. The unit is usually the radian. If normal strain is a ratio of lengths, shear strain is how far a right angle has collapsed. Both are deformation, one a change of length and one a change of angle. The change of angle is the size of the shear, and that is the whole of this picture.
Last, a sense of size. Real strain is surprisingly small. While steel holds in the elastic range, ε is about 0.001, which is about 0.1%. A 1m bar grows by about 1mm. It is so small that the naked eye hardly sees the growth, and you have to raise the magnification M before it shows. That the deformation is tiny is actually a gift. Because it is small you may calculate with the original dimensions, and the awkward terms drop away so the formulas all become linear. The formulas of mechanics of materials are clean proportions because the deformation we handle is this small. The smallness is why the formulas stay linear, and why a magnification is needed before the eye can see the growth.