Strain Is the Ratio of Stretch
Pull a bar and it stretches a little. Call that extra length ΔL. But if it stretched by 1 mm, is that a lot? On a 10 cm ruler it is enormous; on a 100 m bridge it is almost nothing. A stretch only means something compared to the original length. That ratio, ΔL divided by L, is strain. It erases the length and keeps only how much it stretched relative to itself. So strain has no units, and it is usually very small. Sliding deformation is measured by an angle instead of a length ratio. Once you pair this strain with stress, every formula in the lessons ahead opens up.
First, look at the stretch by itself. Drag P to pull the bar, and it grows past its original end. The dashed outline is the original length L, and the part sticking out beyond it is ΔL, the elongation. The harder you pull, the bigger ΔL gets. For now, notice that this ΔL is an absolute length: a real distance you can measure in millimeters. Yet from that one number alone, you cannot tell whether the bar stretched severely or barely at all. The reference to compare against is missing.
That missing reference is exactly the original length. Divide ΔL by L and you get how much it stretched relative to itself. This is strain, ε = ΔLL. Drag ΔL and watch ε change as a fraction, as a percent, and as microstrain, all at once. They are all the same number. The key point: dividing a length by a length leaves no units. It is dimensionless. So strain tells you, fairly and regardless of the bar's size, how much the material stretched. Whether it is 1 m or 1 km long, the same ε means the same proportional stretch.
Nail this difference down with two bars. Give a short bar A and a longer bar B the same strain ε. Drag ε: both stretch by the same ratio, yet their elongations ΔL differ. The long B stretches more. Since ΔL = ε · L, a longer bar gets a bigger absolute stretch for the same ratio. So judging by ΔL alone fools you. Which is more dangerous, a bridge that grew 30 cm or a bolt that grew 3 mm? You cannot tell without the length. Strain removes this trap, because whether a material is near its limit is told by ε, not by ΔL.
Stretching is not the only kind of deformation. There is also the shear you saw in A1, the sliding along a face. This one cannot be measured by a length ratio, because nothing lengthens; the shape skews instead. So shear strain is measured by an angle. Drag γ to push the square element. The once-clean right angle tilts away from 90 degrees. That tilt is the shear strain γ, usually in radians. If normal strain ε is the ratio of stretch, shear strain γ is how far the right angle has collapsed. Both are deformation, but one is captured as a change in length, the other as a change in angle.
Finally, a feel for size. Real strains are surprisingly small. When steel holds in its elastic range, ε is about 0.001, that is, around 0.1 percent. A 1 m bar stretching by 1 mm. It is so small that the stretch is almost invisible to the naked eye. You have to drag the magnification M up before you can see it. This smallness is actually a great gift. Because the deformation is tiny, you can compute with the original dimensions, and dropping the messy higher-order terms makes every equation linear. The formulas of mechanics of materials are clean proportions precisely because the strains we deal with are this small.