In the Elastic Range, σ Is Proportional to ε
So far you met stress σ on the force side and strain ε on the deformation side, separately. Now put them on one graph: ε on the horizontal axis, σ on the vertical. Pull a specimen bit by bit, plotting points, and a single curve appears. This curve is the material's identity card. The same shape always comes out: first a straight line, then a bend somewhere, a rise, and finally a break. That straight part is the key. There, σ is directly proportional to ε, and its slope is the material's stiffness E. σ = E·ε, this one line is the most used equation in mechanics of materials. The rest of the curve is the story of yield, ultimate, and fracture.
Take a specimen and pull it slowly. As you drag the pull, you plot a single point from the current strain ε and the stress σ at that moment. The points connect into the curve. It climbs straight at first, then eases off past a certain point, rises for a long stretch, and at the end the specimen necks down as the curve bends back down. Look at the specimen and you see it stretch, and near the end one spot thins out. This single curve holds the whole story of how that material carries load. Steel, rubber, and concrete each draw a completely different shape.
Zoom into that first straight stretch. This is the elastic range. Drag ε and σ follows in exact proportion. Double the ε and σ doubles too, because it is a straight line. The slope of this line is the Young's modulus E. σ = E·ε. A steeper slope means more stress for the same strain, and that is exactly what it means to be stiff. In the elastic range, release the force and the specimen returns precisely to its original length, like a rubber band pulled gently and let go. σ = E·ε is called Hooke's law, and nearly every calculation in elastic design starts here.
The straight line does not last forever. Past a certain stress, the material yields. That point is the yield point. Once you go past yield and release the force, the specimen does not return to where it started. Drag the peak strain past yield and let go. The path coming down is a straight line parallel to the elastic slope from the way up, and where it touches zero is not the origin. That gap is the permanent set, a stretch that stays forever. Release within the elastic range and it returns cleanly to the origin, but go past yield and a trace remains. That is why design usually keeps a safety margin below yield, staying inside the elastic range.
E is a fixed value, unique to each material. Not the shape or the size, but what the material is decides E. Toggle the material with the buttons. Steel's line is very steep, aluminum is gentler, and concrete lies flatter still. For the same ε, steel takes a far larger σ, and that is what it means for steel to be stiffer. Steel's E is about 200 GPa, aluminum about 70 GPa. One caution: a large E does not mean stronger, that is, harder to break. E is stiffness, how little it bends; breaking is governed by strength, the yield and ultimate. They are different properties.
Follow the story past the straight line all the way to the end. Drag along the curve and you pass three landmarks. First the yield point, where the material starts to take a permanent set. Then the peak after a long rise is the ultimate strength, the maximum stress the material can bear. From there, one spot on the specimen thins into a neck, and the curve stops rising and bends down. And finally fracture, the point where it breaks. A designer usually takes the yield strength as the limit and keeps the room up to the ultimate as a safety margin. Yield strength, ultimate strength, and Young's modulus, these three numbers tell almost the whole story of whether you can use that material.