Hooke's law stress-strain curve
Earlier the stress, on the force side, and the strain, on the deformation side, were met separately. Now they go on one graph together. Strain is the horizontal axis and stress is the vertical one. Pull a specimen a little at a time, plot a point at each step, and a single curve appears. That curve is the material's identity card. Materials differ, but the large outline is the same. First a straight line, then a bend somewhere, then a rise, and at last a break. The straight part is the heart of it. There stress is directly proportional to strain, and the slope is the material's stiffness E. σ = E·ε, and that one line is the formula mechanics of materials uses most. The rest of the curve is the story of yield, ultimate strength, and fracture.
Take one specimen and pull it slowly. Move the pull and each moment's strain and the stress then acting are plotted as one point, and the points join into a diagram. At first it climbs straight, then from some place it flattens, rises for a long while, and at the end the specimen necks and the curve turns down. Look at the specimen beside it and you see the lengthening, and near the end one place grows thin. This one curve holds how that material bears a force. Steel, rubber, and concrete draw entirely different shapes. The shape of the diagram is the character of the material.
Enlarge the first straight stretch. This is the elastic range. Move the strain and the stress follows in exact proportion. Give twice the strain and the stress doubles too, because the line is straight. The slope of this line is Young's modulus E. σ = E·ε. A steeper slope means a larger stress for the same strain, and that is what stiff means. In the elastic range, release the force and the specimen returns exactly to its original length, like a rubber band pulled moderately and let go. σ = E·ε is called Hooke's law, and nearly every elastic design calculation starts here. While the line stays straight, letting go leaves no trace.
The straight line does not go on forever. Past a certain stress the material yields, and that point is the yield point. Release the force after yield and the specimen does not return to where it began. Drag the peak strain past yield and let go. The path down is a straight line parallel to the elastic slope of the way up, and the place it meets 0 is not the origin. That leftover is the permanent set, a lengthening that stays. Release inside the elastic range and it returns cleanly to the origin, but past yield a trace remains. Design therefore usually stays below the yield point, inside the elastic range, with a margin. A trace means the yield has already been passed.
E is a value fixed for each material. It is not the shape or the size that sets E. It is what the material is. Change the material with the buttons and steel's line is very steep, aluminum's is gentler, and concrete lies flatter still. The same strain puts a much larger stress on steel, and that is what it means for steel to be stiffer. Steel's E is about 200 GPa, and aluminum's is about 70 GPa. The caution is that a large E does not mean stronger, in the sense of harder to break. E is stiffness, how little it bends, and breaking is measured by yield and ultimate strength. They are different properties.
Follow the story past the straight line all the way. Move along the curve and it passes three places. First the yield point, where the material begins a permanent set. Next the high summit is the ultimate strength, the largest stress the material can bear. From there necking begins, one place on the specimen growing thin, and the curve stops rising and turns down. At last comes fracture, the point where it breaks. A designer usually takes the yield strength as the limit and keeps the room up to the ultimate strength as a safety margin. Yield strength, ultimate strength, and Young's modulus, these three numbers almost say whether that material can be used. Those three numbers decide whether the material can be used.