 Let's perform a virtual experiment. Consider the rod of tungsten shown here on the left with initial length L0 and cross-sectional area A0. If we apply a force to this rod, we know that it's going to deflect by a given amount, which we will call delta. If we plot the force-deflection behavior of this rod, it will deflect in some given way related to the material behavior. For now, we'll assume it's linear in terms of its relationship up until we reach that force F, giving us a total-deflection delta A. Now if we look at a second rod of material, this time made out of something softer, such as aluminum, and we apply the same force, we intuitively know because it's softer we're going to end up with a larger deflection delta B. However, if we change the geometry of the rod and again use the same soft aluminum material with a shorter length and larger cross-sectional area, we also intuitively know that we will get a smaller deflection. So what does this virtual experiment tell us? It tells us that the stiffness, i.e. the relationship between force and deflection, is affected by both the material as well as the geometry. So how can we separate these effects? Well, we need to look at the intensity of both the loading and the deformation, in other words, the stress-strain behavior. Now in order to plot the stress-strain behavior, we have to know what the stress and strain distribution within this specimen is. For now, for this simple loading case, we're going to assume a uniform stress and strain assumption. Now I know alarm bells are going off in your head right now because I spent a lot of time explaining why you can get into trouble with the uniform stress or uniform strain assumption. However, it is actually quite valid in this case and we're actually going to examine that at a later time. But for now, let's just assume that the normal stress can be approximated by force divided by area and the normal strain can be approximated by the total deflection divided by the initial length. Using these assumptions, we can now plot the stress-strain behavior and reduce stiffness to a material property. If we plot the stress-strain behavior for a typical material until failure, we will get a curve similar to the one shown here. If we look at this curve, we can see that there is a linear portion and a limit in terms of stress to that linear portion. There is also what we call a yield point which is a point at which the stress-strain behavior is fully reversible. There is an ultimate strength or a maximum stress the material can take and there is a rupture or fracture point. Now we can divide this curve up further into a portion that is elastic, i.e. the deformation is reversible and a portion that is plastic. The plastic region can be divided even further into a portion of strain hardening and a portion of necking where necking is a localization of the cross-sectional deformation. Now the full details of a stress-strain curve and all the regions of stress-strain curve are covered in your material science course which you have likely taken by this time already. For the purposes of this course, we are actually only really interested in the linear elastic portion. This is the area we like to design at as engineers and therefore we will restrict ourselves to just looking at material behavior within this region. Because the region is linear, we can see that the stress is proportional to the strain. We call the proportionality constant Young's modulus and denote it by the capital letter E. From Hooke's law, we see that the normal strain in direction of loading is linearly proportional to the normal stress. However, this is not the only deformation that occurs under normal stress loading. We also get a strain in the transverse direction. Observations have shown us that if we take the ratio of transverse strain to strain in the loading direction, we get a constant of all the Poisson's ratio. This ratio can be used to calculate the transverse strain given a normal stress in a loading direction. So far, we've looked at the relationship between normal stresses and normal strains in a material but not shear. If we conduct similar experiments where we apply pure shear loading to a material and measure the shear strain and plot the behavior, we also observe that the shear strain is linearly proportional to the shear stress. There is an analogy to the Young's modulus for shear. We call this the shear modulus and denote it by the letter G. Let's recap the material properties we have looked at so far. We have seen that in the presence of normal stress, we get two normal strain values. The normal strain in the direction of loading is proportional to the normal stress through the Young's modulus. The transverse strain can be related to the strain in the direction of stress using the Poisson's ratio. And in for shear, we see that the shear modulus relates the shear strain to the shear stress. Furthermore, although we did not explicitly say it earlier, these two strains are decoupled, meaning a normal stress does not cause a shear strain and a shear stress does not cause a normal strain. This will be important later when we develop the generalized Hooke's law in the next video lecture.