 Previously, we discussed the various material properties that relate the state of strain in a material to its state of stress, Young's modulus, Poisson's ratio, and Shear modulus. These material constants, however, only relate the strain to mechanical loading within that material. However, there is another form of loading we must contend with in engineering structures, that is thermal loading. If we take an object with initial volume v0 and subject it to a temperature change, it will undergo a volumetric change. It will expand uniformly in all directions. Why does this happen? Well, if you recall what temperature actually is, it's an average measurement of the thermal energy or molecular vibration within a material. So as you increase the temperature, the vibration of these molecules increases, which causes the molecules to push each other apart, thus resulting in the expansion that we observe when a material is subjected to a temperature change. Returning to our object, if it is made of a single isotropic material, the expansion will be uniform in all directions. Thus we will have a uniform thermal strain in all directions of our object. Generally speaking, for most engineering applications, we can consider this expansion to be linear with temperature, and we can define the linear coefficient of thermal expansion, alpha, that we multiply by the temperature change in order to obtain a thermal strain. Why do we concern ourselves with the expansion of materials with temperature anyways? Well the reality is most engineering structures are not entirely free to expand or contract. Take for instance this hypothetical bridge built between the banks of a river. Due to seasonal changes in temperature, the bridge will want to expand due to the heat of the summer, and contract due to the cold of the winter. However, the banks of that river are relatively rigid, they're not going to move. Therefore the bridge is not free to expand or contract, and internal mechanical loads will have to develop to prevent the expansion in the summer, and to prevent the contraction in the winter, and these mechanical loads might ultimately lead to failure. As a result, you can actually look at real bridge structures and see thermal expansion joints that permit some level of expansion and contraction to minimize these thermally induced mechanical loads. Now that we have looked at the strain due to both mechanical loads and thermal loads, let's combine these results and produce a generalized formula relating stress and strain through material properties. We will call this generalized formula the generalized Hooke's Law, and we will look at it first from the perspective of normal strain, then the perspective of shear strain. So far we have seen three contributions to normal strain. We get a direct strain as a result of a normal stress through the Young's modulus. Perpendicular to that applied stress, we get a Poisson effect, a contraction, and if there is a temperature change, we get thermal expansion of the material. If we apply these three contributions for a general state of stress as shown here on the right in the Cartesian coordinate frame, where we have a sigma xx, sigma yy, and sigma zz applied to the material as well as a temperature change delta t, we can begin to calculate the strain in the x direction. As a result of Hooke's Law, we see that we get a component equal to sigma x divided by the Young's modulus. Due to the Poisson effect, we get a contribution due to sigma y, as well as a contribution due to sigma z. And finally, we also get a contribution due to the temperature change. If we collect the terms and simplify this equation, we get the following result for the normal strain in the x direction. If we repeat this procedure for the other two directions, we get the following result for normal strain in the y direction, and the following result for the normal strain in the z direction. Please note that the generalized Hooke's Law for normal strains is often written without the thermal strain component, as a lot of problems do not encounter a temperature change. If we now consider shear strains, we have only seen one contribution to shear strain, and that is Hooke's Law for shear strain. Tau is equal to the shear modulus g times the shear strain gamma. There was no Poisson effect in shear, and there was no temperature effect in shear. As a result for a general state of shear stress and temperature change, the generalized Hooke's Law for shear is as follows. We can combine the general state of normal stress and general state of shear stress with a temperature change as shown on the element on the right. We thus get the normal strains and the shear strains as follows. This is generalized Hooke's Law that will allow you to calculate the normal and shear strains at a point in a material given the normal and shear stresses. We will be using these equations throughout the course in order to examine deformation in everyday engineering structural elements.