In this video, we talk about multiple component elements in a complex assembly and how they can effectively be calculated to produce an overall zero thermal expansion across the entire assembly.

The transcript of this video can be found below.

### Introduction to tailoring to zero thermal expansion or other desired CTE with multiple components.

In this video, we will talk about multiple component elements. In the previous video, we talked about two pieces of material–one of ALLVAR Alloys and one of another material. So, here is the other material – here is ALLVAR Alloys– and effectively this interface did not exist. There was nothing there. It was just the two pieces of material that were butted up against each other. So, you had component one and component two. Now this can happen in special cases. For example, hydroxide bonding or hydroxide catalysis bonding can create an interface that is almost non-existent, so you can ignore it for most practical applications. But if you are outside this special case of a hydroxide binding bond between ALLVAR Alloys and titanium or ALLVAR Alloys and a glass material, you are going to start to see the influence of the interface. So, for example, you can weld ALLVAR Alloys to titanium 6-4, and if you have a weld then you are going to also have a heat affected zone. So, beyond the weld itself you are going to have a region that will be affected by the heat. It will be due to the mixture of the two alloys, or a change in the CTE or potentially phase transformation of the other material and it will create a different CTE in this region. So, in this case you can do a hand wavy thing and say you have effectively three zones or three components that need to be taken into consideration. Something else that we have done on our end is take a piece of ALLVAR Alloys and sandwich it in between two pieces of other material, and this will create a screw joint between the two.

### Defining a characteristic equation for desired CTE of each component

In this case you have the other material’s CTE; you have the screw joint which is like a second component; you have the ALLVAR Alloys; you have another screw joint; and then you have the other material. So, in this case you have five components or five different elements that need to be taken into consideration. So how do we generalize the equation that we had before into a multi-component system? We start from the same position where the change in length, the total change in length, of this whole bar or beam is equal to the individual changes in length for the different components. The change in length of component 1, the change in length of component 2, and the change in length of component 3, and so on until the change in length of component n. This would be thermal displacement.

So then we plug in the definition of thermal expansion as well as strain like we did before and in this case we are going to use average thermal expansion again, so the total average thermal expansion of the total length – or effective thermal expansion – times the original thermal length at temperature one times the change in temperature is equal to the thermal expansion coefficient of component one times the length of component one at temperature one plus a thermal expansion of component 2 times the length of component 2 at temperature 1 plus, all the way to alpha, the thermal expansion of component n times the length of component n at temperature one all times the change in temperature. The change in temperatures cancel as before. This is a uniform temperature change across all components in a region with linear coefficients of thermal expansion for each of these components.

Then you do some rearranging, and you can get the effective total thermal expansion coefficient is equal to one over the original length of the whole beam at temperature 1 times this section over here. Thermal expansion of component 1 times the length of component 1 at temperature 1 – same thing for component 2, all the way to component n.

Now we’ve got a characteristic equation, but as simple as this case was, you could solve very easily for the length of all of the length of the other material for that matter, but now we have a lot of unknowns. The thermal expansion coefficients should be known. The total length should be known. The total thermal expansion coefficient is something that you desire and dictate, but the length you do not know – so you have this equation that you need to solve, but to do that you need to introduce some constraints.

The way that I think about that is these constraints need to restrict the number of unknown elements to “n” minus two; again, where “n” is your total number of elements, or components, in the stack up. So, you need to end up with n minus two unknown lengths, at a minimum in order to solve. So, the constraints that jump to mind is the first one that we used for the previous development for the two-component system is the total length at temperature 1 is equal to the sum of all the lengths at temperature 1 plus L2 at temperature 1 plus “n” at temperature 1. So that is the first constraint, and we have already seen it before, so it should not be a surprise.

Something else that you can take into consideration that will make your job a lot easier is to look for symmetry. For example, in this case there’s symmetry between component 1 and component 5 so in in this case L1 is equal to L5 and component 2 is equal to component 4. L2 is equal to L4 so that would be this the system right here so symmetry can be your friend if you still are not able to reduce it to “n” minus two unknown lengths. Then you should start with this, use performance design requirements to understand, for example, how many threads do you need on the screw to make sure that it will lock in place correctly. That would be a performance requirement or design requirement that would dictate the minimum length of this screw joint.

### Summary

So, these three guiding factors or guiding elements can help you generate the constraints necessary to solve a system of equations so then once you have that you solve the system of equations for a desired thermal expansion.

For alpha [or CTE] total equal to a desired value – this can be zero [CTE] like we talked about in the previous videos and or it could be positive or negative – but again the total thermal expansion coefficient is always going to be bound by your highest thermal expansion coefficient and your lowest thermal expansion coefficient.