Yeah, good morning, everybody, to this little introduction to the topic of material modeling,

which is the wide fields. And I just want to give a little introduction to that, to

the, let's say, most dominant behaviors that we observe in materials and how we model that.

For those of you who are not familiar with the title, this is something that I kind of

stole from the time of the presidential campaign of Bill Clinton from 92, where one of the

main campaign messages was, it's economy, stupid. And here it's the mechanics, stupid.

And we talk about the joy of material modeling. Okay. So maybe first something that is almost

a little bit philosophical, but I think it's still worthwhile to mention that. What is

modeling? Yeah. So in our case, we would observe something maybe in experiments on our materials

that we want to investigate. These phenomena can be all kinds of things. For instance,

that you pull a bar and it becomes longer as one of the, let's say, most obvious phenomena

that we observe when we test materials. Then, of course, we have to somehow agree what for

our applications, which of these phenomena are of relevance, because usually, of course,

reality is so super complex and we will never be able to completely map that in our modeling.

So we have to take decisions. What we think is relevant, whether for instance, the influence

of factors like temperature, whatever, electric field, magnetic fields, chemo influences and

so on, which of these phenomena are really of relevance. And then based on that, eventually,

we set up a model to describe some material. And here, of course, we can distinguish. On

the one hand, we have an idea of what's going on in the material. For instance, that it's

built up from atoms, maybe, or for those of you who are familiar with modeling crystals,

for instance, of how this idea of dislocations and so on. So this would be a physical model,

the concepts of our idea, how this material is built up and how it behaves. And the other

thing is the mathematical model. And this is just equations. It's just equations. And

these equations should, of course, somehow map our concept that we have from the material.

So that once we, let's say, translate these equations, maybe in an algorithm to compute

the answer of the material to a particular loading, for instance, we would then be able

to predict the phenomena. And with that also applies this model to cases that go beyond

the cases that we have tested. So that's the one thing. The other thing is that we can,

let's say, put the various behaviors that materials can show, let's say, in different

draws that we can classify it. And one of the classification that is very powerful is

the sketched here. So if you start, for instance, with this observation, quasi-static hysteresis.

So that refers to the case sigma would be the stress. So that's essentially the force

which you pull on a specimen. Epsilon is a strain. So that's, you can translate it if

you like, in the displacement. And then if you do that very, very slowly, so this rate

goes to 0. And then you load. And then what we say, unloading, you go back here. And if

you do that in this situation, and you go on the same line here, then this is one element

of classifying the material as being elastic. The other thing to classify it is the question

whether this happens, whether there is some rate dependence. So if you can go up and down

this relation between strain and stress, regardless what is the velocity with which you load, and

you have this so-called quasi-static hysteresis, then what we get is elasticity. If you have

this quasi-static hysteresis, you go back on the same line that you climbed up, if you

go very slow. But then if you go with a faster velocity, you observe different behavior,

then the material classifies as viscoelastic. If in this very slow loading, unloading here,

loading, unloading scenario, you would not end up at the same point where you started,

then this would be a quasi-static hysteresis, together with being not rate dependent. This

would qualify as being plasticity. And if you again observe different behavior, different

rates, different velocities of loading, this would qualify as viscoelasticity. So this

is a little bit different arrangement, but the same logic. So these four material behaviors

are, so to speak, the most paradigmatic ones, elasticity, viscoelasticity, plasticity, viscoelasticity,

and I will guide you through these different behaviors in what follows. So elasticity,