How does a tensor work?

To explain

we will start now and will look at what a tensor is

how such a tensor works.

And next time in the lecture we will deal with it a little more

what that means in terms of tension conditions.

And how they affect the material.

Because it is basically the case that a material mechanically reacts to a tension condition.

Now you don't know what a tension condition is, but in principle you can imagine

if we have a mechanical tension on a fixed body

then there are normal forces and shear forces in there.

Plus we have three different directions of space.

And depending on how strong these forces are

the material will react differently.

Tensors are our method of how we represent these tensions.

So to start I usually show a little bit of the finite element program.

So I always show a little bit of the finite element program.

So

does everyone know what a finite element program is?

Who knows what a finite element program is?

No hands up.

So, what is a finite element program?

Does everyone know what a finite element program is?

Who knows what a finite element program is?

No hands up.

So

if we want to numerically calculate which tensions we have in a component

then we typically take a finite element program.

A finite element system simply explains that we take a component

a domain

divide it into small subdomains and then solve the overall distribution of the tension in the whole system.

So

in principle

you can imagine that we do not have a general idea

of how to calculate the total distribution of the tension in the whole system.

So, if I give you a potato and ask you to solve the differential equations for the tension,

then there is no analytical solution for it.

There are only numerical solutions.

The result is that we take finite elements

finally elements

from which we build our components on the computer

and then we solve these differential equations coupled,

because we can solve them for the simple elements.

So

that's maybe a cube or a tetraedron or something similar.

We can solve the equations for that.

And then we couple these equations and solve them together in a linear equation system.

That's what a finite element program does, roughly speaking.

And these are of course commonplace in the world of engineers.

That means, if you have to find something in the company that you cannot calculate by hand,