So both of those components have corresponding strain energies.

So there's a strain energy for the hydrostatic component and

a strain energy for the distortion component.

And what the failure criteria for

the Distortion Energy Theory, or the von Mises theory.

Is that when your component's strain energy per unit volume is greater than or

equal to the distortion energy per unit volume, at yield of a tensile

test specimen that's the same material as your component, you're going to get yield.

So essentially if we take a component, let's take some

sort of component made out of aluminum and we load it in all different directions.

When the distortion energy in that aluminum,

per unit volume, is greater than if I took a bar of aluminum,

loaded it in tension, pulled that bar apart, and see when it yields.

So when the distortion energy in the component due to loading is greater than

the distortion energy per unit volume at yield of a tensile test specimen,

that's when we get yield.

That's when we get failure, so it's kind of an energy comparison.

So the question is, how do you actually compare energies, Right?

And the answer is, there's an equation of course, this is engineering, right?

So we have the equation which is your sigma effective, or

your effective stress, is equal to.

And it includes all the different stresses that could possibly be

applied to your component.

So you have stress in the x direction, y direction, z direction.

And shear stresses in the xy, yz, and zx planes.

And this equation combines all of those stresses into an effective stress.

And you can take that effective stress, sigma prime, and

you can compare it to the yield strength.

And when your sigma prime is greater than or

equal to your yield strength, that's when your component is going to yield.

That's what von Mises theory says.

You can also, so they've simplified the equation down for

principal stresses, remember we talked about principal stresses a bit.

And you can also use this bottom equation for principal stresses.

And then your factor of safety is n, your factor of safety is

equal to your yield strength divided by your effective stress.

And when this is less than 1 you're going to get yield on your component.

So that is the von Mises equation and the Von Mises theory.

So before next time,

what I'm going to have you do is work through this example on your own.

So here we have this perfect aluminum cube.

And note that I've given you stresses, not loads here,

to simplify, this is a very simple example.

We see that the aluminum cube has a yield strength

in tension that's equal to the compression of 75 MPa.

And it has a strain at failure of 0.05, so

it's ductile, it's behaving in a ductile manner.

It has stress in the x direction of 20 MPa, in the y direction of -40 MPa,

in the z direction of 10 MPa, and

in the shear yx plane of 10 MPa.

And so what they're asking is for

the effective stress on the cube and the factor of safety.

So, go ahead and attempt to work through that equation on your own, or

that problem on your own.

And then we'll work through it in the next module.

I'll see you next time.

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