Understanding Strain Invariants in Strength of Materials

[CPP]

Can anyone help me understand the concept of a "strain invariant" given a strain state matrix? or perhaps point me towards something?

thanks for any help.

 

[AlephZero]

This is what the "big idea" of stress and strain invariants is about. The details of the maths is fairly easy to find with Google.

If you have a structure, the way it behaves (the stresses, deflections, failure mode, etc) doesn't depend on where you choose to put the X Y Z axes when you set up model and solve the equations.

But the numbers representing the 6 components of stress and strain at a point (3 direct and 3 shear) DO depend on what direction the axes point in. So, those raw numbers are not always the right things to use, to understand what the solution means.

It turns out there are some functions of the 6 stress/strain components which are independent of the orientation of the axes. There are three of these functions which are called the stress/strain invariants.

If you have an isotropic material, then the physics of the material behaviour is also independent of the orientation of the axes used to describe it. Therefore, the material behaviour (elasticity, plasticity, etc) can be described using the 3 invarants. In fact you must be able to write these formulas using only the invariants, otherwise the material behaviour would not be isotropic.

The first invariant (in the standard numbering convention) is the sum of the direct stresses or strains e_xx + e_yy + e+zz. Physically, that represents the "fluid presssure component" part of a the stress at the point, or the "uniform expansion or compression" part of the strain.

The second invariant is more complicated mathematically, but for stress it's very similar to the Von Mises stress function. It's a measure of the "average amount of shear" at the point. This is why Von Mises stress is a good failure measure for ductile materials which fail because of shear stress not direct stress. The actual formula for VM stress doesn't look much like a "shear stress" at first sight, but if you think about a shear stress, then rotate the axes 45 degrees so the shear is equivalent to tensile and compressive direct stresses, the connection between the VM "difference of principal stresses, squared" formula appears.

The third invariant isn't so simple to understand physically, but the maths says there are three of them, so it is what it is.

Hope that helps get started.