12.520 Geodynamics

Problem set 1

Problem 4 (40%)

For the stress tensor:

a) Find the principal stresses and directions. (Be sure that your principal coordinate system is right-handed)

b) Calculate the three invariants for this stress tensor and for the associated principal stress tensor and compare their values.

c) Find the deviatoric stress tensor s_ij^dev

d) Find the principal stresses and directions of the deviatoric stress tensor. Find the invariants of the deviatoric stress tensor.

e) What are the relations of the principal stresses, directions, shear, invariants, etc. for these tensors (the complete and deviatoric tensors invastigated above)? Can you anything in general about these relations for an arbitrary tensor and its deviator ?

Solution:

a) The principal stresses form a diagonal stress tensor of the form:

If n is a principal direction, and l is its associated principal stress, the following equation is true:

or, in index notation:

This equation has a solution only if:

Hence: l=2, 2, or 0

_ Two conditions are imposed on the principal vectors as a convention:

1) They must be normalized:

2) They constitute a right-handed coordinate system:

The direction associated with s₃=0 is uniquely defined. We only have an uncertainty on the direction. For instance, we chose:

The two other principal values are equal but different from s₃. The directions associated with them are not uniquely determined, but define a plane on which any two perpendicular directions can be selected, as long as the system is right-handed. We decided to impose n1=0 for the direction associated with s₁=2.

Finally, we calculate n₂ to make a right-handed system. Verify that is indeed a principal direction, with principal value s₂=2.

Let's draw a picture of the coordinate system and principal axes :

both of them are right-handed.

b) By definition, invariants do not change under rotation of the coordinate system. Since the principal stress tensor is the original tensor rotated into different coordinate system, their invariants must be the same. It's easier to calculate invariants for the tensor in the principal axis frame. But you should obey the problem set, and check the invariance by calculating the invariants for both the principal and original stress tensors

principal stress tensor original stress tensor

4

-4

0

c) The deviatoric stress tensor is obtained by substracting the (isotropic) pressure:

where pressure is:

d) following the procedure descried in a), we can find out that

under the condition of:

and principal directions are:

the invariants are:

e) As we have seen, the principal stress directions for a stress tensor and its deviator are the same. The principal stress themselves, however, are different because we have removed the pressure from the deviatoric stress tensor. Thus, the principal stresses of the deviator are simply the sum of the original principle stresses and the pressure: where.