12.520 Geodynamics

Problem set 1

Problem 5 (15 %)

a) If there are no body forces, demonstrate whether equilibrium exists for stresses:

b) What are the normal and shear tractions on a plane perpendicular to the y-axis at x=y=1?

Solution:

There is not stress acting on the plane perpendicular to the z-axis, and nor shear in the z-direction. Equilibrium is verified here (we actually are in a 2D case). I'll forget about the z-direction (TA privilege).
If there are no body forces, equilibrium exists in case t_ij,j=0 ; let's check the validity of this expression :
along the x-axis: 
along the y-axis: 

Thus, equilibrium does not exist in general, although it does exist on the plane x=0.
 

The normal vector to the plane perpendicular to the y-axis is:

b) Traction :T^T=s^T.n or for point A(1,1,0): 

and shear traction to this plane is x-component of the vector T , or:

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