Exercises in continuum mechanics

[geotechnique]

Hello
Please forgive me if i am not posting in the correct forum. Also you may find my English a bit rusty since i am basically French

Ok so i want to solve some exercises in continuum mechanics . The first exercise states :
we have a stress tensor in a Cartesian coordinate system with the point O ; origin of the system , such as :

calculate σ_x so that one face of the stress vector is nil at point 0 ( i know it sounds odd)
Then , calculate the components of the vector n^→ on this face .

Another exercise states : the function of a stress for a plain stress condition is given by :

Where a and b are constants and the volume forces are ignored .
Determine the expressions of σ_x and σ_y if for the point M(1,1) the shear stress is equal to 1/2.

For the last exercise , what i have done is calculate the derivative of Φ with respect to x and did that again for y , then calculate the the two derivative functions for the point M so it gave me :
σ₁ =a+3b
σ₂=4a+2b
after that i applied the know functions to calculate σ_x and σ_y for a plain stress problem
it gave me these results :
σ_x =a+3b
σ_y= 4a+2b which equal exactly my first results ( not sure if this is correct tho )
Thanks

 

[geotechnique]

yea well i found out the responses to the question . for the first one , you just need to resolve the serie of nul equations ( the stress tensor multiplied by the n vecteur equal 0)
for the last problem , that function is called the Airy function .