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It is found that the stress tensor at any point (x, y, z)

Chapter 16, Problem 16.20

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QUESTION:

It is found that the stress tensor at any point (x, y, z) in a certain continuous medium has the form (with an unspecified, convenient choice of units) XZ z2 0 = [ z2 y 0 _ (16.145) Find the surface force on a small area dA of the surface x2 + y2 + 2z2 = 4 at the point (1, 1, 1).

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QUESTION:

It is found that the stress tensor at any point (x, y, z) in a certain continuous medium has the form (with an unspecified, convenient choice of units) XZ z2 0 = [ z2 y 0 _ (16.145) Find the surface force on a small area dA of the surface x2 + y2 + 2z2 = 4 at the point (1, 1, 1).

ANSWER:

Step 1 of 3

The stress tensor at a point  

                                                       

At a point  

                                                           

The surface force on a surface element is

                                                             

The surface area of the element  is

                                                                 

Here  is the unit vector.

                                                            

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