Sample Homework Problem 4

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Part (a)

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A solid is subjected to stresses as shown by the arrows in the figure below. Indicate the indices for each of the stress components, and whether the stresses should be positive or negative.

Solution

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Stress components

Part (b)

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Suppose that the stress tensor field in a body is given by

 

Find the body force distribution required to maintain equilibrium. ( ,  , and   are in meters). Show units.

Solution

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The equation of equilibrium is

 

Therefore,

 

or,

 

The required body forces are (in MN/m )

 

Part (c)

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Find the surface tractions at the internal point   in the body on an internal surface with a surface normal  .

Solution

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The surface traction is given by

 

The stress at point   is

 

Therefore,

 

The traction vector is (after converting   into a unit normal)

 

Part (d)

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Find the hydrostatic and deviatoric stress at the point  .

Solution

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The hydrostatic stress is given by

 

In this case,

 

Therefore,

 

The deviatoric stress is given by

 

Therefore,

 

Part (e)

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Find the principal stresses at this point.

Solution

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The principal stresses can be found using the equation

 

where   is a principal stress. In expanded form,

 

Substituting the values of stress into the above equation,

 

Expanding out,

 

or,

 

or,

 

or,

 

Thus, the first possible value of   MPa. Also,

 

Therefore,

 

The principal stresses are (in MPa)

 

Part (f)

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Find the principal direction corresponding to the intermediate principal stress.

Solution

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The directions of the principal stresses can be found using the equation

 

For the principal direction   (corresponding to the principal stress  , we have,

 

Hence,

 

gives us   and

 

Now,  . Therefore,

 

Taking squares of both sides,

 

So we get,

 

Therefore,

 

The direction corresponding to the intermediate principal stress is

 

Part (g)

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The symmetry of the stress tensor can be derived from a certain balance principle. Name the principle and write it down in index notation.

Solution

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The balance principle is

 

In index notation