Sample Midterm Problem 5

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Suppose that, under the action of external forces, a material point   in a body is displaced to a new location   where

 

and   and   are constants.

Part (a)

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A displacement field is called proper and admissible if the Jacobian ( ) is greater than zero. If a displacement field is proper and admissible, then the deformation of the body is continuous.

Indicate the restrictions that must be imposed upon   so that the deformation represented by the above displacement is continuous.

Solution

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The deformation gradient   is given by

 

Therefore, the requirement is that   where

 

The restriction is

 

Part (b)

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Suppose that  . Calculate the components of the infinitesimal strain tensor   for the above displacement field.

Solution

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The displacement is given by  . Therefore,

 

The infinitesimal strain tensor is given by

 

The gradient of   is given by

 

Therefore,

 

Part (c)

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Calculate the components of the infinitesimal rotation tensor   for the above displacement field and find the rotation vector  .

Solution

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The infinitesimal rotation tensor is given by

 

Therefore,

 

The rotation vector   is

 

Part (d)

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Do the strains satisfy compatibility ?

Solution

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The compatibility equations are

 

All the equations are trivially satisfied because there is no dependence on  ,  , and  .

 

Part (e)

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Calculate the dilatation and the deviatoric strains from the strain tensor.

Solution

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The dilatation is given by

 

Therefore,

 

The deviatoric strain is given by

 

Hence,

 

Part (f)

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What is the difference between tensorial shear strain and engineering shear strain (for infinitesimal strains)?

Solution

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The tensorial shear strains are  ,  ,  . The engineering shear strains are  ,  ,  .

The engineering shear strains are twice the tensorial shear strains.

Part (g)

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Briefly describe the process which you would use to calculate the principal stretches and their directions.

Solution

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  • Compute the deformation gradient ( ).
  • Compute the right Cauchy-Green deformation tensor ( ).
  • Calculate the eigenvalues and eigenvectors of  .
  • The principal stretches are the square roots of the eigenvalues of  .
  • The directions of the principal stretches are the eigenvectors of  .