Introduction to Elasticity/Sample midterm5

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Sample Midterm Problem 5[edit | edit source]

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)[edit | edit source]

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[edit | edit source]

The deformation gradient is given by

Therefore, the requirement is that where

The restriction is

Part (b)[edit | edit source]

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

Solution[edit | edit source]

The displacement is given by . Therefore,

The infinitesimal strain tensor is given by

The gradient of is given by

Therefore,

Part (c)[edit | edit source]

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

Solution[edit | edit source]

The infinitesimal rotation tensor is given by

Therefore,

The rotation vector is

Part (d)[edit | edit source]

Do the strains satisfy compatibility ?

Solution[edit | edit source]

The compatibility equations are

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

Part (e)[edit | edit source]

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

Solution[edit | edit source]

The dilatation is given by

Therefore,

The deviatoric strain is given by

Hence,

Part (f)[edit | edit source]

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

Solution[edit | edit source]

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

The engineering shear strains are twice the tensorial shear strains.

Part (g)[edit | edit source]

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

Solution[edit | edit source]

  • 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 .