Continuum mechanics/Strains and deformations

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Strain Measures in three dimensions[edit]

The motion of a body

Initial orthonormal basis:

Deformed orthonormal basis:

We assume that these coincide.

Motion[edit]

Deformation Gradient[edit]

Effect of :

Dyadic notation:

Index notation:

The determinant of the deformation gradient is usually denoted by and is a measure of the change in volume, i.e.,

Push Forward and Pull Back[edit]

Forward Map:

Forward deformation gradient:

Dyadic notation:

Effect of deformation gradient:

Push Forward operation:

  • = material vector.
  • = spatial vector.

Inverse map:

Inverse deformation gradient:

Dyadic notation:

Effect of inverse deformation gradient:

Pull Back operation:

  • = material vector.
  • = spatial vector.
Example[edit]
Push forward and pull back

Motion:

Deformation Gradient:

Inverse Deformation Gradient:

Push Forward:

Pull Back:

Cauchy-Green Deformation Tensors[edit]

Right Cauchy-Green Deformation Tensor[edit]

Recall:

Therefore,

Using index notation:

Right Cauchy-Green tensor:

Left Cauchy-Green Deformation Tensor[edit]

Recall:

Therefore,

Using index notation:

Left Cauchy-Green (Finger) tensor:

Strain Measures[edit]

Green (Lagrangian) Strain[edit]

Green strain tensor:

Index notation:

Almansi (Eulerian) Strain[edit]

Almansi strain tensor:

Index notation:

Push Forward and Pull Back[edit]

Recall:

Now,

Therefore,

Push Forward:

Pull Back:

Some useful results[edit]

Derivative of J with respect to the deformation gradient[edit]

We often need to compute the derivative of with respect the the deformation gradient . From tensor calculus we have, for any second order tensor

Therefore,

Derivative of J with respect to the right Cauchy-Green deformation tensor[edit]

The derivative of J with respect to the right Cauchy-Green deformation tensor () is also often encountered in continuum mechanics.

To calculate the derivative of with respect to , we recall that (for any second order tensor )

Also,

From the symmetry of we have

Therefore, involving the arbitrariness of , we have

Hence,

Also recall that

Therefore,

In index notation,

Derivative of the inverse of the right Cauchy-Green tensor[edit]

Another result that is often useful is that for the derivative of the inverse of the right Cauchy-Green tensor ().

Recall that, for a second order tensor ,

In index notation

or,

Using this formula and noting that since is a symmetric second order tensor, the derivative of its inverse is a symmetric fourth order tensor we have