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Initial orthonormal basis:
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Deformed orthonormal basis:
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We assume that these coincide.
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Effect of :
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Dyadic notation:
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Index notation:
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The determinant of the deformation gradient is usually denoted by and is a measure of the change in volume, i.e.,
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Push Forward and Pull Back
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Forward Map:
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Forward deformation gradient:
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Dyadic notation:
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Effect of deformation gradient:
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Push Forward operation:
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- = material vector.
- = spatial vector.
Inverse map:
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Inverse deformation gradient:
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Dyadic notation:
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Effect of inverse deformation gradient:
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Pull Back operation:
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- = material vector.
- = spatial vector.
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Motion:
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Deformation Gradient:
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Inverse Deformation Gradient:
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Push Forward:
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Pull Back:
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Recall:
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Therefore,
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Using index notation:
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Right Cauchy-Green tensor:
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Recall:
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Therefore,
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Using index notation:
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Left Cauchy-Green (Finger) tensor:
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Green (Lagrangian) Strain
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Green strain tensor:
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Index notation:
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Almansi (Eulerian) Strain
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Almansi strain tensor:
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Index notation:
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Push Forward and Pull Back
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Recall:
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Now,
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Therefore,
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Push Forward:
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Pull Back:
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We often need to compute the derivative of with respect to the deformation gradient . From tensor calculus we have, for any second order tensor
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Therefore,
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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 )
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Also,
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From the symmetry of we have
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Therefore, involving the arbitrariness of , we have
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Hence,
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Also recall that
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Therefore,
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In index notation,
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Derivative of the inverse of the right Cauchy-Green tensor
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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 ,
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In index notation
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or,
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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
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