Using the definition of the line element, for the reference configuration and for the deformation. Say...

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Using the definition of the line element, for thereference configuration and for the deformation. Say how is defined?_{m} associated with m's direction. Describe the different casesfor this deformation (stretching and shortening) and from thiswrite the definition of the unitary elongation ?_{m}.

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The deformation gradient F is the derivative of each componentof the deformed x vector with respect to each component of thereference X vector For xxX thenA slightly altered calculation is possible by noting that thedisplacement u of any point can be defined asuxXand this leads to xXu andIn tensor notationalso written asFijijuijRigid Body DisplacementsAn example of a rigid body displacement isxX5yY2In this case FI is indicative of a lack of deformations As willbe shown next it is also indicative of a lack of rotationsClearly rigid body displacements do not appear in the deformationgradient This is good because rigid body displacements dontcontribute to stress strain etcRigid Body RotationsAn example of a rigid body rotation isThese equations rotate an object counterclockwise See Answer

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