3. Kinematics of Deformation

Section B

Review to Section 3.5 to 3.11

1. Given the deformed and undeformed fields, one can find the displacement field as given in
2. Rigid body motion consists of translation and rotation given by

Note that  is a rotation tensor and is orthogonal,  is the fixed point about which the rotation takes place and  is the translation vector of the origin.

3.      Infinitesimal deformation has very small strain and no rigid body motion. For that purpose we need to define a quantity called displacement gradient given by and an all important quantity deformation gradient

4.      Define infinitesimal strain

It can be shown that the above is affected by rigid body rotation and assumes that displacement gradient is small.

5.      Given the displacement field, calculate displacement gradient and strain as shown in example 3.8.2.

6.      Physical meaning of the components of . The diagonal terms represents normal and off-diagonal terms shear between two lines originally perpendicular in that coordinate system.

7.      How do you compute the deformed geometry given the displacement field (see figure 3.5, page 103). Given a specific point in the deformed (or undeformed) system find the shape of the lines (examples 3.8.3)

8.       The strain tensor  has three principal directions with three scalar invariants.

9.      The first invariant provides volume change per original volume called as dilatation.

10.  The total displacement gradient can be decomposed into

where  is the infinitesimal rotation tensor, with a corresponding dual vector.

See solved problem 3-20

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