1. The Lagrangian (Material) description of deformation for a deforming solid is given as2

3322

3222

311 ),(),1( eXxeeXXxeXXx , where e is a constant.

a) Check if this is an admissible deformation field.

b) If so, determine the Eulerian (Spatial) description of the deformation

c) Obtain the expressions for the displacements ui in both Lagrangian and Eulerian coordinates.

2. The displacement components for a body are given as

23332211 ,,2 XXuXuXXu , determine the following

a) Strain (eA) in the direction 3

1,

3

1,

3

1 (use finite strain tensor and magnification

factor to calculate the strain)

b) Change in angle between the lines whose direction cosines in the un-deformed

medium are (3

1,

3

1,

3

1 ) and (1,0,0).

3. A prismatic bar of length L and having a square cross-section of side, 2a, is hanging under it’s own weight as shown in figure. The connection at the top face (at centroid, A, of the top face) is rigid. The displacement components for this problem with respect to the indicted RCC (origin at the centroid of the lower face) are given as

)(2

)(;; 22

21

12232332123111 XX

ClXCuXXCuXXCu

where C1and C2 are constants.

a) Assuming small strains, determine the strain and rotation components at points (a,-a, l), (a,a,0), (0,0,L).

b) Sketch the final shape and orientation of the three infinitesimal square elements which are parallel to the three coordinate planes (X1X2, X2X3, X1X3) respectively and centered around the above points.

c) Do plane sections perpendicular to X3 remain plane after deformation.

4. The displacement field in a deforming material is given as

33212211 5;23;2 XuXXuXXu

Determine the components of the Lagrangian finite strain tensor, principal strains and principal directions. Also determine the orientation, after deformation, of three line segments originally oriented along the principal axis.

A

X1

X3