Page 179 - Introduction to Continuum Mechanics

164 Problems

(a) Find the components of the strain tensor.

(b) Find the unit elongation of an element initially in the direction of ej + e 2.
3.20. Consider the displacement field

where k = 10 .
(a) Find the unit elongations and the change of angle for two material elements
dX^ = dXi*i and dXs ' = dX^ that emanate from a particle designated by X = «! + e 2-
(b) Find the deformed shape of these two elements.
3.21. For the displacement field of Example 3.8.3, determine the increase in length for the
diagonal element of the cube in the direction of ej + e 2 + 63 (a) by using the strain tensor and
(b) by geometry,
3.22. With reference to a rectangular Cartesian coordinate system, the state of strain at a point
is given by the matrix

(a) What is the unit elongation in the direction 2ej + 2e 2 + 63?

(b) What is the change of angle between two perpendicular lines (in the undeformed state)
emanating from the point and in the directions of 2ej 4- 2e 2 + 63 and 3ej - 663?
3.23. Do the previous problem for (a) the unit elongation in the direction 3ej — 4e2, (b) the
change in angle between two elements in the direction 3ej - 463 and 4e^ + 3e 3 .
3.24. (a)For Prob.3.22, determine the principal scalar invariants of the strain tensor,

(b) Show that the following matrix

cannot represent the same state of strain of Prob.3.22.
3.25. For the displacement field

find the maximum unit elongation for an element that is initially at (1,0,0).
3.26. Given the matrix of an infinitesimal strain field