Basic Continuum Kinematics - Continuum Mechanics of Anisotropic Materials - page 47

Solution : Comparison of the expressions u ( x , t ) obtained in

Example 2.1.4 shows that these two expressions coincide only for very small times
t , only if t 2 is much less than t . In this case

∇ O u ( X , t ) and ∇

∇ O u ( X , t ) ¼ ∇ u ( x , t ) and

2 3

110
110
000

4 5 :

r u ¼ t

From this expression for ∇ u and ( 2.49 ), the rotation tensor is determined to

be Y ¼ 0 , and the strain tensor E is given by

2 3

110
110
000

4 5

E ¼ t

as long as t is small.

Problem

2.3.1. For the motions of the form ( 2.10 ) given in Problem 2.1.1, namely 2.1.1(a)
through 2.1.1(g), determine the conditions under which the motion remains
infinitesimal and compute the infinitesimal strain and rotation tensors, E and
Y . Discuss briefly the significance of each of the seven strain tensors
computed. In particular, explain the form or value of the strain tensor in
terms of the motion.

2.4 The Strain Conditions of Compatibility

Calculating the strain tensor E given the displacement field u is a relatively simple
matter; one just substitutes the displacement field u into the formula ( 2.49 ) for the
strain displacement relations, E ¼ (1/2)(( ∇ u ) T þ ∇ u ). Situations occur in

which it is desired to calculate the displacement field u given the strain tensor E . This
inverse problem is more difficult because the strain displacement relations, E ¼ (1/2)

u ) T

(( u ), become a system of first order partial differential equations

for the displacement field u . Given the significance of the displacement field u in an
object we generally want to insure that the displacement field u is continuous and
single valued. There are real situations in which the displacement field u might be
discontinuous and multiple valued, but these situations will be treated as special cases.
In general it is desired that the integral of the strain-displacement relations, the
displacement field u , is continuous and single valued. The conditions of compatibil-
ity insure this. The conditions of compatibility are equations that the strain tensor
must satisfy so that when the strain-displacement relations are integrated, the
resulting displacement field u , is continuous and single valued. The conditions of
compatibility may be written in the direct notation as

∇ þ ∇