Page 159 - Introduction to Continuum Mechanics

144 Compatibility Conditions for Components of Finite Deformation Tensor

(b) Since e\\ = 0, an element which is in e^ direction in the deformed state (such as B 'C')
,2
had the same length in the undeformed state ( EC in Fig. 3.13). Also since e\i ~ ——, an
element which is in the 62 direction in the deformed state (such as AH' ) had a length AH
given by the equation

from which one obtains

This result checks with the geometry in Fig. 3.13.

3.27 Compatibility Conditions for Components of Finite Deformation Tensor

Whenever the three pathline equations (or equivalently, the three displacement functions)
are given, one can always obtain the six components of e* or C or B or E* etc. by differentiation.
On the other hand, if the six components of e* etc. are given, there exist three displacement
functions corresponding to the given strain components only when compatibility conditions
for the components are satisfied. This is because in general, it is not possible to solve for three
unknown functions from six differential equations. The compatibility conditions can in
principle be obtained by the elimination of the three displacement components U{ from the six
equations relating strain components with the displacement components such as
Eqs. (3,26.12b) by partial differentiation and elimination as was done for the infinitesimal
components (Section 3.16) The procedure is very lengthy and will be omitted. Only the
conditions for e,y* are given below with the super * dropped for convenience: