Page 57 - Mechanics Analysis Composite Materials

42 Mechanics and analysis of composite nzaterials

admissible field. This means that there exist equations that are similar to
Eqs. (2.22), i.e.,

and specify additional strains.
Because additional displacementsare infinitely small, we can assume that external
forces do not change under such variation of the displacements(we do not consider
here special cases in which external forces depend on displacements of the points at
which these forces are applied). Then we can calculate the work performed by the
forces multiplying forces by the corresponding increments of the displacementsand
write the total work of body forces and surface tractions as

(2.38)

Here, V and S are the body volume and external surface of the body in Fig. 2.1.
Actually, we must write the surface integral in Eq. (2.38) only for the surface S,,
on which the forces are given. But since the increments of the displacements
belong to a kinematically admissible field, they are equal to zero on S,, and the
integral can be written for the whole surface of the body. To proceed, we express
px, p,,,and pz in terms of stresses with the aid of Eqs. (2.2) and transform the
surface integral into a volume one using Eq. (2.4). For the sake of brevity,
consider only x-components of forces and displacement in Eq. (2.38). We have in
several steps

a
a
+7 -(dux) +r,--(du,)]dY
yx aY az