Page 55 - Mechanics Analysis Composite Materials

40 Mechanics and analysis of composite materials
2.7. Compatibility equations

Consider strain-displacement equations, Eqs. (2.22), and try to determine
displacements u,, u., and u, in terms of strains E.~, E~,, and yx,,, yxz, y,=. As can be
seen, there are six equations containing only three unknown displacements. In
the general case, such set of equations is not consistent, and some compatibility
conditions should be imposed on strains to provide the existence of the solution. To
derive these conditions, decompose derivatives of the displacements as follows

(2.32)

Here
0 -’(”.-%)

‘-2 ax (x,y,z) (2.33)

is the angle of rotation of a body element (such as the cubic element shown
in Fig. 2.1) around the z-axis. Three Eqs. (2.32) including one and the same
displacement u, allow us to construct three couples of mixed second-order
derivatives of u, with respect to x and y or y and x, x and z or z and x, y and z or
z and y. Because the sequence of differentiation does not influence the result and
since there are two other groups of equations in Eqs. (2.32), we arrive at nine
compatibility conditions that can be presented as

(2.34)

These equations are similar to Eqs. (2.32), Le., t..ey allow us to determine rotation
angles only if some compatibility conditions are valid. These conditions compose
the set of compatibility equations for strains and have the following final form

where
aZE, a%, a2yxy
krJ&,y) =-+---
ax2 axay
ay2 Y,4 7 (2.36)
:
x

Tx(&,7) ==---- -+-- ayJj (x,y,z) .
-
ayaz i (aZ ay
aZEx
ax