Page 47 - Mechanics Analysis Composite Materials

32 Mechanics and analysis of composite materials

Thus, we have three integral equilibrium equations, Eqs. (2.3), which are valid for
any finite part of the body. To convert them into the corresponding differential
equations, we use Green's integral transformation

Y 1'
which is valid for any three continuous, finite, and one-valued functions f(x,y,z)
and allows us to transform a surface integral into a volume one. Taking J; = a,,
= zJT,and = z,, we can write Eqs. (2.3) in the following form

Because these equations hold whatever the part of the solid may be, provided only
that it is within the solid, they yield

Thus, we have arrived at three differential equilibrium equations that could be
also derived from the equilibrium conditions for the infinitesimal element shown in
Fig. 2.2.
However, in order to keep part C of the body in Fig. 2.1 in equilibrium the sum of
the moments of all the forces applied to this part about any axis must be zero. By
taking moments about the z-axis we get the following integral equation

Using again Eqs. (2.2), (2.4) and taking into account Eqs. (2.5) we finally arrive at
the symmetry conditions for shear stresses, i.e.

zxy = zm (x,y,.) * (2.6)
So, we have three equilibrium equations, Eqs. (2.5) which include six unknown
stresses ox, ay,a, and z,,, zxz,z.~=.
Eqs. (2.2) can be treated as force boundary conditions for the stressed state of
a solid.

2.3. Stress transformation

Consider the transformation of a stress system from one Cartesian coordinate
frame to another. Assume that the elementary tetrahedron shown in Fig. 2.3 is