Page 44 - Mechanics Analysis Composite Materials

Chapter 2

FUNDAMENTALS OF MECHANICS OF SOLIDS

Behavior of composite materials whose micro- and macro-structures are much
more complicated than those of traditional structural materials such as metals,
concrete, and plastics is nevertheless governed by the same general laws and
principles of mechanics whose brief description is given below.

2.1. Stresses
Consider a solid body referred to Cartesian coordinates as in Fig. 2. I. The body is
fixed at the part S, of the surface and loaded with body forces qr.having coordinate
components qx, q,., and qr, and with surface tractions ps specified by coordinate
components px,p., and pi.Surface tractions act on surface S, which is determined
by its unit normal n with coordinate components I,, I,,, and I, that can be referred to
as directional cosines of the normal, i.e.,
I, = cos(n,x), />.= cos(n,y), z, = cos(II,z) . (2.1)

Introduce some arbitrary cross-section formally separating the upper part of the
body from its lower part. Assume that the interaction of these parts in the vicinity of
some point A can be simulated with some internal force per unit area or stress r~
distributed over this cross-section according to some unknown yet law. Because the
Mechanics of Solids is a phenomenologicaltheory (see the closure of Section 1.1) we
do not care about the physical nature of stress, which is only a parameter of our
model of the real material (see Section 1.1) and, in contrast to forces F, has never
been observed in physical experiments.Stress is referred to the plane on which it acts
and is usually decomposed into three components - normal stress (a2in Fig. 2.1)
and shear stresses (7, and rZyin Fig. 2.1). Subscript of the normal stress and the first
subscript of the shear stress indicate the plane on which the stresses act. For stresses
shown in Fig. 2.1, this is the plane whose normal is parallel to axis-z. The second
subscript of the shear stress shows the axis along which the stress acts. If we single
out a cubic element in the vicinity of point A (see Fig. 2. I), we should apply stresses
to all its planes as in Fig. 2.2 which also shows notations and positive directions of
all the stresses acting inside the body referred to Cartesian coordinates.

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