Page 270 - Mechanics Analysis Composite Materials

Chapter 5. Mechanics of laminates 255

to complete the analysis, we need to determine stress acting in each layer of the
laminate.
To do this, we should first find strains in any ith layer using Eqs. (5.3) which yield

where ziis the layer normal coordinate changing over the thickness of the ith layer.
If the ith layer is orthotropic with principal material axes coinciding with axes s
and y (e.g., made of fabric), Hooke's law provides the stress we need, i.e.,

where, E$, = E.$,/( 1 - v$v-::) and E.?), E$!",G$,(. vi,!, v!.(i) are elastic constants of the
('
layer referred to the principal material axes. For an isotropic layer (e.g., metal or
polymeric) we should take in Eqs. (5.70) E.? = E,!) = Ej, vi?(' - vJi(i) = vi,
-
G::,, = G; E;/2(1 + vi).
Consider the layer composed of unidirectional plies with orientation angle 4i.Using
Eqs. (4.69) we can express strains in the principal material coordinates as

(5.71)

and find the corresponding stresses, Le.,

where - - EIz/(1 - vyjvt,)) and E?, E!), GYj, vyj, v!/ are elastic constants of a
(;)
unidirectional ply.
Thus, Eqs. (5.69-5.72) allow us to find in-plane stresses acting in each layer or
elementary composite ply.
Compatible deformation of the layers is provided by interlaminar stresses z,,, zjr,
and a,. To find these stresses, we need to attract three-dimensional equilibrium
equations, Eqs. (2.5), which yield

(5.73)

Substituting stresses a,, a,.,and z.~~from Eqs. (5.4) and integrating Eqs. (5.73) with
due regard to forces that can act on the laminate surfaces we can calculate
transverse shear and normal stresses zx=,z,, and a,.