Page 242 - Mechanics Analysis Composite Materials

Chapter 5. Mechanics of laminates 221

In-plane strains of the layer, E,, E,,, and y,,, can be found using Eqs. (2.22) and
(5.1), (5.2) as

au,
ax
Ex = -= E; +ZK,,
3u.v 0
. ay . (5.3)
E,? =-= E, +ZK,,

where

ae, ae, ae,
aex
K, = - KVV'--, IC,,,=-+- .
ax ' aY ay ax
These generalized strains correspond to the following four basic deformations of the
layer shown in Fig. 5.3:
0 in-plane tension or compression (E:, E;),
0 in-plane shear (y:,,),
0 bending in xz- and yz-planes (K~,K,,), and
0 twisting (K,,,).
Constitutive equations for an anisotropic layer Eqs. (4.71), upon substitution of
Eqs. (5.3) yield

where, A,, =A,, are the stiffness coefficients of the material that can depend, in
general, on coordinate z.
As follows from Eqs. (5.4), stresses depend on six generalized strains E, y, and K
which are the functions of coordinates x and y only. To fulfil the derivation of
constitutive equations for the layer under study, we introduce the corresponding
force functions as stress resultants and couples shown in Fig. 5.4 and specified as
(see also Fig. 5.1)

-e -e -e
S S

--e -e -e