ANALYTICAL METHODS FOR TEXTILE COMPOSITES
A A A B B B
A A A B B B
A A A B B B
B B B D D D
B B B D D D
B B B D D D
x
y
xy
x
y
xy
x
y
xy
x
y
xy
12 12 16 11 12 16
12 22 26 12 22 26
16 26 66 16 26 66
11 12 16 11 12 16
12 22 26 12 22 26
16 26 66 16 26 66
−
ε
ε
γ
κ
κ
κ
α
α
α
β
β
β
=
∆T
N
N
N
M
M
M
x
y
xy
x
y
xy
(5.30)
where the axes x and y lie in the plane of the plate;
ε
x
,
ε
y
, and
γ
xy
are mid-plane strains;
κ
x
,
κ
y
, and
κ
xy
are bending curvatures;
α
x
,
α
y
, and
α
xy
are plate thermal expansion
coefficients;
β
x
,
β
y
, and
β
xy
are plate thermal bending coefficients; N
x
, N
y
, and N
xy
are
in-plane force resultants; and M
x
, M
y
, and M
xy
are bending moments.
Table 5.5 summarizes the way the codes presented in this handbook deal with the
application of laminate theory to quasi-laminar textile composites. There are essentially
two approaches to determining the matrices A, B, and D :
(i) The properties of the entire plate can be computed at once, usually by defining a
unit cell spanning the whole thickness; e.g. Fig. 5-11b. The response of the cell is
computed under sufficient sets of boundary conditions representing in-plane loading
and bending to determine all components of the stiffness matrix. This is the approach
followed in the codes µTEX, CCMTEX and SAWC. In a nonperiodic or irregular
textile, the calculation over a unit cell can be replaced by a calculation over some
representative volume element of sufficient size to give consistent results for the
macroscopic stiffness. This is the approach of the code BINMOD.
(ii) The properties of a single ply can be determined first and the plate properties then
built up using laminate theory. The ply properties can be calculated by defining a unit
cell within a single ply (Fig. 5-11a) or by the methods of spatial averaging described in
Section 5.2 under "Modified Laminates." The code TEXCAD follows the unit cell
approach. PW, SAT5, and SAT8 also use unit cells to compute the matrices A, B, and
D for a single 2D layer, but leave to a subsequent application of conventional laminate
analysis the determination of the plate stiffness of a stack of such layers.
