Lamb wave assessment of fiber volume fraction in composites

TABLE I. Destructive test results to determine fiber volume fraction FVF.
fined as were estimated using common matrix values found in
Sample Layers Target FVF Destructive FVF Thickness mm
Q 11 m 4 Q 11 n 4 Q 22 2m 2 n 2 Q 12 4m 2 n 2 Q 66 ,
Q 22 n 4 Q 11 m 4 Q 22 2m 2 n 2 Q 12 4m 2 n 2 Q 66 , 5
40-1 16 0.40 0.380 2.00
50-1
60-1
16
16
0.50
0.60
0.529
0.590
1.78
1.57
Q 12 m 2 n 2 Q 11 m 2 n 2 Q 22 m 4 n 4 Q 12 4m 2 n 2 Q 66 ,
65-1 16 0.65 0.659 1.35 where mcos() and nsin(). The angle is defined as
70-1 16 0.70 0.690 1.35 positive for a counterclockwise rotation from the primed
40-2 32 0.40 0.395 3.84
laminate axes to the unprimed individual lamina axes.
50-2 32 0.50 0.518 3.40
60-2 32 0.60 0.580 3.40 From Eq. 5, the Q ij for the 0° and 90° laminas are given by
65-2 32 0.65 0.660 2.72
70-2 32 0.70 0.690 2.72
Q 11 0 deg Q 11 , Q 11 90 deg Q 22 ,
Q 22 0 deg Q 22 , Q 22 90 deg Q 11 ,
6
II. LAMB WAVE MODEL
Q 12 0 deg Q 12 , Q 12 90 deg Q 12 .
For a laminated composite with the 1-axis defined as the
fiber direction, the 2-axis transverse to the fibers, and the
3-axis being out of the plane of the plate, the stress–strain
relationship in an individual lamina is given by 18
The velocity of the extensional plate mode can be related
to the in-plane stiffness of a composite. 19 For propagation
in the 0° and 90° directions, these stiffnesses are A 11 and
A 22 , respectively. The extensional plate mode velocity is
related to the stiffness by
1 11 Q 12 0
2 Q 12 Q 22 0
6Q 0 0 Q 66 1
2
1 v 1 A 11
h
6,
for propagation in the 0° direction and by
7
where and represent the normal and shear stresses, respectively,
and and represent the normal and shear v 2 A 22
h
strains, respectively. The Q ij are the reduced stiffness components
8
and are defined in terms of the engineering param-
eters as 18
for propagation in the 90° direction. The values for the inplane
stiffnesses A 11 and A 22 can be calculated using Eqs.
2–6 if the engineering stiffnesses of the composite are
Q 11 E 1 /1 12 21 ,
known. If the density, , and overall thickness of the plate, h,
Q 22 E 2 /1 12 21 ,
2
are known as well, then the extensional mode velocity in the
0° and 90° directions can be computed using Eqs. 7 and
8.
Q 12 12 E 1 /1 12 21 ,
The effect of fiber volume fraction on composite lamina
parameters can be derived from a simple rule of mixtures
where E 1 and E 2 are the Young’s moduli in the longitudinal
and transverse directions, respectively, and 12 and 21 are
the major and minor Poisson’s ratios, respectively. The Poisson’s
approach. The density takes the form 20
f V f m V m
9
ratios in Eq. 2 are not independent quantities and are
related to each other by 18
and the elastic constants are given by 18
E 1 E 1 f V f E m V m ,
10
21 E 2
E 12 .
1
3
E m E 2 f
E 2
,
E 2 f V m E m V f
11
The in-plane stiffnesses for the entire plate, A 11 and 12 12f V f m V m ,
12
A 22 , are obtained by integrating the Q ij through the thickness
of the plate. These stiffness values are defined as 19 where is the density, E 1 and E 2 are the longitudinal and
transverse moduli, respectively, 12 is Poisson’s ratio, V f is
the fiber fraction, and V m is the matrix fraction. The f and m
h/2
A ij Q ij k dz, i,j1,2, 4 subscripts represent the constituent properties of the fiber
h/2
and matrix, respectively, and quantities without subscripts
represent the composite lamina properties.
where h is the plate thickness and the subscript k represents
each lamina. The Q ij are the transformed stiffness coefficients
which take into account the orientation of each ply
with respect to the wave propagation direction and are de-
In order to predict the composite properties as a function
of fiber volume, the properties of the fiber and matrix must
be obtained. The elastic modulus of the matrix was obtained
from the manufacturer 21 and the density and Poisson’s ratio
1400 J. Acoust. Soc. Am., Vol. 104, No. 3, Pt. 1, September 1998 Seale et al.: Lamb wave assessment of composites 1400