Geology Reference
In-Depth Information
and
6.6.7
). Linear structures, such as those in fiber-reinforced materials, and
lamellar structures, such as are produced in eutectic and eutectoid systems, will not
be considered specifically (see Kelly
1966
; Kelly and Nicholson
1971
; Le Hazif
1979
; Piatti
1978
).
We first mention some structural aspects of polyphase materials, restricting
consideration mainly to two-phase materials for the sake of simplicity in bringing
out the essential points, although analogous properties will generally apply also to
materials of more than two phases. A preliminary point to note is that, in order for
a more or less equiaxed grain shape of the phases to persist, the interfacial energies
between the phases must not differ greatly from the grain boundary energies in the
pure phases; otherwise, depending on the relative volumes of the phases, there will
be a tendency to effects such as the spreading of one phase along three-grain edges
or the grain boundaries of the phases (Smith
1964
; Waff and Bulau
1979
).
Given an equiaxed grain structure, one of the phases in a two-phase material
can always be expected to be continuous. However, continuity in the second phase
will only exist if its volume fraction is more than about one third. Of course, the
second phase can be continuous at much smaller volume fractions in case of
nonequiaxed grain structure or nonrandom distribution, as illustrated in the con-
nectivity of pore space at low porosities. For the analysis of geometrical mea-
surements of two-phase structures, see Underwood (
1970
), Exner (
1983
) and
Exner and Hougardy (
1988
). A simple ''law of mixtures''
X
¼
X
A
v
A
þ
X
B
v
B
þ
...
ð
6
:
57
Þ
relating an aggregate property X to the phase properties X
A
... and volume frac-
tions v
A
... which is applicable to properties such as density, cannot be applied to
mechanical properties without further considerations. The mechanical properties
depend on the distribution of the phases, which influences the boundary conditions
applicable to the individual grains, and therefore a structure of the aggregate also
has to be specified. However, limits on the mechanical behavior of the aggregate
can be obtained by reference to two extreme cases of ''parallel'' and ''series''
arrangement of the phases (Fig.
6.22
). These arrangements correspond, respec-
tively, to the uniform strain case
r
¼
r
A
v
A
þ
r
B
v
B
þ
......
ð
6
:
58
Þ
and the uniform stress case
e
¼
e
A
v
A
þ
e
B
v
B
þ
......
ð
6
:
59
Þ
where r, e are the macroscopic stress and strain, respectively, for the aggregate and
r
A
;
e
A
;
v
A
;
are the local stress, strain, and volume fraction for the phase A...
Application of Eqs. (
6.58
) and (
6.59
) to linear elastic deformation leads to the
Voigt and Reuss limiting cases, respectively, for the elastic constants of the
aggregate:
E
¼
E
A
v
A
þ
E
B
v
B
þ
......
ð
6
:
60
Þ
