Bounds are here derived for the effective bulk modulus in heterogeneous media, denoted by
k
∗
k*
, using the two standard variational principles of elasticity. As trial functions for the stress and strain fields we use perturbation expansions that have been modified by the inclusion of a set of multiplicative constants. The first order perturbation effect is explicitly calculated and bounds for
k
∗
k*
are derived in terms of the correlation functions
⟨
μ
′
(
0
)
k
′
(
r
)
k
′
(
s
)
⟩
\left \langle {\mu ’(0)k’(r)k’(s)} \right \rangle
and
⟨
[
k
′
(
r
)
k
′
(
s
)
/
μ
(
0
)
]
⟩
\left \langle {\left [ {k’(r)k’(s)/\mu (0)} \right ]} \right \rangle
where
μ
′
\mu ’
and
k
′
k’
are the fluctuating parts of the shear modulus
μ
\mu
and the bulk modulus,
k
k
, respectively. Explicit calculations are given for two phase media when
μ
′
(
x
)
=
0
\mu ’(x) = 0
and when the media are symmetric in the two phases. Results are also included for the dielectric problem when the media are composed of two symmetric phases.