  We report an investigation of the effect of substitution of Y for Tm in
$Tm_{1-x}Y_xVO4$ via low-temperature heat capacity measurements, with the
yttrium content $x$ varying from $0$ to $0.997$. Because the Tm ions support a
local quadrupolar (nematic) moment, they act as reporters of the local strain
state in the material, with the splitting of the ion's non-Kramers crystal
field groundstate proportional to the quadrature sum of the in-plane tetragonal
symmetry-breaking transverse and longitudinal strains experienced by each ion
individually. Analysis of the heat capacity therefore provides detailed
insights into the distribution of local strains that arise as a consequence of
the chemical substitution. These local strains suppress long-range quadrupole
order for $x>0.22$, and result in a broad Schottky-like feature for higher
concentrations. Heat capacity data are compared to expectations for a
distribution of uncorrelated (random) strains. For dilute Tm concentrations,
the heat capacity cannot be accounted for by randomly distributed strains,
demonstrating the presence of significant strain correlations between sites.
For intermediate Tm concentrations, these correlations must still exist, but
the data cannot be distinguished from that which would be obtained from a 2D
Gaussian distribution. The cross-over between these limits is discussed in
terms of the interplay of key length scales in the substituted material. The
central result of this work, that local strains arising from chemical
substitution are not uncorrelated, has implications for the range of validity
of theoretical models based on random effective fields that are used to
describe such chemically substituted materials, particularly when electronic
nematic correlations are present.
