  We study the influence of epitaxial strain on the electronic properties of
InAs/GaSb composite quantum wells (CQWs), host structures for quantum spin Hall
insulators, by transport measurements and eight-band $\mathbf{k\cdot p}$
calculations. Using different substrates and buffer layer structures for
crystal growth, we prepare two types of samples with vastly different strain
conditions. CQWs with a nearly strain-free GaSb layer exhibit a resistance peak
at the charge neutrality point that reflects the opening of a topological gap
in the band-inverted regime. In contrast, for CQWs with 0.50\% biaxial tensile
strain in the GaSb layer, semimetallic behavior indicating a gap closure is
found for the same degree of band inversion. Additionally, with the tensile
strain, the boundary between the topological and nontopological regimes is
located at a larger InAs thickness. Eight-band $\mathbf{k\cdot p}$ calculations
reveal that tensile strain in GaSb not only shifts the phase boundary but also
significantly modifies the band structure, which can result in the closure of
an indirect gap and make the system semimetallic even in the topological
regime. Our results thus provide a global picture of the
topological-nontopological phase diagram as a function of layer thicknesses and
strain.
