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 $mathbf{kcdot p}$ 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 $mathbf{kcdot p}$ 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.
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