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Currently there seem to be two efforts under way:
The latter source (an article in the Popular Mechanics by Anatoly Zak) intimates that structural creep is a major engineering hurdle.
However, I'm not convinced it's the only one.
What are the engineering problems that we have to watch out for while building and deploying inflatable structures in LEO and beyond?
Strong and Flexible generally means heavy. The material has to be strong enough to handle the stresses of both the internal pressure when inflated, but also the localized pressures during inflation.
As the Russians discovered, inflation sequencing can go awry. This can result in a variety of issues, from wrapping around the launching craft, to binding against itself into the wrong shape.
Many projects tested in pressure react differently in the lack of pressure. It must be accounted for. It affects lubricants, expanding pressure-driven components, and a variety of other factors.
Microgravity can allow some aspects to deform in ways that can be anticipated, but in gravity tend to be overlooked.
At present, all such proposals are either for fuel depots or for habitat modules. The fuel depot proposals are for planetary use, either on Luna or Mars, and so have related considerations, but are not "inflatable space modules", and thus beyond the question
Which leaves habitation modules.
Modules must provide protection for occupants from 2 forms of radiation, both high energy particles (solar wind) and electromagnetic radiation (especially US, Gamma, and X). Spacecraft at present do not provide adequate protection from Gamma, X, and particulate radiation, so lifetime space travel limits are imposed by NASA. The particulates are supposedly better stopped by liquids rather than solids, but shipping that much liquid up isn't cheap. It is, however, a 1-time deal for a long-term module.
The resistance to microbody interactions, specifically, being hit by mirco-asteroidal bodies and space-junk, requires a specific set of engineering challenges. Note that space-suits share this requirement, and extensive NASA documentation is available for them.
An inflatable structure still may require additional fittings inside. A habitation module will, at a minimum, require air circulation fittings, while a science lab will require workspaces. How to attach and configure these is a significant issue.
Humans are not well adapted to enclosed spaces. This is part of the reason for inflatable modules; more open spaces are expected to be helpful in long duration spaceflight. Also, view-ports, privacy enclosures, designated work spaces, and similar considerations require engineering consideration.
The proposed inflatable modules are intended to reduce the launch weight per unit volume of habitation modules. Still, the limit on size is a function of weight lofted, and individual launchers still have relatively limited payloads.
While not strictly an engineering consideration, no engineer can work on a project without considering the costs. All projects have budgets, and Engineers who can't keep designs within them don't see their work built.
Spaceflights have very limited crews. If the inflation and assembly require 5 men, then it's going to be impractical for a soyuz launch to inflate, but the ISS could do so at a changeover.
It can burst as did one of the earlier, 1991 attempts to produce a large inflatable object. That has happened because the envelope hitched an antenna. during deployment.
Or as did the IRT -- an even earlier attempt (1984). Because "the lanyards pulled loose from the orbiter instead of pulling out of the canister. The nitrogen inflation cartridge did fire as scheduled, and the balloon burst as it began to inflate inside its canister."
Also it can wrap around your spacecraft during inflation phase. See the picture number three there. Note the Progress space craft completely covered by the thin envelope.
