"There are some merits to filling voids in an automobile's structure with a foam with a high flextural modulus of elasticity (or modulus of elasticity in shear), however as previously mentioned, I would tend to be concerned about moisture entrapment and the life expectancy of a foam subjected to constant mean-plus-alternating stresses.
In my line of work, we do a lot of testing using differing types of foam, even with the highest quality foams, there is a life-expectancy after which the response of the foam cannot be expected to equal to that of new foam. This is especially true with Polyurethane Ether and Ester (different manufacturing processes) foams, which tend to degrade rather rapidly especially in dense formulations. In my industry, typical transit drop events are limited to 10-20 full response cycles before the cushioning material is considered "spent".
As such, I would expect an immediate improvement in the stiffness of the chassis, however I would expect that to degrade as time goes on, where a properly designed stiffening system should have a life expectancy equal to that of the vehicle it is being installed on.
I might try foam filling the chassis on a car that I was planning on running at the "24hr of lemons" races, but not something that I would plan on doing to a car that I was planning on keeping for more than a couple years...
Just my two cents, you can fab or buy a stiffening kit for any car, also could create a 'cage as an integral part of your stiffening system, safety and performance in a single unit...