  This work proves rigorous results about the vanishing-mass limit of the
classical problem to find a shape with minimal elastic compliance. Contrary to
all previous results in the mathematical literature, which utilize a soft mass
constraint by introducing a Lagrange multiplier, we here consider the hard mass
constraint. Our results are the first to establish the convergence of
approximately optimal shapes of (exact) size $\varepsilon \downarrow 0$ to a
limit generalized shape represented by a (possibly diffuse) probability
measure. This limit generalized shape is a minimizer of the limit compliance,
which involves a new integrand, namely the one conjectured by Bouchitt\'e in
2001 and predicted heuristically before in works of Allaire & Kohn and Kohn &
Strang from the 1980s and 1990s. This integrand gives the energy of the limit
generalized shape understood as a fine oscillation of (optimal)
lower-dimensional structures. Its appearance is surprising since the integrand
in the original compliance is just a quadratic form and the non-convexity of
the problem is not immediately obvious. In fact, it is the interaction of the
mass constraint with the requirement of attaining the loading (in the form of a
divergence-constraint) that gives rise to this new integrand. We also present
connections to the theory of Michell trusses, first formulated in 1904, and
show how our results can be interpreted as a rigorous justification of that
theory on the level of functionals in both two and three dimensions, settling
this open problem. Our proofs rest on compensated compactness arguments applied
to an explicit family of (symmetric) $\mathrm{div}$-quasiconvex quadratic
forms, computations involving the Hashin-Shtrikman bounds for the Kohn-Strang
integrand, and the characterization of limit minimizers due to Bouchitt\'e &
Buttazzo.
