  In materials that undergo martensitic phase transformation, macroscopic
loading often leads to the creation and/or rearrangement of elastic domains.
This paper considers an example {involving} a single-crystal slab made from two
martensite variants. When the slab is made to bend, the two variants form a
characteristic microstructure that we like to call ``twinning with variable
volume fraction.'' Two 1996 papers by Chopra et. al. explored this example
using bars made from InTl, providing considerable detail about the
microstructures they observed. Here we offer an energy-minimization-based model
that is motivated by their account. It uses geometrically linear elasticity,
and treats the phase boundaries as sharp interfaces. For simplicity, rather
than model the experimental forces and boundary conditions exactly, we consider
certain Dirichlet or Neumann boundary conditions whose effect is to require
bending. This leads to certain nonlinear (and nonconvex) variational problems
that represent the minimization of elastic plus surface energy (and the work
done by the load, in the case of a Neumann boundary condition). Our results
identify how the minimum value of each variational problem scales with respect
to the surface energy density. The results are established by proving upper and
lower bounds that scale the same way. The upper bounds are ansatz-based,
providing full details about some (nearly) optimal microstructures. The lower
bounds are ansatz-free, so they explain why no other arrangement of the two
phases could be significantly better.
