  The ability to design and fabricate materials with tailored mechanical
properties, combined with immunity to damage, is a frontier of materials
engineering. For example, materials which are characterized by elastic
properties that depend on the position inside a medium are required in
applications where structural stability has to be combined with a soft and
compliant surface, like in impact protection and cushioning. A gradient in the
elastic properties can be built from a single material, varying gradually the
bulk porosity of the material or its geometrical structure. However, if such a
gradient is built into the material at production, damage or wearing over time
might expose unwanted elastic properties. Here, we implement a design principle
for a spatially inhomogeneous material based on topological band-theory for
mechanical systems. The resulting inhomogeneity is stable against wearing and
even cutting the material in half. We show how, by creating a periodic elastic
material with topological properties, one can create an intrinsically polar
behavior, where a face with a given surface normal is stiff while its opposing
face is soft.
