The metric-restricted inverse design problem
Amit Acharya Marta Lewicka Mohammad Reza Pakzad
In Nonlinearity, 29, 1769-1797
We study a class of design problems in solid mechanics, leading to a variation on the
classical question of equi-dimensional embeddability of Riemannian manifolds. In this general new
context, we derive a necessary and sufficient existence condition, given through a system of total
diff erential equations, and discuss its integrability. In the classical context, the same approach
yields conditions of immersibility of a given metric in terms of the Riemann curvature tensor.
In the present situation, the equations do not close in a straightforward manner, and successive
diff erentiation of the compatibility conditions leads to a more sophisticated algebraic description
of integrability. We also recast the problem in a variational setting and analyze the infi mum value
of the appropriate incompatibility energy, resembling "non-Euclidean elasticity". We then derive a
Γ-convergence result for the dimension reduction from 3d to 2d in the Kirchhoff energy scaling
regime. A practical implementation of the algebraic conditions of integrability is also discussed.