Lecture
This lecture discusses the design of architectured materials with elastic energy closely approximating specified continuous functions. It begins by addressing the feasibility of such designs in one dimension, referencing earlier work that demonstrates this possibility. The instructor then transitions to two-dimensional scenarios, exploring the use of concentric helical lattices as springs in layered rectangular lattices. The mathematical foundation is established through the Stone-Weierstrass theorem, which supports the approximation of polynomial energies. The lecture further delves into the implications of Cauchy's relations in nonlinear elasticity, highlighting the significance of energies that can be approximated by pair potentials. The conclusion emphasizes that while the construction is optimal for certain energies, practical challenges remain in designing interpenetrating lattices. Overall, the lecture provides a comprehensive overview of the theoretical and practical aspects of bespoke elasticity in material design.