Fig. 2: Identification of optimal nonlinear constitutive response via DEM simulation for a single impact condition for the case of peak transmitted kinetic energy minimization.
From: Customizable wave tailoring nonlinear materials enabled by bilevel inverse design
a Feasible solutions of nonlinear spring coefficients c2 and c3. The black line represents \({c}_{2}=-\sqrt{3{c}_{3}}\), the red line indicates c2 = − (1 + 3c3)/2, the blue line is c2 = − 3c3, and the green line is the zero strain energy throughout the whole range, c2 = − 3/2 − 3c3/4. b Non-dimensional force-extension relationship of the best performing nonlinear spring (f(Δ x) = Δ x + 5.88Δ x2 + 9.65Δ x3) along with an example of a nearby underperforming (bad) nonlinear spring of f(Δ x) = Δ x − 6.6Δ x2 + 11Δ x3 (blue) and an energy locking bistable spring of f(Δ x) = Δ x − 5.26Δ x2 + 6.75Δ x3 (magenta). These three cases result in KE ratios of 0.0398 (best), 1.2257 (bad), and 0.1448 (energy locking bistable). Circles and triangles indicate the maximum compressive and tensile strain, respectively. c Ratio of maximum kinetic energy of the nonlinear spring to the one of a linear spring at the last particle as a function of nonlinear spring coefficients for the impact condition of M/M0 = 0.05 and V/V0 = 1, with ζ = 0.01. The lines from (a) are overlaid, the star marker denotes the point of best performance, the triangle indicates the nearby case, and the square represents the bistable case. Normalized kinetic energy of the (d) best performing nonlinear, (e) underperforming (bad) nonlinear, and f linear material. Source data are provided as a Source Data file.
