Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimization of a lattice spring model with elastoplastic conducting springs: A case study

Sakshi Malhotra, Yang Jiao, Oleg Makarenkov

Abstract

We consider a simple lattice spring model in which every spring is elastoplastic and is capable to conduct current. The elasticity bounds of spring $i$ are taken as $[-c_i,c_i]$ and the resistance of spring $i$ is taken as $1/c_i$, which allows us to compute the resistance of the system. The model is further subjected to a gradual stretching and, due to plasticity, the response force increases until a certain terminal value. We demonstrate that the recently developed sweeping process theory can be used to optimize the interplay between the terminal response force and the resistance on a physical domain of parameters $c_i.$ The proposed methodology can be used by practitioners for the design of multi-functional materials as an alternative to topological optimization.

Optimization of a lattice spring model with elastoplastic conducting springs: A case study

Abstract

We consider a simple lattice spring model in which every spring is elastoplastic and is capable to conduct current. The elasticity bounds of spring are taken as and the resistance of spring is taken as , which allows us to compute the resistance of the system. The model is further subjected to a gradual stretching and, due to plasticity, the response force increases until a certain terminal value. We demonstrate that the recently developed sweeping process theory can be used to optimize the interplay between the terminal response force and the resistance on a physical domain of parameters The proposed methodology can be used by practitioners for the design of multi-functional materials as an alternative to topological optimization.

Paper Structure

This paper contains 15 sections, 5 theorems, 34 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Algebraic formulas for the optimization functionals
  3. The methods used for numerical simulations
  4. Maximization of multi-functional properties under constraint on the cost of the material
  5. Maximization of strength in the absence of electrical properties
  6. Maximization of the strength-resistance functional $F_R$
  7. Maximization of the strength-conductance functional $F_G$
  8. Minimization of the cost of material under constraint on the multi-functional properties
  9. Minimization of the cost under constraint on the strength-resistance functional
  10. Minimization of the cost under constraint on the strength-conductance functional
  11. Conclusions
  12. Appendix: Conductance of a Wheatstone Bridge
  13. Appendix: Sweeping process framework for finite-time stability of elastoplastic systems with application to the benchmark example
  14. The outline of the general framework
  15. Computations for the benchmark example

Key Result

Proposition 4.1

In order to obtain the maximal strength of the network of Fig. fig1 keeping the cost of the network under a fixed value, only 2 springs connected in serial as shown in Fig. figF are needed.

Figures (11)

  • Figure 1: A system of 5 springs on 4 nodes with a displacement-controlled loading that locks the distance between nodes ① and ⑤.
  • Figure 2: The model of Fig. \ref{['fig1']} with springs 2,3,5 removed (i.e. with $c_2=c_3=c_5=0$). The arrow in the left spring indicates that the left spring will deform plastically as the displacement-controlled loading (double-sided arrow) increases.
  • Figure 3: Maximization of the functional $F_R$. The horizontal and vertical axis stay for $k_1$ and $k_2$ respectively. The radius of the disc centered at $(k_1,k_2)$ refers to the maximum of $F_R$ for the parameters $(k_1,k_2)$. The circles in red denote the points where the response force $F$ takes the value 0.75 and resistance of the system $R$ takes the value 3.33332. The circles in blue denote the points where $F=1$ and $R=2$. The circles in green are the cases where all the optimal values of $c_i^s$ are non zero.
  • Figure 4: Fig. \ref{['fig3a']} zoomed in.
  • Figure 5: The model of Fig. \ref{['fig1']} with spring 3 removed (i.e. with $c_3=0$). The bold springs denote the springs that will deform plastically as the displacement-controlled loading (double-sided arrow) increases.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Remark 1
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 5.1
  • Proposition 5.2