# Understanding memristors and memcapacitors in engineering mechanics applications

- 645 Downloads
- 9 Citations

## Abstract

A significant event happened for electrical engineering in 2008, when researchers at HP Labs announced that they had found “the missing memristor,” a fourth basic circuit element that was postulated nearly four decades earlier by Dr. Leon Chua, who was also instrumental in developing the mathematical theories of memristive, memcapacitive, and meminductive systems, resulting in an entire class of “mem-models” that are the foundation of the present work. By applying well-known mechanical–electrical analogies, the mathematics of mem-models may be transferred to the setting of engineering mechanics, creating the mechanical counterparts of memristors, memcapacitors, etc. However, this transfer is nontrivial; for example, a new concept and state variable called “absement,” the time integral of deformation, emerge. We study these mem-models, which are characterized by a “zero-crossing” property that has interesting implications for nonlinear constitutive modeling, particularly hysteresis, and we identify some examples of “mem-dashpots” and “mem-springs,” which include displacement-dependent and variable dampers, the superelasticity found in shape-memory alloys, and the pinched hysteresis loops associated with self-centering structures. This work adds to the fast-growing body of literature on elements and systems labeled with “mem,” which is a basic branch of study in nonlinear dynamics.

## Keywords

Nonlinear hysteresis Memristor Memcapacitor Memristive system Memcapacitive system## List of symbols

- \(\dot{x}\)
Velocity

- \(x\)
Displacement

- \(a\)
Absement, first time integral of displacement, \(x\)

- \(\sigma \)
Stress

- \(\varepsilon \)
Strain

- \(\varepsilon _t\)
Strain rate

- \(\alpha \)
Strain absement

- \(\dot{r}\)
The first time derivative of \(r\)

- \(r\)
Resisting force or characteristic force of an element

- \(p\)
General momentum, the first time integral of \(r\)

- \(\rho \)
The first time integral of \(p\)

- \({\mathbf {y}}\)
- \({\mathbf {z}}\)
State variables in Table 8

- \(w\)
- \(u\)
- \(M\)
Incremental memristance following Chua [9]

- \(W\)
Incremental memdunctance following Chua [9]

- \(G\)
See Table 2

- \(F\)
See Table 2

- \({\mathbf {g}}\)
See Table 2

- \({\mathbf {f}}\)
See Table 2

- \(e\)
Effort

- \(f\)
Flow

- \(D\)
- \(S\)
- \(K\)
Tangent stiffness, see Property 4

- \(P\)
Power, see Table 3

- \(U\)
Energy, see Table 3

- \(a_0\)
- \(i\)
Current

- \(v\)
Voltage

- \(q\)
Charge

- \(\varphi \), \(\phi \)
Flux linkage

## Notes

### Acknowledgments

This study is partially funded by NSF CMMI 0626401 with Program Office, Dr. S.C. Liu. Part of this work was initiated during the first author’s sabbatical leave; she would like to thank Professor Jim Beck and Professor Jeff Scruggs for their hospitality. The authors would like to thank Professor Jim Beck for providing us with Eq. (48) and helpful comments to earlier drafts, Professor Jeff Scruggs, Professor Dennis Bernstein, and Dr. Giovanni Paziena for referring us to the following references, respectively, Ogata [30], Jeltsema and Scherpen[24], and Strukov [42]. The first author wishes to thank Professor Shirley Dyke for assistance in finding data for the PC4 specimen in Ricles et al. [35].

## References

- 1.ACI Committee 318: Building Code Requirements for Structural Concrete and Commentary. American Concrete Institute, Farmington Hills (2011)Google Scholar
- 2.Applied Technology Council: Evaluation and improvement of inelastic seismic analysis procedures, phase ii work plan. www.atcouncil.org (2001)
- 3.Bellenger, H., Duvel, J.P.: An analysis of tropical ocean diurnal warm layers. J Climate
**22**, 3629–3646 (2009)CrossRefGoogle Scholar - 4.Bernstein, D.S. (ed.): IEEE Control Systems Magazine, vol.
**29**. IEEE Control Systems Society, IEEE (2009)Google Scholar - 5.Caughey, T.K.: Random excitation of a system with bilinear hysteresis. J. Appl. Mech.
**27**, 649–652 (1960a)CrossRefMathSciNetGoogle Scholar - 6.Caughey, T.K.: Sinusoidal excitation of a system with bilinear hysteresis. J. Appl. Mech.
**27**, 640–643 (1960b)CrossRefMathSciNetGoogle Scholar - 7.Chang, T., Jo, S.H., Kim, K.H., Sheridan, P., Gaba, S., Lu, W.: Synaptic behaviors and modeling of a metal oxide memristive device. Appl. Phys.
**102**, 857–863 (2011)CrossRefGoogle Scholar - 8.Christopoulos, C., Tremblay, R., Kim, H.J., Lacerte, M.: Self-centering energy dissipative bracing system for the seismic resistance of structures: development and validation. ASCE J. Struct. Eng.
**134**(1), 96–107 (2008)CrossRefGoogle Scholar - 9.Chua, L.O.: Memrister—the missing circuit element. IEEE Trans. Circuit Theory
**CT–18**(5), 507–519 (1971)CrossRefGoogle Scholar - 10.Chua, L.O., Kang, S.M.: Memristive devices and systems. Proc. IEEE
**64**, 209–223 (1976)CrossRefMathSciNetGoogle Scholar - 11.Damic, V., Cohodar, M.: Bond graph based modelling and simulation of flexible robotic manipulators. In: Wolfgang Borutzky, R.Z., Alessandra, Orsoni. (ed.) Proceedings 20th European Conference on Modelling and Simulation (2006)Google Scholar
- 12.Di Ventra, M., Pershin, Y.V.: On the physical properties of memristive, memcapacitive, and meminductive systems. Nanotechnology
**24**(25), http://arxiv.org/abs/1302.7063 (2013) - 13.Di Ventra, M., Pershin, Y.V., Chua, L.O.: Circuit elements with memory: memristors, memcapacitors, and meminductors. Proc. IEEE
**97**, 1717–1724 (2009)CrossRefGoogle Scholar - 14.Dolce, M., Cardone, D., Marnetto, R.: Implementation and testing of passive control devices based on shape memory alloys. Earthq. Eng. Struct. Dyn.
**29**, 945–968 (2000)CrossRefGoogle Scholar - 15.Farrar, C.R., Worden, K., Todd, M.D., Park, G., Nichols, J., Adams, D.E., Bement, M.T., Fairnholt, K.: Nonlinear system identification for damage detection. Tech. Rep. LA-14353, Los Alamos National Laboratory (2007)Google Scholar
- 16.Ferri, A.A.: Friction damping and isolation systems. ASME J. Mech. Des.
**117**(B), 196–206 (1995)CrossRefGoogle Scholar - 17.Georgiou, P.S., Yaliraki, S.N., Drakakis, E.M., Barahona, M.: Quantitative measure of hysteresis for memristor through explicit dynamics. Proc. R. Soc. Math. Phys. Eng. Sci.
**468**, 1–20 (2012)CrossRefMathSciNetGoogle Scholar - 18.Guckenheimer, J., Holmes, P.: Nonlinear Oscillators, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol. 42. Springer, New York (1983)CrossRefGoogle Scholar
- 19.Hogan, N., Breedveld, P.C.: The Mechatronics Handbook. Chap 15. The Physical Basis of Analogies in Physical System Models. CRC Press (2002)Google Scholar
- 20.Ilbeigi, S., Jahanpour, J., Farshidianfar, A.: A novel scheme for nonlinear displacement-dependent dampers. Nonlinear Dyn.
**70**, 421–434 (2012)CrossRefMathSciNetGoogle Scholar - 21.Inman, D.J.: Engineering Vibration. Prentice Hall, Upper Saddle River (1994)Google Scholar
- 22.Jeltsema, D.: Memory elements: A paradigm shift in Lagrangian modeling of electrical circuits. In: Proceedings of the MathMod Conference, Vienna (2012)Google Scholar
- 23.Jeltsema, D., Dòria-Cerezo, A.: Mechanical memory elements: modeling of systems with position-dependent mass revisited. In: 49th IEEE Conference on Decision and Control, pp. 3511–3516. IEEE, Atlanta (2010)Google Scholar
- 24.Jeltsema, D., Scherpen, J.M.A.: Multidomain modeling of nonlinear networks and systems: energy- and power-based perspectives. IEEE Control Syst. Mag.
**29**, 28–59 (2009)CrossRefMathSciNetGoogle Scholar - 25.Jennings, P.C.: Periodic response of a general yielding structure. J. Eng. Mech. Div. Proc. Am. Soc. Civil Eng.
**90**(EM2), 131–166 (1964)Google Scholar - 26.Kalmár-Nagy, T., Shekhawat, A.: Nonlinear dynamics of oscillators with bilinear hysteresis and sinusoidal excitation. Phys. D
**238**, 1768–1786 (2009)CrossRefzbMATHMathSciNetGoogle Scholar - 27.Madhekar, S.N., Jangid, R.S.: Variable dampers for earthquake protection of benchmark highway bridges. Smart Mater. Struct.
**18**, 1–18 (2009)CrossRefGoogle Scholar - 28.Masri, S.F., Caughey, T.K.: A nonparametric identification technique for nonlinear dynamic problems. J. Appl. Mech.
**46**, 433–447 (1979)CrossRefzbMATHGoogle Scholar - 29.Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley-VCH, Weinheim (1995)CrossRefGoogle Scholar
- 30.Ogata, K.: System Dynamics, 4th edn. Pearson Prentice Hall, Upper Saddle River (2004)Google Scholar
- 31.Oster, G.F., Auslander, D.M.: The memristor: a new bond graph element. ASME J. Dyn. Syst. Meas. Control
**94**(3), 249–252 (1973)CrossRefGoogle Scholar - 32.Paynter, H.M.: Analysis and Design of Engineering Systems: Class Notes for M.I.T. Course 2.751. M.I.T. Press, Cambridge (1961)Google Scholar
- 33.Paytner, H.M.: The gestation and birth of bond graphs. http://www.me.utexas.edu/longoria/paynter/hmp/Bondgraphs.html (2000)
- 34.Priestley, M.J.N., Grant, D.N.: Viscous damping and seismic design and analysis. J. Earthq. Eng.
**9**(2), 229–255 (2010)Google Scholar - 35.Ricles, J.M., Sause, R., Peng, S.W., Lu, L.W.: Experimental evaluation of earthquake resistant posttensioned steel connections. ASCE J. Struct. Eng.
**128**(7), 850–859 (2002)Google Scholar - 36.Rosenberg, R.C., Karnopp, D.C.: Introduction to Physical System Dynamics. McGraw-Hill Series in Mechanical Engineering. McGraw-Hill Inc, New York–(1983)Google Scholar
- 37.Santos, F.P., Cismaşiu, C.: Shape memory alloys in structural vibration control. In: Experimental Vibration Analysis for Civil Engineering Structures (EVACES’07), FEUP, Porto, Portugal (2007)Google Scholar
- 38.Scruggs, J.T., Gavin, H.P.: The control handbook, 2nd edn. Earthquake Response Control for Civil Structures. CRC Press (2010)Google Scholar
- 39.Sivaselvan, M.V., Reinhorn, A.M.: Hysteretic models for deteriorating inelastic structures. ASCE J. Eng. Mech.
**126**(6), 633–640 (2000)CrossRefGoogle Scholar - 40.Sozen, M.A.: Hysteresis in structural elements. Applied Mechanics inEarthquake Engineering, ASME Annual Meeting, Applied Mechanics Division, vol. 8, pp. 63–98 (1974)Google Scholar
- 41.Strogatz, S.H.: Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering. Studies in Nonlinearity. Westview Press, Boulder (1994)Google Scholar
- 42.Strukov, D.: Memristors and their applications (2011)Google Scholar
- 43.Strukov, D.B., Snider, G.S., Stewart, D.R., Williams, R.S.: The missing memristor found. Nature
**453**, 80–83 (2008)Google Scholar - 44.Talasila, V., Golo, G., van der Schaft, A.J.: The wave equation as a port-Hamiltonian system, and a finite dimensional approximation. http://doc.utwente.nl/69144/1/1182.pdf (2002)
- 45.Vaz, A., Maini, A.K.: Modeling of soft materials: integrating bond graphs with finite element analysis. In: 14th National Conference on Machines and Mechanisms (NaCoMM09), NIT, Burgapur, India, pp. 247–252 (2009)Google Scholar
- 46.Visintin, A.: Differential Models of Hysteresis. Springer, Berlin (1994)zbMATHGoogle Scholar
- 47.Willam, K.J.: Encyclopedia of physics science and technology, 3rd edn. Constitutive Models for Engineering Materials, vol. 3, pp. 603–633, Academic Press (2002)Google Scholar
- 48.Williams, R.S.: How we found the missing memristor. IEEE Spectr
**45**(12), 28–35 (2008)CrossRefGoogle Scholar - 49.Wright, J.P., Pei, J.S.: Solving dynamical systems involving piecewise restoring force using state event location. ASCE J. Eng. Mech.
**138**(8), 997–1020 (2012)Google Scholar - 50.Zhu, S., Zhang, Y.: A thermomechanical constitutive model for superelastic SMA wire with strain-rate dependence. Smart Mater. Struct.
**16**, 1696–1707 (2007)CrossRefGoogle Scholar