Nonlinear Dynamics

, Volume 80, Issue 1–2, pp 457–489 | Cite as

Understanding memristors and memcapacitors in engineering mechanics applications

  • Jin-Song Pei
  • Joseph P. Wright
  • Michael D. Todd
  • Sami F. Masri
  • François Gay-Balmaz
Original Paper


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.


Nonlinear hysteresis Memristor Memcapacitor Memristive system Memcapacitive system 

List of symbols






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

\(\sigma \)


\(\varepsilon \)


\(\varepsilon _t\)

Strain rate

\(\alpha \)

Strain absement


The first time derivative of \(r\)


Resisting force or characteristic force of an element


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

\(\rho \)

The first time integral of \(p\)

\({\mathbf {y}}\)

State variables, see Tables 2 and 3

\({\mathbf {z}}\)

State variables in Table 8


Internal state or intermediate variable in Sects. 4 and 5


Driving force, see Eq. (1), Fig. 25 and Table 8


Incremental memristance following Chua [9]


Incremental memdunctance following Chua [9]


See Table 2


See Table 2

\({\mathbf {g}}\)

See Table 2

\({\mathbf {f}}\)

See Table 2






Secant damping, see Table 3 and Fig. 15


Secant stiffness, see Table 3 and Fig. 15


Tangent stiffness, see Property 4


Power, see Table 3


Energy, see Table 3


See Sect. 3.3, especially Table 4







\(\varphi \), \(\phi \)

Flux linkage



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].


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Jin-Song Pei
    • 1
  • Joseph P. Wright
    • 2
  • Michael D. Todd
    • 3
  • Sami F. Masri
    • 4
  • François Gay-Balmaz
    • 5
  1. 1.School of Civil Engineering and Environmental ScienceUniversity of OklahomaNormanUSA
  2. 2.Division of Applied ScienceWeidlinger Associates Inc.New YorkUSA
  3. 3.Department of Structural EngineeringUniversity of California, San DiegoLa JollaUSA
  4. 4.Sonny Astani Department of Civil and Environmental EngineeringUniversity of Southern CaliforniaLos AngelesUSA
  5. 5.CNRS, Laboratoire de météorologie dynamiqueEcole Normale SupérieureParisFrance

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