hysteresis - university of california, san...

J. D. Goddard*

University of California, San DiegoDepartment of Aerospace and Mechanical Engineering

UKC2008US-Korea Conference

onScience, Technology and Entrepreneurship

San Diego, CA14-17 August 2008

Hysteresis

J. D. Goddard*

University of California, San DiegoDepartment of Aerospace and Mechanical Engineering

UKC2008US-Korea Conference

onScience, Technology and Entrepreneurship

San Diego, CA14-17 August 2008

Hysteresis

*referenced herein as “JDG”

Objectives

• Survey briefly a vast literature on phenomenological modeling of rate-independent hysteresis in processes ranging from ferromagnetism to plasticity, and to connect to the literature on polymer viscoelasticity;

• Show that generalized “hypoplasticity”, motivated by the mechanics of granular media and soils, provides a useful framework for hysteresis modeling, and

• Question validity of homogenization according to the classical PIP (Prandtl-Ishlinskii-Preisach) hysteresis modeling which is widely employed for ferromagnetic materials.

Brief history (Ewing to Preisach)• Origins of term: Ewing, J.A. “On the Production of Transient Electric Currents in Iron and Steel Conductors by Twisting Them When Magnetised or by Magnetising Them When Twisted. [Abstract]”, Proc. Roy. Soc. Lond., 33, 21-23,1881.


• Landmark prior work: Boltzmann, L. “Zür Theorie die elastischen Nachwirkung” [On the Theory of the Elastic After Effect] Wien Ber. 275-306, 1876. Boltzmann superposition, which covers a wide range of linear, generally rate-dependent phenomena, represents a first approximation to various nonlinear integral models of viscoelasticity.



• (“Hypoplastic”) scalar ODE model & hysteretic phase transitions: Duhem, P. 1887-1901.



• Influential “after effect”: Preisach, F. Über die magnetische Nachwirkung”, Z. Phys., 94, 277-302, 1935. (references to “elastischen Nachwirkung” but not to Boltzmann)

• (“Hypoplastic”) scalar ODE model & hysteretic phase transitions: Duhem, P. 1887-1901.

Modern Works

Modern Works• Justification and extension of Boltzmann theory, based on fading memory: Coleman, B.D. & Noll, W. ARMA 6, 355-370,1960 & Rev. Mod. Phys. 33, 239- 49,1961.


• White, J.L. & Metzner, A.B. , J. Appl. Polym. Sci. 7,1869-91,1963, and Bogue, D., I&EC Fund. 5, 253-59, 1966, models with stress or strain dependent relaxation that can exhibit rate-independent plastic (glassy) limits at large average strain rates.


• White, J.L. & Metzner, A.B. , J. Appl. Polym. Sci. 7,1869-91,1963, and Bogue, D., I&EC Fund. 5, 253-59, 1966, models with stress or strain dependent relaxation that can exhibit rate-independent plastic (glassy) limits at large average strain rates.

• Related integral models for the mechanics and the ferromagnetism of rate- independent materials:

Pipkin, A.C. & Rivlin, R.S. ZAMP 16, 313-26, 1965, (rheological model with rate-independent history effects), & J. Math. Phys. 8, 878-883,1967 (corresponding integral model for ferromagnetism).

Bouc, R., Acustica 24, 16-25, 1971 proposes a special case of the Pipkin- Rivlin 1967 model for ferromagnetism. Widely cited in the literature on hysteresis, Bouc overlooks Coleman & Noll and Pipkin & Rivlin, referencing older work of Volterra (1928).

JDG 1982-84 employs Pipkin & Rivlin’s (1965) idea for “purely dissipative” materials without characteristic time but with “effaceable memory”. (cf. J. Fluid Mech. 568, 1-17, 2006 - “Parametric hypoplasticity” for granular media.)

Some key modern references


• Differential model for ferromagnetism: Coleman, B.D. and Hodgdon. M, IJES 24, 897,1986, & ARMA 99, 375, 1987


• Visintin, A. “Differential Models of Hysteresis”, Springer, 1994. Excellent survey, including rheological models, is closest in spirit to the present work (but based on set-valued functions, convex analysis and subdifferential calculus).

• Brokate, M. and Sprekels, J. “Hysteresis and Phase Transitions”, Springer, 1996. Mathematical distillation of previous works, including phase transitions.

• Bertotti, G. and Mayergoyz, I. (eds) “The Science of Hysteresis”, Academic Press, 2006 (3 Vols.) Encyclopaedic survey, updating prior works and covering mathematical modelling and a wide range of physical applications. (Vol. I, Chapts. 1-2 provide distillation of above books, with updates.) *



• Visintin, A. “Differential Models of Hysteresis”, Springer, 1994. Excellent survey, including rheological models, is closest in spirit to the present work (but based on set-valued functions, convex analysis and subdifferential calculus).

• Brokate, M. and Sprekels, J. “Hysteresis and Phase Transitions”, Springer, 1996. Mathematical distillation of previous works, including phase transitions.

• Bertotti, G. and Mayergoyz, I. (eds) “The Science of Hysteresis”, Academic Press, 2006 (3 Vols.) Encyclopaedic survey, updating prior works and covering mathematical modelling and a wide range of physical applications. (Vol. I, Chapts. 1-2 provide distillation of above books, with updates.) *

* referred hereafter as B&M


Varieties of hysteresis*

* Brokate & Sprekels 1996

Mode-switching (continuous) State-switching (discontinuous)

– Standard elastoplastic model, (loading-unloading, hardening),– Ferromagnetic B-H curves,– Soil moisture exchange

– Micromagnetic domains,– Shape-memory effects,– Granular force chains?





– Micromagnetic domains,– Shape-memory effects,– Granular force chains? **





– Micromagnetic domains,– Shape-memory effects,– Granular force chains? **

**JDG, Ann. Rev. Fluid Mech. 35, 113, 2003





– Micromagnetic domains,– Shape-memory effects,– Granular force chains?

• Preisach (1935) modeling, which superposes right-hand model to obtain left-hand model, is standard in ferromagnetism.

**

**JDG, Ann. Rev. Fluid Mech. 35, 113, 2003

Elementary hysteresis models*

* Visintin, Chapt. 1 of B&M, who presents rheological analogs for (b) and (c).


(a) “Relay” State switching (Preisach 1935)




(b) “Play” State switching (Ishlinskii 1944)





(c) “Stop” Mode switching (Prandtl 1928)







**





** Works in parentheses propose composite models, with (b) and (c) representing mechanical (Reuss-Voigt) duals.


**

Parametrichypoplasticity


**


**Kolymbas 2000 (Karlsruhe School of Geomechanics, Gudehus, Wu, Bauer et al.), who employ the term “hypoplastic”. Properly dissipative (JDG, Plasticity 2006)?

**

Generalizedhypoplasticityfromviscoelasticity


*


* This transformation and similar plasticity models are obtained from viscoelasto- plastic models proposed by Bogue, White et al. for polymers in ‘60s-’70s, in the limit of large average strain rate.

*


* This transformation and similar plasticity models are obtained from viscoelasto- plastic models proposed by Bogue, White et al. for polymers in ‘60s-’70s, in the limit of large average strain rate.

*

**


* This transformation and similar plasticity models are obtained from viscoelasto- plastic models proposed by Bogue, White et al. for polymers in ‘60s-’70s, in the limit of large average strain rate. **JDG (1982-4), papers on thixotropy and plasticity, overlooking Pipkin & Rivlin 1967

*

**

Example

Single-mode responseτ=0.1:

γ=0.1 1 10

τ=2.0:

γ=0.1 1 10

Two-mode response

τ=[0.01,1], µ=[0.5,0.5] :

γ=0.1 1 10

Two-mode response

τ=[0.01,1], µ=[0.5,0.5] :

γ=0.1 1 10

• The “overshoot” at large τ or γ may be an artifact of description of elasticity using Jaumann stress rate. It does not occur with certain of the other convected (Oldroyd) rates.

Two-mode response

τ=[0.01,1], µ=[0.5,0.5] :

γ=0.1 1 10

• The “overshoot” at large τ or γ may be an artifact of description of elasticity using Jaumann stress rate. It does not occur with certain of the other convected (Oldroyd) rates.

• This model exhibits effaceable memory. Is this a general property of properly dissipative elastoplastic models?

Singular hypoelastic & PIP models

w

v

Comments on PIP modeling

• Despite its prominence in the literature on ferromagnetism and the mathematics of hysteresis, it is not clear the superpositon technique is generally valid.*

• Can anything be salvaged, by connection to contemporary methods of homogenization for non-linear inelastic microstructures?**

• If so, then some suitable modification may be useful in the homogenization of micromechanics and micromagnetics, and in the description of coupling between the two (e.g. piezomagnetism and “magnetoplasticity.”)

* already questioned in prior work, e.g. Visintin, Ch. 1 of B&M** e.g. Fleck and Willis, Mech. Mat. 38, 702, 2004.

Conclusions and questions

• “Generalized hypoplasticity” provides an extremely convenient, possibly universal description of hysteresis.

• Traditional (PIP) homogenization needs revision, for the possible application to a wide range of hysteretic phenomena, including “magnetoplasticity”.

• Is “effaceable memory” guaranteed by a properly dissipative form of hypoplasticity?

• To what extent can rate-independent hysteresis describe cold-working or high-strain rate processing of polymers?

* subject to certain requirements on dissipativity (JDG 2006), which may be easier to formulate for magnetism than for plasticity (since magnetic power H·dB/dt is given explicitly by the usual ODEs).

** Ewing, J.A. 1881

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