Physica B 306 (2001) 132–136

Models of hysteresis in the framework of thermomechanicswith internal variables

Davide Bernardini*

Dipartimento di Ingegneria Strutturale e Geotecnica, Universit !a di Roma ‘‘La Sapienza’’, Via Eudossiana 18, 00184 Roma, Italy

Abstract

The Ziegler–Green–Naghdi approach to the thermomechanics with internal variables is used to model the hystereticbehavior of a one-dimensional system with a single internal variable. In this framework, models of hysteresis arise fromproper assumptions on the dissipation function. Two examples from elastoplasticity and pseudoelastic behavior of

shape memory alloys are presented. r 2001 Elsevier Science B.V. All rights reserved.

PACS: 0.570.Ln; 46.05.+b; 83.10.Ff

Keywords: Thermomechanics; Hysteresis; Internal variables

1. Introduction

The macroscopic behavior of solids is ofteninfluenced by phenomena that take place at finerscales and can give rise to hysteresis [1]. A fewexamples are: phase transformations, dislocationsmotion, microcracking. In the most commonapproach to the thermodynamics with internalvariables, microscopic phenomena are taken intoaccount, in an overall way, by allowing theconstitutive equations to depend on some internalvariables that do not appear in the balanceequations and are prescribed, at the outset, byfurther constitutive equations [2,3].

In this paper, a different framework, henceforthnamed Ziegler–Green–Naghdi approach [4,5] (devel-oped also in Ref. [6]), is used to model the quasi-

static hysteretic behavior of a spatially homoge-neous one-dimensional thermomechanical system B

with a single internal variable a; i.e. a system whosephysical attributes are assumed to be described bythe following time-dependent quantities:

free energy: c rate of energy dissipation: Gstress: s strain: etemperature: W entropy: Zinternal variable: a rate of heat supply: r

The key feature of the approach is to consider, asbasic constitutive ingredients, two functions ex-pressing the free energy and the rate of energydissipation, without assuming at the outset anyevolution equation for the internal variable. It isshown that, by enforcing the validity of thebalance equations for all possible environments,a strong structure is endowed on the possibleinternal variable evolutions. When complementedby a further constitutive assumption, this structure

*Tel.: +39-06-44585276.

E-mail address: davide.bernardini@uniroma1.it

(D. Bernardini).

0921-4526/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 9 9 2 - 9

enables to determine the internal variable evolu-tion. Hysteresis models arise in this frameworkwhen the dissipation function is positively homo-geneous of degree one in the internal variable rate.

2. Basic equations

A set p :¼ c;G;s; e; W; Z; a; rf g; of functions½t1; t2�-R; belonging to a suitable space P is saidto be a thermomechanical process for B on the timeinterval ½t1; t2�: Among the elements of P; onlythose consistent with balance equations, entropyinequality and constitutive equations are candidateto represent actual physical circumstances experi-enced by B and are, for this reason, calledadmissible. An element pAP is said to be consistentwith balance equations and entropy inequality if

’c ¼ s’e� Z ’W� W’Zþ r; ð1Þ

’Z ¼GWþ

r

W; ð2Þ

GX0; ð3Þ

where superposed dot denotes differentiation withrespect to the time t: According to Ref. [7], adistinction between internal and external constitu-tive equations is made. The first ones describe theinternal constitution of B by establishing arelation between two disjoint subsets of p; thesecond ones describe the interactions between B

and the environment via relations between pAPand some functions of time q:

The notation s : ¼ e;W; af gAS ¼ R Rþ�

Ag is introduced for the arguments of the internalconstitutive functions, where ADR is the set ofvalues that the internal variable can take on thebasis of its physical meaning. A special kind ofequations, that enables to recover some importantcases, is considered. Specifically, an element pAPis said to be consistent with the internal constitu-tive equations if it satisfies, for any relevant t:

sðtÞ ¼ #s sðtÞð Þ; ZðtÞ ¼ #Z sðtÞð Þ; ð4Þ

cðtÞ ¼ #c sðtÞð Þ; GðtÞ ¼ #G sðtÞ; ’aðtÞð Þ; ð5Þ

where #s; #Z; #c : S-R and #G : S R-R; are col-lectively called response functions R :¼

#s; #Z; #c; #Gn o

AR: Moreover, p is said to be consis-tent with external constitutive equations if it satisfies

e ¼ a; W ¼ b; ð6Þ

where a : ½t1; t2�-R and b : ½t1; t2�-Rþ; are collec-tively called generalized loads q :¼ a; bf gAQ; withthe assumption that all the environments describedby the model are in one-to-one correspondencewith the set Q: Each choice of aðtÞ in a suitablespace M defines a microstructural evolution for B

on ½t1; t2�; at this juncture, to none of them isassociated a privileged role.

3. Restrictions induced by the balance equations

and entropy inequality

For given R; any choice of q; a; r determines,via (4)–(6), a unique thermomechanical processpR q; a; rð Þ consistent with the constitutive equa-tions defined by R and q: However, in general,pR q; a; rð Þ is not admissible since the fulfilment ofbalance equations and entropy inequality is notguaranteed. Even if, for given R; q; a; the heatsupply %r solving Eq. (2) is used, pR q; a; %rð Þ is stillnot admissible since nothing guarantees the fulfil-ment of Eqs. (1) and (3). While, in principle, forany R and q one can search those a’s that give riseto admissible processes, it is useful to look forspecial classes of response functions R andmicrostructural evolutions a that automaticallyensure the admissibility of all processes consistentwith them for any environment.

A set of response functions R and a micro-structural evolution a are said to be consistent withbalance equations if the processes pR q; a; %rð Þ areconsistent with balance equations for all environ-ments qAQ:

Proposition 1. Let RAR and aAM be a set ofresponse functions and a microstructural evolution,then: ðiÞ R is consistent with balance equations if andonly if the following relations hold true:

#s ¼q #cqe

; #Z ¼ �q #cqW

: ð7Þ

ðiiÞ a is consistent with balance equations and theinternal constitutive equations defined by R if and

D. Bernardini / Physica B 306 (2001) 132–136 133

only if, at any time

#P’a� #G ¼ 0; ð8Þ

where

#P :¼ �q #cqa

ð9Þ

Proof. For the generic choice q ¼ a; bf g then s ¼q; af g and

pR s; %rð Þ ¼ #cðsÞ; #GðsÞ; #sðsÞ; #ZðsÞ; s; %rn o

ð10Þ

with %r solving Eq. (2). Suppose first that pR q; a; %rð Þis consistent with balance equations for any q:Substituting %r into Eq. (1), taking into account(4)–(6) then

q #cqe

� #s

!’aþ

q #cqW

þ #Z

!’bþ

q #cqa

’aþ #G ¼ 0 ð11Þ

holds for any qAQ: Due to the arbitrariness of thefunctions a; b and the affinity with respect to ’aand ’b; Eqs. (7) and (8) follow. Conversely, ifEqs. (7)–(8) are assumed to be fulfilled, thenbalance equations are satisfied for any qAQ: &

Since G is prescribed via the response function#G; entropy inequality will always hold true if #Gyields non-negative values for any choice of itsarguments. Moreover, if the processes with con-stant internal variables involve no dissipation, itmust be #Gð . ; 0Þ ¼ 0: A set of response functions

Radm ¼q #cqe

;�q #cqW

; #c; #G

( )ð12Þ

with #GX0 and #G . ; 0ð Þ ¼ 0 is said to be admissibleand turns out to be completely characterized by #cand #G:

The restrictions (7) among the response func-tions coincide with those that are usually obtainedby the enforcement of the entropy inequality.Here, on the contrary, they follow from balanceequations whereas entropy inequality yields arestriction on #G: The additional condition (8), forfixed #c and #G; defines a constraint among q and athat selects, among all aAM those consistentwith balance equations and with the constitutive

equations defined by R and q

MadmR ðqÞ :¼ aAM7 #Pðq; aÞ’a� #Gðq; a; ’aÞ ¼ 0

� �:

4. Hysteresis

Let us now assume that the dissipation function#G is rate-independent, i.e. it is a positivelyhomogeneous function of ’a: #Gð�; l’aÞ ¼ l #Gð�; ’aÞ forany lARþ: In this simple one-dimensional setting,this assumption, together with those leading to thedefinition of Radm; require that

#G ¼ L1 ’aþ � L2 ’a�; ð13Þ

where xþ :¼ max x; 0f g; x� :¼ min x; 0f g and L1ðsÞ;L2ðsÞ are constitutive functions continuous, non-negative and monotone in a (Fig. 1).

Consequently, the set of the admissible micro-structural evolutions becomes

MadmR ðqÞ :¼ aAM7 #P’a� L1 ’aþ þ L2 ’a� ¼ 0

� �:

For any given s; denoting P :¼ #PðsÞ; the followingclasses of microstructural evolutions are admissi-ble:

* ’a ¼ 0 is admissible irrespective of the value ofP;

* ’aX0 is admissible if P ¼ L1;* ’ap0 is admissible if P ¼ �L2:

This can be written compactly as

’aAFðPÞ ¼

f0g if PAKDR;

Rþ if P ¼ L1;

R� if P ¼ �L2;

8><>: ð14Þ

where K may be any subset of R: The choice of Kis not restricted by the balance equations andhence is a constitutive information characterizing

-Λ2Λ1

Γ)

α&

-Λ2

Λ1

α&

Π

Fig. 1. Rate-independent dissipation function and internal

variable rate versus driving force relationship.

D. Bernardini / Physica B 306 (2001) 132–136134

the features of B: To this end, a natural assump-tion seems (Fig. 1)

K ¼ �L2;L1½ �: ð15Þ

In this case, FðPÞ becomes the normal cone to K atP and (14) can be rewritten

’aANKðPÞ: ð16Þ

If, at any s; the two equations P� L1 ¼ 0 andPþ L2 ¼ 0 can be solved for a giving rise toarðqÞpalðqÞ then Eq. (16) can be written also as

’aA�NCðqðtÞÞðaÞ ð17Þ

with C qðtÞð Þ :¼ ar qðtÞð Þ; al qðtÞð Þ½ �: Given qðtÞ and asuitable initial condition a0; Eq. (17) definesgeneralized play in the terminology of Ref. [8] ora special case of sweeping process in the terminol-ogy of Ref. [9]. The above assumptions induce,therefore, hysteresis in the relation among theinternal variable a and the generalized loads q: Thehysteresis then propagates to the other quantitiesthrough the dependence of the, non-hysteretic,internal constitutive equations (4) on a:

It is remarked that, in analogy to Refs. [10,11],the constitutive hypothesis (15), that lead to (16),can be shown to be equivalent to assume at theoutset the validity of the maximum dissipationprinciple. It therefore provides a mean to pick up,among the admissible actual microstructural evo-lutions Madm

R ðqÞ; the actual one consistent with theenvironment q:

In the next sections, two examples are discussed.Since the aim is simply to highlight the featuresof the above framework, the expressions chosen forthe constitutive functions are purposely rough.Richer models can be obtained, along the same line,by different free energy and dissipation functions orincreasing the number of internal variables.

5. One-dimensional hardening elastoplasticity

In this case, the internal variable a is interpretedas a plastic strain representing the macroscopickinematic effect of the microscopic dislocationpattern. The set of admissible values for a isA ¼ R: The interaction with the environment arespecified by prescribed strain and temperaturehistories while the heat supply %r necessary to

sustain such generalized loads is found fromEq. (2). For the sake of simplicity the attention isrestricted to the isothermal conditions underapplied strain, e.g. q ¼ eðtÞ;W0f g: Constitutivefunctions defining elastoplasticity with linearkinematic hardening are

#cðsÞ :¼ 12Eðe� aÞ2 þ 1

2Ha2 ð18Þ

#Gð’aÞ :¼ Lj’aj ð19Þ

together with assumption (15). Hereabove E > 0 isthe elastic modulus, H > 0 the hardening modulusand L the yield stress in tension for the virginmaterial, so that K ¼ �L;L½ �: Taking into account(7) and the definition (9) then

s ¼ Eðe� aÞ; P ¼ Ee� ðE þHÞa: ð20Þ

Accordingly, Eq. (16) becomes

’aANK PðeðtÞ; aÞð Þ ð21Þ

that turns out to be a play operator e-a [8].Prescribing initial conditions for total and plasticstrain such that Pðeð0Þ; a0ÞAð�L;LÞ the micro-structural evolution aðtÞ follows as the solution ofEq. (21). The stress–strain relationship is thenobtained by substituting aðtÞ in Eq. (20) as shownschematically in Fig. 2.

6. Shape memory alloys

The pseudoelastic behavior exhibited by shapememory alloys provides another example ofmacroscopic response influenced by microscopic

ε

t ε

α

σ

ε

σ

Fig. 2. Elastoplasticity with linear kinematic hardening.

D. Bernardini / Physica B 306 (2001) 132–136 135

phenomena. In this case, the solid phase transfor-mations (PT) activated by mechanical loads areresponsible for the hysteretic macroscopic re-sponse [12]. This behavior can be modeled, in anoverall way, by considering the fraction of one ofthe two phases as internal variable a [13]. In thiscase the physical meaning of the internal variableyields A ¼ ½0; 1�:

The macroscopic strain induced by the PT maybe expressed as ga where g is a positive materialparameter representing the maximum macroscopictransformation strain. Suitable constitutive func-tions are now

#cðsÞ :¼ 12Eðe� gaÞ2 þ Oað1� aÞ

þ BðW� W0Þaþ c0; ð22Þ

#G ’að Þ :¼ LF ’aþ � LR ’a�; ð23Þ

where Oað1� aÞ; with O > 0; represents the elasticenergy of phase interaction, B > 0 and c0 > 0 arefurther material parameters and LF ;LR are posi-tive constants representing the driving forcethresholds to be attained in order to activate thePT. The generalized loads are q ¼ e;Wf g and (7),(9) yield

s ¼ Eðe� gaÞ; ð24Þ

P ¼ bðe;WÞ þ ð2O� Eg2Þa� Oþ BW0 ð25Þ

with bðe; WÞ :¼ Ege� BW: Also

’aANK Pðbðe;WÞ; aÞð Þ ð26Þ

shows how the phase fraction evolution is drivenby the quantity P; often named driving force of the

PT. The play operator now relates the output a tothe input bðe;WÞ that is the e; W-dependent part ofP; i.e. a thermomechanical loading parameterwhose expression is determined by the free energyfunction. Fig. 3 depicts schematically the responseunder isothermal conditions.

7. Conclusions

The Ziegler–Green–Naghdi approach to thethermodynamics with internal variables is basedon the specification of the free energy and the rateof energy dissipation. Balance equations providerestrictions on the possible microstructural evolu-tions. When the dissipation function is rate-independent and the assumption (15) is made,the internal variable evolves according to ageneralized play hysteresis operator having, asinput, a suitable function of the generalized loadsdetermined by the free energy and dissipationfunction. In this way, by appropriate constitutiveassumptions, various hysteresis models can bebuilt on a common thermomechanical basis, as ithas been exemplified by hardening elastoplasticityand shape memory alloys.

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[13] D. Bernardini, T.J. Pence, submitted.

σ

ε

t

σ

ε

ε

α

Fig. 3. Pseudoelastic behavior of SMA.

D. Bernardini / Physica B 306 (2001) 132–136136