'''-,-

Iranian Jownal of Science & Technology Vol. 2 I, No.3, Transaction BPrinted in Islamic Republic of Iran, 1997@ Shiraz University

A THERMODYNAMICALLY CONSISTENT MODEL*

FOR HYSTERETIC MATERIALS

G. AHMADI", F. C. FAN AND M. NOORI'

Department of Mechanical and Aeronautical Engineering, Clarkson University,

Potsdam, NY 13676, U.S.A.'Department of Mechanical Engineering, Worcester Polytechnic Institute,

Worcester, MA01609, U.S.A.

Abstract- Based on the second law of thennodynamics, a set of constitutive

equations for small defonnation of heat conducting, hysteretic materials isproposed. It is shown that the model exhibits eIastic-viscoplastic-fluid behaviorsinlilar to Bingham's materials. Therefore, it may be a suitable candidate formodeling the mechanical behaviors of materials during melting and/orsolidification processes. The corresponding bi-axial and uni-axial fOI;<;e,

displacement models are also developed and their propertjes are discu~s,e+.I~is.,.shown that the resulting force-displacement relations are similar to'ih~se giv'en ,.by the Bouc-Wen ModeJ; however, certain modifications are needed to make the .earlier model thennodynamically consistent. Several nwnerical experiments forthe uni-axial andbi-axial models are also presented.

, '"

..1" .~:.-'(

. '" ~

,---;;::

,,';

Keywords- Hysteretic material, thenno-visc?plastic, macro-modeling .'

1. INTRODUCTION '.'" - """',

For mode:ling the material behavior in inelastic ranges,-the theory of viscoplasticity has

received ..considerable attention in recent years. Based on the theory; of dislocation:in

microstructure, a unified approach was developed which describes,the:material behavior by

constitutive differential equations and a set of internal variables. The foundation of theory

of internal variables in continuum mechanics was reviewed by Maugin, et al., [I). Earlier,

Green and Naghdi [2,3) developed rate-type constitutive equations and described certain

thermodynamic restrictions for elastic-plastic materials. Valanis (4) introduced the

* Received by the editors December 3, 1996

**C;orresponding author "' '.'

G. Ahmadi / et aI.,

endochronic theory of viscoplasticity. Bonder and Partom [5] developed a set of

constitutive equations for elastic-viscoplastic strain-hardening materials. Significant

progress in modeling plastic and viscoplastic materials was reported by Asaro [6], Naghdi

[7], Nemat-Nasser [8], Onat [9] and Chaboche [10]. Constitutive equations for metals at

high temperature were described by Anand [11], Chan, et al., [12] and Ghoneim [13].Extensive reviews on recent developments in modeling plastic and viscoplastic materials

were provided by Krempel [14], Miller [15] and Naghdi [16].To model the mechanical behavior of complex structura1elements, the macro-modeling

approach is usually adapted. Here, the overall behavior of structure instead of that of aninfinitesimal element is modeled. In this approach, it is the force-displacenient relations

rather than the stress-strain relations which are of concern. Along the line of theory of

plasticity, Bouc [17] developed a one-dimensional model which exhibits hysteretic

behaviors. This model was further developed by Wen [18] and was used by Barber and

Wen [19] and Barber and Noori [20] for random vibration analysis of hysteretic str..1ctures.

Recently the model was e>..1endedby Park, et al., [21] to model responses of concrete under

bi-axialloadings.

In this work, starting from the fundamental balance laws and the Clausius-Duhem

inequality and allowing for the internal variables, a set of general constitutive equations for

thermo-viscoplastic material is derived. A special class of the stress-strain relation, which

resembles, the Bouc-Wen hysteretic model, is studied in detail. The corresponding uni-

axial and bi-axial force-displacement relations for macro-modeling are also described. It is

shown that. with certain modifications, these relations become quite similar to those

proposed by Wen [18]. The features of the new model are analyzed and the re!\.llits

presented in several figures and discussed. The presented results also show that the model

is capable of describing fluid and solid-like behaviors and their transition. Hence, it may be

a suitable starting point for modeling the mechanics of melting and solidification

processes.

2. BASIC LAWS OF MOTION

The statement of principles of conservation of mass, balance of linear momentum, balance

of moments ofmomentunlj and.conservation of energy are [22]:

op +(PVi),i =0ot (1)

P u.=t...+ p r., JI,' J, (2)

ti] =tji

pe ={Y'uj,i +qi,i + ph

(3)

(4)

of

lilt

Idi

;at

3].

als

ng

an

illS

of

:tic

e

qi

hnd

es.

ler

A thermodynamically consistent model... 259

In addition to Eqs. (1) through (4), the axiom of entropy leads to the Clausius-Duhem

inequalitygiven as,

(

. e)

I Ip 1]-(j +(jtijv j,i +(jqi(logB),i :2:0

(5)

In Eqs. (1) through (5),the following notations are used:

p density,

velocityvector,

stress tensor,

body force per unit mass,

internal energy per unit mass,

heat flux vector pointing outward of an enclosed volume,

heat source per unit mass,

entropy per unit mass,

absolute temperature.

Vi

t I)

;;

1]

B

forch

Throughout this work, the regular Cartesian tensor notation is used with a dot on the top of

a letter denoting the total time derivative, and a comma in a subscript standing for the

partial derivative with respect to the index following.

Introducing the free energy function,

~m

m-

. IS

)set/J=e-B'7 (6)

Its

:leI

be

as,and using Eq. (4), the Clausius-Duhem inequality may be rewritten in an alternative form

on

. . I

-p(t/J +(17) +tijdji +(jqiB,;:2:0(7)

where d!J is the deformation rate tensor defined by

Id. =- (v + v .)

I} 2 I,J J,1

;e

(8)

Inequality (7) is the appropriate form of the statement of the second law of

thermodynamics for derivation of the constitutive equations.

) 3. CONSTITUTIVE EQUATIONS

In this section, quasi-linear constitutive equations for heat-conducting viscoplastic andhysteretic materials are first developed. The formulation is then reduced to a convenient

specific model for practical applications.

The functional dependence of free energy function is assumed to.be given as,

)

)

:4)

260 G. Ahmadi / et al..

I/J= l/J(eij,dij,zij,(),O./) (9)

where eij is the Eulerian strain tensor and zij is a symmetric traceless (Zii= O)internalvariable tensor corresponding to the inelastic (hysteretic) microscopic deformation. The

constitutive independent variables considered in Eq. (9), including zij, are all frame-

indifferent tensors. According to the principle of equipresence of continuum mechanics,

the stress tensor and the heat flux vector must depend on the same set of independent

constitutive variables. These are

tij =t ij(eij,dij,zij.(),(),i) (10)

(11)qi = qi (eij ,dij ,zij , (), (),i )

Substituting Eqs. (9) through (11) into (7), the results may be restated as,

(OI/J

). ol/J -=-

[

ol/J

J

ol/J. ol/J 1

-p o() +1] (}-p o(). (),i + tij -P oe. dij -P od. dij -P oz. zij +fjqi(},i ~o,/ l} IJ l}

(12)

For the present small deformation model,

Ieij = 2 (Ui,j +Uj,i)' eij = dij (13)

Where Uj is the displacement vector, are used. Inequality (12) cannot be maintained fo~all

variations of the independent thennomechanical processes described by dij,() and(),i'since these are linearly involved in this inequality. Hence the coefficients of these variables

must vanish, i.e.,

ol/J

'7=- o()'(14)

ow =0 ol/J =000 . ' od.

,/ l}

(15)

Inequality (12) now reduces to

ow ol/J. I(tij - p~)dij - P~Zij +- () qi(),i ~ 0

vel} vZl}(16)

Eqs. (1) through (3) supplemented by inequality (16) must be satisfied for any process that

the material undergoes.

In this study, a quadratic free energy function given as,

(9)

:rnal

The

lIDe-

lICS,

dent

(10)

(11)

~o

(12)

~13)

raIl

().i ,

bles

)4)

15)

16)

:hat

'-C" '~,O'..'-' "",",,,',. "'"'C.''-':-''''''='''''

A themlOdynamically consistent modeL.. 261

- So r 2 fiT Al (T) /11(T) a(T)tb---110T-

2TT --ekk +- 2 ekkell +-eijep +- 2 zi]zp

PoP P P P(17)

is considered, where

B=To+T, 1T1«To. To>0, (18)

a> 0, fl1 > O. 2/l1 +3fl1 > 0 (19)

have been used. In these equations, To is the temperature ofth~ natural state, So,P,71o,Y,P

are constants, and ,1.1,fll and a are some 'functions of T. This solid is stress-free in its

natural state (that is, tij = () when T=O and eij = zij = 0). Using Eq. (17), inequality (16)

may be restated as,

(t. +pro. -Illekko' - 2fl1e)d. - az DZ +~qT > 01j 1] . 1] 1j 1j IJ Dt IJ To + T I .1 - .

(20)

Here, D stands for the corotational (Jaumann) derivative defined as.Dt .

D .- zij = zij + Zik(Okj - (i)ikZkjDt

(21)

where wij is the spin tensor given by.

1(i).=-(v. .-v..).

1j 2 I.J J,l(22)

Note that the Jaumann derivative of frame-indifferent tensor is frame-indifferent while its

material time derivative is not.

Now let

dti] =(-[JT+/llekk)liij +2fl1eij +tij (23)

Where /l1and fll are elastic moduli and tt is the dissipative part of the stress tensor which

is assumed to be also traceless (tt =0)- Inequality (20) may then be restated as.

d D j) Il.d.. -az..-z +-qT >01] 1j IJ Dt 1j To+ T I .' -

(24)

where

D d id Iii'di]' = ij - 3' 11/11/}

(25)

is the deviatoric part of the deformation rate tensor. For isotropic materials. the following

constitutive equations which satisfy inequality (24) arc now proposed.

~

262 G. Ahmadi / et al.,

qi =kT,i

td = 2jidf + a (a - blzklZkll; }ij

(26)

(27)

%t zij = -9dBdBI~IZmnZmnln:1zij + (a -bhlZkll~ )df(28)

where the material parameters,u,k,b,a, and c are, in gene[al, functions of temperature,and n is an integer. Combining Eqs. (27) and (23), the explicit constitutive equation forthe stress tensor becomes

tij = (-[JJ + A)ekk )8ij + 2,u) eij + 2,udfl +a( a -blzklZkd~ )zy.(29)

In Eq. (29), the contribution of the thermal, elastic, viscous and hysteretic stresses may be

clearly identified.

For incompressible materials,

ekk =dkk =0, dD- d..ij - 1] (30)

and Eq.(29) is replaced by

tif = (-fJT + P)8iJ + 2j.i)eij + 2j.idij + a( a - *klZkll~ }iJ.(31)

where P is the pressure.

Assuming the material parametersa,A) andj.i)are slowly varying functions of

temperature so that their variation with T may be neglected, the energy equation becomes

[22],

. dporT + PTodkk = (kT, k)' k+tij dij + ph (32)

Eqs.(l) through (3) and (32) supplemented by Eqs. (28) and (29) or (31) provide a set of

complete equations of motion that may be used for analyzing mult-dimensional

deformation of thermo-viscoplastic materials.

4. FIELD EQUATIONS

The field equations for the present thermo-viscoplastic hysteretic model are summarized in

this section. Using the stress constitutive equation given by (29) for compressible materials

(or (30). for incompressible materials) into Eq. (2) th~ equation of motion follows. The

A thermodynamically consistent model... 263

j)explicit heat transfer equation is obtained by using Eq. (27) in (32). The results in vector

notation may be restated as,

7) a) Compressible material

Mass

n(ip + V.(p~) =0{it

e,

)r Momentum

.,

p fd. = p L- fJ VT + (11.1+ /11 ) VV.!!. + /11V 2 !!.+ t J.NV . !!.+ /1 V 2 !!.+

J)

aV[(a-bl"l~ }])e

Energy

J)Port = -1fFo V.~ + V.(k VT) + [2/ldD + a V.[ (a - biZ: ZI~)Z J} dD + ph

Hysteresis

1)

i +z.W -W.Z =-~dD :dDI;lz:zln~1z +( a -blz:zl~)dD

of b) Incompressible materials

es Mass

V.u=o2)

Momentum

of

£11

p,,~ =P" L- fJVT- VP+/l, V2~+/lV2~+aV.[( a-bl"I~}]

Energy

III

lis

heporT =+ V (k VT{2/ld +aV.[(a-blz:~~} ]}d +ph

(33)

(34)

(35)

(36)

(37)

(38)

(39)

",..,..-~-""" =":;..<~;c,._,._,"

264 G. Ahmadi / e/ al..

Eq.(36) governs the evolution of Z for both compressible and incompressible material.

5. MACRO-MODELING

As mentioned before, macro-modeling is frequently used in practice to model a complex,

non-homogeneous structural element. In this section, the bi-axial and uni-axial force-

displacement macro-modeling for hysteretic viscoplastic structural elements is described.

a) Unr-axial macro-model

The uni-axial force-displacement models corresponding to the constitutive eguationsgiven by Eqs. (28) and (29) are given as

p =cu+Ku+a(a-blznz (40)

i = -9u!lzln-l z + (a - blzln)u(41)

where p and u are the force and the displacement, respectively.For isothermal cases, Eqs.

(40) and (41) resemble the hysteretic model developed by Wen [18]. While Eq. (41) is

identical to that proposed in [18). Eq. (40) is somewhat different. The term -blzlnis missingin the force-displacement equation of the original model of Wen. Thus, it may be

concluded that the hysteretic model of [18] is thermodynamically consistent only for b = O.

For a nonzero value of b. slight modifications as given by Eq. (40) are needed. The

hysteretic behavior of the new model is described in the subsequent sections.

b) Bi-axial macro-model

To obtain bi-axial macro-model, it is assumed that the force-displacement relation has

the same form as that of stress-strain constitutive equations given by Eqs. (28) and (29),

That is,

{~:}=[q{::}+[K]{::}+a[[~ :]-b

r

(z;+zn~

(

0

)!:

1]{;:}

- 0 z2 + z2 2. . x y

{:;}=_{

R('; +zJ)";'

0

(42)

0 '{

ZX

}.2 ,,-1 +~ux+u;.(z; +zJ)~ I Zy

A thermodynamically consistent model... 265

lex, [[~ :]-b (,;+,;); 0 J11:;\

0 (z; +Z;)2 ](43)

Irce-

(40)

where Pxand Py are the forces in x and y directions. Equations (42) and (43) provide athermo-dynamically consistent bi-axial hysteretic model.

Recently, Park, et a!., (21) proposeda two-dimensionalhystereticmodelwhichresemblesEqs. (42) and (43) for n ==2.However, there are considerable differences between the twomodels. The nonlinear terms in the evolution equation for the internal hysteretic variable

as given by Eqs. (42) and (43)are quite differentfromthoseproposedbyPark,et al.,

[21]. The nonlinear terms in Eq. (42), which are required for thennodynamical consiste:lcyofthe model, are also missing in the model of[21].

The bi-axial hysteretic model given by Eq. (42) and (43). like that of [21], also exhibits

uni-axial hysteretic behavior. For a path given by,

:d.

1S

(41)

Eqs.

fl) isUx = IICOStjJ, Zx = zcostjJ, Px = pcostjJ,

lIy = IIsintjJ, Zy = zsintjJ, Py = pSll1tjJ,(44)

ssmg

lY be1==0.

. The

Eqs.(42) and (43) reduce to the uni-axial model given by Eqs.(40) and (41). It should be

noted here that, when the bi-axial model of [21] is reduced to uni-axial model, it recovers

only the special case of 11= 2. The present formulation which is for arbitrary integer n is

more general in that it may be used to model various types of elasto-plastic transitions.

6. APPLICATIONS

hasIn tlllS section, the behavior of the tl1ermodynamicallyconsistent hysteretic model under a

variety of conditions is studied. Applications of the present model to solid-liquid phasechange analysis also are described.I).

a) Vibrations of hysteretic spring

(42)For an isothennal condition, uni-axial and bi-axial vibrations of a rigid mass with a

hysteretic spring are studied. The results are described in the following.The equation of vibration of rigid lllass of 1kg with an uni-axial spring is given by

ii+p=sinL (45)

where p is given by Eqs. (40) and (41). For the material parameters, C==0.0, K=O.O,-1 -1 05 -1 d

.a = 1.0Nil/, a = 1.0111,b ==0.2m ,C = . m , an 11 ==2 are used. For a motIOnthat

starts from rest, the resulting force-displacement curves are shown in Fig. I. Thecorresponding response generated by the model of [18).isalso shown in this figure by thedashed line. The hysteresis behavior of the spring is clearly seen from this figure. It is also

"-

266 G. Ahmadi / el al.,

observed that the steady-state hysteresis loop is not symmetric with respect to u =0 and

p =0. This is due to the absence of restoring force for the case considered.

1.5

0.5

:-.:'"~,,:-;,- t ,"

,',' t,' 1" " ,1" 1

" ,'~ .7': 1, ,", -.,,,

II ,

"I

I"

1"

1I

1

,'~

II

I II I

1 I1

11

f .0

~;:.~

0.0

-0.5

-f .01 1 '

" " "" ,,'I I --' ,,'-~--:::;: '

-1.5-3 -2 -f 0 2 3

u(m)Fig. l. Hysteretic loop of force-displacement relations generated by present

uni-axial model C-) and Boue-Wenmodel C---)

For a plane bi-axial oscillator under white noise excitation, the equations of motion are

gIVen as,

{

UX

} {

PX

} {

II! (1)

}iiy + Py = "2 (I)(46)

wherepxandpyare given by Eqs. (42) and (43) and,,}(t) andn2 (I) are two mutuallyindependent Gaussian white noises. It is assumed that [C] =0, [Rl =K[I]with K=O.lm-1,and a = 5.0N m. The rest of the parameters are left unchanged. A spectral intensity of unity

for the excitation noise component is used- The results are shown in Fig. 2 where thehysteretic behavior is clearly observed. This figure also shows the force-displacementbehaviors of the present model are qualitatively similar to those reported in (21).

b) Simple shear motions

In this section, simple shear flows of an incompressible material with PI = 0 under allisothermal condition are studied. In this case, the stress is given by.Eq. (31). The evolutionof the internal (hysteretic) variable tensor is governed by Eq.(28) which may be restated as,

) and

A themlOdYllamically consistent model...

6

~

267

2

.-..;;:. 0ct-

-2}

I

-6-6 --4 -2 0 2 e

Px (N)

a)

6

-4,IiI

mare 2

(46) i 0, '..;:.

ctI

tually -21m-I,unityre the -Iment

i-6

-20 -IS -10 -5 0 5 10

Ux(m)ler an

b)lution

! Fig. 2. The forces and displacements generated by the bi-axialed as, model under a random excitation

---._- ""=, ,---~~~-~~-

I!'

G. Ahmadi / et aI.,268

6I

:1

"-.:.. 0

-2

-4

-6-8 -6 -4 -2 0 2 <4 6 8 10

Uy (m)

c)

to

8

6

'"

2"-.:..

-::;, 0

-2

-<4

-6

-8-20 -15 -to -s 0 5 10

Ux (m)d)

Fig.2. continued

A thermodynamically consistent model... 269

I ,,-\

(

"

)Zij + Zik(()/g"- (()ikZkj=-cldkldkll;IZmnZmnl-;- Zij + a - hizklZkll; dij (47)

For a simple shear flowV = (JY,O,O{anddijand(tJijmay be evaluate:dfromEqs.(8)and

(22). Equation (47) then implies thatzij has only two independent components and isgiven by

[

Zll Zl2 0

]zij = Z12 -Zll 0

0 0 0(48)

The components of Eq.(3I) are given as

'"~(-pr+P)++-*(zI1 +zI2));}"

'" ~ (-pr +P)-+-bWI +zI2));}"

(49)

(50)

t12 = t21 = Pi' +th (51)

where

,h ~+-+(zI1 +zI2));}12

(52)

is the hysteretic shear stress. Similarly, Eq. (47) leads to

. I.[2(

2 2

)]"~l .

Zll + .J2 cy Zll + Z12 zll =YZ12(53)

,,-\

[

n

]

.1. 2 2-;- . l. .22-

Z12 + .J2 cr[ 2(Zll + z12 )] z12 = -Y Zll + 2"y a - b( 2(Zll + z12 )) 2(54)

Here, two problems are studied. The first is concerned with 1he stresses generated in

the material under a given. cpnstant shear rate r. The second is to analyze the motionunder an imposed shear stress t12' For the first problem, Eqs. (53) and (54) are solved andthe shear and normal components of the hysteretic variable are evaluated. The stress statesof the material are then obtained from Eqs. (49) through (52). For the second problem, the

270 G. Ahmadi / e' al.,

differential equations (51), (53), and (54) must be solved simultaneously. Normal com-ponents of the stress tensor are then determined from Eqs. (49) and (50).

For the steady state condition under a constant shear ratey , Eqs. (53) and (54) reduce

to a set of algebraic equations given by,

I 1/-1

J212(Zfl +Zf2)] 2 Zll -Z12 =0(55)

ZII + Jz12(zf,+Zf,)f Z12= i[a-+(Zfl +Zf'))~](56)

It is observed that Zlland Z12(and hence,lI) are independent of shear rate, and the shear

stress '12is a linear function of y. For the case of n =1, Eqs. (55) and (56) can be solved in

closed form. i.e.,

~'0.15

~~~

"-

CI)CI) 0.10

~(J)~Q.)~C1j 0.05

Zll =~ /[I + .Ji.

(.Ji.+~ G":4l

]c2 c c 2 f -r? ) (57)

Z12 = J2;;/[1+ Ii

( J2 + ~~2 + 4 ) ]2c c c 2 c2(58)

0.20

-n=1

- - -n=2//

/////

///

.->-// / Q = 0.5

////

//

//////

0.00

0 60 80?~ 10040

Shear Rate y (sec-1)

Fig. 3. Shear stress-shear rate relation for various a and p

<0'0(:.

!III

iI

I

I

I

I1

II

A thermo(~Vl/amically cOl/sistel/t model... 271

com-0.4

educe

(55)

~~~ 0.3

~CI)CI)

~CI)"- 0.2

m

<55

;g~!b 0.1....

~:t

c(56)

shear

fed in

\ --\-----...--...--...--

_./ -

(57)

b--L---- --- --------

(58)0.0

0.5 1.0 1.5 2.0

a,b,e

Fig. 4. Effects of various material parameters on the steady-state hysteretic shear stress

1~:..

~CI) 0.10

~.E(J)

~\1)

t5.~.....~~~

::t:

T =0.1------------------

~=~~--------

0.000.00 0.05 0.10 0.15 0.20

t (see)"Fig. 5. Time histories oftlysteretic stress under ditTerent suddenly imposed constant stresses.

--- -~- --_uu _n----- n--~ ---

~~~.-~

< I

l ':W-~~-~~i~

-~

1;

272 G. Ahmadi / et al.,

Using (57) and (58) in Eqs. (51) and (52), the expression for the steady-state shear stressfollows. For other values of n, closed form solutions are not available and the components

of z and the steady-state shear stress must be found numerically.Figure 3 shows the shearstress-shear rate relations for different values of a and n. The fixed values of

a =1.0, b =0.5, and e =0.5 are used. A viscosity of Jl=0.00101 N/m see which resemblesthat of water is also considered. It is observed that, for a =0, the material reduces to aNewtonian fluid. For other values of a, the shear stress is still a linear function of y with

finite values at y =0. This stress-shear rate behavior is quite similar to that of a Bingham

fluid. However, it will be seen later that the finite stress aty =0 does not correspond to theyield stress (maximum strength or threshold) in this case. It is simply the minimum stressneeded to maintain the continuous shearing. Figure 3 also shows that, as n increases, theshear stress reduces.

Effects of material parameters a, b, and e on the steady-state hysteretic shear stressare displayed in Fig. 4. Here, the basic parameter values used are a =1.0, b =0.5, e =0.5, n

=1, and a =1.0 N/m2. When a particular parameter is varied, the others are kept fixed attheir basic values. Figure 4 shows that th increases rapidly with a (roughly quadratically).The effects of band e on th are minor. The hysteretic shear stress increases slightly with cand approaches an asymptotic value of about 0.12 N/m2for large c. The hysteretic stress

th, however, decreases graduately with an increase in b as shown in tllis figure.As mentioned before, to detemune the material responses under an imposed shear

stress, Eqs. (51), (53), and (54) must be solved simultaneously. Here, it is assumed that themotion starts from rest and a constant shear stress is suddenly applied. That is, tile shearstress is given by

tl2 = rH(/), (59)

where H(t) is the unit step (Heaviside) function and r is a constant stress intensity. Figuf"e5shows the resulting time histories of the hysteretic stress th for different values of imposed

shear stress. Except for a =0.5 N/m2, the values of parameters used here remainedunchanged. It is observed that, for a high imposed stress, the hysteretic stress increasesrapidly up to a maximum (threshold) value. Once this threshold stress is reached, the

hysteretic stress drops down and the material begins to flow. The differencebetween randthe steady-state th is balanced by the viscous stress fJY. Figure 5 also shows that the

threshold stress of the material (about 0.116 N/n? for the present material parameters) isinsensitive to the magnitude of imposed external stress. Furthermore, the steady-statehysteretic stress is also independent of r and is identical to the limiting stress at zero shearrate (y = 0) as shown in Fig. 3. It is also observed that if the imposed shear stress is lessthan the threshold stress, the material will remain solid with a steady-state th equal to '12 .These latter results are shown by the dashed lines in Fig. 5.

Figure 6 -shows the time variations of the hysteretic shear stress subject to a smoothlyvarying imposed shear stress given by

tl2 =rtanh(fJt). (60)

'--'r~-~~~-'

!

I!

I

i

I

!

I

(

III\I

I

A the17llodynamically consistent model... 273

,tress

nents

shear

es of

nbles

0.05

-n=1

s to awith

~ham0 the

- - - n=2

;tress

" the

;tress.~.....~~~ 0.00

t

>.s,n:edat

ally).th c;tress

-0.050.0 0.2 0.4 0.6 0.8 1.0

;hear

It the t (see);hear

Fig. 6. Time histories of hysteretic stress under different

smoothly varying imposed stresses(59)

Here, the values of a =1.0, b =0.5, c =0.5, a =0.5 N/m2 , l' =0.2 N, and different values of

p are used. The parameter p is a measure of smoothness of the build up of the external

shear stress. It is observed that the time evolution of the hysteretic shear stress varies

significantly with p; however, the threshold stress and the steady-state th remain

unchanged. A comparison between the responses for n =2 and those for n =1 shows that, as

n increases, the threshold stress increases while the steady-state th decreases.

Figure 7 displays the time histories of th for various values of material parameters. The

values of n and a are fixed as 1and 0.5 N/m2, respectively,and a suddenly applied external

stress as given by Eq. (59) with l' =0.2 N is used. It is noticed that the threshold stress, the

steady-state hysteretic shear stress, and the oscillation following the threshold peak are

varied significantly with changes in these parameters. Thus, the present model may be

used to model behaviors of a variety of materials with different solid-liquid transitionforms.

The shear stress-shear strain relations of the model for various values of a and nand

different imposed stress conditions are shown in Fig. 8. Here, the values of a =1.0, b =0.5,

and c =0.5, and Eq. (60) with various,o for imposed stress are used. It is observed that the

ure 5

)osedlined;:ases

t, the- andIt the

rs) isstate:hear; less

) /}2 .

)thly

(60)

274 G. Ahmadi / et al.,

1~:..

~ 0.10CI)CI)

~<;)

~t5(,) 0.05

:<:::;

~~~::t

f', '

\.\.\ I.\

(\ '.I \ \

\ .\\ ,v\ 'J I ~ - - - - 0=0.6.b=0.5.c=0.5\/ --------------

- - - - - - - - _0~1:0.-b:=q.5.::=.}.£>- - -

0= 1.0. b=O.5. c=O.5 ,

. 0= 1.0. b=0.8. c=O.5~ ~----

0.000,00 0.15 0.20"0.05 0.10

I (see)Fig. 7. Time histories of hysteretic stress for various material parameters under a

suddenly different imposed constant shear stress

stress-strain relations are approximately linear for small strain. At large strains, thematerial exhibits softening features. Furthermore, the material behaves as a solid until a

critical stress (the threshold stress) is reached. For the set of parameters used, the solid-

liquid transitions occur at r ==I. Figure 8 also showsthat the stress-strain relation of thematerial is independent offt (smoothness of the loading). An increase in a. or n, however,increases the strength of the material.

Figure 9 shows the responses of the model under a time-varying imposed shear stress.The stress is assumed to be given by

r 0 N /11122t -

\

0.2 N /11112 -

0.02 N /11/2

0.2 N /1112

t<Osee

0 ~ t < 0.4 sec

0.4 ~ t ~ 0.6 sec

0.6 see ~ t.

(61)

The stress variations are shown in Fig. 9a, while the strain and strain rate variatio~lsaredisplayed in Fig. 9b. It is observed that the hystereticshear stress builds up rapidly to thethreshold level and then drops down. After some oscillations, the hysteretic stress and theshear rate become constants. while the shear strain increases linearly. At t =0.4 see, the

imposed stress reduces to 0.02 N/1I12.but the hystereticstress remains unchanged. Thematerial exhibits recoil phenomenon during the time duration of 0.4 to 0.6 see due to the

differencebetween the imposed and the hysteretic stresses. For t >0.6 see when the imposed

10.25

~;:,. 0.20

~~~

(J) 0.15

m

t5 0.10

e

a

.-e

-.10.20

~~CI) 0.15CI)

~(;)

A themlOdy;zamica1Jycollsistell1 model...

-n=i

0.30 - - -n=2

/,/'-- - -'\- -,..,.- - - - - -/ Q = 1 (P =00 andP =2)

//

//

//

/,

/I

I,/-

I.'~

Ii

0.05

0,000.0 1.5 2.00.5 1.0

Shear Strain y

Fig. 8. Shear stress-shear strain relations for various a and 11anddifferent smootlmess parameters

0.25

t'2

th

0.10

0.05

0.000.0 0.4 0.6 0.80.2 1.0

t (see)a)

Fig. 9. Time ev,olntions Qf hysteretic stress, shear rate, and shear strainunder a timc-varying imposed stress

"

I

275

"

276 G. Ahmadi / et,al.,

-50.00.0 0.2 0.4 0.6 0.8 1.0

t (see)b)

Fig. 9. continued

stress is increased to 0.2 N/m2, the material experiences a constant shear rate and the shearstrain increases linearly. It is also noticed that the material continues to behave as a fluid

and no threshold stress appears when the stress level is increased to the original level.

7. CONCLUSIONS

A set of constitutive equations consistent with the second law of thermodynamics for heat

conducting, hysteretic materials is developed. The behavior of the model under a simple

shearing motion is analyzed. It is shown that the present model exhibits solid-liquid

transition phenomenon. The material behaves as a solid until a critical imposed stress is

reached. Beyond this stress, the material begins to flow. The magnitude of the critical

stress and the nature of solid-liquid transition are controlled by the material parameters.

With appropriate temperature-dependent material parameters. the present model may

become a suitable candidate for modeling the mechanical behaviors of materials during

melting and/or solidification processes.

The bi-axial and uni-<txial force-displacement relations corresponding to the present

hysteretic model are i1lso developed and their properties are studied. It is shown that the

resulting force-displacement relations are similar to those given by the Bouc-Wen model;

-'r111"'-

150.0

.S:

(;)

100.0

Q).....50.0

I I.....

Q)

0.0

A thernlO(~Vnamica/ly consistent model... 277

however. certain modifications arc neededto make the Bouc-Wen model thermodynamic-

ally consistent. Applications of these macro-models to vibration analyses of hysteretic

systems are described. Examples of harmonic and random vibrations are presented. The

results show that the formulation may be used for macro-modeling of hysteretic elements.

Acknowledgments- Tbe earlier stages of this work were supported by the NSF through the

National Center for Earthquake Engineering Research, State University of New York atBuffalo.

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