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Based on the second law of thermodynamics, a set of constitutive equations for small deformation of heat conducting, hysteretic materials is proposed. It is shown that the model exhibits elastic-viscoplastic-fluid behavior similar to Bingham's materials. Therefore, it may be a suitable candidate for modeling the mechanical behaviors of materials during melting and/or solidification processes. The corresponding bi-axial and uni-axial force-displacement models are also developed and their properties are discussed. It is shown that the resulting force-displacement relations are similar to those given by the Bouc-Wen Model; however, certain modifications are needed to make the earlier model thermodynamically consistent. Several numerical experiments for the uni-axial and bi-axial models are also presented. - G. Ahmadi, F.-G. Fan, M. Noori
Citation preview
'''-,-
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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