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Iranian Jownal of Science & Technology Vol. 2 I, No.3, Transaction B
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A thermodynamically consistent model for hysteretic materials

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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

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A thermodynamically consistent model for hysteretic materials

  1. 1. '''-,- Iranian Jownal of Science & Technology Vol. 2 I, No.3, Transaction B Printed in Islamic Republic of Iran, 1997 @ Shiraz University A THERMODYNAMICALLY CONSISTENT MODEL * FOR HYSTERETIC MATERIALS F. G. FAN 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, MA 01609, U.S.A. Abstract- Based on the second law of thennodynamics, a set of constitutive equations for small defonnation of heat conducting, hysteretic materials is proposed. It is shown that the model exhibits eIastic-viscoplastic-fluid behavior sinlilar 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 fOI;<;e, ..1" .~:.-'( 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 for the uni-axial andbi-axial models are also presented. 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 "''.'
  2. 2. 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 an infinitesimal 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>..1ended Park, et al., [21] to model responses of concrete under by 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 ot =0 (1) P u.=t...+ r. , JI,' p J, (2) ti] =tji (3) pe ={Y'uj,i +qi,i + ph (4)
  3. 3. A thermodynamically consistent model... 259 of In addition to Eqs. (1) through (4), the axiom of entropy leads to the Clausius-Duhem lilt inequality given as, Idi . e I I ;at (5) p ( 1]-(j ) +(jtijv j,i +(jqi(logB),i :2:0 3]. als In Eqs. (1) through (5),the following notations are used: p density, ng Vi velocityvector, an t I) stress tensor, illS ;; body force per unit mass, of e internal energy per unit mass, :tic qi heat flux vector pointing outward of an enclosed volume, nd h heat source per unit mass, es. 1] entropy per unit mass, ler B absolute temperature. ~m Throughout this work, the regular Cartesian tensor notation is used with a dot on the top of for a letter denoting the total time derivative, and a comma in a subscript standing for the ch partial derivative with respect to the index following. m- Introducing the free energy function, . IS t/J=e-B'7 (6) )se Its and using Eq. (4), the Clausius-Duhem inequality may be rewritten in an alternative form :leI as, be . . I (7) on -p(t/J +(17) +tijdji +(jqiB,;:2:0 where d!J is the deformation rate tensor defined by I d. =- (v I} 2 I,J + v .) J,1 (8) ;e 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 and hysteretic materials are first developed. The formulation is then reduced to a convenient ) specific model for practical applications. :4) The functional dependence of free energy function is assumed to.be given as,
  4. 4. G. Ahmadi / et al.. 260 I/J = l/J(eij,dij,zij,(),O./) (9) where eij is the Eulerian strain tensor and zij is a symmetric traceless (Zii= O)internal variable 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) qi = qi (eij ,dij ,zij , (), (),i ) (11) Substituting Eqs. (9) through (11) into (7), the results may be restated as, OI/J . ol/J -=- ol/J ol/J. ol/J 1 -p ( o() +1]) (}-p o().,/ (),i + [ tij -P oe. l} J dij -P od. IJ dij -P oz. l} zij +fjqi(},i ~o (12) For the present small deformation model, I eij = dij (13) eij = 2 (Ui,j +Uj,i)' 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 (14) '7=- o()' ow =0 ol/J =0 (15) 00 ,/ .' od. l} Inequality (12) now reduces to ow ol/J. I (tij - p~)dij - P~Zij +- () qi(),i ~ 0 (16) vel} vZl} 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,
  5. 5. '-C" '~,O'..'-' "",",,,',. "'"'C.''-':-''''''=''''' A themlOdynamically consistent modeL.. 261 (9) - So r 2 fiT Al (T) /11(T) a(T) tb ---110T- 2T T --ekk +- 2 ekkell +-eijep +- 2 zi]zp (17) PoP P P P :rnal The is considered, where lIDe- B=To+T, 1T1«To. To>0, (18) lICS, dent 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 (10) are constants, and ,1.1 and a are some 'functions ,fll 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) (11) may be restated as, (t. + pro. -Illekko' - 2fl1e)d. - az DZ +~qT >0 (20) 1j 1] . 1] 1j 1j IJ Dt IJ To + T I .1 - . ~o Here, D stands for the corotational (Jaumann) derivative defined as. (12) Dt . D . -Dt zij = zij + Zik(O - (i)ikZkj kj (21) ~13) where wij is the spin tensor given by. 1 raIl (i).=-(v. .-v..). (22) 1j 2 I.J J,l ().i , bles Note that the Jaumann derivative of frame-indifferent tensor is frame-indifferent while its material time derivative is not. Now let )4) d ti] = (-[JT+/llekk)liij +2fl1eij +tij (23) 15) 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) I l.d.. -az..-z +-qT >0 (24) 1] 1j IJ Dt 1j To + T I .' - 16) where D d id Iii' (25) :hat di]' = ij - 3' 11/11/ } is the deviatoric part of the deformation rate tensor. For isotropic materials. the following constitutive equations which satisfy inequality (24) arc now proposed.
  6. 6. ~ 262 G. Ahmadi / et al., qi =kT,i (26) (27) td = 2jidf + a (a - blzklZkll; }ij (28) %t zij = -9dBdBI~IZmnZmnln:1 + (a -bhlZkll~ )df zij 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 for the stress tensor becomes (29) tij = (-[JJ + A)ekk )8ij + 2,u) eij + 2,udfl +a( a -blzklZkd~ )zy. In Eq. (29), the contribution of the thermal, elastic, viscous and hysteretic stresses may be clearly identified. For incompressible materials, ekk =dkk =0, D- .. dij - d1] (30) and Eq.(29) is replaced by (31) tif = (-fJT + P)8iJ + 2j.i)eij + 2j.idij + a( a - *klZkll~ }iJ. 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], . d porT + 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
  7. 7. A thermodynamically consistent model... 263 explicit heat transfer equation is obtained by using Eq. (27) in (32). The results in vector j) notation may be restated as, 7) a) Compressible material Mass n (ip + V.(p~) =0 (33) {it e, )r Momentum ., p fd. = p L- fJ VT + (11.1 /11 ) VV.!!. + /11 V 2 !!.+ + t J.NV . !!.+ /1 V 2 !!.+ J) (34) )e aV[(a-bl"l~ }] Energy J) (35) Port = -1fFo V.~ + V.(k VT) + [2/ldD + a V.[ (a - biZ: ZI~)Z J} dD + ph Hysteresis 1) (36) i +z.W -W.Z = -~dD :dDI;lz:zln~1+( a -blz:zl~)dD z of b) Incompressible materials es Mass V.u=o (37) 2) Momentum of £11 (38) p,,~ =P" L- fJVT- VP+/l, V2~+/lV2~+aV.[( a-bl"I~}] Energy III (39) lis porT =+ V (k VT{2/ld +aV.[(a-blz:~~} ]}d +ph he
  8. 8. ",..,..-~-""" =":;..<~;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 eguations given by Eqs. (28) and (29) are given as (40) p = cu+Ku+a(a-blznz (41) i = -9u!lzln-l z + (a - blzln)u 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 missing in 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, 0 (z;+zn~ (42) {~:}=[q{::}+[K]{::}+a:]-b [~ !: {;:} - [ . r 0 . (z2 + z2) 1] x y 2 R('; +zJ)";' 0 ' ZX {:;}=_ 0 .2 ,,-1 {} + { ~ux+u;.(z; +zJ)~ I Zy
  9. 9. A thermodynamically consistent model... 265 (43) [~ :]-b (,;+,;); J11:; lex, Irce- [ 0 0 (z; +Z;)2 ] :d. where Pxand Py are the forces in x and y directions. Equations (42) and (43) provide a thermo-dynamically consistent bi-axial hysteretic model. Recently, Park, et a!., (21) proposed two-dimensional a hysteretic odelwhichresembles m 1S Eqs. (42) and (43) for n ==2. owever, there are considerable differences between the two H models. The nonlinear terms in the evolution equation for the internal hysteretic variable as given by Eqs. (42) and (43)are quite differentfromthoseproposedbyPark, et al., (40) [21]. The nonlinear terms in Eq. (42), which are required for thennodynamical consiste:lcy ofthe model, are also missing in the model of[21]. (41) 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, Eqs. Ux = IICOStjJ, Zx = zcostjJ, Px = pcostjJ, (44) fl) is lIy = IIsintjJ, Zy = zsintjJ, Py = pSll1tjJ, ssmg lY be Eqs.(42) and (43) reduce to the uni-axial model given by Eqs.(40) and (41). It should be 1==0. noted here that, when the bi-axial model of [21] is reduced to uni-axial model, it recovers . The 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 In tlllS section, the behavior of the tl1ermodynamicallyconsistent hysteretic model under a has variety of conditions is studied. Applications of the present model to solid-liquid phase I). change analysis also are described. a) Vibrations of hysteretic spring For an isothennal condition, uni-axial and bi-axial vibrations of a rigid mass with a (42) hysteretic spring are studied. The results are described in the following. The equation of vibration of rigid lllass of 1 kg 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 -1 a = 1.0Nil/, a = 1.0111,b ==0.2m ,C = 05 m , and 11 ==2 are used. For a motIOnthat . starts from rest, the resulting force-displacement curves are shown in Fig. I. The corresponding response generated by the model of [18).is also shown in this figure by the dashed line. The hysteresis behavior of the spring is clearly seen from this figure. It is also
  10. 10. "- 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 :-.:'"~ f .0 ,,:-;,- t ," ,',' t,' 1 " " ,1" 1 " ,'~ .7': 1 , , -.," , 0.5 , , I I , " I I " ~ ;:. 0.0 1 1 " I ~ 1 ,'~I -0.5 I 1 1 I 1 I 1 I I I 1 1 ' -f .0 "" 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) (46) { iiy } + { Py } = { "2 (I)} wherepxandpyare given by Eqs. (42) and (43) and,,}(t) andn2 (I) are two mutually independent 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 the hysteretic behavior is clearly observed. This figure also shows the force-displacement behaviors 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 all isothermal condition are studied. In this case, the stress is given by.Eq. (31). The evolution of the internal (hysteretic) variable tensor is governed by Eq.(28) which may be restated as,
  11. 11. A themlOdYllamically consistent model... 267 ) and 6 ~ 2 . -..;;:. 0 ct- -2} I -6 -6 --4 -2 0 2 e Px (N) a) 6 -4 , I i I mare 2 (46) i , '..;:. 0 I ct tually -2 1m-I, unity re the -I ment 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-axial ed as, model under a random excitation
  12. 12. ---._- ""=, ,---~~~-~~- I !' 268 G. Ahmadi / et aI., 6 I :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
  13. 13. A thermodynamically consistent model... 269 I ,,- " + Zik(()/g" (()ikZkj = -cldkldkll;IZmnZmnl-;- Zij + a (47) Zij - ( - hizklZkll; ) dij For a simple shear flowV = (JY,O,O{ anddijand(tJij ay be evaluate:dfromEqs.(8)and m (22). Equation (47) then implies thatzij has only two independent components and is given by Zll Zl2 0 zij = Z12 -Zll 0 (48) [ 0 0 0] The components of Eq.(3 I) are given as (49) '" ~(-pr+P)++-*(zI1 +zI2));}" (50) '" (-pr + P)-+-bWI ~ +zI2));}" t12 = t21 = Pi' +th (51) where (52) ,h ~+-+(zI1 +zI2));}12 is the hysteretic shear stress. Similarly, Eq. (47) leads to . I. 2 2 "~l . (53) Zll + .J2 cy [2(Zll + Z12)] zll =Y Z12 ,,- n .1. 2 2-;- . l. .22- (54) Z12 + .J2 cr[ 2(Zll + z12 )] z12 = -Y Zll + 2"y a - b( 2(Zll + z12 )) 2 [ ] 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 motion under an imposed shear stress t12' For the first problem, Eqs. (53) and (54) are solved and the shear and normal components of the hysteretic variable are evaluated. The stress states of the material are then obtained from Eqs. (49) through (52). For the second problem, the
  14. 14. <0'0(:. ! I I 270 G. Ahmadi / e' al., I 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). i I For the steady state condition under a constant shear rate y , Eqs. (53) and (54) reduce I to a set of algebraic equations given by, I 1/-1 I (55) J212(Zfl +Zf2)] 2 Zll -Z12 =0 I I ZII + Jz12(zf, )f Z12 +Zf, = i[ a-+(Zfl +Zf'))~] (56) 1 I It is observed that Zlland Z12(and hence,lI) are independent of shear rate, and the shear I 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., Zll =~ I + .Ji. .Ji.+~ G":4l (57) c2 /[ c ( c 2 f -r ? )] Z12 = J2;; 1+ Ii (58) 2c /[ c 4 ~ c ( J2 + 2 ~2 + c2 ) ] 0.20 -n=1 - - -n=2 // ~'0.15 ~ ///// ~ .->-/ / // ~ "- / / Q = 0.5 CI) /// CI) 0.10 / / ~ / (J) ~ Q.) ////// ~ C1j 0.05 0.00 0 ?~ 40 60 80 100 Shear Rate y (sec-1) Fig. 3. Shear stress-shear rate relation for various a and p
  15. 15. A thermo(~Vl/amically cOl/sistel/t model... 271 com- 0.4 educe ~ ~ (55) ~ 0.3 ~ CI) CI) (56) ~ CI) "- 0.2 c m shear <55 ----- -- fed in ;g ...--...-- ...-- ~ !b 0.1 _./ - .... b (57) ~ :t -- ---- --- L -------- 0.0 0.5 1.0 1.5 2.0 (58) a,b,e Fig. 4. Effects of various material parameters on the steady-state hysteretic shear stress 1 ~ :.. ~ CI) 0.10 ------------------ =0.1 T ~ .E (J) ~=~~-------- ~ 1) t5 .~ ..... ~ ~ ~ ::t: 0.00 0.00 0.05 0.10 0.15 0.20 t (see) "Fig. 5. Time histories oftlysteretic . stress under ditTerent suddenly imposed constant stresses
  16. 16. --- -~- --_uu _n-- --- n--~ --- ~~~.-~ '--'r~-~~~-' < l :W ' I ! -~ ~- ~ ~ I ! i ~ 272 G. Ahmadi / et al., -~ 1 ; Using (57) and (58) in Eqs. (51) and (52), the expression for the steady-state shear stress follows. 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 shear I stress-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 resembles i that of water is also considered. It is observed that, for a =0, the material reduces to a Newtonian fluid. For other values of a, the shear stress is still a linear function of y with I 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 the ! yield stress (maximum strength or threshold) in this case. It is simply the minimum stress needed to maintain the continuous shearing. Figure 3 also shows that, as n increases, the I shear stress reduces. Effects of material parameters a, b, and e on the steady-state hysteretic shear stress ( are 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 at I their basic values. Figure 4 shows that th increases rapidly with a (roughly quadratically). I The effects of band e on th are minor. The hysteretic shear stress increases slightly with c and approaches an asymptotic value of about 0.12 N/m2for large c. The hysteretic stress I 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 the motion starts from rest and a constant shear stress is suddenly applied. That is, tile shear I stress is given by tl2 = rH(/), (59) I where H(t) is the unit step (Heaviside) function and r is a constant stress intensity. Figuf"e5 shows 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 remained unchanged. It is observed that, for a high imposed stress, the hysteretic stress increases rapidly up to a maximum (threshold) value. Once this threshold stress is reached, the hysteretic stress drops down and the material begins to flow. The difference between rand the 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) is insensitive to the magnitude of imposed external stress. Furthermore, the steady-state hysteretic stress is also independent of r and is identical to the limiting stress at zero shear rate (y = 0) as shown in Fig. 3. It is also observed that if the imposed shear stress is less than 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 smoothly varying imposed shear stress given by tl2 = rtanh(fJt). (60)
  17. 17. A the17llodynamically consistent model... 273 ,tress nents shear -n=1 es of - - - n=2 nbles s to a with ~ham 0 the ;tress " the 0.05 ;tress .~ ..... >.s,n :ed at ~ ~ ~ 0.00 ally). th c t ;tress -0.05 0.0 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 ure 5 )osed p are used. The parameter p is a measure of smoothness of the build up of the external lined shear stress. It is observed that the time evolution of the hysteretic shear stress varies ;:ases significantly with p; however, the threshold stress and the steady-state th remain t, the unchanged. A comparison between the responses for n =2 and those for n = 1 shows that, as - and n increases, the threshold stress increases while the steady-state th decreases. It the Figure 7 displays the time histories of th for various values of material parameters. The rs) is values of n and a are fixed as 1 and 0.5 N/m2, respectively, and a suddenly applied external state stress as given by Eq. (59) with l' =0.2 N is used. It is noticed that the threshold stress, the :hear steady-state hysteretic shear stress, and the oscillation following the threshold peak are ; less varied significantly with changes in these parameters. Thus, the present model may be ) /}2 . used to model behaviors of a variety of materials with different solid-liquid transition forms. )thly 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, (60) and c =0.5, and Eq. (60) with various,o for imposed stress are used. It is observed that the
  18. 18. 274 G. Ahmadi / et al., 1 ~ :.. ~CI) 0.10 CI) ~ f' . , ' <;) - - - - - - - - _0~1:0.-b:=q.5.::=.}.£>- - - ~ . I t5 . (,) 0.05 0= 1.0. b=O.5. c=O.5 , :<:::; ~ ( '. .~ 0= 1.0. b=0.8. c=O.5 ~ I ~---- ~ . ::t ,v 'J I ~ - -- - 0=0.6. =0.5. =0.5 b c / -------------- 0.00 0,00 0.05 0.10 0.15 0.20" 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, the material 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 == Figure 8 also showsthat the stress-strain relation of the I. material 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 t<Osee 0.2N /11122 0 N /111 0 ~ t < 0.4 sec t 12 - r - (61) 0.02 N /11/2 0.4 ~ t ~ 0.6 sec 0.2 N /1112 0.6 see ~ t. The stress variations are shown in Fig. 9a, while the strain and strain rate variatio~lsare displayed in Fig. 9b. It is observed that the hystereticshear stress builds up rapidly to the threshold level and then drops down. After some oscillations, the hysteretic stress and the shear rate become constants. while the shear strain increases linearly. At t =0.4 see, the imposed stress reduces to 0.02 N/1I12. ut the hystereticstress remains unchanged. The b material exhibits recoil phenomenon during the time duration of 0.4 to 0.6 see due to the difference between the imposed and the hysteretic stresses. For t >0.6 see when the imposed
  19. 19. A themlOdy;zamica1Jycollsistell1 model... 275 -n=i 0.30 - - -n=2 /,/'-- - -'- -,..,.- - - - - - / / Q =1 (P = 00 andP = 2) / / 10.25 / ~ 0.20 ;:,. / / ~ , / ~ ~ / (J) 0.15 I I, m /- t5 0.10 I.' ~ Ii 0.05 0,00 0.0 0.5 1.0 1.5 2.0 Shear Strain y Fig. 8. Shear stress-shear strain relations for various a and 11and e different smootlmess parameters a 0.25 .- e t'2 10.20 -. ~ ~ CI) 0.15 CI) ~ (;) 0.10 0.05 th 0.00 0.0 0.2 0.4 0.6 0.8 1.0 t (see) a) " " Fig. 9. Time ev,olntions Qf hysteretic stress, shear rate, and shear strain I under a timc-varying imposed stress
  20. 20. 276 G. Ahmadi / et,al., -'r 111"'- 150.0 .S: (;) 100.0 Q) ..... 50.0 I I ..... Q) 0.0 -50.0 0.0 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 shear strain 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;
  21. 21. 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 at Buffalo. REFERENCES I. Mangin, G. A. and Drouat, R., Intemal variables and thenuodynamics of macro-molecule solutions. Int. J. Engng. .'lei.,21, 1'1'.705-724 (1983). 2. Green, A. E. and Naghdi, P. M., Rate-type constitutive equations and elastic-plastic materials, Int. J. Engineering Sci., 11, Pl'. 725-734 (1973). 3. Green, A. E. and Naghdi, P. M., On thennodynamic restrictions in the theory of clastic-plastic materials, Acta Afec/wnica, 30, pp. 157-]62 (1978). 4. Valanis, K. C., On the foundation of the endochronic theory of viscoplasticity, Arc/Is. Rational Alechanics, 27, PI'. 857-868 (1975). 5. Bonder, S. R. and Partom, Y., Constitutive equations for cIastic-viscoplaslic strain-hardening materials, ASAlE J. Appl. Mechanics, 42, PI'. 385-389 (1975). shear 6. Asaro, R. J., Crystal plasticity, ASME J. Appl. Mechanics, 50, PI'. 921-934 (1983). fluid 7. Naghdi, P. M., Constitutive restrictions for idealized elastic-viscoplastic materials, ASME J. Appl. Mechanics, 51, 1'1'.93-101 (1984). 8. Nemat-Na-sser, S., Finite plastic !lows of crystalline solids and geomateriais, AS/'dE J. Appl. Mechanics, 50, PI'. 1114-1126(1983). 9. Onat, E. T., Shear Bows of kinematically hardening rigid-plastic materials, In: Mechanics of r heat Material Behavior (Drucker AIU1iv.Vol., Eds. Dvorak, J and Shield, RT, PI' 311-323, imple Elsevier (1984). liquid 10. Chaboche, J. L., Constitutive equations for cyclic plasticity and cyclic viscoplasticity. 1m. J. 'ess is Plasticity, 5, PI'. 247-3()2 (I I)XI). ritical II. Anand. L., Constitutive equations liJr hot working or metals, Int. .J. Plasticity, 1, PI'. 2] 3-231 leters. ( 1985). may 12. Chan, K. S., Bonder, S. R. and Lindholm, lJ. S., I'henomenologicalmodeling orhanlening and luring thennal recovery in metals, AS,lJE J. h'nKineel'iIIK. ,1aterials Tec"/lolo~'V,110, PI'. 1-8 (1988). 13. Ghoneim, l-I., Ana!'sis and application or a coupled thennoviscoplaslicity theory, ASAfE J. resent ApI'/. ,-feclwl1ics, 57, PI'. X2X-~D5(II)I)()). tat the 14. KrempcI, E., Models of viscopla~llcil':some comments of equilibrium (back) stre~~and drag nodel; stress, Acta Jlecl/(/l1ica, (,1), PI'. 25-42 (1987).
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