- Email: [email protected]
A constitutive model in finite viscoelastoplasticity of rubbery polymers
Mechanics Research Communications, Vol. 26, No. 1, pp. 39--44, 1999 Copyright © 1999 Elsevier Science Lid Prin~d in the USA. All rights reserved 0093-6413/99/S---see front matter
Pergamon
PII S0093-6413(98)00097-4
A C O N S T I T U T I V E MODEL IN FINITE VISCOELASTOPLASTICITY OF RUBBERY POLYMERS
Aleksey D. Drozdov Institute for Industrial Mathematics 22 Ha-Histadrut Street, Be'ersheba 84213, Israel
(Received 19 December 1997; accepted for print 9 November 1998) Introduction The paper is concerned with constitutive equations for the nonlinear response of incompressible rubbery polymers at isothermal loading. To derive the stress-strain relations, we employ a concept of nonaffine transient networks, where sliding of a network of long chains with respect to a bulk medium is described in the framework of finite elastoplasticity. The constitutive equations (developed with the use of the laws of thermodynamics) are applied to determine stresses in a specimen at uniaxial extension. A Model of Transient Networks A viscoelastic medium is modeled as a network of M kinds of long chains connected to junctions [6], where different kinds of chains correspond to different relaxation times. A chain whose ends are connected to separate junctions is treated as an active one. Snapping of an end of a chain from a junction is tantamount to its breakage. When a dangling chain captures a junction, a new active chain is created. Creation and annihilation of chains are characterized by the functions Am(t, T, ho) [the number of active chains of the ruth kind (per unit mass) arising before time r, existing at time t, and having the end-to-end vector ]to at the instant of their formation]. The distribution of initial end-to-end vectors is assumed to be independent of the reformation process,
am(t, ,,e)dVo(e) = "~m(t, , ) P {t~o • dVo(-~)},
(~)
where P is probability, dVo(2) is an elementary volume (in the space of long chains) in the vicinity of a vector z, and ~-,~(t, r) is the number of active chains of the ruth kind (per unit rnass) arising before time r and existing at time t. The quantity -~m(t, 0) equals the number of initial chains of the ruth kind existing at time t; the amount
10.--.m(t,r ) t--T 7re(r)-- -o Or ~rn
39
(--0 = --m(0,0))
(~)
40
A . D . DROZDOV
is the rehd ire rate of reformatiou (the ratio of the number of active chains of the ntth kind created per mill t.ime l.o the initial mmfl)er of aciive chains), and the quantities 1
1'.,,,)(1)=
()~- n,
r 0~-,,g
Z_,,,(I.0) i)l (I.U).
]
l',,,/l,r) . . . . [ ~ ( l . r ) ]
1 ~;)2--~,, i
~(I.T)
(:/)
are the r~'la.ti\'e rates of breakage' (the ratios of t.h(' uumbers of active chains of the ruth kimi amfihilated per unit time to the mmlbers of existing chains). Resolving differential equations (3) with i.he initial conditions (2). we arrive at, the fornmlas 2°exl ) [ - /," l,,,t}(.s)ds ] ,
= (l.()i '
--'"
'~=-'" ---(l.
. ~
[ f' ?m(r)---'2~exp-
Ti
i)T
l,,(.~.r)ds.
]
(1)
. T
The l,oia] uuml)er of active chains of the ruth kind (per unit mass) is calculated as ~,,(I.f)
-_o
{
exp
[
f"
] 1"-;,,,,(r)exp [ -
l',,,~(.~)d.~ -4-
t/
l',,(.~.T)d.~
}
dr
}
.
(5)
\'~'~, confine ourselves to nonaging polymers with constant total numbers of active chains a.d l im~' iudq)emient rates of breakage and mmihJlalio1~ Z,,(t./)
~,.(/)-- .,2-
= Z2,.
1.,.,,(/1 = l '°~,,0,
l,,(z. ~-)
2.
t(i
,Suhstitul.ion of expressions (6) into the balance law (.5) implies that 2,,,o = I,o
-- l,O
(T
Kinematics of a Nonafline Transient Network
Nlotiou of a I)ulk material induces so]n(' pseudo motion in the space of long chaius. Let. h(I) be the e n d q o end vector at time I for a chain created at time r with the initial end-to-eud vector ho. According to the affiniiy hypothesis [8] the deformation gradient ~7(T)h(t) in the space of long chains coincides with the deformation gradient V ( r ) ~ ( t ) in the bulk m a t e r i a l al. a poim with the radius vector /(~). We do not accept this hypothesis and assume t.he polymeric network to slide with respect to the bulk m e d i u m [3,7]. To describe the sliding, we introduce some unloaded configuration (which determines the stress-free state at time I for long chains created at time r) and suppose that the deformation gradient ~7(r)h(t) coincides wii.h the deformation gradient [br lransitiou from the unloaded to actual configuration
~7(r)h(l ) = ~v (/. r)/"(/).
(S)
TIw radius vector r,,(Z, T) in the uuloaded contiguration satisfies the well-known formula
V(7-)r(/) = V(7-).rAI T)- v,,(~, r)7;(l).
(!))
It. follows from E'qs. (8) and (.9) that o v(7)h(t)
= Vtr)h(t).
L(~
L,(t. r ) . v(r)t~(t),
wh('l'('
dr
,It) = ~)-(l).
ih',
,,,{t.r) = ~t/,r).
L(/) = V(l)~:,(t).
I0 )
CONSTITUTIVE MODELS OF RUBBERY POLYMERS
Equation (10) implies that the derivative of the Finger tensor equals
_5_((t,~ . U(t). P(t,~)+P(t,~) . . . L(t) where that
D,,(t, r)
41
['(t, r) = [~'(T)h(t)] v.~7(r)h(t)
2~(~)h(t) b~(t,~).~(~)h(t),
(~1)
= ½[Z~(t, r ) + L~(t, ~-)] and T stands for transpose. It follows from Eq. (11)
0I~
IF(t, r ) :
-
:
where Ik stands for the kth principal invariant, /)(t) = ½[Z(t, r) + Lv(t, r)] is the rate-ofstrain tensor and G(t, r) = V(T)h(t). [~7(T)h(t)]v is the relative Cauchy deformation tensor. T h e r m o d y n a m i c P o t e n t i a l s of a Transient N e t w o r k
Referring to the model of random walk and neglecting the excluded volume effect [2], we trea~ the initial end-to-end vector h0 as a sum of N end-to-end vectors for individual segments. Assuming the segments to be statistically independent, we arrive at the Gaussian formula
where b is the inverse average end-to-end distance. It follows from Eq. (8) that h(t) = ho " ~7(7)]~(t) = ~-7(r)~tT(t). It 0. Combining this equality with Eq. (13) and the incompressibility condition dV(z) = dVo(z), we find the probability for a network of active chains of the ~7~th kind created within the interval It, r + dT] to have an end-to-end vector h located in the unil vicinity of a point 5
dpm(~,T,~)=
(~,T)
exp[--b2z.[2g-l(~,T).z]dT.
(1~}
The specific (per unit mass) entropy dSm(t, r) of a network of active chains created within the interval It, r + dr] and existing at time t differs by a constant dSm. from the average value of logarithm of the distribution function
dS,~(t, r)
= d,5',~. +
kB / dpm(t, T, 2) lnpo(z)dV(5),
(15)
where kB is Boltzmann's constant. Substitution of expressions (13) and (14) into Eq. ([5) results in
dS.,(t,T)
= dS~.+~:~ x In ~
(t,~-) -
b'~9
dt'o(5)dr.
([6)
Summing up expressions (16) with respect to r and neglecting the entropy of interaction between chains, we find the entropy of the network of active links at the current instant l
"re(l) :
Sm*-~- ]¢B
jf ~ ,~-m(~,O)exp t-b2~'~o-l(~)'zj [ ]
+ fot~(t,r)exp[-b22.[~-l(t,,)'2]dT}
[ln(~)3-b252]dVo(2),
42 where
A.D. DROZDOV 1~'.(1) =
~otiT(t) • ~'oh(:). ('~lculating the integral over d~'o(5), we o})ta.ilJ
whet,' "~'2. is a conslant..~ummJng up the specilic c~mtigural.iona.l entropies (17) a.nd addiu~ 1he specific ent.ropy of thermal mot.ion O
,S'therma I
~ In (_)~
wc lind lhe specJ{ic elH.rop3' (per 1Hlil Hlassl .,.(/)
=
,,o + , . ] n ( 9 ( t )
/,',9 £
eo
--i5~ I/, ~1
{z . . . .
01
[1,
3]
(/:'(:.
wh~,re ?,'° is 1he sl)ecitic entropy ill 1]1c nal.ura] sl.al~' al ihe reference temperature (9 °, t, is lhe, specific heal (apacily (per unil nlfLss), and (9 is l.he absolute 4empera.ture. 'T'h~ specific
{'ill rol)y .s' is 'xl)ressed in tel'InS of the specific(per unit, mass) free (ttelmhoh,z) energy qJ hv 1}1(' forI~]~da 5
. . . .
.
(1~})
He
\'Ve accept
l.he following expression for lhv specitic fr~'e energy, which is compa.t.il)le with Efls. (18) and 1.1!)): {')(/}
qJt/:,
::: e ° + ( ~ -
>'°)[O{/I-O°]
,.0(:)1, ~ g - +
/:BO(/)
where ~po i> a material conslant. (!ombining 1he til'st law of t.hermodynamics wil]} lho (:lausius- Duhem inequalil.y~ we arrive al l]le thermodynamic ineflualil~y [1]
(_),IQ,1I .... 's'dOdl :1",11÷ t,-I (.4:I)
~ql • W--(')) _:> (L
(2l)
where Q is lhc sl)e<'ilic (per unit mass) rate ~,1 dissipalion of energy p is a consLanl mass de~Jsit\. ,/ is ihc heal []ux veclol. ,nd /, = do\" ~- is the dc'viatoric COml)onenl of lhe (!am'by ~lv~,ss lel>or b. Sul)sliluling expression (20) inlo Eq. (21) a.nd using Eqs. (}), {7). (12), an,I (IS). we ohtain
,/Q
()7-
= J' v .:. ~ ,::, ~-
/'1 (.4:/)
~)'tl
ve
)
_> 0,
(.,.2~
\\'ll('r~' /~(/)
=
I_,(tj
=
,
.
(:.~)
£[ l')ft.O)+ /,.:-.. i)~-T (t,r)O(t,r): 1~'~0 1) 5[ Z,~(t,O)f:o(:)+ /.--. i~T (t. rlt;'(l, Tjdr j : D(t).
/,H(-)(I)
-~,,,(/,0)(,'o(1) :
-
D~,(I,T)(IT
]
.
,
H*~t
I:~(1) --
m
~
I
.
i
2:/)
CONSTITUTIVE MODELS OF RUBBERY POLYMERS
43
It follows from Eq. (23) that the Clausius-Duhem inequality (22) is fulfilled provided that. (i) the heat flux vector q obeys the Fourier law q(t) = -pA~g(t)O(t) with a nonnegative thermal diffusivity 1, (ii) the Cauchy stress tensor & obeys the constitutive equation a(t) =
-p(t)]+pksO(t) ~
(t,r)f~(t,r)dr ,
E,~(t,0)//'0(t) +
(24)
m=l
where p is pressure, and (iii) the plastic rate-of-strain tensor/)~(t, r) satisfies the equality /)~(t, r)
=/3dev G(t, r),
(95)
where/3 is a nonnegative scalar function. Equation (24) is similar to the Lodge equation [~], (where the tensor fi differs from the conventional Finger tensor), whereas Eq. (25) appears to be novel. It determines the symmetrical part of the tensor L~(t, r). To avoid uncertainty. the skew-symmetrical part of L~ is assumed to vanish. Uniaxial E x t e n s i o n of a Specimen
To verify the constitutive equations (24) and (25), we study uniaxial extension of an incompressible viscoelastoplastic bar X1 = / ~ ( t ) X 1 ,
x: = k-½(t).¥2,
x 3 = k-½(•)X3,
(26)
where k is the extension ratio, and Xi, xi are Cartesian coordinates in the initial and actual configuration, respectively. We assume that transition from the initial to unloaded configuration is described by the formulas similar to Eq. (26) 1
1_
Z2 = k u : ( t , r ) X 2 ,
X 1 = /t?u(t, T ) X 1 ,
Xa = ]~u2(t,T)X3,
(27)
where k,,(t, r) is a function to be found. It follows from Eqs. (26) and (27) that '
V~(t,r)v~(t,r)-
k~(t,r)
(')[~l-
where ~, are unit vectors of the Cartesian coordinates into Eq. (25) implies that
01%~( t , r ) =
{Xi}.
~3[{\k,,(t,r)]k(t) ~2 k~(t,r_)]j k~(t,r)
Substitution of expressions (28)
ku(r,r) = k(r).
(29)
We substitute expressions (28) into the constitutive equation (24), find the unknown pressure p from the boundary condition on the lateral surface of the bar er2 -- 0, use Eqs. (4) and (7), and arrive at the formula for the longitudinal stress
rh(t) = 6'~,~ texp(-r°~t)[\k~(t'O)) +r2 [
gO
exp [ - < ; , ( t -
k(t) ] ,
44
A.D.
DROZDOV
we - ,11 70 where ( , = p/,'B(gE °, q . . . .--- _--'o,,~/~ , and 7~0 = ~,;~=l-,,.The stress oh(/) alld the pla.stic exlension ralio k,(t. 0) are plotted in Figure for leusile test with k ( t ) = e x p ( i t ) . C a l c u l a t i o n s are carried out t})r :11 -- 6, *l* = I0.0, -~ = 1.0. ;,:~ = 0.1. 74 = 0.01, % = (}.00l. 3~; - l).(I (s i) alt(] ql - {I.2. r/_, {).1. ~/a --- 0 . 2 . t/i - 0 . 2 . q-, 0 . 2 . ~/,~ - 0 . 1 .
5.0
().:1 .......-.'}) cr f
1.1}
],'
I()A)
......% .t5 ~'
I.ql
t'
10.0
1-hvoxwnsion ralio/% for transition from lhe iJ~itial to unloaded configuralion (A) and lira longjtudinaI >lress or, (MPa) (B) l'~r~rt.~lho extensiol, ratio k for uniaxial extension of polystyre~e (',EI)EX-1511 al (9 - I.t0 °(' wilb lhe tale-of-strain i = 0.02 (s -1 ). Cireles: experimental dal,:, ()htainod i, [5], solid liltes: pl'edielioli of the model wit}, fl - 0.013 and (' =- 0.037 (MPa) Figure:
Concluding
Remarks
New co1>lil ,tl ix:(' equa/iol,s },a\'c I~eert derived f,,r the viscoelastopla.stic beha.viov uf r u l ) b e , \ 1)olymer~. The, Im)(tel is based o,l l he c o m e p i of mmaffine t r a n s i e n t networks, where a new telal.iotlshi l) is proposed lo link the deformation gradients in a bulk m e d i u m and in a net.work. Stress sl.rain relations are developed based on the laws of t h e r m o d y n a m i c s . T h e c o n s t i t u t i v e equat.ions are employed to del.ermine lhe m a t e r i a l response in lmia.xial tensile t.ests. Results v~t m u m ' r i , a l s i m u l a t i o n d e m o n s t r a t e that the model correctly predicts the viscoelastoplastic response of polyst.yrene al elevaled leml)eral.ures. Acknowledgement
Fi,lancial support l)v lhe lsra.e/i Mi,fisl.rv of Scieu,'e (granl 9611 14)6) is gratefully ackm)wl cdge(I.
References 1. B.]). ('~,h'man and M.I". G u r t i n , 3. ( ' h e m . Phys. 47, 597 (1967). 2. M. 1)oi al~d S.F. Edwards, T h e T h e o r y of P o i y m e r D y n a m i c s , Oxford Universit.y Press. Oxford (19S6). 3. M . \ V . . I o h n s o n and I). S e g a l m a n , 3. N o n - N e w l o n i a n Fluid Mech. 2, 255 (1977). 1. A.S. l,odge. Rheol. Acta 7, 379 (1968). 5. 1~. Muller, l). Froelich, and Y.tl. Zang. 3. Polvm. Sci.: Polym. Phys. Ed. 2 5 , 2 9 5 (19S71. 6. F. :l~mlaka and S.F. Edwards, Macromolecules 25. 1516 (1992). 7. N.P. ' l h i e n a n d tt.]. T a n n e r . . 1 . N o n - N e w t o n i a n Fluid Mech. 2, :/53 (1977). s. IA(.(',. Fveloar, T h e Physics of R u b b e r E l a s t M t y , ( : l a r e n d o n Press, Oxford 1975).
Pergamon
PII S0093-6413(98)00097-4
A C O N S T I T U T I V E MODEL IN FINITE VISCOELASTOPLASTICITY OF RUBBERY POLYMERS
Aleksey D. Drozdov Institute for Industrial Mathematics 22 Ha-Histadrut Street, Be'ersheba 84213, Israel
(Received 19 December 1997; accepted for print 9 November 1998) Introduction The paper is concerned with constitutive equations for the nonlinear response of incompressible rubbery polymers at isothermal loading. To derive the stress-strain relations, we employ a concept of nonaffine transient networks, where sliding of a network of long chains with respect to a bulk medium is described in the framework of finite elastoplasticity. The constitutive equations (developed with the use of the laws of thermodynamics) are applied to determine stresses in a specimen at uniaxial extension. A Model of Transient Networks A viscoelastic medium is modeled as a network of M kinds of long chains connected to junctions [6], where different kinds of chains correspond to different relaxation times. A chain whose ends are connected to separate junctions is treated as an active one. Snapping of an end of a chain from a junction is tantamount to its breakage. When a dangling chain captures a junction, a new active chain is created. Creation and annihilation of chains are characterized by the functions Am(t, T, ho) [the number of active chains of the ruth kind (per unit mass) arising before time r, existing at time t, and having the end-to-end vector ]to at the instant of their formation]. The distribution of initial end-to-end vectors is assumed to be independent of the reformation process,
am(t, ,,e)dVo(e) = "~m(t, , ) P {t~o • dVo(-~)},
(~)
where P is probability, dVo(2) is an elementary volume (in the space of long chains) in the vicinity of a vector z, and ~-,~(t, r) is the number of active chains of the ruth kind (per unit rnass) arising before time r and existing at time t. The quantity -~m(t, 0) equals the number of initial chains of the ruth kind existing at time t; the amount
10.--.m(t,r ) t--T 7re(r)-- -o Or ~rn
39
(--0 = --m(0,0))
(~)
40
A . D . DROZDOV
is the rehd ire rate of reformatiou (the ratio of the number of active chains of the ntth kind created per mill t.ime l.o the initial mmfl)er of aciive chains), and the quantities 1
1'.,,,)(1)=
()~- n,
r 0~-,,g
Z_,,,(I.0) i)l (I.U).
]
l',,,/l,r) . . . . [ ~ ( l . r ) ]
1 ~;)2--~,, i
~(I.T)
(:/)
are the r~'la.ti\'e rates of breakage' (the ratios of t.h(' uumbers of active chains of the ruth kimi amfihilated per unit time to the mmlbers of existing chains). Resolving differential equations (3) with i.he initial conditions (2). we arrive at, the fornmlas 2°exl ) [ - /," l,,,t}(.s)ds ] ,
= (l.()i '
--'"
'~=-'" ---(l.
. ~
[ f' ?m(r)---'2~exp-
Ti
i)T
l,,(.~.r)ds.
]
(1)
. T
The l,oia] uuml)er of active chains of the ruth kind (per unit mass) is calculated as ~,,(I.f)
-_o
{
exp
[
f"
] 1"-;,,,,(r)exp [ -
l',,,~(.~)d.~ -4-
t/
l',,(.~.T)d.~
}
dr
}
.
(5)
\'~'~, confine ourselves to nonaging polymers with constant total numbers of active chains a.d l im~' iudq)emient rates of breakage and mmihJlalio1~ Z,,(t./)
~,.(/)-- .,2-
= Z2,.
1.,.,,(/1 = l '°~,,0,
l,,(z. ~-)
2.
t(i
,Suhstitul.ion of expressions (6) into the balance law (.5) implies that 2,,,o = I,o
-- l,O
(T
Kinematics of a Nonafline Transient Network
Nlotiou of a I)ulk material induces so]n(' pseudo motion in the space of long chaius. Let. h(I) be the e n d q o end vector at time I for a chain created at time r with the initial end-to-eud vector ho. According to the affiniiy hypothesis [8] the deformation gradient ~7(T)h(t) in the space of long chains coincides with the deformation gradient V ( r ) ~ ( t ) in the bulk m a t e r i a l al. a poim with the radius vector /(~). We do not accept this hypothesis and assume t.he polymeric network to slide with respect to the bulk m e d i u m [3,7]. To describe the sliding, we introduce some unloaded configuration (which determines the stress-free state at time I for long chains created at time r) and suppose that the deformation gradient ~7(r)h(t) coincides wii.h the deformation gradient [br lransitiou from the unloaded to actual configuration
~7(r)h(l ) = ~v (/. r)/"(/).
(S)
TIw radius vector r,,(Z, T) in the uuloaded contiguration satisfies the well-known formula
V(7-)r(/) = V(7-).rAI T)- v,,(~, r)7;(l).
(!))
It. follows from E'qs. (8) and (.9) that o v(7)h(t)
= Vtr)h(t).
L(~
L,(t. r ) . v(r)t~(t),
wh('l'('
dr
,It) = ~)-(l).
ih',
,,,{t.r) = ~t/,r).
L(/) = V(l)~:,(t).
I0 )
CONSTITUTIVE MODELS OF RUBBERY POLYMERS
Equation (10) implies that the derivative of the Finger tensor equals
_5_((t,~ . U(t). P(t,~)+P(t,~) . . . L(t) where that
D,,(t, r)
41
['(t, r) = [~'(T)h(t)] v.~7(r)h(t)
2~(~)h(t) b~(t,~).~(~)h(t),
(~1)
= ½[Z~(t, r ) + L~(t, ~-)] and T stands for transpose. It follows from Eq. (11)
0I~
IF(t, r ) :
-
:
where Ik stands for the kth principal invariant, /)(t) = ½[Z(t, r) + Lv(t, r)] is the rate-ofstrain tensor and G(t, r) = V(T)h(t). [~7(T)h(t)]v is the relative Cauchy deformation tensor. T h e r m o d y n a m i c P o t e n t i a l s of a Transient N e t w o r k
Referring to the model of random walk and neglecting the excluded volume effect [2], we trea~ the initial end-to-end vector h0 as a sum of N end-to-end vectors for individual segments. Assuming the segments to be statistically independent, we arrive at the Gaussian formula
where b is the inverse average end-to-end distance. It follows from Eq. (8) that h(t) = ho " ~7(7)]~(t) = ~-7(r)~tT(t). It 0. Combining this equality with Eq. (13) and the incompressibility condition dV(z) = dVo(z), we find the probability for a network of active chains of the ~7~th kind created within the interval It, r + dT] to have an end-to-end vector h located in the unil vicinity of a point 5
dpm(~,T,~)=
(~,T)
exp[--b2z.[2g-l(~,T).z]dT.
(1~}
The specific (per unit mass) entropy dSm(t, r) of a network of active chains created within the interval It, r + dr] and existing at time t differs by a constant dSm. from the average value of logarithm of the distribution function
dS,~(t, r)
= d,5',~. +
kB / dpm(t, T, 2) lnpo(z)dV(5),
(15)
where kB is Boltzmann's constant. Substitution of expressions (13) and (14) into Eq. ([5) results in
dS.,(t,T)
= dS~.+~:~ x In ~
(t,~-) -
b'~9
dt'o(5)dr.
([6)
Summing up expressions (16) with respect to r and neglecting the entropy of interaction between chains, we find the entropy of the network of active links at the current instant l
"re(l) :
Sm*-~- ]¢B
jf ~ ,~-m(~,O)exp t-b2~'~o-l(~)'zj [ ]
+ fot~(t,r)exp[-b22.[~-l(t,,)'2]dT}
[ln(~)3-b252]dVo(2),
42 where
A.D. DROZDOV 1~'.(1) =
~otiT(t) • ~'oh(:). ('~lculating the integral over d~'o(5), we o})ta.ilJ
whet,' "~'2. is a conslant..~ummJng up the specilic c~mtigural.iona.l entropies (17) a.nd addiu~ 1he specific ent.ropy of thermal mot.ion O
,S'therma I
~ In (_)~
wc lind lhe specJ{ic elH.rop3' (per 1Hlil Hlassl .,.(/)
=
,,o + , . ] n ( 9 ( t )
/,',9 £
eo
--i5~ I/, ~1
{z . . . .
01
[1,
3]
(/:'(:.
wh~,re ?,'° is 1he sl)ecitic entropy ill 1]1c nal.ura] sl.al~' al ihe reference temperature (9 °, t, is lhe, specific heal (apacily (per unil nlfLss), and (9 is l.he absolute 4empera.ture. 'T'h~ specific
{'ill rol)y .s' is 'xl)ressed in tel'InS of the specific(per unit, mass) free (ttelmhoh,z) energy qJ hv 1}1(' forI~]~da 5
. . . .
.
(1~})
He
\'Ve accept
l.he following expression for lhv specitic fr~'e energy, which is compa.t.il)le with Efls. (18) and 1.1!)): {')(/}
qJt/:,
::: e ° + ( ~ -
>'°)[O{/I-O°]
,.0(:)1, ~ g - +
/:BO(/)
where ~po i> a material conslant. (!ombining 1he til'st law of t.hermodynamics wil]} lho (:lausius- Duhem inequalil.y~ we arrive al l]le thermodynamic ineflualil~y [1]
(_),IQ,1I .... 's'dOdl :1",11÷ t,-I (.4:I)
~ql • W--(')) _:> (L
(2l)
where Q is lhc sl)e<'ilic (per unit mass) rate ~,1 dissipalion of energy p is a consLanl mass de~Jsit\. ,/ is ihc heal []ux veclol. ,nd /, = do\" ~- is the dc'viatoric COml)onenl of lhe (!am'by ~lv~,ss lel>or b. Sul)sliluling expression (20) inlo Eq. (21) a.nd using Eqs. (}), {7). (12), an,I (IS). we ohtain
,/Q
()7-
= J' v .:. ~ ,::, ~-
/'1 (.4:/)
~)'tl
ve
)
_> 0,
(.,.2~
\\'ll('r~' /~(/)
=
I_,(tj
=
,
.
(:.~)
£[ l')ft.O)+ /,.:-.. i)~-T (t,r)O(t,r): 1~'~0 1) 5[ Z,~(t,O)f:o(:)+ /.--. i~T (t. rlt;'(l, Tjdr j : D(t).
/,H(-)(I)
-~,,,(/,0)(,'o(1) :
-
D~,(I,T)(IT
]
.
,
H*~t
I:~(1) --
m
~
I
.
i
2:/)
CONSTITUTIVE MODELS OF RUBBERY POLYMERS
43
It follows from Eq. (23) that the Clausius-Duhem inequality (22) is fulfilled provided that. (i) the heat flux vector q obeys the Fourier law q(t) = -pA~g(t)O(t) with a nonnegative thermal diffusivity 1, (ii) the Cauchy stress tensor & obeys the constitutive equation a(t) =
-p(t)]+pksO(t) ~
(t,r)f~(t,r)dr ,
E,~(t,0)//'0(t) +
(24)
m=l
where p is pressure, and (iii) the plastic rate-of-strain tensor/)~(t, r) satisfies the equality /)~(t, r)
=/3dev G(t, r),
(95)
where/3 is a nonnegative scalar function. Equation (24) is similar to the Lodge equation [~], (where the tensor fi differs from the conventional Finger tensor), whereas Eq. (25) appears to be novel. It determines the symmetrical part of the tensor L~(t, r). To avoid uncertainty. the skew-symmetrical part of L~ is assumed to vanish. Uniaxial E x t e n s i o n of a Specimen
To verify the constitutive equations (24) and (25), we study uniaxial extension of an incompressible viscoelastoplastic bar X1 = / ~ ( t ) X 1 ,
x: = k-½(t).¥2,
x 3 = k-½(•)X3,
(26)
where k is the extension ratio, and Xi, xi are Cartesian coordinates in the initial and actual configuration, respectively. We assume that transition from the initial to unloaded configuration is described by the formulas similar to Eq. (26) 1
1_
Z2 = k u : ( t , r ) X 2 ,
X 1 = /t?u(t, T ) X 1 ,
Xa = ]~u2(t,T)X3,
(27)
where k,,(t, r) is a function to be found. It follows from Eqs. (26) and (27) that '
V~(t,r)v~(t,r)-
k~(t,r)
(')[~l-
where ~, are unit vectors of the Cartesian coordinates into Eq. (25) implies that
01%~( t , r ) =
{Xi}.
~3[{\k,,(t,r)]k(t) ~2 k~(t,r_)]j k~(t,r)
Substitution of expressions (28)
ku(r,r) = k(r).
(29)
We substitute expressions (28) into the constitutive equation (24), find the unknown pressure p from the boundary condition on the lateral surface of the bar er2 -- 0, use Eqs. (4) and (7), and arrive at the formula for the longitudinal stress
rh(t) = 6'~,~ texp(-r°~t)[\k~(t'O)) +r2 [
gO
exp [ - < ; , ( t -
k(t) ] ,
44
A.D.
DROZDOV
we - ,11 70 where ( , = p/,'B(gE °, q . . . .--- _--'o,,~/~ , and 7~0 = ~,;~=l-,,.The stress oh(/) alld the pla.stic exlension ralio k,(t. 0) are plotted in Figure for leusile test with k ( t ) = e x p ( i t ) . C a l c u l a t i o n s are carried out t})r :11 -- 6, *l* = I0.0, -~ = 1.0. ;,:~ = 0.1. 74 = 0.01, % = (}.00l. 3~; - l).(I (s i) alt(] ql - {I.2. r/_, {).1. ~/a --- 0 . 2 . t/i - 0 . 2 . q-, 0 . 2 . ~/,~ - 0 . 1 .
5.0
().:1 .......-.'}) cr f
1.1}
],'
I()A)
......% .t5 ~'
I.ql
t'
10.0
1-hvoxwnsion ralio/% for transition from lhe iJ~itial to unloaded configuralion (A) and lira longjtudinaI >lress or, (MPa) (B) l'~r~rt.~lho extensiol, ratio k for uniaxial extension of polystyre~e (',EI)EX-1511 al (9 - I.t0 °(' wilb lhe tale-of-strain i = 0.02 (s -1 ). Cireles: experimental dal,:, ()htainod i, [5], solid liltes: pl'edielioli of the model wit}, fl - 0.013 and (' =- 0.037 (MPa) Figure:
Concluding
Remarks
New co1>lil ,tl ix:(' equa/iol,s },a\'c I~eert derived f,,r the viscoelastopla.stic beha.viov uf r u l ) b e , \ 1)olymer~. The, Im)(tel is based o,l l he c o m e p i of mmaffine t r a n s i e n t networks, where a new telal.iotlshi l) is proposed lo link the deformation gradients in a bulk m e d i u m and in a net.work. Stress sl.rain relations are developed based on the laws of t h e r m o d y n a m i c s . T h e c o n s t i t u t i v e equat.ions are employed to del.ermine lhe m a t e r i a l response in lmia.xial tensile t.ests. Results v~t m u m ' r i , a l s i m u l a t i o n d e m o n s t r a t e that the model correctly predicts the viscoelastoplastic response of polyst.yrene al elevaled leml)eral.ures. Acknowledgement
Fi,lancial support l)v lhe lsra.e/i Mi,fisl.rv of Scieu,'e (granl 9611 14)6) is gratefully ackm)wl cdge(I.
References 1. B.]). ('~,h'man and M.I". G u r t i n , 3. ( ' h e m . Phys. 47, 597 (1967). 2. M. 1)oi al~d S.F. Edwards, T h e T h e o r y of P o i y m e r D y n a m i c s , Oxford Universit.y Press. Oxford (19S6). 3. M . \ V . . I o h n s o n and I). S e g a l m a n , 3. N o n - N e w l o n i a n Fluid Mech. 2, 255 (1977). 1. A.S. l,odge. Rheol. Acta 7, 379 (1968). 5. 1~. Muller, l). Froelich, and Y.tl. Zang. 3. Polvm. Sci.: Polym. Phys. Ed. 2 5 , 2 9 5 (19S71. 6. F. :l~mlaka and S.F. Edwards, Macromolecules 25. 1516 (1992). 7. N.P. ' l h i e n a n d tt.]. T a n n e r . . 1 . N o n - N e w t o n i a n Fluid Mech. 2, :/53 (1977). s. IA(.(',. Fveloar, T h e Physics of R u b b e r E l a s t M t y , ( : l a r e n d o n Press, Oxford 1975).