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A comparison of three polymer network models in current use
Computational and Theoretical Polymer Science 9 (1999) 27±33
A comparison of three polymer network models in current use J. Sweeney Department of Mechanical and Medical Engineering/IRC in Polymer Science and Technology, University of Bradford, Bradford BD7 1DP, UK Received 27 October 1997; accepted 2 May 1998
Abstract The relationship between three theories of polymer network deformation is explored. The theories are: the eight-chain model of Arruda and Boyce; the full network model of Wu and van der Giessen; and the crosslink±sliplink model of Edwards and Vilgis. All have a history of use as the network component in theories of solid polymer deformation. Given results from either the eight-chain or full network models, least-squares ®tting of the stresses is used to derive optimal parameters of the Edwards±Vilgis model. Both the eight-chain and the full network models can be closely approximated by an Edwards±Vilgis model, provided the ®nite chain extensibility limit is not approached too closely. The eight-chain model is found to be equivalent to an Edwards±Vilgis model with a small number of sliplinks, whereas the full network model corresponds to an Edwards±Vilgis model with no sliplinks. The physical interpretation of these ®ndings is discussed. # 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Polymer network; Hyperelastic; Multiaxial deformation
1. Introduction There has been much recent interest in mathematical models of molecular chain networks. Some of the motivation for this arises from applications to the mechanical behaviour of solid materials. The network theory may be used directly to simulate the behaviour of rubbers, or alternatively it may form a component in a more complex theory of solid polymer deformation. Among the theories in current use are the eight-chain model of Arruda and Boyce [1], its generalisation in the full network model of Wu and van der Giessen [2], and the model of Edwards and Vilgis [3]. The relationship between these three models is the subject of this paper. In the eight-chain model [1], a representative cubic cell of material has a molecular chain attached to each corner of the cube, and the chains join up at the cube centre. Chains interact only in the sense that they are rigidly connected to one another, in a manner akin to crosslinking. In the full network model [2] this is still the case, but there is a distribution of in®nitely many chains. This model was introduced as a generalisation of the eight-chain model, in the expectation that a model more realistic than the particular geometrical arrangement of the eight-chain model would give better predictions of the behaviour of real material. The Edwards±Vilgis model [2] is more general still, in that it acknowledges that molecular chains may interact in ways other than the rigid crosslink-like interconnections
of the other two models, and allows for a second kind of interaction which has been termed a sliplink. The relative number of crosslinks and sliplinks may be freely speci®ed, and this more complex model network is characterised by more parameters than either of the other two. All three models share the property of ®nite chain extensibility, by which a limit, corresponding to in®nite stress, is imposed on the extent of deformation. The importance of a network component in the deformation of solid polymers has been well established [4]. There are examples of the use of each of the three models to represent the network within a deforming polymer. Typically such a model incorporates a network acting in parallel with a rate-dependent plasticity model, such as that of Eyring [5] or Argon [6]. Thus, Arruda and Boyce [7] made use of the eight-chain network as one of the components in their model of the large deformations of PC and PMMA. G'Sell and Souahi [8] have used the full network model in a similar role in the modelling of PMMA. The Edwards±Vilgis theory has been used by Buckley et al. in their constitutive model of pet [9] and the model of Ball et al. [10], which is a special case of the Edwards±Vilgis model, has been used in the modelling of the high temperature stretching of PVC by Sweeney and Ward [11]. In this ®eld of application, the choice of network model seems to be largely a matter of personal preference. The numerical study presented here is aimed at assessing to what extent there
0189-3156/99/$Ðsee front matter # 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S1089 -3 156(98)00050 -6
28
J. Sweeney / Computational and Theoretical Polymer Science 9 (1999) 27±33
2. The theories
In the full network theory, there is assumed to be an in®nitely large number of chains, which, as the deformation progresses, become oriented according to a chain orientation distribution function. The principal stresses are given by [2]
All three theories are hyperelastic, and the assumption of incompressibility is made, with
iifull
are real eective dierences between the theories, and how signi®cant they are.
l1 l2 l2 1
i 1; 2; 3
1
where CR and llock are material parameters, the latter de®ning the ®nite chain extensibility. p is a hydrostatic pressure arising from the assumption of incompressibility. lchain is the extension ratio of the eight chains, de®ned in terms of the principal extension ratios l1 , l2 and l3 by s l21 l22 l23 2 lchain 3 and llock (the `locking stretch') de®nes the limiting value of lchain , corresponding to in®nite stress, via the inverse Langevin function Lÿ1 , which is itself de®ned with reference to the Langevin function 1 L x coth x ÿ x such that 1 ÿ1 coth x ÿ xL x
2 0
0
Lÿ1
l llock
l2i m0i 2 sin 0 d0 d 0 ÿ p l
i 1; 2; 3
for the principal extension ratios l1 , l2 and l3 . Both the eight-chain and the full network models make use of the inverse Langevin formulation for the stress arising from the stretching chains. They can be viewed as logical developments of earlier Langevin-based models. One such is the three-chain model of Wang and Guth [12] in which a chain is arranged along each principal axis. A later development of Treloar [13] was the four-chain model, in which each of four chains are anchored at one end to the corner of a regular tetrahedron, with the other chain ends joined together at a junction point, the position of which is determined by equilibrium. Similarly, in the eight-chain model the chains are attached at one end to the corners of a cube and joined at the cube centre. The cube is aligned along the principal stretch directions, and on straining becomes a rectangular solid. The eight-chain model is de®ned in terms of the prin8-ch 8-ch 8-ch , 22 and 33 as [1] cipal stresses 11 1 llock l2i ÿ1 lchain L ÿp ii8-ch CR 3 lchain llock
CR llock 4
3
4
5 where CR , llock and p remain as de®ned above. The m0i are de®ned by: m01 sin 0 cos 0 m02 sin 0 sin 0 m03 cos 0 and ®nally l2
X 2 m0i 2li i
Eq. (5) can be evaluated by numerical integration. However, to avoid this cumbersome procedure, Wu and van der Giessen [2], noting that the stress predictions of the full network model always lie between those of the three- and eight-chain models, proposed the following approximation: iifull 1 ÿ ii3-ch ii8-ch
i 1; 2; 3
6
Here ii3-ch are the stresses in the three-chain model of Wang and Guth [12], given by 1 li ÿp ii3-ch CR llock li Lÿ1 3 llock
i 1; 2; 3
and the parameter is de®ned as 0:85lmax =llock where lmax is the largest of the three principal extension ratios. ii8-ch are as de®ned in Eq. (1). The approximation (6) was found to be very accurate up to large stretches, with results for a maximum extension ratio of 8 being demonstrated [2]. The Edwards±Vilgis theory [3] is de®ned in terms of the change in strain energy W per unit volume arising from the imposition of extension ratios l1 , l2 and l3 :
J. Sweeney / Computational and Theoretical Polymer Science 9 (1999) 27±33
"P
#
2 2 X 1 i 1 ÿ li W Nc ln 1 ÿ 2 l2i P 2 2 1 ÿ 2 i li i ! " 2 2 X ÿ 1 li 1 1 ÿ 2 ln 1 li Ns P 2 1 l2i 1 ÿ 2 i l2i i !# X 2 2 li ln 1 ÿ i
7 The quantities Nc and Ns have dimensions of stress, being related to the number of crosslinks per unit volume Nc and number of sliplinks per unit volume Ns by Nc Nc kT and Ns Ns kT: de®nes the degree of slip possible for the sliplinks, with 0 equivalent to a crosslink and otherwise positive. de®nes a limit of extensibility corresponding to a singularity in stress. The constitutive equations in terms of the principal stresses are given by iiEV
@W li ÿ p i 1; 2; 3 @li
8
p is a hydrostatic pressure, with the same role as those in Eq. (1) and Eq. (5). The dierentiations in (8) give rise to lengthy expressions, and are not reproduced here. For computing purposes they were generated using the symbolic algebra package Maple. From Eq. (7) it is clear that the singularity in stress occurs when 1 ÿ 2
X i
l2i 0
1 ÿ 3 2 l2chain 0 Since the inverse Langevin function in Eq. (1) becomes singular at lchain llock , it follows that the eight-chain and Edwards±Vilgis models will produce in®nite stresses at the same strain when llock
3. Numerical methods The relationships between the theories were explored by examining their predictions of stress in various modes of deformation. A pair of computer programs has been written, each of which compares two of the theories. They operate by ®rst producing a set of stress predictions for the ®rst model at ®xed parameter values for a predetermined set of strains; parameter values for the second model are then obtained by nonlinear leastsquares ®tting of the stresses. In all cases three straining modes are used to generate the stresses: uniaxial tension, planar extension (constant width) and equibiaxial tension. It is believed that the use of these three stretching modes should produce comparisons which are de®nitive. All stresses are in plane stress in the 1±2 plane, with uniaxial and planar stretching along the 1 axis, and equibiaxial stretching along the 1 and 2 axes. Thus the principal extension ratios (l1 , l2 , l3 ) are given , lÿ1=2 ) in uniaxial extension, (l1 , 1, lÿ1 by (l1 , lÿ1=2 1 ) in 1 1 planar extension and (l1 , l1 , lÿ2 ) in equibiaxial stretch1 ing. The hydrostatic pressure p occurring in Eq. (1), Eq. (5) and Eq. (8) is eliminated by making use of the fact that 33 0, so that for each theory and stretching mode the stress in the plane in the i direction is given by ii ÿ 33
i 1; 2:
These stresses are calculated by the programs. The ®rst program, denoted program A, compares the eight-chain and Edwards±Vilgis models by ®tting the Edwards±Vilgis model to ®xed eight-chain model stresses. The notations ii8-ch;u ; iiEV;u for principal stresses in uniaxial tension, ii8-ch;p ; iiEV;p for planar extension, and
which in the terminology of the eight-chain model is equivalent to
1 p 3
29
9
There is no such simple relation between the Edwards± Vilgis model and the full network model, since the stress in the latter is not a function of lchain .
ii8-ch;b ; iiEV;b in equibiaxial tension, are adopted. Stresses as de®ned using Eq. (1) and Eq. (8) for all three stretching modes are used, with both axial and lateral stresses ®tted in the case of planar extension. The ®tting is achieved by generating stresses at ®xed intervals l in extension ratio and minimizing the function F
" Nu X 8-ch;u i
#2
" Np X 8-ch;p i
#2
11 11 ÿ1 ÿ1 EV;u EV;p 11 i i1 11 i " #2 " #2 Np 8-ch;p Nb 8-ch;b X X 22 i 11 i ÿ1 ÿ1 EV;p EV;b i1 22 i i1 11 i i1
10
30
J. Sweeney / Computational and Theoretical Polymer Science 9 (1999) 27±33
where the index i corresponds to the extension ratio l1 1 il. Optimal values of Nc , Ns , and are obtained,p subject to the constraints 50 and 04 41 3. The constraint on has its origin in the Edwards±Vilgis theory, in which negative values of are nonphysical. The lower limit of zero for corresponds to no extensibility limit (for which the theory reduces to the Ball et al. model [10]), and the upper limit is necessary to avoid a stress singularity in the undeformed state. The minimisation was carried out using the NAG library routine E04JAF. Similarly, program B ®ts an Edwards±Vilgis model to a full network theory. The full network principal stresses are denoted by iifull;u in uniaxial tension, iifull;p for planar extension, and iifull;b in equibiaxial extension. These stresses are calculated using the approximation (6). An analogous procedure to that above for program A is carried out, with the function to be minimised given by F
" Nu X full;u i i1
11 EV;u i 11
#2 ÿ1
" Np X full;p i 22
i1
EV;p 22 i
#2
ÿ1
" Np X full;p i i1
ii EV;p i 11
#2 ÿ1
" Nb X full;b i 11
i1
EV;b i 11
#2
11
ÿ1
Optimal values of the parameters Nc , Ns , and are obtained within the same constraints and using the same method as for program A. The average proportional error F , de®ned by p F F= Nu 2Np Nb is used as a measure of goodness of ®t. 4. Comparison of theories In this section a general picture of the relationship between the three theories will be built up by looking at a series of cases. The ®rst of these concerns ®ts of both the eight-chain and the full network model to the experimental data on natural rubber-gum of James et al. [14]. The ®tting was done by Wu and van der Giessen
[2], and their parameters are used here to generate eightchain and full network model stresses. Programs A and B are used to ®t Edwards±Vilgis models to the eightchain and full network model stresses, respectively. In all cases stresses are calculated at discrete intervals in extension ratio of l 0:1. In Fig. 1 the stresses generated by the eight-chain model are compared using program A with the ®tted stresses generated by the Edwards±Vilgis model. Table 1 gives the parameters for both theories, together with the average error F . The stresses appear to ®t closely in Fig. 1, corresponding to the 2% value of F in Table 1. The non-zero value of Ns in the table shows that the eight-chain model corresponds to an Edwards±Vilgis model which incorporates some degree of slip, albeit with quite `tight' sliplinks corresponding to 0:02. The value of a corresponds, via Eq. (9), to a locking stretch of 7.42, greatly dierent from the value llock 5 assumed in the eight-chain model. The dierence in shape of the stress-strain curves produced by the two theories ensures that dierent locking stretches are required for an optimum ®t. In Fig. 2, program B has been used to ®t the Edwards±Vilgis model to the full network model, again with the latter as generated by Wu and van der Giessen [2]. Table 2 gives the ®tted Edwards±Vilgis parameters. In Fig. 2 the goodness of ®t is generally worse than in Fig. 1, except in the case of equibiaxial stretching; the average error F has doubled relative to the eight-chain ®t to 4%. It is interesting that the ®tted value for the sliplink number Ns is now zero. This makes sense physically, since sliplinks are explicitly excluded from the full network model. The dierence between the two theories must lie in the use of dierent functions of strainÐthe inverse Langevin of the full network model and the more elementary singular function used by Edwards and Vilgis. The value of now corresponds to a locking stretch of 10.35, again greater than that in the full network model, which corresponds to llock 7:075. A further example of eight-chain parameters obtained by ®tting to experimental data on rubber has been provided by Arruda and Boyce [1]. They took Treloar's data [15] on vulcanised rubber and showed that the eight-chain model gave a better representation of it than the three- or four-chain models. Their eight-chain parameters are given in Table 3, together with Edwards± Vilgis parameters obtained by ®tting using program A. The ®tted stresses are shown in Fig. 3. As in the eight-chain results of Table 1 and Fig. 1, there is a small nonzero Ns term. The greater error F is a result of ®tting to a larger interval in strain in comparison with llock . In contrast with the results of Table 1, in this case the maximum stretches are greater than llock . As the upper limit of the interval in strain over which stresses have been ®tted approaches its maximum value as de®ned by the ®nite chain extensibility, the
J. Sweeney / Computational and Theoretical Polymer Science 9 (1999) 27±33
Fig. 1. Comparison of eight-chain model stresses and the ®tted Edwards±Vilgis model stresses of Table 1: (a) uniaxial stretching; (b) planar extension, axial (1-direction) stress; (c) planar extension, lateral (2-direction) stress; (d) equibiaxial stretching.
31
Fig. 2. Comparison of full network model stresses and the ®tted Edwards±Vilgis model stresses of Table 2: (a) uniaxial stretching; (b) planar extension, axial (1-direction) stress; (c) planar extension, lateral (2-direction) stress; (d) equibiaxial stretching.
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J. Sweeney / Computational and Theoretical Polymer Science 9 (1999) 27±33
goodness of ®t will tend to become limited by the degree of similarity of the functions of strain (Eqs. (1), (5) and (8)) which de®ne the stress. Having studied cases of particular relevance to the behaviour of rubbers, a more systematic comparison will now be made. Without loss of generality, the value of CR is ®xed at unity for both the eight-chain and full network models. The Edwards±Vilgis theory is ®tted to both these models for a series of values of llock up to a maximum of 10, over a range of extension ratio from 1 to 0.9llock . Results from program A for the eight-chain and Edwards±Vilgis models are given in Table 4, and results from program B for the full network model and the Edwards±Vilgis model are given in Table 5. The observations made with the rubber-based results continue to be valid. The eight-chain model is always ®tted by an Edwards±Vilgis model with a small nonzero Ns term, and the full network model always corresponds to an Edwards±Vilgis model with no sliplinks. The ®tting Table 1 Optimal Edwards±Vilgis parameters for ®t to eight-chain model with CR 0:4 MPa and llock 5, for tensile stretching up to l 5 Nc /MPa 0.37
Ns /MPa
F
2.2910ÿ2
1.7010ÿ2
7.7810ÿ2
2.0810ÿ2
Table 2 Optimal Edwards±Vilgis parameters for ®t to full network model with p CR 0:4 MPa and llock 50, for tensile stretching up to l 5 Nc /MPa
Ns /MPa
F
0
±
5.5810ÿ2
4.2610ÿ2
0.405
Table 3 Optimal Edwards±Vilgis parameters for ®t to eight-chain model with p CR 0:09 MPa and llock 65, for tensile stretching up to l 7:6 in uniaxial and planar stretching, and up to 6.1 in equibiaxial stretching Nc /MPa 0.212
Ns /MPa
F
4.17810ÿ2
1.22610ÿ2
8.4110ÿ2
7.810ÿ2
Table 4 Optimal Edwards-Vilgis parameters for ®t to eight-chain model with CR 1 and a series of values of llock for tensile stretching up to l1 0:9 llock for all stretching modes llock
Fig. 3. Comparison of eight-chain model stresses and the ®tted Edwards±Vilgis model stresses of Table 3: (a) uniaxial stretching; (b) planar extension, axial (1-direction) stress; (c) planar extension, lateral (2-direction) stress; (d) equibiaxial stretching.
2 3 4 5 6 7 8 9 10
Nc
Ns
F
0.0 0.0 0.961 0.969 0.966 0.968 0.969 0.970 0.970
1.001 0.997 3.3510ÿ2 2.4210ÿ2 2.4910ÿ2 2.3210ÿ2 2.2210ÿ2 2.1610ÿ2 2.1110ÿ2
7.50410ÿ4 6.94510ÿ4 1.88910ÿ2 2.2410ÿ2 1.9610ÿ2 1.6910ÿ2 1.4410ÿ2 1.2310ÿ2 1.0510ÿ2
0.198 0.126 9.3110ÿ2 7.3910ÿ2 6.1810ÿ2 5.2810ÿ2 4.6110ÿ2 4.0910ÿ2 3.6710ÿ2
6.2610ÿ3 9.3510ÿ3 1.0310ÿ2 1.0610ÿ2 1.1910ÿ2 1.1710ÿ2 1.1610ÿ2 1.1510ÿ2 1.1410ÿ2
J. Sweeney / Computational and Theoretical Polymer Science 9 (1999) 27±33 Table 5 Optimal Edwards±Vilgis parameters for ®t to full network with CR 1 and a series of values of llock for tensile stretching up to l1 0:9 llock for all stretching modes llock 2 3 4 5 6 7 8 9 10
Nc 1.057 1.030 1.013 1.007 1.002 1.000 1.0000 0.999 0.999
Ns 0 0 0 0 0 0 0 0 0
± ± ± ± ± ± ± ± ±
F
0.221 0.144 0.107 8.5310ÿ2 7.1810ÿ2 6.1410ÿ2 5.3610ÿ2 4.7610ÿ2 4.2810ÿ2
9.0910ÿ2 8.4410ÿ2 8.1010ÿ2 7.8810ÿ2 8.3210ÿ2 8.1110-ÿ2 7.9510ÿ2 7.8310ÿ2 7.7410ÿ2
error F is always greater for the full network model; this is probably because the upper limit of the ®tting interval, 0.9llock , is in this case closer to the stretch corresponding to in®nite stress, which is llock for the full network model and greater for the eight-chain model. Increasing the range for the eight-chain ®ts increases the errors to match those in Table 5.
33
be fortuitously captured by the eight-chain model. The numerical results reported here support this view; the eight-chain models are always ®tted by Edwards±Vilgis models which feature some degree of slip, manifested by a nonzero sliplink number Ns . In contrast, the full network models always correspond to Edwards±Vilgis models with no sliplinks. The geometrical arrangement of the eight-chain model is such that there are no chains along the maximum principal stretch directions. Adding chains along the principal directions, as in a three-chain model, would be expected to produce the equivalent of the full network model as the `missing' chains are replaced; this provides a physical interpretation of the approximation (6). Chains along the direction of maximum principal stretch would be most highly stressed and therefore most likely to be subject to slip; omitting them is therefore roughly equivalent to a strain-softening mechanism. This gives an intuitive picture of the reason for the similarity between this model and an Edwards-Vilgis model featuring some degree of slip. References
5. Discussion and conclusion The Edwards±Vilgis model is capable of modelling a much broader range of material behaviour than the other two models, owing to the additional feature of sliplinks. The similarities demonstrated here in this restricted range of rubberlike behaviour show that both eight-chain and full network models could be replaced to a good approximation with Edwards±Vilgis models, provided that the ®nite chain extensibility limit is not approached too closely. In comparing the eight-chain and full network models, Wu and Van der Giessen [2] noted that the eightchain model gave the better performance when modelling the behaviour of rubbers. This was contrary to expectations and they suggested that the non-ideal behaviour of real rubbers, such as chain slippage, might
[1] Arruda EM, Boyce MC. Journal of Mech Phys Solids 1993;41:389. [2] Wu PD, van der Giessen E. Journal of Mech Phys Solids 1993;41:427. [3] Edwards SF, Vilgis TA. Polymer 1986;27:483. [4] Ward IM. Polym Eng Sci 1984;24:724. [5] Eyring H. Journal of Chem Phys 1936;4:283. [6] Argon AS. Philos Mag 1973;28:839. [7] Arruda EM, Boyce MC. International Journal of Plasticity 1993;9:697. [8] G'Sell C, Souahi A. ASME Journal of Engineering Materials and Technology 1997;119:223. [9] Buckley CP, Jones DC, Jones DP. Polymer 1996;37:2403. [10] Ball RC, Doi M, Edwards SF, Warner M. Polymer 1981;22:1010. [11] Sweeney J, Ward IM. Polymer 1995;36:299. [12] Wang MC, Guth EJ. Journal of Chem Phys 1952;20:1144. [13] Treloar LRG. Trans. Faraday Soc 1954;50:881. [14] James HM, Green A, Simpson GM. J Appl Polymer Sci 1975;19:2033. [15] Treloar LRG. Trans Faraday Soc 1944;40:59.
A comparison of three polymer network models in current use J. Sweeney Department of Mechanical and Medical Engineering/IRC in Polymer Science and Technology, University of Bradford, Bradford BD7 1DP, UK Received 27 October 1997; accepted 2 May 1998
Abstract The relationship between three theories of polymer network deformation is explored. The theories are: the eight-chain model of Arruda and Boyce; the full network model of Wu and van der Giessen; and the crosslink±sliplink model of Edwards and Vilgis. All have a history of use as the network component in theories of solid polymer deformation. Given results from either the eight-chain or full network models, least-squares ®tting of the stresses is used to derive optimal parameters of the Edwards±Vilgis model. Both the eight-chain and the full network models can be closely approximated by an Edwards±Vilgis model, provided the ®nite chain extensibility limit is not approached too closely. The eight-chain model is found to be equivalent to an Edwards±Vilgis model with a small number of sliplinks, whereas the full network model corresponds to an Edwards±Vilgis model with no sliplinks. The physical interpretation of these ®ndings is discussed. # 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Polymer network; Hyperelastic; Multiaxial deformation
1. Introduction There has been much recent interest in mathematical models of molecular chain networks. Some of the motivation for this arises from applications to the mechanical behaviour of solid materials. The network theory may be used directly to simulate the behaviour of rubbers, or alternatively it may form a component in a more complex theory of solid polymer deformation. Among the theories in current use are the eight-chain model of Arruda and Boyce [1], its generalisation in the full network model of Wu and van der Giessen [2], and the model of Edwards and Vilgis [3]. The relationship between these three models is the subject of this paper. In the eight-chain model [1], a representative cubic cell of material has a molecular chain attached to each corner of the cube, and the chains join up at the cube centre. Chains interact only in the sense that they are rigidly connected to one another, in a manner akin to crosslinking. In the full network model [2] this is still the case, but there is a distribution of in®nitely many chains. This model was introduced as a generalisation of the eight-chain model, in the expectation that a model more realistic than the particular geometrical arrangement of the eight-chain model would give better predictions of the behaviour of real material. The Edwards±Vilgis model [2] is more general still, in that it acknowledges that molecular chains may interact in ways other than the rigid crosslink-like interconnections
of the other two models, and allows for a second kind of interaction which has been termed a sliplink. The relative number of crosslinks and sliplinks may be freely speci®ed, and this more complex model network is characterised by more parameters than either of the other two. All three models share the property of ®nite chain extensibility, by which a limit, corresponding to in®nite stress, is imposed on the extent of deformation. The importance of a network component in the deformation of solid polymers has been well established [4]. There are examples of the use of each of the three models to represent the network within a deforming polymer. Typically such a model incorporates a network acting in parallel with a rate-dependent plasticity model, such as that of Eyring [5] or Argon [6]. Thus, Arruda and Boyce [7] made use of the eight-chain network as one of the components in their model of the large deformations of PC and PMMA. G'Sell and Souahi [8] have used the full network model in a similar role in the modelling of PMMA. The Edwards±Vilgis theory has been used by Buckley et al. in their constitutive model of pet [9] and the model of Ball et al. [10], which is a special case of the Edwards±Vilgis model, has been used in the modelling of the high temperature stretching of PVC by Sweeney and Ward [11]. In this ®eld of application, the choice of network model seems to be largely a matter of personal preference. The numerical study presented here is aimed at assessing to what extent there
0189-3156/99/$Ðsee front matter # 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S1089 -3 156(98)00050 -6
28
J. Sweeney / Computational and Theoretical Polymer Science 9 (1999) 27±33
2. The theories
In the full network theory, there is assumed to be an in®nitely large number of chains, which, as the deformation progresses, become oriented according to a chain orientation distribution function. The principal stresses are given by [2]
All three theories are hyperelastic, and the assumption of incompressibility is made, with
iifull
are real eective dierences between the theories, and how signi®cant they are.
l1 l2 l2 1
i 1; 2; 3
1
where CR and llock are material parameters, the latter de®ning the ®nite chain extensibility. p is a hydrostatic pressure arising from the assumption of incompressibility. lchain is the extension ratio of the eight chains, de®ned in terms of the principal extension ratios l1 , l2 and l3 by s l21 l22 l23 2 lchain 3 and llock (the `locking stretch') de®nes the limiting value of lchain , corresponding to in®nite stress, via the inverse Langevin function Lÿ1 , which is itself de®ned with reference to the Langevin function 1 L x coth x ÿ x such that 1 ÿ1 coth x ÿ xL x
2 0
0
Lÿ1
l llock
l2i m0i 2 sin 0 d0 d 0 ÿ p l
i 1; 2; 3
for the principal extension ratios l1 , l2 and l3 . Both the eight-chain and the full network models make use of the inverse Langevin formulation for the stress arising from the stretching chains. They can be viewed as logical developments of earlier Langevin-based models. One such is the three-chain model of Wang and Guth [12] in which a chain is arranged along each principal axis. A later development of Treloar [13] was the four-chain model, in which each of four chains are anchored at one end to the corner of a regular tetrahedron, with the other chain ends joined together at a junction point, the position of which is determined by equilibrium. Similarly, in the eight-chain model the chains are attached at one end to the corners of a cube and joined at the cube centre. The cube is aligned along the principal stretch directions, and on straining becomes a rectangular solid. The eight-chain model is de®ned in terms of the prin8-ch 8-ch 8-ch , 22 and 33 as [1] cipal stresses 11 1 llock l2i ÿ1 lchain L ÿp ii8-ch CR 3 lchain llock
CR llock 4
3
4
5 where CR , llock and p remain as de®ned above. The m0i are de®ned by: m01 sin 0 cos 0 m02 sin 0 sin 0 m03 cos 0 and ®nally l2
X 2 m0i 2li i
Eq. (5) can be evaluated by numerical integration. However, to avoid this cumbersome procedure, Wu and van der Giessen [2], noting that the stress predictions of the full network model always lie between those of the three- and eight-chain models, proposed the following approximation: iifull 1 ÿ ii3-ch ii8-ch
i 1; 2; 3
6
Here ii3-ch are the stresses in the three-chain model of Wang and Guth [12], given by 1 li ÿp ii3-ch CR llock li Lÿ1 3 llock
i 1; 2; 3
and the parameter is de®ned as 0:85lmax =llock where lmax is the largest of the three principal extension ratios. ii8-ch are as de®ned in Eq. (1). The approximation (6) was found to be very accurate up to large stretches, with results for a maximum extension ratio of 8 being demonstrated [2]. The Edwards±Vilgis theory [3] is de®ned in terms of the change in strain energy W per unit volume arising from the imposition of extension ratios l1 , l2 and l3 :
J. Sweeney / Computational and Theoretical Polymer Science 9 (1999) 27±33
"P
#
2 2 X 1 i 1 ÿ li W Nc ln 1 ÿ 2 l2i P 2 2 1 ÿ 2 i li i ! " 2 2 X ÿ 1 li 1 1 ÿ 2 ln 1 li Ns P 2 1 l2i 1 ÿ 2 i l2i i !# X 2 2 li ln 1 ÿ i
7 The quantities Nc and Ns have dimensions of stress, being related to the number of crosslinks per unit volume Nc and number of sliplinks per unit volume Ns by Nc Nc kT and Ns Ns kT: de®nes the degree of slip possible for the sliplinks, with 0 equivalent to a crosslink and otherwise positive. de®nes a limit of extensibility corresponding to a singularity in stress. The constitutive equations in terms of the principal stresses are given by iiEV
@W li ÿ p i 1; 2; 3 @li
8
p is a hydrostatic pressure, with the same role as those in Eq. (1) and Eq. (5). The dierentiations in (8) give rise to lengthy expressions, and are not reproduced here. For computing purposes they were generated using the symbolic algebra package Maple. From Eq. (7) it is clear that the singularity in stress occurs when 1 ÿ 2
X i
l2i 0
1 ÿ 3 2 l2chain 0 Since the inverse Langevin function in Eq. (1) becomes singular at lchain llock , it follows that the eight-chain and Edwards±Vilgis models will produce in®nite stresses at the same strain when llock
3. Numerical methods The relationships between the theories were explored by examining their predictions of stress in various modes of deformation. A pair of computer programs has been written, each of which compares two of the theories. They operate by ®rst producing a set of stress predictions for the ®rst model at ®xed parameter values for a predetermined set of strains; parameter values for the second model are then obtained by nonlinear leastsquares ®tting of the stresses. In all cases three straining modes are used to generate the stresses: uniaxial tension, planar extension (constant width) and equibiaxial tension. It is believed that the use of these three stretching modes should produce comparisons which are de®nitive. All stresses are in plane stress in the 1±2 plane, with uniaxial and planar stretching along the 1 axis, and equibiaxial stretching along the 1 and 2 axes. Thus the principal extension ratios (l1 , l2 , l3 ) are given , lÿ1=2 ) in uniaxial extension, (l1 , 1, lÿ1 by (l1 , lÿ1=2 1 ) in 1 1 planar extension and (l1 , l1 , lÿ2 ) in equibiaxial stretch1 ing. The hydrostatic pressure p occurring in Eq. (1), Eq. (5) and Eq. (8) is eliminated by making use of the fact that 33 0, so that for each theory and stretching mode the stress in the plane in the i direction is given by ii ÿ 33
i 1; 2:
These stresses are calculated by the programs. The ®rst program, denoted program A, compares the eight-chain and Edwards±Vilgis models by ®tting the Edwards±Vilgis model to ®xed eight-chain model stresses. The notations ii8-ch;u ; iiEV;u for principal stresses in uniaxial tension, ii8-ch;p ; iiEV;p for planar extension, and
which in the terminology of the eight-chain model is equivalent to
1 p 3
29
9
There is no such simple relation between the Edwards± Vilgis model and the full network model, since the stress in the latter is not a function of lchain .
ii8-ch;b ; iiEV;b in equibiaxial tension, are adopted. Stresses as de®ned using Eq. (1) and Eq. (8) for all three stretching modes are used, with both axial and lateral stresses ®tted in the case of planar extension. The ®tting is achieved by generating stresses at ®xed intervals l in extension ratio and minimizing the function F
" Nu X 8-ch;u i
#2
" Np X 8-ch;p i
#2
11 11 ÿ1 ÿ1 EV;u EV;p 11 i i1 11 i " #2 " #2 Np 8-ch;p Nb 8-ch;b X X 22 i 11 i ÿ1 ÿ1 EV;p EV;b i1 22 i i1 11 i i1
10
30
J. Sweeney / Computational and Theoretical Polymer Science 9 (1999) 27±33
where the index i corresponds to the extension ratio l1 1 il. Optimal values of Nc , Ns , and are obtained,p subject to the constraints 50 and 04 41 3. The constraint on has its origin in the Edwards±Vilgis theory, in which negative values of are nonphysical. The lower limit of zero for corresponds to no extensibility limit (for which the theory reduces to the Ball et al. model [10]), and the upper limit is necessary to avoid a stress singularity in the undeformed state. The minimisation was carried out using the NAG library routine E04JAF. Similarly, program B ®ts an Edwards±Vilgis model to a full network theory. The full network principal stresses are denoted by iifull;u in uniaxial tension, iifull;p for planar extension, and iifull;b in equibiaxial extension. These stresses are calculated using the approximation (6). An analogous procedure to that above for program A is carried out, with the function to be minimised given by F
" Nu X full;u i i1
11 EV;u i 11
#2 ÿ1
" Np X full;p i 22
i1
EV;p 22 i
#2
ÿ1
" Np X full;p i i1
ii EV;p i 11
#2 ÿ1
" Nb X full;b i 11
i1
EV;b i 11
#2
11
ÿ1
Optimal values of the parameters Nc , Ns , and are obtained within the same constraints and using the same method as for program A. The average proportional error F , de®ned by p F F= Nu 2Np Nb is used as a measure of goodness of ®t. 4. Comparison of theories In this section a general picture of the relationship between the three theories will be built up by looking at a series of cases. The ®rst of these concerns ®ts of both the eight-chain and the full network model to the experimental data on natural rubber-gum of James et al. [14]. The ®tting was done by Wu and van der Giessen
[2], and their parameters are used here to generate eightchain and full network model stresses. Programs A and B are used to ®t Edwards±Vilgis models to the eightchain and full network model stresses, respectively. In all cases stresses are calculated at discrete intervals in extension ratio of l 0:1. In Fig. 1 the stresses generated by the eight-chain model are compared using program A with the ®tted stresses generated by the Edwards±Vilgis model. Table 1 gives the parameters for both theories, together with the average error F . The stresses appear to ®t closely in Fig. 1, corresponding to the 2% value of F in Table 1. The non-zero value of Ns in the table shows that the eight-chain model corresponds to an Edwards±Vilgis model which incorporates some degree of slip, albeit with quite `tight' sliplinks corresponding to 0:02. The value of a corresponds, via Eq. (9), to a locking stretch of 7.42, greatly dierent from the value llock 5 assumed in the eight-chain model. The dierence in shape of the stress-strain curves produced by the two theories ensures that dierent locking stretches are required for an optimum ®t. In Fig. 2, program B has been used to ®t the Edwards±Vilgis model to the full network model, again with the latter as generated by Wu and van der Giessen [2]. Table 2 gives the ®tted Edwards±Vilgis parameters. In Fig. 2 the goodness of ®t is generally worse than in Fig. 1, except in the case of equibiaxial stretching; the average error F has doubled relative to the eight-chain ®t to 4%. It is interesting that the ®tted value for the sliplink number Ns is now zero. This makes sense physically, since sliplinks are explicitly excluded from the full network model. The dierence between the two theories must lie in the use of dierent functions of strainÐthe inverse Langevin of the full network model and the more elementary singular function used by Edwards and Vilgis. The value of now corresponds to a locking stretch of 10.35, again greater than that in the full network model, which corresponds to llock 7:075. A further example of eight-chain parameters obtained by ®tting to experimental data on rubber has been provided by Arruda and Boyce [1]. They took Treloar's data [15] on vulcanised rubber and showed that the eight-chain model gave a better representation of it than the three- or four-chain models. Their eight-chain parameters are given in Table 3, together with Edwards± Vilgis parameters obtained by ®tting using program A. The ®tted stresses are shown in Fig. 3. As in the eight-chain results of Table 1 and Fig. 1, there is a small nonzero Ns term. The greater error F is a result of ®tting to a larger interval in strain in comparison with llock . In contrast with the results of Table 1, in this case the maximum stretches are greater than llock . As the upper limit of the interval in strain over which stresses have been ®tted approaches its maximum value as de®ned by the ®nite chain extensibility, the
J. Sweeney / Computational and Theoretical Polymer Science 9 (1999) 27±33
Fig. 1. Comparison of eight-chain model stresses and the ®tted Edwards±Vilgis model stresses of Table 1: (a) uniaxial stretching; (b) planar extension, axial (1-direction) stress; (c) planar extension, lateral (2-direction) stress; (d) equibiaxial stretching.
31
Fig. 2. Comparison of full network model stresses and the ®tted Edwards±Vilgis model stresses of Table 2: (a) uniaxial stretching; (b) planar extension, axial (1-direction) stress; (c) planar extension, lateral (2-direction) stress; (d) equibiaxial stretching.
32
J. Sweeney / Computational and Theoretical Polymer Science 9 (1999) 27±33
goodness of ®t will tend to become limited by the degree of similarity of the functions of strain (Eqs. (1), (5) and (8)) which de®ne the stress. Having studied cases of particular relevance to the behaviour of rubbers, a more systematic comparison will now be made. Without loss of generality, the value of CR is ®xed at unity for both the eight-chain and full network models. The Edwards±Vilgis theory is ®tted to both these models for a series of values of llock up to a maximum of 10, over a range of extension ratio from 1 to 0.9llock . Results from program A for the eight-chain and Edwards±Vilgis models are given in Table 4, and results from program B for the full network model and the Edwards±Vilgis model are given in Table 5. The observations made with the rubber-based results continue to be valid. The eight-chain model is always ®tted by an Edwards±Vilgis model with a small nonzero Ns term, and the full network model always corresponds to an Edwards±Vilgis model with no sliplinks. The ®tting Table 1 Optimal Edwards±Vilgis parameters for ®t to eight-chain model with CR 0:4 MPa and llock 5, for tensile stretching up to l 5 Nc /MPa 0.37
Ns /MPa
F
2.2910ÿ2
1.7010ÿ2
7.7810ÿ2
2.0810ÿ2
Table 2 Optimal Edwards±Vilgis parameters for ®t to full network model with p CR 0:4 MPa and llock 50, for tensile stretching up to l 5 Nc /MPa
Ns /MPa
F
0
±
5.5810ÿ2
4.2610ÿ2
0.405
Table 3 Optimal Edwards±Vilgis parameters for ®t to eight-chain model with p CR 0:09 MPa and llock 65, for tensile stretching up to l 7:6 in uniaxial and planar stretching, and up to 6.1 in equibiaxial stretching Nc /MPa 0.212
Ns /MPa
F
4.17810ÿ2
1.22610ÿ2
8.4110ÿ2
7.810ÿ2
Table 4 Optimal Edwards-Vilgis parameters for ®t to eight-chain model with CR 1 and a series of values of llock for tensile stretching up to l1 0:9 llock for all stretching modes llock
Fig. 3. Comparison of eight-chain model stresses and the ®tted Edwards±Vilgis model stresses of Table 3: (a) uniaxial stretching; (b) planar extension, axial (1-direction) stress; (c) planar extension, lateral (2-direction) stress; (d) equibiaxial stretching.
2 3 4 5 6 7 8 9 10
Nc
Ns
F
0.0 0.0 0.961 0.969 0.966 0.968 0.969 0.970 0.970
1.001 0.997 3.3510ÿ2 2.4210ÿ2 2.4910ÿ2 2.3210ÿ2 2.2210ÿ2 2.1610ÿ2 2.1110ÿ2
7.50410ÿ4 6.94510ÿ4 1.88910ÿ2 2.2410ÿ2 1.9610ÿ2 1.6910ÿ2 1.4410ÿ2 1.2310ÿ2 1.0510ÿ2
0.198 0.126 9.3110ÿ2 7.3910ÿ2 6.1810ÿ2 5.2810ÿ2 4.6110ÿ2 4.0910ÿ2 3.6710ÿ2
6.2610ÿ3 9.3510ÿ3 1.0310ÿ2 1.0610ÿ2 1.1910ÿ2 1.1710ÿ2 1.1610ÿ2 1.1510ÿ2 1.1410ÿ2
J. Sweeney / Computational and Theoretical Polymer Science 9 (1999) 27±33 Table 5 Optimal Edwards±Vilgis parameters for ®t to full network with CR 1 and a series of values of llock for tensile stretching up to l1 0:9 llock for all stretching modes llock 2 3 4 5 6 7 8 9 10
Nc 1.057 1.030 1.013 1.007 1.002 1.000 1.0000 0.999 0.999
Ns 0 0 0 0 0 0 0 0 0
± ± ± ± ± ± ± ± ±
F
0.221 0.144 0.107 8.5310ÿ2 7.1810ÿ2 6.1410ÿ2 5.3610ÿ2 4.7610ÿ2 4.2810ÿ2
9.0910ÿ2 8.4410ÿ2 8.1010ÿ2 7.8810ÿ2 8.3210ÿ2 8.1110-ÿ2 7.9510ÿ2 7.8310ÿ2 7.7410ÿ2
error F is always greater for the full network model; this is probably because the upper limit of the ®tting interval, 0.9llock , is in this case closer to the stretch corresponding to in®nite stress, which is llock for the full network model and greater for the eight-chain model. Increasing the range for the eight-chain ®ts increases the errors to match those in Table 5.
33
be fortuitously captured by the eight-chain model. The numerical results reported here support this view; the eight-chain models are always ®tted by Edwards±Vilgis models which feature some degree of slip, manifested by a nonzero sliplink number Ns . In contrast, the full network models always correspond to Edwards±Vilgis models with no sliplinks. The geometrical arrangement of the eight-chain model is such that there are no chains along the maximum principal stretch directions. Adding chains along the principal directions, as in a three-chain model, would be expected to produce the equivalent of the full network model as the `missing' chains are replaced; this provides a physical interpretation of the approximation (6). Chains along the direction of maximum principal stretch would be most highly stressed and therefore most likely to be subject to slip; omitting them is therefore roughly equivalent to a strain-softening mechanism. This gives an intuitive picture of the reason for the similarity between this model and an Edwards-Vilgis model featuring some degree of slip. References
5. Discussion and conclusion The Edwards±Vilgis model is capable of modelling a much broader range of material behaviour than the other two models, owing to the additional feature of sliplinks. The similarities demonstrated here in this restricted range of rubberlike behaviour show that both eight-chain and full network models could be replaced to a good approximation with Edwards±Vilgis models, provided that the ®nite chain extensibility limit is not approached too closely. In comparing the eight-chain and full network models, Wu and Van der Giessen [2] noted that the eightchain model gave the better performance when modelling the behaviour of rubbers. This was contrary to expectations and they suggested that the non-ideal behaviour of real rubbers, such as chain slippage, might
[1] Arruda EM, Boyce MC. Journal of Mech Phys Solids 1993;41:389. [2] Wu PD, van der Giessen E. Journal of Mech Phys Solids 1993;41:427. [3] Edwards SF, Vilgis TA. Polymer 1986;27:483. [4] Ward IM. Polym Eng Sci 1984;24:724. [5] Eyring H. Journal of Chem Phys 1936;4:283. [6] Argon AS. Philos Mag 1973;28:839. [7] Arruda EM, Boyce MC. International Journal of Plasticity 1993;9:697. [8] G'Sell C, Souahi A. ASME Journal of Engineering Materials and Technology 1997;119:223. [9] Buckley CP, Jones DC, Jones DP. Polymer 1996;37:2403. [10] Ball RC, Doi M, Edwards SF, Warner M. Polymer 1981;22:1010. [11] Sweeney J, Ward IM. Polymer 1995;36:299. [12] Wang MC, Guth EJ. Journal of Chem Phys 1952;20:1144. [13] Treloar LRG. Trans. Faraday Soc 1954;50:881. [14] James HM, Green A, Simpson GM. J Appl Polymer Sci 1975;19:2033. [15] Treloar LRG. Trans Faraday Soc 1944;40:59.