J Elasticity (2007) 87:133–146 DOI 10.1007/s10659-007-9100-x
Constitutive Models for Almost Incompressible Isotropic Elastic Rubber-like Materials C. O. Horgan & J. G. Murphy
Received: 30 November 2006 / Accepted: 31 December 2006 / Published online: 26 January 2007 # Springer Science + Business Media B.V. 2007
Abstract A simple constitutive model is proposed for slightly compressible (or almost incompressible) non-linearly elastic materials that are homogeneous and isotropic. Experimental data for simple tension suggest that there is a power-law kinematic relationship between the stretches for large classes of such materials. It is shown that a common constitutive model for these materials does not, in general, capture this effect. The most general constitutive model giving rise to such a power-law relationship is then obtained. A special case yields the well-known Blatz–Ko model for compressible rubber. The behavior in biaxial tension and pure shear is also discussed. Key words constitutive models . non-linearly elastic materials . rubber . almost incompressible . simple tension . power-law kinematic relation Mathematics Subject Classifications (2000) 74B20 . 74G55
1 Introduction Traditional constitutive models for rubber-like elastic materials assume incompressibility as an idealization. In practice, many such materials exhibit elastic behavior that is almost incompressible. Volume changes that reflect compressibility have been observed, especially in hydrostatic compression tests. The classical experiments of Bridgman [8] and Adams and Gibson [1] clearly demonstrate significant compressibility with the initial volume being reduced by 20% at large pressures. See Ogden [20] and Boyce and Arruda [7] for further discussion of the foregoing experiments. However, the behavior of rubber under large C. O. Horgan (*) Structural and Solid Mechanics Program, Department of Civil Engineering, University of Virginia, Charlottesville, VA 22904, USA e-mail:
[email protected]
J. G. Murphy Department of Mechanical Engineering, Dublin City University, Glasnevin, Dublin 9, Ireland e-mail:
[email protected]
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hydrostatic pressures is not our concern here but rather the response to the basic homogeneous deformations of simple tension, biaxial tension and simple shear where volume changes are generally small. The tensile test is probably the most important test used to characterize materials. For a number of rubbers, including those used in the classical experiments of Rivlin and Saunders [28], Beatty and Stalnaker [3] found a power-law kinematic relationship between the stretch in the direction of the applied force, denoted here by 1T, and the two stretches 1S perpendicular to this, namely 12 þ (
lS ¼ lT
;
ð1:1Þ
where ( is a small (relative to one), positive material parameter (for example, for the urethane used by Beatty and Stalnaker ( =0.007). Ideally then, wide classes of almost incompressible (or slightly compressible) materials should behave in this way. At the very least, it seems reasonable to expect that any kinematic relation for almost incompressible behavior should reduce to the first-order equivalent of (1.1), i.e., 1=2
lS ¼ lT
ð1 þ ( ln lT Þ;
ð1:2Þ
in simple tension. Further evidence to support (1.1) can be found in experimental data of Penn [27], as reported by Boyce and Arruda [7]. A digitized version of these data can be found in the Appendix. If (1.1) is assumed, then if li, i=1, 2, 3, denote the principal stretches, one obtains J 1 ¼ l2T( 1; J l1 l2 l3 :
ð1:3Þ
On fitting (1.3) to Penn’s data, we obtain 2( =0.000337 and a comparison of the predictions from (1.3) with the experimental data is given in Fig. 1. 0.00030
Fig. 1 Comparison of (1.3) and the experimental data of Penn [27]
0.00025
J-1
0.00020
0.00015
0.00010
0.00005 experimental data model predictions 0.00000 1.00
1.50
2.00 stretch
2.50
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An excellent fit is obtained and thus the physical relevance of the assumption (1.1) is further strengthened. Many different constitutive models have been proposed to reflect deviations from incompressibility on assuming that the material is homogeneous, isotropic and hyperelastic. Many share the following general form of the strain-energy function W, introduced by Ogden [20]: W ¼ y ðl1 ; l2 ; l3 Þ þ F ði3 Þ; i3 ¼ l1 l2 l3 ¼ J ;
ð1:4Þ
where y is a symmetric function of the principal stretches and here and henceforth we use the notation i3 = l1l2l3. A discussion of appropriate restrictions on y and F will be given in Section 2. If we formally let i3 → 1 in (1.4) we recover a model for incompressible materials. The choice of y usually reflects the experience and preferences of the individual researchers. One important motivation has been the statistical mechanics of long chain molecules (see, for example, Bischoff et al. [4]). See also Horgan and Saccomandi [14, 15] for an alternative approach. Sometimes the motivation for the choice is the success of the corresponding incompressible model in modeling a wide range of data. This is the approach of Ogden [19, 20], who assumed the following form of the strain-energy function: W ¼
3 X
μn φðan Þ þ F ði3 Þ; φðaÞ la1 þ la2 þ la3 3 a ;
ð1:5Þ
n¼1
with appropriate choices for F. Often, the choice of y has involved the use of the invariants of the Cauchy–Green strain tensors Ik given by I1 ¼ l21 þ l22 þ l23 ; I2 ¼ l21 l22 þ l22 l23 þ l23 l21 ; I3 ¼ i23 ¼ l21 l22 l23 ; and so the strain-energy function (1.4) takes the form pffiffiffiffi I3 : W ¼ y ðI1 ; I2 ; I3 Þ þ F
ð1:6Þ
ð1:7Þ
A major objective in the study of slightly compressible materials is to accurately predict the first-order effects of compressibility. Such a viewpoint was emphasized in the early work of Oldroyd [26] and subsequently by Spencer [29]. Here our concern is with a particular aspect of modeling such materials, namely that potential constitutive models should capture, or closely approximate, the experimental first-order kinematic effects in simple tension described above. It will be shown that a strain-energy function of the form (1.4), subject to additional restrictions given in Section 2, is not, in general, compatible with the power-law behavior (1.1) in simple tension or even with the first-order approximation (1.2). This generalizes the experimental observations of Penn [27] who showed that a general strain-energy function of the form (1.7) does not yield satisfactory predictions of the behavior of nearly incompressible materials in simple tension. See also the discussion in Ehlers and Eipper [11]. Penn’s choice for y made use of invariants, denoted by Ik*, that are the invariants (1.6) written in terms of the modified stretches . 1=6 l*i li I3 :
ð1:8Þ
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Therefore, since I3* 1, the strain-energy function (1.7) becomes pffiffiffiffi W ¼ y I1* ; I2* þ F I3 ;
ð1:9Þ
where, in terms of the classical invariants (1.6), I1* ¼ I1 I3
1=3
2=3 ; I2* ¼ I2 I3 :
ð1:10Þ
See also Ogden [21–25]. Observe that only the second term on the right in (1.9) contributes to the volume change while both terms on the right in (1.4) or (1.7) can do so. Despite the mismatch between models of the form (1.9) and Penn’s data, such models are the basis of most commercial finite element codes developed to simulate compressibility effects in elastomers. To overcome the problem of locking if the idealization of incompressibility is adopted, strain-energy functions of the type (1.9) are first adopted and some limiting procedure is then used to model incompressibility. A motivation for assuming that almost incompressible materials can be modeled as in (1.9) is thus presumably mathematical convenience. An alternative approach is therefore desirable and is provided by Murphy [18] who obtained the most general form of the strain-energy function that is consistent with a powerlaw relationship between the axial and lateral stretches in simple tension. In the present paper, we further elaborate on this approach.
2 Restrictions on the Classical Model All strain-energy functions considered here will be assumed to satisfy the following two conditions: Condition 1 The strain energy is zero in the reference configuration; Condition 2 The stress is zero in the reference configuration. In addition, all strain-energy functions must satisfy Condition 3 On restriction to infinitesimal deformations, the shear and bulk moduli are positive, or equivalently, the strain-energy function is positive definite. Thus for (1.4) it is required that y ð1; 1; 1Þ þ F ð1Þ ¼ 0; y ; i ð1; 1; 1Þ þ F 0 ð1Þ ¼ 0; 3κ ¼ 2y ; ij ð1; 1; 1Þ þ y ; ii ð1; 1; 1Þ þ 2F 0 ð1Þ þ 3F 00 ð1Þ; 2μ ¼ y ; ii ð1; 1; 1Þ y ; ij ð1; 1; 1Þ F 0 ð1Þ;
ð2:1Þ where the prime notation denotes differentiation, a comma denotes partial differentiation, a numerical subscript denotes partial differentiation with respect to the appropriate principal stretch and k, μ > 0 denote the bulk and shear moduli respectively for infinitesimal deformations. In (2.1)3,4 and in (2.2)3,4, (2.5)3,4 below, we take i ≠ j and there is no summation over repeated indices. A fundamental assumption now made is that y and F be independent. This is done in order to distinguish contributions to W arising from change in volume, in an analogous
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fashion to the decomposition (1.9). Thus the function y will be assumed to depend on μ while F is assumed to depend on κ. Equations (2.1) then imply that y ð1; 1; 1Þ ¼ y ;i ð1; 1; 1Þ ¼ 0; 2y ;ij ð1; 1; 1Þ þ y ;ii ð1; 1; 1Þ ¼ 0; y ;ii ð1; 1; 1Þ y ;ij ð1; 1; 1Þ ¼ 2m;
ð2:2Þ
F ð1Þ ¼ F 0 ð1Þ ¼ 0; F 00 ð1Þ ¼ k:
ð2:3Þ
and
Motivated by these restrictions, it will now be assumed that y and all its partial derivatives are of order μ and that F and all its derivatives are of order k. To emphasize this, henceforth the strain-energy function (1.4) will be written as W ¼ m y ðl1 ; l2 ; l3 Þ þ k F ði3 Þ;
ð2:4Þ
where y ð1; 1; 1Þ ¼ y ;i ð1; 1; 1Þ ¼ 0; 2y ;ij ð1; 1; 1Þ þ y ;ii ð1; 1; 1Þ ¼ 0; y ;ii ð1; 1; 1Þ y ;ij ð1; 1; 1Þ ¼ 2;
ð2:5Þ
F ð1Þ ¼ F 0 ð1Þ ¼ 0; F 00 ð1Þ ¼ 1:
ð2:6Þ
and
Thus, the y term in (2.4) is explicitly assumed to make no contribution to the infinitesimal volume change and such volume change is to be reflected by F only. In this sense, the decomposition (2.4) is analogous to that in (1.9). Some specific examples of strain-energy densities of the form (2.4) used to model almost incompressible behavior are given below: (1)
Christensen [10] W¼
(2)
3 2=3 3 i ln i3 2 3 2
m ð3I1 4i3 2 ln i3 5Þ þ k ði3 ln i3 1Þ 6
ð2:7Þ
ð2:8Þ
Levinson and Burgess [17], Burgess and Levinson [9] W¼
(4)
Levinson [16], following Blatz [5] W ¼
(3)
3k m 2=3 þ I1 3i3 2 2
k m 3I1 þ i23 8i3 2 þ ði3 1Þ2 6 2
ð2:9Þ
Ehlers and Eipper [11], following Flory [12] m * 2 * 2 * 2 k l1 þ l2 þ l3 3 þ ðln i3 Þ2 W¼ 2 2 ! m I1 k ¼ 3 þ ðln i3 Þ2 : 2=3 2 i 2 3
ð2:10Þ
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It may be readily verified that the conditions (2.5), (2.6) are satisfied for these four examples. Because we wish to model almost incompressible materials, it is further assumed that h m=k
