1 Introduction

The highly elastic behavior of elastomers is characterized by elastic potentials (also known as elastic strain-energy functions and elastic stored-energy functions). Since the 1940s,many studies have been devoted to obtaining multi-axial elastic potentials for elastomeric materials. However,since there are strong nonlinearities and,in particular,since it may not be realistic to treat all physically possible deformation modes,usually it is required that a reasonable elastic potential sought should match the data from certain benchmark tests,including uniaxial extension,equi-biaxial extension (i.e.,uniaxial compression),and plane-strain extension (i.e., simple shear). Classical test data for these deformation modes were recorded in Treloar’s monogragh^[1] and other relevant references. These data serve as the experimental foundation for determination of elastic potentials.

Two approaches have been applied in obtaining the elastic potentials for elastomers,includ- ing the statistical approach and the phenomenological approach. The former derives specific forms of elastic potentials in terms of certain micro-structural parameters from the statistical mechanics of the network structures of long chainlike macromolecules,while from the viewpoint of continuum mechanics,the latter assumes direct expressions of elastic potentials with a set of adjustable parameters. Reviews of the statistical mechanics treatment were presented by Treloar^[1] and Gent^[2] for earlier results and by Boyce and Arruda^[3],Fried^[4],Ogden et al.^[5], Drozdov^[6],Beatty^[7],and Miehe et al.^[8] for recent results. Results from the phenomenological approach may be found in Ref. ^[9] for earlier results and in Refs. ^[10]-^[16] and references therein for recent results. For both approaches,a number of unknown parameters would be estimated in fitting test data,and usually good agreement with the test data may be achieved only for part of the stretch range.

Most recently,a direct approach^{[17, 18]} has been proposed to derive elastic potentials based on Hencky’s logarithmic strain and interpolation procedures. This new approach presents ex- plicit results that exactly match the finite strain data for the foregoing four benchmark tests, thus bypassing usual cumbersome numerical procedures in estimating a number of unknown parameters.

The biaxial stretch test in a general sense has not been treated in the aforementioned newest study. That is also the case for the existing results in literatures. It is the main objective of this study to construct new elastic potentials in a further sense and compare the predictions of the further result with the extensive data given by Rivlin and Saunders^[19].

The arrangement of the main context is as follows: the main method and the main result are given in Section 2. In Section 3,the theoretical predictions are derived for the biaxial stretch be- havior and then compared with Rivlin and Saunders’ data. The discussion is made in Section 4.

2 Main method and main result

This section presents a short account of the main method and the main procedures as well as the main result. Details may be found in the previous work^{[17, 18]}.

2.1 Hencky strain and its invariants

The elastic behavior of elastomeric materials is characterized by the elastic potential W(F) with the deformation gradient F. From the objectivity principle and the isotropy condition,it may be inferred that W may be reduced to a function of the Cauchy-Green tensor. Generally, an elastic potential may be expressed as a function of any finite strain measure. A stress-strain relation in this broad sense may be derived (refer to Ogden^[10]). Of many finite strain measures, Hencky’s logarithmic strain is of much interest due to its unique,attractive properties^{[20, 21, 22, 23, 24, 25, 26]}.

The starting point of this study is to express the elastic potential W in terms of the Hencky strain,namely,

Here,the Hencky strain is of the form A further expression of h will be given slightly later.

With the elastic potentialW based on the Hencky strain h,as shown in Eq. (1),the following stress-strain relation is derived^{[25, 26]}:

In the above equation,p is a hydrostatic pressure. The above equation indicates that the Cauchy stress (the true stress) may be derived directly from the elastic potential in terms of the Hencky strain.

The potential given by Eq. (1) is an isotropic scalar function of h. Generally,it is expressible as a function for three invariants of h. There are infinite Hencky invariants,but three of them are basic. For the direct approach proposed here,certain Hencky invariants will play essential roles. For the purpose of introducing these invariants,we begin with the basic invariants of h.

Let λ₁,λ₂,and λ₃ be the square roots of the three eigenvalues of the left Cauchy-Green tensor B = F · FT,namely,the three principal stretches,and let n₁,n₂,and n₃ be the corresponding orthonormal eigenvectors. The Hencky strain may also be given in the following spectral form:

The three basic invariants of the Hencky strain are as follows: The deviatoric Hencky strain is given by Here,J is the volumetric ratio, For incompressible deformations,we have

The two basic invariants of the deviatoric Hencky strain are given by

For incompressible deformations,we have (see Eq. (8))

2.2 Bridging invariants and matching invariants

The subsequent development will be based on two groups of new invariants. The first group is composed of the following two invariants:

while the second one is formed by the following equations:

To explain the properties of the above two groups of invariants,two deformation modes, namely,the uniaxial strain and the simple shear,are taken into account. Let λ be the axial stretch and ω be the shear amount (i.e.,the tangent of the shear angle). Then,the following properties hold:

and

The above properties will play substantial roles in the subsequent development. In fact,the two invariants Ψ and β,referred to as the bridging invariants^{[17, 18]},will be used to establish direct relationships between the general multi-axial deformations and the uniaxial strain as well as the shear strain,while the two invariants γ and Χ,referred to as the matching invariants^{[17, 18]}, will be used to match the results derived from the uniaxial case and the shear case.

For the future use,the gradients of the bridging and matching invariants are given as follows^{[17, 18]}:

Here,˘h is used to designate the following deviatoric tensor:

2.3 Multiaxial elastic potentials via rational interpolation

In terms of the bridging invariants and the matching invariants introduced,a direct approach was suggested^{[17, 18]} to obtain multi-axial elastic potentials by means of multi-axial expansion and matching. According to this approach,two one-dimensional potentials are obtained by interpolation procedures based on uniaxial data and shear data,and then a unified form of the multi-axial potential is obtained by expanding the two one-dimensional functions to the multi-axial functions via the two bridging invariants and then matching the latter two via the two matching invariants. Details can be found in the most recent references^{[17, 18]}. Here,only the main procedures and the main result are given below.

First,via rational interpolation,the following two one-dimensional functions are introduced to match the uniaxial data and shear data:

for the uniaxial case and for the plane-strain case. In the above equations,h and σ represent the logarithmic strain and the normal stress in the stretching direction for the two cases,respectively. It would be pointed out that usually the plane-strain test is conducted in replacing the simple shear in a direct sense. For incompressible deformations,the two may be converted to each other using the following relationship:

The stress-strain relationships given by the above two functions are shown in Fig. 1. It may be clear that the two parameters he and hc represent the extension limit and the compression limit for the uniaxial behavior of rubber-like materials,respectively. The plane-strain case (i.e., the shear case) may be given by setting he = hc = h_s in Fig. 1. Accordingly,the curve for the plane-strain case is symmetric with respect to the origin. The parameter h_s represents the limit for the plane-strain case,and the shear limit is given by (see Eq. (27))

Moreover,E is the slope at the origin and supplies Young’s modulus at the infinitesimal strain.

The following two integrations then provide two one-dimensional potentials:

Second,via the two bridging invariants Ψ and β as well as the matching invariant ,the above two potentials are expanded to two multi-axial potentials below:

Third,via the matching invariant Χ,a unified form of the multiaxial elastic potential is even- tually obtained as By means of the properties indicated by Eqs. (16)-(19),it may readily be proven that the last potential given exactly matches the data from the uniaxial extension test,the equi-biaxial extension test (i.e.,the uniaxial compression test),and the plane-strain extension test (i.e.,the shear test). In fact,the multi-axial potential given here reduces to the two one-dimensional potentials. Since the two one-dimensional stress-strain relationships given by Eqs. (25)-(26) may accurately match the uniaxial data and the shear data,respectively,the given multi-axial potential matches the data automatically.

The potential given by Eq. (33) with a nonnegative index m > 0 represents a further devel- opment of the most recent result in Refs. ^[17] and ^[18]. Indeed,the latter may be regarded as the particular case when m = 1.

3 Matching biaxial data

With the new potential given in the last section,we further compare the theoretical pre- dictions with the classical data given by Rivlin and Saunders^[19]. This will be done in this section.

There are 5 parameters E,h_e,h_c,h_s,and m. The four parameters E,h_e,h_c,and h_s are determined directly by matching the uniaxial data and shear data via rational interpolation, separately. The index m will be given slightly later.

3.1 Predictions for biaxial stretch test

The general biaxial extension test is described below. A rectangular sample of the thin block is deformed subjected to two normal stresses σ₁ and σ₂ in two perpendicular directions with a pair of planes free of loading. Let λ₁ and λ₂ be the stretches along the two loading direc- tions. The stretch in the other direction is given by (λ₁λ₂)−1 considering the incompressibility condition. The Cauchy stress and the deformation gradient are of the forms

The Hencky strain and the deviatoric Hencky strain are given by and the invariants j₂ and j₃ are given by

From Eq. (3) and the gradients given by Eqs. (20)-(23),the multi-axial stress-strain rela- tionship is derived as follows^{[17, 18]}:

with the deviatoric tensor ˘h given by Eq. (24) and the two invariant coefficients ζ and ˘ζ given by where Note that the two one-dimensional functions f(h) and z(h) are given by Eqs. (25)-(26).

The two normal stresses σ₁ and σ₂ for the biaxial case at issue are derived as follows:

where

3.2 Determination of parameters

The values of E,h_e,h_c,and h_s may be determined by directly matching the uniaxial data and shear data from the uniaxial extension test and the uniaxial compression test as well as the plain-strain extension test. Two kinds of rubber samples,namely,Rubbers A and B,were used by Rivlin and Saunders^[19]. However,complete data for either of these two rubbers are not available in Ref. ^[19]. In what follows,the data for Rubber A will be considered,but the data for the uniaxial compression are lacking. A remedy for this is to derive the latter from the following relationship:

The data for the right-hand side,namely,for uniaxial extension and plan-strain extension,were given by Rivlin and Saunders^[19]. It has been shown^[1] that the above relationship holds true with high accuracy.

With Rivlin and Saunders’ data for the uniaxial extension test and the plane-strain test for Rubber A as well as the relationship given by Eq. (45),the values of Young’s modulus E,the extension limit h_e,the compression limit h_c,and the shear limit h_s may be found and given below:

The results are shown in Figs. 2 and 3.

3.3 Comparison with test data

The biaxial data of Rivlin and Saunders^[19] were given in terms of the nominal stresses. Here,the data in terms of the true stresses are needed and obtained by converting the former to the latter. The results are listed in the last two columns of Table 1and 2 .

With the predictions given by Eqs. (42)-(43),numerical results may be obtained by using the parameter values given by Eq. (46) and taking m = 0.1. The predicted values for both σ₁ and σ₂ are listed in the 5th and 6th columns of Table 1and 2. Here,the unit for both σ₁ and σ₂ is MPa. Table 1 provides the results for certain constant I₁,while Table 2 supplies the results for certain constant I₂.

In the first two columns of Table 1and 2,the values for two invariants I₁ and I₂ are irrelevant with the numerical treatment here,but they are recorded for the sake of grouping the test data.

The two invariants I₁ and I₂ were used by Rivlin and Saunders in deriving polynomial expressions of elastic potentials. For certain purposes in the theoretical study,Rivlin and Saunders^[19] carried out the biaxial tests with constant values of these two invariants. In terms of the principal stretches,I₁ and I₂ are of the forms

Comparison between the test data and the prediction is shown in Figs. 4 and 5. Note that the latter two correspond with Table 1and 2,respectively. It may be seen from Figs. 4 and 5 that the agreement between theoretical predictions and test data is satisfactory,considering the fact that extensive data are fitted by adjusting only one parameter m,namely,the index m. Note here that the parameters E,h_e,h_c,and h_s are already specified by the uniaxial data and the shear data,and therefore cannot be adjusted for the purpose of fitting the extensive biaxial data.

4 Conclusions

In the previous sections,a further study of rubber-like elasticity is made by developing the approach suggested in a newest study^{[17, 18]}. A new elastic potential in an explicit form is presented. Results show good agreement between the theoretical predictions and the test data for the biaxial stretch behavior. It should be pointed out that usually the test data only for the uniaxial case and the shear case are treated in the existing results in literatures with the biaxial stretch behavior to be considered. It appears that for the first time,results presented here simultaneously match the test data from the uniaxial case and the shear case and,in particular,the biaxial case.

The incompressibility condition is an idealized treatment for the realistic material behavior. In reality,rubber-like materials undergo small volumetric deformation. A direct approach treating compressibility was proposed in a latest contribution^[18]. A further study in a general case of compressible deformation will be made in the succeeding work.