Mathematical models of natural laws are frequently formulated through the concept of symmetry into nonlinear partial differential equations.

For constitutive modeling of isotropic rubberlike materials with nonlinear finite elastic deformations, the second Piola-Kirchhoff stress tensor (8), $S$, has been derived. For work conjugate pairs, a finite deformation measure is needed. The commonly used finite deformation measures are the right Cauchy-Green tensor $C$, the Green-Lagrange strain tensor $E$, and the deformation gradient tensor $F$. The right Cauchy-Green tensor is a deformation tensor that excludes rigid body rotation in $F$. Furthermore, experimental stress-stretch curves of isotropic rubberlike materials are strictly increasing functions, which are injective. Additionally, every stress on the curves must correspond to at least one stretch, which requires a surjective function. A function that is both injective and surjective is also called a bejective function in terms of abstract algebra. For constructing a bejective function, the stretch-based tensor $C$ rather than the strain tensor $E$ is more appropriately selected since stretches are always represented by positive real numbers. More specifically, the strain tensor $E$ could create singularities and imaginary numbers in constitutive modelings. Thus, the second Piola-Kirchhoff stress tensor $S$ and the right Cauchy-Green tensor $C$ as a work conjugate pair have been selected to construct the isotropic CSE functional.

For a physically consistent, mathematically covariant, and geometrically meaningful formulation, the isotropic CSE functional was postulated and balanced with its stress work done based on the conservation of energy by Zhao (2016) [5]

$\Psi =S:\frac{C}{2}.$ (9)

The isotropic CSE functional (9) is covariant to ${\Psi }_E=S:E$ under the transformation of $E=0.5\left(C-I\right)$. Substituting (8) into (9), simplifying, and rearranging yields the partial differential equation of the isotropic CSE functional in terms of three invariants $I_1$, $I_2$, and $I_3$

$\Psi =I_1\frac{\partial \Psi }{\partial I_1}+2I_2\frac{\partial \Psi }{\partial I_2}+3I_3\frac{\partial \Psi }{\partial I_3}.$ (10)

With Lie group methods, the characteristic system of the partial differential Equation (10) takes the form of

$\frac{\text{d}{I}_1}{I_1}=\frac{\text{d}{I}_2}{2I_2}=\frac{\text{d}{I}_3}{3I_3}=\frac{\text{d}\Psi }{\Psi }.$ (11)

Taking its independent first-integrals, ${\psi }_1=I_2/I_1^2$, ${\psi }_2=I_3/I_1^3$, and ${\psi }_3=\Psi /I_1$, the general solution to the isotropic CSE partial differential Equation (10) has been obtained as

$\Psi =I_{1}h\left(I_2/I_1^2,I_3/I_1^3\right),$ (12)

where h as an arbitrary function has been selected as the summation of two arbitrary functions of invariant groups albeit other available combinations

$\Psi =I_1\left[f\left(I_2/I_1^2\right)+g\left(I_3/I_1^3\right)\right],$ (13)

where f and g as two arbitrary functions that can be determined by curvatures of deformations. For normal and shear deformations, the first arbitrary function, f, has been defined as

$f\left(I_2/I_1^2\right)=c_1+c_2\sqrt{I_2/I_1^2}=c_1+c_2\frac{\sqrt{I_2}}{I_1},$ (14)

and for different degrees of ellipsoidal deformations, the second arbitrary function, g, has been generalized as

$g\left(I_3/I_1^3\right)=c_3{\left(I_3/I_1^3\right)}^{-c_4}=c_3\frac{I_1^{3c_4}}{I_3^{c_4}}.$ (15)

Substituting (14) and (15) into (13) augments the isotropic CSE functional by Zhao (2020) [6] . Applying the normalization condition, $\Psi \left(I\right)=0$, gives

$\Psi =c_1\left(I_1-3\right)+c_2\left(\sqrt{I_2}-\sqrt{3}\right)+c_3\left(\frac{I_1^{3c_4+1}}{I_3^{c_4}}-3^{3c_4+1}\right),$ (16)

where the four constitutive parameters, c₁, c₂, c₃, and c₄, will be identified by experimental tests. Thus, the isotropic CSE functional will be used to establish constitutive equations of commonly used deformation modes in experimental tests.

Nominal stress and stretch results are preferably calculated from force and extension measurements with original sample dimensions recorded in experimental tests. The first Piola-Kirchhoff stress tensor or the nominal stress tensor $P$ is related to the second Piola-Kirchhoff stress tensor $S$ through the deformation gradient tensor $F$ by $P=FS$ or more conveniently by

$P^{\text{T}}=\frac{\partial \Psi }{\partial F}=2\left(\frac{\partial \Psi }{\partial I_1}\frac{\partial I_1}{\partial C}+\frac{\partial \Psi }{\partial I_2}\frac{\partial I_2}{\partial C}+\frac{\partial \Psi }{\partial I_3}\frac{\partial I_3}{\partial C}\right)\left(\frac{1}{2}\frac{\partial C}{\partial F}\right)=S{F}^{\text{T}},$ (17)

where the nominal stresses as a function of stretches in indicial notation are generally expressed as

$P_{ji}=\frac{\partial \Psi }{\partial I_1}\frac{\partial I_1}{\partial {\lambda }_{ij}}+\frac{\partial \Psi }{\partial I_2}\frac{\partial I_2}{\partial {\lambda }_{ij}}+\frac{\partial \Psi }{\partial I_3}\frac{\partial I_3}{\partial {\lambda }_{ij}},\left(i,j=1,2,3\right).$ (18)

The three derivatives of the isotropic CSE functional (16) with respect to invariants I₁, I₂, and I₃ are worked out as

$\frac{\partial \Psi }{\partial I_1}=c_1+c_3\left(3c_4+1\right)\frac{I_1^{3c_4}}{I_3^{c_4}},\frac{\partial \Psi }{\partial I_2}=\frac{c_2}{2\sqrt{I_2}},\frac{\partial \Psi }{\partial I_3}=-c_3{c}_4\frac{I_1^{3c_4+1}}{I_3^{c_4+1}}.$ (19)

Thus, the isotropic CSE constitutive equations will be specifically derived from Equations (16) through (19), combining with the Poisson functions based on stretch and stress states at different deformation modes.

Experimental characterizations of rubberlike materials are often conducted by uniaxial tension, uniaxial compression, equibiaxial tension, biaxial tension, pure shear, and simple shear tests. Among them, uniaxial tension, pure shear, and equibiaxial tension tests are the most commonly used testing methods for finite element analyses as reviewed by Charlton et al. (1994) [7] . The experimental characterization requirements along with the constitutive equations are described in the uniaxial tension, pure shear, and equibiaxial tension modes.

In a uniaxial tension (UT) test, the stretch state is ${\lambda }_1=\lambda =L/L_0$ and ${\lambda }_2={\lambda }_3={\lambda }^{-\nu }$ based on the UT Poisson function where the current effective length L and original effective length L₀ must be measured on the specimen away from the clamps by a video extensometer or laser extensometer, respectively. Lateral contractions ${\lambda }_2$ and ${\lambda }_3$ should also be measured as a function of uniaxial stretch ${\lambda }_1$ for compressible materials. The stress state is preferably calculated by nominal stress as $P_1=P/A_0$ and $P_2=P_3=0$ where the applied load P is measured by a load cell and the original cross-section area A₀ can be measured by a caliper. The specimen with the aspect ratio of length to width or thickness should be greater than 10 in order to reduce lateral constraint to thinning. For uniform clamping, a dumbbell-shaped specimen is often used. The three invariants in the uniaxial tension mode are obtained from the stretch state as

$I_1^{ut}={\lambda }^2+2{\lambda }^{-2\nu },I_2^{ut}=2{\lambda }^{2-2\nu }+{\lambda }^{-4\nu },I_3^{ut}={\lambda }^{2-4\nu }.$ (20)

The nominal stress as a function of principal stretch in uniaxial tension mode for rubberlike materials, along with the stress-free condition, can be generally derived as

$\begin{array}{c}{P}_{ut}=2\left[\lambda -2\nu {\lambda }^{-2\nu -1}-\left(1-2\nu \right)\right]c_1+2\left[\frac{\left(1-\nu \right){\lambda }^{1-2\nu }-\nu {\lambda }^{-4\nu -1}}{\sqrt{2{\lambda }^{2-2\nu }+{\lambda }^{-4\nu }}}-\frac{1-2\nu }{\sqrt{3}}\right]c_2\\ +2\{\frac{{\left({\lambda }^2+2{\lambda }^{-2\nu }\right)}^{3c_4}}{{\lambda }^{\left(2-4\nu \right)c_4}}[\left(3c_4+1\right)\left(\lambda -2\nu {\lambda }^{-2\nu -1}\right)\\ \begin{array}{c}\\ {{\displaystyle }}_{}^{}\end{array}-c_4\left(1-2\nu \right)\left(\lambda +2{\lambda }^{-2\nu -1}\right)]-\left(1-2\nu \right){27}^{c_4}\}{c}_3.\end{array}$ (21)

For the assumed incompressible condition, the equation (21) with the Poisson’s ratio $\nu =0.5$ is simplified as

${\tilde{P}}_{ut}=2\left(\lambda -{\lambda }^{-2}\right)c_1+\frac{1-{\lambda }^{-3}}{\sqrt{2\lambda +{\lambda }^{-2}}}{c}_2+2\left(3c_4+1\right){\left({\lambda }^2+2{\lambda }^{-1}\right)}^{3c_4}\left(\lambda -{\lambda }^{-2}\right)c_3.$ (22)

In a pure shear (PS) or planar tension test, the stretch state is ${\lambda }_1=\lambda =L/L_0$, ${\lambda }_2=1$, and ${\lambda }_3={\lambda }^{-\nu /\left(1-\nu \right)}$ with the PS Poisson function where the current and original effective lengths must be measured on the specimen away from the clamps by a video extensometer or laser extensometer, respectively. Thickness reduction ${\lambda }_3$ should also be measured as a function of axial stretch ${\lambda }_1$ for compressible materials. The stress state is preferably calculated by nominal stress as $P_1=P/A_0$, $P_2\ne 0$, and $P_3=0$ where the applied load P is measured by a load cell and the original cross-section area can be measured by a caliper. The specimen with the aspect ratio of width to length should be at least 10 in order to constrain the width and let all specimen thinning occur in the thickness direction. Because the material is nearly incompressible, a state of pure shear exists in the specimen at a 45-degree angle to the stretching direction. The three invariants in the pure shear mode are obtained from the stretch state as

$I_1^{ps}={\lambda }^2+1+{\lambda }^{-\frac{2\nu }{1-\nu }},I_2^{ps}={\lambda }^2+{\lambda }^{\frac{2-4\nu }{1-\nu }}+{\lambda }^{-\frac{2\nu }{1-\nu }},I_3^{ps}={\lambda }^{\frac{2-4\nu }{1-\nu }}.$ (23)

The nominal stress as a function of principal stretch in pure shear mode for rubberlike materials, along with the stress-free condition, can be generally worked out as

$\begin{array}{c}{P}_{ps}=2\left(\lambda -\frac{\nu }{1-\nu }{\lambda }^{-\frac{1+\nu }{1-\nu }}-\frac{1-2\nu }{1-\nu }\right)c_1\\ +\left[\frac{\lambda -\frac{\nu }{1-\nu }{\lambda }^{-\frac{1+\nu }{1-\nu }}+\frac{1-2\nu }{1-\nu }{\lambda }^{\frac{1-3\nu }{1-\nu }}}{\sqrt{{\lambda }^2+{\lambda }^{\frac{2-4\nu }{1-\nu }}+{\lambda }^{-\frac{2\nu }{1-\nu }}}}-\left(\frac{1-2\nu }{1-\nu }\right)\frac{2}{\sqrt{3}}\right]c_2\\ +2\{\frac{{\left({\lambda }^2+1+{\lambda }^{-\frac{2\nu }{1-\nu }}\right)}^{3c_4}}{{\lambda }^{\frac{2-4\nu }{1-\nu }{c}_4}}[\left(3c_4+1\right)\left(\lambda -\frac{\nu }{1-\nu }{\lambda }^{-\frac{1+\nu }{1-\nu }}\right)\\ \begin{array}{c}\\ \\ \\ \end{array}-c_4\frac{1-2\nu }{1-\nu }\left(\lambda +{\lambda }^{-1}+{\lambda }^{-\frac{1+\nu }{1-\nu }}\right)]-\frac{1-2\nu }{1-\nu }{27}^{c_4}\}{c}_3.\end{array}$ (24)

For the assumed incompressible condition, $\nu =0.5$, Equation (24) is simplified as

${\tilde{P}}_{ps}=2\left(\lambda -{\lambda }^{-3}\right)c_1+\frac{\lambda -{\lambda }^{-3}}{\sqrt{{\lambda }^2+{\lambda }^{-2}+1}}{c}_2+2\left(3c_4+1\right){\left({\lambda }^2+{\lambda }^{-2}+1\right)}^{3c_4}\left(\lambda -{\lambda }^{-3}\right)c_3.$ (25)

The biaxial tension (BT) test involves one of the most complicated deformations among the available tests because of the number of independent variable stretches. In general, five variables, ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, ${\kappa }_1$, and ${\kappa }_2$, have to be measured in the deformation gradient tensor

$F_{bt}=\left(\begin{array}{lll}{\lambda }_1\hfill & {\kappa }_1\hfill & 0\hfill \\ {\kappa }_2\hfill & {\lambda }_2\hfill & 0\hfill \\ 0\hfill & 0\hfill & {\lambda }_3\hfill \end{array}\right),$ (26)

When constant ratios of ${\lambda }_1$ to ${\lambda }_2$ are used, the number of independent variables can be reduced, especially for equibiaxial tension tests ${\lambda }_1={\lambda }_2$. In practical applications, however, shear stretches are usually ignored albeit shear stretches are generally two orders of magnitude smaller than normal stretches, i.e. $\kappa \approx {10}^{-2}\lambda$ as studied by Aydin et al. (2017) [8] .

In an equibiaxial tension (ET) test, the simplified stretch state is ${\lambda }_1={\lambda }_2=\lambda =L/L_0$ and ${\lambda }_3={\lambda }^{-2\nu /\left(1-\nu \right)}$ based on the ET Poisson function. The current effective length L and original effective length L₀ must be measured on the specimen away from the clamps by a video extensometer or laser extensometer, similarly. Thickness reduction ${\lambda }_3$ should also be measured as a function of equibiaxial stretches ${\lambda }_1$ and ${\lambda }_2$ for compressible materials. The stress state is $P_1=P_2=P/A_0$ and $P_3=0$ where the applied load P is measured by a load cell and the original cross-section area A₀ can be measured by a caliper. Stress- stretch curves in equibiaxial tension tests of square specimens could be overestimated since parts of applied loads are also used to generate shear stretches rather than just normal stretches. The stress-stretch curves could be underestimated due to branching stability issues from a bifurcation point of view.

The three invariants in equibiaxial tension mode are obtained from the simplified stretch state as

$I_1^{et}=2{\lambda }^2+{\lambda }^{-\frac{4\nu }{1-\nu }},I_2^{et}={\lambda }^4+2{\lambda }^{\frac{2-6\nu }{1-\nu }},I_3^{et}={\lambda }^{\frac{4-8\nu }{1-\nu }}.$ (27)

The nominal stress as a function of principal stretch in equibiaxial tension mode for rubberlike materials, along with the stress-free condition, turns out to be

$\begin{array}{c}{P}_{et}=2\left(\lambda -\frac{\nu }{1-\nu }{\lambda }^{-\frac{1+3\nu }{1-\nu }}-\frac{1-2\nu }{1-\nu }\right)c_1+\left[\frac{{\lambda }^3+\frac{1-3\nu }{1-\nu }{\lambda }^{\frac{1-5\nu }{1-\nu }}}{\sqrt{{\lambda }^4+2{\lambda }^{\frac{2-6\nu }{1-\nu }}}}-\left(\frac{1-2\nu }{1-\nu }\right)\frac{2}{\sqrt{3}}\right]c_2\\ +2\{\frac{{\left(2{\lambda }^2+{\lambda }^{-\frac{4\nu }{1-\nu }}\right)}^{3c_4}}{{\lambda }^{\frac{4-8\nu }{1-\nu }{c}_4}}[\left(3c_4+1\right)\left(\lambda -\frac{\nu }{1-\nu }{\lambda }^{-\frac{1+3\nu }{1-\nu }}\right)\\ \begin{array}{c}\\ \\ \\ \end{array}-c_4\frac{1-2\nu }{1-\nu }\left(2\lambda +{\lambda }^{-\frac{1+3\nu }{1-\nu }}\right)]-\frac{1-2\nu }{1-\nu }{27}^{c_4}\}{c}_3.\end{array}$ (28)

For the incompressible condition, the Equation (28) with the Poisson’s ratio $\nu =0.5$ is simplified as

${\tilde{P}}_{et}=2\left(\lambda -{\lambda }^{-5}\right)c_1+\frac{{\lambda }^3-{\lambda }^{-3}}{\sqrt{{\lambda }^4+2{\lambda }^{-2}}}{c}_2+2\left(3c_4+1\right){\left(2{\lambda }^2+{\lambda }^{-4}\right)}^{3c_4}\left(\lambda -{\lambda }^{-5}\right)c_3.$ (29)

Efforts have been made to reduce shear stretches, increase uniform normal stretches, and minimize branching stability issues by using the bubble inflation technique, square, circular, and cruciform specimens.

For $c_4=\text{A}.\text{BCDEFGH}\in \left(0,10\right)$ example, the TED method starts with ones digit, i.e., $\text{A}=\left\{1,2,3,4,5,6,7,8,9\right\}$. In the nine trials, the number with the smallest error between model and experiment is obtained as A. The number of trials on each of the following digits after the decimal point is 18 instead of 9 since the current number, A.B, could be either larger or smaller than the previous number, A, and a zero digit has been previously tested. Then, there are eighteen trials for $\text{B}=\left\{\pm 0.1,\pm 0.2,\pm 0.3,\pm 0.4,\pm 0.5,\pm 0.6,\pm 0.7,\pm 0.8,\pm 0.9\right\}$ while adding A.B. together. Then, for $\text{C}={10}^{-1}\text{B}$ eighteen trials is conducted while adding up to A.BC. A similar process will continue to be executed for sequentially smaller digits up to the desired digit. In this example, the last desired digit is the ten-millionths $\text{H}={10}^{-6}\text{B}$ while the final decimal number A.BCDEFGH is constructed and solved with only 153 trials.

After the c₄ is obtained at a certain iteration by the TED method, the LLSQ method will be used to resolve the remaining three constitutive parameters c₁, c₂, and c₃ simultaneously. The total error between the model and experiment is given by

$ϵ=\underset{j=1}{\overset{n}{{\displaystyle \sum }}}{\left[P_{ut}\left({\lambda }_j\right)-P_{ut}^{\exp }\left({\lambda }_j\right)\right]}^2,$ (30)

where $P_{ut}^{\exp }\left({\lambda }_j\right)$ represents experimental nominal stress data as a function of stretch ${\lambda }_j$ in uniaxial tension tests and n is the total number of pairs of experimental data. The error, $ϵ$, can be minimized with the following conditions

$\frac{\partial ϵ}{\partial c_k}=0,k=1,2,3.$ (31)

Substituting (21) into (30), and Applying (31) yields the following quasi-linear system

$\underset{j=1}{\overset{n}{{\displaystyle \sum }}}\left[\begin{array}{lll}{A}_j{A}_j\hfill & B_j{A}_j\hfill & C_j{A}_j\hfill \\ A_j{B}_j\hfill & B_j{B}_j\hfill & C_j{B}_j\hfill \\ A_j{C}_j\hfill & B_j{C}_j\hfill & C_j{C}_j\hfill \end{array}\right]\left[\begin{array}{l}{c}_1\hfill \\ c_2\hfill \\ c_3\hfill \end{array}\right]=\underset{j=1}{\overset{n}{{\displaystyle \sum }}}\left[\begin{array}{l}{P}_{ut}^{\exp }\left({\lambda }_j\right)A_j\hfill \\ P_{ut}^{\exp }\left({\lambda }_j\right)B_j\hfill \\ P_{ut}^{\exp }\left({\lambda }_j\right)C_j\hfill \end{array}\right]$ (32)

where the vectors $A_j$, $B_j$, and $C_j$ with the known parameters c₄ and $\nu$ for the isotropic CSE model in uniaxial tension mode are defined as

$A_j=2\left[{\lambda }_j-2\nu {\lambda }_j^{-2\nu -1}-\left(1-2\nu \right)\right],$ (33)

$B_j=2\left[\frac{\left(1-\nu \right){\lambda }_j^{1-2\nu }-\nu {\lambda }_j^{-4\nu -1}}{\sqrt{2{\lambda }_j^{2-2\nu }+{\lambda }_j^{-4\nu }}}-\frac{1-2\nu }{\sqrt{3}}\right],$ (34)

and

$\begin{array}{l}{C}_j=2\{{\left({\lambda }_j^2+2{\lambda }_j^{-2\nu }\right)}^{3c_4}/{\lambda }_j^{\left(2-4\nu \right)c_4}[\left(3c_4+1\right)\left({\lambda }_j-2\nu {\lambda }_j^{-2\nu -1}\right)\\ -c_4\left(1-2\nu \right)\left({\lambda }_j+2{\lambda }_j^{-2\nu -1}\right)]-\left(1-2\nu \right){27}^{c_4}\}.\end{array}$ (35)

The uniaxial tension, pure shear, and equibiaxial tension tests of incompressible S4035A TPE have been conducted. The uniaxial tension test data of S4035A TPE is used to fit the isotropic CSE Model (22). Numerically Solving (32), along with the Vectors (33), (34), (35), and $\nu =0.5$, simultaneously obtains the four constitutive parameters as $c_1=0.0970449\text{MPa}$, $c_2=0.0848708\text{MPa}$, $c_3=5.4486398\times {10}^{-7}\text{MPa}$ and $c_4=0.9251924$. The comparison between the isotropic CSE model and uniaxial tension test of S4035A TPE is shown in Figure 1(a) . To further examine the predictability of the isotropic CSE model, the four constitutive parameters identified from fitting the uniaxial tension test data have been submitted into both the pure shear Model (25) and the equibiaxial tension Model (29). The predictions of pure shear and equibiaxial tension tests are shown in Figure 1(b) and Figure 1(c) , respectively.

The uniaxial tension, pure shear, and equibiaxial tension tests of compressible synthetic rubber, along with the detailed kinematic characterizations, have been conducted by Storåkers (1986) [10] . With the kinematic characterizations: lateral contractions as a function of uniaxial stretch, ${\lambda }_2\left({\lambda }_1\right)$ and ${\lambda }_3\left({\lambda }_1\right)$ data, are used to fit the UT Poisson function. As a result, the Poisson’s ratio, $\nu =0.49122$, is obtained and the fittings of kinematic relationships are shown in Figure 2(b) . The uniaxial tension test data of the synthetic rubber is then used to fit the isotropic CSE Model (21). Numerically Solving (32), along with the Vectors (33), (34), (35), and $\nu =0.49122$, simultaneously obtains the four constitutive parameters as $c_1=0.0066309\text{MPa}$, $c_2=0.0687864\text{MPa}$, $c_3=5.2466927\times {10}^{-5}\text{MPa}$ and $c_4=0.9733049$. The comparison between the isotropic CSE model and the uniaxial tension test of the synthetic rubber is shown in Figure 2(a) . To further examine the predictability of the isotropic CSE model, the five constitutive parameters identified from fitting the uniaxial tension test data have been submitted into both the pure shear Model (24) and the equibiaxial tension Model (28). The predictions of pure shear and equibiaxial tension tests are shown in Figure 2(c) and Figure 2(e) , respectively. Furthermore, The predictions of kinematic charcaterizations by the PS and ET Poisson functions in pure shear and equibiaxial tension modes are shown in Figure 2(d) and Figure 2(f) , respectively.

The experimental characterizations of vulcanized natural rubber with 8 phr sulfur have been conducted by Treloar (1944) [11] . Treloar’s experiments with all specimens taken from a single sheet of the material are usually used as a benchmark for evaluating constitutive models of incompressible isotropic hyperelastic materials. The isotropic CSE constitutive modeling of Treloar’s test data has already been conducted by Zhao (2021) [12] .

The uniaxial tension, pure shear, and equibiaxial tension tests of S4035A TPE have been modeled by the isotropic CSE functional with the constant constitutive parameter $c_4=1$ [5] . The current models with variable constitutive parameter c₄ achieve better fitting on uniaxial tension test data, better prediction on

pure shear test data, and particularly better prediction on equibiaxial tension test data of S4035A TPE.

The uniaxial tension, pure shear, and equibiaxial tension tests of compressible natural rubber, along with the detailed kinematic characterizations, conducted by Storåkers (1986) [10] have been studied with the isotropic CSE functional partially combined with the Poisson functions [6] .

The Poisson functions can be applied to accurately model and predict available kinematic characterization data for compressible rubberlike materials in different deformation modes [4] [10] [13] [14] . The isotropic CSE functional is fully combined with the Poisson functions to formulate general constitutive equations in the three different deformation modes for compressible as well as incompressible rubberlike materials. The uniaxial tension, pure shear, and equibiaxial tension tests of compressible synthetic rubber, along with the detailed kinematic characterizations, conducted by Storåkers (1986) [10] have been studied with the current isotropic CSE constitutive model. The kinematic characterizations of lateral contractions as a function of uniaxial stretch, ${\lambda }_2\left({\lambda }_1\right)$ and ${\lambda }_3\left({\lambda }_1\right)$ data, fit the UT Poisson function accurately, indicating a good isotropic behavior of the synthetic rubber shown in Figure 2(b) . The kinematic characterizations in pure shear and equibiaxial tension modes are accurately predicted by the PS Poisson function and the ET Poisson function shown in Figure 2(d) , and Figure 2(f) , respectively. Furthermore, the isotropic CSE constitutive model accurately fits the uniaxial tension test data and predicts the pure shear and equibiaxial tension test data shown in Figure 2(a) , Figure 2(c) , and Figure 2(e) , respectively. The equibiaxial tension test data, however, is underestimated probably due to branching stability issues.

Uniaxial tension mode along with other deformation modes needed to experimentally characterize compressible rubberlike materials may be optimally selected. With advancements in technologies, experimental characterizations for compressible rubberlike materials with better accuracy, greater deformation range, and better stability are still needed. The benchmark tests for compressible rubberlike materials as good as Treloar’s tests for incompressible rubberlike materials are yet to appear.

The principles of determinism, objectivity, physical connsistency, material symmetry, and local actions with dimensional consistency, existence or well-posedness, and equipresence are required for constitutive equations as summarized by Oden (2011) [15] .

The principle of physical consistency is crucial since constitutive equations cannot violate the physical principles of mechanics. Furthermore, physical models possess predictive capability while empirical models are approximate mathematical fits to a set of experimental measurements and have no power of prediction as emphasized by Ashby (1992) [16] . Therefore, a constitutive model should be formulated based on the physical principles of mechanics such as the conservation of energy, the conservation of mass, and the Clausius-Duhem inequality, making it a physical model.

The partial differential equation of isotropic CSE functional was formulated based on the conservation of energy, generally solved by the Lie group method, and specifically solved by the differential geometry. For modeling physically possible nonlinear elastic deformations, the normal stretch, shear stretch, and ellipsoidal stretch deformations are selected based on the curvatures of deformations. Therefore, a constitutive model should be formulated with geometrically meaningful deformations.

The three terms of the isotropic CSE functional are all in the same order of magnitude, ${\lambda }^2$, since the three terms are asymptotically equal, representing normal stretch, shear stretch, and ellipsoidal stretch deformations. This unique feature of the isotropic CSE functional makes it an isomorphism functional. The thermal deformation does not affect the form of the isotropic CSE constitutive model for structural deformation, which was one kind of material isomorphism assumed by Noll (1972) [17] . Therefore, the finite thermoelastic CSE functional preserves the structure of symmetry in Ψ (16) as studied by Zhao (2023) [18] . In the principle of objectivity, the form of a constitutive model must be invariant under changes in the frame of reference: they must be independent of the observer. As an augmentation, the form of a constitutive model must be invariant under changes in environmental temperature: they must be independent of the environment.

The ideal model is a fully physical model. Constitutive modelings of rubberlike materials are complicated: A fully physical treatment, however, may not be possible. In uniaxial tension deformation as an example, a uniaxial load causes a uniaxial stretch, resulting in laterial contractions. The kinematic relations of two orthogonal lateral contractions with uniaxial stretch cannot be theoretically established based on the conservation of energy since the axial load does not do any work on laterial contraction deformations. For nonlinear finite elastic deformations, the Poisson functions are applied to fit and predict the kinematic relationships at different deformation modes as suggested by Blatz and Ko (1962) [4] . The Poisson’s ratio can be independently determined by fitting the UT kinematic relationship tests by the UT Poisson function. Therefore, the isotropic CSE functional, fully combined with the empirical Poisson functions, is applied to constitutively model and predict nonlinear finite elastic deformations for rubberlike materials.

Last but not least, the existence or well-posedness of practical problems in the principle of local action for a constitutive model is required. A constitutive model should have solutions to properly-posed boundary and initial-value problems. Similarly, the constitutive parameters of a constitutive model should be uniquely identified. Otherwise, multiple solutions are often resolved for the constitutive model with two or more nonlinear constitutive parameters. With the multiple combinations of constitutive parameters, the predictions of constitutive models could be quantitatively widely different as pointed out by Destrade et al. (2022) [9] .