A model for rubber elasticity

Abstract

A constitutive equation for rubber-like materials is developed using the left stretch tensor. This process starts with a model for hyperelastic solids based on a separable energy function. This model accurately fits extensional data for vulcanized natural rubber until the onset of hysteresis at intermediate strains. Better predictions outside the hyperelastic range are obtained by directly modifying this constitutive equation to describe limited extensibility. The resulting model accurately fits biaxial, planar, and uniaxial extension data for a variety of rubber-like materials using three constants. This model also predicts simple shear results derived from planar extension data and characterizes inflation of spherical membranes for elastomers and soft tissue. A final modification accurately describes hardening associated with crystallization at large tensile strains.

Appendices

Appendices

Model conversion

For isochoric deformation, the left stretch tensor V satisfies the Cayley–Hamilton equation (Gurtin 1981 p. 16)

$$ {\mathbf{V}}^3-{\upiota}_1{\mathbf{V}}^2+{\upiota}_2\mathbf{V}=\mathbf{I},{\upiota}_1=\mathrm{tr}\left(\mathbf{V}\right)={\boldsymbol{I}}_1,{\upiota}_2=\left[\mathrm{tr}{\left(\mathbf{V}\right)}^2-\mathrm{tr}\left({\mathbf{V}}^2\right)\right]/2=\left({I}_1^2-{I}_2\right)/2, $$

where the principal invariants (ι₁, ι₂) are expressed in terms of the moment invariants (I₁, I₂). Multiplying this equation by V⁻² and taking the trace implies the second result

$$ \mathrm{tr}\left(\mathbf{B}\right)=\mathrm{tr}\left({\mathbf{V}}^2\right)={I}_2,\mathrm{tr}\left({\mathbf{B}}^{-1}\right)=\mathrm{tr}\left({\mathbf{V}}^{-2}\right)={\left({I}_1^2-{I}_2\right)}^2/4-2{I}_1 $$

relating tr(B⁻¹) to moment invariants of V, where B = V². These results are used to convert function (1) into an expression

$$ w={\mathrm{C}}_1\left[\mathrm{tr}\left(\mathbf{B}\right)-3\right]+{\mathrm{C}}_2\left[\mathrm{tr}\left({\mathbf{B}}^{-1}\right)-3\right]={\mathrm{C}}_1\left({I}_2-3\right)+{\mathrm{C}}_2\left[{\left({I}_1^2-{I}_2\right)}^2/4-2{I}_1-3\right] $$

involving stretch tensor invariants, where C₁ and C₂ are constants. Using this energy function in (8) leads to an alternate form of the constitutive Eq. (2)

$$ \mathbf{T}+\mathrm{p}\mathbf{I}={\mathrm{C}}_2\left({I}_1^3-{I}_1{I}_2-2\right)\mathbf{V}+\left[2{\mathrm{C}}_1+{\mathrm{C}}_2\left({I}_2-{I}_1^2\right)\right]{\mathbf{V}}^2, $$

where the shear modulus μ = 2(C₁ + C₂) is positive. See Treloar (2009 pp. 211–229) for further discussion of the Mooney–Rivlin model.

The left stretch tensor can be expressed in terms of B using the Cayley–Hamilton equation for V. Multiplying this equation by V gives an alternate form

$$ {\mathbf{V}}^4-{\upiota}_1{\mathbf{V}}^3+{\upiota}_2{\mathbf{V}}^2=\mathbf{V} $$

in terms of principal invariants (ι₁, ι₂). Substituting for V³ using the Cayley–Hamilton equation results in the expression

$$ {\mathbf{B}}^2+\left({\upiota}_2-{\upiota_1}^2\right)\mathbf{B}-{\upiota}_1\mathbf{I}=\left(1-{\upiota}_1{\upiota}_2\right)\mathbf{V}, $$

where B = V². Solving for V leads to an expression

$$ \mathbf{V}=\left[{\mathbf{B}}^2-\left({I}_2+{I_1}^2\right)\mathbf{B}/2-{I}_1\mathbf{I}\right]/\left[1-{I}_1\left({I_1}^2-{I}_2\right)/2\right], $$

involving moment invariants I₁ = tr(V) and I₂ = tr(V²) = tr(B). This expression can be written in terms of B⁻¹ and B using a result

$$ {\mathbf{B}}^2={\mathbf{B}}^{-1}+\mathrm{tr}\left(\mathbf{B}\right)\mathbf{B}-\mathrm{tr}\left({\mathbf{B}}^{-1}\right)\mathbf{I} $$

obtained from the Cayley–Hamilton equation for B. A real root of the quartic equation

$$ {\left[{I}_1^2-\mathrm{tr}\left(\mathbf{B}\right)\right]}^2/4-2{I}_1=\mathrm{tr}\left({\mathbf{B}}^{-1}\right) $$

determines I₁ ≥ 3 as a function of invariants tr(B) and tr(B⁻¹). Ogden (1972a) alluded to the complexity of this isotropic function in the context of Varga’s (1966) model. See Hoger and Carlson (1984), Ting (1985), and Sawyers (1986) for a generalization where det(V) ≠ 1.

Constitutive equations for elastic solids determine Cauchy stress T = S(F) as a function of the deformation gradient F. This function satisfies the constraint S(F) = S(FH), where the symmetry transformation H is a constant rotation for isotropic materials. The polar decomposition F = VR leads to a necessary condition S(F) = S(VRH) = S(V) associated with the material rotation R = H^T. The constitutive function also satisfies the constraint S(QVQ^T) = QS(V)Q^T, where Q(t) is a time-dependent rotation associated with a change of frame. This constraint implies the general model (6) for incompressible elastic solids. Substituting a complicated expression for V in terms of B, which is implicit in Gurtin’s derivation (1981 p. 167), leads to an alternate form of the general model. This result suggests that basic constitutive equations involving the natural variable V might provide a better description of isotropic elastic solids than simple models using the substitute variable B.

Hyperelastic solids

An incompressible elastic solid described by the constitutive eq. T = –pI + S(V) is hyperelastic if the symmetric tensor S(V) can be derived from a scalar energy function of the left stretch tensor V. Assume the Cauchy stress T satisfies an isothermal form of Planck’s inequality (Truesdell 1984 pp. 112–115)

$$ \mathbf{T}\cdotp \mathbf{L}-\uprho \dot{\psi}\ge 0 $$

involving the spatial velocity gradient L, density ρ, and a material derivative of the specific free energy function ψ(V), where the scalar product A·B = tr(A^TB) is defined for any tensors A and B in terms of the trace function (Gurtin 1981 p. 5). For incompressible solids, this constraint reduces to

$$ \mathbf{S}\cdotp \mathbf{L}-\dot{w}\ge 0, $$

where w(V) = ρψ is the energy density function. The chain rule implies

$$ \dot{w}=\frac{\partial w}{\partial \mathbf{V}}\cdot \dot{\mathbf{V}}, $$

which can be expressed as

$$ \dot{w}=\frac{\partial w}{\partial \mathbf{V}}\mathbf{V}\cdotp \mathbf{L} $$

using an identity (VanArsdale 2003)

$$ \dot{\mathbf{V}}+{\mathbf{V}\mathbf{W}}_{\mathrm{R}}-{\mathbf{W}}_{\mathrm{R}}\mathbf{V}={\mathbf{L}}_{\mathrm{D}}\mathbf{V},\kern0.5em \mathbf{L}={\mathbf{W}}_{\mathrm{R}}+{\mathbf{L}}_{\mathrm{D}}=\dot{\mathbf{R}}{\mathbf{R}}^{\mathrm{T}}+\mathbf{R}\dot{\mathbf{U}}{\mathbf{U}}^{-1}{\mathbf{R}}^{\mathrm{T}} $$

associated with a polar decomposition F = RU = VR of the deformation gradient in terms of a material rotation R and symmetric stretch tensors U and V with positive principle values. Since the stress S and energy density w are independent of L, the isothermal form of Planck’s inequality is satisfied only if

$$ \mathbf{S}=\frac{\partial w}{\partial \mathbf{V}}\mathbf{V}, $$

where this material does not dissipate energy in any isochoric motion. Since w is an isotropic function of V, the energy density can be expressed as ŵ(I₁, I₂) in terms of the moment invariants I₁ = tr(V) and I₂ = tr(V²) (Gurtin 1981 p. 230). The resulting partial derivative

$$ \frac{\partial w}{\partial \mathbf{V}}=\frac{\partial \hat{w}}{\partial {I}_1}\mathbf{I}+2\frac{\partial \hat{w}}{\partial {I}_2}\mathbf{V} $$

leads to the expression (8) for stress in terms of the moment invariants.

However, only some models described by the general constitutive Eq. (6) have this property. For example, model (12) cannot be derived from an energy function but can be expressed in the alternate form (13). The preceding derivation suggests that one part is determined by an energy function, while the remaining stress can dissipate energy. The dissipation rate associated with this part

$$ \frac{{\mathrm{m}}_2\beta \left({I}_1-3\right){I}_2{\mathbf{V}}^2\cdotp \mathbf{L}}{3\left[1-\beta \left({I}_1-3\right)\right]}=\frac{{\mathrm{m}}_2\beta \left({I}_1-3\right){I}_2{\mathbf{V}}^2\cdotp \dot{\mathbf{V}}{\mathbf{V}}^{-1}}{3\left[1-\beta \left({I}_1-3\right)\right]}=\frac{{\mathrm{m}}_2\beta \left({I}_1-3\right){I}_2{\dot{I}}_2}{6\left[1-\beta \left({I}_1-3\right)\right]}\ge 0 $$

is non-negative for m₂ > 0 if β İ₂ ≥ 0 in dynamic processes. This constraint, which also applies to the constitutive Eq. (14), is consistent with the drop in β during uniaxial unloading from a large tensile strain shown in Fig. 5.

Model predictions

Predictions of model (7) with material constants m₁ and m₂ are listed below, where T_xx, T_xy, T_yy, T_zz, and P_xx denote Cauchy and nominal stress components associated with the orthonormal basis vectors ${\overrightarrow{\mathrm{e}}}_{\mathrm{x}},{\overrightarrow{\mathrm{e}}}_{\mathrm{y}},{\overrightarrow{\mathrm{e}}}_{\mathrm{z}}$. Deformation is characterized by the left stretch tensor V and the deformation gradient F, where symbols γ, λ, λ₁, λ₂ denote the shear strain and principal stretches in coordinate directions. Tensors are expressed as linear combinations of basis tensors defined using a tensor product (Gurtin 1981 p. 4). Predictions of the constitutive Eqs. (12) and (14) are obtained by replacing m₂ with m₂/[1 − β(I₁ − 3)] or m₂/{1 – β[(I₁ – 3) + α(I₂ – 3)³]} in the expressions below.

• Uniaxial extension:

$$ \mathbf{V}=\lambda {\overrightarrow{\mathbf{e}}}_x\otimes {\overrightarrow{\mathbf{e}}}_x+{\lambda}^{-1/2}\left({\overrightarrow{\mathbf{e}}}_y\otimes {\overrightarrow{\mathbf{e}}}_y+{\overrightarrow{\mathbf{e}}}_z\otimes {\overrightarrow{\mathbf{e}}}_z\right),\mathrm{tr}\left(\mathbf{V}\right)=\lambda +2{\lambda}^{\hbox{--} 1/2},\mathrm{tr}\left({\mathbf{V}}^2\right)={\lambda}^2+2{\lambda}^{\hbox{--} 1} $$

$$ {\mathrm{T}}_{\mathrm{yy}}={\mathrm{T}}_{\mathrm{zz}}=0\Rightarrow \mathrm{p}={\mathrm{m}}_1{\uplambda}^{\hbox{--} 1/2}+{\mathrm{m}}_2{\uplambda}^{\hbox{--} 1}\mathrm{tr}\left({\mathbf{V}}^2\right)/3\Rightarrow {\mathrm{T}}_{\mathrm{xx}}={\mathrm{m}}_1\left(\uplambda \hbox{--} {\uplambda}^{\hbox{--} 1/2}\right)+{\mathrm{m}}_2\left({\uplambda}^2\hbox{--} {\uplambda}^{\hbox{--} 1}\right)\mathrm{tr}\left({\mathbf{V}}^2\right)/3 $$

$$ \mathbf{F}=\lambda {\overrightarrow{\mathbf{e}}}_x\otimes {\overrightarrow{\mathbf{e}}}_x+{\lambda}^{-1/2}\left({\overrightarrow{\mathbf{e}}}_y\otimes {\overrightarrow{\mathbf{e}}}_y+{\overrightarrow{\mathbf{e}}}_z\otimes {\overrightarrow{\mathbf{e}}}_z\right)\Rightarrow {\mathrm{P}}_{\mathrm{xx}}={\mathrm{m}}_1\left(1\hbox{--} {\lambda}^{\hbox{--} 3/2}\right)+{\mathrm{m}}_2\left(\lambda \hbox{--} {\lambda}^{\hbox{--} 2}\right)\left({\lambda}^2+2{\lambda}^{\hbox{--} 1}\right)/3 $$

• Planar extension:

$$ \mathbf{V}=\lambda {\overrightarrow{\mathbf{e}}}_x\otimes {\overrightarrow{\mathbf{e}}}_x+{\overrightarrow{\mathbf{e}}}_y\otimes {\overrightarrow{\mathbf{e}}}_y+{\lambda}^{-1}{\overrightarrow{\mathbf{e}}}_z\otimes {\overrightarrow{\mathbf{e}}}_z,\mathrm{tr}\left(\mathbf{V}\right)=\lambda +1+{\lambda}^{\hbox{--} 1},\mathrm{tr}\left({\mathbf{V}}^2\right)={\lambda}^2+1+{\lambda}^{\hbox{--} 2} $$

$$ {\mathrm{T}}_{\mathrm{zz}}=0\Rightarrow \mathrm{p}={\mathrm{m}}_1{\lambda}^{\hbox{--} 1}+{\mathrm{m}}_2{\lambda}^{\hbox{--} 2}\mathrm{tr}\left({\mathbf{V}}^2\right)/3\Rightarrow {\mathrm{T}}_{\mathrm{xx}}={\mathrm{m}}_1\left(\lambda \hbox{--} {\lambda}^{\hbox{--} 1}\right)+{\mathrm{m}}_2\left({\lambda}^2\hbox{--} {\lambda}^{\hbox{--} 2}\right)\mathrm{tr}\left({\mathbf{V}}^2\right)/3 $$

$$ \mathbf{F}=\lambda {\overrightarrow{\mathbf{e}}}_x\otimes {\overrightarrow{\mathbf{e}}}_x+{\overrightarrow{\mathbf{e}}}_y\otimes {\overrightarrow{\mathbf{e}}}_y+{\lambda}^{-1}{\overrightarrow{\mathbf{e}}}_z\otimes {\overrightarrow{\mathbf{e}}}_z\Rightarrow {\mathrm{P}}_{\mathrm{xx}}={\mathrm{m}}_1\left(1\hbox{--} {\lambda}^{\hbox{--} 2}\right)+{\mathrm{m}}_2\left(\lambda \hbox{--} {\lambda}^{\hbox{--} 3}\right)\left({\lambda}^2+1+{\lambda}^{\hbox{--} 2}\right)/3 $$

• Equibiaxial extension:

$$ \mathbf{V}=\lambda \left({\overrightarrow{\mathbf{e}}}_x\otimes {\overrightarrow{\mathbf{e}}}_x+{\overrightarrow{\mathbf{e}}}_y\otimes {\overrightarrow{\mathbf{e}}}_y\right)+{\lambda}^{-2}{\overrightarrow{\mathbf{e}}}_z\otimes {\overrightarrow{\mathbf{e}}}_z,\mathrm{tr}\left(\mathbf{V}\right)=2\lambda +{\lambda}^{\hbox{--} 2},\mathrm{tr}\left({\mathbf{V}}^2\right)=2{\lambda}^2+{\lambda}^{\hbox{--} 4} $$

$$ {\mathrm{T}}_{\mathrm{zz}}=0\Rightarrow \mathrm{p}={\mathrm{m}}_1{\lambda}^{\hbox{--} 2}+{\mathrm{m}}_2{\lambda}^{\hbox{--} 4}\mathrm{tr}\left({\mathbf{V}}^2\right)/3\Rightarrow {\mathrm{T}}_{\mathrm{xx}}={\mathrm{m}}_1\left(\lambda \hbox{--} {\lambda}^{\hbox{--} 2}\right)+{\mathrm{m}}_2\left({\lambda}^2\hbox{--} {\lambda}^{\hbox{--} 4}\right)\mathrm{tr}\left({\mathbf{V}}^2\right)/3 $$

$$ \mathbf{F}=\lambda \left({\overrightarrow{\mathbf{e}}}_x\otimes {\overrightarrow{\mathbf{e}}}_x+{\overrightarrow{\mathbf{e}}}_y\otimes {\overrightarrow{\mathbf{e}}}_y\right)+{\lambda}^{-2}{\overrightarrow{\mathbf{e}}}_z\otimes {\overrightarrow{\mathbf{e}}}_z\Rightarrow {\mathrm{P}}_{\mathrm{xx}}={\mathrm{m}}_1\left(1\hbox{--} {\lambda}^{\hbox{--} 3}\right)+{\mathrm{m}}_2\left(\lambda \hbox{--} {\lambda}^{\hbox{--} 5}\right)\left(\ 2{\lambda}^2+{\lambda}^{\hbox{--} 4}\right)/3 $$

• Biaxial extension:

$$ \mathbf{V}={\lambda}_1{\overrightarrow{\mathbf{e}}}_x\otimes {\overrightarrow{\mathbf{e}}}_x+{\lambda}_2{\overrightarrow{\mathbf{e}}}_y\otimes {\overrightarrow{\mathbf{e}}}_y+{\left({\lambda}_1{\lambda}_2\right)}^{-1}{\overrightarrow{\mathbf{e}}}_z\otimes {\overrightarrow{\mathbf{e}}}_z,\mathrm{tr}\left(\mathbf{V}\right)={\lambda}_1+{\lambda}_2+{\left({\lambda}_1{\lambda}_2\right)}^{\hbox{--} 1}, $$

$$ \mathrm{tr}\left({\mathbf{V}}^2\right)={\lambda_1}^2+{\lambda_2}^2+{\left({\lambda}_1{\lambda}_2\right)}^{\hbox{--} 2};{\mathrm{T}}_{\mathrm{xx}}\hbox{--} {\mathrm{T}}_{\mathrm{yy}}={\mathrm{m}}_1\left({\lambda}_1\hbox{--} {\lambda}_2\right)+{\mathrm{m}}_2\left({\lambda_1}^2\hbox{--} {\lambda_2}^2\right)\mathrm{tr}\left({\mathbf{V}}^2\right)/3, $$

• Simple shear:

$$ \mathbf{V}={\left(4+{\upgamma}^2\right)}^{-1/2}\left[\left(2+{\upgamma}^2\right){\overrightarrow{\mathbf{e}}}_x\otimes {\overrightarrow{\mathbf{e}}}_x+\upgamma \left({\overrightarrow{\mathbf{e}}}_x\otimes {\overrightarrow{\mathbf{e}}}_y+{\overrightarrow{\mathbf{e}}}_y\otimes {\overrightarrow{\mathbf{e}}}_x\right)+2{\overrightarrow{\mathbf{e}}}_y\otimes {\overrightarrow{\mathbf{e}}}_y\right]+{\overrightarrow{\mathbf{e}}}_z\otimes {\overrightarrow{\mathbf{e}}}_z, $$

$$ \mathrm{tr}\left(\mathbf{V}\right)=1+{\left(4+{\upgamma}^2\right)}^{1/2},\mathrm{tr}\left({\mathbf{V}}^2\right)=3+{\upgamma}^2;{\mathrm{T}}_{\mathrm{xy}}={\mathrm{m}}_1\upgamma {\left(4+{\upgamma}^2\right)}^{\hbox{--} 1/2}+{\mathrm{m}}_2\upgamma \left(3+{\upgamma}^2\right)/3, $$

$$ {\mathrm{T}}_{\mathrm{xx}}={\mathrm{m}}_1\left[\left(2+{\upgamma}^2\right){\left(4+{\upgamma}^2\right)}^{\hbox{--} 1/2}\hbox{--} 1\right]+{\mathrm{m}}_2{\upgamma}^2\left(3+{\upgamma}^2\right)/3,{\mathrm{T}}_{\mathrm{yy}}={\mathrm{m}}_1\left[2{\left(4+{\upgamma}^2\right)}^{\hbox{--} 1/2}\hbox{--} 1\right],{\mathrm{T}}_{\mathrm{zz}}=0 $$

Fitting algorithm

The method used here searches for m model parameters that improve goodness of fit (also known as coefficient of determination)

$$ {\mathrm{R}}^2=1-\sum \limits_{\mathrm{i}=0}^{\mathrm{n}-1}{\left({y}_{\mathrm{i}}-{f}_{\mathrm{i}}\right)}^2/\sum \limits_{\mathrm{i}=0}^{\mathrm{n}-1}{\left({y}_{\mathrm{i}}-\overline{y}\right)}^2, $$

where predicted values f_i are compared to n measurements y_i with mean ͞y = Σy_i/n. Residuals are errors y_i − f_i associated with this correlation. The root mean squared error

$$ \mathrm{RMSE}=\sqrt{\frac{1}{n-m}\sum \limits_{\mathrm{i}=0}^{n-1}{\left({y}_{\mathrm{i}}-{f}_{\mathrm{i}}\right)}^2} $$

is essentially the standard deviation of data about a fit with n − m degrees of freedom. This statistic normalizes residuals plotted in most figures, suggesting a confidence interval for predicted values. Root mean squared error is compared with the entire range of measurements y_max − y_min to obtain percent full scale. Relative errors E_i are defined by Ogden et al. (2004) as absolute residuals |y_i − f_i| normalized by the maximum of 0.5 or |y_i|. The largest relative error E and the sum of squared error SSE = Σ(y_i − f_i)² are provided with two significant digits for comparison with results in other papers. The goodness of fit is expressed with four significant digits, while model coefficients and the root mean squared error appear with three digits. A fit with R² > 0.999 and RSME < 1% full scale is considered excellent, while correlations with R² < 0.99 and RSME > 2% full scale are somewhat questionable for data shown in the figures.

The resulting algorithm randomly searches for new coefficients near the initial guess, where the change is usually less than 5% of the value. If a better fit is obtained with R² closer to one, the search resumes from the new set of coefficients. This process stops after a specified number of attempts (typically 1000) fails to improve the fit. Like most methods, this approach is sensitive to an initial guess. However, this guess can usually be refined in real time for m < 4 since the search only involves model evaluations. The resulting coefficients listed in figure captions concurrently fit all data displayed within the specified range except for Figs. 5 and 18. Shear stress predictions in Figs. 2, 10, and 12 do not involve this algorithm, where statistics are provided to characterize correlation with derived data.

An alternate approach is used by Ogden et al. (2004), where fitted values are determined using a least-squares analysis based on the function SSE = Σ(f_i − y_i)². These authors randomly vary their initial guess to improve this solution as determined by SSE and relative errors E_i. Plotting relative errors masks the trend of residuals (Curran-Everett 2011) by mapping all values to the positive quadrant. The significance of relative error is difficult to assess without additional information about experimental uncertainty. Since most test equipment has a fixed resolution, this uncertainty can be a large percentage of small data values. For example, Ogden et al. (2004) obtain relative errors up to 40% for small strains, where models seem to accurately describe extensional data. Smaller errors around 10% can occur at larger strains, where models visually deviate from data. Treloar’s uniaxial and equibiaxial measurements are fit simultaneously by Ogden et al. (2004) to obtain a set of coefficients, while excluding planar extension data. This paper uses an energy function to describe Treloar’s data well outside the hyperelastic range.

The approach in Ogden et al. (2004) is modified by Destrade et al. (2017), where fitted values are determined by minimizing the function Σ[(f_i − y_i)/y_i]² involving relative residuals. These authors must exclude measurements near zero to use this approach, which weights smaller values as more significant in determining a fit compared to a least-squares analysis by Ogden et al. (2004). This weighting is further exacerbated by an apparent choice of units, which stretches the y-axis by a factor of ten. At small strains, Destrade et al. (2017) obtain relative residuals around 15 to 20% for the neo-Hookean model even though fitted values appear to accurately describe data deemed most reliable. They determine model coefficients by fitting just uniaxial data. See referee Martin Kroon’s concern in Supplemental Material as well as observations by Urayama (2006) and Treloar (2009 p. 218) that uniaxial results are insufficient to distinguish between theories. Destrade et al. 2017, Fig. 9) limit predictions in simple shear to γ ≤ 1 and do not compare with values derived from Treloar’s planar extension data. They also use an energy function to describe Treloar’s uniaxial data well outside the hyperelastic range (λ < 5). Neither this paper nor Ogden et al. (2004) provides guidance concerning the significance of any particular relative error.

Fig. 9

A fit of planar and uniaxial data (Meunier et al. 2008) for unfilled silicone rubber using Eq. (12), where coefficients m₁ = 0.262 MPa, m₂ = 0.190 MPa, and β = −0.404 determine curves with correlation R² = 0.9989, relative error E = 10.4%, and SSE = 0.024 (MPa)². Distribution of error is normalized by RMSE = 0.0275 MPa, which is about 0.8% of full scale for 35 data points shown in the figure

Data comparisons

The constitutive Eq. (12) is used to fit extensional data obtained from Mansouri and Darijani (2014) for different rubber-like materials. Correlations in Figs. 9, 11, 14, 15, 16 and 17 have a goodness of fit R² > 0.995 and a root mean squared error RMSE < 2.2% of full scale. Predictions by (12) are similar to the best models evaluated by Mansouri and Darijani (2014, table 4) based on the sum of squared error SSE. In Figs. 10 and 12, simple shear predictions are compared with values derived from planar extension data using the equivalence relations (11). These predictions suggest axial elongation in torsion for silicone rubber, while the porcine liver tissue should exhibit a negative Poynting effect associated with axial contraction. A prediction for inflation of a spherical membrane is shown in Fig. 13 using parameters for porcine liver tissue. This plot is qualitatively similar to measurements on soft tissue by Osborne (1909) and Lari et al. (2012).

Fig. 10

Shear stress values derived from planar extension data for an unfilled silicone rubber in Fig. 9 using the equivalence relations (11). An absolute value is used to map all data to this quadrant. Curves are predictions of Eq. (12) using coefficients from Fig. 9 with correlation R² = 0.9907 to derived data. Deviation from predicted values is RMSE = 0.0209 MPa or about 1.8% of full scale over the entire range of shear strain (− 1.07 → 1.62). The ratio T_xy/γ increases from an initial value μ = m₁/2 + m₂ = 0.321 MPa, while the ratio (T_yy − T_zz)/(T_xx − T_yy) increases from − m₁/8μ = − 0.102 to zero. This material should exhibit axial elongation in torsion since m₁ > 0

Fig. 11

A fit of planar extension data (Gao et al. 2010) for porcine liver tissue using Eq. (12), where coefficients m₁ = − 0.946 kPa, m₂ = 0.479 kPa, and β = 2.14 determine curves with correlation R² = 0.9996, relative error E = 4.4%, and SSE = 0.0012 (kPa)². Distribution of error is normalized by RMSE = 0.00866 kPa, which is about 0.7% of full scale for 29 data points shown in the figure. Note that the shear modulus μ = m₁/2 + m₂ = 0.006 kPa is positive for this soft tissue

Fig. 12

Shear stress values derived from planar extension data for porcine liver tissue in Fig. 11 using the equivalence relations (11). Curves are predictions of Eq. (12) using coefficients from Fig. 11 with correlation R² = 0.9997 to derived data. Deviation from predicted values is RMSE = 0.00545 kPa or about 0.6% of full scale for data shown in the figure. Note that the secondary normal stress difference T_yy − T_zz is positive. This material should exhibit axial contraction in torsion since m₁ < 0

Fig. 13

Scaled pressure P as a function of tangential stretch λ for a spherical membrane using the extended version of Eq. (16) with coefficients from Fig. 11. This curve becomes unbounded for λ > 1.516, suggesting a limit on the size at rupture. This trend is qualitatively similar to measurements by Osborne (1909) on a monkey bladder as well as porcine sclera data obtained by Lari et al. (2012). These authors also observe hysteresis during sclera deflation and uniaxial unloading, which is consistent with β ≠ 0

Fig. 14

A fit of equibiaxial and uniaxial data (Alexander 1968) for a synthetic rubber neoprene using Eq. (12), where coefficients m₁ = 1.90 MPa, m₂ = 0.0597 MPa, and β = 0.0887 determine curves with correlation R² = 0.9958, relative error E = 54%, and SSE = 8.7 (MPa)². The distribution of error is normalized by RMSE = 0.628 MPa, which is about 2.2% of full scale for 25 data points shown in the figure

Fig. 15

A fit of uniaxial data (Fox and Goulbourne 2008) for polyacrylate rubber VHB 4905 using Eq. (12), where coefficients m₁ = 0.144 MPa, m₂ = 0.00135 MPa, and β = 0.0872 determine a curve with correlation R² = 0.9994, relative error E = 1.1%, and SSE = 0.00013 (MPa)². The distribution of error is normalized by RMSE = 0.00308 MPa, which is about 0.7% of full scale for 17 data points shown in the figure. The value of β drops 79% to 0.0183 as the maximum stretch decreases to 6. A fit with β = 0 for λ < 6 has coefficients m₁ = 0.137 MPa and m₂ = 0.00204 MPa with correlation R² = 0.9995, which suggests an upper bound on the hyperelastic range for this material

Fig. 16

A fit of uniaxial data (Yeoh and Fleming 1997) for a vulcanized rubber using Eq. (12), where coefficients m₁ = 0.168 MPa, m₂ = 0.500 MPa, and β = −2.11 determine a curve with correlation R² = 0.9992, relative error E = 11.9%, and SSE = 0.081 (MPa)². The distribution of error is normalized by RMSE = 0.0949 MPa, which is about 1% of full scale for 12 data points shown in the figure

Fig. 17

A fit of uniaxial data (Miehe and Lulei 2001) for carbon black–filled rubber b186 using Eq. (12), where coefficients m₁ = 3.28 MPa, m₂ = 0.0282, and β = 2.37 determine a curve with correlation R² = 0.9979, relative error E = 13.5%, and SSE = 0.11 (MPa)². The distribution of error is normalized by RMSE = 0.0745 MPa, which is about 1.4% of full scale for 23 data points shown in the figure. A better result can be obtained by fitting compressive and tensile stresses separately similar to Fig. 18

Fig. 18

Uniaxial data for carbon black–filled rubber NR 70 obtained by Bechir et al. (2006), where compressive ( ) and tensile ( ) stresses are fit separately using models (7) and (14). Coefficients m₁ = − 0.134 MPa, m₂ = 3.00 MPa determine the compressive stress curve with shear modulus μ = 2.94 MPa and correlation R² = 0.9985 RMSE = 0.140 MPa. Coefficients m₁ = 2.04 MPa, m₂ = 0.0906 MPa, β = 2.27, and α = −0.00713 determine the tensile stress curve with shear modulus μ = 1.11 MPa and correlation R² = 0.9981, RMSE = 0.0681 MPa

Fig. 19

A fit of uniaxial data (Dobrynin and Carrillo 2011) for rubber using Eq. (7), where coefficients m₁ = 1.45 MPa and m₂ = 0.0506 MPa determine a curve with correlation R² = 0.9993, relative error E = 10.1%, and SSE = 0.024 (MPa)². The distribution of error is normalized by RMSE = 0.0311 MPa, which is about 0.7% of full scale for 27 data points shown in the figure. This material should be hyperelastic over the entire range of deformation shown in the figure

Uniaxial data shown in Fig. 18 are obtained from a plot in Bechir et al. (2006, Fig. 8b) for carbon black–filled vulcanized natural rubber. Separate curves for compressive and tensile stress are fit using models (7) and (14). Such an approach appears to be necessary for carbon black–filled rubber due to amplification (Yeoh 1990) at sufficiently large tensile strains, where particle interactions can lead to increased stiffness, tensile strength, and hysteresis (Omnès et al. 2008). While correlation is good, the predicted shear modulus μ = m₁/2 + m₂ is not continuous at λ = 1. An unusual negative value for α in (14) essentially keeps the tensile stress curve from becoming unbounded at large strains. While using six constants to fit uniaxial data seems excessive, Bechir et al. (2006, Fig. 9) obtain worse correlation, especially for compressive stress, with the same number of constants.

Values shown in Fig. 19 are obtained from a plot in Destrade et al. (2017, Fig. 2a), where stresses are about 9.8 times too large in the stated units N/mm² (Sec. 3a). These authors exclude the first 14 data points (λ < 1.1) from their analysis due to a Mooney plot discrepancy that they attribute to possible issues with the experiment. While some points are excluded here for λ < 1.5 due to large, overlapping plot symbols, scanned data do not exhibit the downturn shown in Destrade et al. (2017, Fig. 2c). This material is accurately described by model (7), which suggests hyperelastic behavior over the entire range of deformation shown in Fig. 19. The shear modulus μ = m₁/2 + m₂ = 0.777 MPa predicted for this material is within ± 2% of values (0.761 → 0.791 MPa) obtained by Destrade et al. (2017, tables 2, 3, 5) using 12 different models for hyperelastic solids. However, these authors obtain a 14% variation in μ (0.430 → 0.492 MPa) by fitting the same models to Treloar’s (1944) uniaxial data (Destrade et al. 2017, tables 2, 3, 4), where some correlations involve data outside the hyperelastic range. These values are typically larger with greater variation than the shear modulus (0.396 → 0.431 MPa) obtained for Treloar’s (1944) loading data shown in Figs. 1, 3, 4, and 5.