Uniaxial tension of thin rubber liner sheets and hyperelastic model investigation

Abstract

Uniaxial tension tests were performed on two different thin rubber liner sheets, the silica-filled and the Kevlar-filled EPDM rubber, for the purpose of establishing the constitutive relations via different hyperelasticity models. Due to the fact that the rubber liner sheet can sustain large amount of deformation, over 700 % engineering strain, we used the non-contact, optical technique of digital image correlation to measure local deformation over the sample surface. Three different hyperelastic models were considered for analyzing the experimental measurement, one is the neo-Hookean including the generalized neo-Hookean (GNH), the second is the Rivlin-type models including the special case of Mooney–Rivlin model, and the third is the Ogden model. Model parameters were determined by fitting the hyperelastic model to the experimental data. It was found that the Rivlin model up to the second order and the Ogden model capture the constitutive behavior of both the silica-filled and the Kevlar-filled rubber sheets quite well, while the GNH model can only describe the stress-stretch relation of the Kevlar-filled rubber sheet.

Appendix: Hyperelasticity and hyperelastic constitutive models

A brief description of the hyperelasticity theory and several hyperelastic constitutive models is given in this appendix. We also assume that the material considered is incompressible.

Consider a continuum, which in some configuration occupies a closed region $\mathbb {R}$ in the three-dimensional Euclidean space. A locally volume reserving (incompressible) deformation of the body is characterized by

$$\begin{aligned} {\varvec{y}}({\varvec{x}}) = {\varvec{x}} + {\varvec{u}}({\varvec{x}}), \quad \forall \ {\varvec{x}} \in {\mathbb {R}}, \end{aligned}$$

(9)

with

$$\begin{aligned} {\varvec{F}} = \nabla {\varvec{y}}, \quad \det \,{\varvec{F}} \equiv 1, \quad {\text {on} } \,\mathbb {R}, \end{aligned}$$

in which $\varvec{u}$ and $\varvec{F}$ are the displacement vector field and the deformation gradient tensor field, respectively. We make the assumption that the mapping $\varvec{y}$ is twice continuously differentiable and uniquely reversible on $\mathbb {R}$. Next, let $\varvec{G}$ be the left Cauchy–Green deformation tensor associated with deformation (9) and let $I_{1}$, $I_{2}$, and $I_{3}$ be the scalar invariants of $\varvec{G},$ where

$$\begin{aligned} {\varvec{G}} = {\varvec{G}}^{\top } = {\varvec{F}}{\varvec{F}}^{\top }, \quad \text {on } \,{\mathbb {R}}, \end{aligned}$$

and

$$\begin{aligned} I_{1} = {{\mathrm{tr}}}\,\varvec{G}, \quad I_{2} = \frac{1}{2} \left \{ ({{\mathrm{tr}}}\,\varvec{G})^{2} -{{\mathrm{tr}}}(\varvec{G}^{2}) \right \}, \quad I_{3} = \det \varvec{G} \equiv 1. \end{aligned}$$

Hyperelasticity assumes that the deforming body possesses an elastic potential $W(\varvec{x}) = W(\varvec{F}(\varvec{x}),\, \varvec{x})$, such that the first Piola–Kirchhoff (nominal or engineering) stress field associated with the deformation, $\varvec{\sigma }(\varvec{x})$, is given by

$$\begin{aligned} \varvec{\sigma } = \frac{\partial W}{\partial \varvec{F}}, \quad \text {on } \,\mathbb {R}. \end{aligned}$$

(10)

Further, $\varvec{\sigma }$ is related to the Cauchy (or true) stress field $\varvec{\tau }$ by

$$\begin{aligned} \varvec{\tau } = \varvec{\sigma }\varvec{F}^{\top }, \quad \text {on } \,\mathbb {R}. \end{aligned}$$

(11)

For homogeneous material the elastic potential $W$ will depend on position $\varvec{x}$ only through the deformation gradient tensor $\varvec{F}$, i.e., $W(\varvec{x}) = W(\varvec{F}(\varvec{x}))$. If we further assume that the material is isotropic in the reference configuration, then the elastic potential is only a function of the scalar invariants $I_{1}$ and $I_{2}$. The appropriate constitutive law has the forms

$$\begin{aligned} \varvec{\sigma } = \alpha \varvec{F} - \beta \varvec{G}\varvec{F} - p\,\varvec{F}^{-\top }, \quad \varvec{\tau } = \alpha \varvec{G} - \beta \varvec{G}\varvec{F}^{2} - p\,\varvec{I}, \end{aligned}$$

(12)

where

$$\begin{aligned} \alpha = 2\left (\frac{\partial W}{\partial I_{1}} + I_{1} \frac{\partial W}{\partial I_{2}}\right ), \quad \beta = 2\left (\frac{\partial W}{\partial I_{2}}\right ), \end{aligned}$$

and $p$ represents the arbitrary scalar pressure needed to accommodate the kinematical constraint of incompressibility, and $\varvec{I}$ is the identity tensor. If $\uplambda _{i}$ ($i = 1,\, 2,\, 3$) are the local principal stretches associated with the deformation, their squares are the local eigenvalues of the tensor field $\varvec{G}$. As a result,

$$\begin{aligned} I_{1} = \uplambda _{1}^{2} + \uplambda _{2}^{2} + \uplambda _{3}^{2}, \quad I_{2} = \uplambda _{1}^{2}\uplambda _{2}^{2} + \uplambda _{2}^{2}\uplambda _{3}^{2} + \uplambda _{3}^{2}\uplambda _{1}^{2}, \quad I_{3} = \uplambda _{1}^{2}\uplambda _{2}^{2}\uplambda _{3}^{2} \equiv 1. \end{aligned}$$

(13)

For uniaxial tension, $\uplambda _{1} = \uplambda$, $\uplambda _{2} = \uplambda _{3} = 1/\sqrt{\uplambda },$ where $\uplambda$ is the axial stretch. Therefore we have

$$\begin{aligned} I_{1} = \uplambda ^{2} + \frac{2}{\uplambda }, \quad I_{2} = 2\uplambda + \frac{1}{\uplambda ^{2}}, \quad I_{3} \equiv 1. \end{aligned}$$

(14)

In this investigation, we will study three types of hyperelastic constitutive models. One is the neo-Hookean type [6], including the so-called generalized neo-Hookean [7], the second is the Rivlin type [9], including the so-called Mooney or Mooney–Rivlin model [8], and the third is the Ogden type [10]. The neo-Hookean type models assumes that the elastic potential $W$ depends on the first scalar invariant only, i.e., $W = W(I_{1})$. In particular, the neo-Hookean model has the form

$$\begin{aligned} W = c_{1} (I_{1} - 3), \end{aligned}$$

(15)

where $c_{1}$ is a constant. The elastic potential function of the generalized neo-Hookean (GNH) model has the form

$$\begin{aligned} W = \frac{\mu }{2b} \left \{ \left [1 + \frac{b}{n}(I_{1} - 3)\right ]^{n} - 1 \right \}, \end{aligned}$$

(16)

where $\mu > 0$, $b > 0$, and $n > 1/2$ are constants. Note by setting the exponent $n = 1$ in the GNH model, the model reduces to the original neo-Hookean model.

The Rivlin-type models assume that the elastic potential function can be written in the form

$$\begin{aligned} W = \sum _{i = 0}^{m} c_{i(m-i)} (I_{1} - 3)^{m-i}(I_{2} - 3)^{i}, \quad m = 1,\, 2,\ldots, \end{aligned}$$

(17)

where $c_{ij}$ are constants. If we only consider the Rivlin type of model up to the second order, then

$$\begin{aligned} W&= c_{1} (I_{1} - 3) + c_{2} (I_{2} - 3) \\&\quad +\, c_{3} (I_{1} - 3)^{2} + c_{4} (I_{1} - 3)(I_{2} - 3) + c_{5} (I_{2} - 3)^{2}, \end{aligned}$$

(18)

where $c_{i}$ ($i = 1,\, 2,\, 3,\, 4,\, 5$) are constants. Note that the above expression includes the neo-Hookean model and the Mooney–Rivlin model, which has the form

$$\begin{aligned} W = c_{1} (I_{1} - 3) + c_{2} (I_{2} - 3), \end{aligned}$$

(19)

as special cases.

The final model that we are considering is the Ogden model, which assumes that the elastic potential function $W$ depends directly on the local principal stretches $\uplambda _{i}$ ($i = 1,\, 2,\, 3$) through

$$\begin{aligned} W = \sum _{n=1}^{N} \frac{\mu _{n}}{\alpha _{n}} \left (\uplambda _{1}^{\alpha _{n}} + \uplambda _{2}^{\alpha _{n}} + \uplambda _{3}^{\alpha _{n}}\right ), \quad N = 1,\, 2,\,\ldots, \end{aligned}$$

(20)

where $\alpha _{n}$ and $\mu _{n}$ are constants. The Ogden model also includes the neo-Hookean model ($\alpha _{1} = 2$, $N = 1$) and the Mooney–Rivlin model ($\alpha _{1} = 2$, $\alpha _{2} = -2$, $N = 2$) as its special cases.