A visco-hyperelastic approach to modelling rate-dependent large deformation of a dielectric acrylic elastomer

Abstract

In this work, a new visco-hyperelastic constitutive model is proposed to describe the rate-dependent large deformation behaviour of an acrylic elastomer. The proposed model is based on the Zener model, which consists of a hyperelastic equilibrium spring and a Maxwell element to capture rate-dependent material response. The constitutive equation is based on the history integral of Gent strain energy function. The material constants of the developed model are determined with the aid of available different rate-dependent uniaxial tensile experimental results. The proposed model is validated with the earlier published experimental results of VHB 4910 acrylic elastomer which is considered now as one of the promising actuator materials. Fittings of the proposed model with the earlier published experimental results and present experimental stress-stretch results in different strain rates are found to be in good agreement.

Appendix A

1.1 A.1 Fundamentals of hyperelasticity of elastomer

Study of highly nonlinear elastic materials undergoing large deformation is called as hyperelasticity and the material is special case of Cauchy elastic material (Beatty and Krishnaswamy 2000). However for large deformations, stresses are derived from the strain energy potential function $W$ which is usually expressed in the form of deformation gradient tensor $\varvec{F}$ and the principal invariants. $\varvec{F}$ relates physical quantities from their initial and deformed state. Consider a point initially located at some position X in a material. Displacement of a new position x after deformation gradient $\varvec{F}$ is defined as $\varvec{F} = {{\partial x} \mathord{\left/ {\vphantom {{\partial x} {\partial X}}} \right. \kern-0pt} {\partial X}}$.

Generally deformation of the elastomeric material will be expressed in terms of the left Cauchy-Green deformation tensor $\varvec{B} = \varvec{F} \cdot \varvec{F}^{T}$, or by the right Cauchy-Green deformation tensor $\varvec{C} = \varvec{F}^{T} \cdot \varvec{F}$ which is related to Langrangian strain tensor denoted by $\varvec{E} = {{\left( {\varvec{C} - \varvec{I}} \right)} \mathord{\left/ {\vphantom {{\left( {\varvec{C} - \varvec{I}} \right)} 2}} \right. \kern-0pt} 2}$ where $\varvec{I}$ is the identity tensor.

Simply, we can write $\frac{{\partial \varvec{E}}}{{\partial \varvec{C}}} = \frac{1}{2}$ (A.1)

If a material is said to be path-independent or hyperelastic then the work done by the stresses during deformation process depends only on its initial and final configurations at time 0 and t respectively.

Therefore path-independent strain energy function

$$ W = \int\limits_{\varvec{0}}^{\varvec{t}} {\varvec{S: }} \dot{\varvec{E}}dt\quad {\text{hence}}\,\frac{dW}{dt} = \varvec{S:\dot{E}}\,{\text{or}}\,\dot{W} = \varvec{S:\dot{E}} $$

(A.2)

where $\varvec{S}$ is second Piola-Kirchoff stress tensor and $\dot{\varvec{E}}$ = time rate of change of Lagrangian strain tensor

Assuming W = W( E ( ${\mathbf{X}}$ ), ${\mathbf{X}}$ ) then $\dot{W} = \sum\limits_{i,j = 1}^{3} {\frac{\partial W}{{\partial \varvec{E}_{ij} }}} \dot{\varvec{E}}_{ij}$ (A.3)

On comparing Eq. (A.2) and Eq. (A.3), one gets

$$ \varvec{S} = \frac{\partial W}{{\partial \varvec{E}}} = \left( {\frac{\partial W}{{\partial \varvec{C}}}} \right)\left( {\frac{{\partial \varvec{C}}}{{\partial \varvec{E}}}} \right) $$

(A.4)

Therefore by using Eqs. (A.1) and (A.4), one will get

$$ \varvec{S} = 2\frac{\partial W}{{\partial \varvec{C}}} $$

(A.5)

Strain energy potential function,$W = W\left( {I_{1} ,I_{2} ,I_{3} } \right)$, where $I_{1},\,I_{2}$ and $I_{3}$ are first, second and third invariants of the right Cauchy-Green deformation tensor C.

$$ I_{1} = tr(\varvec{C}) = \lambda_{1}^{2} + \lambda_{2}^{2} + \lambda_{3}^{2} $$

(A.6)

$$ I_{2} = \frac{1}{2}\left[ {\left( {tr\varvec{C}} \right)^{2} - tr\varvec{C}^{2} } \right] = \frac{1}{2}tr\left( {\varvec{CC}} \right) = \left( {\lambda_{1} \lambda_{2} } \right)^{2} + \left( {\lambda_{2} \lambda_{3} } \right)^{2} + \left( {\lambda_{3} \lambda_{1} } \right)^{2} $$

(A.7)

$$ I_{3} = det(\varvec{C}) = \left( {\lambda_{1} \lambda_{2} \lambda_{3} } \right)^{2} = 1 $$

(A.8)

where, $\lambda_{1} ,\lambda_{2} ,$ and $\lambda_{3}$ are principal stretches in three directions.

With the application of chain rule,

$$ \varvec{S} = 2\left( {\left( {\frac{\partial W}{{\partial I_{1} }}} \right)\left( {\frac{{\partial I_{1} }}{{\partial \varvec{C}}}} \right) + \left( {\frac{\partial W}{{\partial I_{2} }}} \right)\left( {\frac{{\partial I_{2} }}{{\partial \varvec{C}}}} \right) + \left( {\frac{\partial W}{{\partial I_{3} }}} \right)\left( {\frac{{\partial I_{3} }}{{\partial \varvec{C}}}} \right)} \right) $$

(A.9)

Since $\frac{{\partial I_{1} }}{{\partial \varvec{C}}} = \varvec{I, }\,\frac{{\partial I_{2} }}{{\partial \varvec{C}}} = I_{1} \varvec{I} - \varvec{C}\,{\text{and}}\,\frac{{\partial I_{3} }}{{\partial \varvec{C}}} = I_{3} \varvec{C}^{ - 1}$ (A.10)

Equation (A.9) can be written as

$$ \varvec{S} = 2\left( {\frac{\partial W}{{\partial I_{1} }}\varvec{I} + \frac{\partial W}{{\partial I_{2} }}\left( {I_{1} \varvec{I} - \varvec{C}} \right) + \frac{\partial W}{{\partial I_{3} }}I_{3} \varvec{C}^{ - 1} } \right) $$

(A.11)

Transformation of the second Piola-Kirchoff stress to Cauchy stress tensor $\varvec{\sigma}$ results

$$ \varvec{\sigma}= \frac{1}{J}\varvec{FSF}^{T} $$

(A.12)

where, $J = det\varvec{F}$, where $\varvec{F} = \left[ {\begin{array}{*{20}c} {\lambda_{1} } & 0 & 0 \\ 0 & {\lambda_{2} } & 0 \\ 0 & 0 & {\lambda_{3} } \\ \end{array} } \right]$

Using Eqs. (A.11) and (A.12), $\varvec{\sigma}$ can be written as

$$ \varvec{\sigma}= \frac{2}{J}\varvec{F}\left( {\frac{\partial W}{{\partial I_{1} }}\varvec{I} + \frac{\partial W}{{\partial I_{2} }}\left( {I_{1} \varvec{I} - \varvec{C}} \right) + \frac{\partial W}{{\partial I_{3} }}I_{3} \varvec{C}^{ - 1} } \right)\varvec{F}^{T} = \frac{2}{J}\left( {\varvec{F}\frac{\partial W}{{\partial I_{1} }}\varvec{IF}^{T} + \varvec{F}\frac{\partial W}{{\partial I_{2} }}\left( {I_{1} \varvec{I} - \varvec{C}} \right)\varvec{F}^{T} + \varvec{F}\frac{\partial W}{{\partial I_{3} }}I_{3} \varvec{C}^{ - 1} \varvec{F}^{T} } \right) = \frac{2}{J}\left( {\frac{\partial W}{{\partial I_{1} }}\varvec{FIF}^{T} + \frac{\partial W}{{\partial I_{2} }}\left( {I_{1} \varvec{I} - \varvec{C}} \right)\varvec{FF}^{T} + \frac{\partial W}{{\partial I_{3} }}I_{3} \varvec{C}^{ - 1} \varvec{FF}^{T} } \right) $$

(A.13)

Since, $\varvec{B} = \varvec{FF}^{T} ,\,\;\varvec{FF}^{ - 1} = \varvec{I}\;and\;\varvec{F}^{ - T} \varvec{F}^{T} = \varvec{I}$

Therefore

$$ \varvec{\sigma}= \frac{2}{J}\left( {\frac{\partial W}{{\partial I_{1} }}\varvec{B} + \frac{\partial W}{{\partial I_{2} }}\left( {I_{1} \varvec{B} - \varvec{BB}} \right) + \frac{\partial W}{{\partial I_{3} }}I_{3} \varvec{I}} \right) $$

(A.14)

Considering incompressibility of the material, $I_{3} = 1\;\;$ and $J = 1$

Cauchy stress at equilibrium condition can be written as

$$ \varvec{\sigma}= - p\varvec{I} + 2\left( {\frac{\partial W}{{\partial I_{1} }} + I_{1} \frac{\partial W}{{\partial I_{2} }}} \right)\varvec{B} - 2\frac{\partial W}{{\partial I_{2} }}\varvec{BB} $$

(A.15)

where, p is hydrostatic pressure and W is strain energy function in equilibrium state.

1.2 A.2 Gent Model

The strain energy density function for the Gent model (Gent, 1996) is

$$ W = - \frac{\mu }{2}J_{m} ln\left( {1 - \frac{{I_{1} - 3}}{{J_{m} }}} \right) $$

(A.16)

where $\mu$ is the shear modulus, J _m is the limiting value for $\left( {I_{1} - 3} \right)$, reflecting limiting chain extensibility.

Since $W = W\left( {I_{1} } \right)$, $\frac{\partial W}{{\partial I_{2} }} = 0$

Cauchy stress tensor shown in Eq. (A.15) will become

$$ \varvec{\sigma}= - p\varvec{I} + 2\frac{\partial W}{{\partial I_{1} }}\varvec{B} $$

(A.17)

Using Eq. (A.16) and (A.17) Cauchy stress for the Gent model is expressed as

$$ \varvec{\sigma}= - p\varvec{I} + \frac{{\mu J_{m} }}{{J_{m} - I_{1} + 3}}\varvec{B} $$

(A.18)

For uniaxial tensile extension deformation gradient

$$ \varvec{F} = \varvec{F}^{T} = \left[ {\begin{array}{*{20}c} \lambda & 0 & 0 \\ 0 & {\frac{1}{\sqrt \lambda }} & 0 \\ 0 & 0 & {\sqrt \lambda } \\ \end{array} } \right] $$

Left Cauchy-Green deformation tensor can be written as

$$ \varvec{B} = \varvec{FF}^{T} = \left[ {\begin{array}{*{20}c} {\lambda^{2} } & 0 & 0 \\ 0 & {\frac{1}{\lambda }} & 0 \\ 0 & 0 & {\frac{1}{\lambda }} \\ \end{array} } \right] $$

where elements of tensor B is as follows;

$$ B_{11} = \lambda^{2} ,B_{22} = B_{33} = \frac{1}{\lambda } $$

For the uniaxial case

$$ I_{1} = tr\varvec{B} = \lambda^{2} + \frac{2}{\lambda } $$

(A.19)

From Eq. (A.18)

$$ \sigma_{11} = - p1 + \frac{{\mu J_{m} }}{{J_{m} - I_{1} + 3}}B_{11} $$

(A.20)

$$ \sigma_{22} = - p1 + \frac{{\mu J_{m} }}{{J_{m} - I_{1} + 3}}B_{22} $$

(A.21)

As $(\sigma_{22} = 0)$

$$ p = \frac{{\mu J_{m} }}{{J_{m} - I_{1} + 3}}B_{22} $$

(A.22)

Using the Eqs. (A.20) and (A.22)

$$ \sigma_{11} = \frac{{\mu J_{m} }}{{J_{m} - I_{1} + 3}}(B_{11} - B_{22} ) $$

(A.23)

Hence

$$ \sigma_{11} = \frac{{\mu J_{m} }}{{J_{m} - I_{1} + 3}}\left( {\lambda^{2} - \frac{1}{\lambda }} \right) $$

(A.24)

Since, $\sigma_{eng} = \frac{1}{\lambda }\sigma_{11}$, Eq. (A.24) can be written in terms of engineering stress as

$$ \sigma_{eng} = \frac{{\mu J_{m} }}{{J_{m} - I_{1} + 3}}\left( {\lambda - \frac{1}{{\lambda^{2} }}} \right) $$

(A.25)