This document summarizes phenomenological modeling of viscoelastic polymers. It discusses how polymers respond mechanically to electrical input via electrostriction and how their viscosity leads to time-dependent effects under large electric fields. The modeling approach uses a multiplicative split of the deformation gradient and an additive split of the free energy contributions. Governing equations are based on Maxwell's equations and a constitutive framework describes the material using energy and free energy functions. Evolution equations for viscosity driving stresses are formulated using a Mandel-type stress tensor. The models are applied to polyurethane elastomers, with good fits found using two viscosity elements and a power law evolution law.

1. Phenomenological modeling of viscous
electrostrictive polymers
L. Harish (AM13D025)
Group meeting

2.  Polymers which respond mechanically to electrical input are
termed as electro active polymers(EAPs)
 Large electric field leads to couloumb forces and it termed as
maxwell effect
Introduction

3.  Electrostatic itself is not enough but also to include the viscosity
of the polymer to consider time dependent effects.
 Large electric field leads to couloumb forces and it termed as
maxwell effect.
Introduction

4.  It is an electro-viscoelastic coupled problem including
electrostriction and time dependence present in PUelastomers.
 It is assumed that the viscosity is related to the deformation of the
body but not directly to the electromagnetic field quantities.
 Electric loading will make the body deform and there by induces
the viscous deformation.
 Viscoelastic modeling will be based on a multiplicative split of the
deformation. And additive split of the longterm and non-
equilibrium viscous contributions of free energy.
Introduction

5.  Deformation gradient is multiplicatively split into its volumetric and isochoric
parts
Deformation Tensors

6.  Consider electro static case
Electromagnetic field quantities

7.  The electric field and displacements are governed by Maxwell’s equations.
Governing equations

8.  Material can be described by an energy function and free energy function can
be split into separate contributions.
Constitutive framework

9. Constitutive models: Energy expressions
 Electric displacements:

10. Total stresses

11. Viscosity driving stresses
Mandel type referential stress tensor:
Referential viscous piola Kirchhoff type stresses:

12. Viscosity driving stresses

13. Evolution Equations
 Mandel type stresses can conveniently be used to formulate a
thermodynamically consistent model.
 The format of evolution law considered here resembles an approach commonly
used in time dependent plasticity theories.
 Considered function or rather potential:

14. Evolution Equations

15. Application to PU elastomer
 For viscous strains the power law type evolution law as well as the Bonet
model are used.
 A good fit to the experimental data can be found by using two viscosity
elements and k=1

16. Application to PU elastomer
 For viscous strains the power law type evolution law as well as the Bonet
model are used.

17. Application to PU elastomer

18. Application to PU elastomer

19. Application to PU elastomer

20. Application to PU elastomer