An approach to model mechanical damage in particulate composites with viscoelastic matrix

Abstract

Particulate composites with polymer matrix are used in engineering applications, such as solid propellants, industrial rubbers and road pavements. The durability of such materials critically depend on the characterization of mechanical property degradation. Mechanical damage studies of these composites thus become important. A phenomenological model describing the mechanical damage in polymer matrix is developed in the present work, where the polymer is modelled as a viscoelastic (VE) solid and the damage is characterized by the progressive degradation of storage modulus. This VE model, with two term Prony series approximation of the relaxation modulus, is then coupled with isotropic damage formulation. Two approaches of isotropic damage evolution, viz. explicit and implicit, are implemented accounting the irreversible thermodynamic considerations. The implicit isotropic damage evolution is derived by a dissipation potential function based on the effective elastic stored energy density. An exponential damage evolution, obtained from experiments, is adapted in explicit approach. The analytical VE-damage framework is numerically implemented as stress integration algorithm, and the corresponding simulation results are illustrated. It is seen from the formulation and results that, the damage implicitly affects the relaxation time of the polymer. The stress hardening behaviour occuring immediately after the damage initiation is captured, and the subsequent softening is also demonstrated.

Appendix

Calculation of stress with N-term Prony series

The formulae needed to calculate stress in case of tensile relaxation modulus aproximated by N-term Prony series is given below :

Current stress $\sigma _{n+1}$ is given by

$$\begin{aligned} \sigma _{n+1} = E_{\infty } \epsilon _{n+1} + \sum _{i=1}^{N} h^{(i)}_{n+1} \end{aligned}$$

where

$$\begin{aligned} h^{(i)}_{n+1} = e^{- \Delta t_{n} / \tau _{i}} h^{(i)}_{n} + E_{i} e^{- \Delta t_{n} / 2\tau _{i}} \Delta \epsilon _{n} \end{aligned}$$

The (linear) elastic strain energy density $W^{o}(\epsilon _{n+1})$ can be approximated by

$$\begin{aligned} W^{o}(\epsilon _{n+1}) = \frac{E_{o} \epsilon ^{2}_{n+1}}{2} \end{aligned}$$

where the instantaneous elastic modulus $E_{o}$ is

$$\begin{aligned} E_{o} = E_{\infty } + \sum _{i=1}^{N} E_{i}, \end{aligned}$$