Understanding decay functions and their contribution in modeling of thermal-induced aging of cross-linked polymers

Highlights

• Different decay functions has been proposed.

• Decay functions were compared in detail to propose the best decay function.

• Micro-mechanical approach was used to achieve constitutive model for aging of elastomers.

• The model can consider the effects of time and temperature.

• The model can consider complex properties such as Mullins effect.

Abstract

Different decay functions to describe the effects of homogeneous thermo-oxidative aging on the constitutive behavior of aged rubber-like materials for longer timescales is presented. Then, a suitable decay function has been introduced and implemented into a micro-mechanical model to study the effects of homogeneous thermo-oxidative aging on the quasi-static mechanical response of rubber-like materials and their inelastic responses such as Mullins effect and permanent set over time. The model describes the aging induced damage with respect to experimental studies on the process of chemical aging which suggests decomposition of original network and creation of a new one. Accordingly, in the course of aging, the strain energy of the polymer matrix is divided from two independent sources, (i) a decomposing original matrix (ii) a newly formed matrix. The model is validated with respect to a comprehensive set of own experimental data as well as data available in literature that were designed to capture thermal induced aging effects of constitutive behavior of polymers. Besides accuracy, the model is relatively simple and easy to fit.

Introduction

Rubber-like materials usage in sensitive applications for damping vibration or to sustain large deformations are increasing. Thus, they have a vital role in many industries such as transportation, automotive, and energy industries. Moreover, in view of their excellent thermal and chemical resistance, use of rubber-like materials is prevalent in extreme applications. However, long time exposure to extreme environmental conditions along with mechanical damages can lead to severe degradation of mechanical performance of rubber-like materials over time. A process which is often referred to as aging.

In regards to the environmental factor, aging has been classified into different types based on their source factor which can be temperature, UV radiation (sunlight), and moisture. Accordingly, major types of aging are thermo-oxidative aging, photo-oxidative aging, and hygrothermal aging, respectively (see Fig. 1. Among these, thermo-oxidative aging is the most prevalent aging condition which is the focus of this paper.

During thermo-oxidative aging, the polymer matrix changes due to chemical reactions of polymer backbone with oxygen [1]. Two types of chemical reactions occur; (1) polymer scission and (2) new cross-linkage which often leads to material embrittlement [2]. Both reactions highly affect the length distribution of polymer chains inside the matrix (see Fig. 2) [3]. Besides, aging also changes the morphology of the matrix. While the original cross-links are facilitated by sulphur links, the cross-links induced by aging have peroxide nature. Therefore, the behavior of aged samples are different than the virgin material. The relative rate of polymer chain scissions and cross-link formation determines whether the material should become ductile or brittle, softer or harder. It is important to understand, even if these rates are equal hypothetically, the aged material would not have the same toughness as of virgin material due to loss of original cross-links.

Elastomeric parts in sensitive components may often be out of reach, where visual inspection is not possible. Hence, exact understanding of their service life reduction due to thermal aging is of utmost importance to prevent catastrophic failure. Such unexpected failures may have significant financial and environmental costs, such as challenger space shuttle disaster. Considering the significant progression of thermal aging at elevated temperatures and its effect on decaying mechanical performance of rubber-like materials, different experimental and theoretical approaches were developed to predict the aging, namely.

In experimental approaches, accelerated aging experiments at elevated temperatures are most popular method to estimate performance decay in service temperatures. Elevated temperatures causes the test to be done in a reasonable time, while emulating service condition may need years to make the effects of aging visible. Then, lifetime estimates at service temperature are often derived based on extrapolating accelerated aging results using Arrhenius relationship [4]. Although Arrhenius relationship proved successful in predicting lifetime of elastomers in many cases [5,6], Gillen et al. [7] raised serious problems regarding using it as a general guideline, since linear Arrhenius relationship cannot describe the observed aging data in many cases due to many factors. In elevated temperature, diffusion limited oxidation (DLO) occurs when the rate of oxygen consumption is higher that oxygen diffusion in the inner layer of matrix. DLO significantly reduces the progress rate of the aging, a feature which cannot be described by Arrhenius function [8]. Second, Arrhenius function is relevant as long as only one mechanism is responsible for the decay. There are cases with different degradation processes in high and low temperatures resulting in variation of activation energy by temperature [[9], [10], [11]]. Thus, predictions based on non-Arrhenius relation is getting higher interest recently.

There are extensive experimental studies exclusively studying the effects of thermo-oxidative aging on the rubber-like materials using accelerated testing. Such works have been thoroughly reviewed in Refs. [4,12] and also several potential problems associated with accelerated testing methods were studied and reported [5,11,13,14]. An experimental study on the effect of aging on viscoelastic relaxation, oxidative scission, and thermal expansion of elastomers has been done by Shaw et al. [15]. Rabanizada et al. [16] examined the aging behavior of natural rubber in different media such as air, sea water, distilled water, and freshwater. They further studied the changes in dynamic behavior by means of dynamic mechanical analysis (DMA) tests. Similarly, Pazur et al. [17] studied the effect of thermal aging on bulk properties of peroxide-cured nitrile butadiene rubber (NBR). Their studies showed that understanding the distribution of the cross-link is necessary for comprehending the decay of bulk properties.

In comparison to experimental approaches, theoretical efforts are limited and can be mainly categorized into two types of phenomenological and micro-mechanical approaches. Phenomenological approaches are mostly based on a thermodynamic framework to describe the degradation of elastomers along with a phenomenological model to characterize mechanical behavior [18,19]. Ha-Anh and Vu-Khanh [20] used an Arrhenius relationship along with Mooney-Rivlin model to predict hyperelastic behavior of aged polychloprene. Lion and Johlitz [21] proposed a three-dimensional phenomenological model to represent the chemical behavior of rubber. They split the Helmholtz free energy to three parts (volumetric material behavior, temperature-dependent hyper-elasticity, and a functional of deformation history) and added it to decomposition of deformation gradient. Their model were able to simulate the experimental data with excellent precision. Furthermore, Johlitz [22] presented a phenomenological model which was rheologically motivated. The model were able to predict both physical and chemical aging behavior. Later on, he presented a model based on finite strain theory and conducted experiments on aged automobile bearing to validate his simulation [23]. Dipple et al. [24] used a coupled chemo-mechanical modelling approach to analyze the aging behavior of an adhesive. They use finite element to simulate geometry dependency of aging between substrate and adhesive. In addition, Naumann and Ihlemann [25] simulated the effects of thermo-oxidative aging on mechanical behavior of rubbers by developing a new model. Their model were based on solving a coupled chemo-mechanical problem. To overcome the time scale difference in chemical and mechanical processes, they proposed an staggered solution algorithm. In their algorithm, the diffusion-reaction problem was solved in large time scale while the mechanical problem was solved intermittently for a defined load spectrum. They also proposed a dynamic network model instead of dual network model a year later [26]. Recently, Musil et al. [27] introduced a continuum mechanical approach to model chemical aging of NBR with regard to its viscoelasticity. Herzig et al. [28] modified reaction-diffusion equation to model heterogeneous aging due to DLO effect. His work showed the high dependency of adsorption, diffusion and reaction mechanisms to temperature. Similarly, Konica and Sain [29] proposed a model in a finite element framework that can analyze numerically the coupled diffusion-reaction and mechanical behavior of polymers undergoing oxidation.

Although phenomenological models can predict the behavior accurately, they can not provide an insight to the effect of material parameters on the behavior. This is due to the fact that their parameters usually comes from mathematical point of view instead of physical one. On the other hand, micro-mechanical models are based on statistical mechanics of polymer structure. Therefore, micro-mechanical model parameters do have physical meaning and come from material properties. Subsequently, a growing interest in micro-mechanical models is developing recently. Schlomka et al. [30] used the staggered solution algorithm presented in Ref. [26] along with a hyperelastic mechanical model of Neo-Hookean type to predict component stiffness. Mohammadi et al. [[31], [32], [33]] used micro-mechanical model based on network evolution and dual network hypothesis to predict the constitutive behavior of elastomers due to thermo-oxidative aging. Their model were able to model inelastic behavior such as Mullins effect and complex loading-unloading profiles. Despite the good agreement of the model with experimental data, the model was based on an empirical decay function with no physical meaning. By a similar approach based on network evolution, Bahrololoumi et al. [34,35] proposed a micro-mechanical model to predict the behavior of polymers going through hydrolytic aging. Recently, Beurle et al. [36] developed a micro-mechanical model for homogeneous aging of elastomers, by classifying the polymer chains into two categories of active and inactive chains. This splitting leaded to a set of coupled, non-linear ordinary differential equations that described the network degradation. They calculated the shear modulus of the material by solving these equations. Moreover, they implemented their concept into the matrix using the micro-sphere concept of Miehe et al. [37].

The main objective of this work is to remedy the empirical decay function of our previous work [31] by offering a suitable kinetic decay function. The proposed decay function is a combination of Arrhenius functions with different decay rates. Moreover, a new distribution function is used to better define the chain length distribution during aging. The proposed model will be based on the two concepts of network decomposition [38] and network evolution [39,40]. We assumed that during aging, the polymer matrix can be divided into two parallel networks, original and newly created aged networks. The newly created aged network is assumed to have shorter chains with higher cross-link density. The model is validated by own designed experimental data as well as continuous relaxation and intermittent data presented in Ref. [23].

The paper is outlined as follows. Statistical mechanics of polymers is presented in section 2. Dual network hypothesis, decay functions, and time-temperature superposition concepts are discussed in section 3. The concept of the model in 1-D dimension is discussed in section 4. Next, in sections 5 Network evolution, 6 Transition to macro-model, energy of a chain and network evolution concepts are presented respectively. In section 7, transition from micro to macro domain is discussed. Parameter sensitivity analysis and validations are represented in sections 7 Parameter sensitivity analysis, 8 Validation respectively. Finally, the conclusion is provided in section 10.