Energy based fracture initiation criterion for strain-crystallizing rubber-like materials with pre-existing cracks

Abstract

Fracture prediction is indispensable for polymers, like rubbers, which have a broad range of applications mainly due to their high extensibility. The phenomenon known as strain-induced crystallization further contributes to the fracture toughness of certain rubbers. In this study, a criterion based on internal bond energy, incorporating the effects of crystallization, is proposed to predict fracture initiation in rubber-like materials with pre-existing cracks. First, a multi-scale mechanical model is developed for characterizing the behavior of rubber when subjected to both uniaxial and biaxial deformation states. At the microscale, both the amorphous and crystalline chain segments are modeled as elastic in order to consider the energy contribution by the molecular bond distortions. This internal energy is considered along with the entropic and crystalline free energy for each chain. In the chain model, the effects of loading condition and the relative orientation of a chain on its crystallinity are taken into account. At the macroscopic scale, an existing crystallinity distribution function is adapted and a mixed finite element formulation with an augmented Lagrangian multiplier is utilized to impose the incompressibility constraint. A non-affine maximal advance path constraint based homogenization model is utilized for bridging the two scales. Its potential to account for anisotropy in the stretched network compels the model to be preferable due to its physical significance, for the purpose of fracture modeling. The rigidity of the crystallites is accounted for by proposing a crystallite distortion energy in addition to the critical bond dissociation energy, for fracture initiation to occur. The model is validated by comparison with existing experimental results for both crystallizing and non-crystallizing rubbers. In addition to its potential to predict the material behavior when subjected to uniaxial and biaxial loading, the capability of the model to quantitatively estimate the effect of crystallization on fracture initiation is also verified.

Introduction

The high stretchability of many polymers prior to their fracture make them a perfect choice in applications such as self-actuators, epidermal electronics and implantable sensors. This has led to a surge in experimental and computational works focusing on the study of such polymers (Li et al., 2020; Wang et al., 2017; Wang and Chester, 2018). Additionally, desirable properties like high toughness, low cost and lower weight result in rubber-based materials having a wide range of industrial applications like tires, seals, hoses and airbags (Loew et al., 2019; Hashemi et al., 2020; Mittal et al., 2011). Thus, predicting the fracture initiation point is paramount to design against fracture or fatigue failures.

In general, the underlying polymer network of the rubber-like materials, whose crosslinks and chain entanglements provide elasticity, are mainly responsible for their stretchability. Interestingly, natural rubber (NR) and certain synthetic rubbers like polyisoprene rubber (IR) exhibit much higher strengths than other similar polymers like styrene–butadiene rubber (SBR) (Clamroth and Kempermann, 1986). In many attempts that have been made to unravel this mystery, the phenomenon called strain-induced crystallization (SIC) has been attributed to be the inducer of the higher fracture resistance (Lake and Thomas, 1967; Mars and Fatemi, 2004; Toki, 2014; Tee et al., 2018).

SIC involves the phase transformation from an amorphous solid to a partially crystalline solid upon the application of a large amount of stretch in certain polymers having conducive chemical composition for the phenomenon to take place. Ever since Katz (1925) discovered the phenomenon of SIC using the state-of-the-art X-ray diffraction method, there have been many experiments studying the effects of crystallization on the material properties (Gehman and Field, 1939, Candau et al., 2014, Toki et al., 2003, Tosaka et al., 2004, Brüning et al., 2012, Chen et al., 2019). The toughening effect of the phenomenon on the fracture properties of crystallizing rubbers has also been experimentally studied (Rublon et al., 2014, Trabelsi et al., 2002, Zhang et al., 2009, Gherib et al., 2010, Brüning et al., 2013a, Brüning et al., 2013b).

One of the earliest theoretical formulations of SIC was done by Flory (1947) wherein Gaussian statistics was used, and the equilibrium condition of the process was considered. Since then, many studies have improved on this by considering non-Gaussian statistics (Smith Jr, 1976), non-equilibrium kinematics (Gent, 1954) and multi-scale bridging (Kroon, 2010; Mistry and Govindjee, 2014; Guilie et al., 2015; Rastak and Linder, 2018; Khiêm and Itskov, 2018; Gros et al., 2019). However, a majority of these studies restrict their attention to describe the SIC phenomenon for the uniaxial tensile loading case. A primary reason for this is the lack of experimental studies focusing on material behavior when subjected to more elaborate deformations. Recently, Chen et al. (2019) experimentally observed a frustration in SIC when a specimen made of NR was subjected to biaxial tension. Thus, there is a need for a more comprehensive model encompassing such experimental findings for different states of deformation.

Though there are copious models for describing the phenomenon of SIC, there is a scarcity of models predicting its effect on the fracture properties. The main reason for this is that all of these models consider the classical representation of the polymer chains, wherein they are composed of freely rotating rigid segments. Thus, these studies concentrate mainly on the free energy due to entropy in the chain and that due to crystallization while neglecting the change in internal energy due to stretching of the constitutive atomic bonds. However, Lake and Thomas (1967) argued that the polymer chain rupture occurs due to the breaking of the molecular bonds and hence is dominated by the internal energy.

Even though good multi-scale theories are developed at both the microscopic and macroscopic levels, their effectiveness highly depends on the link that bridges the scales. The early network theories like the 3-chain model (James and Guth, 1943; Wang and Guth, 1952) assumed affine deformation, which considered the deformation mapping to be the same for both scales. However, this was observed to over-constrain the system, by hindering the process of redistribution of stress by chain reorientation for attaining a lower energy state. This has been overcome by the concept of non-affine deformation which allows for the chain reorientation while satisfying the kinematic compatibility conditions at the same time. The 8-chain model by Arruda and Boyce (1993) considers an affine deformation of representative chains along the diagonals of a unit cubic cell, in order to derive an effective non-affine network response. The non-affine microsphere model by Miehe et al. (2004) adopts a continuous space of chain orientations in a unit sphere and proposes a constraint for polymer network deformation. However, both the models assume isotropic behavior resulting in an equal stretch of deformed polymer chains in all directions. In contrast, another network homogenization model called the Maximal Advance Path Constraint (MAPC) developed by Tkachuk and Linder (2012) has been proven to account for anisotropy during the study of SIC in Rastak and Linder (2018). As chains with larger stretches can rupture and trigger a failure of the network, anisotropic models like MAPC are better suited for fracture applications due to their capability of singling out such chains. However, so far, there is a lack of studies utilizing this theory for modeling fracture in polymers.

The prediction of fracture initiation and complete rupture in rubber-like materials has been mostly phenomenological, using just their macroscale properties. Similar to the Griffith (1921) criterion, Rivlin and Thomas (1953) utilized a characteristic tearing energy based criterion to predict incipient and catastrophic tearing of natural rubber test specimens. This theory was extended from single edge notches to double edge ones using the J-integral by Hocine et al. (2002). In other studies, Kawabata (1973) proposed a criterion for fracture during biaxial stress states determined by a critical stretch, and this was extended to different biaxiality loading ratios by Hamdi et al. (2006). Recently, Kumar et al., 2018, Kumar and Lopez-Pamies, 2020 have developed a phase-transition model for elastomers considering fracture nucleation in regions of large hydrostatic stress concentrations and crack propagation consistent with Griffith’s theory. Although Griffith’s theory works well for specimen with macroscopic flaws, Chen et al. (2017) experimentally demonstrated that its underpinning assumption of flaw-sensitive behavior does not hold good at very small length scales, which range from nanometers to centimeters depending on the material. In addition, fracture models accounting for the micro-mechanical aspects of the polymer material are advantageous for augmenting the effects of complex microscale phenomenon like SIC.

In a landmark fracture study using the microscopic aspects of polymer networks, Lake and Thomas (1967) followed a critical internal energy approach for predicting fracture initiation in polymer networks. Mao et al. (2017) built upon this theory by bridging the micro and macro scales using the eight-chain network model and postulated a fracture initiation criterion for elastomers with flaws, based on molecular bond disruption energy. Unlike other phase-field fracture models for rubbers like Miehe et al. (2004) and Loew et al. (2019) which utilize the macroscopic critical energy release rate theory, Talamini et al. (2018) developed a fracture model employing the microscopic critical internal energy formulation. However, there have been no studies explicitly predicting the crack initiation or propagation through crystallizing rubbers.

Building models to predict crack initiation in rubber-like polymers is paramount due to their wide scope of applications. In particular, crystallizable rubbers have high toughness and hence it is vital to study the effect of SIC on their fracture resistance. Motivated by this, we have postulated an energy-based criterion incorporating the effects of crystallization to predict fracture initiation in rubber-like polymers with pre-existing cracks. The mechanics-based multi-scale model for characterizing the polymer behavior, which is coupled with the criterion, is applicable to even biaxial loading cases. Modeling the crystallized chain segments as elastic by considering the internal energy contribution due to molecular distortions, taking into account the effects of loading condition and the relative orientations of chains on their crystallinity, utilization of an anisotropic network model for fracture prediction, and accounting for the effects of crystallization for predicting fracture initiation, are some of the primary features of this model. Although viscous effects have been observed in rubber-like materials, many studies (Hamed and Park, 1999, Gherib et al., 2010, Hamed, 2005, Zhou et al., 2014, Mars and Fatemi, 2004) highlight the dominant effect of SIC on the fracture properties of crystallizing rubbers. Hence, this study mainly focuses on quantifying the effect of crystallization on their fracture initiation while neglecting viscous effects.

The outline of the paper is as follows. In Section 2, the multi-scale micro–macro approach is presented for modeling the behavior of rubbers. The chains and their segments are modeled at the microscopic level in Section 2.1 and Section 2.2. While the deformation in both the micro- and macroscales are linked together using the MAPC network homogenization model in Section 2.3, the crystallinity variables are linked in Section 2.4. In Section 3, the fracture initiation criterion for non-crystallizing rubbers is first postulated in Section 3.1 and then extended to include crystallizing rubbers in Section 3.2. The numerical implementation of the algorithm using a mixed finite element formulation is explained in Section 4. Experimental results are used to validate the model in Section 5. Finally, after stating the major conclusions from the study in Section 6, some advanced concepts, derivations and additional results are presented in the appendices.