Continuum-micromechanical modeling of distributed crazing in rubber-toughened polymers
Introduction
The ductility and toughness of various polymeric materials – thermoplastics as well as thermosets – can be substantially improved by mixing-in fine dispersed rubber particles; e.g. (Bucknall, 1977). A key effect of the rubber particles is that they enable and initiate inelastic, hence energy absorbing, microscale deformation mechanisms distributed over large regions of the material. The micromechanisms involved in rubber-toughening are shear yielding as well as crazing in the matrix phase, often preceded by cavitation of the rubber particles. The interrelation between these mechanisms and their efficiency (or predominance) in toughening strongly depends on the material at hand. This dependence not only includes the matrix material, the rubber particle size and volume fraction, but also the overall loading conditions (loading rate, temperature and stress triaxiality); e.g. (Beahan et al., 1976, Beguelin et al., 1999, Bernal et al., 1995, Donald and Kramer, 1982, Han et al., 2001, Jar et al., 2002, Steenbrink et al., 1997). Moreover, the “type” (e.g. internal structure) of the rubber particles which may vary with the manufacturing process is known to be of some influence, e.g. (Bucknall, 1977, Giaconi et al., 1998).
Of particular interest in the present work is the role of crazing, i.e. the formation of localized zones in which the bulk polymer is drawn into thin fibrils; e.g. (Kramer et al., 1991). Crazing in homogeneous polymers clearly is a precursor of brittle failure under tensile loading. In rubber-toughened polymers, however, crazes are trapped in the ligament between the dispersed rubber particles and collectively may give rise to macroscopically large inelastic strains prior to failure, e.g. (Ishikawa, 1995). The perhaps most prominent example in this regard is high-impact-polystyrene (HIPS) where overall inelastic deformation is exclusively due to distributed crazing. In rubber-toughened thermoplastics with a less brittle matrix such as acrylonitrile-butadiene-styrene (ABS), some “competition” between shear yielding and crazing is observed. In fact, within the large family of ABS materials, experiments have revealed a wealth of different phenomena – from shear yielding induced by cavitating soft rubber particles all the way to distributed crazing around hard ’salami’ particles, and almost every combination in between (depending on constitution, particle size, manufacturing process and loading conditions). Moreover, microstructural studies in Steenbrink et al. (1997) and Steenbrink (1998) confirm that both shear yielding and distributed crazing may occur in the same ABS specimen depending on the distance to the fracture surface. The present work focuses on the role of spatially distributed crazing as shown in Fig. 1.
While a large number of experimental studies have addressed the complex interplay between microstructure, micromechanisms and resulting overall performance (e.g. fracture toughness), appropriate macroscopic material models for rubber-toughened polymers – and in particular those based on the underlying physical mechanisms – are rare, even to date. Before an all-embracing micromechanical description is feasible, theoretical models first have to deliver a deeper understanding of the individual mechanisms (thereby necessarily making simplifying assumptions). The majority of modeling approaches so far has focused on matrix shear yielding in conjunction with void growth from cavitated rubber particles, e.g. (Danielsson et al., 2007, Lazzeri and Bucknall, 1993, Smit et al., 2000, Steenbrink and Van der Giessen, 1999, Zaïri et al., 2008, Zaïri et al., 2011). Numerical simulations carried out in Pijnenburg et al. (2005), however, suggested that such a modeling approach is unable to reproduce the characteristic shape of the plastic zone at a notch in ABS tensile specimens as it tends to overestimate localization of plastic deformation. It was concluded that the effect of distributed crazing cannot be neglected in the overall inelastic deformation behavior of ABS.
Owing to their localized crack-like appearance, individual crazes in neat glassy polymers have successfully been modeled as cohesive surfaces, e.g. in Estevez and Van der Giessen, 2005, Estevez et al., 2005 and Tijssens et al. (2000). Utilizing such a description, the competition between crazing and matrix shear yielding in the vicinity of a single void, representing a cavitated rubber particle in ABS, was investigated in Seelig and Van der Giessen (2009). The formation of multiple crazes from a rubbery particle in HIPS was modeled numerically using special continuum elements in Sharma and Socrate (2009) and Socrate et al. (2001). To avoid the necessity of having to trace individual crazes, one may adopt a continuum description of distributed crazing, as proposed for neat glassy polymers (i.e. without particles) in Gearing and Anand (2004). This model incorporates the kinematics of craze widening by taking the average spacing between crazes as a characteristic length scale.
The objective of the present work is to complement earlier modeling approaches, e.g. (Pijnenburg et al., 2005, Steenbrink and Van der Giessen, 1999), which considered inelastic deformation due to shear yielding alone by considering here the opposite extreme, i.e. the effect of distributed crazing only. This means that we will ignore shear yielding and make the simplifying key assumptions that, firstly, crazes span the ligament between all the uniformly dispersed rubber particles and, secondly, that (viscoplastic) opening of the crazes is the only source of inelastic deformation. The kinematics of craze opening in the direction of maximum principal tensile stress considered in Gearing and Anand (2004) will be extended to account also for overall shearing as it occurs for instance in the wake of an advancing crack tip. The constitutive model set up in Sect. 2.1 is endowed with scaling relations with respect to microstructural parameters via simple micromechanical considerations in Sect. 2.2. The model is calibrated in Sect. 3 from tensile tests which we performed on a commercial ABS material. In Sect. 4 results of numerical simulations of crack propagation in a notched tensile specimen are presented and analyzed with regard to the influence of the rubber content on the overall fracture toughness. Comparison of model predictions and experimental data will reveal strengths and shortcomings of the present distributed crazing model and provides additional insight into the collective effects of shear yielding and crazing in the overall behavior of ABS.
Throughout the paper, the symbolic bold face notation of vectors a and tensors A is used as well as the index notation ai , Aij with respect to Cartesian base vectors ei (i = 1,2,3). Single and double contraction of indices is represented by the symbols “ ⋅ ” and “: ”, respectively, and the standard summation convention is employed, e.g., a·b = aibi, A·a = Aijajei, A:B = AijBij. The dyadic product ⊗ of two vectors has components (a⊗b)ij = aibj.
Section snippets
Constitutive modeling
This section is concerned with the formulation of a constitutive model for the overall deformation and failure behavior of rubber-toughened materials such as ABS with focus on the effect of distributed crazing. Overall inelastic deformation in this theory results from the specific kinematics of cohesive crack-like opening of crazes, while the volume fraction and size of rubber particles are explicitly accounted for via micromechanical considerations.
Material parameter identification from tensile tests
The material model developed in Sect. 2 addresses the situation of a microstructure with a fine dispersion of rubber particles which cavitate at low stress and give rise to the formation of craze zones (typically not more than one per particle) in the surrounding glassy matrix (Fig. 2b,c). This is found in some ABS materials (see Fig. 1) and rubber-toughened PMMA, e.g. (Beahan et al., 1976, Beguelin et al., 1999, Bucknall, 1977, Donald and Kramer, 1982, Jar et al., 2002, Steenbrink et al., 1997
Fracture behavior of SENT specimen
In order to assess in how far the developed constitutive model is capable of describing the fracture behavior of ABS materials, a single-edge-notch-tension (SENT) specimen, see Fig. 10, is considered in the following. Though not appropriate for the determination of “true” fracture properties (e.g. crack resistance curves) such a specimen type is occasionally utilized in polymer testing, e.g. (Steenbrink et al., 1997, Steenbrink et al., 1998). Some measure of the material's fracture toughness,
Discussion and conclusions
Toughening in ABS materials is mediated by two dissipative micromechanisms, i.e. shear yielding and crazing. In order to analyze the sole effect of spatially distributed crazing a homogenized material model was developed in the framework of large inelastic strains that deliberately ignores shear yielding and assumes crazing to be the only source of inelasticity. In this way the model complements earlier studies, e.g. (Danielsson et al., 2007, Pijnenburg et al., 2005, Smit et al., 2000,
Acknowledgments
Financial support of this work by the German Science Foundation (DFG) under grant no. SE 872/5 is gratefully acknowledged. We would also like to thank the German Academic Exchange Service (DAAD) for financial support of a stay of M.H. at SIMLab/Trondheim where part of the experiments was performed. The authors gratefully acknowledge also the assistance of the laboratory staff at SIMLab and KIT.
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