Computer modelling of crack growth in rubber-toughened polymers.

We present results from a computer modelling study of fracture propagation in a rubber-toughened polymer subject to external uniaxial tension. Using a simple lattice model comprising two bond types (one for rubber and one for polymer), simulations have been performed on both 2- and 3-dimensional systems. Each bond is characterised by two parameters : an elastic constant and a mechanical strength. All bonds behave in a purely elastic manner up to fracture; this limits the model's applicability to impact failure of brittle glassy polymers. We employ a stochastic breaking process based on Eyring's reaction rate theory, and allow the system to equilibrate fully after each bond breakage. It is found that the Conjugate Gradient method is the most efficient technique for calculating these equilibria. The model allows straightforward measurement of the total damage at failure, which is a crude approximation of the material toughness, and the stress distribution during the damage evolution.We show that, for a wide range of bond parameters, as the rubber loading is increased the damage goes through a sequence of behaviour : embrittlement; toughening; saturation; and weakening. This sequence is consistent with experimental data for rubber-toughened Poly (methyl methacrylate) (RT-PMMA). Our results show that the embrittlement and saturation stages are sensitive to the ratio of the two elastic constants but that they disappear as this ratio approaches unity. The toughening achieved is sensitive to both the concentration and the mechanical strength of the rubber bonds, the most damaged systems being those with large rubber mechanical strength at approximately 50% concentration. We also find that the damage is dependent on the spatial distribution of the rubber bonds, the greatest toughness being achieved using a homogeneous rubber distribution.We also find that the evolution of the system's failure goes through four regimes. In the first of these, the damage is evenly distributed throughout the system but showsa propensity for occuring adjacent to rubber bonds. Subsequently, short cracks develop between some of these weakly damaged sites. As more of these develop, a number of long rubber-bridged cracks appear. In the third regime, the stabilising effect of the bridging rubber bonds forces the damage to extend throughout the system, leading to branching and interconnecting of rubber-bridged cracks. Finally, the stress experienced by the crack-bridging rubber bonds becomes unsustainable and catastrophic failure occurs. Zones of material consistent with each of these four regimes can be observed in experimentally fractured RT-PMMA. The main toughening mechanisms in operation in our model are, therefore, crazing (which occurs mainly in the second regime) and rubber-bridging (which occurs in the third). The former is enhanced by using a lower rubber elastic constant, whereas the latter is prolonged by a large rubber mechanical strength.We also apply the model to simulations of 2D and 3D systems containing rubber macro-particles, i.e. large volumes of connected rubber bonds, and we present visual results of the damage and the stress field evolution in such systems. In these, the damage appears to develop in accordance with the four regimes described above. The stress maps of multi-particle 2D systems show the gradual stress transfer which takes place as cracks develop, from initially high-stressed inter-particle polymer regions to the crack tips and, then, to the rubber particles along the crack lines. These visual results also highlight three possible modes of particle-crack interaction : particle debonding, particle splitting and crack reflection.