**Phase-Field damage modeling of rubbery polymers**

Rubber products like seals, hoses and tires are widely used in industry. In order to reduce the financial and time constraints of manufacturing physical prototypes, virtual prototypes can be constructed instead. Virtual prototypes, however, require adequate numerical simulations tools to describe the mechanical responses. Most currently used tools in the industry are limited in their ability to predict fracture.

Miehe et al. [1] published a first phase-field damage model for rubbers. Instead of considering cracks as a sharp discontinuity, phase-field damage models consider cracks in a continuous manner (see *Г*_{l} in Figure 1 b). This entails a damage zone that is governed by a length scale parameter *l _{0}* . From gradient-enhanced damage theory it is known that this parameter must be interpreted as material parameter depending on the microstructure of the underlying material. [2] The main advantage of this continuous representation of fracture is its capability to treat complex crack propagation, branching and coalescence without remeshing or a prior knowledge the crack path. An example explicitly showing the coalescence of many cracks can be found in [3], in which a stochastic analysis of a carbon black filled rubber composite is performed. Phase-field models intrinsically entail mixed methods, because the displacements

*u*, as well as the damage variable

*d*, need to be computed (see Figure 1).

For the time being, we focus on a pre-notched tensile specimen as presented in Figure 2a. A typical crack path predicted by the computations is shown in Figure 2b.

Future work will focus on the extension of the model to incorporate rate- and temperature dependency, fatigue damage as well as the application to industrial relevant examples.

References:

[1] Miehe, C.; Schänzel, L.-M.; *Journal of the Mechanics and Physics of Solids*, 2014, 65, 93-113

[2] Geers, M. G. D.; de Borst, R.; Brekelmans, W. A. M.; Peerlings, R. H. J.; *International Journal of Solids and Structures*, 1999, 36, 2557-2583

[3] Wu, J.; McAuliffe, C.; Weisman, H.; Deodatis, G.; *Computer Methods in Applied Mechanics and Engineering*, 2016, 312, 596-634