A phenomenological model for healing and hysteresis in rubber-like materials P. D’Ambrosio, D. De Tommasi, D. Ferri, G. Puglisi * Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy Received 16 November 2007; accepted 3 December 2007 Available online 28 January 2008 Abstract We propose a predictive model for the dissipative behaviour of rubber-like materials. In the spirit of De Tommasi et al. [D. De Tommasi, G. Puglisi, G. Saccomandi, A micromechanics based model for the Mullins effect, J. Rheol. 50 (2006) 495– 512], we assume that at the scale of the polymeric network the material is constituted by a distribution of links with variable activation and fracture thresholds. By considering the recross-linking effect due to unloading we obtain a three-dimensional, non-linear damage model that describes the rate-independent hysteretic behaviour observed in rubber-like solids. The fea- sibility of the model in treating complex non-homogeneous deformation histories is shown through a numerical application. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Rubber-like materials; Healing; Hysteresis; Mullins effect; Helical shear 1. Introduction The growing attention for mechanical models of rubber-like materials is due to both their wide use in tech- nological applications and their theoretical interest. The typical softening behaviour (usually known as Mul- lins effect [11]) and the complex history dependance of these materials, coupled with the strong non-linearity of their constitutive laws, make the analysis of their mechanical behaviour a challenging task. Particularly impor- tant for many applications (e.g. vehicle tyres, engine mounts, shock absorbers, seismic isolation devices, damp- ing devices [9]) is the possibility of representing the energy dissipation effects. To describe the hysteretic and softening behaviour observed during uniaxial cyclic experiments, we report in Fig. 1 the typical stress–strain diagram of a rubber material (reproduced from [2]). In the figure the curve O– A is followed by the virgin material under monotonically increasing load, and it is usually referred as primary loading path. If the material is unloaded (path A–A 0 –B), then it follows a curve that is always below the pri- mary loading curve, manifesting the typical softening effect. If the material is reloaded, then it follows a third curve (path B–B 0 –C), and describes an internal hysteresis loop. Observe also that the system follows almost the same loop (path C–A 0 –B–B 0 –C) if subjected to the same loading cycle. Finally, the experiments show that if the 0020-7225/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2007.12.002 * Corresponding author. Tel.: +39 080 596 3744; fax: +39 080 596 3719. E-mail address: g.puglisi@poliba.it (G. Puglisi). Available online at www.sciencedirect.com International Journal of Engineering Science 46 (2008) 293–305 www.elsevier.com/locate/ijengsci