A non-monotonous damage function to characterize stress-softening effects with permanent set during inflation and deflation of rubber balloons A. Elías-Zúñiga * , C.A. Rodríguez Tecnológico de Monterrey, Campus Monterrey, Mechanical Engineering Department, Aulas IV-441, Av. Eugenio Garza Sada No. 2501 Sur, Col. Tecnológico C.P. 64849, Monterrey, NL, Mexico article info Article history: Received 28 December 2008 Received in revised form 8 May 2010 Accepted 16 June 2010 Keywords: Stress softening effect Residual strains Rubber elasticity Mullins effect Pseudo-elasticity theory Non-monotonous damage function abstract A non-monotonous stress-softening phenomenological model is applied to study the Mul- lins effect with residual strains to characterize the inflation and deflation of rubber balloons. It is shown that analytical predictions based on our proposed non-monotonous softening function and the modified stress-softening non-Gaussian average-stretch full-network constitutive equation that accounts for residual strains are consistent with experimental data. Also, we use the constitutive equation for equibiaxial extension to predict stress- softening behavior in a kinematically equivalent simple compression deformation state. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction The Mullins effect in equibiaxial extension has been studied both experimentally and analytically [1,2] in relation to bal- loon inflation and uniaxial compression. The study of this effect in equibiaxial deformation is greatly important in balloon inflation procedures encountered in biological tissue and biomedical applications, several of which are described recently by Guo [3]. Johnson and Beatty in [1] and Beatty and Krishnaswamy in [2] have investigated the stress-softening effect in the inflation of spherical membranes by assuming that at each material particle there coexists in the virgin material a fraction of material called soft phase and a fraction called hard phase. The material molecular structure is thus regarded as an amor- phous mixture of soft phase and hard phase. Due to microstructural damage evolution, the hard phase is continuously trans- formed to soft phase; and this gives rise to the stress-softening phenomenon. Beatty and Krishnaswamy [4] applied their stress-softening model to describe the Mullins effect observed in the balloon inflation experiments [1]; but they provided no comparison of their analytical predictions with experimental data. Clément et al. [5] have attributed the Mullins effect to the detachment of slippage of chains having reach their limited extensibility on the filler surface. Just before detaching or slipping, those chains, even if there are a few ones, can withstand large stresses, which gives a high contribution to the elastic modulus but a low contribution to the orientation. Subsequent in situ synchrotron wide-angle X-ray diffraction (WAXD) studies by Toki et al. [6] describe that during stretching, the major- ity of the amorphous chains are isotropic without preferred orientation and that the initial stress in the virgin material is mainly determined by vulcanized chains in the cross-linking network, whereas the final stress in the virgin material is deter- mined by the network of strain-induced crystallization. For the stress-softened material, the effective network structure be- comes different from the structure of the virgin path because some crystallities do not melt immediately. 0020-7225/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2010.06.011 * Corresponding author. Tel.: +52 81 8358 2000x5005 (off.)/+52 81 83496714 (home). E-mail addresses: aelias@itesm.mx (A. Elías-Zúñiga), ciro.rodriguez@itesm.mx (C.A. Rodríguez). International Journal of Engineering Science 48 (2010) 1937–1943 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci