Elasto-plastic analysis of inﬂuences of bond deformability on the mechanical behavior of ﬁber networks J.X. Liu a , Z.T. Chen a,⇑ , H. Wang b , K.C. Li b a Department of Mechanical Engineering, University of New Brunswick,15 Dineen Drive, Fredericton, NB, Canada E3B 5A3 b Department of Chemical Engineering, University of New Brunswick, 2 Garland Court, Fredericton, NB, Canada E3B 6C2 article info Article history: Available online 28 April 2011 Keywords: Fiber network Network modeling Bond deformability Elasto-plasticity Newton–Raphson method abstract Micromechanical modeling of three-dimensional (3D) ﬁber networks is performed by reducing the web structure to ﬁber segments and ﬁber–ﬁber bonds to explore the inﬂuence of ﬁber–ﬁber bond deformabi- lity on ﬁber network’s elasto-plasticity. The ﬁber segment between every two adjacent bonds is described by a Timoshenko beam element, while ﬁber–ﬁber bonds are taken as a different two-node element due to the extremely large height-to-span ratio. This overcomes the rigid-bond assumption employed by most previous network models, providing the feasibility to build the relationship between bonding condition and global mechanical properties, resulting in a 3D network structure and able to accommodate curled ﬁbers. Both ﬁbers and bonds are assumed to be elasto-plastic and described by the bilinear kinematic hardening model. Deformation of the network with load increasing is simulated by the Newton–Raphson method. Numerical tensile tests show reasonable results and agree qualitatively with experiments. The inﬂuence of bond deformability on the mechanical behavior of the network is discussed in detail. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Fiber networks composed of randomly distributing ﬁbers [10,20] are studied numerically with necessary simpliﬁcations. Typical examples are paper sheets and some woven composites, as shown in Fig. 1 [1]. Each ﬁber is connected with some other ﬁ- bers, and intersection points between ﬁbers are called ﬁber–ﬁber bonds. In other words, it is composed of interacting material lines, obviously different from the classical continuum composed of material particles. Investigations of mechanical properties of such ﬁber networks have been increasingly attractive [8–10,19–24], however, the exact properties of both ﬁbers and bonds are still un- clear. For example, in a paper structure, every single wood ﬁber is actually multi-layered, and intersecting wood ﬁbers are actually connected with each other through hydrogen bonds [21,27]. Gen- erally, ﬁbers and bonds have extremely complex compositions, which has been an independent research topic [25,27]. Neverthe- less, our attention here is focused on the global mechanical prop- erties of the network composed of a large number of ﬁbers and bonds, then both ﬁbers and bonds are simpliﬁed as homogeneous and isotropic continua [3,10,22]. Based on this simpliﬁcation, a new network model is established to relate this special network- like microstructure to the macro-level material behavior. Lots of network models with rigid bonds have been developed for simulating random ﬁber webs [3,10,22]). A systematic review about network models or equally lattice models for different mate- rials can be found in [23]. For paper sheets composed of random wood ﬁbers, ﬁber segments have been generally simpliﬁed as beam elements, while the ﬁber–ﬁber bond has been popularly ta- ken as rigid [3,10,22,29], which means two ﬁbers intersecting with each other have exactly the same deformation at their intersection region, resulting that every bond can be represented by a single point, namely one node when the ﬁnite element method is adopted, as shown in Fig. 2b. This rigid-bond assumption has been attractive due to its simplicity and its expected capacity of reﬂect- ing properly the initial responses of the network under low-level external loads. However, the rigid-bond assumption can be questionable and lead to loss of important information of real material structures. A rigid-bond network model cannot work correctly once consider- able plastic deformation happens [3]. As we can imagine, bond material parameter should be of the same magnitude as those of ﬁbers because the bond is composed of ﬁbers. Furthermore, if ﬁ- bers are not bonded very well, bonding stiffness is much lower than ﬁber stiffness, and then considerable relative sliding deforma- tion arises in bond under external loadings. As we will show later deformability of ﬁber–ﬁber bonds will markedly inﬂuence the macroscopic behavior even though the load level is low. Further- more, a bond is actually the neighborhood around the small con- tact region between ﬁbers (Fig. 1), therefore bond size along the 0167-8442/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2011.04.003 ⇑ Corresponding author. Tel.: +1 506 458 7784; fax: +1 506 453 5025. E-mail address: ztchen@unb.ca (Z.T. Chen). Theoretical and Applied Fracture Mechanics 55 (2011) 131–139 Contents lists available at ScienceDirect Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec