13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- adopted to calibrate the material properties of the network based on those of the continuum paper. To characterize the heterogeneity of the microstructure of fibrous materials, the yield strength of the elasto-plastic fiber element is considered to follow a correlated random distribution. Nonlinear finite element method is utilized to study the structure response of the lattice under external tension loading. The lattice network can be considered as a stochastic representative volume element (SRVE), which transports the local effects into global solutions with uncertainty information, e.g. probability distribution. The effect of the correlation length on the strength of the SRVE is studied. 2. Random lattice model To simulate the damage of fibrous materials, one of the most effective numerical approaches is to use the lattice model that allows disorder to be introduced naturally. Since fibrous material microstructure is extremely complicated, as shown in Figure 1, it is very hard to construct a numerical network exactly the same as the true physical one. Therefore, the idea of adopting a regular lattice equivalent to macro-level continuum in terms of strain energy [8, 9, 15, 16], is applied to study the damage of fibrous materials. For simplicity, the regular triangular network, as shown in Figure 2, is used in this work to study the failure properties of fibrous material structures. The fibers are distributed in three different directions with an increment of 3π , which leads to the isotropic properties of the structure when all fibers are assigned the uniform properties [8, 9]. In the framework of finite element method, the nodes correspond to fiber-to-fiber bonds, while two-node elements are formed by fiber segments between every two neighboring nodes. To simulate the randomness of the microstructure of fibrous materials, we need to generate random field (RF) samples according to given probability distributions. In this study, non-Gaussian RF samples are generated from underlying Gaussian RF samples by the so-called translation method. An overview of the random field simulation is presented in [17]. For simplicity, bar-elements are used to construct the regular lattice network, and there are two translation degrees of freedom for each node. The bar elements are considered elasto-plastic and their yield strength is assumed to follow a correlated random Weibull distribution [1]. A Weibull RF sample Y can be generated from an underlying Gaussian RF sample X via [1] ( ( )) 1 Y F F X g W − = (1) where ( )⋅ gF is the standard normal cumulative density function (cdf), and ( ) 1 ⋅ − WF is the inverse of the Weibull cdf. The correlation function of the underlying Gaussian RF is assumed to be ( ) [ ]2 2 2 ( , ) exp x y d x y = − + ρ (2) where d indicates the correlation length. As a rule of thumb, an average tensile strength of fibrous materials can be chosen as equal to elastic modulus times (1.0 0.1)% ± [13]. The generated random Weibull distribution with a set mean value is mapped to the regular lattice network to characterize the heterogeneity of the microstructure. For example, two of the Weibull RF samples are shown in Figure 3 for 1= d and 4= d , respectively. In Figure 3, the strength values are normalized by the mean value of the
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