13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0 0.5 1 1.5 2 2.5 3 Strain Stress (MPa) d=1 d=2 d=3 { } { } [ ]{ } { } [ ] [ ]0 1 ( ) u u K u u F K i i i i − = − + (24) where [ ]0K is a secant stiffness matrix, which is kept unchanged within each load cycle, and the subscript i represents the equilibrium iteration. The criterion for stopping the numerical iteration is { } { } { } δ < − + i i iu u u 1 (25) where denotes a norm and δ is a tolerance value. 5. Numerical example As paper is a material which has a fibrous network structure consisting of wood fibers, it is convenient to apply the random lattice model to study the failure process of the fibrous structure of paper. Without loss of generality, the Young’s modulus of paper can be chosen as E GPa c 2= [19], and the corresponding Young’s modulus of the bar element (within the limit of elasticity) can be obtained using Eq. (17), provided by choosing the length and the cross section of the element as t mm l mm, 0.1 10 = = . It is noted that fibers in paper maybe not exactly perfect-plastic as assumed in the present model, the aim of the study is to find the effect of correlation length on global strength when plasticity is considered. Figure 6. Stress-strain curves of the random elasto-plastic lattice. The stress-strain curves of the lattices of size (8 8× ), which denotes the number of nodes of the system is 8 8× , are shown in Figure 6. Each curve for a correlation length d is the mean value of stress-strain curves of 100 random lattice samples. The strength of the lattice increases as the
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