Stress-strain curve of paper revisited - Innventia.com

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$Engineering fracture mechanics analysis of paper ... - Innventia.com$

PAPER PHYSICS Material and Methods Network modeling In our model, we considered a three-dimensional network of fibers. The network was created with the help of a deposition technique described elsewhere (Kulachenko and Uesaka 2010). Each fiber is represented as a series of Timoshenko quadratic beam elements with a tubular cross-section. The beam element has three translational and three rotational degrees of freedom at each node. The following parameters can be varied during construction of the network: fiber length, width, wall thickness, curl, basic weight and other material properties of fibers. All the listed parameters can vary according to a specified distribution law. We accounted for large deflections, rotations and strains. A 4 x 4 mm 2 snippet of a typical fiber network is shown in Fig 1. Fiber-to-fiber bonds were modeled by a point-wise contact with friction (Zavarise and Wriggers 2000). Fiber bonds can break and separate during loading. This requires a specific description. A relation between force and displacement of an individual bond is schematically demonstrated in Fig 2. There are three principal characteristics of the curve above: stiffness, strength and work of separation (dark area below the descending part of the curve). We captured this behavior with a bilinear cohesive zone model, Fig 3, where F bs is the bond strength expressed through force, K c is the contact stiffness which describes the compliance of the bonding regions, d f is the fracture distance, and d s is the separation distance. A mixed I + II debonding mode was assumed, in which the bond separation depends on both normal and tangential contact forces. A power law energy criterion was used to define the completion of debonding, E E n cn Et + = 1 (1) E ct where E n and E t are the computed normal and tangent fracture energies respectively. The inputs for the model are the maximum traction in both directions, and the critical energy release rates E cn and E ct . When a bond fails, the contact between corresponding fibers is described with a frictional contact. This is probably the first threedimensional model which is capable of simulating the fracture process of paper accounting for nonlinearities at the fiber level and bond failures. Material characterization Nonlinear stress-strain behavior of a single fiber was described by bilinear isotropic hardening plasticity, Fig 4. Selecting reasonable values for the fiber material properties was a difficult task. Most of the experiments reported in the literature were performed on the untreated fibers with relatively low micro-fibril angles. These fibers behave almost elastically before failure. It was shown that fibers dried under compressive strains or having a high fibril Fig 1. A typical simulated fiber network, 4 mm x 4 mm with 27 g/m 2 , where MD is machine direction, CD – cross-machine and ZD – thickness directions (ZD coming out of the network plane). Fig 2. Schematic diagram of a force-displacement curve measured during fiber joint testing in either normal or tangent direction. Joint strength is the maximum point of the curve, joint stiffness is the tangent of the linear region of the curve and work of separation is the area below the curve from the point of maximum load to the separation distance. Fig 3. Bilinear cohesive zone model used for modeling of contact debonding. Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 319
PAPER PHYSICS Fig 4. Bilinear isotropic hardening plasticity fiber material model, where σy is yield stress, E is Young’s modulus and Etan is tangent modulus. For numerical values see Table 1. angles exhibit stronger nonlinear behavior with a distinct hardening region (Page and El-Hosseiny 1983; Groom et al. 2002). In this respect, drying inside the sheet affects the fibril angles, reduces the elastic modulus of the fiber and promotes distinct non-linear hardening response through introducing compressive strains. Fig 5 shows the material data found in the literature on a fiber dried under compressive restraint (Seth and Page 1983) and an untreated fiber (Groom et al. 2002). These two measurements exemplify probably the upper and lower bounds for the fiber behavior inside the network. For our numerical experiments, we assigned the following average fiber properties, Table 1, which were measured for the pulp used in the experimental sheets preparation; described in the next section. The fiber width and length were varied according to a Gaussian distribution with cut-off values of 6 μm in fiber width and 100 μm in fiber length. Fiber bonding properties are widely available in the literature (Lindström 2005). However, it is often left unspecified whether normal or shear strength is measured since a clear distinction is difficult to make in the experimental analysis. There are at least eight experimental methods including in-plane, shear, peel or z-directional loading modes for determination of fiber-fiber bond strengths, which are often reported in stress units (Retulainen 1993). There has been a debate about the relative bonded area, as it is involved in calculations. Based on the previously published results of measured bond strengths, Joshi 2007 and Batchelor 2010 stated that most of the reported shear bond strengths are far too low compared to reality. However, since we specify the bonding strength in force units, we stay unaffected by the uncertainty related to the relative bonded area. For the reference case, we selected the data for summerwood fibers (Aulin et al. 2010; Niskanen 1998) summarized in Table 2. The friction coefficient which acted between the fibers after debonding was assumed to be 0.5. 320 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 Fig 5. Stress-strain curve for a latewood fiber, adopted from Groom et al. 2002 and for holocellulose summerwood fiber, adopted from Seth and Page 1983. Table 1: Fibers properties used during network construction. Fiber length mean±std [mm] Fiber width mean±std [μm] Young’s modulus [GPa] Tangent modulus [GPa] 3.20±1.38 32.45±2.45 20 10 100 Yield stress [MPa] Table 2 Parameters used for the control case in simulation: separation distance (SD), bond strength (BS), contact stiffness (CS). Norm BS [mN] Tang BS [mN] Norm SD [μm] Tang SD [μm] Norm CS [GPa] 25.5 4.2 0.73 0.60 40 8 Tang CS [GPa] Experimental setup The purpose of the experimental setup was to get a qualitative comparison between the numerical network model and the physical fiber network. With the help of a sharp die, we extracted and tested small rectangular pieces from isotropic handsheets made of unbleached softwood sulfate pulp with removed fines. Fines were removed by a screening technique. Fiber length analysis was performed prior to fractionation. The measured fiber length distribution followed a bimodal Gaussian mixture distribution. Analyzing the mode which corresponds to longer fiber fraction retained after screening gave the mean fiber length of 3.2 mm with standard deviation of 1.38 mm. This fiber length data was used during numerical network generation. Laboratory handsheets of the specified pulp were prepared according to ISO 5269/2 using Rapid Köthen sheet former equipment in a climate-controlled room. Sheets were dried under restraints at 93°C in the dryer for 15 minutes. After preparation, the handsheets were conditioned at 50% RH and 23 °C until testing. Tensile tests were performed on a INSTRON ® 5944 system
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