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Figure2. Analysis flowchart. 2.2. Material Model
It is not possible to adopt the isotropic behavior for tissue paper if the kind of paper has different behaviors in the machine and cross directions [8], and ABAQUS TM doesnot have a native constitutive law to model plasticity for orthotropic materials. Hence, a user material subroutine for explicit simulations (VUMAT) was implemented to simulate the orthotropic elastic–plastic behavior for the paper sheet. The material model, proposed by Mäkelä and Östlund [13], allows the paper anisotropic behavior to be accounted for, since the paper response is highly dependent on the fiber orientation. The model assumes the decomposition of the strain tensor into an elastic strain tensor and a plastic strain tensor (Equation (2)) while conserving the volume.
p ij
e ij + ε
(2)
ε ij = ε
where ε ij is the total strain, ε e ij is the plastic strain. The material model adopts the concept of an isotropic plasticity equivalent mate- rial [14], a fictitious material that relates the orthotropic stress state to the isotropic stress state. Equation (3) gives the relation between the Cauchy stress tensor and the isotropic plasticity equivalent (IPE) deviatoric tensor. s ij = L ijkl σ kl (3) where s ij is the deviatoric IPE stress tensor, σ kl is the Cauchy stress and L ijkl is the fourth order transformation tensor shown in Equation (4) for plane stress. L = ⎡ ⎢⎢ ⎣ 2 A C − A − B 0 C − A − B 2 B 0 B − C − A A − B − C 0 0 0 3 D ⎤ ⎥⎥ ⎦ (4) ij is the elastic strain, and ε p
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