a b s t r a c tLaminated paperboard is often used as a packaging material for products such as toys, tea and frozenfoods. To make the paperboard packages appealing for consumers, the fold lines must be both neat and undamaged. The quality of the folds depends on two converting processes: the manufacture of fold lines (creasing) and the subsequent folding. A good crease contains some delamination, initiated during creasing, to reduce the bending stiffness and to prevent the board from breaking during folding. However, for boards of high grammage breaking of the top layer is nevertheless a frequent problem. The mechanisms that operate in the creasing zone during creasing and folding, and that may thus result in breaking of the top layer, are studied in this contribution on the basis of idealized small-scale creasing and folding experiments. However, since experimental observations are only limited means to study the paperboard's behavior, a mechanical model is proposed to obtain more detailed insight. Although the material and delamination descriptions used in the mechanical model are both relatively straightforward, comparisons between the model and the experimental data show that the model predicts the paperboard's response well. The mechanical model shows -in combination with experimental strain fields -that multiple delaminations are initiated in the shear regions. Moreover, only the mechanical model reveals the mechanism that is responsible for the failure of the top layer if a crease is too shallow. Finally, the model also demonstrates that not only delamination but also plastic behavior must occur during creasing if breaking of the top layer is to be avoided.

Phase-field models have the advantage in that no geometric descriptions of cracks are required, which means that crack coalescence and branching can be treated without additional effort. Miehe et al. [1] introduced a rate-independent phase-field damage model for finite strains in which a viscous damage regularization was proposed. We extend the model to depend on the loading rate and time by incorporating rubber's strain-rate dependency in the constitutive description of the bulk, as well as in the damage driving force. The parameters of the model are identified using experiments at different strain rates. Local strain fields near the crack tip, obtained with digital image correlation (DIC), are used to help identify the length scale parameter. Three different degradation functions are assessed for their accuracy to model the rubber's rate-dependent fracture. An adaptive time-stepping approach with a corrector scheme is furthermore employed to increase the computational efficiency with a factor of six, whereas an active set method guarantees the irreversibility of damage. Results detailing the energy storage and dissipation of the different model constituents are included, as well as validation results that show promising capabilities of rate-dependent phase-field modeling.

SUMMARYMany studies in different research fields use lattice models to investigate the mechanical behavior of materials. Full lattice calculations are often performed to determine the influence of localized microscale phenomena on large-scale responses but they are usually computationally expensive. In this study the quasicontinuum (QC) method (Phil. Mag. A 1996; 73:1529-1563) is extended towards lattice models that employ discrete elements, such as trusses and beams. The QC method is a multiscale approach that uses a triangulation to interpolate the lattice model in regions with small fluctuations in the deformation field, while in regions of high interest the exact lattice model is obtained by refining the triangulation to the internal spacing of the lattice. Interpolation ensures that the number of unknowns is reduced while summation ensures that only a selective part of the underlying lattice model must be visited to construct the governing equations. As the QC method has so far only been applied to atomic lattice models, the existing summation procedures have been revisited for structural lattice models containing discrete elements. This has led to a new QC method that makes use of the characteristic structure of the considered truss network. The proposed QC method is, to the best of the authors' knowledge, the only QC method that does not need any correction at the interface between the interpolated and the fully resolved region and at the same time gives exact results unlike the cluster QC methods. In its present formulation, the proposed QC method can only be used for lattice models containing nearest neighbor interactions, but with some minor adaptations it can also be used for lattices with next-nearest neighbor interactions such as atomic lattices.

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Lattice models and discrete networks naturally describe mechanical phenomena at the mesoscale of fibrous materials. A disadvantage of lattice models is their computational cost. The quasicontinuum (QC) method is a suitable multiscale approach that reduces the computational cost of lattice models and allows the incorporation of local lattice defects in large-scale problems. So far, all QC methods are formulated for conservative (mostly atomistic) lattice models. Lattice models of fibrous materials however, often require non-conservative interactions. In this article, a QC formulation is derived based on the virtual-power of a non-conservative lattice model. By using the virtual-power statement instead of force-equilibrium, errors in the governing equations of the force-based QC formulations are avoided. Nevertheless, the non-conservative interaction forces can still be directly inserted in the virtualpower QC framework. The summation rules for energy-based QC methods can still be used in the proposed framework as shown by two multiscale examples.