Multiscale quasicontinuum modelling of fibrous materials

Structural lattice models and discrete networks of trusses or beams are regularly used to describe the mechanics of fibrous materials. The discrete elements naturally represent individual fibers and yarns present at the mesoscale. Consequently, relevant mesoscale phenomena, e.g. individual fiber failure and bond failure, culminating in macroscopic fracture can be captured adequately. Even macroscopic phenomena, such as large rotations of yarns and the resulting evolving anisotropy, are automatically incorporated in lattice models, whereas they are not trivially established in continuum models of fibrous materials. Another advantage is that by relatively straightforward means, lattice models can be altered such that each family of discrete elements describes the mechanical response in one characteristic direction of a fibrous material. This ensures for a straightforward experimental identification of the elements’ parameters. In this thesis such an approach is adopted for a lattice model of electronic textile. A lattice model for interfiber bond failure and subsequent fiber sliding is also formulated. The thermodynamical basis of this lattice model ensures that it can be used in a consistent manner to investigate the effects of mesoscale parameters, such as the bond strength and the fiber length, on the macroscopic response. Large-scale (physically relevant) lattice computations are computationally expensive because lattice models are constructed at the mesoscale. Consequently, large-scale computations involve a large number of degrees of freedom (DOFs) and extensive effort to construct the governing equations. Principles of the quasicontinuum (QC) method are employed in this thesis to reduce the computational cost of large-scale lattice computations. The advantage is that the QC method allows the direct and accurate incorporation of local mesoscale phenomena in regions of interest, whereas substantial computational savings are made in regions of less interest. Another advantage is that the QC method completely relies on the lattice model and does not require the formulation of an equivalent continuum description. The QC method uses interpolation to reduce the number of DOFs and summation rules to reduce the computational cost needed to establish the governing equations. Large interpolation triangles are used in regions with small displacement fluctuations. In fully resolved regions the dimensions of the interpolation triangles are such that the exact lattice model is captured. Summation rules are used to sample the contribution of all nodes to the governing equations using a small number of sampling nodes. In this thesis, one summation rule is proposed that determines the governing equations exactly, even though a large reduction of the number of sampling points is obtained. This summation rule is efficient for structural lattice models with solely nearest neighbor interactions, but it is inefficient for atomistic lattice computations that include interactions over longer ranges. Therefore, a second ’central’ summation rule is proposed, in which significantly fewer sampling points are selected to increase the computational efficiency, at the price of the quality of the approximation. The QC method was originally proposed for (conservative) atomistic lattice models and is based on energy-minimization. Lattice models for fibrous materials however, are often non-conservative and energy-based QC methods can thus not straightforwardly be used. Examples are the lattice model proposed for woven fabrics and the lattice model to describe interfiber bond failure and subsequent frictional fiber sliding proposed in this thesis. A QC framework is therefore proposed that is based on the virtual-power statement of a non-conservative lattice model. Using the virtual-power statement, dissipative mechanisms can be included in the QC framework while the same summation rules suffice. Its validity is shown for a lattice model with elastoplastic trusses. The virtual-power-based QC method is also adopted to deal with the lattice model for bond failure and subsequent fiber sliding presented in this thesis. In contrast to elastoplastic interactions that are intrinsically local dissipative mechanisms, bond failure and subsequent fiber sliding entail nonlocal dissipative mechanisms. Therefore, the virtualpower-based QC method is also equipped with a mixed formulation in which not only the displacements are interpolated, but also the internal variables associated with dissipation.