The computational analysis of fiber network fracture is an emerging field with application to paper, rubber-like materials, hydrogels, soft biological tissue, and composites. Fiber networks are often described as probabilistic structures of interacting one-dimensional elements, such as truss-bars and beams. Failure may then be modeled as strong discontinuities in the displacement field that are directly embedded within the structural finite elements. As for other strain-softening materials, the tangent stiffness matrix can be non-positive definite, which diminishes the robustness of the solution of the coupled (monolithic) two-field problem. Its uncoupling, and thus the use of a staggered solution method where the field variables are solved alternatingly, avoids such difficulties and results in a stable, but sub-optimally converging solution method. In the present work, we evaluate the staggered against the monolithic solution approach and assess their computational performance in the analysis of fiber network failure. We then propose a hybrid solution technique that optimizes the performance and robustness of the computational analysis. It represents a matrix regularization technique that retains a positive definite element stiffness matrix while approaching the tangent stiffness matrix of the monolithic problem. Given the problems investigated in this work, the hybrid solution approach is up to 30 times faster than the staggered approach, where its superiority is most pronounced at large loading increments. The approach is general and may also accelerate the computational analysis of other failure problems.

Abstract To model fiber failures in random fiber networks, we have developed an elastoplastic Timoshenko beam finite element with embedded discontinuities. The method is based on the theory of strong discontinuities where the generalized displacement field is enhanced by a jump. The continuum mechanics formulation accounts for a fracture process zone and a bulk material while retaining traction continuity across the discontinuity. The additional degrees of freedom that are associated with the discontinuity are represented by a midpoint node, which is statically condensed to enable the implementation in commercial software through the user element interface . We propose a quasi-brittle fracture model, where the failure-related deformation is uncoupled from the plastic deformation in the bulk material. To retain the positive definite finite element stiffness matrix of the bulk material, we neglect the fracture-related softening of the discontinuity and employ a modified Newton iteration in the strain softening domain. Our implementation facilitates the integration into commercial finite element software and examples illustrate the robustness of the method. The FORTRAN source code is freely available to benchmark our model. We show that fiber failures contribute to the nonlinear stress–strain response of paper. Together with fiber–fiber bond failures, they can potentially explain the nonlinear stress–strain response of paper and nanopaper.

We develop a thermodynamically consistent continuum damage micromechanics model for the compressive failure of flax fiber composites. We used a micromechanics-based constitutive model reported recently [1]. It describes the microstructure of a unidirectional composite and captures the material behavior of the fiber and matrix constituents, respectively. The description has been formulated in the reference configuration (i.e. the undeformed state of the composite) and is therefore independent of fiber rotations that may appear during the deformation of the composite. A hyperelastic finite deformation plasticity with power law hardening [3] mimics the compressive elastic-plastic stress-strain response of the fiber (reported in [2]) and the matrix. The model has been extended to account for fiber damage, resulting in a thermodynamically consistent continuum damage micromechanics model. Our results indicate that fiber damage plays an utmost role in the compressive failure of flax fiber composites – it is a major determinant of the material’s compressive stress-strain response. X-ray Computed Tomography and Scanning Electron Microscopy show that fiber damage can be attributed to intra-fiber splitting and elementary fiber crushing.

Paper materials are natural composite materials where fibers are almost randomly distributed in a fiber network. Mechanical properties of fiber networks are known to be strongly controlled by fiberfiber interactions and single fiber properties. A fiber network is often modeled as a beam network where beam-to-beam interactions are treated as cohesive zones and single beams stretch indefinitely without breaking. The latter assumption is not physically correct and leads to an overprediction of the mechanical response of the beam network. In this work, we present a computational modeling framework for simulating beam failures and thereby closing the gap to physically based micromechanical modeling of paper and packaging products. Modeling beam failure is a challenging engineering problem. At the onset of failure, the tangent stiffness tensor projected in a direction normal to the surface of discontinuity (commonly referred to as the localization tensor) is singular, i.e. we have a bifurcation point and the problem is ill-posed. Another implication of ill-posedness for the numerical simulation after a spatial discretization is a pathological mesh dependency of the computed result. We use the ED-FEM where a failure process zone (FPZ) is introduced into a multi-scale continuum mechanics formulation (i.e. the material is split into a small scale and a large scale defining the FPZ and the bulk material, respectively), making the computed result mesh independent. The multi-scale nature of the ED-FEM enables an operator splitting implementation method as opposed to carrying out the computations of the nodal displacement vector and the displacement discontinuity vector simultaneously with the global loop where the global stiffness matrix would be singular at the onset of failure. We show that fiber failures and fiber-fiber bond failures can contribute to the observed elastoplastic stress-strain response of paper.

Abstract In this study, the back calculated compressive properties of flax fibers utilizing the Impregnated Fiber Bundle Test (IFBT) were investigated. The back calculated stress-strain response can be described by the Ramberg-Osgood model. The compressive modulus of the fiber is similar to its tensile modulus. The compressive strength of the fiber is approximately 45 % of its tensile strength. Considering the presence of local fiber kinking within the elementary fibers as well as global fiber kinking due to fiber misalignments and plastic shear deformation in the matrix material, this is a remarkably high value for the compressive strength. Our results indicate that local fiber kinking precedes global fiber kinking. We show that IFBT is a promising method for determining the compressive properties of flax fibers and provides necessary input data for finite element analysis of the compressive failure mechanisms in unidirectional flax fiber reinforced composites.

Abstract Compression failure by fiber kinking limits the structural applications of fiber composites. Fiber kinking is especially prevalent in laminates with holes and cutouts. The latter behavior is characterized by strain localization in the matrix material and fiber rotations. To study fiber kinking on the level of the individual constituents, a homogenization of fiber composites is presented. It is based on a total Lagrangian formulation, making it independent of fiber rotations. It accounts for the microstructure of the composite, including fiber-matrix interfacial decohesion, and enables all types of material behavior of the constituents. The response of each constituent of the composite is modeled separately and the global response is obtained by an assembly of all contributions. The model is implemented as a user-defined material model (UMAT) in ABAQUS and used for multiscale modeling of notched unidirectional plies subjected to compression. The model performs well in agreement with a finite element model of an explicit discretization of the microstructure and literature results. The simulations predict the formation of a kink band in near 0-degree plies and show that the open-hole compression strength is sensitive to fiber-matrix interfacial decohesion. The present work suggests a convenient and computationally efficient tool for simulating the elastic-plastic behavior of fiber composites on the fiber-matrix level and predicting the compressive strength of laminates.

Abstract The micromechanics of kink band formation in open-hole fibre composites under compressive loading is described. The objective being the development of a methodology for designing of structural components with open-holes. Our results explain why failure by kink band formation propagates from the edges of an open-hole in a direction almost perpendicular to the loading direction and why the 0 degree plies govern the compressive failure of an open-hole laminate. The proposed design methodology accounts for the microstructure, including the fibre/matrix bonding, and the nonlinear behaviour of the constituents, enabling it to prevent local failure at the hole edges, or global failure, by kink banding of a laminate containing stress concentrations.