We investigate the compressive failure mechanisms in flax fiber composites, a promising eco-friendly alternative to synthetic composite materials, both numerically and experimentally, and explain their low compressive-compared-to-tensile strength, the compressive-to-tensile strength ratio being 0.28 -0.6. We present a novel thermodynamically consistent continuum damage micromechanics model capturing events on the fiber-matrix scale. It describes the microstructure of a unidirectional composite and includes the instantaneous constitutive behavior of matrix and fibers. We show that flax fibers behave as elastic-plastic-damaged solids in compression. Furthermore, we show 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. Using X-ray Computed Tomography (XCT) and Scanning Electron Microscopy (SEM), we identify the fiber damage as intra-technical fiber splitting and elementary fiber crushing. Due to micro -structural similarities among natural fibers, the same micro-mechanisms are likely to appear in other bio-based fibers and their composites.

We introduce a three-dimensional geometrically nonlinear Reissner beam theory with embedded strong discontinuities for the modeling of failure in structures and discuss its finite element implementation. Existing embedded beam theories are geometrically linear or two-dimensional, motivating the need for the present work. We propose a geometrically nonlinear beam theory that accounts for cracks through displacement discontinuities in 3D. To represent the three modes of fracture inside an element, we enrich each displacement field with an incompatible mode parameter in the form of a jump, an additional degree of freedom. We then eliminate these nodeless degrees of freedom through static condensation and evaluate them within the framework of inelasticity by utilizing an operator split solution procedure. Seeing that the coupled strain-softening problem is non-convex, we present an alternating minimization (staggered) algorithm, thus retaining a positive definite stiffness matrix. Finally, the four-parameter representation by quaternions describes a three-dimensional finite rotation. We demonstrate a very satisfying and robust performance of these new finite elements in several numerical examples, including the fracture of random lattice structures with application to fibrous materials. We show that accounting for geometrical nonlinearity in the beam formulation is necessary for direct numerical simulations of fiber networks regardless of the density.

This thesis summarizes seven appended papers dealing with: (1) The fracture of fibrous materials, e.g., paper and paperboard, toward understanding the upper limits of paper products and eventually optimizing packaging performance in its endeavor to replace plastics with recyclable packaging; (2) The compressive failure of flax fiber composites, a promising eco-friendly alternative to synthetic composite materials, toward understanding the low compressive-compared-to-tensile strength of biocomposites, a design-limiting feature, and ultimately engineer better performing natural fiber composites for sustainable structures. 

(1) In Paper I, we consider an elastoplastic Timoshenko beam finite element formulation with embedded strong discontinuities in the description of multi-fracturing fibers in fiber networks, a deficiency in previous studies. Seeing that the coupled (monolithic) problem is non-convex, materializing through poor robustness and undesirable material instabilities, we present an alternating minimization (staggered) algorithm for this class of problems and thus retain a positive definite stiffness matrix. In Paper II, we propose a hybrid of monolithic and staggered solution methods for robust and computationally efficient fracture simulations, with an up to 30-fold performance gain compared to the staggered approach in the benchmark exercises. The hybrid method represents a matrix regularization technique that retains a positive definite stiffness matrix while approaching the tangent stiffness matrix of the monolithic problem. In Paper III, we develop a geometrically nonlinear Simo-Reissner beam theory with embedded strong discontinuities based on the method of incompatible modes, capturing the activation of additional fibers during loading. We show that accounting for geometrical nonlinearity in the beam formulation is necessary for direct numerical simulations of fiber networks regardless of the density. 

(2) In Paper IV, we formulate a multi-scale homogenization framework for layered composite materials, where we model the instantaneous constitutive behavior of the matrix and the fiber separately utilizing a combined Voigt and Reuss approximation, followed by an upscaling to the composite. Advantages include the independence of fiber rotations because it is fully defined in the known initial configuration of the composite. In Paper V, we back-calculate the compressive stress-strain response of the flax fiber from the Impregnated Fiber Bundle Test (IFBT) in compression using the rule of mixtures, necessary input data in the micromechanical description of flax fiber composites. In Paper VI, we formulate hyperelastic models for deformation plasticity into the finite strain range. One application includes mimicking the stress-strain response of the fiber and the matrix in the homogenization of layered composite materials, which we numerically verify against a micromechanical model. In Paper VII, we extend the hyperelastic model to account for fiber damage. We show numerically and experimentally through X-ray Computed Tomography (XCT) and Scanning Electron Microscopy (SEM) that fiber damage plays the utmost role in the compressive failure of flax fiber composites – it is a major determinant of the material’s compressive stress-strain response. The micromechanisms include elementary fiber crushing and intra-technical fiber splitting.

Avhandlingen sammanfattar sju bifogade artiklar om (1) fiberbrott i nätverksbaserade material som t.ex. papper och kartong, och (2) kompressionshållfastheten hos linfiberkompositer, vilket är ett lovande miljövänligt alternativ till syntetiska kompositmaterial.

(1) I Paper I betraktas en finit elementformulering av en elastiskt-plastisk Timoshenko balk med en inbäddad stark diskontinuitet för att beskriva multipla fiberbrott i fibernätverk. Detta har inte varit möjligt i tidigare studier. Eftersom det kopplade (monolitiska) problemet är icke-konvext, materialiserat genom dålig robusthet och oönskade materialinstabiliteter, presenteras en sekventiell minimeringsalgoritm för denna klass av problem som medför att styvhetsmatrisen förblir positivt definit. I Paper II föreslås en hybrid av monolitiska och sekventiella lösningsmetoder för robusta och beräkningseffektiva simuleringar av multipla fiberbrott i fibernätverk. Jämfört med det sekventiella tillvägagångssättet erhålles en upp till trettiofaldig prestandavinst för ett antal i testexempel. Hybridmetoden representerar en matrisregulariseringsteknik som bibehåller en positivt definit styvhetsmatris samtidigt som tangentstyvhetsmatrisen närmar sig det monolitiska problemet. I Paper III utvecklas en geometriskt olinjär Simo-Reissner balkteori med inbäddade starka diskontinuiteter baserad på metoden för icke kompatibla deformationsmoder, som fångar aktiveringen av ytterligare fibrer under belastningen. Dessutom visas att beaktande av geometriska olinjäriteter i formuleringen av balkteorin ger betydande bidrag till responsen vid direkta numeriska simuleringar av fibernätverk oavsett nätverkets densitet.

(2) I Paper IV formuleras ett flerskaligt ramverk för homogenisering av kompositlaminat. Det momentana konstitutiva beteendet hos matrisen och fibern modelleras separat med hjälp av en kombinerad Voigt och Reuss approximation. Kompositlaminatets egenskaper fås därefter genom en uppskalning. En fördel med detta tillvägagångssätt är att det är oberoende av fiberrotationer eftersom det är helt definierat i kompositens referenskonfiguration. Linfiberns spännings-töjningsrespons i kompression är nödvändig i den mikromekaniska beskrivningen av linfiberkompositer. I Paper V beräknas den med data från kompressionsprovning av impregnerade fiberbuntar, det s.k. Impregnated Fiber Bundle Test (IFBT), och blandningslagarna för kompositer. I Paper VI formuleras hyperelastiska modeller för deformationsplasticitet för stora töjningar. En applikation inkluderar att efterlikna fiberns och matrisens spännings-töjningsrespons vid homogenisering av skiktade kompositmaterial, vilket numeriskt verifieras med en mikromekanisk modell. I Paper VII utökas den hyperelastiska modellen för att ta hänsyn till fiberskador. Det visas både numeriskt och experimentellt, genom röntgendatortomografi (XCT) och svepelektronmikroskopi (SEM), att fiberskador har en avgörande betydelse för linfiberkompositers kompressionsstyrka. Typiska mikroskopiska fiberskador är krossning av elementärfibrer och splittring av tekniska fibrer.

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.

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 propose a staggered solution scheme for the embedded strong discontinuity finite element method applied to fracture of beam structures, e.g., see our recently accepted paper [1]. We demonstrate the robustness of our implementation for modeling multi-fracturing fibers in random fiber networks loaded in tension. Our implementation enables a user-friendly integration into commercial finite element software. The FORTRAN source code is freely available to benchmark our implementation. 

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.

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.