Random Fiber Networks Research Articles

Multiscale Computational Model Predicts Mouse Skin Kinematics Under Tensile Loading.

Skin is a complex tissue whose biomechanical properties are generally understood in terms of an incompressible material whose microstructure undergoes affine deformations. A growing number of experiments, however, have demonstrated that skin has a high Poisson's ratio, substantially decreases in volume during uniaxial tensile loading, and demonstrates collagen fiber kinematics that are not affine with local deformation. In order to better understand the mechanical basis for these properties, we constructed multiscale mechanical models (MSM) of mouse skin based on microstructural multiphoton microscopy imaging of the dermal microstructure acquired during mechanical testing. Three models that spanned the cases of highly aligned, moderately aligned, and nearly random fiber networks were examined and compared to the data acquired from uniaxially stretched skin. Our results demonstrate that MSMs consisting of networks of matched fiber organization can predict the biomechanical behavior of mouse skin, including the large decrease in tissue volume and nonaffine fiber kinematics observed under uniaxial tension.

Computational study of the geometric properties governing the linear mechanical behavior of fiber networks

Materials whose microstructure is formed by random fiber networks play an important role both in biology and engineering. So far, it still remains unclear which geometric properties of the fiber network determine the macroscopic mechanical properties of such materials. This paper presents a computational study based on a large number of representative volume elements of random fiber networks. Our study reveals that the linear mechanical properties of fiber networks (i.e., Young’s modulus and Poisson’s ratio) are largely determined by only four scalar key descriptors. These are the number of fibers per volume, the mean node valency, the mean fiber length, and the mean direction cosine between fibers adjacent to the same node. Number of fibers per volume and node valency were found to be responsible for around 80% of the variance of the mechanical properties, making them the two by far dominant microstructural descriptors. In the part of the configuration space covered by our study, we observed a linear or quadratic relationship between the above four scalar microstructural descriptors and the Young’s modulus. For the number of fibers per unit volume we propose a theoretical explanation for this simple relation.

Modeling multi-fracturing fibers in fiber networks using elastoplastic Timoshenko beam finite elements with embedded strong discontinuities — Formulation and staggered algorithm

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.

Multi-scale modeling study on fibrous network of cellulose separator for lithium-ion battery

Separator is an essential component of lithium-ion battery, affecting the electrochemical performance and safety of battery. The material-structure-tensile property relationship is significant, herein, a multi-scale modeling method is proposed to explore cellulose separator. Theoretical calculation is utilized to study the composite material in material scale. A multi-layered model is established to survey single fiber in micro scale. Random fibrous networks are applied as representative volumetric element (RVE) models to investigate the fibrous network in meso scale. The multi-scale modeling method manages to conduct reliable simulation on tensile property of cellulose separator, and provide behavior predicting function on further study. A multi-scale modelling method is proposed for cellulose separator Scales including composite materials, single fiber, and fibrous network are studied One-step method is realized combining modelling and simulation of RVE models Relationship between structural parameters. A multi-scale modelling method is proposed for cellulose separator Scales including composite materials, single fiber, and fibrous network are studied One-step method is realized combining modelling and simulation of RVE models Relationship between structural parameters and tensile properties are predicted and tensile properties are predicted.

Mechanical response of composite fiber networks subjected to local contractile deformation

The extracellular matrix (ECM), forming the cell microenvironment, is a random fiber network whose heterogeneous microstructure serves a vital role in cellular functions such as migration, proliferation, and differentiation. Although prior computational studies used a network of randomly crosslinked fibers for investigating the ECM mechanical response due to the cell contraction, we hereby investigate the cell-induced deformation by considering the interplay between the collagen fibers and soft proteoglycan matrix domain. For this purpose, we represent the complex ECM microstructure by a double network model in which a random network of relatively stiff fibers (representing collagens) is embedded in a homogeneous network (representing the matrix domain). We demonstrate that the interaction between the network of stiff fibers and the network of matrix fibers plays a significant role in the propagation of cell-induced displacements. In particular, as the stiffness of the matrix domain increases, the long-range displacement propagation becomes suppressed. Furthermore, the degree of inhomogeneity in the displacement propagation is controlled by the relative stiffness of fiber and matrix networks. The influence of matrix stiffness becomes prominent for composite networks with random networks having a low average network connectivity and/or composed of fibers with low bending rigidity.

Theoretical prediction of effective stiffness of nonwoven fibrous networks with straight and curved nanofibers

The nonwoven random fibrous network materials have been widely used in many fields due to their excellent physical and chemical properties. To understand the mechanical behavior of these nonwoven fibrous networks, a theoretical model is developed based on the network microstructure and microstructure evolution in the loading process. It is revealed that the effective stiffness of nonwoven fibrous networks is not only contributed by the fiber stretching and bending, but also depending on the rotation deformation of crosslinks between fibers. Since the nonwoven fibrous networks are constructed by straight or curved fibers, especially at nanoscale, the curvature effects on the effective stiffness are also studied systematically. Furthermore, finite element simulation is conducted to verify the theoretical predictions on the elastic properties of nonwoven fibrous networks consisting of straight and curved fibers, respectively. The theoretical model presented here is a general model, enabling to capture the elastic mechanical behavior of many complex nonwoven network systems from micro to nano length scales.

Parametric Finite Element Model and Mechanical Characterisation of Electrospun Materials for Biomedical Applications.

Electrospun materials, due to their unique properties, have found many applications in the biomedical field. Exploiting their porous nanofibrous structure, they are often used as scaffolds in tissue engineering which closely resemble a native cellular environment. The structural and mechanical properties of the substrates need to be carefully optimised to mimic cues used by the extracellular matrix to guide cells’ behaviour and improve existing scaffolds. Optimisation of these parameters is enabled by using the finite element model of electrospun structures proposed in this study. First, a fully parametric three-dimensional microscopic model of electrospun material with a random fibrous network was developed. Experimental results were obtained by testing electrospun poly(ethylene) oxide materials. Parameters of single fibres were determined by atomic force microscopy nanoindentations and used as input data for the model. The validation was performed by comparing model output data with tensile test results obtained for electrospun mats. We performed extensive analysis of model parameters correlations to understand the crucial factors and enable extrapolation of a simplified model. We found good agreement between the simulation and the experimental data. The proposed model is a potent tool in the optimisation of electrospun structures and scaffolds for enhanced regenerative therapies.

Nonlinear Mechanical Properties of Prestressed Branched Fibrous Networks

Random fiber networks constitute the solid skeleton of many biological materials such as the cytoskeleton of cells and extracellular matrix of soft tissues. These random networks show unique mechanical properties such as nonlinear shear strain-stiffening and strain softening when subjected to preextension and precompression, respectively. In this study, we perform numerical simulations to characterize the influence of axial prestress on the nonlinear mechanical response of random network structures as a function of their micromechanical and geometrical properties. We build our numerical network models using the microstructure of disordered hexagonal lattices and quantify their nonlinear shear response as a function of uniaxial prestress strain. We consider three different material models for individual fibers and fully characterize their influence on the mechanical response of prestressed networks. Moreover, we investigate both the influence of geometric disorder keeping the network connectivity constant and the influence of the randomness in the stiffness of individual fibers keeping their mean stiffness constant. The effects of network connectivity and bending rigidity of fibers are also determined. Several important conclusions are made, including that the tensile and compressive prestress strains, respectively, increase and decrease the initial network shear stiffness but have no effect on the maximal shear modulus. We discuss the findings in terms of microstructural properties such as the local strain energy distribution.

Effect of random fiber networks on bubble growth in gelatin hydrogels.

In hydrodynamics, the event of dynamic bubble growth in a pure liquid under tensile pressure is known as cavitation. The same event can also be observed in soft materials (e.g., elastomers and hydrogels). However, for soft materials, bubble/cavity growth is either defined as cavitation if the bubble growth is elastic and reversible or as fracture if the cavity growth is by material failure and irreversible. In any way, bubble growth can cause damage to soft materials (e.g., tissue) by inducing high strain and strain-rate deformation. Additionally, a high-strength pressure wave is generated upon the collapse of the bubble. Therefore, it is crucial to identify the critical condition of spontaneous bubble growth in soft materials. Experimental and theoretical observations have agreed that the onset of bubble growth in soft materials requires higher tensile pressure than pure water. The extra tensile pressure is required since the cavitating bubble needs to overcome the elastic and surface energy in soft materials. In this manuscript, we developed two models to study and quantify the extra tensile pressure for different gelatin concentrations. Both the models are then compared with the existing cavitation onset criteria of rubber-like materials. Validation is done with the experimental results of threshold tensile pressure for different gelatin concentrations. Both models can moderately predict the extra tensile pressure within the intermediate range of gelatin concentrations (3-7% [w/v]). For low concentration (∼1%), the network's non-affinity plays a significant role and must be incorporated. On the other hand, for higher concentrations (∼10%), the entropic deformation dominates, and the strain energy formulation is not adequate.

Fibrous mesoporous polymer monoliths: macromolecular design and enhanced photocatalytic degradation of aromatic dyes

A bottom-up design provides polymer monoliths comprising two monomers that form mesoporous, compressible, random fiber networks, as in marine sponges.

Statistical analysis of the crack sensitivity of fiber networks

Fracturing two-dimensional random fiber networks of different densities (porosities) were statistically analyzed using both high-resolution finite element models and image analysis algorithms. Under small strains, the finite element fracture models revealed that networks with high relative densities were able to localize evolving fractures to small cracks while surprisingly larger cracks were required to localize fractures in networks of lower density. Further, it is indicated that the pore size distribution in fiber networks is rather diverse and can be captured using two mixed Gamma distributions; one part describing the background size distribution containing the vast majority of pores, and a second part describing the size distribution of larger pores and open regions. The second part covers less than 5% of the total network area but seems to be of paramount importance for the network’s global fracture behavior. It seems as a fiber network’s crack sensitivity is related to the average pore size in the second part of the mixed Gamma distribution. The analysis is remarkably consistent with reported experiments.

Constitutive models for random fiber network materials: A review of current status and challenges

A large number of materials have a stochastic fiber network as their main structural component. In some cases, the network is embedded in a matrix, while in some other cases materials are made exclusively from fibers. The field of homogenization and constitutive modeling of fibrous composites with solid matrix is well developed. However, situations in which the fibers form a crosslinked network (with or without matrix) have not been studied extensively. This article presents a summary of the state of the art in this area and outlines the challenges still to be met in the quest for the development of reliable and representative constitutive models for this type of materials. The discussion focuses on fibrous networks without matrix since this represents a limit case of network mechanics. A brief review of the available microstructure-informed constitutive models is presented. The goal of the article is to outline the limitations of current constitutive formulations for fiber network materials and to define the physical aspects important for the mechanics of such materials and which need to be captured by new models.

Size effects in random fiber networks controlled by the use of generalized boundary conditions

Materials with a stochastic fiber network as the main structural constituent are broadly encountered in engineering and in biology. These materials are characterized by multiscale heterogeneity and hence their properties evaluated numerically or experimentally are generally dependent on the size of the sample considered. In this work we evaluate the size effect on the linear and non-linear mechanical response of three-dimensional stochastic fiber networks and determine its dependence on material parameters and on the degree of affinity of network deformation. The size effect is more pronounced in non-affine networks than in affine networks and decreases slowly when the model size increases. In order to eliminate this effect, models lager than can be effectively solved with current computers have to be considered. To address this issue, we propose a method that allows using relatively small models, while accurately predicting the small and large strain behaviors of the network. The method is based on the generalized boundary conditions introduced in (Glüge 2013, Computational Materials Science 79, 408–416), which are being adapted here to the requirements imposed by fibrous materials.

Understanding extensibility of paper: Role of fiber elongation and fiber bonding

The tensile tests of individual bleached softwood kraft pulp fibers and sheets, as well as the micro-mechanical simulation of the fiber network, suggest that only a part of the elongation potential of individual fibers is utilized in the elongation of the sheet. The stress-strain curves of two actual individual pulp fibers and one mimicked classic stress-strain behavior of fiber were applied to a micromechanical simulation of random fiber networks. Both the experimental results and the micromechanical simulations indicated that fiber bonding has an important role not only in determining the strength but also the elongation of fiber networks. Additionally, the results indicate that the shape of the stress-strain curve of individual pulp fibers may have a significant influence on the shape of the stress-strain curve of a paper sheet. A large increase in elongation and strength of paper can be reached only by strength-ening fiber-fiber bonding, as demonstrated by the experimental handsheets containing starch and cellulose microfi-brils and by the micromechanical simulations. The key conclusion related to this investigation was that simulated uniform inter-fiber bond strength does not influence the shape of the stress-strain curve of the fiber network until the bonds fail, whereas the number of bonds has an influence on the activation of the fiber network and on the shape of the whole stress-strain curve.

In-plane and out-of-plane stiffness of 2D random fiber networks: Micromechanics and non-classical stiffness relation

2-dimensional (2D) random fiber networks are extremely thin plate-like structures constructed by randomly distributed fibers, and have been widely used in many fields from nano to macroscales due to their excellent properties. To make a comprehensive understanding on their in-plane and especially out-of-plane elastic properties, a mechanics model is established and analytical expressions for the in-plane and out-of-plane stiffnesses are derived based on micromechanics analysis. It is revealed that the fiber networks carry in-plane loads by axial stretching/compression of the fibers in the networks and resist out-of-plane loading by bending and torsion of the fibers. Due to the discreteness and randomness of the networks, the in-plane and out-of-plane stiffnesses present a non-classical relation, which can be described by a newly proposed formula. This work gives an insight into the understanding and designing of the mechanical properties of discrete materials/structures ranging from nanoscale to macroscale.

Effect of microstructure on porosity of random fibrous networks

Porosity of fibrous networks plays an important role in many processes, especially in filtration and separation applications. The paper aims to statistically analyze characteristics of porosity for random fibrous networks employing analytical and numerical approaches. An analytical model evaluating the number of pores related to microstructural parameters (e.g. fiber diameter) is proposed and validated with a numerical approach. An image processing of numerically generated fibrous networks was first performed to reveal pore-size distributions and then to calculate the number of pores with high accuracy. Additionally, analyses on orientation and crimp of fibers indicate that as more fibers are aligned in a principal direction, the number of pores decreases, and this induces the increase in the average pore size. The investigations also demonstrate that crimp does not affect pore characteristics in random distributions considerably; however, the number of pores decreases with fiber crimp as alignment increases.

Strain-stiffening and strain-softening responses in random viscoelastic fibrous networks: interplay between fiber orientation and viscoelastic softening

ABSTRACT Collagen networks are usually modeled as an inelastic material that accentuates the role of bond dynamics in their viscoelastic response. However, permanent cross-links can be used to fix these fibers to a large degree. Here, we model a network in which the fibers are assumed to maintain their viscoelasticity nature while the cross-linking bonds between them are fixed. Our models show that the interplay between fiber orientation and energy dissipation due to the viscoelasticity of fibers gives rise to a rich mechanical response. The network can stiffen, soften, or have an approximately constant stiffness as a function of strain when different loading rates are imposed. Using the order parameters of Q-tensor, we verify that the stiffening is due to the orientation of fibers, which is a well-established behavior in fibrous networks. In contrast to inelastic networks where there is always a linear initial response, the present network exhibits a constant stiffness only for a small range of loading rates. Also, it is shown that this duality is independent of the fiber concentration and only depends on the loading rate. Consequently, the existence of both strain-softening and strain-stiffening responses is a general behavior that is in common in fibrous networks with viscoelastic components and permanent cross-links.

Mechanical properties of subisostatic random networks composed of nonlinear fibers.

Fibrous protein networks provide structural integrity to different biological materials such as soft tissues. These networks display an unusual exponential strain-stiffening behavior when subjected to mechanical loads. This nonlinear strain-stiffening behavior has so far been explained in terms of the network microstructure and the flexibility of constituting fibers. Here, we conduct a comprehensive computational study to characterize the importance of material properties of individual fibers in the overall nonlinear mechanical response of random fiber networks. To this end, we consider three nonlinear material models, ranging from an almost linear form to a highly nonlinear one, for the fibers of subisostatic disordered networks. We characterize the amount of strain-stiffening as a function of bending rigidity of the fibers, the amount of nonlinearity of the fibers, and the connectivity of random networks. We find that networks composed of highly nonlinear fibers exhibit much more strain-stiffening than networks made up of linear fibers. Furthermore, the local strain distribution becomes more homogenous as the amount of nonlinearity in the material models increases. Increasing the network connectivity signifies the importance of the nonlinear material response of individual fibers in the overall mechanical behavior of networks. The constitutive behavior of fibers plays an important role in defining the failure response of networks particularly in the damage initiation and evolution. These important findings for how the mechanical response of individual fibers affects the overall mechanical properties of random networks could find applications in designing new biomimetic materials and could help scientists better understand the mechanical properties of biological materials.

Random fiber networks with inclusions: the effect of the inclusion stiffness

The mechanical behavior of random fiber networks containing inclusions is studied in this work. Inclusions are considered spherical and of radius comparable to the network mean segment length. Their stiffness is varied in a broad range, from very soft to rigid. It is observed that the presence of inclusions modifies the small strain stiffness of the network but does not change the functional form of the non-linear part of the stress-strain curve which, therefore, remains independent of the inclusion stiffness. Soft, bending-dominated networks can be reinforced more efficiently than the stiffer and more affinely deforming axially dominated networks. The variation of the composite modulus with the stiffness of inclusions does not follow the prediction of continuum homogenization theory. Numerical data that allow predicting the homogenized stiffness are presented.

Random auxetics from buckling fibre networks

Auxetic materials have gained increasing interest in the last decades, fostered by auspicious applications in various fields. While the design of new auxetics has largely focused on meta-materials with deterministic, periodically arranged structures, we show here by theoretical and numerical analysis that pronounced auxetic behaviour with negative Poisson’s ratios of very large magnitude can occur in random fibre networks with slender, reasonably straight fibre segments that buckle and deflect. We further demonstrate in experiments that such auxetic fibre networks, which increase their thickness by an order of magnitude and more than quintuple their volume when moderately extended, can be produced by electrospinning. Our results thus augment the class of auxetics by a large group of straightforwardly fabricable meta-materials with stochastic microstructure.