Fibers Of Circular Cross Section Research Articles

Tangent second-order homogenisation estimates for incompressible hyperelastic composites with fibrous microstructures and anisotropic phases

This work presents new estimates for the effective mechanical response of composites with fibrous microstructures and hyperelastic, anisotropic, incompressible phases. The composites are two-phase materials consisting of aligned cylindrical fibres of circular cross section that are randomly distributed in the transverse plane of the matrix phase. The proposed homogenisation is based on the tangent second-order method proposed by Avazmohammadi and Ponte Castañeda (2013), which utilises variational principles and a suitably chosen “linear comparison material” to provide estimates for effective strain energies of nonlinear materials. To take into account the incompressibility of the fibre phase, we employed an appropriate asymptotic expansion of the fibre deformation field. By doing so, a constraint pressure is identified as the consistent limit of emerging indeterminate forms in the dilatational part of the fibre stress tensor. Furthermore, we performed an asymptotic analysis to obtain regular expressions of the underlying equations in the incompressible matrix limit. While we addressed the general case of incompressible and generally anisotropic phases, we put particular emphasis on composites with transversely isotropic phases whose symmetry axes are collinear with the fibre direction. For such materials, we derived estimates for the effective strain energies in terms of macroscopic strain invariants. In addition, for composites that are described by isotropic phase energies augmented with J4-invariant-based anisotropic terms, which is the case for many biological materials, we show that the anisotropic part of the energy can be easily homogenised by means of a simple Voigt-type average. Moreover, we present closed-form results for composites with isotropic Neo-Hookean phases augmented by transversely isotropic energy contributions. Finally, it is shown that the new estimates correspond to certain exact results under longitudinal shear deformation and that they properly linearise for small strains by recovering the corresponding linear-elastic estimates.

Refinement of the Maxwell formula for composite reinforced by circular cross-section fibers. Part I: using the Schwarz alternating method

The effective properties of the fiber-reinforced composite materials with fibers of circular cross section are investigated. The novel estimation for the effective coefficient of thermal conductivity refining the classical Maxwell formula is derived. The method of asymptotic homogenization is used. For analytical solution of the periodically repeated cell problem, the Schwarz alternating process is employed. The principal term of the refined formula coincides with the classical Maxwell formula. On the other hand, the refined formula can be used far beyond the area of applicability of the Maxwell formula. It can be used for dilute and non-dilute composites. It is confirmed by comparison with known numerical and asymptotic results.

Asymptotic and numerical homogenization methods applied to fibrous viscoelastic composites using Prony’s series

The paper focuses on the evaluation of the effective properties of linear viscoelastic composites with a periodic structure, containing long cylindrical fibers of circular cross-section and for two different cell arrangements: square and hexagonal unit cells. For this purpose, we use the two-scale asymptotic homogenization method (AHM) and a numerical homogenization method (NHM). Based on the correspondence principle, the local functions and the relaxation overall properties are obtained in explicit form by the AHM using the Prony series. Additionally, the NHM is established for a three-dimensional representative cell, and the problem is solved under appropriate boundary conditions, by using the Finite Element Method. The numerical results obtained by the AHM and NHM are compared and verified with other theoretical approaches. The comparisons show a good agreement and a benchmark for further experimental and theoretical investigations.

Computation of the relaxation effective moduli for fibrous viscoelastic composites using the asymptotic homogenization method

A two-phase parallel fibre-reinforced periodic viscoelastic composite is considered wherein the constituents are isotropy. Simple closed-form formulae are obtained for the effective properties of composites with square and hexagonal cells by means of the two-scale asymptotic homogenization method. The computation of the effective properties of non-ageing linear viscoelastic composites with periodic structure containing long cylindrical fibres of circular cross-section is performed. The local problems and overall viscoelastic properties are obtained in explicit form using the elastic-viscoelastic correspondence principle and assuming perfect contact conditions at the interface between constituents. Comparison with different viscoelastic models allowing explicit inverse Laplace transforms such as, traditional Maxwell and Kelvin models and Rabotnov-Scott Blair fractional exponential model are shown. The analytical results are verified by comparison with computational ones.

Performance of clay soil reinforced with fibers subjected to freeze-thaw cycles

In order to study the strength properties of the weak soil reinforced by fibers, a series of unconsolidated undrained triaxial compression tests of fine-grained soil subjected to freeze-thaw cycles were conducted. Experimental samples were made by adding various contents of basalt and polypropylene fibers (0, 0.25, 0.5, 0.75% of the dry weight of the soil) into clay soil; freeze-thaw cycles were performed in a closed system (0, 2, 5, 8, 10, 15 cycles). Reinforcing fibers of circular cross-section were employed. A decrease in the strength of the test samples was observed with an increase in the freeze-thaw cycle, by carrying out triaxial compression experiments. A comparison of the strength properties of clay soil between unreinforced samples and reinforced ones with different fibers was made. The greatest decrease in the strength of clay was observed in samples after 15 freeze-thaw cycles, but it was noted that the using of fibers helps to prevent a decrease in the strength of clay subjected to the freeze-thaw cycle. After 15 freeze-thaw cycles, the strength of all unreinforced and fiber-reinforced samples decreases approximately by 30–35%. When reinforcing 0.75% with polypropylene fibers, the strength increases by 70%. Inclusion of 0.75% basalt fiber increases the strength by 41.2% before freezing and by 27.1% after the 15th cycle. Cohesion of unreinforced soil after the 15th cycle is reduced from 56.9% of unreinforced soil to 50.4% with reinforcement 0.25%, up to 54.4% at 0.5% and up to 54.0% with reinforcement 0.75% basalt fiber. The resilient modulus of all samples decreases with increasing freeze-thaw cycles, however, samples which were reinforced with fiber showed an increase in the resilient modulus as compared with not reinforced samples both before and after freezing-thawing.

On quantitative expression in fibrous composites based on an exemplary distribution of roving glass-fibers

Abstract The paper deals with analysis of relationships between the microgeometry and mechanical properties of fiberglass i.e. epoxy resin composites with random fibers distribution. The mechanical properties (e.g. a flexural modulus E f , taking into account shear effects) of the material were determined in bending by Short Beam Shear Tests (SBST). Microgeometry of the structure was characterized on the basis of microscopic images of test specimens. The paper proposes a new measure of irregular reinforcement distribution expressed as an area ratio of the matrix surrounding the single fiber to the circuit of the fiber. The measure, denoted by G AB , is expressed in microns. In particular the relationship between the results obtained in the test samples with random distribution of fibers and theoretical regular structures of square (K) and hexagonal (H) type is presented. Specimens of the investigated composite (E-glass fibers and Epidian 5 epoxy resin), supplied by industrial partner, were cut out from raw material of helicopter blades girders. The obtained experimental results for the analyzed composite corresponded to approximation by the theoretical model of the square (K) type. The results can be used to estimate the flexural elastic modulus E f of the unidirectional composites with irregular arrangement of fibers, made manually or using the molding apparatus. Based on those results, it can be concluded that the square distribution model (K) is appropriate for numerical estimation of the strength composites properties reinforced with glass fibers of circular cross-section, having the diameter in the range of 10–16 μm.

Computation of the Effective Conductivity of Fibrous Composites with Imperfect Thermal Contact by Combination of Asymptotic Homogenization, Domain Decomposition and Finite Elements Methods

The methods of asymptotic homogenization, domain decomposition and finite elements are combined for the computation of the effective thermal conductivity of periodic biphasic fibrous composites with interfacial thermal resistance. The asymptotic homogenization method is used to obtain the so-called local problems on the periodic cell whose solution allows the calculation of the effective conductivity tensor. The numerical solution of the local problems requires a special treatment because of the temperature discontinuity on the interfaces due to the thermal barrier. In the present work these problems are decomposed into two, one for each phase, linked via a coupling condition. The finite element method, implemented in the software FreeFEM++, is employed to solve the resulting problems on each phase. FreeFEM++, which is based on the variational formulation of the problems, potentially allows to consider arbitrary shapes for both the fiber and the periodic cell cross sections. Numerical results for a square periodic cell with fibers of circular cross-section are presented and compared with results from other reported approaches.

Exact connections between the effective elastic moduli of fibre-reinforced composites with general imperfect interfaces

Transversely isotropic composites consisting of an elastic matrix reinforced by unidirectional elastic fibres of circular cross-section are studied. The displacement and traction vectors at the interface between a fibre and the matrix are assumed to be simultaneously discontinuous and governed by a general isotropic imperfect interface model. It is first proved that piecewise uniform strain fields transversely isotropic about the fibre direction can be generated inside such a composite. The existence of these piecewise uniform strain fields is then exploited so as to derive two exact connections between the five effective elastic moduli of the composite. These two exact connections are finally shown to cover as particulars cases all the relevant results reported in the literature for fibre-reinforced composites.

Effective elastic moduli of fiber-reinforced composites with interfacial displacement and stress jumps

The generalized self-consistent method (GSCM) is applied to obtain the closed-form estimates for the 5 elastic moduli of a transversely isotropic composite consisting of an isotropic matrix reinforced by unidirectional isotropic fibers of circular cross-section. The interfaces between the matrix and the fibers in this composite are taken to be imperfect and characterized by the general elastic isotropic model which includes as particular cases the widely used elastic spring-layer and membrane-type imperfect interface models. The displacement, strain and stress fields in an infinite homogeneous medium containing a composite cylinder with a general imperfect interface and subjected to each of 4 elementary remote uniform loadings are specified and used in deriving the estimates for the effective elastic moduli. These results extend and bridge the special relevant results reported in the literature for fiber-reinforced composites with imperfect interfaces characterized by the spring-layer or membrane-type imperfect interface model. Numerical examples are also given to illustrate some results.

Overall thermal conductivity of a fiber reinforced composite with partially debonded inhomogeneities

A homogenization problem for conductivity of a composite containing infinite fibers of circular cross-section with interface arc cracks is solved with the account of the orientation distribution of cracks. Solution for a single inhomogeneity with debonded zone is based on the known result on heat flux intensity factor and utilizes the connection between this quantity and resistivity contribution tensor. Maxwell's scheme is used for the homogenization procedure. The results are compared with exact numerical solution for the case when all the fibers are debonded and all the interfacial cracks have the same length and orientation. The application of the proposed approach is illustrated on the example of the recovery of information on damage accumulated in the process of hydrothermal aging.

Numerical and analytical analyses for active fiber composite piezoelectric composite materials

This study consists of the calculation of the effective properties for active fiber composites made of either circular or square cross-section fibers not only by using finite element analysis and representative volume elements, but also based on the asymptotic homogenization method. Thus, there is an investigation about different approaches, which have specific mathematical formulations and unique characteristics. The comparison between numerical and analytical approaches shows that the numerical results are in good agreement with investigations performed by both analytical and semi-analytical methods, mainly the predictions for loading applied in fiber direction. For active fiber composites made of circular cross-section fibers, the maximum difference between asymptotic homogenization method and finite element analysis is from 1.29% to 5.49% for mechanical and piezoelectric effective properties, respectively, considering representative volume element in square arrangement. However, for active fiber composites made of square cross-section fibers, the maximum difference between semi-analytical method and finite element analysis is from 2.15% to 17.09% for mechanical and piezoelectric effective properties, respectively, considering representative volume element in square arrangement.

Effective properties evaluation for smart composite materials

The purpose of this article is to present a method which consists in the development of unit cell numerical models for smart composite materials with piezoelectric fibers made of PZT embedded in a non-piezoelectric matrix (epoxy resin). This method evaluates a globally homogeneous medium equivalent to the original composite, using a representative volume element (RVE). The suitable boundary conditions allow the simulation of all modes of the overall deformation arising from any arbitrary combination of mechanical and electrical loading. In the first instance, the unit cell is applied to predict the effective material coefficients of the transversely isotropic piezoelectric composite with circular cross section fibers. The numerical results are compared to other methods reported in the literature and also to results previously published, in order to evaluate the method proposal. In the second step, the method is applied to calculate the equivalent properties for smart composite materials with square cross section fibers. Results of comparison between different combinations of circular and square fiber geometries, observing the influence of the boundary conditions and arrangements are presented.

On Maxwell's concept of equivalent inhomogeneity: When do the interactions matter?

The concept of Maxwell's equivalent inhomogeneity is re-evaluated in the context of the effective elastic properties of unidirectional multi-phase composite materials with the fibers of circular cross-section. The following key questions are addressed in this paper. Is the concept of equivalent inhomogeneity valid only for the materials with low volume fractions, as originally suggested by Maxwell? When do the interactions among the fibers matter? Could Maxwell's concept be generalized to allow for accurate estimates of the effective properties? The approach advocated in the paper provides an unique opportunity to study, in the framework of one model, the effects of the interactions among fibers on the effective transversely isotropic elastic properties of multi-phase composite materials.

Equivalent inhomogeneity method for evaluating the effective elastic properties of unidirectional multi-phase composites with surface/interface effects

A new technique is presented for evaluating the effective properties of linearly elastic, multi-phase unidirectional composites. Various effects on the fiber/matrix interfaces (perfect bond, homogeneously imperfect interfaces, uniform interphase layers) are allowed. The analysis of nano-composite materials based on the Gurtin and Murdoch model of material surface is also included. The basic idea of the approach is to construct a circular inhomogeneity in an infinite plane whose effects on the displacements and stresses at distant points are the same as those of a finite cluster of inhomogeneities (fibers of circular cross-section) arranged in a pattern representative of the composite material in question. The elastic properties of the equivalent inhomogeneity then define the effective elastic properties of the material. The volume ratio of the composite material is found after the size of the equivalent circular inhomogeneity is defined in the course of the solution procedure. This procedure is based on a semi-analytical solution of a problem of an infinite plane containing a cluster of non-overlapping circular inhomogeneities subjected to loading at infinity. The method works equally well for periodic and random composites and – importantly – eliminates the necessity for averaging either stresses or strains. New results for nano-composite materials are presented.

Fibrous encapsulation of single polymer microfibers depends on their vertical dimension in subcutaneous tissue.

The purpose of this research was to investigate possible explanations for why small-diameter microfiber implants do not experience encapsulation in subcutaneous tissue as do large-diameter fiber implants. Single polypropylene microfibers of approximately rectangular cross-section with rounded edges were twisted about their longitudinal axes and affixed at their ends to polycarbonate frames. The frames were implanted in rat subcutaneous dorsum for a 5-week period, then removed and processed for light microscopy analysis. Fibrous capsule presence/absence and thickness around the implants were assessed, and their relationships to geometric features of the fibers investigated. A logistic regression analysis between presence/absence of a fibrous capsule and geometric features of interest demonstrated strong predictive ability (92.4% correct predictions) for implant height and a well-defined threshold separating the presence and absence of a fibrous capsule at 5.9 microm (p < 0.001). Implant height was defined as the vertical distance between the most superficial and deepest level of the implant. This 5.9-microm threshold value of implant height is comparable to the 6.0-microm diameter threshold for capsule presence/absence in fibers of circular cross-section [Sanders et al. J Biomed Mater Res 2000; 52(1):231-237]. Fiber major axis length, minor axis length, aspect ratio, surface area per unit length, implant width, and implant angle did not show similar predictive ability or a well-defined threshold separating the presence and absence of a fibrous capsule. It is reasoned that for fibers greater than the threshold height of 5.9 microm, separation of collagen fibers in the extracellular matrix creates dead space regions adjacent to the fibers that attract inflammatory cells and stimulate fibrous capsule formation.

Quadruple suspension design for Advanced LIGO

In this paper, we describe the conceptual design for the suspension system for the test masses for Advanced LIGO, the planned upgrade to LIGO, the US laser interferometric gravitational-wave observatory. The design is based on the triple pendulum design developed for GEO 600—the German/UK interferometric gravitational wave detector. The GEO design incorporates fused silica fibres of circular cross-section attached to the fused silica mirror (test mass) in the lowest pendulum stage, in order to minimize the thermal noise from the pendulum modes. The damping of the low-frequency modes of the triple pendulum is achieved by using co-located sensors and actuators at the highest mass of the triple pendulum. Another feature of the design is that global control forces acting on the mirrors, used to maintain the output of the interferometer on a dark fringe, are applied via a triple reaction pendulum, so that these forces can be implemented via a seismically isolated platform. These techniques have been extended to meet the more stringent noise levels planned for in Advanced LIGO. In particular, the Advanced LIGO baseline design requires a quadruple pendulum with a final stage consisting of a 40 kg sapphire mirror, suspended on fused silica ribbons or fibres. The design is chosen to aim to reach a target noise contribution from the suspension corresponding to a displacement sensitivity of 10−19 m Hz−1/2 at 10 Hz at each of the test masses.

Novel developments in mid-IR fiber-optic spectroscopy for analytical applications

Mid-infrared spectroscopy using the attenuated total reflection (ATR) technique plays an important role in analytical chemistry. Among others, silver halide fibers are promising for the development of remote sensor probes. Studies on long-term stability under various storage conditions were carried out, which demonstrate excellent stability for appropriately encased fibers. For the first time, fibers of square cross-section shape are presented and their advantages for infrared spectroscopic applications in comparison to fibers of circular cross-section are demonstrated. Various probes, employing such fibers, were developed for measurements of liquid and inhomogeneous solid samples. A short, u-shaped fiber was directly coupled to an MCT-detector, enabling ATR-measurements of solid samples with a minimum spot size of 20 μm×60 μm. Using the same optical setup, lowest concentration detection limits were achieved for aqueous solutions. For routine analysis of liquid samples, two types of probes were developed with the use of silver halide fibers of square cross-section. One ATR-probe contained an exchangeable, u-shaped piece of fiber, whereas the second probe was made of a fiber coupled diamond micro-prism; the latter is especially advantageous for small sample volumes to be measured or when extreme chemical inertness of the ATR-crystal is required. Measurements of different beverages were performed with high spectral reproducibility. For routine analysis demanding ultimate sensitivity, exchangeable sensor elements with an Ag 2S coating were tested successfully to give an ATR-absorption enhancement by more than a factor of 10 compared to uncoated fibers.

Icosahedral quasiperiodic packing of fibres

Methods for constructing icosahedral packing of fibres are reported. Closest packing constructions were realised using straight fibres of circular cross-section and threaded parallel to five- or three-fold icosahedral axes, or both. Their symmetry point group and related 2D tiling as obtained by cut and projection from periodic hyperspace, are presented in the case of the closest packing of fibres only parallel to the six five-fold axes. As a result of the analysis of the related 2D tiling, constructions can also be made using simple 2D building rules. Future experiments on elasticity properties of composite materials with a quasiperiodic fibrous reinforcement are discussed.

An Elliptic Regularity Result for a Composite Medium with "Touching" Fibers of Circular Cross-Section

In this paper we consider the elliptic equation $\nabla \cdot a\nabla u= 0$ in a two dimensional domain $\Omega$, which contains a finite number of circular inhomogeneities (cross-sections of fibers). The coefficient, a, takes two constant values, one in all the inhomogeneities and one in the part of $\Omega$ which lies outside the inhomogeneities. A number of the inhomogeneities may possibly touch, but in spite of this we prove that any variational solution u (with sufficiently smooth boundary data) is in $W^{1,\infty}$. For this very interesting, particular type of coefficient, our result improves a classical regularity result due to DeGiorgi and Nash, which asserts that the solution is in the Hölder class $C^\gamma$ for some positive exponent $\gamma$.

Analysis of Liquid Crystalline Fiber Coatings

A phenomenological model for the spreading of nematic liquid crystals on fibers of circular cross section is presented. The model is used to calculate the wetting thresholds of nematics on fibers for three chracteristic director orientations: axial (undistorted), radial (splay), and azimuthal (bend): the wetting thresholds are given in terms of critical film thickness hc and critical spreading parameter Sc. When the spreading parameter S is greater than the critical value Sc, nematic liquid crystal fiber wetting is similar to that of viscous fluids when director distortions are absent. In other cases when nematic liquid crystals spread on fibers the distortion due to the cylindrical geometry gives rise to Frank elasticity. For symmetric director boundary conditions the characteristic modes of splay and bend increase the magnitude of critical spreading parameter Sc. It is found that when spreading occurs (S>Sc) the equilibrium film thickness decreases with increasing elastic storage. The film thickness for rod-like and disc-like nematics differs because the splay-bend elastic anisotropy is reversed.