Halpin-Tsai Equation Research Articles

Mechanical properties of acrylate-grafted henequen cellulose fibers and their application in composites

The properties of cellulose fiber and PMMA- or PBA-grafted cellulose fibers are investigated as a function of the initiator (ceric-ammonium nitrate) concentration and the amount of grafted polymer onto cellulose fiber. The molecular weight of cellulose decreases while the crystallinity increases with an increment of initiator concentration because of the partial degradation of the amorphous zone of the fibers exposed to the oxidation by the initiator. This results in a reduction of the elastic modulus and tensile strength at high initiator concentrations. Degradation of cellulose is partially inhibited during the grafting process and, therefore, the effect of initiator on the mechanical properties is less notorious in the grafted cellulose fiber. The grafting of PMMA or PBA on the fiber results in lower mechanical properties than those of the ungrafted cellulose fiber. The reduction of the elastic modulus is independent of the amount of grafted PMMA or PBA, but the tensile strength decreases with the PBA content on the PBA-grafted fiber. Either the grafted or the ungrafted cellulose fibers improve the mechanical properties of plasticized PVC composites, and the best results are obtained for PMMA-grafted cellulose fibers because of the better fiber–matrix adhesion. The Halpin–Tsai equation seems to better agree with the experimental data when there is a good fiber–matrix adhesion. In contrast, for poor fiber–matrix adhesion the experimental data has a better agreement with the parallel arrangement equation.

A comparative study ofin-situ composite fibers reinforced with different rigid liquid crystalline polymers

Poly(ethylene terephthalate) (PET) and 2 thermotropic liquid crystalline polymers (LCPs) with different chain rigidity were blended to make in-situ composite fibers on a conventional melt spinning equipment. The addition of the LCP-1 (60PHB–PET) with a less rigid chain has been found to lower the orientation of the as-spun fibers while the LCP-2 (VectraA900) with whole aromatic rigid chain has a reverse effect, as evidenced from the birefringence results. Both kinds of composite fibers with 5 wt % LCP have a good drawability. There is a diffraction peak characteristic of intermolecular packing of LCP on the wide-angle X-ray diffraction curve for the as-spun fibers containing LCP-2 but not the case for LCP-1. The morphology formed during elongational flow is highly dependent on the composition and rigidity of LCP. For the dispersed phases of LCP-1, it is relatively difficult to be elongated, whereas LCP-2 dispersed phases will be easily deformed into fibrils during melt spinning. The mechanical properties of the blend fibers containing the LCP-1 component are inferior to those containing the LCP-2 component. For the fibers with discontinuous fibril morphology, a Halpin–Tsai equation could well be used to describe the elastic modulus of in-situ composite fiber with LCP-2. © 1998 John Wiley & Sons, Inc. J. Appl. Polym. Sci. 70: 1035–1045, 1998

Simple formulae for the effective moduli of unidirectional aligned composites

AbstractSimple closed form expressions for the effective tensile moduli of an unidirectionally aligned two phase composite reinforced by fiber‐like or flake‐like inclusions of aspect ratio α are obtained. The case of α ≫ 1 corresponds to fiber‐like inclusions whereas the case of α ≪ 1 corresponds to flake‐like inclusions. These expressions are based on the exact results obtained by Tandon and Weng, who used the Mori‐Tanaka method to obtain the effective moduli of such composites. These expressions are then compared with various empirically based formulae such as the Halpin‐Tsai equation and the Modified Rule of Mixtures equation. It is found that very large discrepancies result when the modulus ratio E0/E1 is sufficiently small compared with the aspect ratio of the flake‐like inclusions, where E1 and E0 are the Young's modulus of the inclusions and the matrix, respectively. Simple criteria for the validity of the rule of mixtures relation are also given. Also, the predictions of the simple closed form expressions are compared to available experimental results and favorable agreements are obtained.

Structure-modulus relationships for injection-molded long fiber-reinforced polyphthalamides

Abstract The structural information needed for modulus evaluation of long fiber-reinforced composites includes the fiber volume fraction, the modulus and the Poisson's ratio of the constituents, the fiber orientation and the fiber length distribution. In this work, the fiber orientation distribution is expressed in terms of three parameters which may vary with spatial location on the part: the average orientation angle on the surface, the average orientation angle on the mid-plane and the thickness of shell-core layers. With the independent evaluation of the fiber length distribution, composite models are used to evaluate mechanical property distribution such as modulus and Poisson's ratio as a function of spatial location on the part. The approach used in this work combines the Halpin-Tsai equations and a stiffness modeling scheme which follows a combination of iso-stress and iso-strain averaging techniques to determine the anisotropic material behavior of these composites. The predictions are compared with experimental data from injection-molded plaques fabricated at different processing conditions. Modulus variation with location is measured by fabricating dog-bone tensile samples from the plaques at four in-flow and four cross-flow positions. Modulus variability is shown to be high with location and the model captures this non-uniformity over the whole range of tested samples.

A Comparison of homogenization, Hashin-Shtrikman bounds and the Halpin-Tsai Equations

In this paper we study a unidirectional and elastic fiber composite. We use the homogenization method to obtain numerical results of the plane strain bulk modulus and the transverse shear modulus. The results are compared with the Hashin-Shtrikman bounds and are found to be close to the lower bounds in both cases. This indicates that the lower bounds might be used as a first approximation of the plane strain bulk modulus and the transverse shear modulus. We also point out the connection with the Hashin-Shtrikman bounds and the Halpin-Tsai equations. Optimal bounds on the fitting parameters in the Halpin-Tsai equations have been formulated.

Influence of Short Glass Fiber Addition on the Mechanical Properties of Sisal Reinforced Low Density Polyethylene Composites

This paper presents the evaluation of enhancement in the mechanical properties of short sisal fiber reinforced polyethylene composites by the incorporation of short glass fiber as an intimate mix with sisal. Intimately mixed short glass-sisal hybrid fiber reinforced polyethylene composites (GSRP) were prepared by solution mixing technique. The effects of fiber orientation and alkali treatment on sisal fiber in GSRP were studied. Addition of relatively small volume fraction of glass (0.03) to the sisal reinforced polyethylene matrix (SRP) enhances the tensile strength of longitudinally oriented composites by about 80%. Addition of the same volume fraction of glass to the alkali treated sisal incorporated SRP enhances the tensile strength by more than 90%. The flexural strength of the longitudinally oriented composites was also studied. The incorporation of glass fiber ( V f = 0.03) to SRP enhances the flexural strength by more than 60%. The effect of hybridization on water absorption tendency of the sisal fiber was studied by immersing SRP and GSRP in boiling water for 3 hours. It was observed that water uptake of GSRP was 2 to 4 times less than that of SRP composites. Halpin-Tsai equation for composites was tried for calculating the tensile modulus of longitudinally oriented GSRP.

Mechanical performance of the extruded blends of liquid-crystalline copolyester and modified polyphenylene oxide

Polymer blends of modified polyphenylene oxide (PPO) and thermotropic polyester were prepared by extrusion followed by drawing. The morphology and mechanical properties of the blends were examined by scanning electron microscopy, static tensile tests and dynamic mechanical analysis. The static mechanical tests showed that the liquid-crystalline polymers (LCP) additions led to an increase in both modulus and tensile strength of the blends, while elongation at break was generally decreased. The magnitude of the reinforcing effect depended significantly on the LCP content, but also on draw ratio. The Tsai-Halpin equation was used to estimate the aspect ratio of LCP microfibrils. The storage modulus showed that the LCP phase tended to reinforce the PPO matrix over a wide range of temperature but not near the glass transition zone.

Development of the ferrule for optical fiber connectors with short fiber polymer composites

In this work, the thermal expansion behavior of short carbon fiber/epoxy composites was investigated with respect to fiber length and fiber volume fraction. The effective coefficient of thermal expansion of short carbon fiber/epoxy composites was calculated based on the Halpin-Tsai equation for moduli and the linear coefficient equation. From the investigation, it has been found that the coefficient of thermal expansion is more dependent on fiber volume fraction than fiber length. Therefore, the ferrule for optical connector that has the compatible coefficient of thermal expansion with the optical fiber was manufactured with short carbon fiber/epoxy composite by controlling the fiber volume fraction.

Morphology and mechanical properties of thermoplastic elastomers from nylon-nitrile rubber blends

Nylon-nitrile rubber blends having different plastic-rubber component ratios (100/0, 80/20, 70/30, 60/40, 50/50, 40/60, 30/70, 20/80, and 0/100) were prepared by melt mixing technique in a Rheocord-90 at a temperature set at 180°C. The mixing characteristics of the blends have been analyzed from the rheographs. The morphology of the blend was studied using optical and electron microscopies, with special reference to the effect of blend ratio. The micrographs indicate a two-phase system where the component having lower proportions was found to disperse in the major continuous phase. A cocontinuous morphology was observed for 50/50 composition. Mechanical properties of the blends have been measured according to standard test methods. The effect of blend ratio on the mechanical properties like tensile strength, tear strength, elongation at break, stress-strain behavior, and hardness has been analyzed. The influence of the strain rate on the mechanical properties has also been analyzed. The mechanical properties were found to have a strong dependence on the amount of nylon in the blend. It is found that the blends with higher proportions of nylon have superior mechanical properties. The observed changes in mechanical properties are explained on the basis of the morphology of the blend. Various theoretical models such as Series, Parallel, Halpin-Tsai, and Coran's equations have been used to fit the experimental mechanical data. © 1996 John Wiley & Sons, Inc.

The Mechanical Properties and Deformation of Shear-Induced Polymer Liquid Crystalline Fibers in an Engineering Thermoplastic

In this paper, through the morphological study of surface-treated samples and by using the well-known Halpin-Tsai equations, the aspect ratio and the elastic modulus of the shear-induced fibers from a thermotropic liquid crystalline polymer (LCP, Vectra A950) in a polycarbonate (PC) matrix were determined. Based on Cox's classic theories about the development of disperse liquid drops in suspending medium, the deformation mechanism of the LCP phase in the thermoplastic matrix was quantitatively or semi-quantitatively studied. The relationships among the viscosity ratio of LCP and PC, Weber number and the fiber development are given.

Acousto-ultrasonic non-destructive evaluation of fatigue damage in steel-belted radial tires dos Reis, H.L.M. and Warmann, K.A. Insight (Dec. 1994) 36 (12), 958–963

A finite element tire model was developed in this study. In the model, the rubber compounds in tires were simulated by the incompressible element which was treated with based on the Lagrangian multipliers method. The non-linear mechanical properties of the elastomers were modeled by the Mooney–Rivilin model. The corresponding material constants of elastomers were obtained from experimental data. Belts, carcass and bead were modeled by the equivalent orthotropic material model in which the effective moduli were determined from the individual material properties of the rubber compound and cord based on the Halpin–Tsai equations. The contact constraint of a radial tire structure with flat foundation and rigid rim was treated with using the variable constraint method. For the large deformation description of tires, the Lagrangian method was used here. Numerical results showed that the model’ reliability and convergency are fairly good.

Tearing behavior of blends of isotactic polypropylene and nitrile rubber: influence of blend ratio, morphology and compatibilizer loading

Tearing behavior of blends of isotactic polypropylene (PP) and nitrile rubber (NBR) has been investigated with special reference to the effect of blend ratio and addition of compatibilizing agents. It has been observed that the tear strength of blends depends on the rubber concentration. Attempts have been made to correlate the tear strength with the morphology of the blends. Various composite models such as parallel model, series model, Halpin-Tsai equation and Coran's model have been used to fit the experimental values. The effect of addition of phenolic modified polypropylene (Ph-PP) as a compatibilizer on the tear strength of PP/NBR blend was investigated. The observed increase in tear strength with increase of Ph-PP concentration has been explained in terms of the reduction in particle size of dispersed NBR domains.

Tearing behavior and recyclability of nitrile rubber/poly(ethylene-co-vinyl acetate) blends

The tear strength of nitrile rubber/poly(ethylene-co-vinyl acetate) (NBR/EVA) blends was measured as a function of blend composition in an Instron universal testing machine. An increase in tear strength was observed with increasing EVA content. The variation in this property was correlated with the microstructural morphology of the system. The tear failure mechanism of these blends as a function of blend composition was studied by examining the failure surfaces and the fracture patterns correlated with the strength and failure of these materials. Various models such as parallel model, series model and the Halpin-Tsai equation were used to fit the experimental tear strength values. In order to analyze the reprocessability of these blends, the blends were recycled up to three times and the mechanical properties were measured. It was found that the properties of the blends were little affected by recycling.

Blends of isotactic polypropylene and nitrile rubber: morphology, mechanical properties and compatibilization

Morphology and mechanical properties of blends of isotactic polypropylene (PP) and nitrile rubber (NBR) have been investigated with special reference to the effects of blend ratio. Morphological observations of blends showed a two-phase system, in which the rubber phase was dispersed as domains in the continuous PP matrix at lower proportions of NBR (≲50%). The 30/70 PP/NBR blend was found to exist as a co-continuous system. Attempts have been made to correlate the changes in morphology with properties. The mechanical properties of blends were found to depend on the blend ratio. Various composite models, such as the series model, the parallel model, the Halpin-Tsai equation and Coran's model, have been used to fit the experimental mechanical properties. The effect of concentration of maleic-modified polypropylene (MA-PP) and phenolic-modified polypropylene (Ph-PP) as compatibilizers on the morphology and mechanical properties of the blend was also investigated. The compatibilizer concentrations used were 1, 5, 10 and 15 wt%. The domain size of the dispersed NBR particles decreased with the addition of a few per cent of the compatibilizer followed by a levelling off at higher concentrations. The levelling off was an indication of interfacial saturation. The mechanical properties of the blends were improved by the addition of the compatibilizer followed by a levelling off at higher concentrations. The levelling off was an indication of interfacial saturation. The mechanical properties of the blends were improved by the addition of the compatibilizers. The experimental results were compared with the current compatibilization theories of Noolandi and Hong.

Concise encyclopedia of composite materials

Aircraft and aerospace applications of composites. Applications of composites: an overview. Artificial bone. Asbestos fibers. Automobile tires. Automobile components: fabrication. Boron fibers. Carbon-carbon composites. Carbon-fiber-reinforced plastics. Carbon fibers. Composite armor. Continuous-filament glass fibers. Creep of composites. Dental composites. Failure of composites: stress concentrations, cracks and notches. Fatigue of composites. Fiber-reinforced cements. Fiber-reinforced ceramics. Fiber-reinforced polymer systems: extrusion. Fibers and textiles: an overview. Fibrous composites: thermomechanical properties. Friction and wear applications of composites. Glass-reinforced plastics: thermoplastic resins. Glass-reinforced plastics: thermosetting resins. Halpin-Tsai equations. Helicopter applications of composites. High-modulus high-strength organic fibers. High-performance composites with thermoplastic matrices. Hybrid fiber-resin composites. In situ composites: fabrication. In situ polymer composites. Joining of composites. Kirchhoff assumption (Kirchhoff hypothesis). Laminates: elastic properties. Long-term degradation of polymer-matrix composites. Manufacturing methods for composites: an overview. Metal-matrix composites. Molded fiber products. Multiaxial stress failure. Multilayers, metallic. Multiple fracture. Nanocomposites. Natural composites. Natural-fiber-based composites. Nondestructive evaluation of composites. Nonmechanical properties of composites. Oxide inorganic fibers. Paper and paperboard. Particulate composites. Percolation theory. Plywood. Polymer-polymer composites. Recycling of polymer-matrix composites. Silicon carbide fibers. Silicon nitride fibers. Smart composite materials systems. Solid fiber composites as biomedical materials. Strength of composites. Strength of composites: statistical theories. Structure performance maps. Thermosetting resin matrices. Three-dimensional fabrics for composites. Toughness of fibrous composites. Whiskers. Wood-polymer composites. Wood strength. Woven-fabric composites: properties

Dynamic mechanical properties of multi-functional epoxy resin filled with pitch-based carbon short fibers

Three types of multi-functional epoxy resins cured with six types of hardeners were filled with pitch-based carbon and graphite short fibers for 22.2 wt%. It was found by SEM observation that the distribution of short fibers in the resin was random. Dynamic mechanical properties were measured for short fiber filled composites in the temperature range of -150 to 300° C. The glass transition temperature (Tg) for the composites became higher than that for the matrices, and the dynamic storage moduli, E', were also increased. The values of E' for the graphite filled composites in the glassy region at low temperature (-140° C) and room temperature (25° C) were larger than those for the carbon filled composites but at high temperature (200° C), E' for the carbon filled composites became larger as the surface activation of carbon was higher than that of graphite. The value of E' for the composites at 25° C was larger than the estimated values from the rule of mixture for both types of composites, thus demonstrating the filling effect. The experimental values were in good agreement with the Halpin-Tsai equation modified by Nielsen. However, some variations on the increase of Tg and E' were seen depending on the combination of epoxy resin and hardener as well as the difference between carbon and graphite as the pitch-based carbon fibers. The filling effect on flexural strength was evident in the composites whose matrices were brittle. It was found that heat resistant and high modulus composites can be developed by using multi-functional epoxy resin as the matrix filled with functional short fibers.

Microscopic Simulation Technique for Composite Meterials Using the Homogenization Method Based on Fixed Grid.

Recently, the homogenization method is becoming a matter of interest as a new analysis method of composite materials. This method aims at stricter modeling and more precise simulation for any kind of reinforcement. Discretization of complicated microstructures of such reinforcement, however, becomes the bottleneck in the analysis procedure. Therefore, a new simulation technique based on a fixed grid is proposed in this paper. Fundamental investigation of the accuracy of the homogenization method based on a fixed grid for unidirectional FRP is presented. The obtained elastic constants show good agreement with the solution of the Halpin-Tsai equation. Microscopic stress distribution has also been evaluated well compared with conventional smooth finite-element discretization.

Mechanical properties of LDPE/granular starch composites

AbstractThe mechanical properties of composites of granular starch and low density polyethylene (PE) have been studied as functions of starch volume fraction ϕ, granule size, and presence of compatibilizer. Property–volume fraction relationships were interpreted using various theories of composite properties. The dependence of elongation (ϵ ∼ ϕ1/3) and tensile strength (σ ∼ ϕ2/3) agree with theoretical predictions, although the proportionality constants are less negative than theoretical values. The addition of compatibilzer (ethylene‐co‐acrylic acid copolymer, EAA) did not significantly affect the elongation or tensile strength, but significantly increased the composite tensile modulus. The cornstarch/PE moduli could be described by the Kerner or Halpin‐Tsai equations. Analysis of the composite moduli data using the Halpin‐Tsai equation allowed the estimation of the modulus of granular starch. The value obtained, 15 GPa, is considerably greater than most unfilled synthetic polymers of commercial importance, but significantly lower than the modulus of cellulose. It is also greater than a previously reported value of 2.7 GPa. © 1994 John Wiley & Sons, Inc.

The mechanical properties of ternary composites of polypropylene with inorganic fillers and elastomer inclusions

The effect of elastomer volume fraction and phase morphology on the elastic modulus of ternary composites polypropylene (PP)/ethylene-propylene rubber (EPR)/inorganic filler containing 30 vol % of either spherical or lamellar filler has been investigated. Phase morphology was controlled using maleated polypropylene (MPP) and/or maleated ethylene-propylene elastomer (MEPR). As revealed by SEM observations, composites of MPP/EPR/filler exhibit separation of the filler and elastomer and good adhesion between MPP and the filler, whereas composites of PP/MEPR/filler exhibit encapsulation of the filler by MEPR. Composite models were utilized to estimate upper and lower bounds for the elastic modulus of these materials, which is strongly dependent on the morphology of the ternary composite. A model based on the Kerner equation for perfect separation of the soft inclusions and rigid fillers gives a good prediction of the upper limit for relative elastic modulus as a function of filler and elastomer volume fractions. The lower limit, achieved in the case of perfect encapsulation, depends significantly on the particle shape. Good agreement was found between experimental data and lower limits predicted using the Halpin-Tsai equation for lamellar filler and the Kerner-Nielsen equation for spherical filler. In order to calculate reinforcing efficiency of the core-shell inclusions, the finite element method (ANSYS 4.4A, GT STRUDL) has been used.

Mechanical properties of in situ composites based on polycarbonate and a liquid crystalline polymer

The mechanical properties of in situ composites based on blends of polycarbonate and a liquid crystalline copoly(ester amide) (Vectra B950) have been studied as functions of the liquid crystalline polymer (LCP) concentration and the draw ratio, a processing parameter. It is shown that both the elastic modulus and the tensile strength of the in situ composites increase steadily with the LCP concentration and the draw ratio. However, the ultimate tensile strain decreases with these two parameters. A model is proposed for the longitudinal elastic modulus of the in situ composites, which is based on the Halpin-Tsai equation and Northolt's model for the LCP phase. The experimental elastic moduli of the in situ composites are found to conform fairly well with the theoretical values derived from the model.