UNIVERSITY OF WISCONSIN SYSTEM SOLID WASTE RESEARCH PROGRAM Student Project Report

Transcription

1 UNIVERSITY OF WISCONSIN SYSTEM SOLID WASTE RESEARCH PROGRAM Student Project Report Characterization of Regenerated Cellulose for Bio-based Epoxy Fibrous Composites May 2013 Student Investigator: Issam Qamhia Advisor: Professor Rani El-Hajjar Department of Civil Engineering & Mechanics; Materials Science & Engineering University of Wisconsin-Milwaukee 1

2 Introduction: The objective of this research is to characterize regenerated and nano-cellulose fibers and their composites with epoxy for mechanical properties, and to evaluate manufacturing techniques for these composites. Cellulose is the most abundant natural material and is the main component of many solid waste products of households, businesses and construction projects. Experimental and computational fracture mechanics and strength approaches are used to suggest an optimum architecture of the reinforcement to produce the desired mechanical properties. The resultant optimum design of the reinforcement architecture is expected to give rise to a new method of producing high-strength, bio-based and sustainable composites for different regenerated cellulose fibers and cellulose nano-fibers. The results of this research are presented and published in the following conference and journal publications: 1. Qamhia, I., El- Hajjar, R. F., Processing of Nanocellulose Scaffolds for Increased Fiber Content Thermosetting Composites. 1st International Conference on Natural Fibers, Guimarães, Portugal, 2013 June Qamhia, I., Shams, S. S., El- Hajjar, R. F., Analytical Prediction of Elastic Properties In Triaxially Braided Regenerated Cellulose Composites. 1st International Conference on Natural Fibers, Guimarães, Portugal, 2013 June Qamhia, I., El- Hajjar, R. F., Preparation and Thermomechanical Characterization of Nanocellulose Scaffolding / Thermoset Composites, Advancements in Fiber- Polymer Composites: Wood Fiber, Natural Fibers and Nanocellulose Conference; 2013 May 6-7; Milwaukee, WI, USA 4. El- Hajjar, R. F., Qamhia, I., Shams, S. S., Analytical Characterization Of The Mechanical Properties In Triaxially Braided Regenerated Cellulose Composites, Advancements in Fiber- Polymer Composites: Wood Fiber, Natural Fibers and Nanocellulose Conference; 2013 May 6-7; Milwaukee, WI, USA 5. Qamhia, I. I., El- Hajjar, R. F., Mechanical and Thermal Properties of Nanocellulose Scaffolding/Epoxy Composites Prepared by a Vacuum Assisted Heated- Press Approach,. Poster session presented at: Sustainable Cities and Infrastructure. 10th Annual Sustainability Summit and Exposition; 2013 March 6-7; Milwaukee, WI, USA. 6. El- Hajjar, R. F., and Qamhia, I. I., Modeling and Characterization of the Moisture Dependent Bilinear Behavior of Regenerated Cellulose Composites, Journal of Wood Science, Accepted for Publication

3 Part I: Lyocell fibers and composites: Objectives: The goal of this research is to characterize regenerated cellulose fibers as a potential natural reinforcement for composite materials. Understanding the characteristics of cellulose fibers and composites means that potential applications can be deducted and the fibers can be produced from wood waste products to help reduce waste in Wisconsin. The objectives of the research are: Perform initial characterization of the properties of Lyocell fibers strands. Evaluate the effect of moisture on the mechanical properties of Lyocell fibers and Lyocell fibers composites. Examine the adhesion of thermosetting resins, especially epoxies, to regenerated cellulose fibers. Epoxy is chosen since some of its types are bio-based, increasing the appeal for green celluloseepoxy composites. Produce and evaluate the mechanical properties composites from regenerated cellulose fibers. Run a finite element analysis (FEA) model to correlate experimental data to it and use it to predict the behavior of Lyocell/epoxy composites at conditions different from those set in experiments. Experimental procedure: The tension behavior of the dry and wet Lyocell fibers (Tencel; Lenzing Fibers Inc., Axis, Alabama, USA) and composites made from epoxy and different fiber volume fractions of Lyocell were studied. Lyocell fibers used were in an uncrimped and unbleached state with staple fibers form. Experiments were carried out at room temperature and a humidity level between 30-35%. The first experiment was to characterize the mechanical properties of wet and dry Lyocell composites. The ASTM ID: D3822 standard was used for determining the fiber properties. Tow strands of the same weight and gauge lengths of 25.4 mm (1.0 in) to 254 mm (10 in) were attached to cardboards using a small amount of epoxy (Figure 1.a). Samples were tested under dry conditions, wet conditions by soaking in a water bath for a 120 ± 5 minutes period and wet conditions by extending the soaking period to 240 ± 5 minutes. Samples were tested under direct tension by applying load from an electro-mechanical loading machine and using a load cell of maximum capacity of 2220 N (500 lb) as shown in figure 1.b. 3

4 For the second experiment, composite specimens were prepared using a wet layup and resin infusion method. The Lyocell Fibers were weighted to back-calculate the fiber volume fraction. The epoxy chosen for this study (Super Sap 100 Epoxy; Entropy Resins Inc., Gardena, California, USA) contains bio-renewable materials sourced as co-products or from waste streams of other industrial processes, such as wood pulp and bio-fuels production. Samples with low fiber volume fractions (5-10%) were prepared in a dumbbell shaped form according to the ASTM ID: D638 standard. For higher fiber volume fractions, preparing panels using a resin-infusion and manual layup followed by degasing processes was introduced. Panels produced using wet layup resulted in a maximum fiber volume fraction of 0.33 whereas the ones with resin infusion resulted with a volume fraction of Testing of composite coupons was performed under direct tension according to the ASTM ID: D3039 standard. Strain properties were measured across a 25.4 mm (1.0 inch) gage length in the middle of coupons by the use of an extensometer. (a) (b) (c) (b) Figure I.1. (a) Lyocell tows attached to cardboards (ASTM D3822) (b) Testing of Lyocell Tows (c) Testing of Lyocell/epoxy composite Unit cell finite element model: The constitutive response of the regenerated fibers was found to be dependent on the moisture condition of the fibers. This suggests the need for a suitable analysis framework for these composites. Regenerated Lyocell fibers possess a bilinear material behavior with elastic-plastic tendencies. Under moisture exposure, the elastic response gradually dissipates and results in a largely plastic unrecoverable behavior. For the purpose of this research, a multi-scale analysis approach for simulation of Lyocell/bio-based epoxy composites that uses a representative unit cell is used. The behavior and mechanical properties of the constituents is independently recognized. This modeling approach is implemented in a p-version finite element analysis (p-fea) approach. The primary advantage of this approach is the ability to check the convergence of the solutions with increasing element order reducing the dependency on the mesh size. StressCheck (Stress Check V9.0; ESRD, St. Louis, Missouri, USA) software was used to create and run 4

5 the model. The typical unit cell model used for the hexagonal packing contained 178 elements consisting of 72 hexahedral and 106 pentahedral elements. A hexagonal packing geometry was used to represent the packing of the Lyocell fibers in the matrix. A bilinear elastic-plastic stress-strain relationship was used to correlate the model to experimental data. The model was calibrated for one set of fiber volume fraction (FVF) and was thereafter ran with different FVF for validation purposes of the modeling capability. Loading was applied in the form of a constant displacement on one face in the fibers direction. In and outof-plane displacements on the opposing face were restricted. The results of p-fea modeling were compared to those obtained by experiments in the fiber direction by calculating the average stress resulting on the loaded surfaces. Figure I.2. Unit cell FEA model for Lyocell/epoxy composites (micro-mechanical structure) Significant results Dry and wet Lyocell fibers The results show a bilinear elastic-plastic response in the mechanical behavior of the dry fibers. The dent in the loading curves between the two regions can be attributed to a more preferable molecular orientation of the fibers which is produced by stretching the fibers and stabilized by interconnections between crystalline regions and by hydrogen bonding (Morton, 2008). The load versus crosshead displacement results for the tow testing is shown in figure I.3. The behavior of the wet specimens shows a change in the constitutive response. For surface dried fibers, an increase in weight by 1.8 times the original weight was tracked for the two hours soaking and a 1.9 times the original weight for the four hours soaking. The loss of modulus as the fibers are wetted is clearly shown by the loading curves. Some variability in the mechanical behaviour is shown for 2 hours soaking but the results are more consistent for 4 hours soaking 5

6 and show a drastic loss in modulus and a non-linear behavior of the fibers. A reduction in the failure stress accompanied by a higher strains to failure is also observed for the wet fibers compared to the dried ones. Additionally, the knee seen in the loading curves for the dry fibers is shifted toward the (0, 0) point of the curves as the level of moisture increases in the fibers due to the removal of the hydrogen bonds (Morton, 2008). The behavior of the wet fibers is plastic. Figure I.3. Load-displacement curve for dry and wet Lyocell tows showing the effect of moisture on the behavior of the fibers. Lyocell/epoxy composites Panel preparation by wet layup followed by degasing was found to produce better samples than other examined procedures. Pilling and fibrillation are inherent in the structure of Lyocell and adds difficulties to the manufacturing process. Using resin infusion, thinner samples with higher FVF were produced. However, good infusion of the resin with the fibers could not be achieved for the whole panel and some areas were not properly wetted. Equal distribution of fibers among the panel and waviness prevention were two major challenges of the wet layup method. By using tape to stretch and fix the fibers in position, the effect of these challenges was reduced. Degasing the samples with a vacuum pump after wet layup also insured minimal porosity. Finite element model The p-fea model proposed was calibrated for one set of experimental values using the specimens having a fiber content of 33%. Error analysis is performed on the results by examining the global energy norm versus the polynomial order and degrees of freedom. The predictive properties of the stress strain graphs 6

7 are shown in figure I.4 superimposed with the FE calibration curve. The p-fea model was successfully used to capture the experimental stress-strain behavior of the composites and the FE model was calibrated to average the differences seen in the plastic region for the samples. The results show that it is possible to reproduce accurate results for different fiber volume fractions. Changes in the elastic and plastic modulus of Lyocell-epoxy composites with different fiber contents is shown figure I.5. A linear change is seen which can be attributed to the use of a bilinear stress-strain curve for Lyocell and a linear one for epoxy. This linear mechanical behavior is expected to change with the wet composites where the non-linearity of the Lyocell loading curve is expected to lead to nonlinearity in the stress-strain curve of the composites. Some of the coupons with the high fiber volume fraction were tested for the effect of moisture on Lyocell composites by soaking for two hours investigate how they compare to the model of dry Lyocell composites. The result of these are shown in figure I.5. Figure I.4. Experimental and FEA model results for stress-strain curves of Lyocell/epoxy composites for different fiber volume fractions (19% and 33%). 7

8 Figure I.5. Experimental and FEA model results for the change in elastic and plastic moduli with fiber volume fraction. Part II: Cellulose nanofibers and composites: Objectives: The goal of this research is to lay the groundwork for using Cellulose NanoFibers (CNFs) Aerogels as reinforcements in epoxy-based composites. This research is aimed to compliment ongoing efforts at Forest Products Laboratory (FPL) to produce reinforcements from CNFs. Specifically, the objectives are to: Perform initial investigations into the swelling and liquid flow into and through CNF scaffolds. Examine the adhesion of thermoset resins, especially epoxies, to CNFs. Investigate different fabrication techniques for CNF/epoxy composites to produce composites with the highest possible CNF content and minimal porosity. Investigate the effect of the presence of cellulose on resin curing behavior. Experimental procedure: The material used for this study was a CNF scaffold provided by Forest Products Laboratory; Madison. The experimental part of this research was to characterize the nanocellulose scaffolds, investigate the wetting of the nanocellulose by different resin systems and to investigate different fabrication procedures for the composites. Two resin systems were used to prepare composites, namely: Super Sap 100 Epoxy (Super Sap 100 Epoxy ; Entropy Resins Inc., Gardena, California, USA) and Embed-It epoxy (Embed- It Low Viscosity Epoxy; Polysciences, Inc, Warrington, Pennsylvania, USA). Composites were prepared from the two resin systems by manual layup followed by degasing. The Samples with Super-Sap epoxy showed very little wetting due to the high viscosity of the resin. Samples made with Embed-it epoxy showed better wetting. Thus, Super Sap epoxy was eliminated and Embed-it epoxy was used due to its low viscosity (65 cps). The fabrication techniques investigated were as follows: 1. Wet Layup followed by degasing Samples in Embed-It epoxy were degased by a vacuum pump for 2 hours at 610 mm Hg (-24 in. Hg) over a hot plate, and were cured for 16+ hours at 175 ºF. These samples showed good impregnation. Embed-It resin-only samples were prepared as a control point. Samples prepared by this technique showed proper wetting but resin content was very high, leading to low fiber content. 8

9 Figure II.1 Sample preparation of CNF composites by wet layup (left) and cured CNF/epoxy composites (right) 2. Resin infusion The high viscosity resin only was used to fabricate composites by resin infusion. Samples produced were highly pressed; yet a large void content was observed. Nanocellulose was not well impregnated by resin, especially at midsections due to the high viscosity of the resin. 3. Wet Layup followed by degasing and hot pressing This fabrication method was adopted to increase Nanocellulose content in the prepared composites. Samples were prepared using Embed-It epoxy by manual layup followed by degasing for 30 minutes. Samples were then inserted in a hot-press for 7 hours; then removed and left to complete curing overnight at 175 ºF. Hot press was used to apply a pressure (100 lb load) as well as heat (160 ºF) to the samples and led to 4-5 times higher FVF than the previous procedure. Fiber volume fractions up to 5-7% by volume were obtained using this technique. The mechanical testing part of this study included carrying out a three points bending test on the prepared composites using the recommendations of ASTM D790-07standard. Sample dimensions fell within acceptable limits recommended by the standard. Stresses and strains were calculated according to standard formulas for three point bending tests:!! = 3!" 2!!! (1) ɛ! = 6!"!! (2) Where σ f and ɛ f are the flexural stress and strain respectively, P is the applied load, L, b and d are the span length, width and depth of the specimen respectively, and D is the maximum deflection at the center of the beam. The bending test setup is shown below. 9

10 Figure II.2 Sample preparation of CNF composites by hot pressing (left), an image of the hot press (center) and the three points bending test setup (right). Other Experiments carried out to characterize the thermal behavior of the composites upon heating were Differential scanning calorimetry (DSC) and Thermogravimetric analysis (TGA). DSC is a thermoanalytical technique in which the difference in the amount of heat required to increase the temperature of a sample and reference is measured as a function of temperature. Round samples of around 25mg in weight were cored out from CNF/epoxy composites and pure Embed-IT epoxy resin samples for testing. The main aim of the test was to see if cellulose is changing the glass transition temperature of the resin. However, more important information about decomposition of the composites, and curing were obtained. TGA is a type of testing performed on samples that determines changes in weight in relation to a temperature program in a controlled atmosphere. Such analysis relies on a high degree of precision in three measurements: weight, temperature, and temperature change. The main use for this technique was to find the decomposition temperature of CNF/epoxy composites. Experiments were carried out in Argonne gas to prevent oxidation. Significant Results: Transparency and density analysis reveal that the Super-Sap epoxy resin did not infuse well with the aerogels for both resin infusion and wet layup techniques. Resin infusion produced better results but the percentage of voids was still high. By the use of low viscosity epoxy resin (Embed-It), the curing time for manual layup was relatively high (around hours), but good results of infusion were obtained. The use of the hot press enabled obtaining higher cellulose content in the composites (up to 7.5% by volume) and ensured minimal porosity. Samples prepared by this technique showed higher elastic modulus in addition to higher strength to failure when compared to the pure resin and to lower enforcement levels of cellulose prepared by manual layup. 10

11 The Embed-It resin was found to be sensitive to curing temperature and curing cycles. When the hot pressing preparation method was used, the mechanical properties of the pure resin samples and the nanocellulose composites were improved in terms of modulus and failure stress. Results for samples prepared by this technique shows that the nano-cellulose is not significantly improving the mechanical properties (figure II.3). A 15% average increase in modulus is observed for reinforcement levels of 5-7.5%. However, nano-cellulose samples are more brittle and fail at a lower stress. DSC results reveal the importance of proper resin curing to the properties of the composites. No exothermic peaks are observed for samples prepared by hot pressing indicating that the resin is well cured. Glass transition temperature (T g ) for the pure resin was found to be around 265 ºC. When nanocellulose is added, data on T g is inconclusive. TGA results indicate that the decomposition temperature of cellulose/epoxy composites is 270 ºC; where a noticeable mass drop is observed (figure II.4). Figure II.3 Flexural stress-strain behavior of CNF/epoxy and pure epoxy composites 11

12 Figure II.4 Sample preparation of CNF composites by wet layup (left) and cured CNF/epoxy composites (right) Part III: BioMid fibers and composites: Objectives: The goal of this research is to characterize BioMid regenerated cellulose fibers composites for potential use as naturally reinforced composites in industries. Understanding the mechanical behavior of these composites means that potential applications can be deducted and the fibers can be produced from wood waste products to help reduce waste in Wisconsin. The objectives of the research are: Examine different test methods (Mechanical, Acoustic Emission) for testing regenerated cellulose fiber composites. Evaluate notched and un-notched test specimens for determining the mechanical behavior of the Regenerated Cellulose Fibers/Bio-Epoxy specimens. Investigate the applicability of a proposed Modified Classical Lamination Plate Theory Model to capture the mechanical properties relating them to the undulations in the bias yarns and fiber content. Experimental procedure: A triaxially braided Regenerated cellulose BioMid fiber system with [-60, 0, 60] fiber orientation was used for reinforcement in test specimens. A&P Technology, OH performed the braiding of the materials used in this study. Fiber architecture is produced by having the bias yarns alternating two over and two under the axial yarns. The epoxy used in this study is a Super Sap 100/1000 (Entropy Bio-Resins, Gardena, California, USA). The resin also has a total calculated biomass of 50%. The actual fiber volume fraction of the composite is determined by using the volume measurements of the matrix and fiber constituents, resulting in an average fiber volume fraction of 0.60 with a sample standard deviation of The spacing between the bias yarns was at 26 mm, and between the axial yarns the center-to-center spacing was 5.2 mm. The axial bundle would fill a space of approximately 2.5 mm in width between the bias yarns. 12

13 Figure III.1 Braided BioMid sample showing yarns orientation For the early trials, a wet layup technique was used to prepare the composites, but the fibers weren t fully infused, especially the axial yarns. The early trials of resin infusion produced medium levels of porosity, a modification of the technique was implemented; in which silicon connectors were inserted inside the vacuum bag and the pipes were attached to them and insulated with tape from outside the bag. With this modification, the levels of porosity reduced drastically. The panels were infused under room temperature conditions. BioMid fibers were stored at room temperature and a humidity level of 30%. Resin infusion was carried out at room temperature and a humidity level of 30%. Panels were cut into 305 mm long by 38 mm wide coupons in transverse and longitudinal directions. Some panels were also cut into 152 mm by 76mm wide samples. For these samples, a 19mm notch was cut on both sides of the sample towards the middle. It is difficult to obtain reliable strength properties due to failure mode inconsistencies. Notched specimens are more suited to strength measurements because all bias fiber tows are gripped. Notched samples are also easy to fabricate and are expected to generate higher measured loads. The elastic properties were obtained by testing the specimens in tension according to ASTM ID: D3039. Standard for tensile properties of polymer reinforced fiber composites. The experiments were performed in a displacement-controlled mode at a displacement rate of 1.3 mm/min. 13

14 (a) (b) Figure III.2 (a) preparation of BioMid/epoxy composites by resin infusion (b) tensile testing of notched and un-notched samples. For one longitudinal coupon and one transverse coupon, simultaneous mechanical testing, and acoustic emission (AE) tests were carried out. Acoustic emission was used to investigate damage initiation and damage propagation through the energy bursts levels of events and the timing of these events and relating these to changes in the loading curve obtained by mechanical testing. A Physical Acoustics WS-alpha transducer was used for measuring the AE signals. The sensor has an operating frequency of khz with a 63.74dB peak sensitivity occurring at khz. In order to improve the quality of the captured signals; a pre-amplifier coupled with a bandpass filter adjusted to pass frequencies in the range of khz was used. A Mistras 2001 data acquisition system was used for recording the data, including waveforms. Analytical model: An analytical model was used to correlate experimental results for mechanical properties and provide a solution to determine the sensitivity of the braided materials to braiding properties such as crimp angle and bias yarns angles as well as the composites properties such as fiber content. A discrete 3-layer analytical model was evaluated for this purpose. The analytical model uses a representative volume element (RVE) or a representative unit cell to predict the macroscopic properties of triaxial composites from constituent microscopic material properties. The model presents yarns architecture as a total equivalent stiffness matrix of the triaxially braided composite in a unit cell. El-Hajjar et al. (2013) examined various analytical methods and found consistent results with experiments on glass fibers using this approach. The global stiffness of the composites is given as (Shokrieh and Mazloomi, 2010): [ C RUC ] = t [ C ] + t [ C ] + t [ C ] (3) + θ + θ θ θ 0 0 global global global where t + θ, t θ and 0 t are the thickness of each layer to the thickness of the laminate. In this research we use the actual thickness values obtained from microstructural examinations of the cross sections of the prepared composites as inputs for the model. Significant Results: 14

15 The stress versus strain curves for longitudinal and transverse specimens loaded in tension shows that are shown in figures III.3 and III.4. Two distinct regions can be identified for the stress-strain curve in which the first is linear elastic and the second is the region with a lower stiffness. The loading properties are similar to other regenerated cellulose composites, eg. Lyocell/epoxy composites (El-Hajjar and Qamhia 2013). The loading behavior and modeling of Lyocell composites was discussed in part I. The results of cumulative acoustic emissions (figure III.5) show that the initiation of damage for transverse specimens starts at a much lower stress level compared to the early emissions observed in the longitudinal specimens. The initial emissions are most likely attributed to the matrix cracking. Higher energy emissions before failure are observed for the longitudinal specimens where the final cumulative energy at failure is 35% higher. The reason for this is thought to be associated with the fracture of the axial fibers oriented in the direction of loading. The incidence of high acoustic emissions can be used as an indication of the point of detrimental irreversible damage. For the fracture study, the analysis of the data is still in progress and will be assessed in future work. Figure III.3 Stress versus strain plots of BioMid/epoxy quasi-isotropic composites (longitudinal) 15

16 Figure III.4 Stress versus strain plots of BioMid/epoxy quasi-isotropic composites (transverse) Figure III.5 Acoustic emission cumulative energy plots of Biomid/epoxy quasi-isotropic composites The results of the discrete three-layers model show a good corellation to the experimental data (figures III.3 and III.4). The variation of longitudinal and transverse elastic modulus with respect to the fiber volume fraction is shown in Figure III.6(a). The elastic moduli in both in-plane directions are very 16

17 sensitive to the change in reinforcement content. The effects of changing the crimp angle (or the aspect ratio) of the bias yarns in the braided fiber architecture on the elastic modulus of the composites are also investigated. Figure III.6(b) indicates a high sensitivity of the transverse modulus to the change in crimp angle. On the other hand, there is no significant variation in the predicted longitudinal stiffness as the behavior in the axial direction is mainly dominated by the longitudinal fibers that have no undulations. Figure III.6 Prediction of the elastic modulus versus fiber volume content Aspect ratio Conclusion The objective of this research was to characterize regenerated cellulose fibers and their composites with epoxy for mechanical properties, and to evaluate manufacturing techniques for these composites. Cellulose is the most abundant natural material and is the main component of many solid waste products of households, businesses and construction projects. Experimental and computational fracture mechanics and strength approaches will be used to suggest an optimum architecture of the reinforcement to produce the desired mechanical properties. The study performed an initial investigation into the swelling and liquid flow into and through regenerated cellulose fibers, and examined the adhesion of thermoset resins, especially epoxies. Three different cellulose based materials (Lyocell, CNF and BioMid) where characterized for their mechanical properties. The results reveal a good potential for the use of regenerated cellulose fibers as reinforcement in composite materials to replace other man-made fibers. The main issue is to recognize the moisture dependent behavior of regenerated cellulose fibers. The potential of the use of CNF scaffolds needs to be further assessed to understand its behavior. Part of my future work will include making and testing composites from CNFs with uniformly oriented nanofibers which are expected to give higher mechanical properties than the randomly oriented fibers scaffolds used in this study. 17

18 References: El-Hajjar, R.F. Shams, S.S. and Kehrl, D.J. Closed form solutions for predicting the elastic behavior of quasi-isotropic triaxially braided composites. Composite Structures, In press. El-Hajjar, R.F. and Qamhia, I. I., Modeling and characterization of the moisture-dependent bilinear behavior of regenerated cellulose composites, Journal of Wood Science, In press. Morton, W.H., J, Physical properties of textile fibres, 4 ed. Woodhead Publishing. Mottershead, B., Eichhorn, S.J., Deformation micromechanics of model regenerated cellulose Shokrieh M.M., Mazloomi M.S. An analytical method for calculating stiffness of twodimensional tri-axial braided composites. Composite Structures. 2010;92: