Predicting the elastic modulus of natural fibre reinforced thermoplastics

Transcription

1 Composites: Part A 37 (006) Predicting the elastic modulus of natural fibre reinforced thermoplastics Angelo G. Facca a, Mark T. Kortschot a, *, Ning Yan b a Department of Chemical Engineering and Applied Chemistry, University of Toronto, 00 College Street, Toronto, Canada, M5S 3E5 b Faculty of Forestry, University of Toronto, 33 Willcocks Street, Toronto, Canada, M5S 3B3 Received 16 March 005; received in revised form 6 October 005; accepted 18 October 005 Abstract Natural fibre reinforced thermoplastics (NFRT) are increasingly used in a variety of commercial applications, but there has been little theoretical modeling of structure/property relationships in these materials. In this study, micromechanical models available in the short fibre composites literature were used to predict the stiffness of some commercially important natural fibre composite formulations. Also included are equations that correct the YoungÕs modulus of natural fibres for changes in moisture content and density that occur as a result of processing. Hemp fibres, hardwood fibres, rice hulls, and E-glass fibres were blended into high-density polyethylene in mass fractions of wt%. The YoungÕs modulus of these composites was compared to theoretical values generated by the rule of mixtures, Halpin Tsai, NairnÕs generalized shear-lag analysis and Mendels et al. stress transfer (micromechanical) models. Based on a sum of errors squared criterion, the Halpin Tsai equation was found to predict the experimental data most accurately for the NFRT created for this study. Ó 005 Elsevier Ltd. All rights reserved. Keywords: A. Discontinuous reinforcement; A. Thermoplastic resin; B. Mechanical properties; C. Analytical modelling 1. Introduction Short fibre reinforced thermoplastics (SFRT) have been successfully used in a variety of applications due to their exceptional mechanical properties and a high strength to weight ratio. Molded and extruded SFRTs have much lower production costs than conventional long fibre/thermosetting resin composites. Early SFRTs were composed of traditional synthetic fibres such as glass or carbon. Recently, there has been an emphasis on replacing these synthetic fibres with natural, cellulose-based reinforcing materials [1]. Natural fibres were historically added to plastics as fillers rather than as reinforcing components. The advantage of natural fibres over inorganic materials has traditionally been associated with reduced cost but other attributes include low density, high specific strength to density ratio, * Corresponding author. Tel.: ; fax: address: kortsch@chem-eng.utoronto.ca (M.T. Kortschot). low abrasiveness, biodegradability and the fact that they are produced from a renewable resource. Natural fibre composites can be mixed and molded using high intensity, high volume production machinery such as extrusion and injection molding with minimal strength degradation. During the production of natural fibre reinforced thermoplastics (NFRT) the processing temperature is typically kept below 00 C due to the possibility of lignocellulosic degradation []. Consequently, thermoplastics such as polyethylene, polypropylene and polystyrene, which can be processed below the 00 C threshold, are commonly used as the matrix in NFRT. The major problem associated with integrating natural fibres into commodity thermoplastics is the chemical incompatibility between the hydrophilic fibres and the hydrophobic polymer [3]. This incompatibility leads to poor adhesion and a reduction in the ability of the matrix to transfer stress to the fibres. The adhesion is often improved in practice through the addition of coupling agents, which are chemicals that contain polar and non-polar X/$ - see front matter Ó 005 Elsevier Ltd. All rights reserved. doi: /j.compositesa

2 A.G. Facca et al. / Composites: Part A 37 (006) groups in their structure to form a bridge between fibres and matrix. The hydrophilic natural fibres are also responsible for potential dimensional changes and/or failure of the composite material caused by the undesirable uptake of moisture. Compared to synthetic fibres, natural fibres exhibit large variations in properties, even for fibres derived from the same species. These variations exist due to constant fluctuations in the environmental conditions (moisture, soil, temperature, etc.) in which these natural materials grow [4]. Consequently, a statistical approach is generally used to characterize the properties of these materials [5]. In the literature, there are many experimental studies of the mechanical properties of NFRT created using different fibre sources and processing techniques [3,6 9]. There is an extensive literature devoted to modeling the properties of polymers reinforced with short synthetic fibres. Such modeling is extremely important and can shed light on the relationship between composite microstructure (e.g. fibre aspect ratio and orientation) and the properties of the composite. Surprisingly, there seems to have been only one study applying established models to NFRT. Biagiotti et al. used a semi-empirical model to determine the theoretical stiffness of a polypropylene +0 wt% flax fibre composite [5]. Furthermore the authors have not found in the short fibre literature work that relates how changes in moisture content and density of natural fibres that occur during processing influences their mechanical properties. The objective of the current study was to predict the stiffness of a variety of NFRT materials over a range of mass fractions (0 60 wt%) using analytical micromechanical model available in the short fibre literature. In addition the objective is to correct the YoungÕs modulus of natural fibres for changes in moisture content and fibre density that occur during composite manufacturing.. Background to micromechanical models The elastic properties of SFRT can be experimentally determined or derived from a variety of mathematical models. The advantage of a comprehensive mathematical model is it reduces costly and time-consuming experiments. Furthermore, a mathematical model may be used to find the best combination of constituent materials to satisfy material design considerations. Lastly, a physical (as opposed to empirical) model can yield insight into the fundamental mechanisms of reinforcement. Micromechanical composite models are derived based on the properties of the individual components of the composite and their arrangement [10]. Properties such as the elastic modulus (E), PoissonÕs ratio (m) and the relative volume fractions (V) of both fibre and matrix are the fundamental quantities that are used to predict the properties of the composite. In some cases, fibre aspect ratio and fibre orientation are also included. In this report the following models are applied to NFRT: 1. Rule of mixtures (ROM). Inverse rule of mixtures (IROM) 3. Halpin Tsai equation [11,1] 4. NairnÕs generalized shear-lag analysis [13] 5. Mendels et al. stress transfer model for single fibre and platelet composites [14].1. Rule of mixtures and the inverse rule of mixtures equations The simplest available model that can be used to predict the elastic properties of a composite material is the rule of mixtures (ROM). To calculate the elastic modulus of the composite material in the one-direction (E 1 ), Voigt assumed that both the matrix and fibre experience the same strain (e 1 ) as shown in Fig. 1. This strain is a result of a uniform stress (r 1 ) being applied over a uniform crosssectional area, A [15]. The ROM equation for the apparent YoungÕs modulus in the fibre direction is [10]: E 1 ¼ E F V F þ E M V M where E F, E M, V F and V M are the moduli and volume fractions of the fibre and matrix materials respectively. This model works extremely well for aligned continuous fibre composites where the basic assumption of equal strain in the two components is correct. The elastic modulus of the composite in the two-direction (E ) is determined by assuming that the applied transverse stress (r ) is equal in both the fibre and the matrix as shown in Fig. (ReussÕs assumption) [15]. As a result, E is determined by an inverse rule of mixtures equation that is given as [10]: E F E M E ¼ V M E F þ V F E M For all composites with well-bonded reinforcements, YoungÕs modulus in the principle fibre direction will be somewhere in between the extreme values predicted by either the ROM or the IROM equations. σ 1 L Matrix Matrix 1 Fibre Δ L ð1þ ðþ Fig. 1. Simple representation of a fibre and matrix stressed in the onedirection. σ 1

3 166 A.G. Facca et al. / Composites: Part A 37 (006) W σ σ.. Halpin Tsai equation Matrix Matrix The semi-empirical equations developed by Halpin and Tsai are widely used for predicting the elastic properties of SFRT [11]. The following form of the Halpin and Tsai equation is used to predict the tensile modulus of SFRT [1]: 1 þ ngv F E 1 ¼ E M ð3þ 1 gv F In Eq. (3) the parameter g is given as [1]: ð g ¼ E F=E M Þ 1 ð4þ ðe F =E M Þþn where n in Eqs. (3) and (4) is a shape fitting parameter to fit the Halpin Tsai equation to the experimental data. The significance of the parameter n is that it takes into consideration the packing arrangement and the geometry of the reinforcing fibres [10,15]. A variety of empirical equations for n are available in the literature, and they depend on the shape of the particle and on the modulus that is being predicted [1]. If the tensile modulus in the principle fibre direction is desired, and the fibres are rectangular or circular in shape, then n is given by the following equation [1]: n ¼ L or n ¼ L ð5þ T D where L refers to the length of a fibre in the one-direction and T or D is the thickness or diameter of the fibre in the three-direction. In Eq. (5), as L! 0, n! 0 and the Halpin Tsai equation reduces to the IROM equation. In contrast when L!1, n!1 the Halpin Tsai equation reduces to the ROM equation...1. Shear-lag theory One of the first analytical models to describe the reinforcing effect of short fibres on the strength and modulus of composite materials was derived by H.L. Cox in 195 [16]. The work of Cox is commonly referred to in the literature as shear-lag theory. A complete derivation of the shear-lag equation with a method of determining CoxÕs parameter H for either hexagonal or square fibre packing was later provided by Piggott [17]. Fibre Fig.. Simple representation of a fibre and matrix stressed in the twodirection. 1 Traditional shear-lag theory assumes that the reinforcing fibres are both aligned and packed in an orderly manner [17,18]. Furthermore, it is assumed that both fibre and matrix are perfectly elastic and isotropic, and that stress is transferred between the two constituents without yielding or slip [17]. The transfer of an applied load from matrix to fibre is thought to occur by interfacial shear stresses and the tensile stress at the ends of the fibres is assumed to be zero [19]. As a result, the maximum tensile stress occurs at the middle of the fibre whereas the maximum shear stress occurs at the ends of the fibre. These assumptions lead to the classic shear-lag equation for the axial tensile modulus of a composite as [16,17]: 0 1 tanh gl E 1 ¼ E F 1 gl C A V F þ E M V M In Eq. (6), g is given as [16,17]: 0 11= g ¼ 1 E M B E F ð1 þ t M Þln P C A F V F where m M is PoissonÕs ratio of the matrix material, P F refers to the packing factor of the fibres and R represents the radius of the fibres that are assumed cylindrical in shape. The value of the packing factor (P F ) depends on the type of fibre packing that is assumed. Typically, either a simple square or hexagonal fibre packing arrangement is used. For square packing P F = p and for hexagonal packing P F = p/ p 3. From Eq. (6) it can be shown that as the aspect ratio (L/r)!1, the shear-lag equation reduces to the ROM equation as expected. Although CoxÕs shear-lag equation is frequently referenced in the short fibre literature its accuracy in predicting stress transfer and energy in axisymmetric fibre/matrix problems has been shown to be unacceptable [13]. Nairn derived an improved shear-lag parameter g, based on exact elasticity equations for axisymmetric stress states with transversely isotropic materials [13]. The shear-lag parameter (g) derived by Nairn is identical to previous equations presented by Nayfeh (1977) and McCartney (199) when a two-cylinder fibre/matrix problem is considered which gives the following equation [13,0,1]: 33 E F V F þ E M V M g ¼ 6 6 r E F E M V M þ ln 1 V M 4G F G M V M where G F and G M refer to the shear modulus of the fibre and matrix respectively. The shear-lag parameter shown in Eq. (8) is referred to as the optimal shear-lag analysis for concentric cylinders [13]. Nairn extended the optimal shear-lag analysis to a generalized case by enabling shear stresses to be described by shape functions []. The capa- V F ð6þ ð7þ 1= ð8þ

4 A.G. Facca et al. / Composites: Part A 37 (006) bilities of the generalized shear-lag analysis was extended to include an imperfect interface by the addition of a interface parameter D S which changes Eq. (8) to the following: 33 g ¼ 6 6 E F V F þ E M V M 4r E F E M 4 V M þ ln 4G F G M V M V F þ v 1 V M þ 1 rd S = ð9þ The parameter v in Eq. (9) is a constant added to correct for a problem in the use of shape functions which gives a ln(1/v F ) term in the denominator of Eq. (8) []. Without v the denominator of Eq. (8) would approach infinity as V F! 0. Excellent agreement between shear-lag predictions and a finite element analysis for both isotropic and anisotropic fibres at various E F /E M ratios were determined with a universal value of v = []. If perfect adhesion is assumed, D S = 1 and Eq. (9) reduces to Eq. (8). As a result of the presence of an interface parameter D S in the shearlag equation, it is possible to characterize the improvement in interfacial adhesion with the addition of coupling agents. A stress transfer model for single fibre and platelet composites by Mendels et al. is also available in the short fibre where w C is stated as [14]: w C ¼ E M re F ð13þ If the specimen radius (r S ) is smaller than the matrix radius influenced by stress transfer (r I ) which is determined using Eq. (11), then r I is replaced by r S in Eqs. (10) and (1) [14]. Mendels et al. also derived a two-dimensional stress transfer equation for platelet reinforced thermoplastics similar to the cylinder/fibre case shown in Eq. (10) [14]. The platelet with dimensions L (length) by W (width) by T A (half thickness) is embedded in a specimen with half thickness T S. The matrix boundary where the shear stress is zero is labeled T I. The shear-lag parameter by Mendels et al. for platelet geometry is stated as follows [14]: 1= T A E F þðt I T A ÞE M g ¼ j P ð14þ T A E F ð1 þ m M Þ where T I and j P are defined by the following equations [14]: T I ¼ T AE F E M " # þ w j P ¼ P ðt I þ T A Þ 1ðT I T A Þþw P ðt I T A T IT A Þ ðt I T A Þþw P ðt I T A T IT A Þ 8ðT I T A Þ þ w P ð5t 3 I 7T AT I T A T I 5T 3 A Þ ð15þ ð16þ literature [14]. This model has no adjustable parameters and avoids a common limitation of many models that require the determination of an equivalent matrix radius [14]. The model proposed by Mendels et al. uses a similar approach by Hsueh to calculate stress fields with the exception that a decay function was used to model previously neglected changes of axial stress in the radial direction [14,3,4]. The shear-lag parameter presented by Mendels et al. for a single fibre with a radius of r embedded in a r S radius specimen is stated as [14]: r E F ðr I g ¼ j 1= r ÞE M C ð10þ re F ð1 þ m M Þ where r I represents the radius where the shear stress is zero and the matrix is not influenced by stress transfer between the fibre and matrix. The matrix radius influenced by stress transfer is determined by the following equation [14]: r I ¼ re F E M In Eq. (10) j C is defined as [14]: j C ¼ ð11þ where w P present in Eq. (16) is stated as [14]: w P ¼ E M ð17þ T A E F If the specimen half thickness (T S ) is smaller than the matrix boundary influenced by stress transfer (T I ) determined using Eq. (15), then T I is replaced by T S in Eqs. (14) and (16) [14]. The tensile modulus of the platelet reinforced composite is determined by substituting Eq. (14) into Eq. (6).... Modification to shear-lag equation for rectangular fibres In this paper, a minor modification is presented that would enable micromechanical models (NairnÕs generalized shear-lag analysis) that were derived for cylindrical fibres to be extended to fibres that have rectangular cross-sections. To achieve the transition from a circular to a rectangular cross-section, the concept of an equivalent radius is proposed. If the rectangular fibre has dimensions L (length) by W (width) by T (thickness) then the equivalent radius of the fibre (R EQ ) is defined as the radius of a circular fibre with the same perimeter, and is therefore given as: 1 ðr I rþþw C ðr I n r Þ r 4rðr I r Þ 16ðr 3 I r3 Þ 3w C ðr r I Þ þ 6r I ð þ w C r I Þ r I ln o ð1þ r I r ðr I r Þ

5 1664 A.G. Facca et al. / Composites: Part A 37 (006) R EQ ¼ W þ T ð18þ p Conventional shear-lag theory for cylindrical fibres is then applied using R EQ. 3. Applying micromechanical models to composites containing natural fibres During processing, both natural and synthetic fibres will experience some form of fibre degradation. Natural and brittle synthetic fibres tend to undergo significant size reduction caused by shearing forces that are imparted to the fibre during processing. In addition natural fibres can suffer considerable densification because of their porous cellular structure. For wood species, the maximum fibre density that can be achieved is approximately 1.5 g/cm 3, which corresponds to the density of the cell wall material [4]. The moisture content in the natural fibres will also decrease from their preprocessed equilibrium moisture content due to elevated temperatures during composite manufacturing. The moisture content (MC) in the fibres is calculated as follows: MCð%Þ ¼100 M W M O ð19þ M O where M O and M W refer to the oven dry and wet masses of the fibres respectively. The volume of cell wall that is present (V CW ) can be determined as follows: V CW ¼ M O q CW ¼ q CW M W ð0þ MC% 100 þ 1 where q CW is the cell wall density. The mass of dry cell wall material does not vary during processing, and as a good approximation the modulus of wood scales linearly with densification. It is therefore possible to apply the micromechanical models using the volume fraction (V F ) of dense cell wall, together with the modulus of the cell wall. The volume fraction of fibre that is present on a cell wall basis (V F,CW ) is calculated as follows: V CW V F;CW ¼ ð1þ V V þ V M þ V CW where V V refers to the total volume of voids that are present in the processed composite, including the fibre lumens and other fibre pores that remain. If the density of the composite material (q C ) is known, the total void space can be approximated using Eq. (). V V ¼ M O þ M M q C ðv CW þ V M Þ ðþ q C where M M refers to the mass of plastic that is present in the composite. The moisture content of natural fibres also has an influence on their mechanical properties [4]. The mechanical properties of wood fibres will increase as the moisture content decreases below the fibre saturation point (FSP). The fibre saturation point is defined as the point at which the cell wall of the fibre is saturated with water [4]. For most wood species this occurs at a moisture content of approximately 30%. As the moisture content decreases below 30%, the fibre shrinks. Above 30%, the dimensions of the fibre no longer change as additional moisture is stored within the lumens of the fibres. The relationship between the moisture content of wood and its mechanical properties below the FSP is determined as follows [4]: E 1 E MC ¼ E 1 E G 1 MC M T 1 ð3þ where E MC, E 1 and E G are the moduli of wood at the moisture content (MC) of interest, at a 1% moisture content and at the green condition respectively. The moduli of wood at the 1% and at the green conditions are available for a variety of wood species in the Wood Handbook [4]. The parameter M T refers to the threshold moisture content. It is the moisture content below which the mechanical properties of wood will begin to change. For most wood species M T can be assumed to be 5% [4]. An additional fibre modulus correction is required to account for fibre densification that occurs during processing. The simplest approach is to assume that the modulus linearly increases with an increase in fibre density. Since it is assumed that the apparent density of the fibre approaches its cell wall density (q CW ) after processing, the modulus of the fibre (E F ) is predicted as follows: q E F ¼ E CW MC q MC ð4þ where q MC is the initial density of the fibre prior to processing, and is assumed to be at the final moisture content that the fibre would have in the composite. If the specific gravity of green wood (G B ) is known, then the density of wood at a particular moisture content can be calculated as follows [4]: 0 1 B C q MC ðkg=m 3 1 0:65 G B 30 MC 30 G B C A 1 þ MC 100 ð5þ The equivalent modulus of cell wall material is computed by substituting Eq. (5) in Eq. (4). This value is used in conjunction with the volume fraction of cell wall from Eq. (1) in the micromechanical models (Eqs. ((1) (3) and (6))) to predict the modulus of a composite containing densified, dried fibres embedded in a thermoplastic material. 4. Experimental materials and procedure The thermoplastic selected for this study was high-density polyethylene (Formolene HB550B) from CCC Plas-

6 A.G. Facca et al. / Composites: Part A 37 (006) tics. It has a reported specific gravity of and a melt index of 0.35 [5]. To this plastic the following natural fibres were added: 1. Hemp fibre 1/ in. staple length. Hardwood A 0 mesh oak wood flour 3. Rice hulls 4. Hardwood B 40 mesh oak wood flour To compare the results of NFRT to traditional reinforcing materials, plastic composites were also produced containing 1/4 in. E-glass fibres. For each of these fibres composites were created with a fibre mass fraction from 10 to 60-wt% in increments of 10 wt%. The fibres were blended into the plastic using a twin-screw Brabender mixer that has a batch capacity of 00 g. Compounding was carried out at a temperature of 160 C and a speed of 60 RPM for a period of 10 min. A constant compounding time of 10 min was selected as it was observed that it took approximately 5 6 min for the motor current (load) to reach steady state conditions. Once compounding was completed, the contents of the mixer were removed and placed in a mold to produce a puck shaped specimen with dimensions mm. The puck was then reheated in an oven at a temperature of 160 C for 60 min before being placed in the center of a lubricated compression mold (Fig. 3) that was designed to produce one-dimensional, extensional flow. The mold was placed between the heated platens of a 50-ton hydraulic press and the contents of the mold compressed to the final rectangular profile shown in Fig. 3. Each of the rectangular specimens shown in Fig. 3 has dimensions of mm. These specimens were cut in two to produce 10 specimens, which were then used to conduct tensile tests using the ASTM D638 standard. The tensile tests were performed at an average room temperature of 0 C on a Sintech 0 tensile tester. The initial jaw separation distance of the tensile tester was mm with a test speed of 10.0 mm/min. A.0-in. extensometer was mounted to the side of the specimens to accurately measure specimen strain. The composite density was determined in accordance with ASTM D79, with the exception that mercury rather than distilled water was used as the immersing liquid. The use of mercury was necessary due to the hydrophilic nature of the NFRT materials. The initial moisture content of the fibres was obtained by drying the fibres at a constant temperature of 110 C for a period of 6 h. The results obtained after drying are presented in Table. The average post-processed dimensions of these fibres, obtained using either an optical microscope or a scanning election microscope (SEM), are shown in Table 1. Scaled photographs of the various composite specimen surfaces enabled the fibre dimensions to be quickly determined without having to desolve the polymer away from the Compression Mold 4.5 in 3.5 in in 14.0 in 3.65 in 1/" 1/8" 1/" 1/4" 1/8" 1/" Composite Specimen Fig. 3. Cross-sectional view of compression mold.

7 1666 A.G. Facca et al. / Composites: Part A 37 (006) Table 1 Average dimensions of post-processed natural fibres Fibre name Length (mm) Width or diameter (mm) Thickness (mm) E-glass ± ± Hardwood A 1.64 ± ± ± Hardwood B 1. ± ± ± 0.08 Hemp ± ± 0.04 Rice hulls 1.30 ± ± ± 0.01 Density (kg/m 3 ) Table Initial moisture content of natural fibres Fibre name Moisture content (%) Hardwood A 7.9 Hardwood B 5.47 Hemp 8.60 Rice hulls Mass fraction fibers (oven dry basis) E-glass hardwood A hemp rice hulls hardwood B Fig. 5. Composite density vs. mass fraction fibres. fibres. In the case of composites containing E-glass or hemp fibres the composite specimens were reheated and gently pressed into thin films to allow better light penetration and visibility of the contours of the encased fibres (Table ). 5. Results and discussion Surface photos provided in Fig. 4 of HDPE reinforced with hardwood A and B fibres illustrate that the compression mold shown in Fig. 3 will produce composites containing predominantly well-aligned fibres. As a result the effect of fibre misalignment on composite modulus was assumed to be negligible and was not accounted for in this study. The densities of composites made using the fibres listed in Table 1 are shown in Fig. 5 and the corresponding numerical data are presented in Table 3. Fig. 5 illustrates that as the mass fraction of fibre increases, the composite density generally increases. The composites containing E- glass fibres have the highest density because E-glass fibres have a density of.54 g/cm 3 compared to natural fibres, which have a maximum fibre density of 1.5 g/cm 3. Of the natural fibres, the hemp fibres would have the highest percentage of cellulose and therefore produced a composite with a greater density. Although the hardwood fibres (hardwood A, hardwood B) contain on average slightly more cellulose then rice hulls, the silica (density. g/ cm 3 ) content in rice hulls apparently makes up for this difference. There is an initial reduction in density at a fibre loading of 10% where problems in the dispersion and packing of fibres begin to produce voids. At higher percentages, the effect of voids is offset by the presence of more dense fibrous material and the density increases. For each composite, the void fraction was calculated using the known density of the fibres (either the glass or cell wall material) and the composite and this data is presented in Fig. 6. Interestingly, the void fraction seems to peak at 10% loading for the longer hemp and glass fibres. Higher volume fractions might allow air to migrate out of the specimens more readily during compression molding (see Table 4). For composites containing E-glass fibres, the void space obviously exists external to the fibre, however with natural fibres it is not possible to determine whether the void fraction is external to the fibres or due to uncollapsed lumens. Unfortunately, cross-sections prepared by polishing or Fig. 4. Orientation of hardwood A and hardwood B fibres in high-density polyethylene at a fibre loading of 30-wt%.

8 A.G. Facca et al. / Composites: Part A 37 (006) Table 3 Density of composites shown in Fig. 5 Mass fraction Composite density (kg/m 3 ) E-glass Hardwood A Hardwood B Hemp Rice hulls ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Void fraction Young's modulus (GPa) Mass fraction fibers (oven dry basis) E-glass hardwood A hemp rice hulls hardwood B Mass fraction fibres (oven dry basis) Fig. 6. Volume fraction voids vs. mass fraction fibres. E-glass hardwood A hemp rice hulls hardwood B Fig. 7. Composite modulus vs. mass fraction fibres. Table 4 Typical chemical composition of fibres [4,6] Fibre name Cellulose Hemi-cellulose Lignin Ash Silica Hardwoods 38 49% 19 6% 3 30% <1% Hemp 57 77% 14 17% 9 13% 0.8% Rice hulls 8 48% 3 8% 1 16% 15 0% 9 14% microtoming do not help because the resin is quite soft, and hence the fine void structure cannot be observed directly using scanning electron microscopy. For this reason, in this paper, micromechanical modeling is based on the modulus and volume fraction of the pure cell wall material rather than using the fibre as a basis for the calculation. The density of the cell wall material does not depend on the manufacturing operating conditions used to fabricate the fibre reinforced composite. When the micromechanical model is based on the volume of collapsed cell wall, only the moisture content of the fibre and the density of the cell wall material is required. Both of these quantities are easy to measure or can be quickly found in the literature. In Fig. 7, the stiffness of the composites created with fibres listed in Table 1 is shown to increase when fibre content is increased. For the natural fibres used in this study, at a fibre mass fraction of 60%, the tensile modulus is increased by a factor of between 3 and 6. The addition of 60 wt% E-glass fibres yields an increase in modulus of a factor of 1, almost twice the improvement obtained with hemp fibres. This difference is not due to the modulus of the fibres themselves, which is almost identical (see Table 5). E-glass and hemp fibres have post-processed aspect ratios of 31.3 and 3.4 respectively (see Table 1). Hemp Table 5 Material properties required for micromechanical models Material property Value Measured HDPE density ± 5.3 kg/m 3 HDPE shear modulus [7] 0.39 GPa a HDPE PoissonÕs ratio [7] 0.34 a Measured HDPE tensile modulus 1.07 ± GPa E-glass tensile modulus [17] 7.0 GPa a Hemp tensile modulus [8] 69.0 GPa a Hardwood A and B tensile 9. GPa a modulus at green condition[4] Hardwood A and B tensile 1.0 GPa a modulus at 1% MC [4] Hardwood A and B tensile 15.3 GPa modulus at 0% MC (Eq. (17)) Hardwood A and B tensile 3.7 GPa modulus at 0% MC, q = 1.5 g/cm 3 (Eq. (18)) Assumed rice hull tensile modulus.0 GPa a a Symbolizes average reported values.

9 1668 A.G. Facca et al. / Composites: Part A 37 (006) fibres have a lower density then E-glass fibres, producing a larger volume fraction of fibres at equivalent mass fraction. According to the shear-lag equation (Eq. (6)) the larger volume fraction of hemp fibres should offset the small differences in aspect ratio and modulus of these fibres and produce composites with similar stiffnesses. It is possible that fibre degradation during processing (see Table 1) alters the stiffness of processed hemp fibres. Alternatively, the hemp may be comparatively poorly bonded to the matrix, producing composites with moduli lower than expected. The moduli of the remaining natural fibre composites are lower than that of the hemp fibre composite. This is primarily due to the low aspect ratio and cell wall moduli of the other natural fibres. A lower aspect ratio means that less of the fibre is fully stressed, since the tensile stress in the fibre must build from zero at the ends through the shear stresses from the matrix. This principle is embodied in the shear-lag equations (either the original Cox equation, or the modified version), where the modulus is expressed as a function of L/r. With one exception, the properties of the fibre and matrix needed to use the micromechanical models are known. Unfortunately, the size of the rice hull fibre makes it extremely difficult to measure YoungÕs modulus experimentally. As a result the tensile modulus of rice hulls was estimated using the fibres with the closest geometry, the hardwood B fibres. It was assumed that a fitting parameter a in a modified rule of mixtures equation (Eq. (6)) would apply to both rice hulls and hardwood B fibres. E 1 ¼ E M V M þ ae F V F ð6þ Rearranging Eq. (6) enables the fitting parameter a i at each data point to be determined for hardwood B fibres as follows: a i ¼ E 1 E M V M ð7þ E F V F The parameter a i was then applied to Eq. (6) using rice hull data. This method produced an estimated average rice hull modulus of GPa. None of the theoretical models presented earlier has been applied to natural fibre composites and it was not known which of them would predict the results with the greatest accuracy. Hence each of the models was applied using experimental data found in Tables 1 and 5. The results obtained for high-density polyethylene (HDPE) reinforced with the fibres listed in Table 1 are shown in Figs. 8 1, respectively. In Figs. 8 1 it is observed that the ROM and IROM equations form upper and lower bounds for the experimental data, as expected. The ROM equation assumes that the fibres are oriented and fully strained along their length. The IROM equation assumes that the fibres and matrix are equally stressed. Except for the composite reinforced with E-glass fibres, the Halpin Tsai equation, when used with the computed value of n (Eq. (5)), overestimated the tensile modulus of Young's modulus (GPa) Volume fraction E-glass fibres Experimental data Inverse rule of mixtures Nairn shear-lag Rule of mixtures Halpin Tsai Mendels et al. Fig. 8. Determination of composite modulus containing E-glass fibres. Young's modulus (GPa) Volume fraction hardwood A fibres Experimental data Inverse rule of mixtures Nairn shear-lag equiv. radius Rule of mixtures Halpin Tsai Mendel et al. Fig. 9. Determination of composite modulus containing hardwood A fibres. the composite (Figs. 8 1). Values of n adjusted to provide the best-fit using least squares analysis are found in Table 6. Table 6 illustrates that there are fairly large discrepancies between the best-fit and values of n computed using Eq. (5). These discrepancies occur because the relationship for n given in Eq. (5) was empirically determined for synthetic fibres with well-defined cylindrical or rectangular cross-sections. As a result the Halpin Tsai equation produced good predictions for the modulus of the glass fibre reinforced composites (Fig. 8). In contrast, natural fibres with complex geometric cross-sections are apparently not well represented by the relationship given in Eq. (5). The shear-lag equations by Nairn (Eq. (8)) or Mendels et al. (Eq. (10) or Eq. (14)) were found to consistently overestimate the axial tensile modulus (see Figs. 8 1) of com-

10 A.G. Facca et al. / Composites: Part A 37 (006) Young's modulus (GPa) Young's modulus (GPa) Volume fraction hemp fibres Experimental data Inverse rule of mixtures Nairn shear-lag Rule of mixtures Halpin Tsai Mendels et al Volume fraction hardwood B fibres Experimental data Inverse rule of mixtures Nairn shear-lag equiv.radius Rule of mixtures Halpin Tsai Mendels et al. Fig. 10. Determination of composite modulus containing hemp fibres. Fig. 1. Determination of composite modulus containing hardwood B fibres. 1.0 Young's modulus (GPa) Volume fraction rice hulls Experimental data Inverse rule of mixtures Nairn shear-lag equiv.radius Rule of mixtures Halpin Tsai Mendels et al. Fig. 11. Determination of composite modulus containing rice hulls. posites containing either cylindrical (E-glass, hemp) or platelet (hardwood A and B, rice hulls) fibres. Furthermore, Figs. 8 and 10 show that although Nairn and Mendel et al. shear-lag models for cylindrical fibres (Eqs. (8) and (10)) are quite different, both would predict similar results for HDPE containing either E-glass or hemp fibres. However, differences were observed (see Figs. 9, 11 and 1) for composites containing fibres with an approximately rectangular cross-section (hardwood A or B, rice hulls) using NairnÕs shear-lag equation (Eq. (8)) with an equivalent fibre radius (Eq. (18)) compared to Mendel et al. (Eq. (14)) stress transfer model for platelet composites. Table 6 Calculated values of Halpin Tsai parameter n Fibre name Calculated value of n using Eq. (5) Best-fit n E-glass Hardwood A Hardwood B Hemp Rice hull Recall that the equivalent radius method transforms a rectangular fibre to a cylindrical fibre of equivalent perimeter and hence surface area. By choosing an equivalent radius based on similar cross-sectional areas (Eq. (8)) it was discovered that there was closer agreement between the two micromechanical models to determine the tensile modulus of platelet reinforced composites. R EQ ¼ WT 1= ð8þ p The fact that there were significant discrepancies between experimental and results predicted using shearlag analysis must be due to either flaws in the assumptions of the models, or through errors in the estimation of model parameters. The shear-lag models of Nairn and Mendels et al. have been validated using a variety of analytical methods such as finite element analysis or Raman spectroscopy [13,14,]. However, it is important to note that the original shear-lag models are based on an analysis of a two component concentric cylinder arrangement. Although the more recent models agree closely with finite element analysis of this configuration, neither the analytical nor the finite element models properly represent a composite with staggered fibres distributed throughout a matrix. This is

11 1670 A.G. Facca et al. / Composites: Part A 37 (006) Table 7 Interface parameter D S required in Eq. (9) to fit experimental results to shear-lag predictions Fibre name D S (GPa/mm) E-glass 5.36 Hardwood A Hardwood B Hemp 0.4 Rice hull expected to lead to an overestimation of the modulus of real composites, which was observed here. A second, significant source of discrepancies is due to the assumption of perfect adhesion, which would also cause the shear-lag analysis to overestimate the tensile modulus of the composites shown in Figs Poor adhesion is a particular problem in composites with hydrophilic natural fibres and hydrophobic polymer matrices, and the resultant slip at the interface will reduce the average stress in the fibres. To overcome poor adhesion a coupling agent can be added that improves chemical compatibility at the fibre/matrix interface. The simple shear-lag models assume perfect adhesion, and this assumption is generally acceptable for modeling the modulus at very low stresses. To quantify the quality of the fibre/matrix adhesion, the generalized shear-lag equation (Eq. (9)) was fit to the experimental tensile modulus data by varying the interface parameter D S. The best-fit values of D S for the various fibres selected for this study are shown in Table 7. Recalling that D S = 1 implies perfect adhesion, the values shown in Table 7 suggest that poor adhesion is present in the composites studied here. The performance of the various micromechanical models in predicting the tensile modulus of each of the composites presented in this study was determined by using the sum of errors squared criterion. The results shown in Table 8 indicate that the semi-empirical Halpin Tsai equation produced minimum total error. All of the micromechanical models presented in this study predict that the stiffness of a composite can be increased by 1. increasing the mass/volume fraction of stiff fibres in the composite,. selecting fibres with large moduli and aspect ratios, and 3. using dry natural fibres to maximize their stiffness. Further improvement in the models could probably be obtained if factors such as fibre angle and length distributions corrections were included. Here, we assume that the fibres are perfectly aligned and of uniform length and aspect ratio. In practice, the alignment is quite good, but there is a significant distribution in length and aspect ratio produced during processing. Furthermore, no account has been taken for the stress transfer that occurs through the ends of the fibres. Although a model is available in the literature that includes this effect, the added complexity is not justified for the present purposes [19]. The effect of stress transfer through the ends of the fibre would become less significant as the fibre aspect ratio (L/T or L/D) increases. The approach adopted here is to use simple models that are capable of capturing the main experimental trends without undue complexity. 6. Conclusions The addition of a natural fibre component to the polyethylene resulted in an increase of stiffness by a factor of between three and six, while not resulting in a significant weight increase. It was found that standard micromechanical models, which have been used successfully to predict the stiffness properties of traditional synthetic fibre composites, can be applied to natural fibre systems with mixed success. To apply micromechanical models to composites containing natural fibres, the volume fraction of fibres should be computed and used on a cell wall basis. It was necessary to use a correction to the fibres tensile modulus to reflect the increased cell wall density. An additional correction was presented to account for the relationship between the mechanical properties of wood fibres and their moisture content. It was found that although the density of natural fibres does change, the three lower aspect ratio natural fibres were not significantly degraded during processing. This is regarded as a benefit of using natural fibres as opposed to synthetic fibres, which are susceptible to significant fibre degradation during processing. Table 8 Comparing the performance of the various micromechanical models by using the sum of errors squared criterion Micromechanical model Fibre type Total error E-glass Hardwood A Hardwood B Hemp Rice hulls Rule of mixtures Inverse rule of mixtures Halpin Tsai NairnÕs shear-lag (Eq. (8) with/without Eq. (18)) Mendels et al. (Eqs. (10) or (14))

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