《纺织复合材料》课程参考文献（Principles of the Manufacturing of Composite Materials）CHAPTER 6 Pultrusion

Copyrighted Materials Copyright 2009 DEStech Publications Retrieved from www.knovel.con CHAPTER6 Pultrusion 1. GENERAL Pultrusion is a manufacturing process that combines many steps in the manufacturing of composites into two steps. Refer to Figure 1.7(a) of Chapter 1. That figure shows four steps in the manufacturing process. Figure 6.1 shows a representation of the pultrusion process. For pultrusion, one goes directly from step a to step d, and one does not have intermediate products in between. . Step a: Beginning from the left-hand side, the fiber tows drawn from fiber racks are routed through a series of guides. The fibers then traverse through a bath of low viscosity resin for impregnation. . Step d: Upon exiting from the resin bath, fibers are collimated into an aligned bundle before entering into a heated die. While inside the die, the following takes place: -The resin flows and wets the fibers. -The ensemble of fibers and resin is compacted. -The resin cures and the fiber/resin system becomes solid. Upon exiting from the die, the composite structural component is pulled by a puller. It is then cut to length and is ready for ship- ment. Since the pultrusion process combines many steps into one, the oppor- tunity for checking the quality is reduced. Also the requirements for con- 233

CHAPTER 6 1. GENERAL Pultrusion is a manufacturing process that combines many steps in the manufacturing of composites into two steps. Refer to Figure 1.7(a) of Chapter 1. That figure shows four steps in the manufacturing process. Figure 6.1 shows a representation of the pultrusion process. For pultrusion, one goes directly from step a to step d, and one does not have intermediate products in between. • Step a: Beginning from the left-hand side, the fiber tows drawn from fiber racks are routed through a series of guides. The fibers then traverse through a bath of low viscosity resin for impregnation. • Step d: Upon exiting from the resin bath, fibers are collimated into an aligned bundle before entering into a heated die. While inside the die, the following takes place: —The resin flows and wets the fibers. —The ensemble of fibers and resin is compacted. —The resin cures and the fiber/resin system becomes solid. Upon exiting from the die, the composite structural component is pulled by a puller. It is then cut to length and is ready for ship￾ment. Since the pultrusion process combines many steps into one, the oppor￾tunity for checking the quality is reduced. Also the requirements for con- 233

234 PULTRUSION trol of quality are increased to ensure good quality.However,pultrusion offers a fast production rate.The production rate can be in terms of me- ters per minute.It can also produce composite parts at low cost.This is because fibers in tow form are less expensive than in prepreg or woven form.The rule is that if more processing is involved in transforming the fiber,either by wetting with the resin or in changing the form from unidi- rectional tows to woven or braids,cost and time are involved.As such. pultrusion has been used to produce many low cost composite structural elements,if strict quality is not a requirement.However there are also limitations.One of the main limitations is that the fibers are mostly unidi- rectional. By pultrusion,many lightweight,corrosion resistant and low electrical conductivity components can be made.These include standard shapes such as rods,angles,clips,I beams,panels,plates,and rebars for con- crete reinforcement.Other components include side rails for ladders, fishing rods,tool handles,bus components,sign posts,and sucker rods for oil drilling rigs. Due to the low cost of the process,E glass fibers are mostly used even though applications include fibers such as S glass,carbon,and Kevlar. Resins are usually low cost polyester or vinyl ester even though other res- ins such as epoxy,phenolic and thermoplastic have also been used. There are two forms of pultrusion products.The first category con- sists of solid rod and bar stock produced from axial fiberglass rein- forcements and polyester resins.These are used to make fishing rods and electrical insulator rods,which require high axial strength.The second category is structural profiles,which use a combination of axial Fiber racks cloth racks material guides pulling mechanisms pultrusion moving engaged disengaged die cutofl saw hydraulic rams finished prelorming guides resin tank product FIGURE 6.I Schematic of the pultrusion process(reproduced from"Pultrusion of com- posites,"by J.P.Fanucci,S.Nolet,and S.McCarthy,in Advanced Composites Manufac- furing by T.G.Gutowski,1997,with permission from John Wiley and Sons)

trol of quality are increased to ensure good quality. However, pultrusion offers a fast production rate. The production rate can be in terms of me￾ters per minute. It can also produce composite parts at low cost. This is because fibers in tow form are less expensive than in prepreg or woven form. The rule is that if more processing is involved in transforming the fiber, either by wetting with the resin or in changing the form from unidi￾rectional tows to woven or braids, cost and time are involved. As such, pultrusion has been used to produce many low cost composite structural elements, if strict quality is not a requirement. However there are also limitations. One of the main limitations is that the fibers are mostly unidi￾rectional. By pultrusion, many lightweight, corrosion resistant and low electrical conductivity components can be made. These include standard shapes such as rods, angles, clips, I beams, panels, plates, and rebars for con￾crete reinforcement. Other components include side rails for ladders, fishing rods, tool handles, bus components, sign posts, and sucker rods for oil drilling rigs. Due to the low cost of the process, E glass fibers are mostly used even though applications include fibers such as S glass, carbon, and Kevlar. Resins are usually low cost polyester or vinyl ester even though other res￾ins such as epoxy, phenolic and thermoplastic have also been used. There are two forms of pultrusion products. The first category con￾sists of solid rod and bar stock produced from axial fiberglass rein￾forcements and polyester resins. These are used to make fishing rods and electrical insulator rods, which require high axial strength. The second category is structural profiles, which use a combination of axial 234 PULTRUSION FIGURE 6.1 Schematic of the pultrusion process (reproduced from “Pultrusion of com￾posites,” by J. P. Fanucci, S. Nolet, and S. McCarthy, in Advanced Composites Manufac￾turing by T.G. Gutowski, 1997, with permission from John Wiley and Sons)

Materials 235 fibers and multidirectional fiber mats to increase a set of properties that meet the requirements of the application in the transverse and longitu- dinal directions. 2.MATERIALS 2.1.Fibers 2.1.1.Unidirectional Rovings Fiberglass is the most commonly used fiber for pultrusion.Unidirec- tional fibers are the least expensive reinforcements available.Most pultruded profiles are used in structural applications where loading is highly oriented along the length of the profile.Unidirectional loading minimizes friction drag in the die,provides the highest pulling strength possible,and simplifies the design and fabrication of forming guides at the entrance of the die. The combination of cost,design applicability,and manufacturing ease make unidirectional rovings the most widely used reinforcement in pultrusion processing.Unidirectional filaments in the form of rovings, tows or threads are the building blocks for virtually all other reinforcement forms. For most practical applications,the use of all-unidirectional rovings is unrealistic.The highly orthotropic behavior of the unidirectional com- posite results in transverse properties that are much too low.Parts con- structed this way might have unacceptable resistance to crushing or splitting parallel to the fiber direction.Some means of providing strength in the transverse direction is often mandatory. 2.1.2.Woven and Nonwoven Broad Goods Low cost commercial pultrusions made of unidirectional glass rovings often include inexpensive forms of nonwoven broad goods called contin- uous strands and chopped strand mat.The random orientation of these materials provides some degree of off-axis strength and stiffness en- hancement at very low cost.Woven materials used in the pultrusion pro- cess must be placed between more stable forms such as layers of unidirectional rovings.Figures 6.2 and 6.3 show the incorporation of mats along with unidirectional rovings. When more complicated laminates are required,cloth,random mats, and preplied fabrics can be added to obtain transverse and off-axis rein-

fibers and multidirectional fiber mats to increase a set of properties that meet the requirements of the application in the transverse and longitu￾dinal directions. 2. MATERIALS 2.1. Fibers 2.1.1. Unidirectional Rovings Fiberglass is the most commonly used fiber for pultrusion. Unidirec￾tional fibers are the least expensive reinforcements available. Most pultruded profiles are used in structural applications where loading is highly oriented along the length of the profile. Unidirectional loading minimizes friction drag in the die, provides the highest pulling strength possible, and simplifies the design and fabrication of forming guides at the entrance of the die. The combination of cost, design applicability, and manufacturing ease make unidirectional rovings the most widely used reinforcement in pultrusion processing. Unidirectional filaments in the form of rovings, tows or threads are the building blocks for virtually all other reinforcement forms. For most practical applications, the use of all-unidirectional rovings is unrealistic. The highly orthotropic behavior of the unidirectional com￾posite results in transverse properties that are much too low. Parts con￾structed this way might have unacceptable resistance to crushing or splitting parallel to the fiber direction. Some means of providing strength in the transverse direction is often mandatory. 2.1.2. Woven and Nonwoven Broad Goods Low cost commercial pultrusions made of unidirectional glass rovings often include inexpensive forms of nonwoven broad goods called contin￾uous strands and chopped strand mat. The random orientation of these materials provides some degree of off-axis strength and stiffness en￾hancement at very low cost. Woven materials used in the pultrusion pro￾cess must be placed between more stable forms such as layers of unidirectional rovings. Figures 6.2 and 6.3 show the incorporation of mats along with unidirectional rovings. When more complicated laminates are required, cloth, random mats, and preplied fabrics can be added to obtain transverse and off-axis rein￾Materials 235

236 PULTRUSION ROVING RESIN BATH CONTINUOUS MAT PULLER MANDREL PREFORMING DIES- RESIN BATH HEATED DIE- CONTINUOUS MAT FIGURE 6.2 Introduction of mats into the pultruded products (reproduced from Hand- book of Pultrusion Technology,by R.W.Meyer,with permission from Springer). forcing.Fiber tension is not a severe problem for commonly pultruded constructions composed primarily of unidirectional tows.Problems be- gin to occur in more sophisticated applications when broad goods and other off-axis reinforcements are included in the laminate.If not prop- erly handled,these laminates tend to distort and deform as they are folded during assembly outside the die,and can be further distorted when dragged along the tooling surfaces inside the die. 2.2.Resins The necessary characteristics for a resin to be used to make pultruded products are that it have low viscosity and that gel time and cure time SURFACING MAT CONTINUOUS STRAND MAT- Xg人XA ROVING 2002p99900 CONTINUOUS STRAND MAT au0Q史 ROVING PULTRUSION COMPOSITE CONTINUOUS STRAND MAT 海YX效 URFACING MAT FIGURE 6.3 Exploded view of materials in pultrusion (reproduced from Handbook of Pultrusion Technology,by R.W.Meyer,with permission from Springer)

forcing. Fiber tension is not a severe problem for commonly pultruded constructions composed primarily of unidirectional tows. Problems be￾gin to occur in more sophisticated applications when broad goods and other off-axis reinforcements are included in the laminate. If not prop￾erly handled, these laminates tend to distort and deform as they are folded during assembly outside the die, and can be further distorted when dragged along the tooling surfaces inside the die. 2.2. Resins The necessary characteristics for a resin to be used to make pultruded products are that it have low viscosity and that gel time and cure time 236 PULTRUSION FIGURE 6.3 Exploded view of materials in pultrusion (reproduced from Handbook of Pultrusion Technology, by R. W. Meyer, with permission from Springer). FIGURE 6.2 Introduction of mats into the pultruded products (reproduced from Hand￾book of Pultrusion Technology, by R. W. Meyer, with permission from Springer)

Materials 237 are short to allow for the high rate of production.If,for example,a rate of production of 20 cm/minute is desired,for the length of a die of 100 cm long,the duration of the resin inside the mold is 5 minutes.The resin should flow through the interstices of the tows,wet the fibers,gel and cure during this time.When the resin gels,it also shrinks,which helps to release the composite from the die wall.This,in turn,will reduce the pulling force.The resin normally used to make pultruded products is polyester resin,due to its low cost and low viscosity.Table 6.1 shows the viscosity of polyester resin along with gel time and cure time.When better corrosion resistance is required,vinyl ester resins are used. When a combination of superior mechanical and electrical properties is required,epoxy resin is used.Epoxy resins are expensive materials in a number of aspects.The resins are three to six times more expensive than polyesters and have a number of process-related costs not found with polyesters.Because they are cured by a stepwise reaction rather than an addition reaction,as with polyester resins,their reaction rate is very slow.The gelation of epoxy resins occurs at a later stage of reac- tion,and it is critical that the exotherm developed be contained within the die.This dictates a slow reaction rate,which results in a high labor. Because the epoxy begins to react slowly as soon as it is mixed,the pot life is short.The resin scrap rate is potentially higher if viscosity buildup affects wet-out to the extent that the bath must be recharged. The die temperature profiles used for epoxy are typically hotter than polyesters,and the drip-off at the entrance must often be discarded rather than recirculated to the bath.Because of the tendency for the ep- oxy resin to bond strongly to the die wall,epoxy products often display surface defects,such as exposed fibers,chipping,or loss of dimension, all of which increase finished-product scrap rate.These additional costs place epoxy resins in a class in which the end-use requirements must justify the high price [1]. TABLE 6.1 General Properties of Polyester Resins Used for Pultrusion. Property Value Viscosity at 25C(cP) 500-2000 Specific gravity 1.1 Gel time (minutes) 3-8 Cure time (minutes) 5-18 Peak exotherm (max temperature 415-470℉(213-243°C) during the process)

are short to allow for the high rate of production. If, for example, a rate of production of 20 cm/minute is desired, for the length of a die of 100 cm long, the duration of the resin inside the mold is 5 minutes. The resin should flow through the interstices of the tows, wet the fibers, gel and cure during this time. When the resin gels, it also shrinks, which helps to release the composite from the die wall. This, in turn, will reduce the pulling force. The resin normally used to make pultruded products is polyester resin, due to its low cost and low viscosity. Table 6.1 shows the viscosity of polyester resin along with gel time and cure time. When better corrosion resistance is required, vinyl ester resins are used. When a combination of superior mechanical and electrical properties is required, epoxy resin is used. Epoxy resins are expensive materials in a number of aspects. The resins are three to six times more expensive than polyesters and have a number of process-related costs not found with polyesters. Because they are cured by a stepwise reaction rather than an addition reaction, as with polyester resins, their reaction rate is very slow. The gelation of epoxy resins occurs at a later stage of reac￾tion, and it is critical that the exotherm developed be contained within the die. This dictates a slow reaction rate, which results in a high labor. Because the epoxy begins to react slowly as soon as it is mixed, the pot life is short. The resin scrap rate is potentially higher if viscosity buildup affects wet-out to the extent that the bath must be recharged. The die temperature profiles used for epoxy are typically hotter than polyesters, and the drip-off at the entrance must often be discarded rather than recirculated to the bath. Because of the tendency for the ep￾oxy resin to bond strongly to the die wall, epoxy products often display surface defects, such as exposed fibers, chipping, or loss of dimension, all of which increase finished-product scrap rate. These additional costs place epoxy resins in a class in which the end-use requirements must justify the high price [1]. Materials 237 TABLE 6.1 General Properties of Polyester Resins Used for Pultrusion. Property Value Viscosity at 25°C (cP) 500–2000 Specific gravity 1.1 Gel time (minutes) 3–8 Cure time (minutes) 5–18 Peak exotherm (max temperature during the process) 415–470°F (213–243°C)

238 PULTRUSION 3.COMBINATION OF OTHER PROCESSES WITH PULTRUSION As part of the process of automation,other types of processes have been combined with pultrusion to produce parts with enhanced proper- ties.These include in-line filament winding or in-line braiding,together with the pultrusion process. 3.1.In-Line Winding In situations such as the case of pultruded rods used for the reinforce- ment of concrete,it is necessary to provide roughness on the surface of the putruded rods to enhance the mechanical interlock between the rein- forcement rod and concrete.Filament wound strips can be placed on the outer surface of the pultruded rods for this purpose.At the exit of the pultrusion machine,spools of unidirectional tows loaded onto two coun- ter-rotating disks can be circumferentially wound around the exited pultruded rod. 3.2.In-Line Braiding In in-line braiding,a vertical braider is positioned in front of the pultrusion die.As the braider pays off the material,the material is drawn forward through the braiding ring and laid down on the mandrel.The re- sulting fiber angle is a function of the braider speed and pultrusion line speed.Impregnation of the thin walled braids is accomplished via con- tinuous resin transfer molding,called direct resin injection.The impreg- nated braid is drawn into the pultrusion die and the resin is polymerized. The cured tube is manufactured continuously. 4.FACTORS AFFECTING THE PULTRUDABILITY OF A COMPOSITE COMPONENT Two important characteristics affecting the pultrudability of a com- posite product are:the pulling force required to move the product steadily through the system and the pulling speed.The pulling force has to be sufficient to overcome the resistance at the different stages of the pultrusion process,and the pulling speed determines the productivity of the process.These are discussed below

3. COMBINATION OF OTHER PROCESSES WITH PULTRUSION As part of the process of automation, other types of processes have been combined with pultrusion to produce parts with enhanced proper￾ties. These include in-line filament winding or in-line braiding, together with the pultrusion process. 3.1. In-Line Winding In situations such as the case of pultruded rods used for the reinforce￾ment of concrete, it is necessary to provide roughness on the surface of the putruded rods to enhance the mechanical interlock between the rein￾forcement rod and concrete. Filament wound strips can be placed on the outer surface of the pultruded rods for this purpose. At the exit of the pultrusion machine, spools of unidirectional tows loaded onto two coun￾ter-rotating disks can be circumferentially wound around the exited pultruded rod. 3.2. In-Line Braiding In in-line braiding, a vertical braider is positioned in front of the pultrusion die. As the braider pays off the material, the material is drawn forward through the braiding ring and laid down on the mandrel. The re￾sulting fiber angle is a function of the braider speed and pultrusion line speed. Impregnation of the thin walled braids is accomplished via con￾tinuous resin transfer molding, called direct resin injection. The impreg￾nated braid is drawn into the pultrusion die and the resin is polymerized. The cured tube is manufactured continuously. 4. FACTORS AFFECTING THE PULTRUDABILITY OF A COMPOSITE COMPONENT Two important characteristics affecting the pultrudability of a com￾posite product are: the pulling force required to move the product steadily through the system and the pulling speed. The pulling force has to be sufficient to overcome the resistance at the different stages of the pultrusion process, and the pulling speed determines the productivity of the process. These are discussed below. 238 PULTRUSION

Factors Affecting the Pultrudability of a Composite Component 239 4.1.Pulling Forces in Pultrusion Controlling the buildup of pulling loads and developing ways to deal with their inevitable presence are major concerns for all pultruders.Pre- form compaction and the effects of fiber packing in the die contribute the most to pulling force generation during pultrusion.This is particularly true in the processing of epoxies where fiber and filler must be kept high to prevent sloughing or resin adhesion to die surfaces,and to maintain good surface finish. One way to analyze the parameters that contribute to the pulling force is to examine the contribution of the resistance from each step of the pro- cess.As can be seen from Figure 6.1,the resistance should be considered from the four different steps:Force required to collimate the fiber tows from the creel to the entrance to the die,force required to compact the fi- ber tows into the cavity in the die,force required to overcome the viscos- ity of the liquid resin,and force required to overcome the friction between the wall of the die and the solid composite product.This can be written as: Fotal=FcolFcompaction +FviscousFfrction (6.1) 4.1.1.Force Due to Collimation The collimation force Fo depends on the loading of the fibers relative to the size (diameter)of the die.There is a limit as to the difference be- tween the diameter of the collimated fiber bundles to the diameter of the die.Within limits,the larger the amount of fibers,the larger the FOne other aspect is the ease with which the fiber bundles are introduced into the die.Figure 6.4 shows the arrangement of a tapered entrance into the die.The taper configuration facilitates the introduction of the fibers into the die and reduces the F 4.1.2.Force Due to Compaction In Equation (6.1),the compaction force Fm is a force along the pull (axial)direction of the process.However,compaction is occurring along the radial direction of the die,which is normally the pull direction.Nor- mally,the greater the number of fibers that are squeezed into the cavity of the die,the larger will be the compaction.If too much fiber is put in,the compaction will be too large and pulling may not be possible.However, if too few fibers are put into the die,insufficient compaction will occur and voids may appear.In addition,there is resin shrinkage,which can

4.1. Pulling Forces in Pultrusion Controlling the buildup of pulling loads and developing ways to deal with their inevitable presence are major concerns for all pultruders. Pre￾form compaction and the effects of fiber packing in the die contribute the most to pulling force generation during pultrusion. This is particularly true in the processing of epoxies where fiber and filler must be kept high to prevent sloughing or resin adhesion to die surfaces, and to maintain good surface finish. One way to analyze the parameters that contribute to the pulling force is to examine the contribution of the resistance from each step of the pro￾cess. As can be seen from Figure 6.1, the resistance should be considered from the four different steps: Force required to collimate the fiber tows from the creel to the entrance to the die, force required to compact the fi￾ber tows into the cavity in the die, force required to overcome the viscos￾ity of the liquid resin, and force required to overcome the friction between the wall of the die and the solid composite product. This can be written as: F FF F F total col =+ + + compaction viscous friction (6.1) 4.1.1. Force Due to Collimation The collimation force Fcol depends on the loading of the fibers relative to the size (diameter) of the die. There is a limit as to the difference be￾tween the diameter of the collimated fiber bundles to the diameter of the die. Within limits, the larger the amount of fibers, the larger the Fcol. One other aspect is the ease with which the fiber bundles are introduced into the die. Figure 6.4 shows the arrangement of a tapered entrance into the die. The taper configuration facilitates the introduction of the fibers into the die and reduces the Fcol. 4.1.2. Force Due to Compaction In Equation (6.1), the compaction force Fcomp is a force along the pull (axial) direction of the process. However, compaction is occurring along the radial direction of the die, which is normally the pull direction. Nor￾mally, the greater the number of fibers that are squeezed into the cavity of the die, the larger will be the compaction. If too much fiber is put in, the compaction will be too large and pulling may not be possible. However, if too few fibers are put into the die, insufficient compaction will occur and voids may appear. In addition, there is resin shrinkage, which can Factors Affecting the Pultrudability of a Composite Component 239

240 PULTRUSION Z=la Fibers =bL Pulling Z=-lL Direction Z =-lig FIGURE 6.4 Tapered entrance facilitates the introduction of fibers into the die. also affect the compaction pressure.One way to get an estimate of the pressure in the material inside the die is to use Equation (3.13)in Chapter 3,repeated here as: (6.2) where, e,o1=the strain and stress along the pull direction e=the strain and stress in the radial direction Fi=the compliance coefficients of the fiber bundles The strain e is governed by the geometry of the die.If R is the radius of the die entrance with an initial volume fraction V,and R,is the die ra- dius at the position corresponding to the fiber volume fraction V,then the strain e is calculated to be: RJ-R。 e6= R。 (6.3) The axial strain depends on the degree of waviness of the fibers in the fiber bundle.Refer to Figure 3.19(Chapter 3),assuming that the length of the stretched fiber can be calculated by: 12L2 +a (6.4) 44

also affect the compaction pressure. One way to get an estimate of the pressure in the material inside the die is to use Equation (3.13) in Chapter 3, repeated here as: e e F F b F F b b bb b 1 11 1 1 1       =             σ σ (6.2) where, e1, σ1 = the strain and stress along the pull direction eb, σb = the strain and stress in the radial direction Fij = the compliance coefficients of the fiber bundles The strain eb is governed by the geometry of the die. If Ro is the radius of the die entrance with an initial volume fraction Vo, and Rf is the die ra￾dius at the position corresponding to the fiber volume fraction Vf , then the strain eb is calculated to be: e R R R b f o o = − (6.3) The axial strain depends on the degree of waviness of the fibers in the fiber bundle. Refer to Figure 3.19 (Chapter 3), assuming that the length of the stretched fiber can be calculated by: l L a 2 2 4 4 = + (6.4) 240 PULTRUSION FIGURE 6.4 Tapered entrance facilitates the introduction of fibers into the die

Factors Affecting the Pultrudability of a Composite Component 241 where, I=the length of the stretched fiber over one cycle L the length of the unstretched fiber over one cycle a the amplitude of the waviness as shown in Figure 3.19 (Chapter 3) The strain e can be calculated to be: 1-L +1 4 e1= (6.5) L where B=L/a.If B=200,one has e=0.00005. Recall the expressions for F from Chapter 3 as: ,41 =元E521+24s-1 (6.6a) E6=F1=-16B 元3Es(s-1) (6.6b) and E(S-1)4 (6.6c) 3πE One can calculate the stresses with information on the strains. Example 6.1 A pultrusion machine having a radius R 12.7 mm and R,=6.35 mm is used to pultrude fiberglass/polyester rod.The desired volume fraction is 0.60.Determine the contribution to pull force due to compaction.Assume B=200 and E=70GPa,V=0.4 andV.=0.785 From Equation ).6.35-12.7 =-0.50 12.7 0.785 V=V06 1144

where, l = the length of the stretched fiber over one cycle L = the length of the unstretched fiber over one cycle a = the amplitude of the waviness as shown in Figure 3.19 (Chapter 3) The strain e1 can be calculated to be: e l L L 1 2 1 4 = 1 − =+ − β (6.5) where β = L/a. If β = 200, one has e1 = 0.00005. Recall the expressions for Fij from Chapter 3 as: F E 11 4 1 2 1 = +− 12 1 π ς ς [ ( )] (6.6a) F F E 1 1 b b 3 2 16 3 = =− −1 π β ς ς( ) (6.6b) and F E bb = − β π ς 4 4 3 ( )1 (6.6c) One can calculate the stresses with information on the strains. Factors Affecting the Pultrudability of a Composite Component 241 Example 6.1 A pultrusion machine having a radius Ro = 12.7 mm and Rf = 6.35 mm is used to pultrude fiberglass/polyester rod. The desired volume fraction is 0.60. Determine the contribution to pull force due to compaction. Assume β = 200 and E = 70 GPa,Vo = 0.4 and Va = 0.785. From Equation (6.3), eb = − = − 6 35 12 7 12 7 0 50 . . . . ς= = = V V a f 0 785 0 60 1144 . .

242 PULTRUSION The compliance coefficients can be calculated using Equations(6.6)as =若+24s-l1=会70deua47m+20u4-1=0o25G π70GPa 。=F。1=sBg2(S-12=-162002 π370GPa 1144)1144-1)2=-1.009GPa F6= B4 s-1)= 2004 (1144-1)=1043GPa- 3πE 3π(70GPa) Equation(6.2)can be inverted to be written as: 1 (6.7) For this particular case: [10431.0090.00005 26.08-1.021.0090.025 0.5 1 「0.5521 「0.022 T22 GPa= MPa 25.060.0125 0.0005 0.5 Note that due to the small value ofe.its contribution to o is about 10%and it has lit- tle contribution to o It can be seen that the contribution of compaction to the axial load is quite significant.Note also that the above calculations are only approximate due to the application of linear assumption to a situation of large deformation. The above example shows a simplified estimate for the compression stress o,due to existence of the compressive strain e.The compressive strain in the example is a function of the reduction in geometry of the die from the entrance to the point of interest.This,in turn,is related to the change in volume fraction of the fibers at the different positions.There are other parameters that also have an effect on the relation between the fiber volume fraction and the compressive stress as discussed below. 4.1.3.Parameters Affecting the Compression Stress Figure 6.5 shows the relation between fiber volume fraction and com- pressive stress.It can be seen that even though the shape of the curve be- tween compression stress and volume fraction is similar,Different types of fiber forms show different curves but the shapes of the curves are simi- lar,the actual values of the stresses depend on the type of fibers.Apart from these,there are many parameters that affect the compressibility of

The above example shows a simplified estimate for the compression stress σb due to existence of the compressive strain eb. The compressive strain in the example is a function of the reduction in geometry of the die from the entrance to the point of interest. This, in turn, is related to the change in volume fraction of the fibers at the different positions. There are other parameters that also have an effect on the relation between the fiber volume fraction and the compressive stress as discussed below. 4.1.3. Parameters Affecting the Compression Stress Figure 6.5 shows the relation between fiber volume fraction and com￾pressive stress. It can be seen that even though the shape of the curve be￾tween compression stress and volume fraction is similar, Different types of fiber forms show different curves but the shapes of the curves are simi￾lar, the actual values of the stresses depend on the type of fibers. Apart from these, there are many parameters that affect the compressibility of 242 PULTRUSION The compliance coefficients can be calculated using Equations (6.6) as F E 11 4 1 2 2 2 12 1 4 1 70 = +−= 1144 1 2 1144 + − π ς ς π [ ( )] (. )[ (. GPa 1 0 025 2 1 )] . = − GPa F F E 1 1 b b 3 2 2 3 3 2 16 1 16 200 70 = = − =− 1144 11 π β ς ς π ( ) ( . )( . GPa 44 1 1 009 3 1 − =− − ) . GPa Fbb = −= −= β − πΕ ς π 4 4 4 4 3 1 200 3 70 ( ) 1144 1 1043 ( (. ) GPa) GPa 1 Equation (6.2) can be inverted to be written as: σ σ 1 11 1 2 1 1 11 1 1 b bb b bb b FF F b b F F F F e e       = − − −             (6.7) For this particular case: σ σ 1 1 26 08 1 02 1043 1 009 1 009 0 025 0 0 b       = −       . . . . . . 0005 0 5.       =       =       = 1 25 06 0 552 0 0125 0 022 0 0005 22 . 0 . . . . GPa .5       MPa Note that due to the small value of e1, its contribution to σ1 is about 10% and it has lit￾tle contribution to σb. It can be seen that the contribution of compaction to the axial load is quite significant. Note also that the above calculations are only approximate due to the application of linear assumption to a situation of large deformation