5 Manufacturing A key ingredient in the successful production application of a material or a component is a cost-effective and reliable manufacturing method.Cost- effectiveness depends largely on the rate of production,and reliability requires a uniform quality from part to part. The early manufacturing method for fiber-reinforced composite structural parts used a hand layup technique.Although hand layup is a reliable process,it is by nature very slow and labor-intensive.In recent years,particularly due to the interest generated in the automotive industry,there is more emphasis on the development of manufacturing methods that can support mass production rates.Compression molding,pultrusion,and filament winding represent three such manufacturing processes.Although they have existed for many years, investigations on their basic characteristics and process optimization started mostly in the mid-1970s.Resin transfer molding(RTM)is another manufac- turing process that has received significant attention in both aerospace and automotive industries for its ability to produce composite parts with complex shapes at relatively high production rates.With the introduction of automa- tion,fast-curing resins,new fiber forms,high-resolution quality control tools, and so on,the manufacturing technology for fiber-reinforced polymer compos- ites has advanced at a remarkably rapid pace. This chapter describes the basic characteristics of major manufacturing methods used in the fiber-reinforced polymer industry.Emphasis is given to process parameters and their relation to product quality.Quality inspection methods and cost issues are also discussed in this chapter. 5.1 FUNDAMENTALS Transformation of uncured or partially cured fiber-reinforced thermoset poly- mers into composite parts or structures involves curing the material at elevated temperatures and pressures for a predetermined length of time.High cure temperatures are required to initiate and sustain the chemical reaction that transforms the uncured or partially cured material into a fully cured solid.High pressures are used to provide the force needed for the flow of the highly viscous resin or fiber-resin mixture in the mold,as well as for the consolidation of individual unbonded plies into a bonded laminate.The magnitude of these two important process parameters,as well as their duration,significantly affects the 2007 by Taylor&Francis Group.LLC
5 Manufacturing A key ingredient in the successful production application of a material or a component is a cost-effective and reliable manufacturing method. Costeffectiveness depends largely on the rate of production, and reliability requires a uniform quality from part to part. The early manufacturing method for fiber-reinforced composite structural parts used a hand layup technique. Although hand layup is a reliable process, it is by nature very slow and labor-intensive. In recent years, particularly due to the interest generated in the automotive industry, there is more emphasis on the development of manufacturing methods that can support mass production rates. Compression molding, pultrusion, and filament winding represent three such manufacturing processes. Although they have existed for many years, investigations on their basic characteristics and process optimization started mostly in the mid-1970s. Resin transfer molding (RTM) is another manufacturing process that has received significant attention in both aerospace and automotive industries for its ability to produce composite parts with complex shapes at relatively high production rates. With the introduction of automation, fast-curing resins, new fiber forms, high-resolution quality control tools, and so on, the manufacturing technology for fiber-reinforced polymer composites has advanced at a remarkably rapid pace. This chapter describes the basic characteristics of major manufacturing methods used in the fiber-reinforced polymer industry. Emphasis is given to process parameters and their relation to product quality. Quality inspection methods and cost issues are also discussed in this chapter. 5.1 FUNDAMENTALS Transformation of uncured or partially cured fiber-reinforced thermoset polymers into composite parts or structures involves curing the material at elevated temperatures and pressures for a predetermined length of time. High cure temperatures are required to initiate and sustain the chemical reaction that transforms the uncured or partially cured material into a fully cured solid. High pressures are used to provide the force needed for the flow of the highly viscous resin or fiber–resin mixture in the mold, as well as for the consolidation of individual unbonded plies into a bonded laminate. The magnitude of these two important process parameters, as well as their duration, significantly affects the 2007 by Taylor & Francis Group, LLC
quality and performance of the molded product.The length of time required to properly cure a part is called the cure cycle.Since the cure cycle determines the production rate for a part,it is desirable to achieve the proper cure in the shortest amount of time.It should be noted that the cure cycle depends on a number of factors,including resin chemistry,catalyst reactivity,cure tempera- ture,and the presence of inhibitors or accelerators. 5.1.1 DEGREE OF CURE A number of investigators [1-3]have experimentally measured the heat evolved in a curing reaction and related it to the degree of cure achieved at any time during the curing process.Experiments are performed in a differential scanning calorimeter (DSC)in which a small sample,weighing a few milligrams,is heated either isothermally (i.e.,at constant temperature)or dynamically (i.e., with uniformly increasing temperature).The instrumentation in DSC monitors the rate of heat generation as a function of time and records it.Figure 5.1 schematically illustrates the rate of heat generation curves for isothermal and dynamic heating. The total heat generation to complete a curing reaction (i.e.,100%degree of cure)is equal to the area under the rate of heat generation-time curve obtained in a dynamic heating experiment.It is expressed as dt. (5.1) where HR heat of reaction (do/dt)d rate of heat generation in a dynamic experiment le =time required to complete the reaction H 0 Time Time (a) (b) FIGURE 5.1 Schematic representation of the rate of heat generation in (a)dynamic and (b)isothermal heating of a thermoset polymer in a differential scanning calorimeter(DSC). 2007 by Taylor Francis Group,LLC
quality and performance of the molded product. The length of time required to properly cure a part is called the cure cycle. Since the cure cycle determines the production rate for a part, it is desirable to achieve the proper cure in the shortest amount of time. It should be noted that the cure cycle depends on a number of factors, including resin chemistry, catalyst reactivity, cure temperature, and the presence of inhibitors or accelerators. 5.1.1 DEGREE OF CURE A number of investigators [1–3] have experimentally measured the heat evolved in a curing reaction and related it to the degree of cure achieved at any time during the curing process. Experiments are performed in a differential scanning calorimeter (DSC) in which a small sample, weighing a few milligrams, is heated either isothermally (i.e., at constant temperature) or dynamically (i.e., with uniformly increasing temperature). The instrumentation in DSC monitors the rate of heat generation as a function of time and records it. Figure 5.1 schematically illustrates the rate of heat generation curves for isothermal and dynamic heating. The total heat generation to complete a curing reaction (i.e., 100% degree of cure) is equal to the area under the rate of heat generation–time curve obtained in a dynamic heating experiment. It is expressed as HR ¼ ðtf 0 dQ dt d dt, (5:1) where HR ¼ heat of reaction (dQ=dt)d ¼ rate of heat generation in a dynamic experiment tf ¼ time required to complete the reaction HR Rate of heat generation Rate of heat generation 0 H (b) t Time 0 tf (a) Time FIGURE 5.1 Schematic representation of the rate of heat generation in (a) dynamic and (b) isothermal heating of a thermoset polymer in a differential scanning calorimeter (DSC). 2007 by Taylor & Francis Group, LLC.
The amount of heat released in time t at a constant curing temperature T is determined from isothermal experiments.The area under the rate of heat generation-time curve obtained in an isothermal experiment is expressed as H=a} (5.2) where H is the amount of heat released in time t and (do/dt);is the rate of heat generation in an isothermal experiment conducted at a constant temperature T. The degree of cure ac at any time t is defined as H dc= (5.3) HR Figure 5.2 shows a number of curves relating the degree of cure ac to cure time for a vinyl ester resin at various cure temperatures.From this figure,it can be seen that ac increases with both time and temperature;however,the rate of cure,dac/dt,is decreased as the degree of cure attains asymptotically a max- imum value.If the cure temperature is too low,the degree of cure may not reach a 100%level for any reasonable length of time.The rate of cure dac/dt, obtained from the slope of ac vs.t curve and plotted in Figure 5.3,exhibits 0.8 65℃ 0.6 5✉回SD 6 0.4 4AA△△△bP50C 1 35C 0.2 000 00 0 .1 1 10 100 Cure time(min) FIGURE 5.2 Degree of cure for a vinyl ester resin at various cure temperatures.(After Han,C.D.and Lem,K.W.,J.Appl.Polym.Sci.,29,1878,1984.) 2007 by Taylor&Francis Group.LLC
The amount of heat released in time t at a constant curing temperature T is determined from isothermal experiments. The area under the rate of heat generation–time curve obtained in an isothermal experiment is expressed as H ¼ ðt 0 dQ dt i dt, (5:2) where H is the amount of heat released in time t and (dQ=dt)i is the rate of heat generation in an isothermal experiment conducted at a constant temperature T. The degree of cure ac at any time t is defined as ac ¼ H HR : (5:3) Figure 5.2 shows a number of curves relating the degree of cure ac to cure time for a vinyl ester resin at various cure temperatures. From this figure, it can be seen that ac increases with both time and temperature; however, the rate of cure, dac=dt, is decreased as the degree of cure attains asymptotically a maximum value. If the cure temperature is too low, the degree of cure may not reach a 100% level for any reasonable length of time. The rate of cure dac=dt, obtained from the slope of ac vs. t curve and plotted in Figure 5.3, exhibits 65C 60C 50C 45C 40C 35C 0 0 .2 0 .4 0 .6 0 .8 Degree of cure Cure time (min) 0.1 1 10 100 FIGURE 5.2 Degree of cure for a vinyl ester resin at various cure temperatures. (After Han, C.D. and Lem, K.W., J. Appl. Polym. Sci., 29, 1878, 1984.) 2007 by Taylor & Francis Group, LLC.
0.7 0.6 65℃ 0.5 ⊙ ⊙ 60C 0.4 50℃ 0.3 ⊙ 0.2 45C 40C 0.1 35C A 0.0 0 0.2 0.40.6 0.8 1.0 Degree of cure (ac) FIGURE 5.3 Rate of cure for a vinyl ester resin at various cure temperatures.(After Han,C.D.and Lem,K.W.,J.Appl.Polym.Sci.,29,1878,1984.) a maximum value at 10%-40%of the total cure achieved.Higher cure temperatures increase the rate of cure and produce the maximum degree of cure in shorter periods of time.On the other hand,the addition of a low-profile agent,such as a thermoplastic polymer,to a polyester or a vinyl ester resin decreases the cure rate. Kamal and Sourour [4]have proposed the following expression for the isothermal cure rate of a thermoset resin: dae=(k+kzae)(1-ae)", (5.4) dt where k and k2 are reaction rate constants and m and n are constants describ- ing the order of reaction.The parameters m and n do not vary significantly with the cure temperature,but k and k2 depend strongly on the cure temperature. With the assumption of a second-order reaction (i.e.,m+n=2),Equation 5.4 has been used to describe the isothermal cure kinetics of epoxy,unsaturated polyester,and vinyl ester resins.The values of k1,k2,m,and n are determined by nonlinear least-squares curve fit to the dac/dt vs.ac data.Typical values of these constants for a number of resins are listed in Table 5.1. 2007 by Taylor Francis Group,LLC
a maximum value at 10%–40% of the total cure achieved. Higher cure temperatures increase the rate of cure and produce the maximum degree of cure in shorter periods of time. On the other hand, the addition of a low-profile agent, such as a thermoplastic polymer, to a polyester or a vinyl ester resin decreases the cure rate. Kamal and Sourour [4] have proposed the following expression for the isothermal cure rate of a thermoset resin: dac dt ¼ (k1 þ k2am c )(1 ac) n , (5:4) where k1 and k2 are reaction rate constants and m and n are constants describing the order of reaction. The parameters m and n do not vary significantly with the cure temperature, but k1 and k2 depend strongly on the cure temperature. With the assumption of a second-order reaction (i.e., m þ n ¼ 2), Equation 5.4 has been used to describe the isothermal cure kinetics of epoxy, unsaturated polyester, and vinyl ester resins. The values of k1, k2, m, and n are determined by nonlinear least-squares curve fit to the dac=dt vs. ac data. Typical values of these constants for a number of resins are listed in Table 5.1. 65C 60C 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 0.2 0.4 0.6 Degree of cure (ac) 0.8 1.0 50C 45C 40C 35C Rate of cure ( ac) (per minute) FIGURE 5.3 Rate of cure for a vinyl ester resin at various cure temperatures. (After Han, C.D. and Lem, K.W., J. Appl. Polym. Sci., 29, 1878, 1984.) 2007 by Taylor & Francis Group, LLC.
TABLE 5.1 Kinetic Parameters for Various Resin Systems Kinetic Parameters in Equation 5.4 Temperature, Resin ℃(F) k (per min) k2 (per min) m n Polyester 45(113) 0.0131 0.351 0.23 1.77 60(140) 0.0924 1.57 0.40 1.60 Low-profile polyester 45(113) 0.0084 0.144 0.27 1.73 (with 20%polyvinyl 60(140) 0.0264 0.282 0.27 1.73 acetate) Vinyl ester 45(113) 0.0073 0.219 0.33 1.76 60(140) 0.0624 1.59 0.49 1.51 Source:Adapted from Lem,K.W.and Han,C.D.,Polym.Eng.Sci.,24,175,1984. 5.1.2 VisCOsITY Viscosity of a fluid is a measure of its resistance to flow under shear stresses. Low-molecular-weight fluids,such as water and motor oil,have low viscosities and flow readily.High-molecular-weight fluids,such as polymer melts,have high viscosities and flow only under high stresses. The two most important factors determining the viscosity of a fluid are the temperature and shear rate.For all fluids,the viscosity decreases with increas- ing temperature.Shear rate does not have any influence on the viscosity of low- molecular-weight fluids,whereas it tends to either increase (shear thickening) or decrease (shear thinning)the viscosity of a high-molecular-weight fluids (Figure 5.4).Polymer melts,in general,are shear-thinning fluids since their viscosity decreases with increasing intensity of shearing. The starting material for a thermoset resin is a low-viscosity fluid.However, its viscosity increases with curing and approaches a very large value as it transforms into a solid mass.Variation of viscosity during isothermal curing of an epoxy resin is shown in Figure 5.5.Similar viscosity-time curves are also observed for polyester [3]and vinyl ester [5]resins.In all cases,the viscosity increases with increasing cure time and temperature.The rate of viscosity increase is low at the early stage of curing.After a threshold degree of cure is achieved,the resin viscosity increases at a very rapid rate.The time at which this occurs is called the gel time.The gel time is an important molding parameter,since the flow of resin in the mold becomes increasingly difficult at the end of this time period. A number of important observations can be made from the viscosity data reported in the literature: 2007 by Taylor&Francis Group.LLC
5.1.2 VISCOSITY Viscosity of a fluid is a measure of its resistance to flow under shear stresses. Low-molecular-weight fluids, such as water and motor oil, have low viscosities and flow readily. High-molecular-weight fluids, such as polymer melts, have high viscosities and flow only under high stresses. The two most important factors determining the viscosity of a fluid are the temperature and shear rate. For all fluids, the viscosity decreases with increasing temperature. Shear rate does not have any influence on the viscosity of lowmolecular-weight fluids, whereas it tends to either increase (shear thickening) or decrease (shear thinning) the viscosity of a high-molecular-weight fluids (Figure 5.4). Polymer melts, in general, are shear-thinning fluids since their viscosity decreases with increasing intensity of shearing. The starting material for a thermoset resin is a low-viscosity fluid. However, its viscosity increases with curing and approaches a very large value as it transforms into a solid mass. Variation of viscosity during isothermal curing of an epoxy resin is shown in Figure 5.5. Similar viscosity–time curves are also observed for polyester [3] and vinyl ester [5] resins. In all cases, the viscosity increases with increasing cure time and temperature. The rate of viscosity increase is low at the early stage of curing. After a threshold degree of cure is achieved, the resin viscosity increases at a very rapid rate. The time at which this occurs is called the gel time. The gel time is an important molding parameter, since the flow of resin in the mold becomes increasingly difficult at the end of this time period. A number of important observations can be made from the viscosity data reported in the literature: TABLE 5.1 Kinetic Parameters for Various Resin Systems Resin Temperature, 8C (8F) Kinetic Parameters in Equation 5.4 k1 (per min) k2 (per min) m n Polyester 45 (113) 0.0131 0.351 0.23 1.77 60 (140) 0.0924 1.57 0.40 1.60 Low-profile polyester (with 20% polyvinyl acetate) 45 (113) 0.0084 0.144 0.27 1.73 60 (140) 0.0264 0.282 0.27 1.73 Vinyl ester 45 (113) 0.0073 0.219 0.33 1.76 60 (140) 0.0624 1.59 0.49 1.51 Source: Adapted from Lem, K.W. and Han, C.D., Polym. Eng. Sci., 24, 175, 1984. 2007 by Taylor & Francis Group, LLC
Newtonian Shear- thickening JeayS Shear-thinning Shear rate FIGURE 5.4 Schematic shear stress vs.shear rate curves for various types of liquids. Note that the viscosity is defined as the slope of the shear stress-shear rate curve. 1.A B-staged or a thickened resin has a much higher viscosity than the neat resin at all stages of curing. 2.The addition of fillers,such as CaCO3,to the neat resin increases its viscosity as well as the rate of viscosity increase during curing.On the 500 400 ● 300 O S 200 L。gopoce84oo00f 100 0000 0 0 10 20 30 40 Time(min) FIGURE 5.5 Variation of viscosity during isothermal curing of an epoxy resin.(After Kamal,M.R.,Polym.Eng.Sci.,14,231,1974.) 2007 by Taylor Francis Group,LLC
1. A B-staged or a thickened resin has a much higher viscosity than the neat resin at all stages of curing. 2. The addition of fillers, such as CaCO3, to the neat resin increases its viscosity as well as the rate of viscosity increase during curing. On the Shearthickening Newtonian Shear-thinning Shear rate Shear stress FIGURE 5.4 Schematic shear stress vs. shear rate curves for various types of liquids. Note that the viscosity is defined as the slope of the shear stress–shear rate curve. 500 400 127C 117C 107C 97C 300 200 100 0 0 10 20 Time (min) Viscosity (poise) 30 40 FIGURE 5.5 Variation of viscosity during isothermal curing of an epoxy resin. (After Kamal, M.R., Polym. Eng. Sci., 14, 231, 1974.) 2007 by Taylor & Francis Group, LLC.
other hand,the addition of thermoplastic additives (such as those added in low-profile polyester and vinyl ester resins)tends to reduce the rate of viscosity increase during curing. 3.The increase in viscosity with cure time is less if the shear rate is increased. This phenomenon,known as shear thinning,is more pronounced in B-staged or thickened resins than in neat resins.Fillers and thermo- plastic additives also tend to increase the shear-thinning phenomenon. 4.The viscosity n of a thermoset resin during the curing process is a function of cure temperature T,shear rate y,and the degree of cure ae n=n(T,y,ac). (5.5) The viscosity function for thermosets is significantly different from that for thermoplastics.Since no in situ chemical reaction occurs during the processing of a thermoplastic polymer,its viscosity depends on tempera- ture and shear rate. 5.At a constant shear rate and for the same degree of cure,the n vs.1/T plot is linear (Figure 5.6).This suggests that the viscous flow of a 1000 500 200 100 50 0 20 40 10 0 5 H=30 kcal 2.4 2.5 2.6 2.7 2.8 2.9 1000 T (perK) FIGURE 5.6 Viscosity-temperature relationships for an epoxy resin at different levels of cure.(After Kamal,M.R.,Polym.Eng.Sci.,14,231,1974.) 2007 by Taylor Francis Group.LLC
other hand, the addition of thermoplastic additives (such as those added in low-profile polyester and vinyl ester resins) tends to reduce the rate of viscosity increase during curing. 3. The increase in viscosity with cure time is less if the shear rate is increased. This phenomenon, known as shear thinning, is more pronounced in B-staged or thickened resins than in neat resins. Fillers and thermoplastic additives also tend to increase the shear-thinning phenomenon. 4. The viscosity h of a thermoset resin during the curing process is a function of cure temperature T, shear rate g_, and the degree of cure ac h ¼ h(T, g_, ac): (5:5) The viscosity function for thermosets is significantly different from that for thermoplastics. Since no in situ chemical reaction occurs during the processing of a thermoplastic polymer, its viscosity depends on temperature and shear rate. 5. At a constant shear rate and for the same degree of cure, the h vs. 1=T plot is linear (Figure 5.6). This suggests that the viscous flow of a 70 60 50 40 H =30 kcal 1 2 5 10 20 50 100 200 500 1000 2.4 2.5 2.6 2.7 2.8 2.9 Viscosity (poise) 1000 T (per K) FIGURE 5.6 Viscosity–temperature relationships for an epoxy resin at different levels of cure. (After Kamal, M.R., Polym. Eng. Sci., 14, 231, 1974.) 2007 by Taylor & Francis Group, LLC.
thermoset polymer is an energy-activated process.Thus,its viscosity as a function of temperature can be written as 7=noex (5.6) where n viscosity*(Pas or poise) E flow activation energy (cal/g mol) R universal gas constant T cure temperature (K) no =constant The activation energy for viscous flow increases with the degree of cure and approaches a very high value near the gel point. 5.1.3 RESIN FLOW Proper flow of resin through a dry fiber network (in liquid composite mold- ing [LCM])or a prepreg layup (in bag molding)is critical in producing void-free parts and good fiber wet-out.In thermoset resins,curing may take place simultaneously with resin flow,and if the resin viscosity rises too rapidly due to curing,its flow may be inhibited,causing voids and poor interlaminar adhesion. Resin flow through fiber network has been modeled using Darcy's equa- tion,which was derived for flow of Newtonian fluids through a porous med- ium.This equation relates the volumetric resin-flow rate q per unit area to the pressure gradient that causes the flow to occur.For one-dimensional flow in the x direction, q= (5.7) where g =volumetric flow rate per unit area(m/s)in the x direction Po=permeability (m m viscosity (Ns/m) dp =pressure gradient (N/m),which is negative in the direction of flow (positive x direction) Unit of viscosity:I Pas 1 Ns/m2=10 poise (P)=1000 centipoise(cP). 2007 by Taylor Francis Group,LLC
thermoset polymer is an energy-activated process. Thus, its viscosity as a function of temperature can be written as h ¼ ho exp E RT , (5:6) where h ¼ viscosity* (Pa s or poise) E ¼ flow activation energy (cal=g mol) R ¼ universal gas constant T ¼ cure temperature (8K) ho ¼ constant The activation energy for viscous flow increases with the degree of cure and approaches a very high value near the gel point. 5.1.3 RESIN FLOW Proper flow of resin through a dry fiber network (in liquid composite molding [LCM]) or a prepreg layup (in bag molding) is critical in producing void-free parts and good fiber wet-out. In thermoset resins, curing may take place simultaneously with resin flow, and if the resin viscosity rises too rapidly due to curing, its flow may be inhibited, causing voids and poor interlaminar adhesion. Resin flow through fiber network has been modeled using Darcy’s equation, which was derived for flow of Newtonian fluids through a porous medium. This equation relates the volumetric resin-flow rate q per unit area to the pressure gradient that causes the flow to occur. For one-dimensional flow in the x direction, q ¼ P0 h dp dx , (5:7) where q ¼ volumetric flow rate per unit area (m=s) in the x direction P0 ¼ permeability (m2 ) h ¼ viscosity (N s=m2 ) dp dx ¼ pressure gradient (N=m3 ), which is negative in the direction of flow (positive x direction) * Unit of viscosity: 1 Pa s ¼ 1 Ns=m2 ¼ 10 poise (P) ¼ 1000 centipoise (cP). 2007 by Taylor & Francis Group, LLC.
The permeability is determined by the following equation known as the Kozeny-Carman equation: P0= d眼(1-v)3 (5.8) 16Kv2 where d fiber diameter Vr fiber volume fraction K Kozeny constant Equations 5.7 and 5.8,although simplistic,have been used by many investiga- tors in modeling resin flow from prepregs in bag-molding process and mold filling in RTM.Equation 5.8 assumes that the porous medium is isotropic,and the pore size and distribution are uniform.However,fiber networks are non- isotropic and therefore,the Kozeny constant,K,is not the same in all direc- tions.For example,for a fiber network with unidirectional fiber orientation, the Kozeny constant in the transverse direction(K22)is an order of magnitude higher than the Kozeny constant in the longitudinal direction(Ku).This means that the resin flow in the transverse direction is much lower than that in the longitudinal direction.Furthermore,the fiber packing in a fiber network is not uniform,which also affects the Kozeny constant,and therefore the resin flow. Equation 5.8 works well for predicting resin flow in the fiber direction. However,Equation 5.8 is not valid for resin flow in the transverse direction, since according to this equation resin flow between the fibers does not stop even when the fiber volume fraction reaches the maximum value at which the fibers touch each other and there are no gaps between them.Gebart [6]derived the following permeability equations in the fiber direction and normal to the fiber direction for unidirectional continuous fiber network with regularly arranged, parallel fibers. In the fiber direction:Pu 2(1-v) (5.9a) C1 v? Normal to the fiber direction:P22=C2 Vf.max (5.9b) 41 where C=hydraulic radius between the fibers C2 a constant Vr.max maximum fiber volume fraction (i.e.,at maximum fiber packing) 2007 by Taylor Francis Group.LLC
The permeability is determined by the following equation known as the Kozeny-Carman equation: P0 ¼ d2 f 16K (1 vf) 3 v2 f , (5:8) where df ¼ fiber diameter vf ¼ fiber volume fraction K ¼ Kozeny constant Equations 5.7 and 5.8, although simplistic, have been used by many investigators in modeling resin flow from prepregs in bag-molding process and mold filling in RTM. Equation 5.8 assumes that the porous medium is isotropic, and the pore size and distribution are uniform. However, fiber networks are nonisotropic and therefore, the Kozeny constant, K, is not the same in all directions. For example, for a fiber network with unidirectional fiber orientation, the Kozeny constant in the transverse direction (K22) is an order of magnitude higher than the Kozeny constant in the longitudinal direction (K11). This means that the resin flow in the transverse direction is much lower than that in the longitudinal direction. Furthermore, the fiber packing in a fiber network is not uniform, which also affects the Kozeny constant, and therefore the resin flow. Equation 5.8 works well for predicting resin flow in the fiber direction. However, Equation 5.8 is not valid for resin flow in the transverse direction, since according to this equation resin flow between the fibers does not stop even when the fiber volume fraction reaches the maximum value at which the fibers touch each other and there are no gaps between them. Gebart [6] derived the following permeability equations in the fiber direction and normal to the fiber direction for unidirectional continuous fiber network with regularly arranged, parallel fibers. In the fiber direction: P11 ¼ 2d2 f C1 1 v3 f v2 f , (5:9a) Normal to the fiber direction: P22 ¼ C2 ffiffiffiffiffiffiffiffiffiffiffi vf, max vf r 1 5=2 d2 f 4 , (5:9b) where C1 ¼ hydraulic radius between the fibers C2 ¼ a constant vf,max ¼ maximum fiber volume fraction (i.e., at maximum fiber packing) 2007 by Taylor & Francis Group, LLC.
The parameters Ci,C2,and Vr.max depend on the fiber arrangement in the network.For a square arrangement of fibers,C=57,C2=0.4,and Vr.max =0.785.For a hexagonal arrangement of fibers (see Problem P2.18), C=53,C2=0.231,and Vr.max =0.906.Note that Equation 5.9a for resin flow parallel to the fiber direction has the same form as the Kozeny-Carman equation 5.8.According to Equation 5.9b,which is applicable for resin flow transverse to the flow direction,P22=0 at Vr Vr.max,and therefore,the transverse resin flow stops at the maximum fiber volume fraction. The permeability equations assume that the fiber distribution is uniform, the gaps between the fibers are the same throughout the network,the fibers are perfectly aligned,and all fibers in the network have the same diameter. These assumptions are not valid in practice,and therefore,the permeability predictions using Equation 5.8 or 5.9 can only be considered approximate. 5.1.4 CONSOLIDATION Consolidation of layers in a fiber network or a prepreg layup requires good resin flow and compaction;otherwise,the resulting composite laminate may contain a variety of defects,including voids,interply cracks,resin-rich areas,or resin-poor areas.Good resin flow by itself is not sufficient to produce good consolidation [7]. Both resin flow and compaction require the application of pressure during processing in a direction normal to the dry fiber network or prepreg layup.The pressure is applied to squeeze out the trapped air or volatiles,as the liquid resin flows through the fiber network or prepreg layup,suppresses voids,and attains uniform fiber volume fraction.Gutowski et al.[8]developed a model for consolidation in which it is assumed that the applied pressure is shared by the fiber network and the resin so that p=+Pr, (5.10) where p=applied pressure o =average effective stress on the fiber network pr=average pressure on the resin The average effective pressure on the fiber network increases with increasing fiber volume fraction and is given by O=A 1- (5.11) (-)1 2007 by Taylor Francis Group,LLC
The parameters C1, C2, and vf,max depend on the fiber arrangement in the network. For a square arrangement of fibers, C1 ¼ 57, C2 ¼ 0.4, and vf,max ¼ 0.785. For a hexagonal arrangement of fibers (see Problem P2.18), C1 ¼ 53, C2 ¼ 0.231, and vf,max ¼ 0.906. Note that Equation 5.9a for resin flow parallel to the fiber direction has the same form as the Kozeny-Carman equation 5.8. According to Equation 5.9b, which is applicable for resin flow transverse to the flow direction, P22 ¼ 0 at vf ¼ vf,max, and therefore, the transverse resin flow stops at the maximum fiber volume fraction. The permeability equations assume that the fiber distribution is uniform, the gaps between the fibers are the same throughout the network, the fibers are perfectly aligned, and all fibers in the network have the same diameter. These assumptions are not valid in practice, and therefore, the permeability predictions using Equation 5.8 or 5.9 can only be considered approximate. 5.1.4 CONSOLIDATION Consolidation of layers in a fiber network or a prepreg layup requires good resin flow and compaction; otherwise, the resulting composite laminate may contain a variety of defects, including voids, interply cracks, resin-rich areas, or resin-poor areas. Good resin flow by itself is not sufficient to produce good consolidation [7]. Both resin flow and compaction require the application of pressure during processing in a direction normal to the dry fiber network or prepreg layup. The pressure is applied to squeeze out the trapped air or volatiles, as the liquid resin flows through the fiber network or prepreg layup, suppresses voids, and attains uniform fiber volume fraction. Gutowski et al. [8] developed a model for consolidation in which it is assumed that the applied pressure is shared by the fiber network and the resin so that p ¼ s þ pr, (5:10) where p ¼ applied pressure s ¼ average effective stress on the fiber network pr ¼ average pressure on the resin The average effective pressure on the fiber network increases with increasing fiber volume fraction and is given by s ¼ A 1 ffiffiffiffiffi vf vo q ffiffiffiffi va vf q 1 4 , (5:11) 2007 by Taylor & Francis Group, LLC.