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.