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The fallacy of high volume fraction
The fallacy of high volume fraction G. R. FROUD*
There are theoretical as well as practical reasons why a high glass content in glass fibre reinforced plastics is not always desirable. An optimum glass content is calculated to give maximum flexural strength and stiffness for simple beams and plates, this content being governed by similar rules to those that apply for the optimum configuration in sandwich beams. The argument leads to the surprising result that, in particular instances, volume fractions as low as 13% can be required theoretically to give optimum construction. It is concluded that, in many bending applications, hand lay-up can compete fully with hot moulded construction, and in some cases even prove superior.
Because the strength and stiffness of glass fibre reinforced plastics (GRP) derive mainly from the glass fibre itself, it is quite usual to endeavour to obtain the maximum possible volume fraction of glass in the finished product. This is justifiable on purely theoretical grounds when considering the material under tensile forces, because of the high specific tensile strength (tensile strength per unit weight) and relatively high specific Young's modulus of glass fibre. Practically speaking this philosphy leads to several other problems, one of which is the necessity to use potentially expensive and often unwieldy compression moulding techniques in order to obtain high glass volume fractions. In considering simple GRP beams and plates under the action of bending forces, however, theory shows that, on a weight basis, there is an optimum value for volume fraction to give maximum bending resistance and flexural rigidity. For whereas the parameters of interest for specific strength and stiffness are (f/p) and (E/p) respectively (where f is tensile strength, E is Young's modulus and p the density) the important parameters in bending are (f/p2) and (E/p3). This is because flexural strength is proportional to (thickness) 2 and flexural rigidity is proportional to (thickness) 3,(reference 1) and for a given length, width and weight of plate or beam the thickness is inversely proportional to material density. hr what follows the ccnditions for optimum flexural strength and stiffness in GRP are derived; these conditions are found (not surprisingly) to be similar in principle to the conditions necessary for optimum flexural strength and stiffness in sandwich beams. With these theories the necessity for high volume fraction can be critically examined, always remembering the fringe benefits that can accrue with lower glass volume fraction laminates. *86 Smugglers Lane, Christchurch, tlampshire, UK
COMPOSITES. MARCH 1973
FLEXURAL R I G I D I T Y As already stated, the parameter of interest in determining the relative efficacy of different materials in the bending stiffness of simple beams and plates is not the specific modulus but the ratio (E/p 3) [or more correctly when considering weight, (E1/3/p)]. Were this not so GRP would not be the competitive material it is in the yacht and car body field, for the specific modulus of GRP is low compared with steel or aluminium alloy (see Table 1). Because the density of GRP is so much lower than that of the engineering metals however, and because of the influence of the term p3, the flexural rigidity (bending stiffness) of GRP per unit weight is much higher than steel and marginally higher than aluminiun] alloy (see Table 1). This fact is very well known and appreciated by engineers and designers who need to minimize weight. What is not generally realised is that the fairly large difference between the density (specific gravity) of glass (2-54) and the density of most laminating resins (1"0 1-2) leads to important increases in the average density of GRP as its glass content is increased. Consequently an optimum value exists for the volume fraction of glass to give maximum bending stiffness. This is obtained as folh)ws: Consider a simple beam under the action of forces to produce a bending moment M. The resistance of the beam to bending is proportional to the product E1 (where I is the second moment of area about axis of bending). This product is called the flexural rigidity of the beam, and this must be maximized for maximum bending stiffness. For a simple beam of unit width I = t3/12: so for maximum stiffness Et 3 must be a maximum. Now using a modified form of the mixture law, E = BVGE G + VRER where B is the efficiency of reinforcement, V is volume fraction and the subscripts G and R signify glass and resin respectively.
65
Table 1 Specific modulus (E/p) and flexural stiffness per unit weight (E/p 3) for various materials compared with steel Material
E/p
E/p 3
Steel Aluminium alloy GRP unidirectional GRP bidirectional
1"0 1.1 0"9 0.5
1 10 21 12
Table 2 Specific strength (f/p) and flexural strength per unit weight (f/p2) for various materials compared with high strength steel
aV a = p/3. That is, the weight of reinforcing fibre = 1/3 of the total beam weight. This is the same as the skin weight criterion for maximum flexural rigidity in sandwich beams when core modulus is ignored 2.
FL E X U R A L STRENG TH Similar arguments apply as before in the comparison of bending strength per unit weight with specific tensile strength (see Table 2). In the bending of a simple beam M = l i l y where f is the extreme fibre stress and y is the distance of the extreme fibre from the neutral axis. For the glass fibre
M = fGEI/EGy Material Steel Aluminium alloy GRP unidirectional GRP bidirectional
f/p
f/p2
1.0 0-9 3.0 1-5
1-0 2"7 14.0 7.0
= eGEI/y where e G is the strain in the glass fibre. Failure occurs when this exceeds the ultimate value. This assumes that the resin is more ductile than the fibre.
For a given mass per unit area (m) I-C
m = PGt G + PRtR = pt where t a and tR are equivalent thicknesses of glass and resin, t = total thickness of the laminate and p = average density. Now t G = VGt and t R = VRt.
O.E
Thus
O.~
P = PG V(; + (1 -- VG)PR =(PG --PR)VG +PR
puta=PG-PR
-}
.
O.,
= aVG + PR
=(BVGEG+ER)(;)
B
0-2
Approz~qmating,Et 3 = (BVGE G + ER )t 3
3
O O
a
l
0.2
I
O'4
I
O.6
v~
I
OB
18(mot)
I
IO
(BVGE G + E R ) m 3 0'8
(aV G + OR) 3 For maximum or minimum
0.6
dEt 3
dye,
-0 O.z
ie -3(a V(; + PR )-4a(B V G E G + E R ) + BE G (a V c + PR )'3 = 0 whence
0-:
EG/E R = 3a/B(OR - 2aV G)
This expression is plotted as a function of V G for three different forms of GRP in Fig 1. It is seen that the maximum stiffness per unit weight is obtained at V G = 25% for unidirectional (B = 1), 18% for bidirectional (B = ½), and 13% for random reinforcement (B = 3/8), when taking EG/E R = 20 (a typical figure) and PR = 1 '0 and PG = 2-5. Note that when E R tends to zero, the condition for maximmn flexural rigidity is that PR = 2aVG giving
66
B= 3/81mot)
0 b
I
I
0'2
0.4
[
0.6
vg
I
0.8
I
I.O
FIG I(a) Flexural rigidity per unit weight as a function o f volume fraction for GRP (b) Flexural strength per unit weight as a function o f volume fraction for GRP
COMPOSITES. MARCH 1973
Following previous arguments:
M= eG(BVGEa + E R)
m3
aVG + PR
12(a V G + PR )3
½m
For maximum M, dM -0
unidirectional, 47% for bidirectional and 40% for random reinforcement, although the curves are quite 'flat', and any figure between 30% and 80% would be fairly satisfactory for all three cases. Referring to both Fig la and Fig lb, if both maximum strength and maximum stiffness are desirable, glass volume fraction should be chosen in the region 30-40%. Such ratios are those normally obtained by straightforward hand lay-up, among other processes.
CONCL USIONS =(aV G + pR)'2BEG - ( a V G + pR)'32a(BVGEG + ER)
whence
EG/ER = 2a/B~R - aVG)
This function is plotted in Fig 2. Note that as E R tends to zero, PR = aVG ie a VG = p/2. The maximum bending strength will be obtained when the weight of reinforcement equals one half of the total weight. This again corresponds with the optimum skin weight fraction for maximum bending strength of sandwich beams 2. Using the same values as before for material parameters, maximum bending strength is obtained at VG = 57% for
COMPOSITES. MARCH 1973
It is seen that for many bending purposes the gains obtainable by use of high volume fraction GRP composites are at best marginal, and in many practical cases actually negative on a 'total weight' basis. Thus the notion that high volume fraction GRP laminates are always desirable is fallacious when considering bending applications.
REFERENCES 1 Timoshenko, S. 'Strength of Materials', Chapters 4 and 5, van Nostrand (1940) 2 Engel, H. C., Hemming, C. B. and Merriman, H. R. 'Structural Plastics', p 139 et seq, McGrawHill (1950)
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There are theoretical as well as practical reasons why a high glass content in glass fibre reinforced plastics is not always desirable. An optimum glass content is calculated to give maximum flexural strength and stiffness for simple beams and plates, this content being governed by similar rules to those that apply for the optimum configuration in sandwich beams. The argument leads to the surprising result that, in particular instances, volume fractions as low as 13% can be required theoretically to give optimum construction. It is concluded that, in many bending applications, hand lay-up can compete fully with hot moulded construction, and in some cases even prove superior.
Because the strength and stiffness of glass fibre reinforced plastics (GRP) derive mainly from the glass fibre itself, it is quite usual to endeavour to obtain the maximum possible volume fraction of glass in the finished product. This is justifiable on purely theoretical grounds when considering the material under tensile forces, because of the high specific tensile strength (tensile strength per unit weight) and relatively high specific Young's modulus of glass fibre. Practically speaking this philosphy leads to several other problems, one of which is the necessity to use potentially expensive and often unwieldy compression moulding techniques in order to obtain high glass volume fractions. In considering simple GRP beams and plates under the action of bending forces, however, theory shows that, on a weight basis, there is an optimum value for volume fraction to give maximum bending resistance and flexural rigidity. For whereas the parameters of interest for specific strength and stiffness are (f/p) and (E/p) respectively (where f is tensile strength, E is Young's modulus and p the density) the important parameters in bending are (f/p2) and (E/p3). This is because flexural strength is proportional to (thickness) 2 and flexural rigidity is proportional to (thickness) 3,(reference 1) and for a given length, width and weight of plate or beam the thickness is inversely proportional to material density. hr what follows the ccnditions for optimum flexural strength and stiffness in GRP are derived; these conditions are found (not surprisingly) to be similar in principle to the conditions necessary for optimum flexural strength and stiffness in sandwich beams. With these theories the necessity for high volume fraction can be critically examined, always remembering the fringe benefits that can accrue with lower glass volume fraction laminates. *86 Smugglers Lane, Christchurch, tlampshire, UK
COMPOSITES. MARCH 1973
FLEXURAL R I G I D I T Y As already stated, the parameter of interest in determining the relative efficacy of different materials in the bending stiffness of simple beams and plates is not the specific modulus but the ratio (E/p 3) [or more correctly when considering weight, (E1/3/p)]. Were this not so GRP would not be the competitive material it is in the yacht and car body field, for the specific modulus of GRP is low compared with steel or aluminium alloy (see Table 1). Because the density of GRP is so much lower than that of the engineering metals however, and because of the influence of the term p3, the flexural rigidity (bending stiffness) of GRP per unit weight is much higher than steel and marginally higher than aluminiun] alloy (see Table 1). This fact is very well known and appreciated by engineers and designers who need to minimize weight. What is not generally realised is that the fairly large difference between the density (specific gravity) of glass (2-54) and the density of most laminating resins (1"0 1-2) leads to important increases in the average density of GRP as its glass content is increased. Consequently an optimum value exists for the volume fraction of glass to give maximum bending stiffness. This is obtained as folh)ws: Consider a simple beam under the action of forces to produce a bending moment M. The resistance of the beam to bending is proportional to the product E1 (where I is the second moment of area about axis of bending). This product is called the flexural rigidity of the beam, and this must be maximized for maximum bending stiffness. For a simple beam of unit width I = t3/12: so for maximum stiffness Et 3 must be a maximum. Now using a modified form of the mixture law, E = BVGE G + VRER where B is the efficiency of reinforcement, V is volume fraction and the subscripts G and R signify glass and resin respectively.
65
Table 1 Specific modulus (E/p) and flexural stiffness per unit weight (E/p 3) for various materials compared with steel Material
E/p
E/p 3
Steel Aluminium alloy GRP unidirectional GRP bidirectional
1"0 1.1 0"9 0.5
1 10 21 12
Table 2 Specific strength (f/p) and flexural strength per unit weight (f/p2) for various materials compared with high strength steel
aV a = p/3. That is, the weight of reinforcing fibre = 1/3 of the total beam weight. This is the same as the skin weight criterion for maximum flexural rigidity in sandwich beams when core modulus is ignored 2.
FL E X U R A L STRENG TH Similar arguments apply as before in the comparison of bending strength per unit weight with specific tensile strength (see Table 2). In the bending of a simple beam M = l i l y where f is the extreme fibre stress and y is the distance of the extreme fibre from the neutral axis. For the glass fibre
M = fGEI/EGy Material Steel Aluminium alloy GRP unidirectional GRP bidirectional
f/p
f/p2
1.0 0-9 3.0 1-5
1-0 2"7 14.0 7.0
= eGEI/y where e G is the strain in the glass fibre. Failure occurs when this exceeds the ultimate value. This assumes that the resin is more ductile than the fibre.
For a given mass per unit area (m) I-C
m = PGt G + PRtR = pt where t a and tR are equivalent thicknesses of glass and resin, t = total thickness of the laminate and p = average density. Now t G = VGt and t R = VRt.
O.E
Thus
O.~
P = PG V(; + (1 -- VG)PR =(PG --PR)VG +PR
puta=PG-PR
-}
.
O.,
= aVG + PR
=(BVGEG+ER)(;)
B
0-2
Approz~qmating,Et 3 = (BVGE G + ER )t 3
3
O O
a
l
0.2
I
O'4
I
O.6
v~
I
OB
18(mot)
I
IO
(BVGE G + E R ) m 3 0'8
(aV G + OR) 3 For maximum or minimum
0.6
dEt 3
dye,
-0 O.z
ie -3(a V(; + PR )-4a(B V G E G + E R ) + BE G (a V c + PR )'3 = 0 whence
0-:
EG/E R = 3a/B(OR - 2aV G)
This expression is plotted as a function of V G for three different forms of GRP in Fig 1. It is seen that the maximum stiffness per unit weight is obtained at V G = 25% for unidirectional (B = 1), 18% for bidirectional (B = ½), and 13% for random reinforcement (B = 3/8), when taking EG/E R = 20 (a typical figure) and PR = 1 '0 and PG = 2-5. Note that when E R tends to zero, the condition for maximmn flexural rigidity is that PR = 2aVG giving
66
B= 3/81mot)
0 b
I
I
0'2
0.4
[
0.6
vg
I
0.8
I
I.O
FIG I(a) Flexural rigidity per unit weight as a function o f volume fraction for GRP (b) Flexural strength per unit weight as a function o f volume fraction for GRP
COMPOSITES. MARCH 1973
Following previous arguments:
M= eG(BVGEa + E R)
m3
aVG + PR
12(a V G + PR )3
½m
For maximum M, dM -0
unidirectional, 47% for bidirectional and 40% for random reinforcement, although the curves are quite 'flat', and any figure between 30% and 80% would be fairly satisfactory for all three cases. Referring to both Fig la and Fig lb, if both maximum strength and maximum stiffness are desirable, glass volume fraction should be chosen in the region 30-40%. Such ratios are those normally obtained by straightforward hand lay-up, among other processes.
CONCL USIONS =(aV G + pR)'2BEG - ( a V G + pR)'32a(BVGEG + ER)
whence
EG/ER = 2a/B~R - aVG)
This function is plotted in Fig 2. Note that as E R tends to zero, PR = aVG ie a VG = p/2. The maximum bending strength will be obtained when the weight of reinforcement equals one half of the total weight. This again corresponds with the optimum skin weight fraction for maximum bending strength of sandwich beams 2. Using the same values as before for material parameters, maximum bending strength is obtained at VG = 57% for
COMPOSITES. MARCH 1973
It is seen that for many bending purposes the gains obtainable by use of high volume fraction GRP composites are at best marginal, and in many practical cases actually negative on a 'total weight' basis. Thus the notion that high volume fraction GRP laminates are always desirable is fallacious when considering bending applications.
REFERENCES 1 Timoshenko, S. 'Strength of Materials', Chapters 4 and 5, van Nostrand (1940) 2 Engel, H. C., Hemming, C. B. and Merriman, H. R. 'Structural Plastics', p 139 et seq, McGrawHill (1950)
67