- Email: [email protected]
Vibration analysis and finite element modeling for determining shear modulus of pultruded hybrid composites
1359-8368(95)00018-6
EI,SEVIER
Composites: Part B 27B (1996) 329-337 Copyright ~(') 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/96/$15.00
Vibration analysis and finite element modeling for determining shear modulus of pultruded hybrid composites
Chandrasekhar V. Nori, Tyrus A. McCarty* and P. Raju Mantena Department of Mechanical Engineering, The University of Mississippi, MS 38677, USA (Received January 1995; acceptedJune 1995) An impulse-frequency response vibration technique was employed for determining the shear modulus of glass/epoxy, graphite/epoxy and hybrid (glass-graphite/epoxy) pultruded cylindrical composite rods in torsion. The distribution of the fibers and matrix in the shell-core regions were examined microscopically and the volume fractions of the various constituents determined using the stereology point counting technique. Based on the examined cross-section finite element meshes were generated and analyzed for predicting the shear modulus of such composite rods. It was observed that there was close agreement between the finite element predictions and the experimentally obtained data for only some of the hybrid configurations. The nature of the pultrusion manufacturing process which causes variations in the fiber packing geometry in the shell and core regions of hybrids has been found to significantly influence the accurate prediction of shear modulus, using either analytical or finite element methods. (Keywords: pultruded composites; shear modulus; impulse-frequency response technique; finite element method; glass/epoxy; graphite/epoxy; hybrid composites)
1 INTRODUCTION High modulus fibers such as graphite are widely used in aerospace, automotive, and structural applications because of their exceptionally high specific modulus and strength. Unfortunately, high cost and low impact strength are two disadvantages which limit their use. Hybridization or the mixing of high cost, high modulus, low impact fibers with low cost, low modulus fibers such as glass provides a possible solution worth exploring further. In this way, it is possible to take advantage of the combined action of individual fibers, thus optimizing the properties of the composite. There are basically two ways in which two or more types of fibers can be combined to formulate a hybrid composite: either by intermingling the fibers in a common matrix, or laminating layers of each type of fiber I. The mechanical analysis of conventional materials focuses mainly on predicting their behavior in structural applications. These applications assume that the fundamental material properties can be determined by testing them experimentally. Various analytical methods have been developed which are suitable for determining the properties of conventional materials primarily because of their isotropic nature. Hence, composite materials being * To whom correspondence should be addressed
anisotropic makes it very difficult to predict their material properties analytically. Summerscales and Short 2, in an extensive literature survey, discussed the results obtained by various researchers on the determination of material properties. It was reported that many investigators have found the experimental data to be consistently higher than that predicted by the law of mixtures. The four basic properties which are important for composite design are the longitudinal Young's modulus (ELa-), major Poisson's ratio (ULT), transverse Young's modulus (ET), and the shear modulus (GLT). Of these properties the experimental values of the transverse modulus and the shear modulus are known to disagree with the values predicted by the inverse law of mixtures. Gibson I verified that this is mainly due to the fact that these two properties depend upon the fiber packing geometry and also highlighted the importance of numerical techniques for predicting the transverse and shear moduli of composite specimens. When pultruded composite rods are subjected to pure torsion, the torque applied varies linearly with the angle of twist for small angles. Outwater 3 employed the initial slope of the torque v e r s u s angle of twist for computing the torsional shear modulus of such composites. Suarez e t al. and Mantena e t al. 5 demonstrated non-destructive vibration test techniques to determine the dynamic
329
Impulse-frequency response technique: C. V. Nori et al. flexural, extensional and shear moduli of composites. One of these methods employs a variation of the impulsefrequency response technique for testing cylindrical specimens subjected to a torsional mode of vibration. Static and dynamic techniques were employed by Kumar 6 to determine the shear properties of pultruded cylindrical composite rods in torsion. Innovative developments have been made on pultruding 'hybrids' in the Composite Materials Research Laboratory at the University of Mississippi using a commercial scale pultrusion machine. The improved characteristics of these hybrids demonstrate the potential for designing stiff, lightweight, and well damped composite structures having a number of practical applications. In this paper, the details of the nondestructive torsional vibration technique and a finite element model used for determining the shear modulus of pultruded composites are presented. The experimental technique evaluated the dynamic shear modulus and the finite element method predicted the static shear modulus. Previous research on similar pultruded hybrid cylindrical composite specimens showed close agreement in the static and dynamic shear moduli 6. Hence, in this investigation, the static slaear modulus computed by finite element method was used for comparison with the dynamic shear modulus obtained from the low strain level torsional vibration technique.
2 TEST SPECIMENS An in-house industrial scale pultrusion machine (Pulstarr 804) was used to manufacture unidirectional glass/ epoxy (glass/e) and graphite/epoxy (graphite/e) composites along with glass-graphite/epoxy (glass-graphite/e) pultruded composite rods. The schematic representation of the pultrusion process is shown in Figure 1. A typical pultrusion machine consists of roving/mat creels, resin impregnating tank, resin controller, shape preformer, heated dies, pullers, and cut-off saw. The manufacturing process involves the pulling of continuous roving/mats through a resin impregnating tank. The fiber impregnated resin is then passed through a resin control unit to remove any excess resin followed by passing it through a shape preformer of desired shape. The pullers are continuously operated hydraulically to pull the res.ulting product through a heated die where it is cured
120VIN(3/~D~T CF~EEL
RESIN
MC I ~%EGhA l TING C[]~TR
5
p~
~ATED
DIE
PULLER
SAV
\ P~IX£T
Figure 1 Schematic representation of the pultrusion manufacturing process
330
continuously. A cut-off saw is used to cut the desired length of the pultruded composite. With specially designed pre-form plates, hybrids were pultruded by intermingling glass and graphite fibers in a common matrix 7. The pre-form plates guided the fibers into the heating dies and helped maintain the constant crosssection of the shell and core regions of each hybrid during the curing process. Various combinations of hybrids were manufactured with all other pultrusion process conditions maintained constant. Each hybrid combination was prepared in a single pass operation of the pultrusion manufacturing process. The fibers used were E-glass and AS-4W-12K graphite, and the matrix was Epon 862/W resin (Shell Oil Company). The diameter of all the composite rods was 9.53mm (3/8inch), with overall fiber volume fraction maintained at 60% and the remaining 40% was matrix. Three samples of glass/e, graphite/e, and various glass-graphite/e hybrid composite rods were first tested dynamically to determine their torsional shear modulus. Table 1 shows various combinations of the round composites which were subjected to the dynamic tests. These specimens include glass-graphite/e hybrid combinations along with mono-fiber glass/e ('A') and graphite/e ('B') composites. Hybrids 'C', 'E' and 'G' have glass fibers in the shell and graphite fibers in the core, and hybrids 'D', 'F' and 'H' have graphite fibers in the shell and glass fibers in the 'core'. Only types 'A' through 'D' were modeled using the finite element method because of observed inconsistency in the fiber packing geometry of types 'E'-'H'. The typical fiber packing of these cross-sections were observed under an optical microscope. Based on an approximation of the observed distribution of fibers and matrix, the glassgraphite/e hybrids were modeled by the finite element method to obtain an estimate of their shear modulus which was then compared with experimental data. The rods were modeled assuming that the distribution of fibers and matrix in the shell and core regions of the hybrids were in the same proportion as that in the glass/e and graphite/e composites. Further analysis of the hybrid cross-sections was performed by applying the stereology point counting microscopy method for Table I Cross-sections and fiber volume fractions of hybrid combinations pultruded Percent fiber volume fraction Specimen*
Glass (E-glass)
Graphite (AS-4W-12K)
Cross-section
Glass/epoxy (A) Graphite/epoxy (B) Hybrid C Hybrid D Hybrid E Hybrid F Hybrid G Hybrid H
60 0 30(s)t 30(c) 37(s) 37(c) 48(s) 48(c)
0 60 30(c) 30(s) 23(c) 23(S) 12(c) 12(s)
() • o, O ~e~ O 'e, O
* Volume fraction of epoxy in all the above composites is 40% t (s) refers to the shell region and (c) refers to the core region of the hybrids
Impulse-frequency response technique: C. V. Nori et al. determining the true fiber packing in the shell and core regions.
3 EXPERIMENTAL SETUP AND PROCEDURE The impulse-frequency response technique involves the excitation of all the modes of vibration of sample specimens with an electromagnetic impulse hammer having a piezoelectric force transducer in its tip. The vibrational response is measured with a non-contacting eddy current displacement transducer. The frequency response function displayed on the spectrum analyzer consists of sharp peaks depicting the various modes of vibration. The resonant frequency of interest is used along with the appropriate characteristic equation for determining material properties. The experimental set-up shown in Figure 2 has been used to obtain the fundamental torsional mode of vibration of the cylindrical specimens. The set-up consists of dual channel spectrum analyzer (HP 35665A), an impact hammer, a non-contacting eddy current probe with a displacement measuring system,
and a computer. The test specimen, which is fixed at the top of an L-plate, has a lumped cylindrical mass with two diametrically opposite 'ears' attached to its lower end. The purpose of the end mass is to sustain torsional oscillations of the specimen after initial excitation. One of the ears on the end mass is impacted with the impulse hammer and the response is picked up by the noncontacting eddy current displacement transducer placed at the other ear. The input and output signals are fed to the spectrum analyzer which displays the frequency response function of the modes of vibration. The computer, interfaced with the analyzer, is used for data transfer and instantaneous computation of the dynamic shear modulus and damping of the candidate materials being tested. Off-axis impact of the end mass, attached to the base of the pultruded rods, results in the simultaneous excitation of both the bending and torsional modes of vibration. These are displayed as two closely spaced sharp peaks within the first 100 Hz frequency span of the spectrum analyzer. It is extremely important that the correct torsional peak is analyzed and incorporated into the characteristic equation for dynamic shear
Adjustable L-Plate Lathe Chuck (ToholdSpecimen)
Triangular
Plate / ~E~ Cross-sectioSpeci n mE~-~'~en II ~ i
Chuck ('With"ears"attached)
Lathe
Impact Hammer
I Displacemmt Measuring
.
Non-Contacting EddyCurrentProbe .
.
-I -I
Plotter
PowerSupply
.
liP 35665A
Computer
I-
Pfiatt~"
Figure 2 Experimentalset-upfor torsionalshearmodulusdetermination
331
Impulse-frequency response technique. C. V. Nori
e t al.
modulus for each specimen. Incorporation of the redundant bending peak into this equation will give erroneous data. The dynamic shear modulus, G', computed from the torsional frequency fn is given by: G' - 47r2f2nLZp A2 n
(1)
along with the characteristic equation: cot A -
J)~n plpL
in shear than graphite fibers (Ggraphit e = 14.14GPa), the inverse law predicts only a marginal difference between the glass/e and graphite/e composites. Hence the shear moduli predicted by this law show that the matrix (Gmatrix = 0.77GPa) is the dominant constituent for these composite specimens. It has also been shown that the experimentally obtained shear modulus was consistently higher than the shear modulus predicted by the inverse law of mixtures I .
(2) 5 MATHEMATICAL MODEL
where L, p, and Ip are the length, mass density, and polar area moment of inertia of the specimen, respectively. An is the eigenvalue for torsional modes and J is the polar mass moment of inertia of the attached lower mass. For calibration, a 2024-T4 aluminium sample of known shear modulus was first tested to determine the polar mass moment of inertia of the lumped lower mass because of its complex geometry. The fundamental torsional frequency of vibration of the 0.127 m (5 inches) long aluminium sample was found to be 26 Hz, and the mass polar moment of inertia of the lower mass (J) was determined to be 4.4 × 10-3 Kg-m 2 (0.035 lb in s2).
The solution of the following two dimensional boundary value equation has been considered for the torsion of cylindrical rods 9
02
(4)
The boundary conditions associated with this equation are given by • (x,y)=0
onF
(5)
and 0@ nx~+
4 ANALYTICAL MODEL: INVERSE LAW OF MIXTURES
in ~
0@ ny~y = 0
onF
(6)
The properties of composite materials depend on the properties of its constituents, their distribution and the physical and chemical interactions between them. The equation used to obtain the shear modulus of composite materials using the inverse law of mixtures is given by 8
where G(= GLT) is the shear modulus, @ the Prandtl's stress function, 0 the angle twist per unit length, ~ refers to the interior, and F refers to the boundary of the crosssection of the rods. n x and n r are the cosines of the angles between x and y axes and the normal to the boundary, respectively. The non-zero stress components are given in terms of the stress function by the following expressions
1 -- Vglass "k Vgraphite ~ Vep°xy GET Gglass Ggraphite Gepoxy
0@ O@ qz =Oyv and ryz - Ox
(3)
where GLT is the longitudinal shear modulus of the composite rod, V and G are the volume fraction and shear modulus of the constituents of the composite rod, respectively. The inverse law of mixtures predicts that the shear modulus of a composite can be raised to five times the matrix modulus only by providing 90% fibers 8. Producing a composite having 90% fibers is not practical because it does not provide proper bonding between the fibers and matrix. Fibers do not contribute much to the shear modulus unless the percent of fibers is very high. Hence, the use of high modulus fibers does not significantly enhance the shear modulus. Further, the inverse law of mixtures predicts shear modulus of a composite based on the assumption of uniform distribution of fibers in the matrix. For glass/e and graphite/e composites, the inverse law of mixture predictions for shear modulus are 1.85 and 1.77GPa, respectively. Although, glass fibers (Gglass = 30.13 GPa) are approximately two times stiffer
332
(7)
and the torque transmitted, T, is given by T
=
f 2 / @dA J
(8)
In the present problem, the Garlerkin weighted residual method is employed in deriving the finite element equations m. In this technique, the governing" differential equation is first put into an equivalent weighted integral form given by I~ ~ IW-TS,,2+
2GO dxdv
0
(9)
where w is the weighing function. The resulting equation is integrated by parts to obtain the weak form as
LOx Ox +
bTJ dxdy+ 2J, wOOaxdy
Impulse-frequency response technique: C. V. Nori et al.
- ~ r W ( n x O~b-~x+ ny ~ v0) d S, =
(10)
From the inspection of the boundary condition, it may be noted that the specification of • constitutes the essential boundary condition and hence q~ is the primary variable. Specifying the coefficient of weighing function in the boundary expression constitutes the natural boundary condition, and hence it is the secondary variable of the formulation. The weak form obtained in the above equation forms the basis of the finite element model.
6 FINITE ELEMENT MODEL The finite element equation corresponding to the weak form of the governing differential equation is obtained by choosing an approximation for ~. The variable, ~, should be at least linear in both x and y, so that there are no zero terms in equation (10). Hence to obtain the finite element equation, ~ and w are approximated over an element by the following relations
where A is the area of the triangular element, i ~ j ~ k and i, j and k permute in a natural order. The torque or twisting moment acting on the rod can be computed as: N
T=2I
~dxdy=ZZ(~i+~/+~k)(e)
(17)
e=l
where N is the number of elements, ~i, ~j and ~k denote the values of stress function at nodes i, j and k, respectively, of element 'e'. Assuming symmetry, only a quarter section of the rod was modeled. The domain of the composite discretized using three-noded triangular elements (with linear interpolation function). The domain (an area of 0.713cm 2) was divided into 2 592 elements using 1 354 nodes. By maintaining the volume fractions of the individual components, each element was assigned the properties of either the appropriate glass, graphite fiber or epoxy matrix. Each of these constituents was assumed to be homogeneous, isotropic and linearly elastic. It was also assumed that there is perfect bonding between the matrix and fibers and that the interface between them is continuous. The longitudinal shear moduli assigned to the glass fibers, graphite fibers and the epoxy matrix were 30.13 GPa, 14.14 GPa and 0.77 GPa, respectively.
m
/e) (x,>),
• (x, v)
(ll)
j=l
and w=~i
( i = l, 2 , . . . , m )
(12)
where ~ and qJ~ are the stress and interpolation functions at the jth node of the element and m is the number of nodes per element. On substituting equations (11) and (12) into equation (10), the finite element expression is obtained as:
"
{0 ,ow,
}
LOx ox + o>,
~idxdy
+2 [~ [GO~i](e)dxdy - ~r [k~iqn](e)ds = 0 (i = 1, 2 .... ,m)
(13)
where 0~
0(I,
qn = nx o~[~v+ ny --fffy = O
(14)
The coefficient of the ~j in the first term of equation (13) represents the element stiffness matrix, and the boundary term vanishes. Linear interpolation of the form ~b(x, y) = cl + CeX + c3y has been considered for a three node triangular element. The interpolation functions are given by:
c,~i= x/yk - -
~i =
)) - Yk and 7 / = xk - Xi (15)
where
~le/
1 = 2--7(c*i + ~qix
i= 1, 2, 3
7 STEREOLOGY: POINT COUNTING METHOD The stereology point counting method (ASTM Specification E562) is widely used for determining the volume fractions of various constituents in a material ~1. In this method an optical microscope with a CRT monitor is used to view the micro-structure of various composite cross-section at appropriate magnifications. A typical composite specimen observed under the microscope displays the fibers (circular in shape) distributed in the epoxy matrix. In the stereology point counting method, a test grid is first placed on the micro-structure displayed on the CRT monitor. Then the number of points lying on the fibers is determined. The intersection of two crossed lines of the test grid is considered as a point. If the intersection lies completely inside the phase, it is considered to be one count and if it lies on the phase boundary it is counted as one-half count. If Po is the number of grid points and P,, the point count for each of the 'n' number of randomly selected fields, the volume fraction Pp of the constituent of interest is given by the relation: ~ P~
pp -
nPo
(is)
There are various factors involved in the selection of an appropriate grid size for determining volume fractions using the stereology point counting method II. These include volume fractions of the constituents, choice of magnification and resolution limits of the optical microscope used. It was found that consistent results were obtained for the hybrid composites analyzed with an 11 × 11 grid size at a magnification of 60×. Six
333
Impulse-frequency response technique." C. V. Nori et al. randomly selected fields were selected over the crosssection of each sample for determining the fiber to matrix ratio. Similar analysis was performed in the glass/e and graphite/e regions of the glass-graphite/e hybrid composites. The ratio of fiber to epoxy in the glass/e and graphite/e regions was computed.
5,0
8
~
~
~o
]116
-0.96
0 (~
8 RESULTS AND DISCUSSION
Percent Graphite
?
0
-0.75 "~
Olasst¢shell, graphile/¢~r¢] OraphiCe/eshell, glasst¢c o r e ~c~erimenu~Sm~er Em~rim¢ nlalResutt~ / Inverse law ofMixture~ J
]
-0.6s -0.55 ~ -O.45 ~ ~
The shear moduli of all the experimentally tested composite specimens were found to be higher than the values predicted by the inverse law of mixtures. The experimental method determines the dynamic shear modulus, whereas both the inverse law of mixtures and finite element method predict the static shear modulus. Previous research conducted on these hybrid composite specimens showed that the dynamic and static shear moduli were approximately the same 6. Hence, the static modulus predicted by inverse law of mixtures and finite element method are compared to the experimentally determined dynamic moduli in this paper. Figure 3 shows the comparison of static modulus predicted by the inverse law of mixtures and the dynamic modulus determined by experimental means. It can be observed that the experimental values are higher than the values predicted by the inverse law of mixtures. This may be attributed to the fact that the inverse law of mixtures predicts shear modulus for composites having uniform distribution of fibers in matrix. The predicted shear modulus obtained using the inverse law of mixtures was marginally lower for graphite/e specimens as compared to the glass/e specimens. The shear moduli for all other hybrid combinations were intermediate to that of the glass/e and graphite/e composite specimens. It can also be observed that the inverse law of mixtures predicts identical shear modulus for hybrids having similar volume fractions of glass and graphite fibers, irrespective of the type of fiber located in core and shell regions. In contrast, experimental results showed higher shear modulus for the hybrids having glass/e shell-graphite/e core, compared to hybrids with glass/e core-graphite/e shell. The graphite fibers have lower density than glass fibers. Experimental results show that by replacing a portion of glass fibers in the glass/e composite with graphite fibers changed the overall stiffness to weight ratio. A hybrid composite with glass/e shell-graphite/e core had marginally higher torsional stiffness than the glass/e composite, suggesting a synergistic effect. In addition, the density of this hybrid was less than the glass/e composite. When a hybrid with graphite/e shellglass/e core was tested with the same volume fraction of constituents, it exhibited a lower specific stiffness. Although the density of the two hybrid composite rods was the same, the stiffness varied with the location of glass and graphite fibers. All the hybrid combinations with glass/e shell-graphite/e core exhibited higher specific torsional stiffness than the graphite/e shell-glass/e core
334
6 ~o
~o
Jo Percent Glass
o
Lo.~
o /o
-0.35
0,15 60
~o .-~
Figure 3 Shear modulus: comparison of experimental law of mixtures
versus
inverse
Figure 4 Finite element mesh for quarter section of composite cross-section
hybrids and/or the glass/e and graphite/e composite rods. When the cross-sections of the hybrid composites were examined under an optical microscope (at magnification of 10x), the concentrated fiber and matrix regions were observed, implying non-uniform fiber distribution over the cross-section. Hence it was extremely difficult to model such a composite by any of the available analytical methods. Inputing the test. data obtained for glass/e and graphite/e composites (60% fiber and 40% epoxy) into the core and shell regions of the hybrid models (for predicting their properties) appeared to have serious shortcomings. Three special cases have been considered for predicting the shear modulus of pultruded composite rods using the finite element method.
Case 1." Uniform distribution of fibers Finite element analysis was performed assuming uniform distribution of fibers in matrix. Properties of
Impulse-frequency response technique: C. V. Nori et al. Table
2 Resultsfrominverselaw of mixturesand finiteelementanalysisbased on uniformdistributionof fibers Torsional shear modulus(GPa)
Composite
Law of mixtures*
Numerical (FEM)*
Experimentalt
60% Glass/40% Epoxy 60% Graphite/40% Epoxy 30% Glass/20% Epoxy(Shell)30% Graphite/20% Epoxy(Core) 30% Glass/20% Epoxy(Core)30% Graphite/20% Epoxy(Shell)
1.85 1.77 1.82
1.96 1.86 1.93
7.35 6.11 7.63
1.82
1.89
6.04
* Based on uniformdistributionmodel t Actuallypultrudedspecimens non-uniformdistribution) fibers and epoxy were assigned to each of the elements in the domain of the composite by maintaining constant volume fractions of its constituents. The properties were also assigned in such a way that the fibe~ elements were uniformly distributed in the entire domain. Figure 4 shows the discretized domain of a quarter section of the composite cross-section. The static shear modulus of glass/e, graphite/e and the two core and shell composites (hybrids C and D) were obtained. Table 2 shows the shear modulus obtained from the finite element analysis. The results were in good agreement with the values predicted by the inverse law of mixtures for mono-fiber glass/e and graphite/e composites. Closer examination of the hybrids having identical volume fractions of glass and graphite fibers indicates that similar shear moduli are predicted by the inverse law of mixtures. In contrast, the finite element method predicts marginally higher shear modulus for the hybrid having glass fibers in the shell region as compared to the hybrid having graphite fibers in the shell region (for uniform distribution of fibers). Hence, it can be inferred that the inverse law of mixtures can only predict shear moduli for those hybrids which have both glass and graphite fibers distributed uniformly throughout the composite domain. It fails to predict the properties for core-shell type hybrids even with uniform distribution of fibers in each region. Also, the experimental results showed that the shear moduli of graphite/e, glass/e and hybrid composite rods were higher than that predicted by the finite element model based on uniform distribution of fibers in the respective regions.
Case 2." Glass-epoxy and graphite-epoxy composites (non-uniform fiber distribution model) The pultrusion process involves pulling of resin impregnated fibers through a heated die by setting the pull speed and die temperature. Although this is an automated manufacturing process and all care was taken to manufacture a perfectly uniform composite, there was a high probability that the fibers were not distributed uniformly in the matrix. A little concentration of fibers in one region over the cross-section would change the overall property of the composite. Hence, the shear modulus as well as other mechanical properties of these composites depend mainly upon their fiber packing geometry.
Depending upon the location of fibers in the matrix, the overall shear modulus of a composite can be varied. Microscopic observation of the composite cross-sections revealed that the fibers were not uniformly distributed in the matrix. Finite element meshes were generated based upon the observed distribution for the glass/e and graphite/e composites. The meshes for these composites were approximated with alternate layers of epoxy and fiber, starting with epoxy in the outermost layer. The thickness of the fiber layers was maintained at twice the thickness of the epoxy layers. Additionally, microscopic observations also confirmed the existence of lumped regions of fibers and matrix about the cross-section. Based upon these observations, regions of concentrated fibers and epoxy were also incorporated in the finite element mesh. These concentrated fiber and epoxy regions were different for the graphite/e and glass/e composites. This was mainly due to the large diameter ratio of glass to graphite fibers, which is approximately 4 : 1. Although the area of an individual element in the mesh is very large compared to the area of a single fiber, the overall mesh is a close approximation representing the fiber-matrix distribution in the domain. Figure 5 shows the fiber distribution in glass/e and graphite/e regions which were used for modeling glass/e and graphite/e mono-fiber composites, respectively.
Case 3." Modeling of core-and-shell type hybrids A special kind ofpre-form plate was designed to guide and maintain a constant circular cross-section of fibers constituting the core of the hybrids. Every care was taken to produce hybrids with cylindrical cores. When a glass/e shell-graphite/e core hybrid rod was pultruded, perfect cylindrical cores were produced. However, this was not the case with the glass/e core-graphite/e shell hybrids where the cores were observed to be not quite as cylindrical, which may be attributed to the large difference in the diameters of glass and graphite fibers. Two types of core and shell type hybrids, one with glass/e shell-graphite/e core (Hybrid C) and the other with graphite/e shell-glass/e core (Hybrid D) were modeled. Both of these hybrids contained 30% glass fibers, 30% graphite fibers and 40% epoxy. The two meshes generated for glass/e and graphite/e composites were in turn used for modeling the hybrid composites.
335
Impulse-frequency response technique. C. V. Nori et al.
Figure 5 Finite element mesh for quarter section of composite crosssection showing fiber distribution in the glass/epoxy and graphite/ epoxy regions
The meshes for the fiber distribution of the glass/e and graphite/e composites were used for the glass/e and graphite/e regions of the hybrids, respectively. The finite element meshes for a hybrid having glass/e shell and graphite/e core is shown in Figure 5. In order to model a hybrid having glass/e core and graphite shell, the glass/e and graphite/e regions in the finite element mesh shown in Figure 5 were interchanged. The results from the analysis of these combinations were found to be in close agreement with the experimental data as shown in
Table 3
Table 3. In these hybrid finite element models, it was assumed that the cores are cylindrical in shape. It may also be noted that the numerical value for the shear modulus of a 30% graphite/e shell-30% glass/e core hybrid composite is higher than the value obtained from the experiment. This is believed to be mainly due to the irregular core shape of this hybrid (formed by the larger diameter glass fibers, as observed under the optical microscope), which has not been taken into account in the finite element models. When the stereology point counting method was used to determine the volume fractions of fibers and matrix, it was observed that only the fiber volume fraction of the glass/e and the graphite/e regions of only the two hybrids modeled above were very close to the typical fiber volume fractions present in the glass/e and the graphite/e composites. For the other combinations of hybrids that were manufactured, the stereology point counting technique revealed that they contained different volume fractions and/or distribution of epoxy in the glass/e and graphite/e regions, when compared to that present in glass/e and graphite/e composites. Table 4 shows the ratio of fibers to epoxy in the glass/e and graphite/e regions of all the manufactured hybrid combinations. For hybrids having graphite fibers in the shell region (hybrids F and H), it was found that the ratio of fibers to epoxy was very low, i.e. larger amount of epoxy was found concentrated in the shell region than in the core region. In contrast, hybrids with graphite fibers in the core (hybrids E and G) exhibited very low epoxy concentration in the core region. Only the two composites modeled above exhibited similar distribution of epoxy in both the shell and core regions, hence there was agreement between finite element method and experimental data.
Results from experimental technique and finite element analysis based on non-uniform distribution of fibers Torsional shear modulus (GPa)
Composite
Experimental
Numerical (FEM)
60% 60% 30% 30% 30% 30%
7.35 6.11 7.63
7.41 6.32 7.60
6.04
6.93
Glass/40% Epoxy Graphite/40% Epoxy Glass/20% Epoxy (Shell)Graphite/20% Epoxy (Core) Glass/20% Epoxy (Core) Graphite/20% Epoxy (Shell)
Table 4
Stereological determination of volume fractions of constituents in the glass/epoxy and graphite/epoxy regions of hybrid composites Glass/epoxy region
Graphite/epoxy region
Composite
% Glass
% Epoxy
Glass/epoxy ratio
% Graphite
% Epoxy
Graphite/epoxy ratio
Glass epoxy (A) Graphite/epoxy (B) Hybrid C Hybrid D Hybrid E Hybrid F Hybrid G Hybrid H
60 -30(s) 30(c) 37(s) 37(c) 48(s) 48(c)
40 -20.22 19. l I 28.18 20.35 35.25 24.98
1.5:1 -1.48:1 1.56:1 1.31:1 1.81:1 1.36:1 1.92:1
-60 30(c) 30(s) 23(c) 23(s) 12(c) 12(s)
40 19.78 20.89 11.82 19.65 4.75 15.06
1.5:1 1.51:1 1.44:1 1.94:1 1.17:1 2.52:1 0.80:1
336
Impulse-frequency response technique: C. V. Nori et al. 9 CONCLUSIONS This paper presents both an efficient non-destructive experimental technique and a numerical approach for determining the shear modulus of pultruded cylindrical composite rods in torsion. An impulse-frequency response vibration technique was employed as the experimental technique and the finite element method was utilized as the numerical approach. For numerical modeling purposes, the distribution of fibers and matrix in the pultruded composite rods cross-sections were approximated through the use of a stereology point counting method and an optical microscope. The test specimens included glass/e, graphite/e, and hybrid glass-graphite/e pultruded composites. The results of the study indicated that the elastic behavior of pultruded composites in shear could.effectively be predicted by finite element analysis if the approximate location and distribution of the fibers in the matrix over the cross-section is known. The shear modulus as well as other mechanical properties of composites depend on their fiber packing geometry. Depending upon the location of fibers in the matrix, the overall shear modulus of mono-fiber type and hybrid composites can be varied. ACKNOWLEDGMENTS
REFERENCES 1 2 3
4 5
6
7 8 9 10 11
Gibson, R.F. 'Principles of Composite Material Mechanics' McGraw-Hill, 1992 Summerscales, J. and Short, D. Carbon fiber and glass fiber hybrid reinforced plastics. Composites 1978, 9, 157 Outwater, John O. The Torsional Failure and Fracture Energy of Pultruded Rod--Test Methods for Design Allowables for Fibrous Composites, In A S T M STP 1003, (Ed. C.C. Charmis), Am. Soc. of Test. and Mat., Philadelphia, Vol. 2, 1989, pp. 224-230 Suarez, S.A. and Gibson, R.F. Improved impulse response techniques for measurement of dynamic mechanical properties of composite materials. J. qf Test. and Eva/. 1987, 15, 114 Mantena, P.R., Gibson, R.F. Place, T. and Alan. A torsional impulse-frequency response technique for evaluting the dynamic mechanical properties of structural materials, ln'Proc, of the 6 th Annual ASM/ESD Advanced Comp. Conf ", Detroit, MI, 1990, pp. 477 485 Kumar, Sujit S. and Mantena, Raju P. Charcterization of dynamic mechanical properties of pultruded hybrid cylindrical composite rods in the torsional mode of vibration. In '35th AIAA/ASME/ASCE/AHS/ASC Struct., Struct. Dynamics, and Mat. Con[i '. Hilton Head, SC, 1994 pp. 1223 1232 Kumar, N.V.S. Papineni, 'Study of Static Mechanical Properties of Pultruded Hybrid Graphite/Glass-Epoxy Composites', M.S. Thesis, The University of Mississippi, 1994 Agarwall, B.D. and Broutman, L.J. ~Analysis and Peformance of Fiber Composites', Wiley, New York, 1992 Rao, S.S. "The Finite Element Method in Engineering', 1st Edn. Pergamon Press, UK, 1982 Reddy, J.N. "An Introduction to Finite Element Method'. 1st Edn. McGraw-Hill, 1985 Voot, V. 'Metallography Principles and Practice', McGraw-Hill, 1984
This research was supported by the National Science Foundation/EPSCoR (Grant No. OSR-9108767), the State of Mississippi, and the University of Mississippi.
337
EI,SEVIER
Composites: Part B 27B (1996) 329-337 Copyright ~(') 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/96/$15.00
Vibration analysis and finite element modeling for determining shear modulus of pultruded hybrid composites
Chandrasekhar V. Nori, Tyrus A. McCarty* and P. Raju Mantena Department of Mechanical Engineering, The University of Mississippi, MS 38677, USA (Received January 1995; acceptedJune 1995) An impulse-frequency response vibration technique was employed for determining the shear modulus of glass/epoxy, graphite/epoxy and hybrid (glass-graphite/epoxy) pultruded cylindrical composite rods in torsion. The distribution of the fibers and matrix in the shell-core regions were examined microscopically and the volume fractions of the various constituents determined using the stereology point counting technique. Based on the examined cross-section finite element meshes were generated and analyzed for predicting the shear modulus of such composite rods. It was observed that there was close agreement between the finite element predictions and the experimentally obtained data for only some of the hybrid configurations. The nature of the pultrusion manufacturing process which causes variations in the fiber packing geometry in the shell and core regions of hybrids has been found to significantly influence the accurate prediction of shear modulus, using either analytical or finite element methods. (Keywords: pultruded composites; shear modulus; impulse-frequency response technique; finite element method; glass/epoxy; graphite/epoxy; hybrid composites)
1 INTRODUCTION High modulus fibers such as graphite are widely used in aerospace, automotive, and structural applications because of their exceptionally high specific modulus and strength. Unfortunately, high cost and low impact strength are two disadvantages which limit their use. Hybridization or the mixing of high cost, high modulus, low impact fibers with low cost, low modulus fibers such as glass provides a possible solution worth exploring further. In this way, it is possible to take advantage of the combined action of individual fibers, thus optimizing the properties of the composite. There are basically two ways in which two or more types of fibers can be combined to formulate a hybrid composite: either by intermingling the fibers in a common matrix, or laminating layers of each type of fiber I. The mechanical analysis of conventional materials focuses mainly on predicting their behavior in structural applications. These applications assume that the fundamental material properties can be determined by testing them experimentally. Various analytical methods have been developed which are suitable for determining the properties of conventional materials primarily because of their isotropic nature. Hence, composite materials being * To whom correspondence should be addressed
anisotropic makes it very difficult to predict their material properties analytically. Summerscales and Short 2, in an extensive literature survey, discussed the results obtained by various researchers on the determination of material properties. It was reported that many investigators have found the experimental data to be consistently higher than that predicted by the law of mixtures. The four basic properties which are important for composite design are the longitudinal Young's modulus (ELa-), major Poisson's ratio (ULT), transverse Young's modulus (ET), and the shear modulus (GLT). Of these properties the experimental values of the transverse modulus and the shear modulus are known to disagree with the values predicted by the inverse law of mixtures. Gibson I verified that this is mainly due to the fact that these two properties depend upon the fiber packing geometry and also highlighted the importance of numerical techniques for predicting the transverse and shear moduli of composite specimens. When pultruded composite rods are subjected to pure torsion, the torque applied varies linearly with the angle of twist for small angles. Outwater 3 employed the initial slope of the torque v e r s u s angle of twist for computing the torsional shear modulus of such composites. Suarez e t al. and Mantena e t al. 5 demonstrated non-destructive vibration test techniques to determine the dynamic
329
Impulse-frequency response technique: C. V. Nori et al. flexural, extensional and shear moduli of composites. One of these methods employs a variation of the impulsefrequency response technique for testing cylindrical specimens subjected to a torsional mode of vibration. Static and dynamic techniques were employed by Kumar 6 to determine the shear properties of pultruded cylindrical composite rods in torsion. Innovative developments have been made on pultruding 'hybrids' in the Composite Materials Research Laboratory at the University of Mississippi using a commercial scale pultrusion machine. The improved characteristics of these hybrids demonstrate the potential for designing stiff, lightweight, and well damped composite structures having a number of practical applications. In this paper, the details of the nondestructive torsional vibration technique and a finite element model used for determining the shear modulus of pultruded composites are presented. The experimental technique evaluated the dynamic shear modulus and the finite element method predicted the static shear modulus. Previous research on similar pultruded hybrid cylindrical composite specimens showed close agreement in the static and dynamic shear moduli 6. Hence, in this investigation, the static slaear modulus computed by finite element method was used for comparison with the dynamic shear modulus obtained from the low strain level torsional vibration technique.
2 TEST SPECIMENS An in-house industrial scale pultrusion machine (Pulstarr 804) was used to manufacture unidirectional glass/ epoxy (glass/e) and graphite/epoxy (graphite/e) composites along with glass-graphite/epoxy (glass-graphite/e) pultruded composite rods. The schematic representation of the pultrusion process is shown in Figure 1. A typical pultrusion machine consists of roving/mat creels, resin impregnating tank, resin controller, shape preformer, heated dies, pullers, and cut-off saw. The manufacturing process involves the pulling of continuous roving/mats through a resin impregnating tank. The fiber impregnated resin is then passed through a resin control unit to remove any excess resin followed by passing it through a shape preformer of desired shape. The pullers are continuously operated hydraulically to pull the res.ulting product through a heated die where it is cured
120VIN(3/~D~T CF~EEL
RESIN
MC I ~%EGhA l TING C[]~TR
5
p~
~ATED
DIE
PULLER
SAV
\ P~IX£T
Figure 1 Schematic representation of the pultrusion manufacturing process
330
continuously. A cut-off saw is used to cut the desired length of the pultruded composite. With specially designed pre-form plates, hybrids were pultruded by intermingling glass and graphite fibers in a common matrix 7. The pre-form plates guided the fibers into the heating dies and helped maintain the constant crosssection of the shell and core regions of each hybrid during the curing process. Various combinations of hybrids were manufactured with all other pultrusion process conditions maintained constant. Each hybrid combination was prepared in a single pass operation of the pultrusion manufacturing process. The fibers used were E-glass and AS-4W-12K graphite, and the matrix was Epon 862/W resin (Shell Oil Company). The diameter of all the composite rods was 9.53mm (3/8inch), with overall fiber volume fraction maintained at 60% and the remaining 40% was matrix. Three samples of glass/e, graphite/e, and various glass-graphite/e hybrid composite rods were first tested dynamically to determine their torsional shear modulus. Table 1 shows various combinations of the round composites which were subjected to the dynamic tests. These specimens include glass-graphite/e hybrid combinations along with mono-fiber glass/e ('A') and graphite/e ('B') composites. Hybrids 'C', 'E' and 'G' have glass fibers in the shell and graphite fibers in the core, and hybrids 'D', 'F' and 'H' have graphite fibers in the shell and glass fibers in the 'core'. Only types 'A' through 'D' were modeled using the finite element method because of observed inconsistency in the fiber packing geometry of types 'E'-'H'. The typical fiber packing of these cross-sections were observed under an optical microscope. Based on an approximation of the observed distribution of fibers and matrix, the glassgraphite/e hybrids were modeled by the finite element method to obtain an estimate of their shear modulus which was then compared with experimental data. The rods were modeled assuming that the distribution of fibers and matrix in the shell and core regions of the hybrids were in the same proportion as that in the glass/e and graphite/e composites. Further analysis of the hybrid cross-sections was performed by applying the stereology point counting microscopy method for Table I Cross-sections and fiber volume fractions of hybrid combinations pultruded Percent fiber volume fraction Specimen*
Glass (E-glass)
Graphite (AS-4W-12K)
Cross-section
Glass/epoxy (A) Graphite/epoxy (B) Hybrid C Hybrid D Hybrid E Hybrid F Hybrid G Hybrid H
60 0 30(s)t 30(c) 37(s) 37(c) 48(s) 48(c)
0 60 30(c) 30(s) 23(c) 23(S) 12(c) 12(s)
() • o, O ~e~ O 'e, O
* Volume fraction of epoxy in all the above composites is 40% t (s) refers to the shell region and (c) refers to the core region of the hybrids
Impulse-frequency response technique: C. V. Nori et al. determining the true fiber packing in the shell and core regions.
3 EXPERIMENTAL SETUP AND PROCEDURE The impulse-frequency response technique involves the excitation of all the modes of vibration of sample specimens with an electromagnetic impulse hammer having a piezoelectric force transducer in its tip. The vibrational response is measured with a non-contacting eddy current displacement transducer. The frequency response function displayed on the spectrum analyzer consists of sharp peaks depicting the various modes of vibration. The resonant frequency of interest is used along with the appropriate characteristic equation for determining material properties. The experimental set-up shown in Figure 2 has been used to obtain the fundamental torsional mode of vibration of the cylindrical specimens. The set-up consists of dual channel spectrum analyzer (HP 35665A), an impact hammer, a non-contacting eddy current probe with a displacement measuring system,
and a computer. The test specimen, which is fixed at the top of an L-plate, has a lumped cylindrical mass with two diametrically opposite 'ears' attached to its lower end. The purpose of the end mass is to sustain torsional oscillations of the specimen after initial excitation. One of the ears on the end mass is impacted with the impulse hammer and the response is picked up by the noncontacting eddy current displacement transducer placed at the other ear. The input and output signals are fed to the spectrum analyzer which displays the frequency response function of the modes of vibration. The computer, interfaced with the analyzer, is used for data transfer and instantaneous computation of the dynamic shear modulus and damping of the candidate materials being tested. Off-axis impact of the end mass, attached to the base of the pultruded rods, results in the simultaneous excitation of both the bending and torsional modes of vibration. These are displayed as two closely spaced sharp peaks within the first 100 Hz frequency span of the spectrum analyzer. It is extremely important that the correct torsional peak is analyzed and incorporated into the characteristic equation for dynamic shear
Adjustable L-Plate Lathe Chuck (ToholdSpecimen)
Triangular
Plate / ~E~ Cross-sectioSpeci n mE~-~'~en II ~ i
Chuck ('With"ears"attached)
Lathe
Impact Hammer
I Displacemmt Measuring
.
Non-Contacting EddyCurrentProbe .
.
-I -I
Plotter
PowerSupply
.
liP 35665A
Computer
I-
Pfiatt~"
Figure 2 Experimentalset-upfor torsionalshearmodulusdetermination
331
Impulse-frequency response technique. C. V. Nori
e t al.
modulus for each specimen. Incorporation of the redundant bending peak into this equation will give erroneous data. The dynamic shear modulus, G', computed from the torsional frequency fn is given by: G' - 47r2f2nLZp A2 n
(1)
along with the characteristic equation: cot A -
J)~n plpL
in shear than graphite fibers (Ggraphit e = 14.14GPa), the inverse law predicts only a marginal difference between the glass/e and graphite/e composites. Hence the shear moduli predicted by this law show that the matrix (Gmatrix = 0.77GPa) is the dominant constituent for these composite specimens. It has also been shown that the experimentally obtained shear modulus was consistently higher than the shear modulus predicted by the inverse law of mixtures I .
(2) 5 MATHEMATICAL MODEL
where L, p, and Ip are the length, mass density, and polar area moment of inertia of the specimen, respectively. An is the eigenvalue for torsional modes and J is the polar mass moment of inertia of the attached lower mass. For calibration, a 2024-T4 aluminium sample of known shear modulus was first tested to determine the polar mass moment of inertia of the lumped lower mass because of its complex geometry. The fundamental torsional frequency of vibration of the 0.127 m (5 inches) long aluminium sample was found to be 26 Hz, and the mass polar moment of inertia of the lower mass (J) was determined to be 4.4 × 10-3 Kg-m 2 (0.035 lb in s2).
The solution of the following two dimensional boundary value equation has been considered for the torsion of cylindrical rods 9
02
(4)
The boundary conditions associated with this equation are given by • (x,y)=0
onF
(5)
and 0@ nx~+
4 ANALYTICAL MODEL: INVERSE LAW OF MIXTURES
in ~
0@ ny~y = 0
onF
(6)
The properties of composite materials depend on the properties of its constituents, their distribution and the physical and chemical interactions between them. The equation used to obtain the shear modulus of composite materials using the inverse law of mixtures is given by 8
where G(= GLT) is the shear modulus, @ the Prandtl's stress function, 0 the angle twist per unit length, ~ refers to the interior, and F refers to the boundary of the crosssection of the rods. n x and n r are the cosines of the angles between x and y axes and the normal to the boundary, respectively. The non-zero stress components are given in terms of the stress function by the following expressions
1 -- Vglass "k Vgraphite ~ Vep°xy GET Gglass Ggraphite Gepoxy
0@ O@ qz =Oyv and ryz - Ox
(3)
where GLT is the longitudinal shear modulus of the composite rod, V and G are the volume fraction and shear modulus of the constituents of the composite rod, respectively. The inverse law of mixtures predicts that the shear modulus of a composite can be raised to five times the matrix modulus only by providing 90% fibers 8. Producing a composite having 90% fibers is not practical because it does not provide proper bonding between the fibers and matrix. Fibers do not contribute much to the shear modulus unless the percent of fibers is very high. Hence, the use of high modulus fibers does not significantly enhance the shear modulus. Further, the inverse law of mixtures predicts shear modulus of a composite based on the assumption of uniform distribution of fibers in the matrix. For glass/e and graphite/e composites, the inverse law of mixture predictions for shear modulus are 1.85 and 1.77GPa, respectively. Although, glass fibers (Gglass = 30.13 GPa) are approximately two times stiffer
332
(7)
and the torque transmitted, T, is given by T
=
f 2 / @dA J
(8)
In the present problem, the Garlerkin weighted residual method is employed in deriving the finite element equations m. In this technique, the governing" differential equation is first put into an equivalent weighted integral form given by I~ ~ IW-TS,,2+
2GO dxdv
0
(9)
where w is the weighing function. The resulting equation is integrated by parts to obtain the weak form as
LOx Ox +
bTJ dxdy+ 2J, wOOaxdy
Impulse-frequency response technique: C. V. Nori et al.
- ~ r W ( n x O~b-~x+ ny ~ v0) d S, =
(10)
From the inspection of the boundary condition, it may be noted that the specification of • constitutes the essential boundary condition and hence q~ is the primary variable. Specifying the coefficient of weighing function in the boundary expression constitutes the natural boundary condition, and hence it is the secondary variable of the formulation. The weak form obtained in the above equation forms the basis of the finite element model.
6 FINITE ELEMENT MODEL The finite element equation corresponding to the weak form of the governing differential equation is obtained by choosing an approximation for ~. The variable, ~, should be at least linear in both x and y, so that there are no zero terms in equation (10). Hence to obtain the finite element equation, ~ and w are approximated over an element by the following relations
where A is the area of the triangular element, i ~ j ~ k and i, j and k permute in a natural order. The torque or twisting moment acting on the rod can be computed as: N
T=2I
~dxdy=ZZ(~i+~/+~k)(e)
(17)
e=l
where N is the number of elements, ~i, ~j and ~k denote the values of stress function at nodes i, j and k, respectively, of element 'e'. Assuming symmetry, only a quarter section of the rod was modeled. The domain of the composite discretized using three-noded triangular elements (with linear interpolation function). The domain (an area of 0.713cm 2) was divided into 2 592 elements using 1 354 nodes. By maintaining the volume fractions of the individual components, each element was assigned the properties of either the appropriate glass, graphite fiber or epoxy matrix. Each of these constituents was assumed to be homogeneous, isotropic and linearly elastic. It was also assumed that there is perfect bonding between the matrix and fibers and that the interface between them is continuous. The longitudinal shear moduli assigned to the glass fibers, graphite fibers and the epoxy matrix were 30.13 GPa, 14.14 GPa and 0.77 GPa, respectively.
m
/e) (x,>),
• (x, v)
(ll)
j=l
and w=~i
( i = l, 2 , . . . , m )
(12)
where ~ and qJ~ are the stress and interpolation functions at the jth node of the element and m is the number of nodes per element. On substituting equations (11) and (12) into equation (10), the finite element expression is obtained as:
"
{0 ,ow,
}
LOx ox + o>,
~idxdy
+2 [~ [GO~i](e)dxdy - ~r [k~iqn](e)ds = 0 (i = 1, 2 .... ,m)
(13)
where 0~
0(I,
qn = nx o~[~v+ ny --fffy = O
(14)
The coefficient of the ~j in the first term of equation (13) represents the element stiffness matrix, and the boundary term vanishes. Linear interpolation of the form ~b(x, y) = cl + CeX + c3y has been considered for a three node triangular element. The interpolation functions are given by:
c,~i= x/yk - -
~i =
)) - Yk and 7 / = xk - Xi (15)
where
~le/
1 = 2--7(c*i + ~qix
i= 1, 2, 3
7 STEREOLOGY: POINT COUNTING METHOD The stereology point counting method (ASTM Specification E562) is widely used for determining the volume fractions of various constituents in a material ~1. In this method an optical microscope with a CRT monitor is used to view the micro-structure of various composite cross-section at appropriate magnifications. A typical composite specimen observed under the microscope displays the fibers (circular in shape) distributed in the epoxy matrix. In the stereology point counting method, a test grid is first placed on the micro-structure displayed on the CRT monitor. Then the number of points lying on the fibers is determined. The intersection of two crossed lines of the test grid is considered as a point. If the intersection lies completely inside the phase, it is considered to be one count and if it lies on the phase boundary it is counted as one-half count. If Po is the number of grid points and P,, the point count for each of the 'n' number of randomly selected fields, the volume fraction Pp of the constituent of interest is given by the relation: ~ P~
pp -
nPo
(is)
There are various factors involved in the selection of an appropriate grid size for determining volume fractions using the stereology point counting method II. These include volume fractions of the constituents, choice of magnification and resolution limits of the optical microscope used. It was found that consistent results were obtained for the hybrid composites analyzed with an 11 × 11 grid size at a magnification of 60×. Six
333
Impulse-frequency response technique." C. V. Nori et al. randomly selected fields were selected over the crosssection of each sample for determining the fiber to matrix ratio. Similar analysis was performed in the glass/e and graphite/e regions of the glass-graphite/e hybrid composites. The ratio of fiber to epoxy in the glass/e and graphite/e regions was computed.
5,0
8
~
~
~o
]116
-0.96
0 (~
8 RESULTS AND DISCUSSION
Percent Graphite
?
0
-0.75 "~
Olasst¢shell, graphile/¢~r¢] OraphiCe/eshell, glasst¢c o r e ~c~erimenu~Sm~er Em~rim¢ nlalResutt~ / Inverse law ofMixture~ J
]
-0.6s -0.55 ~ -O.45 ~ ~
The shear moduli of all the experimentally tested composite specimens were found to be higher than the values predicted by the inverse law of mixtures. The experimental method determines the dynamic shear modulus, whereas both the inverse law of mixtures and finite element method predict the static shear modulus. Previous research conducted on these hybrid composite specimens showed that the dynamic and static shear moduli were approximately the same 6. Hence, the static modulus predicted by inverse law of mixtures and finite element method are compared to the experimentally determined dynamic moduli in this paper. Figure 3 shows the comparison of static modulus predicted by the inverse law of mixtures and the dynamic modulus determined by experimental means. It can be observed that the experimental values are higher than the values predicted by the inverse law of mixtures. This may be attributed to the fact that the inverse law of mixtures predicts shear modulus for composites having uniform distribution of fibers in matrix. The predicted shear modulus obtained using the inverse law of mixtures was marginally lower for graphite/e specimens as compared to the glass/e specimens. The shear moduli for all other hybrid combinations were intermediate to that of the glass/e and graphite/e composite specimens. It can also be observed that the inverse law of mixtures predicts identical shear modulus for hybrids having similar volume fractions of glass and graphite fibers, irrespective of the type of fiber located in core and shell regions. In contrast, experimental results showed higher shear modulus for the hybrids having glass/e shell-graphite/e core, compared to hybrids with glass/e core-graphite/e shell. The graphite fibers have lower density than glass fibers. Experimental results show that by replacing a portion of glass fibers in the glass/e composite with graphite fibers changed the overall stiffness to weight ratio. A hybrid composite with glass/e shell-graphite/e core had marginally higher torsional stiffness than the glass/e composite, suggesting a synergistic effect. In addition, the density of this hybrid was less than the glass/e composite. When a hybrid with graphite/e shellglass/e core was tested with the same volume fraction of constituents, it exhibited a lower specific stiffness. Although the density of the two hybrid composite rods was the same, the stiffness varied with the location of glass and graphite fibers. All the hybrid combinations with glass/e shell-graphite/e core exhibited higher specific torsional stiffness than the graphite/e shell-glass/e core
334
6 ~o
~o
Jo Percent Glass
o
Lo.~
o /o
-0.35
0,15 60
~o .-~
Figure 3 Shear modulus: comparison of experimental law of mixtures
versus
inverse
Figure 4 Finite element mesh for quarter section of composite cross-section
hybrids and/or the glass/e and graphite/e composite rods. When the cross-sections of the hybrid composites were examined under an optical microscope (at magnification of 10x), the concentrated fiber and matrix regions were observed, implying non-uniform fiber distribution over the cross-section. Hence it was extremely difficult to model such a composite by any of the available analytical methods. Inputing the test. data obtained for glass/e and graphite/e composites (60% fiber and 40% epoxy) into the core and shell regions of the hybrid models (for predicting their properties) appeared to have serious shortcomings. Three special cases have been considered for predicting the shear modulus of pultruded composite rods using the finite element method.
Case 1." Uniform distribution of fibers Finite element analysis was performed assuming uniform distribution of fibers in matrix. Properties of
Impulse-frequency response technique: C. V. Nori et al. Table
2 Resultsfrominverselaw of mixturesand finiteelementanalysisbased on uniformdistributionof fibers Torsional shear modulus(GPa)
Composite
Law of mixtures*
Numerical (FEM)*
Experimentalt
60% Glass/40% Epoxy 60% Graphite/40% Epoxy 30% Glass/20% Epoxy(Shell)30% Graphite/20% Epoxy(Core) 30% Glass/20% Epoxy(Core)30% Graphite/20% Epoxy(Shell)
1.85 1.77 1.82
1.96 1.86 1.93
7.35 6.11 7.63
1.82
1.89
6.04
* Based on uniformdistributionmodel t Actuallypultrudedspecimens non-uniformdistribution) fibers and epoxy were assigned to each of the elements in the domain of the composite by maintaining constant volume fractions of its constituents. The properties were also assigned in such a way that the fibe~ elements were uniformly distributed in the entire domain. Figure 4 shows the discretized domain of a quarter section of the composite cross-section. The static shear modulus of glass/e, graphite/e and the two core and shell composites (hybrids C and D) were obtained. Table 2 shows the shear modulus obtained from the finite element analysis. The results were in good agreement with the values predicted by the inverse law of mixtures for mono-fiber glass/e and graphite/e composites. Closer examination of the hybrids having identical volume fractions of glass and graphite fibers indicates that similar shear moduli are predicted by the inverse law of mixtures. In contrast, the finite element method predicts marginally higher shear modulus for the hybrid having glass fibers in the shell region as compared to the hybrid having graphite fibers in the shell region (for uniform distribution of fibers). Hence, it can be inferred that the inverse law of mixtures can only predict shear moduli for those hybrids which have both glass and graphite fibers distributed uniformly throughout the composite domain. It fails to predict the properties for core-shell type hybrids even with uniform distribution of fibers in each region. Also, the experimental results showed that the shear moduli of graphite/e, glass/e and hybrid composite rods were higher than that predicted by the finite element model based on uniform distribution of fibers in the respective regions.
Case 2." Glass-epoxy and graphite-epoxy composites (non-uniform fiber distribution model) The pultrusion process involves pulling of resin impregnated fibers through a heated die by setting the pull speed and die temperature. Although this is an automated manufacturing process and all care was taken to manufacture a perfectly uniform composite, there was a high probability that the fibers were not distributed uniformly in the matrix. A little concentration of fibers in one region over the cross-section would change the overall property of the composite. Hence, the shear modulus as well as other mechanical properties of these composites depend mainly upon their fiber packing geometry.
Depending upon the location of fibers in the matrix, the overall shear modulus of a composite can be varied. Microscopic observation of the composite cross-sections revealed that the fibers were not uniformly distributed in the matrix. Finite element meshes were generated based upon the observed distribution for the glass/e and graphite/e composites. The meshes for these composites were approximated with alternate layers of epoxy and fiber, starting with epoxy in the outermost layer. The thickness of the fiber layers was maintained at twice the thickness of the epoxy layers. Additionally, microscopic observations also confirmed the existence of lumped regions of fibers and matrix about the cross-section. Based upon these observations, regions of concentrated fibers and epoxy were also incorporated in the finite element mesh. These concentrated fiber and epoxy regions were different for the graphite/e and glass/e composites. This was mainly due to the large diameter ratio of glass to graphite fibers, which is approximately 4 : 1. Although the area of an individual element in the mesh is very large compared to the area of a single fiber, the overall mesh is a close approximation representing the fiber-matrix distribution in the domain. Figure 5 shows the fiber distribution in glass/e and graphite/e regions which were used for modeling glass/e and graphite/e mono-fiber composites, respectively.
Case 3." Modeling of core-and-shell type hybrids A special kind ofpre-form plate was designed to guide and maintain a constant circular cross-section of fibers constituting the core of the hybrids. Every care was taken to produce hybrids with cylindrical cores. When a glass/e shell-graphite/e core hybrid rod was pultruded, perfect cylindrical cores were produced. However, this was not the case with the glass/e core-graphite/e shell hybrids where the cores were observed to be not quite as cylindrical, which may be attributed to the large difference in the diameters of glass and graphite fibers. Two types of core and shell type hybrids, one with glass/e shell-graphite/e core (Hybrid C) and the other with graphite/e shell-glass/e core (Hybrid D) were modeled. Both of these hybrids contained 30% glass fibers, 30% graphite fibers and 40% epoxy. The two meshes generated for glass/e and graphite/e composites were in turn used for modeling the hybrid composites.
335
Impulse-frequency response technique. C. V. Nori et al.
Figure 5 Finite element mesh for quarter section of composite crosssection showing fiber distribution in the glass/epoxy and graphite/ epoxy regions
The meshes for the fiber distribution of the glass/e and graphite/e composites were used for the glass/e and graphite/e regions of the hybrids, respectively. The finite element meshes for a hybrid having glass/e shell and graphite/e core is shown in Figure 5. In order to model a hybrid having glass/e core and graphite shell, the glass/e and graphite/e regions in the finite element mesh shown in Figure 5 were interchanged. The results from the analysis of these combinations were found to be in close agreement with the experimental data as shown in
Table 3
Table 3. In these hybrid finite element models, it was assumed that the cores are cylindrical in shape. It may also be noted that the numerical value for the shear modulus of a 30% graphite/e shell-30% glass/e core hybrid composite is higher than the value obtained from the experiment. This is believed to be mainly due to the irregular core shape of this hybrid (formed by the larger diameter glass fibers, as observed under the optical microscope), which has not been taken into account in the finite element models. When the stereology point counting method was used to determine the volume fractions of fibers and matrix, it was observed that only the fiber volume fraction of the glass/e and the graphite/e regions of only the two hybrids modeled above were very close to the typical fiber volume fractions present in the glass/e and the graphite/e composites. For the other combinations of hybrids that were manufactured, the stereology point counting technique revealed that they contained different volume fractions and/or distribution of epoxy in the glass/e and graphite/e regions, when compared to that present in glass/e and graphite/e composites. Table 4 shows the ratio of fibers to epoxy in the glass/e and graphite/e regions of all the manufactured hybrid combinations. For hybrids having graphite fibers in the shell region (hybrids F and H), it was found that the ratio of fibers to epoxy was very low, i.e. larger amount of epoxy was found concentrated in the shell region than in the core region. In contrast, hybrids with graphite fibers in the core (hybrids E and G) exhibited very low epoxy concentration in the core region. Only the two composites modeled above exhibited similar distribution of epoxy in both the shell and core regions, hence there was agreement between finite element method and experimental data.
Results from experimental technique and finite element analysis based on non-uniform distribution of fibers Torsional shear modulus (GPa)
Composite
Experimental
Numerical (FEM)
60% 60% 30% 30% 30% 30%
7.35 6.11 7.63
7.41 6.32 7.60
6.04
6.93
Glass/40% Epoxy Graphite/40% Epoxy Glass/20% Epoxy (Shell)Graphite/20% Epoxy (Core) Glass/20% Epoxy (Core) Graphite/20% Epoxy (Shell)
Table 4
Stereological determination of volume fractions of constituents in the glass/epoxy and graphite/epoxy regions of hybrid composites Glass/epoxy region
Graphite/epoxy region
Composite
% Glass
% Epoxy
Glass/epoxy ratio
% Graphite
% Epoxy
Graphite/epoxy ratio
Glass epoxy (A) Graphite/epoxy (B) Hybrid C Hybrid D Hybrid E Hybrid F Hybrid G Hybrid H
60 -30(s) 30(c) 37(s) 37(c) 48(s) 48(c)
40 -20.22 19. l I 28.18 20.35 35.25 24.98
1.5:1 -1.48:1 1.56:1 1.31:1 1.81:1 1.36:1 1.92:1
-60 30(c) 30(s) 23(c) 23(s) 12(c) 12(s)
40 19.78 20.89 11.82 19.65 4.75 15.06
1.5:1 1.51:1 1.44:1 1.94:1 1.17:1 2.52:1 0.80:1
336
Impulse-frequency response technique: C. V. Nori et al. 9 CONCLUSIONS This paper presents both an efficient non-destructive experimental technique and a numerical approach for determining the shear modulus of pultruded cylindrical composite rods in torsion. An impulse-frequency response vibration technique was employed as the experimental technique and the finite element method was utilized as the numerical approach. For numerical modeling purposes, the distribution of fibers and matrix in the pultruded composite rods cross-sections were approximated through the use of a stereology point counting method and an optical microscope. The test specimens included glass/e, graphite/e, and hybrid glass-graphite/e pultruded composites. The results of the study indicated that the elastic behavior of pultruded composites in shear could.effectively be predicted by finite element analysis if the approximate location and distribution of the fibers in the matrix over the cross-section is known. The shear modulus as well as other mechanical properties of composites depend on their fiber packing geometry. Depending upon the location of fibers in the matrix, the overall shear modulus of mono-fiber type and hybrid composites can be varied. ACKNOWLEDGMENTS
REFERENCES 1 2 3
4 5
6
7 8 9 10 11
Gibson, R.F. 'Principles of Composite Material Mechanics' McGraw-Hill, 1992 Summerscales, J. and Short, D. Carbon fiber and glass fiber hybrid reinforced plastics. Composites 1978, 9, 157 Outwater, John O. The Torsional Failure and Fracture Energy of Pultruded Rod--Test Methods for Design Allowables for Fibrous Composites, In A S T M STP 1003, (Ed. C.C. Charmis), Am. Soc. of Test. and Mat., Philadelphia, Vol. 2, 1989, pp. 224-230 Suarez, S.A. and Gibson, R.F. Improved impulse response techniques for measurement of dynamic mechanical properties of composite materials. J. qf Test. and Eva/. 1987, 15, 114 Mantena, P.R., Gibson, R.F. Place, T. and Alan. A torsional impulse-frequency response technique for evaluting the dynamic mechanical properties of structural materials, ln'Proc, of the 6 th Annual ASM/ESD Advanced Comp. Conf ", Detroit, MI, 1990, pp. 477 485 Kumar, Sujit S. and Mantena, Raju P. Charcterization of dynamic mechanical properties of pultruded hybrid cylindrical composite rods in the torsional mode of vibration. In '35th AIAA/ASME/ASCE/AHS/ASC Struct., Struct. Dynamics, and Mat. Con[i '. Hilton Head, SC, 1994 pp. 1223 1232 Kumar, N.V.S. Papineni, 'Study of Static Mechanical Properties of Pultruded Hybrid Graphite/Glass-Epoxy Composites', M.S. Thesis, The University of Mississippi, 1994 Agarwall, B.D. and Broutman, L.J. ~Analysis and Peformance of Fiber Composites', Wiley, New York, 1992 Rao, S.S. "The Finite Element Method in Engineering', 1st Edn. Pergamon Press, UK, 1982 Reddy, J.N. "An Introduction to Finite Element Method'. 1st Edn. McGraw-Hill, 1985 Voot, V. 'Metallography Principles and Practice', McGraw-Hill, 1984
This research was supported by the National Science Foundation/EPSCoR (Grant No. OSR-9108767), the State of Mississippi, and the University of Mississippi.
337