Analytical Model for Prediction of the Damping Loss Factor of Composite Materials

ROGER M. CRANE

David Taylor Research Center Annapolis, Maryland 21 402

and

JOHN W. GILLESPIE, J R .

Center for Composite Materials University of Delaware

Newark. Delaware 1971 6

Numerous approaches have been undertaken to determine the damping of composites. These approaches can be grouped into micromechanical, macrome- chanical, and structural approaches. This paper describes a macromechanical approach that has been experimentally validated using various S-2 glass1350 1-6 laminates. Our approach is an extension of the elastic-viscoelastic approach, which accounts for the frequency dependence of the loss factor. The experimen- tally determined material loss factor for the glass/epoxy determined in a previ- ous investigation is used as input to the model. The material complex moduli are then determined and used as input to the model. The loss factor of a quasi- isotropic configuration is analytically determined in the frequency range of the experimental data. The loss factors for these beams are then experimentally determined using a cantilever beam configuration set into vibration with an impulse excitation. The loss factor at various frequencies are determined using the half power band width technique. The analytical values are within 15% of the experimental values in the frequency range of test. In addition, a parametric study is given on the effect of fiber orientation on loss factor. The analytically determined loss factor using the proposed model shows that inconsistencies documented in the literature on the fiber orientation at which a maximum in loss factor occurs can be resolved by incorporating the frequency dependence of the composite loss factor.

INTRODUCTION

he numerous approaches that have been uti- T lized to determine the mechanical vibration damping of composites can be grouped into mi- cromechanical. macromechanical, and structural approaches. The most fundamental approach based on micromechanics would derive the damping re- sponse of composites from the material constituents in an analogous manner to that in which the elastic behavior of composites is determined. This ap- proach would provide the most utility. However, in addition to the bulk material constituent properties, their interaction and physical characteristics, such as void content, fiber diameter, and interfacial char- acteristics, would also need to be incorporated since numerous experimental investigations have shown that these effects will influence the damping re-

sponse (1-8). The incorporation of these effects, as well as the necessary experimental validation of a micromechanical approach, would require the de- velopment of new test procedures, which at best would entail extensive testing.

In the structural analysis of composites, the fun- damental building block is the composite lamina. The elastic response of the lamina incorporates the elastic response of the material constituents as well as their interaction and physical characteristics. Because of this, most composite design utilizes material characteristics of the lamina in lieu of the characteristics of the individual constituents. In ad- dition, the lamina characteristics can be readily de- termined through numerous standardized test pro- cedures.

In a similar vein, the determination of the damp- ing of a composite from knowledge of the composite

POLYMER COMPOSITES, JUNE 7992, Vol. 73, No. 3 179

Roger M. Crane and John W. Gillespie, J r

350 1

laminae damping characteristics will automatically incorporate the material constituent characteristics, e.g., the fiber diameter, fiber matrix interface, fiber and matrix loss factor, and fiber volume fraction. With knowledge of the anisotropic damping loss fac- tors of the laminae, the damping response of a gen- eral laminated composite could then be determined. In addition, experimental techniques have been de- veloped that can determine the laminae material loss factors. Therefore, one should be able to experi- mentally determine the material characteristics re- quired for such a model. For these reasons, a macromechanical approach is, in the authors’ opin- ion, the most utilitarian approach at this time.

There have been two approaches previously un- dertaken to model the macromechanical damping response of composites: the elastic viscoelastic cor- respondence principle and the strain energy ap- proach. In the former approach, the fully populated complex reduced elastic stiffness matrices are deriv- able from the in-plane material complex elastic properties. However, as the model currently appears in the literature, the frequency dependence, or vis- coelastic characteristic of the composite, is typically omitted from the analysis. This paper presents an extension of the elastic viscoelastic correspondence principle, incorporating this frequency dependence of the loss factor into the analysis.

The polymer matrix material utilized in the com- posites materials discussed herein possesses tem- perature and frequency dependent damping loss factors, as illustrated in Fig. 1. Because of this, the composite material should also exhibit a frequency and temperature dependent loss factor. In fact, in the discussion of the elastic viscoelastic correspon- dence principle, many authors refer to this fact yet omit its effect from the subsequent analysis. To provide a more accurate description of the damping loss factor of composites, their frequency depen-

300 4 A

0

350 -

300 - h

0

2 150 4 loo 50 1

0 100 200 300 400 500 6 0 0 700 8 0 0 900 1000

Frequency (hz)

Fig. 1 . Loss factor us. frequency for 5208 neat epoxy resin determined using the cantilever beam technique.

dence should be determined. The analytical model presented herein is a modification of the elastic vis- coelastic correspondence principle, which incorpo- rates the frequency dependent anisotropic loss fac- tor of the laminae. These material loss factors are then used to analytically determine the damping loss factor of a general laminated composite.

MACROMECHANICAL MODEL DEVELOPMENT

The development of the elastic viscoelastic corre- spondence principle for composites has been previ- ously presented in the literature. A detailed discus- sion of the development of this model for use with laminated plate theory has been previously pre- sented (9, 10). This is necessary to ensure that the principles associated with the elastic viscoelastic correspondence principle are not violated in the ap- plication to laminated plate theory.

The correspondence principle states that if the elastic solution for any dependent variable having a time varying component exists, then the viscoelas- tic problem can be solved by replacing the equations of the elastic material by the equations that describe the viscoelastic material. The principle can be ap- plied if 1) the elastic solution is known; 2) no opera- tion in obtaining the elastic solution has a corre- sponding operation in the viscoelastic solution that involves separating complex modulus into real and imaginary parts, with the exception of the final de- termination of that response; and 3) the boundary conditions for the elastic and viscoelastic cases are identical (1 1). These conditions are satisfied for the case of the vibrating beam.

Hashin (12) utilized this theory with further sim- plifications for composite materials. Letting F repre- sent one of the elastic moduli of the composite, it can be described in terms of N phase properties as F = F( p,) where p j are the phase properties for the composite. These phase properties form the set of all elastic constants of all phases that are needed to predict the effective property F. The effective com- plex viscoelastic property is then given as

F* = F( pj*)

where F is still the elasticity solution. Its argu- ments, however, are now complex phase properties. These complex phase properties can also be written as

pi* = p j [ 1 f i 7 J j ] (2)

where pj* is the complex modulus. The effective modulus is then expanded in a multiple, complex Taylor series, using the real part of the complex phase property pj as the set of values about which the expansion is made. Assuming that all con- stituent loss factors are sufficiently small, the sec- ond and higher order terms in qj can be neglected. This results in the Taylor expansion of F* as

( 3 )

180 POLYMER COMPOSITES, JUNE 1992, Vol. 13, No. 3

Analytical Modelfor Prediction of the Damping Loss Factor

where F' is the elastic solution in terms of the real part of the phase properties. This development shows that with small damping, the effective com- plex properties can be derived directly from analyti- cal elasticity solutions. If the constituent loss factors are not small, this same procedure can be used with a minor modification. In this case, second or possi- bly higher order terms in the Taylor series would need to be included. In tl-e development that fol- lows, then, the substitutior. of the viscoelastic com- plex moduli in the form giv1:n in Eq 3 will be made.

An additional assumptio 1 made in this develop- ment is that the composite s linearly viscoelastic. A viscoelastic material posses 3es a time and frequency dependent elastic and viscous response to either an applied stress or strain loading. A linearly viscoelas- tic material is a subset of 'h is having the property that the time and frequenc r dependent mechanical properties are independent of the level of stress or strain loading.

The local material behatiior is governed by the elastic constitutive law as

(4)

The elastic viscoelastic correspondence principle can be applied to Eq 4 since this equation meets the conditions stated in the principle. For the specific case of a vibrating beam, the stress and strain have time varying components. As such, aI, and eLJ are given as

E * = eLJelWt (5)

It is also assumed that the complex moduli are frequency dependent, consisting of a real and imagi- nary term, called the storage and loss modulus, respectively. It will be assumed that the storage modulus is independent of frequency. In the fre- quency range of interest, up to 1000 Hz, this is a very good approximation based on the work of other investigators (12- 15). The frequency dependence of the complex modulus is therefore the result of the frequency dependence of the loss modulus. This frequency dependence of the loss modulus is the result of the frequency dependence of the loss mod- ulus of the matrix material, which has been re- ported by various investigators (15- 18). As such, using the elastic viscoelastic correspondence princi- ple, E q 4 is given as

Two assumptions are now made. First is that the fiber composite material can be approximated as a homogeneous material with orthotropic material properties. This reduces the number of elastic con- stants given by Eqs 4 and 7 from 36 to 9. A second assumption is that the material is in a state of plane stress, where the stresses normal to the plane of the

plate are assumed to be zero, i.e., ag = T~~ = T~~ = 0. Without loss in generality, the time-varying compo- nents of the stress, a3eIwt = 723eiwf = 7,3eiwt can like- wise be assumed to be zero. This assumption fur- ther reduces the number of elastic constants to four independent values. The viscoelastic constitutive re- lationships can now be expressed as

Equations 9 through 1 1 reduce to the elastic case if there is no frequency variation in stresses and strains.

The flat plate may be acted upon by applied mo- ments, M, distributed applied loads, q , in-plane loads, N , and point loads, P . I t is assumed that the plate consists of multiple layers of composite lami- nae, with the fibers in each plate being parallel to the plane of the plate. Kirchhoff s hypothesis is then applied to the plate, i.e., lines that are straight and normal to the laminate's geometric mid-surface re- main straight and normal to this geometric mid- plane and do not change length. This means that the lamina interfaces remain parallel to each other after application of the applied loads. I t is therefore possible to express the displacement of the material points that lie along a line perpendicular to the laminates geometric midsurface in terms of the dis- placement and rotation of the point on the line lo- cated at this midplane. The implication of this hy- pothesis is that the displacement a t any point (x, y, z) depends linearly on z. The time dependent displacement in the x, y, and z directions, given as u*, u*, and w*, respectively, can be written in terms of the midplane displacements, denoted by the "0"

superscript, as

w*( x, y , z ) = w"( x, y )eLW* (14) In the case where these displacements do not vary with time, Eqs 12 through 14 reduce to the elastic case, again making use of the fact that eiwt = 1 for w = 0.

The effect of Kirchhoff's hypothesis on the strain response, using the definition of the strain in the x direction on Eq 12, results in

a U * ( x, y, 2 ) a U o ( x, y)eiwt - E* , = - ax ax

POLYMER COMPOSITES, JUNE 7992, Yo/. 13, No. 3

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In Eq 15, the first term in the last equality is due to the extensional strain of the reference surface. The second term is the curvature of the reference surface in the x direction since the material is limited to small rotations, which will henceforth be given by K O . Using this information, Eq 15 can be rewritten as

E : ( x, y . z ) = c;( x , y)e lwt + z ~ : ( x, y)e'"' (16)

This again reduces that the classical case when the frequency is set equal to zero.

In an analogous manner, the time dependent strain in the y direction can be determined using Eq 16 in the same manner that the strain in the x direction was determined. This development results in the transverse strain given as

au*( x, y , Z ) a U o ( x, y ) e L w f - E * y = -

a Y a Y

Again, the resultant frequency independent trans- verse strain can be obtained from Eq 17 by using w = o .

In Eq 17, the first term in the last equality is the result of the extensional strain of the reference sur- face. The second term is the curvature of the refer- ence surface in the y direction since the material is limited to small rotations. Using this information, Eq 17 can be rewritten as

~ * y ( x , y , z ) = ~ ; ( x . y ) e ~ ~ ~ + z ~ O , ( x , y ) e ' " ~ (18)

Similarly, using the definition of the in-plane shear strains, the complex in-plane shear strains are given as

Using the strains obtained using the Kirchhoff hy- pothesis, Eqs 9 through 11 can be written in the alternative form

where u: are the frequency dependent stresses as given in E q 6. These relationships again clearly reduce to the general elastic case when the fre- quency dependence is taken to be zero (9). The aij, both real and complex, are the reduced stiffnesses for a composite with arbitrary fiber orientation.

The frequency dependent force and moment re- sultants are now defined as

h

- 2

= 1- (a,, uy ,7 , , ) eiwt dz

2 = 1 _ - h(uxX'uy .7 ,y )e iwrzdz

2

where h is the laminate thickness. These reduce to the analogous elastic case by letting w equal zero. The substitution of Eq 20 into Eq 21 yields for the time varying or complex normal force in the x direction

h

which reduces to the classical force resultant for the frequency independent case.

The integrand in Eq 23 can be distributed over the six resultant terms. In addition, the strains and curvatures can be taken outside the integral since they are not functions of position z . Further simpli- fications can be made by considering each of the integrals separately. The first integral in Eq 23 is given as

h

The reduced stiffnesses are material properties that vary from layer to layer but are constant within any given layer. Since the reduced stiffnesses are piece- wise constant, the integral can be expanded through the thickness to give

h

Since each of the a:j are only functions of frequency within the integral, they can be taken out of the integration. The integration then becomes simply the thickness of the particular layer of the material.

182 POLYMER COMPOSITES, JUNE 1992, Vol. 13, No. 3

Analytical Model f o r Prediction of the Damping Loss Factor

Equation 25 can be written as a finite summation, given as

h

2

In the classical elastic case, this is typically denoted as A l l . As such, this term will be denoted as AT,.

The same procedure can be applied to the other terms of Eq 23 with similar reductions being made. The mathematical manipulations performed do not violate any fundamental principles. With the utiliza- tion of Eq 8, it can be shown that when there is no frequency dependence, the e j ' s reduce to the clas- sical elastic case (9).

Continuing this process for the moment resul- tants, the frequency dependent complex general laminated plate theory reduced stiffness matrix can be determined. Since the development is straightfor- ward, involving only simple mathematical manipu- lations, all that will be given here is the summary of the results in contracted notation.

n= 1

(27)

In matrix form, the above frequency dependent or complex reduced elastic constants are used to relate the in-plane stresses to strains. The specific relation- ship is given as

Eq 3, the relationship of the Qij's in terms of elastic constants can be written as

where the * denotes a complex value. Equations 31 through 34 will reduce to the classical elastic case when there is no frequency dependence.

In Eqs 31 through 34, all of the elastic constants can be readily determined through various experi- mental procedures. The real part of the complex moduli is determined using standardized ASTM or SACMA type tests. The complex part of the elastic moduli can be determined using the procedure given by Crane (9). with the exception of the complex Poisson's ratios. Since there have been no experi- mental techniques identified to determine the com- plex Poisson's ratios, two assumptions will be made concerning them. First, it is assumed that the Pois- son's ratios are independent of frequency. Second, it is assumed that they are real. The consequences of these assumptions have been shown to result in errors that are less than 5% (9, 10).

These stiffness terms are the basic building blocks for the analysis of the frequency response of a gen- eral laminated plate, Using this information devel- oped for the unidirectional material characteristics,

the response of a general laminated composite mate- rial can be determined using conventional tensor transformations. The transformed reduced stiff- nesses can be determined as

The AT,, B;",, and D: terms are functions of the Q , terms. The QtJ terms are functions of the elastic constants. This can be seen by using E q s 9 through 1 1 with a uniaxial composite laminate. Using the relationship given in Eq 3 that the loss factor 9 is the ratio of the imaginary to real part of the elastic moduli, and the development of Hashin (12) given in + n 4 ~ ; , ( f ) (35)

POLYMER COMPOSITES, JUNE 1992, Vol. 13, No. 3 i a3

Roger M. Crane and John W. Gillespie, J r .

where m is the cosine of 0 and n is the sine of 9, and 0 is the angle the fibers make to the principal loading direction of the laminate. For the general laminate, the extensional, coupling, and bending stiffness matrices are determined from the G;", terms as was given in Eqs 27 through 29.

From the reduced stiffness matrix, the loss factor of a general laminated plate can be determined. This is readily done by utilizing the relationship given in Eq 3. The relationship states that the loss factor is the ratio of the imaginary to the real part of the complex moduli. Other investigators have utilized a similar development. In some cases, however, the frequency dependent loss modulus is not accounted for. Another assumption that is made in their efforts is in the determination of the laminated composite loss factors. Their development indicates that the loss factor associated with the complex laminate extensional stiffness in tension, for example, is given as

where A';, i s the imaginary part of AT, and A;, is the real part of AT, (19). In the traditional definition of the extensional loss factor in tension, the value of q x is given as the ratio of the imaginary to real part of the complex elastic moduli. The effective moduli for a composite can be determined from the ABD inverse matrix. The inverse ABD matrix takes into account the stress couplings that may occur from the various orientations of the fibers used in the laminate. A s such, the effective moduli is given as

Using this formulation of the effective extensional moduli of the composite, the material effective loss factor is given as

A" ;I1

9 x = -7 11

(43)

It should be noted that this is different than the formulation given in Eq 41. In addition, it should be noted that the values for q x are positive, since the complex matrix inversion results in a negative value for the ratio of A";,' to A';:. Similarly, the trans- verse modulus is again given in terms of the inverse ABD matrix as

Using the relationship for the loss factor from E q 3, the transverse loss factor is then given as

In a similar manner, the other components of the inverse ABD matrix can be used to determine the loss factor in shear, q,, of the laminate.

In the typical cantilever beam testing of a general laminated composite sample, the effective bending stiffness will govern the beam motion. For the 0" and 90" composite orientations, the stiffness of the beam in bending is equal to the modulus of the material in that direction, or can be described as E x . However, in the case of the general laminated beam, the beam motion will be governed by the effective bending stiffness of the specific construction. It has been shown analytically and experimentally that the effective bending stiffness of a general laminated beam is given as (20)

where h is the beam thickness. Following the same lines of reasoning as given above, in complex nota- tion, the complex effective bending stiffness is given as

(47)

With the effective bending stiffness determined, the loss factor as a function of frequency can be deter- mined as the ratio of the complex to real parts of the value, or

where

For a general laminated beam that incorporates multiple fiber orientations and/or material systems, then, the loss factor can be analytically determined

I a4 POLYMER COMPOSITES, JUNE 1992, Vol. 13, No. 3

Analytical Model for Prediction of t h e Damping Loss Factor

in the following manner. First, the material complex elastic moduli is determined using the test proce- dure described by Crane (9) on cantilever beam specimens with fiber orientation of 0". 90" and 45" over the frequency range of interest. The complex reduced stiffnesses, given by the ABD matrix, can then be determined as was presented above. The longitudinal, transverse, or bending loss factor can then be determined using Eqs 43, 45, and 48, re- spectively.

Although the use of the elastic viscoelastic corre- spondence principle has been proposed for use with composites to predict the loss factor, the explicit incorporation of the frequency dependent damping loss factor has not appeared in the literature. The most probable reason for this is twofold. First, there does not appear to have been a complete loss factor characterization performed on any composite sys- tem, as was evident from surveys of the literature (5, 2 1-23). Typically, the investigators reported a loss factor at a specific frequency. The frequency of test was dependent on the specific geometry and mate- rial used. To carry out a complete characterization is both time-consuming and expensive. Second, the mathematics of complex matrix manipulation is quite difficult and is not easily done by hand. Matrix manipulation using computers has been feasible, using real variables. However, the incorporation of complex terms into the matrix manipulations has until recently been difficult, if not impossible. With the advent of symbolic manipulators such as MathematicaTM, this type of manipulation can now be readily performed.

Parametric Studies of the Flexural Damping Loss Factor of S-2 Glass/3501-6

Some general comments can be made about the information that can now be generated using the above analytical model. The complete characteriza- tion of the damping loss factor of a composite as a function of frequency is difficult to obtain. Typi- cally, the loss factor that results from a specific beam configuration, or at a specific frequency only, is reported in the literature. As such, the analytical determination of the loss factor of a general lami- nated composite over a given frequency range is rarely reported. This model has the utility of being able to determine the effect of stress couplings on the loss factor. It has been proposed by some inves- tigators that the difference in loss factor as a func- tion of fiber orientation between the off-axis and angle-ply laminates is due to the stress coupling terms (24. 25). These stress couplings should then lend themselves to loss factors that are higher than that achievable in pure shear. For an angle-ply com- posite, the material is balanced and symmetric, which results in minimal values of D,, and D,,, i.e., minimal stress coupling effects. For the off-axis configuration, however, since the material is unbal- anced, significant stress couplings are present.

Numerous investigators have experimentally in-

vestigated the effect of fiber orientation on the loss factor of composites. Adams and Bacon (2) showed that the off-axis configuration has a peak in loss factor at an orientation of approximately 30", whereas the angle-ply configuration showed a loss factor maximum at a fiber orientation of approxi- mately +45", as seen in Figs. 2 and 3. Calculating the magnitude of the coupling terms D,, and DZ6, a

200 100

150 75 b 0 h

3 100 8 50

z:

i 50 2 25

I

W

0 0

I I I I I I I I I

I 10 20 30 40 50 60 70 80 90

e (DEGREE)

Fig. 2. Variation of flexural modulus, E,. and damping loss factor, 7,. of unidirectional high tensile strength graphite fibers embedded in D X 2 1 0 epoxy resin with fiber uolume fraction of 50%. as a function of angle that the fibers make with the longitudinal axis of the beam tested in flexure with a maximum bending moment of 2 lbf-in. (0.226 N-m); 0 E , qf. 1.0 in (25.4 mm) wide specimen, 0 Er. vJ 0.6 in (12.7 mm) wide specimen. Prediction for free flexure ~ E,,,---vff [after Adarns and Bacon (2)].

0 I Ib 2b 3b do 5b do 70 sb 9b ? e (DEGREE)

Fig. 3. Variation of flexural modulus, E,, and damping loss factor. 7,. of cross-ply, + 8 , high tensile strength graphite fibers embedded in D X 2 1 0 epoxy resin with fiber uolumefraction of 50%, as afunction of angle that the fibers make with the longitudinal axis of the beam tested in flexure with a maximum bending moment of I lbf-in. (0.1 13 N-rn); 0 El, 0 v,, 1 .O in. (25.4 mm) wide specimen, 0 E,. 7, 0.5 in (12.7 mni) wide specimen. Prediction for free flexure - E,, ,---qJr [after Adams and Bacon (211.

POLYMER COMPOSITES, J U N E 1992, Vol. 13, No. 3 1 a5

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peak occurs at a fiber orientation of approximately 30". This gives the intuitive indication that the loss factor of a general laminated composite can be greater than the material's shear loss factor, by tak- ing advantage of the flexibility in material design. It should be pointed out the loss factor information presented in Figs. 2 and 3 was not determined at the same frequency for all orientations.

The loss factors in bending of 16 ply angle-ply and off-axis S-2 glass/350 1-6 beams were analytically determined over a frequency range of 50 to 1000 Hz. The fiber orientations used were from 0" to 90" in increments of 15". The analytical model pre- sented above was used to analytically determine the complex inverse ABD matrix for each orientation and at frequencies of 50 Hz and from 100 to 1000 Hz in increments of 100 Hz. The loss factor was then determined using Eq 48. The results from this modeling are presented in Figs. 4 and 5 for the angle-ply and off-axis material, respectively.

Several results are evident from comparing the loss factors as a function of fiber orientation for these two configurations. The major difference in the two configurations is that the magnitude of the bending-twisting coupling for the off-axis configura- tion is approximately a factor of 7 higher than for the angle-ply configuration. The first obvious differ- ence in the two configurations is that the loss factor for a specific orientation over the entire frequency range is greater for the off-axis orientation, with the exception of the 0" and 90" orientations, where the loss factors are identical. Second, the effect of the stress coupling on loss factor is more pronounced for the 15" and 30" orientations than any of the other configurations. At a frequency of 1000 Hz, the 15" off-axis loss factor was 28.5% greater than the angle-ply loss factor, while the 30" off-axis loss fac- tor was 24.8% greater than the angle-ply loss factor.

I - . 'J 1

I ' C 4 .

C I ;m: t I

Fig 4 Flexural loss factor us f r equency f o r angle-ply S-2 G~CSS I3501 -6.

This increase is attributed solely to the stress cou- pling terms. Third, the ranking of the loss factors as a function of material fiber orientation is different in the two configurations, as is evident in the magni- tude of the loss factors at 1000 Hz.

In addition, there are also some similarities that can be pointed out. First, the general shapes of the curves are similar. Second, the maximum value of the loss factor-in both cases this occurs for the 45" orientation-is on the same order of magnitude. Third, in all cases, in the frequency range of 50 to 1000 Hz, the flexural damping loss factor increases as the frequency increases.

Some interesting results can be obtained using the analytical model and the information given above. For example, in the experimental determina- tion of the loss factor for an angle-ply and off-axis S-2 glass/3501-6 specimen, beams with the same dimensions are typically used. For a given orienta- tion, the beam stiffness and therefore the resonant frequency will vary. Numerous investigators have not acknowledged the frequency dependence of the loss factor of composite materials and therefore ig- nore the frequency at which the test is conducted. The information that they present, then, gives the loss factor as a function of orientation at different frequencies.

A s an exercise, the effect of fiber orientation on loss factor will be analytically determined in the same manner in which previous investigators have experimentally determined the loss factor. Specifi- cally, an arbitrary beam length of 15.2 cm (6.0 in) will be assumed. In addition, all beams will be as- sumed to have a thickness of 5.0 mm (0.2 in). To use the analytical model, the effective beam stiff- ness is first determined. This was done using a lamination plate theory routine to determine the ABD inverse matrix. The beam effective bending

186 POLYMER COMPOSITES, JUNE 7992, Yo/. 73, No. 3

Analy t ica l Model for Prediction of the D a m p i n g Loss Factor

stiffness is then determined by substituting the D;: term into Eq 46 for each of the orientations used. It should be noted that the effective beam stiffness of an off-axis beam is different from that of its angle-ply counterpart by virtue of the stress couplings that occur. Using this beam stiffness, the first resonant frequency of each beam is then determined using the following equation:

f i = /-x x Ci w, L4

The loss factor for each orientation at each of the frequencies is then determined using the analytical model. Tables 1 and 2 present the results for the angle-ply and off-axis beams, respectively. These Tables show the variation in resonant frequency that occurs when the beam dimensions are kept constant.

The results of both the off-axis and angle-ply loss factor given in Tables 1 and 2 are presented graphi- cally in Fig. 6. This graph shows trends that are similar to those obtained by other investigators, such as was seen in Figs. 3 and 4. First, there is a difference in the rate of increase in loss factor for the two configurations. The off-axis material shows a more rapid increase in loss factor than the angle-ply configuration. The explanation for this is that addi- tional losses are present by virtue of the stress cou- plings in the off-axis material, whereas the angle-ply configuration has no stress couplings, since it is both balanced and symmetric. At fiber orientations greater than 45". the loss factor of the two configu- rations are within 2%.

Figures 4 and 5 can be used to visually determine the effect of orientation at a constant frequency. What is evident in these Figures is that it is impor-

Table 1. Analytical Determination of Loss Factor vs. Fiber Orientation for Angle-Ply S-2 Glass/3501-6.

Frequency Loss Factor Orientation (Hz) ( x I o - ~ )

0 186.36 61.42 t 1 5 174.17 62.90 t 30 139.71 66.91 ? 45 107.39 68.04 t 60 101.55 73.79 2c 75 106.64 79.02

90 109.19 80.43

Table 2. Analytical Determination of Loss Factor vs. Fiber Orientation for Off-Axis 5-2 Glass/3501-6.

Frequency Loss Factor Orientation (Hz) (x10 4)

0 186.36 61.42 15 155.14 68.05 30 1 19.73 69.55 45 103.93 69.66 60 101.39 72.83 75 105.86 77.85 90 109.19 80.43

6 0 1 . I , I . I . , . I . 0 1 5 3 0 4 5 6 0 7 5

Fiber Orientation 3

Fig. 6. Flexural lossfactor us. fiber orientation for off-axis and angle-ply configurations for a 6.0 in. long, 0.2 in. thick beam.

tant to specify the frequency of interest. The shape of the curves of loss factor vs. fiber orientation at constant frequency will vary, showing maxima for different orientations at different frequencies. This helps to explain the inconsistency in the results of several investigators who report varying orienta- tions at which the maximum in loss factor occur.

It should also be pointed out that the model has the capability of determining the effect of hybridiza- tion on resultant loss factor. This effect has not been considered in the literature but would provide an alternative means of analytically attaining specific damping levels that would currently be achievable only through fiber orientation variations.

ANALYTICAL MODEL VALIDATION

Test Specimens

There is an infinite variety of possible laminated configurations that could be used to validate the analytical model. One material configuration that is often used in structural applications is the quasi- isotropic configuration, [0/90/45/-45],,. Although this configuration results in material properties that are isotropic in the plane of the material, the bend- ing stiffness is a function of the orientation tested. In addition, there is a variation in the magnitude of the stresses or strains through the thickness of the ma- terial when it is subjected to a bending moment. A s such, there should be a variation in the loss factor of this material as a function of the outer ply orientation.

A 16-ply laminate with the configuration [0/90/45/-45] ns was fabricated. After fabrication, the panel was nondestructively inspected using an ul- trasonic C-scan inspection system. In addition, the fiber volume fraction was determined using the ASTM D-2584 procedure. This testing indicated that the panel was free of manufacturing defects and had

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Roger M. Crane and John W. Gillespie, J r .

thc same fiber volume fraction as the glass material that had been previously tested, 63%.

Two sets of specimens were machined from this panel using a diamond-impregnated blade attached to a milling machine. One set had an outer fiber orientation of 90". while the second set had an outer fiber orientation of 45". The two beam configura- tions tested then were [90/0/-45/45],, and [45/-45/90/0],,. In all cases the specimens had a width of 25.4 mm (1.0 in).

Test Procedure

The material was tested using the apparatus and the robust testing methodology described by Crane (9). The material was to be characterized in the frequency range of 50 to 1000 Hz, since this is the range in which the material loss factor for the com- posite was previously determined. This is accom- plished by testing beams of various length. Five specimens of a specific length were used for each of the frequencies tested. Initial beam lengths for each configuration were chosen so that its first resonant frequency was approximately 50 Hz. The beam lengths were then reduced to obtain loss factors at frequencies that were multiples of 100 Hz. Thc min- imum beam length tested was 50.8 mm (2.0 in), to minimize shear and rotary inertia effects on the loss factor. The results of the experimental testing are presented in Table 3.

Several features concerning the loss factor of these two configurations are evident in Table 3. First, for both configurations, there is an increase in loss fac- tor with increasing frequency. This follows the gen- eral trend that was evident in the testing of the unidirectional composite samples. Second. the quasi-isotropic configuration results fall within the range of values previously obtained for the angle-ply configuration. Intuition tells us that this should oc- cur, since the loss factor of the quasi-isotropic con- figuration should be some combination of the loss factors from the orientations used. Finally, the (451-45/90/0],, configuration had a higher loss fac- tor than the [90/0/45/-45],, configuration. Intuition again indicates that this should occur. since the former configuration has the higher damping mate- rial subjected to a higher stress level.

Table 3. Flexural Loss Factor Determination of S-2 Glass/3501-6.

Frequency Loss Factor (Std. Dev.) Configuration (W ( ~ 1 0 - 4,

~

[45/-45/90 / 0 ] Z s 5 3 4 61 52 [6 251 78 0

115 1 208 8 326 8 441 4

98 6 213 8 487 4

76 96 [l 81 65 90 [ 5 0) 6 5 39 [2 31 77 68 [6 01 78 78 (8 01

[90/0 /45/ 45],, 54 7 55 47 [6 51 64 31 [8 6 ) 60 19 [3 61 67 33 [ l 81

Analytical Determination of Flexural Loss Factor

The input required for the analytical model is the material loss factors as a function of frequency, which is utilized to determine the loss modulus, and the material elastic properties, or storage modulus. This loss factor information as well as the experi- mental technique for determining these values has been previously given by Crane (9). The various loss factors are reported below for S-2 glass/3501-6:

(51) v1 = 5.15 x 10-7f+ 5.99 x

q2 = 8.37 x h(f) + 4.07 X (52)

q12 = 25.5 x 1n(f) - 50.37 x (53)

Using these loss factors along with the values of the storage modulus, which were determined using con- ventional testing techniques, the material complex moduli for the S-2 glass/3501-6 are given as

Er = 57.85( 1 + i(5.15 x 10-7f+ 59.9 x 10-4))GPa

EZ = 19.86

x (1 + i(8.37 x 10-41nf+ 40.7 x 10-4))GPa

GT2 = 6.1( 1 + i (25.5 x 10-41nf- 50.4 x 10-4))GPa

v12 = 0.264 v Z 1 = 0.0913

The analytical model was written into a computer program to determine the complex ABD matrix. The program that was utilized was called Mathe- maticaTM, a symbolic manipulator program. The program was written so that various frequencies could be used as input. The loss factor as a function of frequency was determined for three different outer-ply orientations of the quasi-isotropic configu- ration. These orientations were (45/-45/90/0],,, [90/0/45/-451 ,,, and [0/90/45/-451 ,,. These analyti- cally determined loss factors are presented graphi- cally in Fig. 7.

The loss factors shown in Fig. 7 show trends similar to the material loss factors. First, the loss factor increases with increasing frequency. Second, the specific stacking sequence affects the loss factor of the laminate. For the quasi-isotropic laminate configuration, the orientations can be given in order of increasing loss factor as [0/90/45/-45],,, [90/0/ -45/45],,, and [45/-45/90/0],,. This shows that to achieve the maximum damping loss factor, the stacking sequence used should have the orientation with the highest loss factor located near the surface of the laminate. The analytically determined loss factors for the quasi-isotropic configuration fall in the range of loss factors previously determined for the angle-ply configuration.

Comparison of the Analytically Determined Loss Factor With the Experimental Results

For comparison purposes, the experimental and analytically determined loss factors will be pre-

1 aa POLYMER COMPOSITES, JUNE 1992, Vol. 13, No. 3

Analy t ica l Model for Prediction of the D a m p i n g Loss Factor

sented graphically. Figure 8 shows the comparison for the [90/0/-45/45],, configuration. The two curves shown in Fig. 8 are curve fits to the analyti- cal and experimental loss factor values. The analyti- cally determined loss factor is shown to follow the same trends as the experimental values. In general, the analytical values are on the upper side of the scatter or as a maximum are on the order of 10% greater than the experimental values. This may be due to the manner in which the values of the experi- mental loss factor are determined. The linear ex- trapolation to zero displacement may not be appro- priate. Instead, a nonlinear fit, such as a logarithmic fit, may be more appropriate.

Fig. 7. Flexural loss factor us. frequency for quasi- isotropic S-2 glass/3501-6 beams with varying outer ply orientations.

, , --F%v,l: , ill

o 'YYTj:)? 4 i ; O Y E ' Y Z m ' o o F F E . ~ ' . I C ? ~ C ) ' ( H z i

Fig. 8. Analytical us. experimentally determinedflexural loss factor for (90/0/-45/45),, laminate as a function of frequency.

Figure 9 shows the comparison of the experimen- tal and analytically determined loss factors for the [45/-45/90/0],, configuration. For this configura- tion the analytical model provides an accurate de- scription of the experimentally determined values of loss factor. The analytically determined loss factor falls within the scatter of the experimental values in the frequency range in which the experimental val- ues were determined.

In general, the analytical model based on the elas- tic viscoelastic correspondence principle appears to provide an adequate prediction of the damping loss factor of a general laminated composite configura- tion. Trends that are shown to occur experimentally in the material are shown to occur using the analyti- cal model. The analytical model has been shown to provide a loss factor that is within 15% of the exper- imentally determined values in the frequency range of 50 to 500 Hz.

CONCLUSIONS

An analytical model is developed based on the elastic-viscoelastic correspondence principle. In this research. the frequency dependence of the damping loss factor is included, thereby extending the model as it is currently used in the literature. The incorpo- ration of this frequency dependence has been shown to analytically explain the discrepancies in the liter- ature on the loss factor as a function of fiber orienta- tion. Different investigators have obtained conflict- ing experimental results for the composite laminate fiber orientation that results in the maximum loss factor. Using the analytical model developed in this research, all of their results can be shown to be valid. This occurs because for a given investigation, the test specimen dimensions are typically kept con- stant. When the fiber orientation of the test speci-

) & T T T T 7- m r r , - r 7 r - - 7 7

J I ",> 5 iL I 1 1 1 1 f - -

Ftg 9 Analytical u s expertmentally deterrnmedflexural loss factor for (45/-45/90/0),, laminate as afunctton of frequency

POLYMER COMPOSITES, JUNE 1992, Yo/. 13, No. 3 189

Roger M . Crane a n d J o h n W. Gillespie, J r

men is changed, the beam stiffness also changes. This results in a change in the first resonant fre- quency of the beam. The results of the model show that for a given frequency, the fiber orientation that has the highest loss factor is different.

The model has shown that the stress couplings that occur in unbalanced laminated constructions have a pronounced effect on the loss factor. The loss factor was shown to increase proportionally to the magnitude of the bending twisting coupling terms. The loss factor therefore showed the most pro- nounced increase for orientations of 15“ and 30“.

The loss factors determined using the analytical model have shown good agreement to the experi- mental results. This model can therefore be used as an analytical tool to determine the material loss factor of composite materials at any frequency of interest. In addition, the loss factor in any direction can also be determined. This will enable the bound- ing of the structural loss factor to be obtained. This would be accomplished by using the following pro- cedure. First, the maximum and minimum values of the material !oss factor in the frequency range of interest are determined for the specific material con- figuration. These minimum and maximum values would then be used as input to one of the various finite element routines, since most finite element routines allow only a single value of damping loss factor as input. The finite element model would then determine the strain energy dissipated and stored in the structure. The loss factor is then determined using the ratio of the energy dissipated to the stored energy. In addition, the structural loss factor at spe- cific frequencies can also be determined by deter- mining the specific material loss factor at a given frequency and using this as input to the finite ele- ment routine. This capability, to the authors’ knowl- edge, has not been previously available to the struc- tural designer.

The analytical model, as well as the experimental investigation, indicates that to achieve the highest flexural loss factor for a given set of laminae orienta- tions, the orientations that have the highest loss factor should be positioned near the surfaces of the specimen. This is demonstrated in the analytical determination of the loss factor for the quasi- isotropic configurations shown in Figs. 5 and 6.

ACKNOWLEDGMENTS

The authors would like to acknowledge the finan- cial and administrative support of James Kelly of ONT, Joseph Crisci and Dr. Bruce Douglas of DTRC, and Dr. A. K. Vasudaven of ONR. The authors would also like to thank Dr. Thomas Juska, Harry Tele- gadas, Sandro D’Agaro, Darin Castro, Dr. Vincent Castelli, and Dr. Eugene Fischer for their technical support in this effort.

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