-. 

. 1 

NASA Technical Memorandum 102107 

. Unified Micromechanics of Damping 

for Unidirectional Fiber 

Reinforced Composites 

I1 i 

I1 

11 D.A. Saravanos and C.C. Chamis 

Cleveland, Ohio 

n -LxwlkResearch Center 

i August 1989 

- 

(H&SA-TP!-?02707) UIIPIBD HICBOHECE1BICS OF N89-269 19 

DALnPfNG FOB UWIDIBECTIONAL FIBEB REIYFCECED 

COCIPOSXTES (NASA., Lewis Research Center) 28 P CSCL 11D Unclas 

G3/24 0224114 
UNIFIED MICROMECHANICS OF DAMPING 

FOR UNIDIRECTIONAL AND OFF-AXIS FIBER COMPOSITES 

D. A. Saravanos*, and C. C. Cham's 

wational Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, OH 44135 

ABSTRACT 

-4n integrated m'cromechanics methodology for the prediction of damping capacity 

in fiber-reinforced polymer matrix unidirectional composites has been developed. Explicit 

micromechanics equations based on hysteretic damping are presented relating the on-axis 

damping capacities to the fiber and matrix properties and volume fraction. The damping capacities of unidirectional composites subjected to off-axis loading are synthesized 

from on-axis damping values. Predicted dues correlate satisfactorily with experimental 

measurements. The hygro-thermal effect on the damping performance of unidirectional 

composites due to temperature and moisture variations is also modeled. The damping 

contributions from interfacial friction between broken fibers and matrix are incorporated. 

Finally, the temperature rise in continuously vibrating composite plies is estimated. Application examples illustrate the significance of various parameters on the damping performance of unidirectional and off-axis fiber reinforced composites. 

~ 

* Xational Research Council - NASA Research Associate 
1. INTRODUCTION 

The significance of material damping to the dynamic performance of structures is 

broadly recognized. Passive damping has been proved to be a significant design parameter for vibration control, fatigue endurance: and impact resistance. It is well known that 

fiber/polymer-matrix composites may provide one or two orders of magnitude higher material damping than common metals. in addition toeother superior elastic properties, such 

as high specific moduli and specific strength. An additional appealing design factor is the 

possibilitj- to tailor the composite damping, together with other mechanical properties, by 

controlling the anisotropy of the composite material.. This combination of high damping 

and advanced mechanical characteristics makes fiber/polymer matrix composites materials 

ideal to a range of high-performance light-weight structures where passive vibration control 

is critical. such as space and aerospace structures, engine blades, high-speed mechanisms, 

etc. 

In order to realize significant design benefits from the inherent damping capacity of 

composite materials. suitable mechanics theories should be formulated which will analytically correlate the damping of the composite structure to the properties of the basic 

constituents, ply stacking sequence, hygro-thermal conditions, existing damage, and structural configuration. Research at Lewis Center has led to the development of an integrated 

micromechanics theory for composite materials including the hygro-thermal (moisture, 

temperature) environmental effects [l]. The research has been being currently extended 

to incorporate prediction of damping. The current paper presents an int.egrated micromechanics theory for the on-ais and off-axis damping capacity of unidirectional composites. 

Much but limited in scope research has been reported on the micromechanics of composite damping. Hashin [2] has provided analytical expressions for the on-axis ply damping, 

and Shultz and Tsai i3,4] have conducted experimental studies on the damping capacity 

of composite plies. -\dams and co-workers [5-81, Bert and co-workers [9-10], and Gibson 

and co-workers [ll-131 have performed analytical and experimental investigations on the 

micromechanics of composite damping. However. many reported theories on the micromeb 
chanics of composite damping have been mainly restricted to ply damping associated with 

the longitudinal normal stress clI1 and/or the in-plane shear stress 4ll2. The current work 

presents a complete micromechanics theory for all 6 damping coefficients of a composite ply associated with the following local stresses: (1) longitudinal. (2) transverse, ( 3) 

through the thickness normal stress, (4) in-plane shehr, (5) through the thickness (1-3) 

shear, and (6) through the thickness (2-3) shear. In contrast to many previous works 

which include only the hysteretic contribution of matrix, these six composite damping capacities are synthesized from the hysteretic damping of matrix and fibers. and interface 

friction. Isotropic dissipative properties are assumed for the matrix and anisotropic (but 

transversely isotropic) properties for the fibers. Temperature and moisture variations are 

known to have a significant impact on the properties of polymer matrix composites, hence, 

the influence of these two important factors on composite damping is modeled. In addition 

to hysteretic damping. the present method includes the contribution of interfacial friction 

damping due to broken fibers to the overall composite ply damping. 

Off-axis specific damping capacities (SDC's). ie. the SDC's of a ply loaded at an 

off-axis angle. are also synthesized from the on-axis SDC's. The transformation for this 

purpose has been developed. Further, as a result of strain energy dissipation. temperature rise has been observed within vibrating plies [14]. Temperature variations affect the 

performance of composite materials: therefore, knowledge of the developed temperature 

profiles in vibrating coniposite plies is desirable. Foi this reason, the temperature rise in 

composite plies subjected to cyclic vibration is also predicted. 

In the rest of the paper. the micromechanics for ply damping properties is described. 

application studies are presented. and most important conclusions are summarized. 

NOMENCLATURE 

-4d Area of debonded fiber. 

df : Fiber diameter. 

E: Kormal modulus. 

/E,] : Off-axis ply stiffness matrix. 

3 
q: 

T: 

T, : 

T,, : 

t: 

I' : 

€: 

v: 

0: 

11' : 

Ld: 

Subscripts 

c: 

D: 

f: 

On-axis ply stiffness matrix. 

Shear modulus. 

Volume ratio. 

Thermal conductivity. 

hloisture. 

Generated heat rate. 

Number of broken fibers through the ply thickness. 

Stress transformation matrix. 

Temper at ure. 

Temperatlure of composite at the center. 

Temperature of composite at the surface. 

Ply thickness. 

Stored strain energy per cycle. 

Stored specific strain energy per cycle. 

l'olume. 

Effective fiber length. 

Dissipated strain energy per cycle. 

Dissipated specific strain energy per cycle. 

Engineering strain. 

Friction coefficient between fiber and matrix 

Poisson's rat io. 

Stress. 

Specific damping capacity. 

Forcing angular velocity. 

Ply (off-asis). 

Damping. 

Fiber. 

4 
fr : 

yd : 

gw : 

I: 

m: 

M: 

0: 

rad : 

Direction 

11 : 

.-).-) . &_ . 

33 : 

12 : 

23 : 

13 : 

xz : 

YY : 

ss : 

n: 

s: 

Fiberlmatrix friction. 

Glass transition, dry. 

Glass transition, wet. 

Ply (on-axis). 

Matrix. 

Mechanical. 

Reference. 

Radial. 

Xormai longit cldinal. 

Xornict~ in-plane transverse. 

Piorma1 out-of-plane. 

Shear in-plane. 

Shear out-of-plane. 

Shear out-of-plane. 

In-plane normal, off-axis, x-direction. 

In-plane normal, off-axis, y-direction. 

In-plane shear. off-axis. 

Normal. out -of-plane. z-direction. 

Xormal. 

Shear. 

2. COMPOSITE DAMPING 

This section presents the damping micromechanics theory for continuous fiber composite material systems. In particular, the theory is focused on the damping of unidirectional 

composites subjected to : (1) on-axis cyclic loading, and (2) off-axis cyclic loading (ie. 

loading at an angle with respect to the fibers). 'The theory also addresses the impact of 

hygro-thermal effects and damage on the damping of composites. 
The specific damping capacity (SDC) of a material is defined as the ratio of dissipated 

strain energy over the stored strain energy during one cycle of vibration. 

In the remaining section, the specific damping capacity (SDC) is consistently used: and all 

references to damping will imply SDC unless otherwise noted. 

2.1. On--4xis Ply Damping 

Fiber composites are nonhomogeneous materials, therefore, candidate sources of composite damping would be: (1) hysteretic damping in the polymer matrix, (2) hysteretic 

damping in the fibers, and (3) friction damping at the fiber-matrix interface as a result 

of bonding imperfections, broken fibers, or debonding. Most available theories on the micromechanics of composite damping have mainly provided closed-form solutions for the ply 

damping associated with the longitudinal normal stress nil1 and the in-plane shear stress 

glI3. The current work presents simple explicit expressions of ply SDC's associated with all 

6 engineering stresses illustrated in Figure la, namely, the normal longitudinal stress glll , 

normal transverse stress 0~22, normal through- the-thickness stress 6133: in-plane shear 

stress nl12. and out-of-plane shear stresses u233 and 6113. 

For fiberlpolymer matrix composites. the damping capacity of the matrix is usually 

significantly higher than the damping capacity of the fibers, for this reason: many micromechanics theories 12,571 have neglected the contribution of fiber damping. This assqmption, 

may be satisfactory for the transverse normal or shear specific damping capacities, but is 

certainly less accurate for the longitudinal normal damping. Moreover, aramid fibers typically exhibit longitudinal SDC about 1.8%, which is not negligible compared to reported 

SDC values for epoxies. In order to remedy this discrepancy, the presented micromechanics 

theory includes the contribution of the fibers into the damping of the compcjsite. 

Another drawback of most micromechanics theories is their restriction to fibers and 

In general, this hypothesis is 

The present theory remedies also this 

matrix with isotropic elastic and dissipative properties. 

incorrect, especially for carbon/graphite fibers. 

6 
restriction by assuming linear anisotropic viscoelastic properties for the fibers and linear 

isotropic viscoelastic properties for the polymer matrix. Experimental work on glass and 

carbon composites :6] has shown low dependence of composite damping to the strain 

values, indicating that the latter assumption would be a reasonable approximation for 

most engineering applications. The present micromechanics theory assumes square packing 

arrays of fibers as this type of packing represents the average value of randomly distributed 

fibers. Severtheless: if necessary. similar micromechanics theories can be developed for 

other packing patterns. 

Longitudinal Normal Damping. The square array of a ply consists of one fiber 

and the surrounding matrix as shown in Figure la. -4ssuming that a uniform cyclic 

longitudinal normal stress of amplitude glll is applied to the ply, then the dissipated 

strain energj' within the representative fiberlmatrix segment would be: 

Because damping is assumed to be independent, of stress and strain. 

The longitudinal SDC' c7ll1 is defined as: 

Further. the stored strain energy during one vibration cycle will be 

The last equation represents the rule of mixtures in strain energy form. It should be 

recognized that based on the assumed stress/strain uniformity: equation (4) leads to the 

broadly accept,ed rule of inixtures for the longitudinal modulus: 
Combination of equations (2), (3), and (4) will provide the following expression for lonqit,udinal SDC, 

Transverse Xormal Damping. If a cyclic transverse normal stress of amplitude 

oI2? is applied to the constitutive fiber/matrix section of Figure la. then the dissipated 

and stored strain energy during a vibration cycle may be similarly described by: 

and 

The stress distribution in the matrix and fiber is not uniform mainly due to the curvature of the fiber. however. uniform stress distributions may be assumed within the fiber 

arid matrix provided that the shape and the volume of the fiber are properly corrected. 

This approach has been successfully utilized by Chamis :l5] in the development of micrornechanics equations for the transverse and shear elastic moduli Elll. El22: El33. Grl2. 

Gl13. and G123 which are also used herein. The validity of the assumption has been verified 

by three-dimensional finite element analysis [16:. Combination of equation5 (7) and (8) 

results in the following equation for the transverse normal SDC of the composite: 

w- h er e , 

Because of the transverse isotropy of the fibers, and consequently of the composite 

material: the transverse damping capacit.y of the ply along the material 3-axis would be. 
In-Plane Shear Damping. Based on the same assumptions and following a similar 

procedure as in case of transverse damping, the in-plane shear damping capacity is given 

by : 

Through-the-Thickness Sheai. Damping. Due to the transverse isotropy of the 

composite ply: the interlaminar SDC ~‘1; is equal to the in-plane SDC of the ply, 

i2n analogous approach also provides the out-of-plane shear SDC Gp3, 

where, 

(1.5) 

2.2. Hygro-Thermal Effect 

I’ariations in the temperature and moisture content of a polymer matrix composite will 

primarily affect the properties of the plastic mat,rix. Previous studies [l’i] have shown t,hat 

the hygro-thermal impact on most mechanical properties of the matrix can be expressed 

as 

Based on the fact that for t,emperature ranges below the glass transition the matrix damping is increasing with temperat,ure, an inversed form of equation (18) is proposed for the 

hygro-thermal effect on the matrix damping: 

9 
The wet, glass transition temperature is, 

The exponent q can be easily correlated to experimental data of each individual polymer matris. In this study, the value q=O.5 is assumed. The hygro-thermal effect on the 

ply properties is related to the effect on the matrix via the micromechanics equations of 

Section 2.1. The hygro-thermal effects are expected to dominate the transverse and shear 

damping properties of a ply, as they are predominantly controlled by the matrix, The 

results presented in Section 3 illustrate this trend. 

2.3. Off-Axis Ply Damping 

During off-axis cyclic loading (Figure lb), more than one of the on-axis SDC's will 

contribute to the overall ply damping. Based on this observation, one may anticipate that 

the off-axis ply damping will be related to both on-axis damping properties and the orientation of the fibers. -4 transformation law has been derived to describe this relationship 

based on the fact that strain energy loss is invariant to stress/strain transformations. 

By definition. the hysteretic strain energy loss per unit volume in the structural (offaxis j coordinate system is. 

while in the material (on-axis) coordinate system it will be: 

(22) 

Transforming the stress and strain from the material coordinate syst.em to the structural 

coordinate system, t,he specific strain energy loss becomes, 

(23) 

The stress t.ransformation matrix [Ru] is given in Appendix. Equating the right hand sides 

of equations (21) and (23) the following ply damping transformation law results, 

(24) 

10 
In the previous expression the on-axis damping matrix is diagonal: 

while the off-axis ply damping is fully populated. 

(2.5 J 

(26) 

The non-diagonal terms indicate coupling between the axial and shear stresses. -4pparently. 

off-axis loading will affect the overall dan;ping capacity of a ply in two distinct ways: (1) 

by altering the values of the diagonal terms: which is equivalent to altering the dissipative 

capability of the ply directly associated to normal and shear strain. and (2) by- inducing 

and altering non-diagonal terms, which control the amount of strain energy dissipated by 

coupled deformation modes. 

2.4. Interfacial Fyiction Damping 

Relative motion between broken debonded fibers and the surrounding matrix is the 

source for additional frictional damping in a composite. During one vibration cycle: friction 

will be developed between the debonded fiber and matrix when the radial stress Urad is 

compressive. The dissipated energy per cycle from a debonded fiber will be: 

where. p is the friction coefficient between fiber and matrix? urad the average radial stress 

around the fiber, and r the distance from the point where debonding starts. The area of 

where 6: is the effective fiber length [18j, 

(29) 

11 
For Kfb broken fibers through the thickness, the specific energy loss per cycle will be: 

Integration of equation (27) over the surface of the debonded fiber. and combination of 

equations (27-30) will lead to the following equsticii for the specific strain energy loss per 

cycle due to friction: 

For sinusoidal stressing equation (31) becomes, 

Taking into consideration that friction exists only when Q,,d < 0, equation (32) reduces 

to: 

The total SDC of a ply may be ensembled from the frictional loss and the hysteretic loss 

described in equations 

equivalent longitudinal 

(21-22). When only longitudinal stress is applied on the ply, the 

friction damping will be, 

Based on a previous micromechanics analysis reported by Chamis [19]? the radial stress 

is related to the ply microstresses. As shown in ref. 1191, ply microstresses are directly 

related to mechanical ply stresses, temperature variations, and moisture variations: therefore, the current analysis for composite friction damping readily incorporates the effect of 

mechanical loads, residual thermal microstresses: and hygral microstresses. 

2.5. Temperature Rise 

12 
-4s a result of damping, a portion of stored strain energy is lost and is converted to 

heat during the vibration of the composite. The generated heat is transferred to the surrounding environment and temperature gradients are developed in the material. During 

continuous dynamic excitation, significant temperature rise may occur within the composite. As stated in the previous section, the mechanical and thermal properties of polymer 

matrix composites are temperature dependent. hence. as the temperature increases. the 

mechanical and thermal properties of the composite. as well as. the generated heat continuouslj. change. In case of a steady-state excitation. the composite will eventually reacl, 

thermal and vibrational stability. The actual temperature rise depends on the SDC of the 

composite material. the conductivities of the composite material and surrounding media. 

the laminate configuration. and the induced excitation power. Fourier’s law may be utilized to estimate the steady-state temperature rise. For plies of infinite dimensions being 

subjected to steady-state escitation; 

For cyclic excitation. the heat generation rate is described by equation (32) for unidirectional composites or by equation (31 jfor off-axis composites. Solutions are provided for 

the mid-surface temperature T, assurning equal temperatures T,, at both surfaces of the 

coinposi te. 

On - asi s 1o;ttli rig: 

(36j 

Off-axis loading: 

3. DISCUSSION OF RESULTS 

13 
Results are presented in this section for various composite systems. Whenever possible. the predicted values are compared to available experimental data. It is pointed out. 

however, that limited experimental data has been reported correlating constituent damping 

properties to ply damping (on-axis or off-axis). In the following applications experimental 

measurements performed by -4dams and co-workers [3-8,201 are utilized. 

3.1. On-Axis Ply Damping 

Figures 2, 3. 4 and 5 demonstrate the varidtion of longitudinal SDC $ill: in-plane 

shear SDC u'll?. transverse SDC Gz22. and out-of-plane SDC <~~23 with respect to the fiber 

volume ratio. The different curves represent different fiber/matrix combinations. The pljSDC's have been normalized by the normal or shear SDC of the matrix. The SDC's for 

the intermediate modulus high strength (IMHS) matrix and intermediate modulus low 

strength (IXILS) matrix are shown in Table 1. and they represent values for DE10 epoxy 

and polyester respectively. provided in Refs. [5.8]. Non-reported fiber/matrix SDC's were 

backcalculated from on-axis composite damping data reported in reference ['i]. 

-1s seen in Figure 2. the agreement between analytical predictions and experimental 

measurements for longitudinal SDC is good. For E-Glass/IMLS the agreement is escellent. The predicted values for the high modulus surface treated graphite/polyester (HMS/IMLS) underestimate the esperimental ones, but an uncertainty exists about the actual 

vdlues of the fiber properties. In Figure 3. the analytical predictions for shear in-plane 

SDC overpredict the measured values for the glass fiber, but they are in acceptable agreement for the graphite fiber. The same comments apply for the correlation of transverse 

SDC with experimental data in Figure 4. Overall? the present micromechanics theory has 

yielded acceptable damping predictions for fiber volume ratios within the range of interest 

to most engineering applications. The study has also revealed the need for systematic 

collection of esperimental data regarding on-axis composite damping. -41~0, the present 

micromechanics is expected to yield better underst anding regarding the design of suitable 

experiments for measuring composite damping. 

The longitudinal damping is fiber controlled, while the transverse and shear damping 

14 
are matrix controlled. The fibers have also significant impact on the transverse and shear 

ply damping, as they control the amount of strain energy stored in the matrix. The fiber 

effect is clearly demonstrated in Figures 2,3:4 and 5. In general, fibers with high moduli 

will decrease the damping of the ply, as they will reduce the amount of strain energy stored 

in the matrix and hence the overall dissipated strain energy. 

3.2. Hygro-Thermal Effect on Ply Damping 

The hygro-thermal effect on the longitudinal, in-plane shear, and transverse SDC's is 

shown in Figures 6: 7, and 8 respectively for an HM-S/IMHS composite system. The various curves represent different temperature/moisture increments above the reference values. 

The matrix glass transition temperature is 420 degrees F, the reference temperature 70 

degrees F, and the reference moisture content 0%. The longitudinal SDC has shown insensitivity to temperature and moisture variation, because it is mainly a fiber controlled 

property. However, the transverse and shear SDC's indicate dramatic dependence to temperature or moisture variations. The combined effect of temperature and moisture is also 

illustrated in the same Figures. The out-of-plane shear SDC 1J123 has shown analogous 

hygro-thermal dependence. 

3.3. Off-Axis Ply Damping 

The predicted values for off-asis SDC's are shown in Figure 9 for a 50% fiber volume 

fraction HM-S,/DX210 graphite/epoxy ply. For a coherent assessment on the accuracy of 

the transformation law, the on-axis SDC's were assumed equal to those measured by Xi 

and Adams [20]. These damping measurements were conducted on cantilever off-axis composite plate beams harmonically excited at their first bending vibration mode. Thus. the 

measured values represent the modal damping at individual vibration modes and include 

the effect of coupling terms. Therefore, it is appropriate to compare the measured values 

with the equivalent axial ply SDC 7i'z,azta[ defined as the combined damping capacity of 

plies subjected to a single cyclic normal stress ocZz. The axial damping includes the coupling effect and its calculation is straight forward by utilizing equation (31) and Hook's 

law (equation (A3) in Appendix) for the ply. The resultant axial damping (i~z~azza~) values 
17s. the fiber angle B are plotted in Figure 9 and are in excellent agreement with the experimental results. The remaining error has been attributed to the fact that global modal 

damping is compared to predicted local values. Overall, the agreement of the damping 

transformation is very good. 

The diagonal SDC terms in Equation (26) yczz, @c.cyy, and d~~~~ are also plotted as 

functions of the fiber angle 8 in Figure 9. The normal SDC is minimum at 0 degrees 

and maximum near 60 degrees, while the shear SDC is symmetric beinq maximum at 0 

and 90 degrees and minimum at 45 degrees. Figure 9 also vividly demonstrates the effect 

of coupling damping terms on the axial damping capacity of an off-axis ply. Coupling 

damping terms are the predominant source of axial damping for ply angles between 5 and 

30 degrees and their contribution progressively decreases for ply angles greater than 30 

degrees. 

3.4. Friction Effect on Ply Damping 

The interfacial friction damping contribution due to broken fibers is shown in Figure 

10. Figure 10 illustrates the increase in longitudinal SDC PS. the fiber volume fraction for 

various percentages of broken fibers of a 0.02in thick HM-S/IMHS unidirectional composite 

subject to unidirectional cyclic stress ulll of 20 ksi amplitude. The effect of any thermal 

curing residual microstresses was neglected, although this capability readily exists in the 

method, in order to assess the effect of pure mechanical loads on the friction damping. 

The coefficient of friction between fiber and matrix was assumed to be 0.30. Although 

the friction effect is comparable to the hysteretic damping values, the resultant friction 

damping is rather low due to the small radial stress involved in longitudinal loading. 

In the case of off-axis loading, the friction damping will be higher because the transverse and shear stresses will increase the radial stress. The trend is illustrated in Fi, nure 

11 where the axial SDC's of two 50% HM-S,'IhlHS off- axis plies, the first without broken 

fibers and the second with 20% broken fibers, are plotted as functions of the ply angle. 

Both plies are excited by a cyclic axial stress ucrz. Interestingly, the friction damping is 

low at 0 degrees, then increases and peeks near 25 degrees, is reduced again to zero as the 

16 
Ilongitudinal strain is reduced due to Poisson's effect, and then is increased again. At, fiber 

angles near 90 degrees, the friction damping is mainly induced by the Poisson's effect. 

3.5. Temperature Rise 

The temperature rise at the outer surface and mid-plane of a @.04in, 50% HT-S/IXIHS 

unidirectional composite, excited by a sinusoidal uniaxial off-axi? stress Q,,,, with 10 ksi 

amplitude and 50 Hz frequency, is shown in Figure 12 for variou, off-axes angles. The 

surrounding media of the composite is assi;mc-d to be motionless ail with convection rate 

0.007 BTU/(hr-in'-"F). The actual convection rate around a vibrating composite would 

be higher, hence, Figure 12 presents an upper bound for the temperature rise. Estimation of 

the actual heat convection rate is beyond tte scope of the paper, but regardless of this issue, 

a significant temperature rise is predicted as the off-axis angle is increased. Consequently-. 

variations in the elastic and damping characteristics of composite plies should be expected 

in off-axis cyclic loading. In view of the sensitivity of composite damping to temperature 

variations. differences in predicted and measured values for composite damping in off-axis 

cyclic loading map occur. 

4. SUMMARY 

A unified micromechanics theory for the damping capacity of composites has been 

presented. The theory includes simple approximate equations for synthesizing specific 

damping capacities based on: (1) elastic and dissipative constituent properties: (2) interface 

properties, (3) ply temperature and moisture, (4) off-axis loading, and (5) temperature 

rise due to continuous vibration. The micromechanics theory has been augmented into the 

ICA?; computer code [l]. 

1 

Predicted results compare well with available experimental measurements reported in 

open literature. The correlation has added confidence to the validity of the theory. but 

additional experimental work is needed in some areas. The results illustrate the following 

important trends regarding the specific damping capacity (SDC) of composite materials: 

i. The longitudinal on-axis SDC is controlled by the fibers and is rather insensitive 

to fiber volume fractions of practical interest. All other SDC's are predominantly 
controlled by matrix and are sensitive to fiber volume ratio. The fiber moduli also 

influence the SDC of the composite, in that, they control the distribution of strain 

energy into the matrix. In view of this wide sensitivity of composite damping to 

micromechanics parameters, the present theory is a valuable approach for selecting 

suitable fiber/matrix combinations and fiber volume ratios in order to satisfy particular damping requirements. 

ii. Off-axis ply SDC’s have demonstrated wide variation with respect to fiber angle, indicating that, ply angles may be an effective design parameter for tailoring the damping 

capacity of composite plies. 

iii. Temperature and moisture variations have shown significant influence on the damping 

capacity of composite plies. 

vi. The temperature rise within continuously vibrating composite plies was found significant for off-axis loading and disproportional to the respective ply damping. The 

latter observation indicates that the issue of temperature rise requires special design 

consideration, since it significantly influences the damping capacity. 

v. Int,erfacial friction damping contributes to the overall damping capacity of unidirectional composites and its contribution is more significant in case of off-axis loading. 

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18 
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19 
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Three-Dimensional Finite-Element Analysis, ” Journal of Composites Technology and Research, 

Vol. 8, No. 3, Fall 1986, pp. 77-83. 

17. Chamis, C. C., Lark, R. F., and Sinclair, J. H., “Integrated Theory for Predicting the 

Hydrothermomechanical Response of Advanced Composite Structural Components, ” Advanced 

Composite Materials-Environmental Effects, J. R. Vinson, ed., ASTM STP-658, American 

Society for Testing and Materials, Philadelphia, PA, 1978, pp. 160-192. 

18. Chamis, C. C., “Micromechanics Strength Theories, Fatigue and Fracture, Vol. 5, 

L. J. Broutman, ed., Academic Press, 1974, pp. 93-151. 

1 9 . Chamis, C. C., “!;implified Composite Micromechanics for Predicting Microstresses,” Journal 

of Reirlforced Plistics and Composites, Vol. 6, No. 3, July 1987, pp. 268-289. 

2 0 . Ni, R. G. and Adams, R. D., “The Damping and Dynamic Moduli of Symmetric Laminated 

Composite Beams - Theoretical and Experimental Results, ” Journal of Composite Materials, 

Vol. 18, No. 2, Mar. 1984, pp. 104-121. 

20 
APPENDIX 

On-axis ply stiffness: 

Off-axis ply stiffness: 

[Ec;-' = iRu]T;Ei]-'[Ru! 

1 cos29 sin'@ sin28 

sin' B cos'8 -sin28 

-0.5sin28 0.5sin29 cos29 

1 1 vcyz rz 

[R,] = 

(A1 j

(-42) 
Table 1 Specific damping capacities for matrix sad fibers. 

Matrix 

IMLS (Polyester) 8.5 13.8 

IMHS (Epoxy) 6.6 6.9 

Fiber qfll +f22 $f12 $f23 

(%) (%) (%) (%) 

E-Glass 1.1 1.1 0.6 0.6 

HT-S 0.7 0.7 0.6 0.6 

HM-S 1 .o 1 .o 1 .o 1 .o 

22 
I. 

(a) ON-AXIS. 

(b) OFF-AXIS. 

FIGURE 1. - TYPICAL CONTINUOUS FIBER COMPOSITE 

PLIES. 

HM-S/IMLS 

HT-S/IMLS I= ----- E-GLASS/IMLS 

.o# 0 HM-S/IMLS. MEASUREMENT 

(REF. 5) 

I \ 0 E-GLASS/IMLS. MEASUREMENT 

.4 - 

.2 - 

0 .2 .4 .6 .8 1 .o 

VOLUME FRACTION 

FIGURE 3. - IN-PLANE SHEAR SPECIFIC DAMPING 

CAPACITY . 

1.01 

.8 

.6 

T: 7- 

+- .4 

.2 

0 

HM-S/ IMLS 

I---- HT-S/IMLS 

E-GLASS/IMLS 

(REF. 5) 

(REF. 5) 

----- 'I, HM-S/IMLS. MEASUREMENT 

0 E-GLASS/IMLS. MEASUREMENT 

y\ '\\ 0 

.2 .4 .6 .8 1.0 

VOLUME FRACTION 

FIGURE 2. - LONGITUDINAL NORMAL SPECIFIC 

DAMPING CAPACITY. 

HM-S/IMHS 

--- HT-S/IMHS 

----- E-GLASS/ IMHS 

0 E-GLASS/IMLS, MEASURENENT 

(REF. 8) 

1 .o c 

.8 

.6 

3 

x 

2 

m .4 

.2 

2 .4 .6 .8 1 .o 

0 .2 

VOLUME FRACTION 

FIGURE 4. - TRANSVERSE NORMAL SPECIFIC DAMPI NG CAPACITY . 

23 
HM-S/ I MLS 

--- HT-S/I MLS 

----- E-GLASS/IMLS 

1.0 

.8 

g .6 

.4 

3 

'M 

3 

cu 

.2 

.14 DT , M. 

OF PERCENT 

0 0 

100 0 

200 0 

--- *..E .10 ----- c .08 

-06 

c 

.04 

.02 

0 .2 .4 .6 .8 1.0 

VOLUME FRACTION 

FIGURE 5. - THROUGH-THE-THICKNESS SHEAR SPECIFIC DAMPING CAPACITY. 

0 .2 .4 .6 .8 1 .o 

VOLUME FRACTION 

FIGURE 6. - HYGROTHERMAL EFFECT ON LONGITUDINAL NORMAL DAMPING. HM-S/IMHS GRAPHITEEPOXY, 

DT , M. 

OF PERCENT 

0 0 

100 0 

200 0 

200 2 

--- 

----- 

--- 

.12 

.10 

2 3 .08 

.06 

.04 

.02 

0 .2 .4 .6 .8 1 .o 

VOLUME FRACTION 

FIGURE 7. - HYGROTHERMAL EFFECT ON IN-PLANE 

SHEAR DAMPING. HM-S/IWHS GRAPHITE-EPOXY. 

.16 

.14 

.12 

.10 M .. 

3 .08 

.06 

* 04 

.02 

DT. M, 

OF PERCENT 

0 0 

100 0 

200 0 

200 2 

--- - 

----- - --- 

I\ 

0 .2 .4 .6 .8 1.0 

VOLUME FRACTION 

FIGURE 8. - HYGROTHERMAL EFFECT ON TRANSVERSE 

NORMAL DAMPING. HM-S/IMHS GRAPHITE-EPOXY. 

24 
.10 

t. 

U 

a 

t, .08 

e, DEGREE 

FIGURE 9. - SPECIFIC DAMPING CAPACITIES OF A 

50 PERCENT HM-S/DX210 OFF-AXIS COMPOSITE. 

- 

Z X - 

6 

BROKEN FIBERS. 

PERCENT 

0 

20 

50 

--- 

----- 

3 

c 

- ___--------- - - - - - =-=-:-=-=-=- 

I I 

0 .2 .4 .6 .8 1 .o 

VOLUME FRACTION 

FIGURE 10. - EFFECT OF FRICTION DAMPING ON 

THE ON-AXIS LONGITUDINAL SDC OF A HM-S/ 

IMHS COMPOSITE. 

BROKEN FIBERS, 

PERCENT 

r rMID-PLANE 

SURFACE 

LL 

0 

0 20 40 60 80 100 

e. DEGREE 

FIGURE 11. - EFFECT OF FRICTION DAMPING ON 

THE AXIAL SDC OF A 50 PERCENT HM-S/IMHS 

OFF-AXIS COMPOSITE. 

2 X 

0 20 40 60 80 100 

0. DEGREE 

FIGURE 12. - TEMPERATURE RISE IN A 50 PERCENT 

HT-S/IMHS OFF-AXIS COMPOSITE SUBJECTED TO 

CONTINUOUS SINUSOIDAL EXCITATION. 

25 
I 

1 ,II c r I, A,,.oiiai 11, 3 an.’ Report Documentation Page I 

1 -,, ;$ ,Ira 1 1 - __ . - 

1 1 Rep011 No 1 NASA TM-102107 

2 Government Accession No -I--- - - -- 

I I Temperature effects; Moisture effects; Off-axis 

~ 

damping; Broken-fiber effects; Temperature rise 

I 

I Fjhrr Reinforced Composites 

Vilified Microtnechartics of Damping for Unidirectional 

;7 Key Words (Suggested by Author@)) 

Composites: In-plane damping; Out-of-plane damping; 

D.A. Saravanos and C.C. Chamis 

18. Distribution Statement 

Unclassified - Unlimited 

. __ ___ 9 P-.rform ng Organization Name and Address 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135-3191 

12 Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D C. 20546-0001 

- 

15. Supplementary Notes 

D.A. Saravanos. National Research Council---NASA Research Associate. 

3. Recipient’s Catalog No. 

5. Report Date 

August 1989 

6. Performing Organization Code 

_- 

8. Performing Organization Report I 

E-4874 

10. Work Unit No. 

505-63- 1 1 

11. Contract or Grant No. 

13. Type of Report and Period Cover 

Technical Memorandum 

14. Sponsoring Agency Code 

An integrated micromechanics methodology for the prediction of damping capacity in fiber-reinforced polyme 

matrix unidirectional composites has been developed. Explicit micromechanics equations based on hysteretic 

damping are presented relating the on-axis damping capacities to the fiber and matrix properties and volume 

fraction. The damping capacities of unidirectional composites subjected to off-axis loading are synthesized frc 

on-axis damping values. Predicted values correlate satisfactorily with experimental measurements. The hygrothermal effect on the damping performance of unidirectional composites due to temperature and moisture 

variations is also modeled. The damping contributions from interfacial friction between broken fibers and ma 

are incorporated. Finally, the temperature rise in continuously vibrating composite plies is estimated. Applica 

examples illustrate the significance of various parameters on the damping performance of unidirectional and c 

axis fiber reinforced composites. 

I Subject Category 24 

119 Security Classif. (of this report) 

! Unclassified Unclassified 

20. Security Classif. (of this page) 21. No of pages 

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