I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

1 

I 

I 

I 

I 

~I 

A Study of Material Damping 

in Large Space Structures 

Final Technical Report 

by 

Dr. A.L. Highsmith 

Dr. D.H Allen 

Department of Aerospace Engineering 

Texas A&M University 

Submitted to 

NASA Johnson Space Center 

Grant No. NAG-9-192 

April 1989 
Table of Contents 

1 INTRODUCTION ....................................................... 1 

2 CONTINUUM DAMAGE MODEL ............................................. 2 

2.1 Theoretical Development ....................................... 2 

2.2 Experimental Program .......................................... 6 

2.3 Comparison of Theory and Experiment ........................... 10 

3 MICROMECHANICS MODEL ............................................... 15 

3.1 Theoretical Development ....................................... 15 

3.2 Comparison with Experiment .................................... 20 

4 CONCLUSIONS ........................................................ 25 

5 REFERENCES ......................................................... 26 

6 BIBLIOGRAPHY ....................................................... 29 

7 THESIS AND DISSERTATION ABSTRACTS ....................... ...... ..... 30 

8 TECHNICAL REPORTS .................................................. 35 

i 
1 INTRODUCTION 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

Because they provide high stiffness and strength and are relatively light 

weight, continuous fiber reinforced laminated composite materials are candidates for use in large space structures. One of the significant differences 

between these composites and more conventional engineering materials is that 

laminated composites develop a considerable amount of large-scale damage. 

Numerous ply cracks and delaminations may develop in a laminate early in its 

load history. 

Unlike fiber breaks and fiber-matrix debonds, which tend to influence 

only a small volume of material, ply cracks and delaminations can cause significant changes in the bulk properties of a laminate. Numerous researchers 

have studied and modelled the inf-luence of ply cracking [1,2,5] and 

delamination [3,6] on the elastic behavior of laminated composites. But while 

significant increases in damping have been observed as a result of damage 

development [7,8], and modelling efforts have predicted that significant 

changes in the dynamic response of structures can result from changes in material damping (9,101, existing models of damping in composites [11,12] have 

not included the effects of damage on damping. 

This report summarizes the work completed under NASA Grant NAG-9-192. 

This research effort had as its objective the development of a damage dependent constitutive model for predicting the influence of damage on the damping 

properties of laminated composites. In particular, the beam bending behavior 

of cross-ply laminates was studied. 

cracking. The primary effort was to 

develop a continuum damage model based on internal state variables to describe 

the viscoelastic behavior of laminates as a function ply cracking. In support 

of the continuum modelling, an experimental program was implemented to determine stiffness and damping as a function of damage. A second effort involved 

The damage mode of interest was ply 

The research proceeded along two fronts. 

the development of a micromechanics model of damage dependent damping. 

1 
I 

I 

I 

1 

2 CONTINUUM DAMAGE MODEL 

2.1 Theoretical Development 

The constitutive model used in this study follows t,.e mode developed $1 

and used by Allen, et al. to predict the damage dependent elastic response of 

laminated composites containing ply cracks [5] and delaminations [6]. For a 

given ply, the uniaxial constitutive equation has the form 

where E is the modulus of the material, and aI is an internal state variable 

which reflects the effect of ply cracking on ply behavior. In general, there 

would be one internal state variable (ISV) for each different damage mode. 

Multiaxial behavior requires a tensor quantity to characterize the effect of 

damage. 

In the earliest version of the model [13,14,15], it was assumed that a 

ply could be modelled as a Voigt material, which has the mechanical analog 

shown in Fig. 1. For such a material, the constitutive equation has the form 

ox, = E€,, + l& I (2.2) 

where the term qkxx represents the viscous part of the material response. 

This equation can be made to fit the form Eqn. In 

fact, two ISV's, one representing stiffness changes due to ply cracking and 

the other representing damping changes due to ply cracking, were used in the 

analysis. Strictly speaking continuous fiber reinforced materials do not 

behave like Voigt materials, within a limited range of frequencies and amplitudes appropriate Voigt material parameters can be chosen to provide a reasonable representation of ply response. 

(1) by taking aI=-2&,, . 

2 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

1 

I 

I 

I 

I 

'I 

I 

I 

~' 

& 

Figure 1.The Voigt model. 

3 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

While the Voigt model was useful in the early stages of the development 

of the damage dependent constitutive model, it was somewhat restrictive. For 

example, Voigt type ply response led to a linear frequency dependence on laminate damping [13,14,15]. This frequency dependence was not observed in subsequent experimental results. 

developed which uses a complex modulus representation of viscoelastic ply 

properties [16]. 

In this formulation, the damage dependent, elastic constitutive relation has 

the form 

A more general formulation has since been 

It is this second formulation which will be discussed here. 

uxx -EE,,+ la, , (2.3) 

where now the term la, captures the damage dependence. For a linear viscoelastic material under sinusoidal loading, the constitutive relation takes the 

form 

.- .- Z,,-E E,,+I a, . 

where 

I = 1'+ il' 

E,, = €,,eiw' 

Q, = a,Ae'w' . 

4 
Using the viscoelastic constitutive relation above, it can be shown that 

the energy dissipated by a ply per cycle of loading is given by 

Aw = n E' E: + n I' a I A , 

and the stored energy during the peak displacement is given by 

II - [E'€: + IOEAa IA]/~ . 

It then follows that the loss factor is 

E'E:+I'QIAEA 

" E'€~+I'Cll,€A . 

With the additional assumptions that 

I*a,A = AE'E~ , 

we arrive at the expression for the loss factor 

The complex modulus of a laminate may be determined from the ply level 

constitutive relations using classical lamination theory [17] to be 

(2.10) 

where E' is the complex modulus of the laminate, h is the thickness of the 

laminate, N is the number of plies in the laminate, E; is the damage dependent 

5 
I 

I 

I 

'I 

I 

I 

I 

I 

I 

'I 

'I 

I 

I 

,I 

II 

complex modulus of ply i, and the z,are the coordinates of the ply interfaces. 

00 and 90" plies, the expression above reduces to 

For a symmetric cross-ply laminate, a symmetric laminate consisting of 

E'- aEL+ bE; (2.11) 

if the laminate is undamaged. In Eqn. (2.11), a and b depend on laminate 

geometry, and E; and E; are the complex moduli of the 00 and 900 plies, respectively. The flexural damping is given by 

aE' + b E', 

aE',+bE', * 

tan+, = q - (2.12) 

In the present study, it was assumed that ply cracking would affect E; 

but not E; . 

given by 

It follows that for a damaged laminate, the flexural damping is 

aEmL+ bE',+ CAE', 

"aE',+bE',+CAE', ' (2.13) 

where CAE', and CAE',are functions of the total crack surface area and location of the cracks. Both AE', and AE', can be determined from experimental 

data from a single laminate, and then can be used to predict behavior in other 

laminates via Eqn. (2.13). 

2.2 Experimental Program 

In order to evaluate the analytical model, damping was measured as a 

function of damage in a variety of graphite/epoxy cross-ply laminates [14,15]. 

Damage was introduced into straight sided coupon type specimens under uniaxial 

tensile loading in an MTS servohydraulic testing machine. Edge replication 

6 
was used to monitor the development of 900 ply cracks during tensile loading. 

At various stages of damage development, the tensile test was interrupted, so 

that the specimen could be removed and its flexural damping measured. 

The experimental setup used in performing the damping measurements is 

shown schematically in Fig. 2. 

housed in a vacuum chamber. 

motorized wheel with a contacting rod was used to deflect the specimen and 

excite its first mode of free vibration. Care was used to insure that the 

initial deflection was small, so that no new damage would be introduced during 

the damping test. A strain gage mounted on the specimen near the cantilevered 

support was used.to monitor the surface strain in the specimen during free 

vibration. A personal computer with an A/D board was used to start and stop 

the motor, and to record strain data. 

The specimen was cantilevered from a support 

All damping data was obtained in vacuum. A 

After capturing the strain data, the peaks in the strain versus time 

curve were obtained. 

cycle number is shown in Fig. 3. The decay of strain amplitude is seen to be 

nearly linear on a semi-log plot. The slope of the best fit straight line to 

the log strain amplitude data was taken to be the logarithmic decrement, which 

is directly related to damping. 

A representative plot of log strain amplitude versus 

7 
I 

I 

I 

I 

bridre/meter oscilliscopc 

I I 

Figure 2. Schematic diagram of the damping measurement system. 

8 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

1 

I 

~I 

I 

I 

1 

.. 

.1 

.01 

0 10 20 30 40 

Cyde Number 

50 60 

Figure 3. Representative plots of strain amplitude versus 

cycle for 00 and 900 graphite/epoxy laminates. 

9 
I 

I 

I 

I 

I 

I 

I 

2.3 Comparison of Theory and Experiment 

From measured values of the stiffness and damping of laminates containing 

all 00 and all 900 plies, the parameters E'L, E'T, E"L, and €IT can be determined. 

laminates consisting of combinations of Oo and 90° plies. A comparison of the 

predicted and observed damping in several cross-ply laminates is presented in 

Fig. 4. Agreement between theory and experiment is quite reasonable. 

plies exhibit matrix dominated behavior, and hence are the primary contributors to damping in laminates. 

with distance from the midplane. 

These parameters can in turn be used to predict the damping of other 

The 900 

In addition, the effect of a 90° ply increases 

In order to predict the influence of damage on damping, it was necessary 

to determine the parameters AE'T and AE'T as a function of crack surface area. 

Since the elastic properties of the graphite/epoxy laminates are dominated by 

the Oo plies, AE'Tterm was assumed to have a negligible effect on the denominator of the right hand side of Eqn. (2.13). Further, damage dependent damping data from [0/90/0], specimens was used to determine 

A E',(s) = 57.97 X 1 O-6[0.0002(~) -1.7 108 x 1 O-'s(~2)] , 

where s is the crack surface area in the 90° ply. Using this expression in 

Eqn. (2.13), damping was predicted as a function of damage in,other cross-ply 

laminates. 

Figures 5, 6, and 7 present a comparison between predicted and observed 

damping values in [90/0/90],, [0/90],,, and [90/0/90,], laminates at two different damage states. Generally, agreement between theory and experiment is 

quite good. The effect of damage on the behavior of the [0/90],, is smaller 

than the effect of damage on the behavior of the other laminates because the 

90° plies are located closer to the midplane. Thus, any influence of the goo 

plies on the flexural damping is reduced. This stacking sequence dependence 

is also reflected in a smaller damping value for the [0/90],, laminate. 

10 
I 

I 

4 

3 

0 

I 

0 

4 

x, 

02 

I 

L 

3 

V 

0 

L 

v) 

v) 

0 

2 

I 

Figure 4. Damping values for various undamaged graphite/epoxy 

laminates. 

I 

I 

~I 

I 

I 11 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

Damage State I Damage State 11 

Figure 5. Damage dependent damping values for [90/0/90], graphice/epoxy laminates. 

12 
I 

I 

2 

1.5 

1 

0.5 

0 

b 1 

(l;s"l Experiment 1 

Damage State I Damage State I1 

Figure 6. Damage dependent damping values for [0/90],, graphite/epoxy laminates. 

13 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I1 

I 

I 

1B Experiment I 

Damaae State I Damage State 11 

Figure 7. Damage dependent damping values for [90/0/902], graphite/epoxy laminates. 

14 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

‘I 

I 

I 

lB 

I 

I 

‘I 

3 MICROMECHANICS MODEL 

3.1 Theoretical Development 

In order to better understand the details of damage-induced damping, a 

micromechanics model of damaged crossply laminates was developed [19]. The 

model is based on shear lag theory, and explicitly includes 90° ply cracks. 

In fact, two versions of the model were developed. In the first, each ply is 

treated as an Euler-Bernoulli beam, while in the second, each ply is treated 

as a Timoshenko beam. The Timoshenko beam version will be described below. 

For the Timoshenko model, the axial displacement in layer i is given by 

where 

dw 

dx k,(x)=pi(X)-- . 

In Eqn. (3.2) w is the deflection of the ply, which is assumed to be the same 

as the deflection of the laminate. The Euler-Bernoulli model does not include 

the ply shear term, which is represented by (3. 

Ply equilibrium equations are obtained by performing force and moment 

summations on a differential element taken from the ply. 

shown in Fig. 8. The interlaminar shear stresses shown in this figure are 

determined using the shear lag assumption, i.e., they are assumed to be proportional to the difference in ply displacements across the interface. 

Such an element is 

In fact, the shear strain in the interfacial region is given by the difference in the axial displacements of the two plies divided by the thickness of 

the shear transfer region. 

appropriate shear modulus for the shear transfer region. Following this procedure, the shear lag assumption yields 

Stresses are obtained by multiplying by an 

15 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

-1 

1 

PlY i - 1 

z 

> ply i 

-. v -1 

ply i - 1 

Leeend 

a; = Thickness oi ply i 

b = Thickness of shezr tiznsier region 

Figure 8. Equilibrium element from ply i for the Timoshenko 

micromechanics model. 

16 
Axial equilibrium for the ply has the form 

equilibrium in the z direction yields 

a,,,'= 0 . (3.5) 

and moment equilibrium has the form 

Exterior plies have slightly different equilibrium equations owing to the fact 

that on one side of such a ply there is no neighboring ply, and thus there are 

no interlaminar shear stresses. 

The equilibrium equations for the plies within a laminate can be 

assembled into a system of coupled, second order differential equations in the 

form 

17 
I 

I 

I 

I 

I 

I 

I 

I 

I 

where the column vector (y) has as its components the midply displacements u,i 

and the ply rotations k,. These governing equations are applied to a region 

of the laminate which corresponds to one-half of the crack spacing. Such a 

region is shown in Fig. 9. At x-0, the displacement in the uncracked plies is 

zero, while in the cracked plies, the crack surfaces are stress-free. A 

symmetry condition at x-1 requires that ply displacements be equal at the ply 

interfaces. Applied loads then determine the midplane displacement and rotation at this end. 

The system of linear ordinary differential equations was solved using 

techniques commonly used for analyzing dynamic systems [20]. This approach 

requires the determination of eigenvalues and eigenvectors of the coefficient 

matrix [B]. The solution takes the form of a .linear combination of "mode 

shapes" : 

where the a,are the eigenvalues and the {vi) are the eigenvectors. 

conditions determine the coefficients Ai in the solution. 

Boundary 

The viscoelastic problem was also solved in order to determine the effect 

of cracking on damping. 

analysis described above, and obtaining the corresponding solution. Note that 

only material viscoelastic behavior was included, and that other dissipation 

mechanisms, such as friction, were ignored. 

This was accomplished using complex moduli in the 

I 

I 

18 
n 

Figure 9. Representative volume within a damaged laminate. 

19 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

II 

I 

I 

3.2 Comparison with Experiment 

Experimental data for bending stiffness as a function of crack density 

was obtained for [0/90,], and [90,/0,), glass/epoxy laminates. A glass/epoxy 

system was chosen because it is a less fiber dominated system than graphite/epoxy, and thus should exhibit larger property changes due to damage development. 

damage, which was monitored using edge replication. Four-point bending tests 

were used to determine the bending stiffness at a variety of damage states. 

This data was compared to model predictions. 

Specimens were loaded in quasi-static tension in order to introduce 

Figures 10 and 11 present the theoretical and experimental bending stiffness data for the [0/90,], and [90,/0,], laminates, respectively. Note that in 

Fig. 10, there is little bending stiffness reduction observed. This is 

because the 00 plies are located at the exterior of the laminate. 

this small stiffness reduction, the predicted responses are in reasonable 

agreement with experiment. 

dictions than the Euler-Bernouli model. The [90,/0,], laminate (Fig. 11) 

exhibits larger stiffness changes. The Timoshenko model provides a reasonable 

estimate of laminate response up to crack densities of about 13 cracks/in., at 

which point delaminations begin to develop. 

In light of 

The Timoshenko model provides slightly better preFig. 12 shows the predictions of loss factor as a function of crack density in a [90z/OzJ, laminate. At low crack densities, the damping increases 

with cracking. But at larger crack densities, damping decreases with 

cracking. It is believed that this behavior is a result of excessive load 

transfer from the relatively viscoelastic 900 plies to the relatively elastic 

Oo plies that occurs at high crack densities. At lower crack densities, 

stress concentrations in the shear transfer region are able to generate an 

increase in damping. This decrease in damping is not observed in the laboratory. At present, it is believed that other dissipation mechanisms, such as 

friction between rubbing crack faces, account for the discrepancy. With 

20 
B 

B 

I 

B 

B 

B 

I 

B 

B- 

- 

- 

1.05 - 

- 

v) 

t 

- 

; 1.00 - SA- - C4 

.- - 

m - 

C 

73 

C - 

W - 

~0.95 -

- .- 

0.90 - 

m 

Specimen 1 0 Specimen 2 

Specimen 3 

(0/9 o,>* 

Eu I er - Be r nou Ili 

A 

0 0 - a Ti rn os h e n ko A A 

0 

AD 

i 73 

Q) 

N 

0 

.- -

E 0.85 

0 

Z 

25 

1 l"i"iiii l'~'iii'l'l 

0 5 10 15 20 

0.80 liiiiiiililiiiiiitii~iiiiil1ii' 

Cracks per Inch 

Figure 10.Danage dependent bending stiffness in [0/90,], 

glass/epoxy laminates. 

21 
I ~' 

~I 

~' 

,I I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

1 .O 4 

2 0.9 4 

(r 

-0 

C 

e, 

5 0.8 

m 

.e, 

N 

0 

.- 

E 

L 0.6 

0 

Z -I 

Specimen 4 . Specimen 5 - A Specimen tj 

A 

e r- Bern o u iii 

A 0 

Timoshenko A -A A 

A 

Figure 11.Damage dependent bending stiffness in [90,/0,], 

glass/epoxy laminates. 

22 
I 

I 

I 

I 

I 

II 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

additional effort, it should be possible to use the micromechanics model to 

estimate frictional dissipation. 

23 
k 

0 

4 

0 1.2 

7 

=r( 0.4 

g 0.0 1 I 1 I 1 1 I 

I I I I 0 10 20 3'0 40 i 

Crack Density (Cracks/in. j 0 

Figure 12.Predictions of loss factor as a function of damage 

using the Timoshenko model. 

24 
I 

~I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

~I I 4 CONCLUSIONS 

A constitutive model.has been developed for predicting damping as a function of damage in continuous fiber reinforced laminated composites. 

age model is a continuum formulation, and uses internal state variables to 

quantify damage and its subsequent effect on material response. 

sensitive to the stacking sequence of the laminate. 

line data from unidirectional material, and damping as a function of damage in 

one crossply laminate, damping can be predicted as a function of damage in 

other crossply laminates. 

quite good. 

The damThe model is 

Given appropriate baseAgreement between theory and experiment has been 

A micromechanics model has also been developed for examining the influence of damage on damping. 

The model provides reasonable predictions of bending stiffness as a function 

of damage. 

thought to be a result of dissipation mechanisms such as friction, which are 

not presently included in the analysis. 

This model explicitly includes crack surfaces. 

Damping predictions are not in agreement with experiment. This is 

25 
I 

I 

I 

I 

'I 

'I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

iI 

5 REFERENCES 

[l] Highsmith, and Reifsnider, K.L., "Stiffness Reduction Mechanisms in 

Composite Laminates," Damaee in CQnuxsite Mat-. ASTMSTP- , K.L. 

Reifsnider, Ed., ASTM, 1982, pp. 103-117. 

[2] Norvell, R.G., "An Investigation of Damage Accumulation in Graphite/Epoxy Laminates," Texas A&M University Thesis, August, 1985. 

[3] O'Brien T.K., "Characterization of Delamination Onset and Growth in a 

Composite Laminate," Damaee in Cwosite Materials. ASTM STP 775 , K.L. 

Reifsnider, Ed., ASTM, 1982, pp. 140-167. 

(41 Allen, D.H., Allen, C.E., and Groves, S.E., 'A Thermomechanical Constitutive Theory for Elastic Composites with Distributed Damage - Part I. 

Theoretical Development," JnternatiQnal Journal of Solids and Structures 

[5] Allen, D.H., Allen, C.E., and Groves, S.E., "A Thermomechanical Constitutive Theory for Elastic Composites with Distributed Damage - Part I. 

Application to Matrix Cracking in Laminated Composites,"International 

JJ 

[6] Allen, D.H., Groves, S.E., and Harris, C.E., "A Cumulative Damage Model 

for Continuous Fiber Composite Laminates with Matrix Cracking and 

Interply Delaminations," GTM STP , 8th Symposium on Composites: 

Design 

[7] Plunkett, R., "Damping in Fiber Reinforced Laminated Composites at High ~ 

Strain, 'I J . Cow Materials Sup- , Vol. 14, pp. 109-117, 1980. 

26 
I 

II 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

Edberg, D.L., "Material Damping of Sample Structures in a Simulated 

Space Environment, , Journal of SDwft , Vol. 23, pp. 

288-296, 1986. 

Kalyanasundaram, S., Lutz, J.D., Haisler, W.E., and Allen D.H., "Effect 

of Degradation of Material Properties on the Dynamic Response of Large 

Space Structures," Journal of SD- , Vol. 23, pp. 

297-302, 1986. 

Chang, H.T., and Allen, D.H., "Predicted Dynamic Response of a Composite 

Beam with History Dependent Damage," Texas AM University Mechanics and 

Materials Center, MM 4875-86-17, July, 1986. 

Gibson, R.R., and Plunkett, R., "Dynamic Mechanical Behavior of FiberReinforced Composites: Measurement and Analysis," Journal of ComposiLe 

Materials, Vol. 10, pp. 325-341, 1976. 

Nil R.G., and Adams, R.D., "The Damping and Dynamic Moduli of Symmetric 

Laminated Composite Beams - Theoretical and Experimental Results," Jour-, 

rial of Composite Materials, Vol. 18, pp. 104-121, 1984. 

Allen, D.H., Harris, C.E., and Highsmith, A.L., "Prediction and Experimental Observation of Damage Dependent Damping in Laminated Composite 

Beams,'' The Role of DamDing in Vibration and Noise Control , DE-VOI. 5, 

L. Rogers, ed., ASME, 1987, pp. 253-263. 

Smith, S.A., "An Experimental Investigation of Damage-Dependent Material 

Damping of Laminated Composites," Texas A&M University Thesis, May 1988. 

27 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

~I 

[15] Smith, S.A., Allen, D.H., and Harris, C.E., "An Experimental Investigation of Damage-Dependent Material Damping in Laminated Composites," 

DOC. AIAA/ASME/ASCE/AHS 29th structures. structural- and 

Conference, pp. 1501-1510, 1988. 

[16] Kalyanasundaram, S., and Allen, D.H., "A Model for Predicting Damage 

Dependent Damping in Laminated Composites," -and S-, 

Vol. 30, NO. 1/2, pp. 113-118, 1988. 

[17] Jones, R.M., Mechanics of C- , Hemisphere, New York, 

1975. 

[18] Kalyanasundaram, S., Allen, D.H., and Schapery, R.S., "Dynamic Response 

of a Viscoelastic Timoshenko Beam," proc. AIAA/ASME/ASCE/AHS 28th Structures. Structu,,rd D-mmics. and Materials Conference, 1987. 

[19] Frailey, J.A., "Matrix Cracking and Bending Stiffness Reduction in Composite Laminates," Teexas AM University Thesis, December, 1988. 

[20] Junkins, J.L., and Turner, J.D., -a1 Spacecraft Rotational Maneu- 

-, Elsevier, Amsterdam, 1986. 

28 
~ 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

1 

I 

I 

I 

6 BIBLIOGRAPHY 

[l] Allen, D.H., Harris, C.E., and Highsmith, A.L., "Prediction and Experimental Observation of Damage Dependent Damping in Laminated Composite 

Beams," Role of ~-idLik,~ , DE-Vol. 5, 

L. Rogers, ed., ASME, 1987, pp. 253-263. 

[2] Smith, S.A., Allen, D.H., and Harris, C.E., "An Experimental Investigation of Damage-Dependent Material Damping in Laminated Composites,'' 

hoc. -S 79th Structures. Structural Dyna~ics. and 

Uterials Conference, pp. 1501-1510, 1988. 

[3] Kalyanasundaram, S., and Allen, D.H., "A Model for Predicting Damage 

Vol. 

Dependent 

30, NO. 

Damping 

1/2, pp. 

in Laminated 

113-118, 

Composites, 

1988. 

'I u, 

[4] Kalyanasundaram, S., Allen, D.H., and Schapery, R.S., "Dynamic Response 

of a Viscoelastic Timoshenko Beam," Proc. AIAA/ASME/ASCE/AHS 78th Strtures. Structural Dvmmics. and Materlals Conference, 1987. 

[5] Smith, S.A., "An Experimental Investigation of Damage-Dependent Material 

Damping of Laminated Composites," Texas A&M University Thesis, May 1988. 

[6] Frailey, J.A., "Matrix Cracking and Bending Stiffness Reduction in Composite Laminates," Teexas AM University Thesis, December, 1988. 

29 
I 

I I I 

~I 

11 

‘I I 

~I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

7 THESIS AND DISSERTATION ABSTRACTS 

30 
I 

ABSTRACT 

. An Experimental Investigation of Damage-Dependent Material Damping 

I 

I 

I 

of Laminated Composites. (May 198s) 

Scott Andrew Smith, B.S., Texas A&M University 

Co-Chairs of Advisory Committee: Dr. Alton L. Highsmith 

Dr. Charles E. Harris 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

1 

I 

I 

An experimental program was developed in which the material damping 

of laminated graphite/epoxy specimens was measured as a function of matrix 

cracking. The damping test apparatus w2s designed to meisure the first mode 

free vibration response..of a cantilevered beam. Matrix cracks were introduced 

into cross-ply laminates, and damping, calculzted by the logarithmic 

decrement method, was measured at severd levels of damage. 

The 90 degree plies were found to have a giezter damping czpacity than 

the 0 degree plies, and placement of the 90 degee plies awzy from the midplane of the lminzte thickness resulted in 2 higher d2mping measurernen:. 

Matrix crzcks in the 90 degree plies were seen to significantly increzse the 

damping of the cross-ply laminates. In addition, the matrix cracks which 

fomed in 90 degree plies near the specimen SUiface, caused 2 greater increzse 

in damping than matrix cracks near the mid-Tlane. rinzlly, a dzmagedependent material damping model w2s applied, which closely predicted fhe 

measured damping values. 

-. 

31 
I 

I 

I 

I 

I 

I 

I 

I 

1 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

AB STRICT 

Matrix Cracking and Bending Stiffness Reduction 

in Composite Laminates. (December 1988) 

James Alan Frailey, B.S., Texas XkM University 

Chair of Advisory Committee: Dr. Alton L. Highsmith 

Shear-lag theory was used in the development of two mathematical models 

to predict the behavior oi cracked, cross-ply, laminated composites. One model used 

Euler-Bernoulli bezm theory to model the deformation of each ply while the other 

used the Timoshenko beam theory. Solutions to the governing system of difierentill 

equations provide displacenents and stresses throughout a rei)resenlzti\ve solume 

of the lzminzte. The models were used to predict the response oi lzminates under 

two loading conditions: axid: pure bending. 

Xnalpical results were compared to e.upe;imental dzta io: Z?cial 2nd ben2- 

ing stifiness reductions with matrix crack density. Gltssjepoxy :0/90,], znd 

!9O2/O?], laminates \\-ere selected for the andytical-expeimentd compzrison. 

32 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

X B S T RACT 

An Investigation of Damping Behavior of 

Fiber Reinforced Composite Materials. (May 1989) 

Shankar Kalyanasundaram, B.Tech., Madras Insti tnte of Technology 

hf.S., Oklahoma State Gnirersity 

Co-Chairmen of Advisory Committee: Dr. David H. Xllen 

Dr. Walter E. Haisler 

Advanced fibrous composite materials are known to eshibit viscoelastic behavior, which in turn contributes to the damping of these materials. This study 

hzs investigated the efiect of shear deformations, rotary inertia and matris crack 

damage on the damping characteristics of composite mzterizls. For a composite 

bezm the infiuence of shear deformation 2nd rotmy inertia depend on the ratios of 

.€'IC' Tan@~/Tan@~, the length to thickness iaxio and mode number. For 'nigiJJanisotropic beams the loss factors in shear 2re substantidly higher thzn longitudinzl 

loss factor. This leads to large diEeiences between the dzmping predictions obSned 

from Euler-Bernoulli and Timoshenko beam theories. The results of this study show 

that for highly znisotropic slender bezrcs in which the ratio of Tan@G/Tan@E and 

€'IC' are high, the influence of shear dzmping is irnpoitant even in low yibiz5on-d 

modes. -4 constitutive model hzs been developed for predicting the eilcect of microstructural damage on the dynzmic response of composite mzterials. This model 

is valid for a fixed damage state 2nd smzll amplitude vibrztions. The theorexical 

formulztion hzs been verified ior crossply grzphite-eposy lzminates n-ith tiznsver~e 

m2:ix cxick dzmage. Foi crosspl!- iaminares the loss factor in shezi is of the same 

33 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

order as the longitudinal loss factor and Euler-Bernoulli theory has been found to 

be adequate for obtaining dynamic properties. Damage dependent material constants obtained from [0/90/0], laminates have been used to predjct the increase 

in damping in !90/0/90],, [0/90j2, and [90/0/90,], laminates for different matrix 

crack damage states. Damping has been found to be more sensitive to matrix crack 

damage than the stiiness loss. Thus, damping measurements hold some promise for 

further studying the damage development in composite structural components. 

34 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

8 TECHNICAL REPORTS 

35 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

8.1 Dynamic Response of a Viscoelastic Timoshenko Beam 
DYNAMIC RESPONSE OF A VISCOELASTIC TIMOSHENKO BEAM 

bY 

5. Kal yanasund arm* 

D.H. Allenand 

R.A. Schapery** 

Texas MU University 

College Station, TX 77843 

I 

I 

I 

I 

I 

I 

I 

I 

Abstract 

The analytical determination of the' steady 

state and transient flexural vibrations of beams 

is often based on the Euler-Bernoulli equation. 

In this theory what are normally secondary 

effects, such as shear deformations and rotary 

inertia are not included; but these effects may 

significantly influence the vibrational response 

under certain conditions. 

Advanced fibrous composite materials are 

known to exhibit viscoelastic behavior, which in 

turn contributes to the damping of these 

materials. Tne analysis presented in this study 

deals with the vibratory response of viscoelastic 

Timoshenko beams under the assumption of small 

material loss tangents. The appropriate method 

of analysis employed here may be applied to mare 

complex sxruc:ures. 

This study compares the damping ratios 

obtained from the Timoshenko and Euler-Bernoulli 

theories for a Given viscoelzstic material 

system. From this study the effect of shear 

deformation and rotary inertia on damping ratios 

can be identified. 

'Research Assistant, Aerospace Engineering 

Department 

Student Member AIAA 

**Associate Professor, Aerospace Engineering 

Department 

Member AIAA 

***PrOfESSor, Civil and Aerospace Engineering 

Deparme nt 

Member AIM 

E* 

G' 

A 

0 

W 

v 

L 

I 

k 

n 

E' 

G' 

TamE 

TancG 

h' 

b 

Nomenclature 

Complex Longitudinal Modulus 

Complex Shear Modulus 

Cross Sectional Area 

MESS Density 

Total Deflection 

Bending Slope . 

Length of the Beam 

Area Moment of Inertia of Cross 

Section 

Numerical Shape Factor for Cross 

Section 

Mode number 

Longitudinal Storage Modvlus 

Shear Storage Modulus 

Longitudinal Loss Factcr 

Shear Loss Factor 

Thickness of the Beam 

Width of the Beam 

Introduction 

The study of vibration damping properties of 

advanced composites is important for several 

reasons. An automatically controlled flexible 

space structure may act either docile or 

unmanageable depending on the level of damping in 

higher vibration modes . It is also known that 

damping is sensitive to microstructural detail so 

that damping me?.surements can be used to study 

damage development in composite members . 

1 

2 

The analytical determination of the oynamic 

properties of beams generally employs the EulerBernoulli equation. For highly anisorropic 

beams, neg1ec:ing shezr oeformations and rctary 

1 
I 

1 

I 

1 

I 

I 

I 

I 

I 

I 

1 

I 

I 

I 

I 

I 

I 

I 

I 

inertia can result in relatively large errors in 

the predicted dynamic properties. For long 

slender beams composed of undirectional composite 

laminae in which the ratio of longitudinal 

modulus (El1) to the shear modulus (GI2) is high, 

transverse shear deformat ions Can become 

significant even in lower vibrational modes. 

In experiments3 conducted on unidirectional 

E-gldss composite beams, the measured values of 

longitudinal loss modulus are typically 50 to 80 

percent higher than the prediction obtained from 

Euler-Bernoull i beam theory. Adams, et 

have also indicated that the measured damping 

values may depend on shear effects. Schultt and 

T~ai~*~ have used the damping ratios measured for 

three fiber orientation angles to predict the 

damping ratio for any angle. The damping ratios 

exnibired qualitative but not quantitative 

agreement with the predicted properties. 

There is a need to develop an improved model 

for the analysis of large space structures which 

is capable of describing global structural 

response including accurate representation of 

damping characteristics of st ru ctur a1 

components. The global dynanic response of space 

s;ructures can be m3delled with equivalent 

con:inuum beam and plate models. Sun, et C1.9 

have presented procedures for constructing 

several equivalent continuum models. It was 

found that tne Timoshenko btam and Mindlin plate 

theories were suitable for equivalent continuum 

modeilinc. Mazerial damping'' of structural 

members can be included in such formulations. 

For small material loss tangents the dynamic 

steady-state and transient response of a 

viscoelastic Timoshenko beam and other structures 

can be predicted by simple techniques described 

by Schapery . The Timoshenko beam theory1*, which includes 

the effects of shear deformation and rotary 

inertia, his been used to predict flexural 

resonance frtcuencies obtained for anisotropic 

specimens slrch as w00d13 and fiber reinforced 

11 

*. 

compositecL4. Although the Timoshenko theory was 

originally CeveloDed for iso;ropic materials, it 

may be applied ;a anisotropic beams, providing 

the apprcpriate longitudinal and shear moduli and 

associated loss faczors are employed in the 

model. When the wave lengths of the deformation 

response are relatively small, a higher order 

shear deformable theory may be necessary in order 

to accurately predict dynamic properties of 

composite beams . The goal of the present investigation is to 

compsre the free-vibration dampjng obtained from 

Timoshenko and Euler-Bernoulli beam theories for 

a simply supported beam. From this study the 

effect of shear deformation and rotary inertia on 

damping can be identified. The results of the 

study can be used to interpret the difference 

between experimental and predicted damping values 

for beams and to guide a more general study of 

passive damping in large space structures. 

IS, 16 

Analysis 

The governing differential equation of the 

free vibrations of an elastic Edler-Bernoulli 

bean is given by 

The solution for free vibrations can De wric;en 

in the form 

w(x,t) = W(x) exp(iwt) 

where U(x) is ths displacement amplitude at any 

point in the beam. For sufficiently small 

damping, free vibrations of a viscoelastic 

structure decay very slowly, which permits the 

use of an approximation wherein stress-strain 

equations for steady-state harmonic vibration are 

employed. Thus, by the correspondence principle 

of linear viscoelzsticity, tht elastic Young's 

modulus E is replaced by the corresjonding 

complex modulus E*, where 

= 

E = E'(1-i Tans,) L 

- 

2 

(3) 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

which, in general, depends on frequency. The 

boundary condltions for the simply supported 

Eu ler-Bernoull i bem are 

at x=O, L (4) 

The deflection of the nth mode of vibration is 

giben by 

nn x wn(x.t) = An sin - Le 

The frequency equation 

(~1.2, ...) (5) ut n 

s given by 

I* 

2 w;, W'l n wn TamE = , "2 '2 = - 0 n 2(1 - wn /Un 1 Wn 

I1 I 

which shows that bn /Wn is small when 

material loss tangent tamE fs small. 

The coupled equations for the 

deflection w and bending slope 41 for 

Timoshenko beam17 are as' f 01 lows : 

2 

ax at 

( 3w a* 0 2 

EI~+~(~-w)AG-Io~= 

the 

otal 

the 

0) 

2 iw OA 7 3t - k . (7 ix - E) AG = 0 2 aw 

Since E* is complex, the frequency values are 

also complex and can be written 2s 

Eliminating 3 from equations (10) and (11). the 

following equation in w is obtained: 

I II 

, =L' +iu 

nn n (7) 

I, 

Observe that when wn 0, the displacement, 

equation (5). is predicted to decay 

exponentially. Separating the real and imaginary 

parrs of equarion (6) resulrs in 

E=- ' 0AL4 '2 

nn I 4 4 wn 

A1 so, 

4 

aw 

ax at 

2 li W' I 4 E1 i w + 7 - (1+E/kG) - 22 -- 

2 oA ax 

+L-= 0 2 4w 

kGA .,4 

Ob 

Tne boundary conditions for the simply supported 

Timoshenko beam Ere given by 

0 at x=O,L (i3) ae 

'OX .w=-= 

The deflection of the nth node of vibration iS 

given by 

wn(x,t) = Ansin - nnx eiWnt 

L 

As in the iuler-Bernoulli theory for smtll 

damping, tile complex frequency equarion is 

obtained by using Equarion (14) in (12) and by 

replacing E and G by :ne complex moauli Ex and 

G , where t 

3 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

and 

Therefore, 

2 

n 

- f 41 EIE 

OA L4 kG 

+cI y4'0 

kG*A 

(15) or 

NO\, csnsider again equation (7) and the small 

loss tsngent approximation, such that 

"2 

'2 

n 

(18) W 2 2 L- ~c 1, Tan tE << 1. Tan oG << 1 U 

I1 

Using (19) in (17) and separating the real and 

imaginary parts results in two equations. 

Setting the real part equal to zero gives the 

f ol 1 owing equat i on: 

- 

1 44 EInDA L4 n kG 

I1 

n2n2 < (TansE - TancG) >]I n 

A kG wn 

+I, 

L 

I1 

whicn is identical to the I frequency I equation for 

an elastic beam except E and G appear in place 

of elastic Constants. Similarly, by setting the 

imaginary part equal to zero, the following 

equation is obtained: 

kG A 

I 

Observe that un can be obtained from eauation 

(21) for each n. This value can then be 

substituted into equation (22) to obtzin 

w n . wn can be easily estimated 

approximation, it can be shown that the last 

ten in eauation (21) is small compared 

to . n B I/L A. Neglecting the last ten, the 

frequency un is given by 

I, I 

when n 22 n I/L 2 A is a small quantity. Using this 

22 2 

I 

E 22 22 1GI = JG' nr. t 1-- a I 1 -71) (23) 

n L2 2L kG 

Tano for the viscoelastic Tirnosnenko beam is 

defined as, 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I1 

2wn 

wn 

Tane - - 

I I1 

where wn and wn are found from equation (22). 

This loss tangent provides a measure of the decay 

per cycle of free vibration, which can be shown 

by substituting equation (7) into (5). and 

replacing wn in favor of tano by means of 

equation (24). The ratio of displacement at 

time t + 2r/un to that at time t Is found to 

be exp(-ntanc). Note that tane = tamE for the 

Euler-Bernoulli theory, equation (9). 

,I 

I 

Discussion of Results 

The influence of shear deformation on the 

damping will depend on the ratios of 

E /G , TanGG/TanGE, the length to thickness 

ratio, and mode number. For unidirectional 

It 

.. composites, the ratios E /G and TancE/TanoG 

depend on the volume fraction of fibers and. 

matrix and the moduli of fibers and matrix. 8y 

considering different material systems and 

different orientation angles it is possible to 

have uide variaxions in the rctios 

In the continuum model of 

large space structures, the effective shear and 

longitudinal moduli can be varied independently 

of each other by changing the configuration of 

the individual cells. All the results presented 

in this study will be given in a dimensionless 

form. 

The material properties and dimensions of 

the beam are given in Table 1. For the purpose 

of illustration the loss factors and storage 

modulus are assumed to be independent of the 

frequency . 

II 

E/G and TanoE/TanGG. 

Table 1 

Typical Properties of A Composite Beam 

E' = 20w106psi E- 10-40 

G 

4 = 20-100 

TanoG 

- = 100-1 TamE 

L = 50 in 

TanoG = 0.05 

p 5 0.065 lb/in width b = 1.0 in 3 

shape fsctor k = 5/6 

I, 

The influence of E /G, TancG/TamEE' L/h and mode 

numbers on the ratio of damping predicted by the 

Timoshenko and Euler-6ernoulli theories are shown 

in Figs. 1 through 8. 

Figure 1 is a plot of Tanc/TancE for 

The first 

mode of vibration (n=l) is considered. The 

length to tnickness ratio of the beam is 100. I: 

is found that for large values 

of TancG/TaneE (100) the damping values predicted 

by Timoshenko theory are 9% to 38% higher than 

the prediction by Euler-Bernoulli theory 

depend i ng on the ratio of E /G. 

Increasing E /G 

TanbG/Tanci varying from 100 to 1. 

II 

*I 

increases the shear effects. 

The third mode of vibrztion (n=3) is 

considered in Fig. 2. The effect of shear 

damping is more pronounced than in the first 

mode. Figures 3 and 4 illustrate similar trends 

for a beam of smaller length to thickness ratio 

(50). The damping Tano predicted by the 

Timoshenko theory is substantially higher than 

Tan;E (38% to 1300% for TanGG/TancE = 100). 

It is interesting to no:e that when the 

difference between TcnZG and TansE becomes small 

tne difference in prediczed amping values 

obtained from the Timoshenko and Euler-Sernouiii 

theories is very small . 

in Figs. 5 and E damping is predicted for 

different modes wnile keeping the ratio of 
I 

ID 

,I 

ID 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

,I 

E /G constant. For higher modes the shear 

effects become important and this effect is 

enhanced as the ratio of Tanei/7aneE is 

increased. 

Figures 7 and 8 illustrate the effect of the 

variation of length to thickness ratio on 

damping. For small L/h ratios the damping 

predicted by Timoshenko theory is higher than the 

predictior, by Euler-Bernoulli beam theory. 

Conclusions 

This study has investigated the effects of 

shear deformations and rotary inertia on the 

damping characteristics of beams. For typical 

longitudinal ly reinforced composite beams, the 

loss factors in shear ( T a m G ) . a r e substantially 

higher than TancE. This leads to large 

differences between the predictions obtained from 

Euler-Bernouil i and Timoshenko beam theory. The 

results of this study indicate that for long 

slender o2anis in which the ratio of 

Tan.: /Tan;E and E'/G' is large, the influence of 

shear damping is important even in low 

vibrational modes. The methods and results from 

this inves:iga:ion can be used to study the role 

of Snear damping in higher modes of vibration of 

large space strucfures. 

G 

Acknowl edaement 

The authors gratefully acknowledge the 

support provided for this research by NASA 

Johnson Space Center under grant no. NAG 9-140. 

References 

111 Golla D.F., and Hughes P.C., "Dynamics of 

V i s co e 1 as t i c 5 t ruc tu r e s , Ti me -D oma i n , Finite Element Formu lati on" , Journal of 

Amlied Mechanics, Vol. 52, Dec. 1985. 

121 Scnultz. A.B., and Uarwick, D.N., "Vibration 

Response: A Non-Destructive Test for 

Fatioue Crack Damaae in Fiiment-Reinforced 

A 

Composites," Journil of Cornarsiie Materials, 

Vol. 5.. July 1071. 

6 

131 

141 

151 

61 

71 

IS I 

19 1 

Gibson, R.F.. and Plunkett R., 'Dynamic 

Mechanical Behavior of Flber-Reinforced 

Cmwsi tes: Measurement and Analysis.' 

Journal of ComDosite Materials, Vol. lo, 

Oct. 1976. 

Adam, R.D., and Bacon D.G.C., 'Effect Of 

Fibre Orientation and Laminate Geometry on 

the Dynamic Properties of CFRP,' Journal of 

Composite Materials, Vol. 7, Oct. '1973. 

Adams. R.D.. Fox, M.A.O., Flood R.J.L., 

Friend. R.T., and Heuitt. R.L., "The Dynamic 

Properties of Unidirectional Carbon and 

Glass - ___ Fibre ~ ~ Reinforced Plastics in Torsion 

and Flexure," Journal of com~o;i;e 

Materials. Vol. 3, 1969. 

Ni, R.G.. and Adams. R.D., "The Damping and 

Dynamic Moduli of Symnetric Laminated 

Composite Beams - Theoretical and 

Experimental Results ." Journal of ComDosi te 

Materials, Vol. 18, March 1984. 

Schultz. A.B., and Tsai, S.W., "Dynamic 

Moduli and DamDinq Ratios of Fibre- -. - 

Reinforced Composi'tes," Journal of ComDosi te 

Materials, Vol. 2, July 1966. 

Schul tz. A.E., and Tsai, S.W., "Measurements 

of Complex Dynamic Moduli for Laminated 

Fibre-Reinforced Composites," Journal of 

Comosite Materials, Vol. 3, July 1565. 

Sun, C.T., Kim, B.J., and Bogdanoff. J.L., 

"On The Derivation of Equivalent Simple 

Models for Beam and Plate Like Structures in 

1101 Sun, C.T., and Juang, J.N., "Modeling Globrl 

Structual Damping in Trusses Using Simpie 

Continum Models," Proceedinas AIM 24th 

Structures, Structural Dvnamics, and 

Mazeriels Conference, Lake Tahoe, Nevadc, 

May 1953. 

1111 Schapery, R.A., "Viscoelastic Behavior and 

Analysis of Composite Materials," Chapter 4. 

ComDosite Materials, Vol. 2. G.P., Sendecky, 

Ed., Acaaemic Press, tiew York, 1974. 

1121 Timoshenko. S.P., Vibration Problems in 

Enaineerinq. Third Edition, 0. Van tiostrand 

Company, inc., New York, 1055. 

1131 Hearmon. R.F.S., "The Influence of Shear and 

Rotary Inertia on the Free Flexural 

Vibrations of Wooden Beams." Eritish Journal 

of Aoolied Phvsics, Vol. 9, 10%. 

1141 Dudek, T.J., "Ycung's and <hear Moduli of 

Unidirectional Composites by a Resonant Bean 

Method," Jcurnal of ComDcsite Materials, 

Vol. 4, 1570. 

1151 Eiot, M.A.. "A Kew Approach to the Mechenics __ 

of Anisotropic Multileyered Plates." ict. J. 

Solids Structures, Vol E., !972. 
I 

I 

-I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

Biot. H.A., "Simplified Dynamics of 

Hultilayed Orthotropic Viscoelastic Plates," 

Int. J. Solids Structures. Vol. 8, 1972. 

Huang, T.C., "The Effect of Rotary Inertia 

and of Shear Deformation on the Frequency 

and Normal Mode Equations of Uniform Beams 

With Simple End Conditions." Journal of 

Applied Mechanics, Vol. 28, No. 4, 1961. 

1.40 

1.30 

4 

t 

2 1.10 

C 1.20 

0 

-8- 

t 

1 .oo 

0.90 ' 1 

0.60 25.b0 5.0.00 75.00 . lO0:OO 

Fig. 1 Effect of shear deformation on dempin5 

z function of Tan;-/Tanc, for variolrs 

values of E'/G' (LFh = 100, n=l) 

5.00 

4.00 

W 

h 

c 3.00 

0 

t * 

c 

2.00 

1 .oo 

0.00 

C 

E'/G' 

0 10 / 0 20 A 30 

1 25.b0 5,O.bO 75.b0 1OC 

Tan@, lb /Tan?, - 

25 

.oo 

Fjo. 2 Effect 0: shear deformtion on den?ing 2s 

a function of TanL./Tan<E for various 

values of E'/G' (i/h = 100, n=3) 

7 
I 

I 

I 

1 

I 

I 

I 

2.50 

c 2.00 

0 

-8- 

t 

-c 

1.50 

1 .oo 

0.50 ' 

0.60 25.b0 50.bO 75.b0 lOO!OO 

E 

Ton6 /Ton? 

IG 

Fig. 3 Effect of shear deformation on damping as 

e function of Tan: /Tan;, for various G values 0: E'/G' (L/h = 50, n=l) 

-1 12.00 E'/G' 

10.00 

Y 

-W 

c 

o 8.00 

t 

-8- 6 6.00 

k. 

4.00 

2.00 

0.00 ' 

0.60 25-00 50.b0 75.b0 101 

Tcnb /Tandl IG IE 

)O 

Fig. 4 Effect of shear deformation on damping 25 

a function of TancG/jane, for various 

values of E'/G' (L/h = 50, n=3) 

8 

lsO 

1.40 

1.30 e# 

c 

0 

I21.20 

c 

U 

t1.10 

1.00 

I I 0.90 4 1 1 

N 

I 1 

1 .o 2.0 3.0 4.0 5.0 

Fig. 5 Effect of shear deformation on dampino 2s 

a function of n for various values of 

TantE/TaneG (L/h = 100, E'/G' = 20) 

2-50 I / 

/ 

2.00 

4 

\1.50 

c 

0 

I-- 

c 

0 

I1 .oo 

A 0.3 

* (2.4 

Fig. 6 Effect 0: shear deformation on aacping 2S 

a iunczion of n for various values of 

Tanc,/TancG L (L/h = 50, E'/G' = 20) 
I 

M 

I 

I 

I 

I 

I 

I 

I 

'I 

I 

I 

I 

I 

I 

I 

~I 

I 

1 

1 .so 

1.40 

€JsO C 

0 

tl 20 

43-* 

C 

0 

I1.10 

1 .oo 

0.90 

( 25.0 50.0 75.0 100.0 

L/h 

Fig. 7 Effect of shear deformation on damping as 

a function of L/h for various values of 

TZntE/TantG (n=1, E'/G' = 20) 

4.00 

3.50 

3.00 

w * 

t 

6 2.00 

2.50 

I1 SO 

1 .oo 

0.50 

0 0.2 A 0.3 

0.4 \ Q 

"\ 

2510 501.0 7510 10 

L/h 

0 

Fis. 8 Effect of shear deformztion on damping 2s 

a function of L/h for various values of 

TanCE/TanCG L (n=3, E'/G' = 20) 

9 
I 

I 

I 

I 

I 

I 

'I 

I 

8.2 Prediction and Experimental Observation of Damage 

Dependent Damping in Laminated Composite Beams 
I 

I 

I 

I 

I 

I 

I 

PREDICTION AND EXPERIMENTAL OBSERVATION OF DAMAGE DEPENDENT 

DAMPING IN LAMINATED COMPOSITE BEAMS 

D. H. Allen. C. E. Hanir. and A L. Hi#vmirh 

Amrorpacr Enginrwinp Dqwtmmt 

Trxar ALM Univrnin 

Collrpc Station, Texm 

1- 

:I 

I 

I 

I 

'I 

'I 

I 

I 

h3STRACT 

The equations of motion are developed herein 

for laminated composite beams with load-lnduced 

mttrix cracking. The damage is accounted for by 

u:illzing Internal State variables. The net 

rcsu.lt of -there varAables on the fleld equations 

fs the introductlon of both enhanced dani?lnS. and 

degraded stiffness. Both quantities are history 

drpendent and spa:ially variable, thus resulting 

in nonlinear equatlons of notion. It 1s 

explained brlefly hw these equations may be 

outs 4-1 1 nearized for lam1 ncted polymeric composites under certain types of structural 

loading. The coupled heat conductfon equation is 

developed. and It is shown that an enhanced Zener 

damping effect Is produced by the introduction of 

cfcrostructural damage. The resulting equations 

tre ut( ltzed to demonstrate how dunage dependent 

material properties may be obtained from dynamic experimen:s. Finally, experimental results are compared to model predictions for several 

conpos 1 t e 1 cyupr . HOHEHCUTURE 

k - beam cross-sectional area 

Ai - cross-sectional area of lth lamina 

A+ - modulus weighted beam cross- 

$* - coefficients in cantilever beam em 

b - beam width 

C - viscoelastic modulus 

C" 

sectional area 

displ acement f ield 

- specific heat L: constant volume 

- coefficients in cantilever beam 

displacement field Cl* c2 

2.53 

Eo, El, n1 

E - axlal modulus of elasticity 

- viscoelastic material parameters 

- axial modulus of elastlclty of lth 

- reference axlal modulus of 

lamina E1 

El elasticity 

end of cantilever beam 

- transverse load anplitude 

F - transverse load applted to free 

FA 

h - beam depth 

- modulus weighted moment of inertia 

IYY about y axis 

k - material constant in internal 

+ 

state variaDle growth law 

- coefficient of heat conduction 

- loss in heat conduction 

ki j 

coefficient due to damage kDi j 

L' - beam length 

- beam moment resultant about y axis ' % 

- beam moment resultant about y axis 

due to temperature change 

H - number of cycles of applied load ' 

n - number of plies in laminate 

- material constant in tnternal 

state varieble growth law "2 

P - beam axial resulten: 

P' - beem exial resultant due to 

temperrture change 

- 
I 

I 

I 

I 

I 

- beam uial load per unit length 

- beam transverse load per unit 

- heat flux vector 

- heat source 

- term used to obtain lminate 

length 

emping coefflclent 

local volume element 

- surface area of aratrlx cracks In 

- t wpe rature 

- reference temperature 

- tlme 

I 

z 

ax1 a1 d Ispl acement 

crack opening displacement 

axial displacement of neutral 

surf ace 

Internal energy dissipation per 

unit volume due to matrix cracklng 

total Internal energy disslpatlon 

due to matrlx cracklng 

beam shear resultant In t 

d ire:tlon 

transverse displacement of neutral 

surf ace 

beaQ axial coordinate directton. 

beam transverse coordlnate 

a I r ec ti on 

z component of modulus weighted 

centroidal axis 

2 component of distance from 

centrold Of lth ply to modulus 

welghted centrold 

coefflclent of therm1 expansion 

Internal state variable (1%) due . to damage 

axial component of matrix cracklng 

damage 1SV 

Cxla1 component of matrlx cracklng 

image ISV In ith lamlna 

Initial displacement applled to 

'ree end of cantllever beam 

ixlal strain component 

iverage axinl strain In ?:h lamina 

- dmplng Coefficient 

"A 1 

"B 1 

"hi 

"B i 

P 

OA 

OXX 

- axial damping coefficient in Ith 

- bending dunping coefficient in ith 

- axial stiffness loss coefflclent 

- bending stiffness loss coefficient 

- modal frequency of mth resonant 

- m&ss density 

- stress amplitude of cyclic loadlng 

- axlal stress component 

lamina 

lsnina 

in ith lamlna 

in ith lamina 

mode 

- shear stress component OXZ 

e - temperature mlnus reference 

temperature 

- beam rotation about y axis 

Y e 

INTRODUtTlON 

Microstructural damage in lamlnated flbrous 

composltes has been studied by several 

researchers in the pest decade. It hcs been 

observed that thls dauage can nSUli In 

significant stiffness loss 11-31. Furthermore. 

experlmental results indicate that aterial 

damplng may increase by an order of magnitude or more as a result of damage 14.51. Recent 

research has Indlcated that this damace'lnduced 

stlffness loss may have a substantial effecs on 

the dynamic response of both truss structures 161 

and beams in bending 17). However, to these 

authors' knowledge, no concerted effort has been 

previously reported to model the effect of damage 

induced damping on dynamic structural response. 

While there have been models Ii?.9] oeveloped for 

damping in laminated composi.tes. these models do 

not include the effects of microstructural 

damage. The prlmary objective of the research 

described herein is to develop models that 

expllcitely account for the effect of 

microstructural damage on the stiffness and 

damping In the structural dynamics equations of 

mtion. 

In this paper a model Is developed for 

predlcting the structural dynamics response of 

composlte beams In both bending and extension, 

where mlcrostructural damage induces both damping 

and stlffness loss. The equatlons of motion are 

formulated using the theory of Internal state 

variables (ISV's) to account for damage 110- 

12). The resulting equations are ccst In such 2 

way that experimental data can be readily 

implemented to :he model. 

DEVELOPMENT OF THE EQ'JATIOHS OF K3TIOH 

Conslder a free body Ciagram of a luninated 

compos;te bem. as shown In Fig. 1. 
I 

I 

I 

I 

I 

I 

I 

II 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I: I 

L 

pX 

v 

z 

Fls. 1. Free Body DIagrarn of Lamlnald Cornpoille Bairn 

The governing equations are 

1) resultant loads 1131 

P P I oxxdA (la) 

k 

equilibrium 1131, neglecting rotatory inertla 

1141 

an 

2-V 

ax z 

3) kinematic I13.151 

au 

xx ax t =- 

awO e =- Y ax 

u = uo+ e z. u = uo(x). e - e,,(x) (3c) Y .Y 

where equation (3c) Is the Euler-Bernoulli 

as sumpti on. 

4) thermrechanlcal cons:itution IlO.11J 

TR uxx = E(cxx - a1 - E e). B E T - 

si = -(kj, - kDij1T.j 

255 

i1 - nl(cU(. 7, el) (4c) 

kDIj - kDlj('l) (44 

where a1 Is an Internal state varlable 

representing matrix cracklng In each ply 191. It 

can be shown that al Is a kInematIc descrlptlon 

of the locally averaged mlcrocrack geometry 

llO*lll: that is. 

(5) 

where VL Is a local characterlstlc volume which 

is large enough to encompass StatlStkA1ly 

homogeneous damage. Furthermore. uc Is the 

normal component of crack opening displacement, 

and 5, Is the surface area of cracks in VL* as shown in Flg. 2. V, may typfcally be taken IS 

one square inch in the x-y plane for laminated 

polymeric composites [ 31. For matrix cracking 

the 2 Coordinate Is taken to be one ply thickness 

1111. Equation (4c) is termed the internal State 

varlable grodLh law. 

A= 

I 

L- sc "c LV L 

Fig. 2. Klnenatics of Matrix Cracking 

5) thermodynamics [ 16.17 1 

(6) 31 'It -ocv E - q,,, + b = o 

which Is a statement of conservation of energy 

for the material described by equationsc (4). 

Although specifIc forms may be Given for uL .cs a 

function of cXx and al. thls Is covered In a 

later sectlon. Hote that although the mechanical 

fornulation Is one-dimensional in the spatl&l 

coordinate x, nulti-dimensionality nus: be 

retained in thermdynamlc equation (6). 

Kou subst1:u:e (3c) into (3a) to obtain 

(7) 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

1 

I 

I 

1 

I 

Substituting the above result into (sa) gives 

Now substitute (8) into (la) and (IC) to obtain 

where all is assumed to be independent of y and 

2 in the Ith ply 1111. and 

F A* i I f- dh (loa) 

A1 

Furthermore, 

T P t EGedh (11~) 

A 

The term conzaining a In equation (9b) does not 

go to zero even lifor a symnetric layup 

because ali Is not symtric (due to the crack 

opening displacements in equation (5)) during bending. 

Now. suppose the modulus weighted centroidal 

€Xis is utilized, such that, 

Note that -since E Is the inItla1 undamaged 

modulus. 2 represents the Initial modulus 

weighted centroidal axis. which Is not damage 

dependent. Therefore, using equations (3b) and 

(12). equations (sa) and (9b) simpllfy to the 

following for symnetric layups 

The term containing all in equation (13b) 

represents a shlft in thE centroidal Lxis due to 

matrix cracking, which is nonzero when the matrix 

crack displacements Ere not symnetric through the -. 

256 

thickness. 

ad with asymetric l~~~ps 1151. 

This is generally the case In bending 

Nor substitute (l3a) Into (za) to obtain 

which is the governing equation of motlon for 

axial dl splacemcnt . Similarly, substitute (13b) into (2c) to 

obtain 

Substituting (15) into (2b) thus gives 

whtch is the governin: equation of motion for 

transverse di sp1 acement. 

Now consider a simple examle for the 

constitutive behavior. Suppose thzt the stressstrain relation can be mdelled by a Voic: 

element, shown in Fig. 3. 

The governing differenzial equction for the 

analog is 

0 = ECxx + GXX - Eo8 (17) xx 

Suppose we let 

Then equation (17) may be written 

oxx = E(cxx - all - &e) (19) 

which Is equivalent to (4~). and (18) replaces 

(4c); that is. 

t1 A -A "1i(tl) = -- s Q1(Cxxi' CXXI' T. all)dt = 

Substltuting (7) and (3b) into (1E) gives 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I. 

I 

I 

I 

I 

I 

I 

~' 

I 

U 

i 

FIo. 3. Volot Model 

1 

Therefore, substituting (21) into (14) plves 

Furthennore, substituting (21) into (16) gives 

. 

-P, + 2 

ax 

i 

L 

Therefore, if nAi and nei are constants in 

time t. the Lbove reduces to lineer theory. 

Experlmenttl evidence indicctes thet 

pol-m-ric composites may not be represented by 

simple mechaniccl analogs such as the Voip: model 

1181. For small displacements c single integrcl mdel of the fofloving form hcs been shown to be 

accurate 119 ] : 

(24) 

c(t-7) K(cxx a - oe)dr 

- 

where the kernel function C is a power law in 

time, end is also P weak function of damge (for 

neer-constant amplitude vibrations); that is, 

here f, - to(.,). El - El(al) and nl - nl(al)- 

In general . the constitutive del 

represented by equation (46) Is not in agreerrnt 

vith equltion (24). However, it Is our Intent to 

shw that this assumption is at letst 

Approximately correct. To do this, first recall 

that equation (24) nay be represented by an inflnlte order differential equbtion: 

fquatlon (26) may be written in the same fonn LS 

equation (17). whert ln this Case 

n n(ixX. cxx, ---, 9) (27) 

Therefore, epuations (22) and (23) mty still 

be used except that the damping coefficients are 

not constant. For near-constant amp1 itude load 

histories. it is nor essumed that 

Furthennore, experimentcl evidence 1201 indicctes 

that for constant amplitude cyclic loading 

Al = koA "2 (29) 

where k and nz ere materiel constants. 

For n2>>1, the above may be integrated in time to 

plve 

n1 = y(H. oA) (30) 

n = n(cA, L, (31) 

so that equation (27) mcy be simplified to 

How, consider L more genere1 representctlon 

of equation (21) given by 

where 

Substitutin; (52) into (14) cives 

257 
1 '. 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I The last tern in each of equations (34) and 

(37) vi11 produce Zener-like daaping due to 

therromtchanlcal coupling in equation (6). To 

conslder these two terns, the fully coupled field 

problem must be solved. That is. esuations (6). 

(34). and (37) ccnnprlse a set of three coupled 

equatlons In three unknowns. uo, wo, and T. 

Determination of Material Damotno Constants 

at least first order in damage [lo]: that Is, 

The energy dissipatlon due to damage nust be 

a '"0 a pT - doA -) at - -Px + )x (34) 

where 

- r" 1-1 'AiCiAi 

'A0 E EIA. 

- f 'BiElG 1-1 

'B1 hEIA' 

Therefore. assuming that Zener damping may 

be neglected, a flrst order approximation Is 

glven by [lO.ll] 

q - E CXXDl ' (41) 

Thus, the total energy dlsslpatlon durlng I 

simple cycle in a specimen of length L is given 

by 

t.+2./u , 

Substituting equation (3) for the ccse of 

n pure bendlng thus gives 

Thus, substituting (32) Into the rbove 

results in 

Substituting (32) Into (16) gives 

4 

-- 'B( )dxdt (44) Ei 2l -.2. 2 d6. .X 

The cbove neglects energy losses due to shearing . 1211, which may not be negligible in laminared 

composites 1181. (37) 

Hw consider the case where the dmage is 

independent of x and no new damage Is introduced 

during the cycle of Interest. It follws that 

for symnetric vlbrationS. She first term in :he 

rbove equation 1s zero, and equation (44) reduces 

.' 

(384 to 

where 

(45) 

u'e2 f 

t1+27/u 2 4 

t. o ax2 rt2ax2 

i wo a wo SE 8 J - - dxdt ($6) 

and 

(398) 

(39b) 

1 

Substituting equation (39b) and rearranging thus 

resulrs in 

(47) 

IYY 

258 
Therefore, for r berm of glven geometry rnd 

boundary condltlons. eouatlon (47) can be used to 

obtrln nB2. fuurtlon (390) crn then be 

utlllred to extract nB1 for ccrtaln slnpllfled 

strcklq sequences. This lnformtlon nay then be 

ulllized -to prcdlct the damping coefflclents 

hS 6 functlon of dsnrgc ln other 

strcklng seauences by using equation (39b) for 

other condltlonr. 

klthoqh It 1s In principal posrlble to 

obtain the other coefflclents descrlbed by 

equations (35). (36). (38) and (39). It 1s 

difflcult to accurctely measure U: durlhg 

dynamlc axlsl tests, so thet xAo and ;;kl arc 

not ecslly cvrlueted by the method descrlbed 

here. - Howper. the- stiffness loss coefficients 

pal, pA1, and uB2 mty be obtclned from 

direct static testlng 111,12]. 

OB1 "BZ 

'A0 * 

Exper1mn:cl Determinetton of Dcmlno 

6eceuse of its relative simpllctty, e 

ccntilevered beac hcs been se1e:ted cs the 

current test conf lguration for studylng materiel 

dampinc. (1: should be noted thct the 

theoretlcal development descrlbed In the previous 

section Is valid for ell beam bounCrrr 

cord1:ions. WE are also lnveszlgtttng other becm 

boundcry conditions such as the free-free been 

but will conflne the present discussion to the 

behavior of L cantilever bean). The laboretory 

setup is shorn schematically in Fig. 4. X coupan 

1.0 in. (25.5 mn) wide snd 12.0 In. (332.6 m) 

long Is clamped In c bean bending fixture to 

echieve c ccntilever of :he desired leneth. The 

be-, is set Into vlbrctory n3tion by c motorized 

meel which rotctes until c pas; engages the free 

end of the beu;.. cchieves c presole:ted 

oeflection end then relecses the free enc!. 

Therefore. :+e frequency End em?litUoe of the 

bo,&,, ere cor.:tolied by vcryin; the length of the 

be.% end the initjal def1ec:iofi. The freo de:€)' 

response is recordeC by e scrrin gage momzed on the bean near the c'lmp. The bend fixture is 

housed in c vacuurr, Chambet end the test is 

Initiated externally by c corputer h%Ich elso 

acouires the strain gece de&? in Cig:tal fwr. 

The OULLPU: of the strcin pcc? is recorded et 1503 

ZctC uoinrs uer seconl. Tne easllfied signrl 

from the' 1000 ohm strcin gcoe is filterei to 

Improve the deta reduction accuracy. The dmping 

of the beac is cc1cule:ed by :he 1oGarithic 

decrement method. 

Ir, thc present study the outer fiber strein 

ucplitude wcs selected so thet the logarithmic 

decemnt response of the bem, is llnear. The 

system, (join:) dmpino hcs been estimated from 

tests on several well knoirn metericls and hcs 

been subtracted from the experimental dmping 

values meesured from the composite. It iS 

recognized by the hriters thc: the centilever 

bean may no: be :he nos; desircble test 

configuretion for e study of metericl daa7in;. 

However, our primcry objective is to stuay 

orn;in; cs c function of nicros:ruc:urtl 

denage. inerefore. by construc:ing carefull)' -. 

controlled laboratory expcrlrnts. the rrlrtfvc 

change in drrglnp due to a progrtssion of 

mlcrostructurrl damage will be rclrtivcly 

In6epedent of the system dunping whlch should 

rewin constant. 

An*l_vsis of t Ccntileveret Eecm 

This section providos the speciclizction of 

the dunage-depenoer.t bema theory for the 

ccr,tilever beu;. The analyticel procedure fot 

oetemining the material damping constents frm, 

the experiwntal dczc is also descr'ibed. 

For en undmped prismatic centilever bEm, 

undercoin: free vibrc:ions, the solution is of 

:ne fox, 122) 

where 

fr(t) = C.sin LI. c t - C2cos t~~,t. 

and 

(slnh cr,x - sin cnx) 

A1 so 

E! L = 1.675 r= 1 

( 6- 1 /2 ) 7 r~ i li, 

259 

ORIGINAL PAGE IS 

OF POOR QUALtTy 

(51) 
1 

I 

'I 

I 

I 

I. 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

and 

Furthermore, depends on the mkxlmun 

deflection and%, and C depend on the in1 

conditions. Equatlon (52) may be used 

tlP 

i a1 

to 

determine the iamlnate bendl ng 

coefficient yB2 when wm can be obtained from an 

experimental test. 

In our initial phase of research, the 

damage-dependent constants are mre easily 

determined if we consider only the first mde of 

vibration and confine our experiments to the 

first mode. (Later research will include other 

modes of vibration.) Considering only the first 

mode and a beam with the fol'lorlng lnltial 

conditions: 

wc(o.L) = 6 (53a) 

(53b) - a "0 (0.x) = 0 ;t 

Then Cl = 0 and C2 = 1. Furthermore. 

el(x) - A1[(G.73:l)(sinh 1.875 f - sln 1.875 4) 

+ cosh 1.S75 f - cos 1.675 (54) 

Substituting the above into (48) gives 

wo(t.x) - Altos ult I(0.73Sl)(slnh 1.875 f 

X - sin 1.875 5 + cosh 1.875 L 

- cos 1.875 f] (55) 

Satisfying initial conditlon (53) gives 

6 

5.2734 Al = - 

Therefore, 

wo(t.x) = j.2734 cos 10.7341 (sinh 1.875: 

X - sin 1.875 i) + COS 1.E75 1 

X - COS 1.875 TI 

Substltuting (57) into (46) thus gives 

(57) 

10.7341 (slnh 1.875 + sin 1.875 f ) 

XZ - + co'sh 1.875 f + COS 1.875 i] dxdt 

substitutlng Into equation (47) thus glves 

(58) 

Integrating the above equation &nd 

The energy dlsslpated per cycle, U;, of a 

vlbratlng beam is related to the experlmenzally 

determined logarlthmic decrement, t., by the 

folloding relazionship 

where Us Is the maxiwm strain energy of the 

beam. The strain energy of the beam ls'given by .I 

5 'W 

U - $ ,JL E (+)2 :I 2' dAdx (61) .. and for a cantilever vibratlng in the first -de 

with an initla1 tip deflerzion, A , 

Substituting equations (109). (60). and (62) into 

(59) gives 

- tlA 

nE2 = 0.31627 (7) 

1 

The procedure used herein to determine the 

damplng constants, qE1, is to experimentally 

measure the logarittmic decrement for a 

-. [0/90/0]S laminate and calculaxe f rm 

.-- equations (63) and (298). Since wen%? only 

considering maxrix crack damage at present, we 

will assum that all the damping may be 

axtributed to the 90' plies of the 10/90/O]s 

laminate. This will provide an experimental measure of the dunping in a single 90' ply ulZh 

matrix crack damage. Substituting equation (398) 

Into equczion (63) and rearranging gives the 

folloving expression for the logarlthmic 

decrement 

.- I 

Once r,21 is determined IS a functlon of dunage, 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

couation (64) can be used to predict the damping 

of a cantilever beam of rny cross-ply lamirate 

stacking ietwence. Whlle tquatlon (64) appears 

linear In damping this Is not the case because 

the damping constants, nBi are also function of 

frequency. (See equstlon 33). 

Comparison of Ixperimntel ad Analytical Results 

Experimental results have been obtained fra 

cross-ply lamlnates ulth varlous combinatlons of 

0’ and 90’ plles of AS-4/3502 grdphite/epoxy. 

The lmlnates were fabricated using a standard 

prepreg tape IWUP dnd hot press curing process 

follwlng the curing procedure recomnended by the 

prepreg tape vendor. The approximate per Dly 

thickness of the cured laminetes wcs 0.005’ 

(0.127 mm) and the- fiber volume fraction was approximately 62:. The mecsurtd lamine 

propertles are gjven ln Table 1. These 

properties are typictl of this materiel system. 

The theory presented herein res developed 

only for materiel damping. Therefore, it ucs 

essentls’l to seperote the system damging from the 

messured values of loperithmlc decrement to 

detennine mctertcl darnvjnp constan:$ and to 

comare theoretlccl pred1c:lons to experimental 

vclues. Using test results for t vrrlety of 

mctericls, the sysren dcqing for the current 

conflgurition of c ccntilevered beam, vibrc:ing 

predonlncntly in the firs: rode hcs been 

determine6 to bc rpproxlmctely 3.52~10”. 

Teble 1. P,c:erirl Properties for Hercules 

This 

M4/?502 

Lccinr Proaer:ies 

r 21.0 x 1C’ 5 2.0: psi (i44.6 GPc) 

c.634 X 16 6 psi (4.75, ;?e) 

‘il 

1.2s X IO6 2 2.:: PSI (f.55 tP2) 522 

-1 c1.e iL 

0.210 2 3.7: 

326003 2 2.5: FSi (2.246 Hot?) Fz” 

110S5 2 S.E% psi (76.4 HPc) 

0.0144 2 4.G in/ln 

0.03773 2 E.?: in/in 

fU 

tu 

i 

CtU 

Theoreticslly calculated from test ti&tc’for c 

12-45 ]Zs lami ncte. .-- 

velue of system damping wrs considered to be c 

constan: for a11 laninetes in this study. 

Considercble ettention wcs civen to the 

experimen:cl test paremete-s thet yielded :he 

m~st reDertcb1e results 2nd held consten: :hOse 

varamefers which influenced dm7ing such zs 

frepuency en:: stress .m>i irude. This procedture 

ellows for the n~;t direct con2erison between 

theory and experinen: for the lm,inrtes discussed 

herein. All test results reported herein ycrc 

obtrtned rt the freouency of the first &e of 

vlbration. The inltlal tip Ocf1ection was 

stlccted so that the outer flber stress near the 

root was ulthin the stress mplitude range where 

Oamping was constant and the free decay war 

llnear when plotted on a log scale. as shon In 

Flg. 5. 

rep1 iczte tests. 

h complete charccterizction of the damage 

depenoent mctericl dem;iing of grcphitelepoxy 

laniinctes is in progress. The experimentel 

results confirm e stron? relrtionship between 

rnzterial dampin? and cicrostructurcl ~~+tge. 

Figure 6 shovs tne relrtionship between rnztorir? C. 

tianpins 2nd citrcsrructurel aamege. ,-:~~re 6 

shods the reicrionsnip between nrtericl GzyJin; 

end mz.trix creck ob!;ace in the OS’ lcye- of E 

26 1 

ORlGfNAL PAGE IS 

OF POOR QUALITY 
I. 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

Table 2. 

LAHI HATE , STACKING 

SEQUENCE 

I 0/9 0 1 2s 

I 0/9 03 1 5 

I o/902 13s 

lo/903 1, developed 

Prcdlcted and Experlaentrl Values of 

Darpplng Detemlned by the 

Log ar 1 t hml c Decrement Met hod 

FIRST EXPERIMENTAL PREDICTED 

MOD E OAHP I NG DAHPING 

FREQUENCY 

18.5 4.5~~10-3 4 .2oX10-3 

18.7 5.11~10'~ 5.56~10'~ 

25.3 

laminate. (The damage states were in uniaxial tens lon-tens ion fat igue. ) 

As can be seen In flg. 6. damping, nannallzed by 

the initlal value of damping for the undamaged 

laminate. lncreases by about a factor of 2 as the 

matrix crack damage state progresses to the crack 

saturation state. 

Sumnarv cnd Conclusions 

Equations of motion for a vibrating beam 

have been developed for composlte materials which 

account for the influence of microstructural 

damage on beun stiffness and material damping. 

This new theoretical approach introduces damagedependence i nto the cons ti tu ti ve re 1 at ionshi pS 

via lnternal stite variables whlch characterlze 

the current micros:ructural dmage state. Since 

the constitutive re1a:ionship is time dependen:, 

the equations of motion contaln explicit terms 

representing the dunage-dependent material 

dampins. The equations of motion are developed 

for a lminated composite beam by enploying 

standard lamination thecry and assuming thaf 

InClviducl ply contributions to material dunping 

can be represented by a rule-of-mixtures 

formulation. The equrtions of motion may be 

solved to determine the dunage-dependent natural 

frequencies and mde shapes of thc bean;. The 

beam theory Is also used to construct a 

mathematlcal mdel of damage-dependent material ' 

damping at the lamlnate level whlch is the 

primary subject of this paper. 

Damage-depend en t mat erl a1 dampi ng constants 

Ere obtained from basellne experlmental data. 

Once the material constants are determined for P 

particular material system, the mathema; ical 

mdel of the lamlnate material damping is used to 

predict the damping of any other lamlnate 

stacking sequence. 

In order to verify the validity of the 

theoretical formlation. experimental results were obtained from cross-ply graphlte/epoxy 

laminates with a variety of stacking sequences. 

A Cantilevered beam was selected for experimental 

Study and the test results to date are confined 

to the first mde of vibration and at initial tip 

deflections that produced damping values 

relatively Indepenoent of stress unplitude. 

Because our sfudy is cvrrently only addressing 

the behavior of cross-ply laminates with the 

single dmage rode of matrix cracks in the 90' 

layers. we have assumed that all the material 

262 

daqing can be attrlbuted to the 90' layers. The 

-1 

I. 

undamaged damping constants for a 90' ply were detennlned from the experlmen:al data of the 

[0/90/0], laminate. The undmaged damping of the 

10/90] , 10/93 ] and [0/9O2]3, laminates was 

predictid by th3e 'zheoretical mael of damping. 

The predicted values for these laminates were 

within 28: of the measured values. While dunagedependent data are rut fully developed at this 

rim?. =he excellent agreement between the theory 

and experiment for the undamaged dampins provides 

verification of the theoretical formulation. 

ACKti OW LED; EHENTS 

The writers wish to acknowledge the 

financial support provided by a grant from HA% 

Johnson Space Center. 

REFERENCES 

1. Highsmith. A.L.. Stinchcomb. U.U.. End 

Reifsnider, K.L.. 'Stiffness Reduction 

Resulting from Transverse Cracking in 

Fiber-Reinforced Composite Laminates,' 

Virglnia Polytechnic Institute and State 

Unlversity. "PI-E-81.33, November, 1981. 

2. O'Brlen. T.K., "An Evaluation of 

Stiffness Reduction as a Oamaoe Parameter 

and Criterion for Fatigue Fallure In 

Composite Materials.' Ph.0. Dissertztlon. 

Vlrglnia Polytechnic Institute and State 

University. October. 1S7e. 

Damage Accuwlation In Graphite/Epoxy . L&minates; Texas AM Unlversity Thesis. 

Augus:, l9E5. 

4. Plunkett. R.. 'Dunping in Flber 

Reinforced Laminated Composltes at High 

4. 2 Norvell. R.G., 'An Investigztion of 

Strain.' J. Comosite Hatericls 

Sumlement. V- 
I 

I 

I 

I. 

I 

I 

I 

~I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

a 

5. 

6. 

7. 

8. 

9. 

ID. 

11. 

12. 

13. 

14. 

15. 

Ldkrg, D.L., *Material Duping of Srsle 

Structures in a Slulrted Spree 

Lnvlromnt,’ Journal Of Spacemrft rd 

Rockets, VOI. 23, pp. 286-296, 1986. 

Kalymrsundarrn, S., Lutz, J.D., Hrlsler, 

W.E., and Allen, D.H., ‘Effect of 

Degrrdrtion of Mstetlrl Properties on the 

Dynanlc Response of Large Sp6ct 

Structures .* Journal of Spacecraft and 

Rockets, Vol. 23, Pp. 297-302, 1986. 

chang, H.T., rnd Allen, D.H.. 'Predicted 

Dynamic Response of A caposlte Bern vlth 

History Dependent Omage.’ lexas AW 

University Hechanfcs and Haterials 

Center, H4 4875-86-17, July, 1986. 

Gibson, R.R.. and Plunkett, R., ’Dynarr,lc 

Mechanical Behavlor of Fiber-Reinforced 

Composltes: Heasurment and Analysis,’ J. 

Cowoslte Hatcrials, Vol. 10, pp. 3232 

341. 1976. 

Hi, R.G. and Adams. R.D., ’The Dmplng 

and Dynamic Hodull of Symmetric Laminated 

Composite 6eams - Theoretical and 

Experimental iiesults,’ J. cOmDOS1te 

Hateriels, Vol. le, pp. 104-121. 1584. 

Allen, D.H., Hcrris, C.E., and troves, 

S.E., ”A Thermmechanical Constitutive 

Theory for Elcstlc Connosltes with 

Distributed Damase - Part 1. Theoretical 

Development.” to appear in Internetionel 

Journcl of Solids and Structures, 1967. 

Allen. O.H.. Harris, C.E., and Groves, 

S.E., ”A Thermmchanical Cons:ltutive 

Theory for Elastic Connosites with 

Dlszributed Darntoe - Part 11: App1lca:ion 

to Hatrix Cracking in Laminated 

CoVosi:es,’ to cppecr In 1n:ernrtionrl 

Journcl of Solies end Structures, 1567. 

Allen. D.E., Harris, C.E.. Groves, S.E.. 

and h‘orvell, E.G., “Cnarecterfzetion of 

Stiffness Loss in Crossply Lminrtes 4th 

Curved Matrix Cracks,” to appear in 

Journcl of Composite Mcteriels, 1967. 

Allen. 0.E.. end Heisier. U.E.. 

Introduction . to AfY@SDECe Stmcturci 

Ansivs7s. Wiieg. hew lork. 1985. 

Timshenko, S.P., Vibrction Problems in 

Enoineerinc, 2nd edition, Van Hostrand, 

hew boric, 1937. 

Allen, 0.H.. Groves S.E., and Harris, 

C.E.. “A CunrJlatlve Damage Model for 

Continuous Fiber Composite Laminates with 

Matrix Cracking and Interp ly 

Delaminatlons,” to appear in MTH SDeclcl 

Technicel Publicetion, 8th Symposium on 

Composite Mazeriais: Testing and Design, 

American Society for Testing and 

Rsterials. 1987. 

263 

16. 

I?. 

18. 

19. 

20. 

21. 

22. 

Allen. D.H.. ‘A Prediction of k&t 

Allen, D.H., ’Predicted Axlrl Tenperature 

frrdient in r Viscuplastlc Unlrxltl Bar 

he tO Thernamcchsnlcsl bupllng,’ 

Internationel dournal for Numerical 

kthoas in Lmineerl~, Vol. 23, ho. 5, 

pp. 903-917. 1966. 

Schapcry, R.h., ‘Vl scoelastlc Behavlor 

rnd Analysls of Conposlte Haterlals.’ 

&chsnlCs of Composite Waterlalr, Vol. 2, 

t.P. Senaecryj, Ld., At~dmit Press, 

Schultz, A.8.. and Uswick. D.H.. 

’Vlbratlon Response: A tion-Destructive 

lest for Fatigue Crack Damage ln 

F I 1 ament -Re I nf orced Compos 1 tes , ’ J ou rna 1 

406. 1571. 

Chou, P.C.. WanS, A.S.D., and Hlller. H.. 

’Cumu1a:lve Damage hdel for Mvenced 

Composite Heterlals,’ AFWAL-TR-GZ-t-E?, 

September, 1962. 

Adams, E.D., and Bacon, D.G.’C., ”The 

Dynamic Properties of Unidirectloncl 

Fibre Reinforced Canposltes In Flexure 

and Torsion,’ J. Cmoosite Haterlc’ls, 

vol. 7, pp. 53-67, 1575. 

LY., pp. e5-167, 1974. 

Of COmDOSftf H.3tCrftlS, VOl. 5, PF. 334- 

Blocs, J.R., Introduction to Structure1 

Dvnarrics, HcGraw-Hili, heu \ark, 1964. 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

8.3 An Experimental Investigation of Damage-Dependent 

Material Damping in Laminated Composites 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

'I I 

A": EXPERIME!r\"TAL L'\1E:SnGATJON OF 

DAMAGE-DE?EVDE!!T MATERIAL DAMPLVG Lq 

LAhC3N ATED COMPOSITES 

S. A. smith ' 

D. H. Allen ... C. E. Hurir - 

Texas AM Unhersity 

College Station, Texas 

An cxpcrimcntal program is reported in which the 

matcrid damping of laminated graphrie/epoxy specimens 

was mcasured as a function of mrtrir cracking. Thc damping 

tcst apparalus is desiped IO measure the first mode free 

vibration response of a cantilevered beam. Damping is 

calculated by the logarithmic decrement method. Matrix 

crac'rs arc introdurcd into the material, and damping i5 

measurcd at several levels of damage. 7he results from six 

differenl laminate stacking sequences are recorded and 

cumpared to each other. Damping is found to be sipificmtly 

aflectcd by the laminate ply sucking sequence, and is dso 

Leen IO increasc dramatically with the presence of matrix 

cracks. Finally, mathematical model predictions are 

mmparcd to thc experimental results. 

Nomtnciat+x 

- bcan mss-sectional area 

- coss-serdonrl area of the ith Lmi~a 

- axial modujus of elasticity 

- mference axial modulus o! Jrsdory 

- axial modulus o! ehsdor). o! ith lamire 

-modulus weight& mmen: of inda 

- khn mom: dw: about y axis cue to 

- nmk- o! ?des o! a??lie? load 

- numk- o! plies in Lrrrinate 

- beam %we= ioal y-- cni: lmg5 

- :e- cs& to obtzin la~rate drrn?ing 

- to,-;l<X~?xe 

- the 

- tohl intemzl energy dissipation due to 

- mrdmum srrain energy o! a defiectd 

- ~n.sve.-se di~lamnen: o! neural scriace 

- bezrrr axial coortirate direction 

- bccm t~we-se coordirate Cirecdon 

- : co-pone::! o! duaace from cenzoil o! 

about y axis 

tem.rcsam hge 

coefii 5 en: 

mzix tacking 

bZT. 

. 

i5 pi, to moiuim weighted cenb-oil 

- coe!xicien! o! the& expasion 

- zxial component o! matrix cracking 

- axizl coixponen: of mz:rix cracking 

cama,oe inte.T,ai sute vzriabie (!S\') 

Czrnqo 5; in i5 12%a'2 

- bet?. ti? cc-2e5on 

- axial swain component 

- bending damping coerrldeni in ith lamm 

- bending stiffness loss coefficicnt in 1th 

- modal frequency of the Is! rexmant mode 

- streu amplimde of rydic loading 

rl - damping coefficient 

TBi 

p6i 

Jamina 

9 

P - mas5 densig 

OA 

Jnrrcdu m nn 

Miuost~CNral damge in lamiruled fibrous composite 

materials has been found to aller several material 

properties']-5. I Extensive modeling and testing has been 

pcrlormed to deternine stiffness loss in Jamjrutd composites as c result o! matrix cracking and interpi? oelaminzt~ons. h 

addilior., ~veral invesdgaton have notel md measu-ed tne 

changes in damping causec by camage4'L. fiOW\*e:, no 

atlcmp: has 'mn made to quantify 0: model tne e!iec: o! 

damage on matcizj dam?ing. Tine objective o! this saay was 

to define a direct reiationship between micros:ructuraI 

damage in i laminate ant tire resulting change in the damping 

capacity o! that Ianinatc. 

Tnis pzvr presents ml disxsses tne results o! UI 

uprimend program whkh 'has investipard the eSecn o! 

nr:rix cracking on material damping of laminated 

pphite/eFox). s7_P=imens. Tne &ec: o! piy satkg +uc..=e 

on naterh.1 damping is alu &i~-sseL A mat.rerna5=rl nmd 

aevelo,xC by Alier., Harris, 2nd ";+~srii.'-~ - -c 'h 

z?piicC, zne the resulting predic5ons O! c.zmge-de.xndent 

&rn?ing are corn.& to the ex+hsd reds. 

).!zthtnaticr! Moirtl 

h!any ma91e~a!id moads hve 'bee7 deveioF5 wLhich 

rttenrp: to precic: the marerizl crming 0:' co.iposite 

nateria1s'@l5. However, none of hese rrr&eis zre zjie to 

predic: the rnrteriai camping o! corrr-sites LS z fun:don o! 

nicrosrruns-a1 ctmage. Tne =ode1 cisassel i7 his setjon 

was developel by Mler., Hz+s, rnl Fighs6ni:hD a! the 

Mechanics ani' Materjzls Center a: iexrs hkh5 UnivexirJ. 

~ inis section provides only a brief ove.?iew. of this modd, as 

. detailel derivations mZ eqiL?rdons z?p- ir. Reiermce C. 

 ne node1 uses :ne concep: o! con:inuurn crma,gt 

me.z.z.znis, using intend state vaiajles t~, accour.: fo: &map 

in tne consritu5ve equations. Tne interrid s:ak rrrirbies 

coz~site. Tne coas5i-5ve qurtion io: he mte5ri ?sten 

cwf in :his szfy is 5zseC: on i noaiiner: Voi~ rnDdei 

- 

c 

c- ..e? k ccfinei fo: erci ?~-6rhr fox of cz~age i:: the 
IC ' 

I 

where q is a mnsunt nutcrul papy,b ir Ow dhcunc d 

thcrmrl expnsmn, and AT IS the change an temperature. 

Exprrmmul evrdence h~s sktm that pol.ymen~ composites 

are not accurately reprccentd by lhrr runpk model. In thc 

model dnclopmmt, it ii shown that a Voigr del in which q 

is not wumd to be constant for .U loadings, buc is inrcrad a 

funclion of initid soerr, oA, hque~y, op, ud the numkr d 

cydn, N, will cbsdy approximate the actd mteri.1 

aDnstirution of polymeric composites for rmrll disphcanna. 

model drvelopment AIS0 ACQDIVII) for Arid dupbamsrt, 

but will not k &dad in this study since only the bend% 

w hu been invntigatcd experimcnully. I Subni~uting (5) into (0 yields 

I 

I 

I 

I 

I 

I 

I 

(6) 

whcre 

E and I 

n Figure 1. Voight Model 

' Ai f;' 

The general thennomechanical constitutive relationship 

for a material. including the internal state variable R], is : 

The determination of the laminate Camping coefficients 

is accomplished by alorlaring the enerp dissipiled pu cyde 

of vj~ntion, L?. Assuming that zmer and she= damping 

effects can be neglected, that the damage distribution is 

unifom ovc the Im@ of the @mar, and that additiorul 

damage k not introduced during the cyde of interest, the 

kncing damping c~n~tant 3~2, is hpprodmated by 

Equating (1) and E), the intend state variable 

for E single ply within the 1L7JMte is determind. 

The equation of motion for msvene displacemen! 

.-ding) is developed using the constitutive equations, 

equilibrium. ant strain displacemen: relationships for M 

Eul~-lje?l~ulli ben 

(7) I 

I 

I 

I 

I 

I 

I 

I 

I I 

where n 

where 

1 and 

3 

ORIGINAL PAGE BS 

OF POOR QUALITY 
I 

1 

I 

I 

I 

I 

I 

I 

II 

19 

I' 

(I 

P 

I 

I 

I 

I 

I 

Y A2 s =2Q.12- L* (9) 

where A is the initial kam tip dcfkcaton and L is the knph 

of the beam SubWtuting (91 into (71 yclds the lollowing 

. explusion: 

(10) 

The loganth~nk decrement. 8, is defined 10 b onthalf of the 

enera dissipatcd per cyclc. L! , divided by the maximum 

strain energy of the deflected beam, Us. For the given 

mnfipration, 

n 

Therefore, 

ill 

. which is c first order approximation of the loprithmic 

decrement for a symmetric, ontijcvcrd Aamirute;vionting in 

the first mode. Additionally, this form assumes that the 

displacements are small, that the beam is not shear 

deformable, urd that lamina prom qgj is a function o! OA, 

Tie values of qgj for t given &mage sate, initial sues 

level, and frequency are determined by measuring the 

loparirhmic otcraen: md exeacting qgi for each ply. 1; is 

dx, rssumed i.. this model t!!k for oos~ply lzmlnata, qgj 

for 0 degree plies dw not change whn matrix aacks LT~ 

innocucd i.7 the 93 dqe plies. inerefore, for this may, 

~)gi(~) will be assumed corman: for rll damage level% 

tioweve:, ~)gj(90) will be defined as c function O! matrix 

tacking, and will be determined apuimentally. 

0, h', and ply orientafions. 

. 

Emirnrn%? 9- 

Szdrnm Prm+r+5on 

in ordc to met the objectives of this study-to idate 

ma& cracking as the only Cunepe mooe-cros.-ply laminates 

wee chosen to be tested. 12 inch quare plates were pr@ 

from Hercules ASUS502 Graphite/Epov preimprepted 

tzpe, and 1 inch wide specimens were subsequently cu: from 

these piates using 2 hjph speed diamond-dge cutting wheel. 

Table I lists the ply pro-pehes e! the material which have 

teen obtained from appropriate testing methods Table 2 lists 

the laminates sekte;' for testing in this study. 

To minimize undesirable damage caused by the grips of 

' the loading machine. z tfri7 coatihg o! e-y WLS spread ovethe ends o! :ne s?efh.-nex . Tne s.x$;men eips were :hm 

Fiishei to insme high qurlity edge re?licas. 109331, single 

ejemen: smin gzges, dignet ~5th the kan &s, were bon5d 

to the s??=irnens to mezF2re the respnse si,4 dcrinE the 

crm?izg m=svxne..s. Tne fi-al s.p.5~~~ co~5p-ation is 

 seer^ iri 5pre 2 

. 

.. panen? M?Z~J~Z 

' because o! its simplicity, c canfilevered bcax 

i 

. codiplration wLs cirosen-for the damping rppk-am. A 

support WLS aesipned which minimiid e= siippgt a: 

' the boundq, and avoided indudng damage in the wm 

Ape S is a de drawing of the funve To conduc: dam- .* tests, the speamen was placed in the slot with a 

predetermined Imp?? extending from the sup.- This length 

was chosen to pro5uce a ad beun kpen~. 

Kany experimental teinniqua hzve bear ~lployed in 

the pas: to study the material dLm?ing of compositer One 

concern in these smdies 'has been the ecixt on damping cad 

by external fac;o.&bv1&1C. Some o! tle undesi~ble damping 

contributions encountered in past investigations are 

asnoghrzi: presnue, friction cause2 by oontactihg insrurncnts 

- and/or boundaries, and additional mss Rep-diess o! the 

... method usel for damping meanueme~~t, considedie di%culty 

has been encountered by almost all experimer.:zlists in 

ohining re,peatabie and accurate results. Since this study 

focused on the efiect of damage on material dam?in& all 

o:her d~~+ig cor.=ibutions were rnhimi~ed url hdd constan: 

to es:zXish zn exa:: corre1z:ion betwetin darnFng and 

czziage. .4.:ncs?h.,i: p:ess.zre hzs bee.. founC to i.di~cn~ 

Canping. io eiiahte efie:: iron nwuret drz.?i?& iLu 

tests were conducre;' insijle a vaLvz.m G:?tn'hr. To smdy the 

die:: of the eTxy ent tabs on m;?terirl dan?ing, tests we-e 

condu:teci firs: ~<*hou: - the en2 tzbs, 2nd then w*,:i, the mC 

tabs on the sxiimer.. in? nersurel cty.?ing for 2 gven 

OlRIGiNAL PAGE cs 

OF POOR QUALITY 
In order to a6te be bum at a pr&M amplitude. J 

mechanical system was developed ID 'pluck' the free end of 

the beam to a given ddlectior.. A-I el&: m:o: is c:achd 

to an aluminum mun! which slides in c groove cut into the 

baseplate of the specimen supwri Mso atoched to the mount 

is a routing wheel which is powd e the mto:. An ann 

exten& from the wheel ant is used to cuntact the fm end of 

the r- Tne excitation source Urc suppt, M shown in ' 

Kpve 4. Upon initiation of J damping wmt the whcJ 

is rotated una the ~m contacts the specimen TM urn then 

conanus to rotate, causing the bum to de!!&., and wmtudy 

rr]ersa the Fdmen into a free \Torating mo5on. Tne tip 

ddihon o! the beam is adjusted moving the contact pin: 

o! the Lm to the 

' 

: 

hkazr 

I 

I I I 

4.66' 

J Easeplate I, 

J . + i25- I 

- ine strdn response sipal WLS transrnjttd to 2 haL' 

bricp co.::zinel i;: i Mko Measxenms SAM-i, whkh WLS 

also uxd to indicate st25: szcin vrlu~s. In orde- to o?xz4r, UI 

iniiizl s:rain ievel :he: procuces c si@ a! su!iiciez: 

To armmtely masw the -ping and frequenq of he 

specimen, r ruffiden! numbc of cycla had to be apmd. 

data poinu muhr in recording 180 cydes. This numba of 

cycles weas adequate to describe the decry o! the -&men 

' rapoonx in vacuur.. With the nunk of datr points md the 

sampling rate pres=riberi, the computc was prrprred to stan 

the dunping ucperiment Xn initial zcro load smjn reading 

was nude which was subnnd from the output si@. Mrrr 

this due was mrded, L 5 volt analog signal, pent& by 

the D/A interface board, was supplid to m external circuit 

board. r)rir orcuit then dirmel A 12 voh power Supply to 

tnnsmi: a si,d to the motor, callsing the mechanical am^ to 

rotate ht this point, the computer began to rud the dru 

signal from the stationery km As the rotating arm 

ontactd the *en and dc'lectd downwaxC, the resulting 

tde s=ain was rud by the computer as L positive volagc 

Wncn the am relerse2 the spcknen fre en.' mC the h 

defimed upward, the smin approa~d the initial unioadd 

value. Onre the beam cos& this initial ade&on plane, the 

suain beame compressive, and was read LS a negative vahc 

kr son as the computer recognized the negative value, it 

& to supply a volmge to the extcnal kit, causing the 

wheel to stop rotating. Tnen the compute- cokctd cam a: 

the prescnicd vmpiing ntc Sy collecting d2tc only atrer 

the motor had been turned of!, :ne elec~ical signal and 

vior;ltion caused by the motor, would not influence the data. 

. Since &IJ was rohct.eC at SO points per +e. mpmg 

One mezsure of the )opriL:Ynic desem~s:, . -. 6, is given by 

05) 

.where xi is the peak szain ~mplituude o! the ith cyde For 

most mteials, howwer, the peak decry of conse=rrSve cydes 

is exa-eemely srnal!, and &%cult to mlve from exrp-imerral 

Bat2. Another zpproach is to measure tne logzrith-ric 

aecrcmen: teween P finite ncmber o! cyds. iine 10)rj*~nic 

aese-rrs: beoms 

4 

ORIGINAL PAGE IS 

OF POOR QUALITY 
~~ 

pressure (AS druurwd carlrer), the fm decay will k 

exponenual, and vdcpndcnt of stram kvcls. By ~m~ng 

thm diccu lo hr negligibk the lqanthrmc decrement Wab 

mcasured Irtting a straight 1rnc through thc natural log of 

thc pcak valucr. fhr5 W.5 done for thc pitrvc (tenuon) and 

~C~IIVC (comprcssmn) pak values. Figure 6 u the poutiw 

puk value decay for a 10/%/0& specimen pbncd on a sedlog uak. thc degree of %m~riry', ud the vmll unount of 

aperimenul data yatter of the dcuy can asUy k seen in 

his graph 

delrminations may fox depending upon the cross-ply 43 

stacking sequence. Since this study was focused on ma?& 

catking, all tests were conducted below the matrix nack 

szturation level, determined for each ltminate, minimizing 

the presence o! tc.g:7~- 6~=gt mods. 

-dum 1. 

Tie objectives in the damage inducement phase o! this 

study were to intzoduce mah cr;r&g in the w. and 

to ountif?. the dting Camap. A 20 kip h!l5 SI0 testing 

rrw%ne w.s wd to load the spccimau AU of the wens 

WR loaded at L rate o! 20 pounds pr second as recummended 

e AS?4 Standard 303-70. X uniaxial aterwmete: which 

WILS amchet! to the s+men with Tdix douse-stick ".a 

wrsd towasurcstraininthesp5mens 

Edge re?lication was lrsed to docurnm: ma*x cracking. 

Since matrl: cracics gmeraliy extent awss the width o! the 

@men, zs indicated by x-ray radiopphs, \isll3ie crack 

M be aetectd on both sides o! the specimen Mge repliicl~ 

indicate the location of cradcs with to the thickness 

. dimension, inlormztion which is no: ezsy to ob'ain using 

' radiography. However, edge re?lias do not reveal ma& 

cack in the 0 degree plies (axial splits), or creas o! 

delamination ,Previous studies of xSC/9502 moss-ply 

laminates show P distinc! damage progression for monotonic 

loadinp, in which ma5i cracks form only in the 90 degree 

plies at relativeiy low loads. )J the load is increased, the 

number of matrix cracks in the 90 degree plies grows 

suSs:antially, until a matrix cack saturation levei is reach& 

~ 

wu maswed again. kouw mrtria cracks AR not dwan 

Se Udarm)!, ah6 tk kn@h Of SpRIm, and YM 

bpClng bc difkrmt Bn nOn(OnLCCutiW dv @BS. thC 

number of cracks per inch in each ply was avcragrd ovn a 2 

inch g~g~ length. fhc gage yctmn was the two rncha Cbt 

10 the root of the specrmcn when in the clamped damprng 

apparatur thc ply dimmriOnr v~ricd slightly from spedlo specimen. Therdorc, damage was qruntificd by crack 

surface arm pcr inch of length. A nudul V~UC was 

detennined by multiplying the cracks per in& of a ply thr 

(thickness x width) of the ply eonuihg 

tk md; The ~rfarr AIU is approximrtely rwia this VAIUC 

kner uch mck hrr two faces. A diagram of a mtrh UJCk in 

a single ply appears in Figure 7, which indicates the 

dimensions of uack sudacc am. 

r -- I 

The firs: step in conducting the danphg and damage 

uperimentr was to measure canping for lOj6 LX! 190j6 

laminates, which would be used to dwelop the undamged 

damping m~unts for 0 and 90 dqec plies. Using the avenge 

dunping and frequenq values o! 5 tests, IS well as the 

Wmtn dimensions and projxrth, the model constants for 0 

and 90 degree plies wen dcuhted. The mWn5 ?.re in units 

of lbsec/in?, and are 

q~j(0) = lG.625 

qKi(F33) = 1%.%6 . 

io illustate the difkrencc in Ole d~mping O! these kmiates, 

the peak value decays are pion& ro@-er in Figure E. 

3 1 -i c. . 

ORIGINAL PAGE IS 

OF POOR QUALITY 
I. 

D 

I 

I 

I 

This p~+culu stacking -mu was chosen to ddop 

the dam~gcde~ndent damping relationship for mrtrix 

~~cidng in a single W OF ply bec~wc the 93 depc ply was 

no: adjacent to any othc 90 dvee plk, therej. re~siccing 

the sire of the maaix crack to one ply thih-. Once a 

rclations'hip was obtainel for the damping O! the s?eipm. 

the mathemtical model pcrametc, 15 u'zs calculatd as a 

funson o! natrix cacicing , and suSxquently t~ predict 

damping for other &/3502 Graph.ite/Epoxy cross-ply 

laminates A: various levels of mat% crack dmge. Figurn 10 

is a plo: of the measure& logarithic decrement vs. clck 

rcrffrce area for this Lminate. Tne %due qsi(90) AS a 

function o! cack satace E. WLS CrlcJlated kom the nodel at 

each damrge state for this s-xdmer, and is seen to follou* A 

curve of fie form y=&. A powc law curve fit WLS ap?lie~, 

an& is shown in Figure 11. Tnis approximation is vrlid for 

crack surface ziea values greater than 0.005 in2, which 

corresponds to 1 crack per 12 inches o! specimen lensh 

Damage below this 'level u'zs rssumed to indicate an 

'undzmaped" specimen. Tiemfore, for the case in w*hiCir 

ma& tach could no: be detectd, q~i(93) wzs ~~snuned m k 

the value exxacted from the damping test of the [SOjs 

laminate. This approximation was then usel to predic: 

&=ping o! all o! the cross-ply qxcimens a: any given mtix 

cack cunrge szte, inciuding tire mdampel szte. 

ine reluction in stiffness was exzernely sxzll for tj-is 

Izxinzte. 'oecr-se the catz xatte: was great in the s:','hless 

mersure.nenS, and the redudon b*as minim!, no qurntkatjve 

rela5onsbi? be:ween axial sdf.%ess ani mz~x cracking wzs 

es:r5lissh&. FiFire 12 is 2 $0: o! no.Tzli:d s$;!.'nesr as c 

function of ni:5x ciack scrface aiez for :%s 10/90/0!, 

~ejxen. h!z:5x c:a&ing i: i izsrste is M to czcse onjy a 

-. 

6 

slight decrease the aual modulus (iccr than 1% for ths 

LMNIC at urnranon), wherus the dampng vl~use IS wry 

opihcant (37% lor the @me level of damage). Thedore, 

damping can h- conridercd US &. J su~or method Of rwnaa~ruaivdy waluating the IIym cracbng In a Lrrurutc. 

i 

b 

1' 

! 

L 1 

SI 

ORIGINAL PAGE IS 

OF POOR QUALITY 
I 

I 

~I 

I 1 

I' 

I' 

I' 

I' 

I' 

I' 

" 

I 

I 

I 

I 

I 

i A 

Ab 

AI I 

DJI I 

Wtp E. I?D/@/O& 

The next laminate tested was also a six ply laminate, 

Wjth a single 0 degree ply between two 90 dcgrec plies in Uch 

hlf-?he. Tne ~nl~xgd bexLn~ sSness was preOiad to 

k significantly less than that 0: the [0/90/0], laminate, 

which resulted in a smaller fundamental frequency lo: the 

me length hm linerefore, the length of the 190/0/90& 

beams was shonened to increase the fmpency, in m attempt 

to match it with that o! the [0/90/0j, laminate- tiowevc, tiW 

specimen had to k long enough to mhhk * the presence of 

shear defonnationls. Tne frequenr). for the undarmgd 

lo/90/0], specimen wzs 20.7 FA, and the frequency of the 

>e nithematical fomula5on for the logarithmic 

dmcn: in (10 indudes c term whih defines the ddm,?hg 

D be a liner- &?don o! fmquenq, for a cons=! value O! q~i. 

Goxr inspsion o! the model awelopmen! reveals, howm, 

that q~i is not Constant with frequency, and in fact, the 

' lopriithmic decrement is predicted to follow A Zener like 

awe ~5th rew. to freque?~. Tne darnping ter: appu-atw 

does not pro\iae for tesdng L sinpie -&nen a! a wiae rrnp 

o! frequenoes. Sinx the diffe-en- in frequcxy for all of the 

tests are not dramatic Oess than 6 Et difference), the 

differences in camping for the vzrious frequencies are approximated by the linear relationship of the model. 

Tnerefore, for t? study, qzj was ZSS~L~& to be constant w-itS 

respct to frequenq. L! mens a: much.highher frepenoa, 

Le., greater than 50 iiz had bee mecsurel, this =sumption . would no: neceszrily be valiL 

Tine measured and predicted damping values for the 

1 undarrrrged 190/0/90& lzrninate CM k sem in Apre 9. hfazix 

crack initiztion wcs found to OCCLT- rt approximately 1200 IDS. 

for this sacicing sequence. L7 t'his lami-rzte the inkial c~=ge 

szte wzs one in w;zith r;rzt.x czcb were zixnizn: L7 the 

octer 90 &Fee plies, ani rrL-.irnzI z: the r;rid-?izne 2: z load 

of lCC5 ijs. A: I sligh9y higher 1028 of '14% I&, the cads in 

tire inner 90 aepe plies ixreased by z fznor o! 12, while the 

czcks in tle o-te- Diies in=&- by less tkn z fzctor o! 5. At 

the szc~:z:ion ieve!, all of the 90 cevee plies hzc 

. _._ 1w/o/P3:s bexn w7s 14-iF2 . -. --- . 

#~JCillUW)y thC Ym Crack spang (about x) kb& 

inch). 

Thr mrlhermtical del predicts Out the damping will 

k af8ectd more by mrtrii cracks far away from thr mdplane, than matrix crach which are near the mid-plane. 

Thrrcforc, it would not k an accurate dcpimon of darmgt 

dependent damping to rJIn thc damping strictly to the IOUJ 

crack SUI~AQ AN in J Lminrtc me btion of thc mckr 

with "pa to the mid-flane must k 1J;m into amnt whm 

reponing the dau. ThC tcponed Crack surfact arc4 Was 

therefore weighted by nondirnensional factor ( Ej/t)2, 

where Ei is the dirtan- from Ihe centroid of the crack to Ihe 

mid-plane, and I is the laminale thickness. The equation for 

(his weighted crack surlam arm, 5'. iS 

N 

ill . where ci is the numb of cracks pcr inch of length in the ith 

ply, and tj is the thickness of the ith ply, and w is the width 

of the specimen. The factor of 2 accaunts for the numb of 

crack fam per manix mck 11 is spased that this VAIUC was 

not used in the model IO make damping predictiom, but is 

simply used to identify the amount and location of matrix 

cracks in a laminate. Figure 15 indiotes the ~ktio~hip of 

mwd ud pdcad dunping to s' fur olir rpecimaL -- 

A 

1 

A 

t 

ORIGINAL PAGE IS 

OF POOR QUALITY 
,;i E'] -1 

1 I 

bminatr w-@J2 

This hTtiMtC wries had the smallest bending stiffness of 

a11 of the laminates considered. Ihe natural frequency for the 

undamped swmens was 127 Hz., which uw also lower than 

Out of rll of the other laminatu. The undamaged damping 

value was by far the highs1 of the spedncns taied (the 190/6 

laminate was higher, but was only testel to-obtain model 

constants), and was closely predictet by the mathematical 

model as seen in Fipre S. 7ne plo: of the logaritivnic 

decrement vs. 9 this laminate zppecs in Figure IS, with the 

corresponding model predictions. ?he initial and final 

predictions cormpnd very well to the rnusurcd values. Tne 

measured value at the middle dunage state is slightly higher 

thrn the fim! mernuuIIent and is attibuted to orpeimcntal 

-0:. 

=- 1 

Ei1t A 

A 

A 

t 

t 

L 

s 

t 

1 

u' 4.' 8.'. . , , . . . , , . , 

rpr4 ra cp CP Ut rs 

WuEh1.C tnek AN? (v') 

Tqun 16. IqdUunir -en: n. WuEhld hlnx Cnk Nu 

PlS& 

--- 

bmimte F. I90/0/9C& 

i'e sucking sequence for kminate F was sindrc to the 

10/90;Js series, acmt for one o! the 9G dp plies bing 

outside the 0 depee ply. This diffemce hzs no efiec: on axial 

s*.ness, but it does dim the h&g s&%esr Tne Cmeg 

o! the undamaged [90/0/902Is sFrimen is seen to be 

sifi!icantiy nigh^ than the [0/93;ls in Fipre 5. For tht 

190/0/9~), sacking sequence, the ma& mcjcs gnw vcy 

qujddy in the out=- 90 age pEes, and more slowly in the 

inn- 90 degree plies This laminate showed I very Lrp 

increase in Camping with inueased damage, which was 

closely predicted by the mode!. Tnis resulhg function is 

plotted in Figure 17. . . . - . . .. . . . . 

hrninztp Cornx5so~ 

Tne results which have been presented inccate a swong 

dependexe o! disn?ing on lamirate stacking sequence and 

ma& cacking. 1: was postuiatel in the m~d devdopmcnt 

iha: the damping o! a hnhite would k p.Zuy de;mdent 

-uFn the 93 der plies, ax! only slighily oe.mden: 2.m the 

. 0 degree plies. Tnis assum?tion is nap.mpned by the fac: that 

the damping of Lhe [93]6 1cnhte was found to be 7 times 

Fete; than the czrnping o! the [Oj, laminate. tiowever, the 

extensive testing o! COSS-ply lzminates indicated that the 

percc:,:rge o! 00 degree plies in a spedzen hzs very EVJe 

e!iez on he rate+d CzmnpinE o! 'he ILmLTate. Fir2-e 16 is a 

=io: o! the zeasx& logz%:>n': decrexen: for 211 of ';ne 

unczraget s?p-dmsx rgins: tire correspnting .v-~ce~tzp O! 

93 deBee p!ies LT tire swrnez Fo: exrm?ie, the 10/03/G& 

ltrrlnzre con*& I:t5 Free..: SD dgee plies. Xo -en6 CLT 

.be sen ir. t+s ?lo: m*:?icn wodC inciczte 52: 5e Caz,?ing 

chnges m$';7 'he ~:cn~~e o! 03 dme Diies. 

ORIGINAL PAGE IS 

OF POOR QUALITY 
I' 

I' 

I' 

I' 

I' 

I' 

I' 

1' 

@ 

1 

1 

I 

I 

I 

L 

A 

I 

I 

1 A 

J ,* I- "I' *'I ,'a** *I'.'I'. *I-'* 1'''I~'~l~'' 

I 

e ro s u e s E x r e5 rm 

h.trCYnEC C!* ?&a t?'d~UtC 

FyVn 16. h4-d *-&tkrk &nn! 8%. Z Sco?b- 

- ine major factor contn'buting to the relative damFing 

naplitudes for the undamaged +mens was seen to 'be the 

placement of the 90 ovee plies with respect to the midplane. k plot similar to Figure IS was created which 

weighted each 90 degree ply by the distance from the midpiane, anti related this value to the measured Ioprithmic 

deumen:. Tne value used for the domain is L weighted 

yxexzge of 90 degree plies given by the following foxuia: 

J 

i=l 

whe:e J is the nuzber of SO depe piies ani N is the :otz! 

n-dk of piies in the izniratt A l~mkrte wiLSi a sigzikan: 

irac5on o! the 93 depe plis fzr cwry horn :ne mio-?!me 

k-n! hrve c high due 0: ?, w:u'le a it~.L-ate wi:h Lqe szne 

perce:.:a,ne of 00 6cpe plies ciose; to the miTlic'-?iane, will 

hrve z low value o! Y. Tne ~rph o! this reiationshj? is 

A 

1 

Figure 20 is I plot o! the incrux in the loprithmic 

decrunen: for each stacking sequence, from the undamaged 

state, to !he matrix crack saturation damrge state. The 

corresponding increase predined by the model is also shown 

for uch specimen. h d of the speimens a- the 1~/01, 

spamen, the predicted inmu is lower than the mead 

increase. This difference is possibly due to the p-cc of 

additional damage which may have kcn inrroduced during 

the loading proms. Tie stadking euences which we= 

grer:ly underprecictec, [93/O/901s (25 Fcentap points 

rmriez!, L.. the [?3/0/9+), (25.7 permr~g? points crrdc), bo5 

had 93 degee plies on the outc Lyer. XS the nL-k- o! 

. mat% cab direcily unde the sclin p,ge ind fo: both 

o! thse kmirutes, the movement of the crack faces a@: 

the s& gagz resulted in an inouwC danping murumnmt 

Tnis did no: om: in the [9q/O], laminate bemuse the 

dam?ing ws no: mersured a: a damage level which contained 

c pr.! nu& o! matrix -9s. From the damagede-pendent 

dampg plots of the individual larinates given throughout 

the results section, i: M be sen that for thse sjxdmsrs with 

the 90 depe plies a: the oute: surface, the model makes 

accurate pre6ictions r: earlier matrix crack drmape states. 

For speomens with 90 degee plies at the outer surfac, 

containing a krge nmkr of na?rL: cab, the modei does no: 

seem to cur=eqwnd well with the enc,inen.sl mersllremenn 

For all of the lamirates with 0 degree plies a: the out&- 

s-cface, the nodel provides reasonable prelic5ons of the 

inuezse in Iarping at all mat5 czck lev& 

fiXIv5iO?~ 

ine nzte5zI c~~$ng o! severcl p+.ite/e.ppxz. SOSCply la.?jnztes ~*zs mezsured z.s a f-mction o! rrrrtir czcking. 

i ne tes:ing con5pration u-as r crntiieveiel bezn, 

meckni:zlly ex=itd to vikete in the 5:s: mode. Tne free 

res?o?.se wrs cz?:xeC, ani 'he Iop5i--+: deczerien: was crjL-ia:eC ko.7 :he peak decry o! Lie szain a~.?li~dc. Six 

li!!erei.: 1z:inatrs were teste2 a: severrl leveis o! ma-x 

-. 

-. 

c 

ORIGINAL PAGE IS 

OF POOR QUALITY 
1. 

2. 

3. 

4. 

5. 

6. 

i 

10 

~~ ~ ~~ ~ ~ ~~ 

7. A.T. Dikndetto, J.V. Gauche!, R.1 Thomar, and J.W. 

hrbr. %ordestructive Iictrrmnrtion of Fatigue Crack 

Damp in Composites Uung Vibratmn feu.' bumr! pi 

P.:atcri&, JMLSA, 7211-215 (lune 1Y2L 

6. Dahsin Liu, C.T. Sun, and L.E. Malvcrn, 'structural 

Dqndrtion of lmpaaed Crsphitc/Epxy Luninrtn,'- 

wv- . Bulktin 56, Part 2, 514 

(Aupsl1986). 

D. H. Anm, CE. Hams. and A.L Highsmith, Tdiction 

and Expcrimcntal Observation of Damage Dependent 

Damping in Lmiruted Composilc beams,' Texas A & M 

Univeraity Mechanicr and Mataids Center Uune 1967). 

IO. Albcrl 8. Schultz and Stephen Tui,'D,ynamic Moduli and 

Damping Ratios in Fiber-Reinforced Composites,'L 

9. 

crrnmsttr Ma a 2366-379 (July 1968). 

11. R.D. Adam and D.C.C Bacon, 'Ella of Fibre Orientation 

and Lminatc Geometry on the Dpmic Properties of 

CFRP,' 1. Cornmsit? Matcriak, 7:402426 (hober 

1973). 

12 DX Lin, R.C. Ni, and R.D. Adam, 'Prediction and 

Mururment of the Vibrational Damping Parameten of 

trbon and Glass Fibre-Reinforced Plastic Plates.' L 

Cornmsitr Matcrials. 16:132-153 (March 19W. 

15. CW. bert and C.C Sin, 'Vibration and Damping of 

kminatd bmposittMaterial Plates Inciuiiing Shear 

Effects,' School of herospce, Meduniul, and Nuclear 

En@n&nL The Univusiy of OWom (March 1SZL 

14. FJ. Gibson and Robet. Piunkett, Pynamic Mechanical 

Behavior of Fiber-Reinford bmposites: Measurement 

and hnalysis,']. Cornmyin MEteriak IC33341 (bok 

1916). 

' 15. S. Kalyanasundaram. D.K. Mlen, and F-h. Schapev, 

"Dynamic Response o! a Virwli&c Timoshenko -.' 

Tars A br K University Mechanics and Materirls Centc 

(1 967). 

16. Albert E. Schul- 2nd Stephen 'is& wnk Mc&d an:! 

Drx?ing btios i=. Fiber-Reinforce< Composites,' L 

Co-=sit? Mrte5rl~ 136E-575 Oufy 196). 

i7. R.D. Adam and D.G.C Bacon, 'Measuremen: o! the 

Fkud Dwr?ing Gpoy and Dynamic YOU~E'S Moddlius 

of Metals an& Reinfoxed Rzstics,' 1. Fnrs. D. Aa5. 3% 

6'??41 (1SZ). 

. 

16. S.A. Suarez and R.F. Gibson, 'improvement and 

Optimization of Internal Damping in FTwr Reinforced 

Coznposite hbterials,' Mechanical Engineering D?p:., 

Unjvemty of Idaho (March 1966). 

ORIGINAL PAGE IS 

OF POOR QUALfTV 
I 

I 

11 

I' 

I 

I 

I 

~I 

'I 

I 

'I 

I 

I 

I 

I 

I 

I 

I 

1 

8.4 A Model for Predicting Damage Dependent Damping in 

Laminated Composites 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

A MODEL FOR PREDICTING DAMAGE DEPENDEh'T 

DAMPING IN LAMINATED COMPOSITES 

s. KALYWASU~DARM and D. H. ALLEN 

Aerospace Engineering Depamnent, Texas A&M University. College Station, TX 77643, U.S.A. 

Abstnc+A constitutive model has been developed for predicting the dynamic properties of 

composite beams which accounu for the inAuencc of microsrrucrural damage. The model assumes the 

damage state to be fixed. The model can account for both the damping increase and suff ness loss due 

IO microsvucrural damage. Damage dependent material constanu obtained from a baseline dam sei 

can be used to predict damping changer for several composite layups. The validiry of the model has 

being verified by comparing the experimental resulu 10 model predictions for crossply laminates with 

a variety of stacking sequences. 

NOTAT109 

cross sectional area of the beam 

\'oung's modulus of elasticiy 

toul number of plies in the laminate 

iota! su*ace area o! matrix cracks 

ma.ximum strain energy of dcficcred beam 

OCfiCCtiOn of the beam 

IoSarithnic occremeni 

mas densiry/unit length 

lcnph c! rhc beam 

com?lex cunarurc 

frequency paramere: 

loss factor 

compicx frequency 

conplcx exlcnsional modulus of the cenval ply 

area moment o! incrrja of the cross section 

conplcx Young's modulus 

sforace modulus 

loss rnoauiu 

complex bending moment 

complex extensional modulus of jth ply 

extcnsiona! storage modulus o! the jth ply 

extensional loss modulus of the jth piy 

complex moaulu of 0' plies 

complex modulus of 90' plies 

ficxural loss tangenr 

ISTRODUCTION 

The process of ultimate failure of advanced composite marerizis is preceded by a sequence of 

microstructural and macrostructural events termed 

as damage. These evenu may be due to transvene 

matrix cracking, delamination, fiber breaking and 

fiber mztrix debnding[l4]. This damage results 

in stiffness loss which in turn may have a significant 

effect on ;he dynamic response of structural 

componenu[5J. Experimental results indictte that 

material damping may be more sensitive to dam2ge 

de\.elopmenr thzn stiffness loss. Damping in the 

presence oi damage hu been observed 2s high 2s 

Z50y0 greater than ihe reference undamaged 

ezrnping airnough the corresponding reducrion in 

resonan: frequencies is only a'wut 5%. 

Scnulu 2nd U:zrwick[6] used oscillarory flexural 

l?; 

loads of zero mean value to faugue cantilever 

beam specimens of scotchply laminates. During 

this fatiguing, the vibration response of fhc specimens was measured to deternine if the response 

changes were indicarive of fatigue. It was found 

that the increvc in dampins correlated fairly u*ell 

wirh damzge. For specimens which exhibited 

significant crack damage the damping ratio 

increased by 100010. In 2 closely related in- - 

vestigation[7] similar resulu were reported for oscillatory tensile loads of non-z:ro mean value. h 

torsional pendulum w'zs used to measure the torsional srorage modulus and damping capaci? of 

carbon fikr r:inforctd bars before and 2ftei 

crzcks were introduced through s:atic torsion 

testing [SI. The spec5c dampin_r capacity increased 

by nearly 80% after the introduction of damage. 

Chandrz et n1.[9] mevurcd dzmping in crossply 

gizsslepoxy composite beams beicre and zfter the 

introduction of damage due to s:atic and fatigue 

loads. It u'as found rh2: there was 2 considerable 

increase in damping with the increve in numkr of 

cycles of fatigue loading and the level of prestrescing. For the fatigue ioad the reduction in storage 

modulus was very small after five cycles, but the 

damping showed 2 monotonic increase with increasing cycles. The damping was found to k independent of amplitude of oscillzrion up to about 

0.04Sb strain. 

Plunkett [ 101 measured damping factors of cantilever bezms made of crossply glusle~oxy 

laminates as they were vibrated at difierent strzin 

levels in the fin: and second bending mode simultaneously. 11 was found tn2: the dzmping factor 

was subsranrjall\ inireased by Ia~ge st-zin induced 

matrix cracking in laminae ui:h t'2ns~'ese fi'oeis. 

ine damping factor was indepcndent of cczen: 

srrzin riis:o~ or mode shape 2nd u'zs 2 function of 

damage. Ii w2s conciudec th2: this kf12vior makes 

ir possible to use a linezr supqmsition mehod for 

-. 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

1- 

I 

I 

I 

I 

I 

I 

I 

1 

114 

dynamic analysis. Smith et a1.(11] have measured 

material damping of laminated graphitelepoxy 

specimens as a function of matrix cracking. It was 

found that darnping increased rignihcrntly for 

various stacking sequencer due to transverse 

matrix cracks. AIlen cf al.[IZ] hsve developed a 

theoretical model for predicting the damage 

dependent damping based on the theor). of internal 

state variables IO account for damage. Weitsman[13] has developed a continuum damage 

model for viscoelastic material. The special cases 

of uniaxial damage under uniaxial stress and interaction of damage with moisture diffusion have 

been considered. 

In this paper a model is developed for predicting 

the structural dynamic response of composi~e 

beams in the presence of transverse matrix cracks. 

The validity of the theoretical formulation is 

verified by comparing the theoretical results uith 

the experimental results obtained in 1111 for 

diff went stacking sequences. 

MATH EhlATI C AL MODEL 

The determination of damping propenks of 

beams usually employs the Euler-Bernoulli 

assumption. In this theoq it is assumed that the 

beam element is in pure bending and shear efiecu 

arc neglected. For highly anisotropic materials and 

in the presence of interply delaminations, the 

dynamic properties obtained on the basis of 

the Euler-Bernoulli beam theon' may exhibit 

significant errors. Since the Current study will concentrate on crossply laminated beams with v2nsverse matrix crack, the Euler-Bernoulli beam 

theoq is assumed to be adequate for obtaining the 

dynamic propenies. The governing difierential 

quarion for the dynamic response of an elastic 

Euler-Bernoulli beam is given by 

The solution for free vibrations can be wrinen in 

the form 

For sufficiently small damping, free vibrations are 

also approximately harmonic and from the correspondence principle of viscoelasticity, the clastic 

Young's modulus may be replaced by the corresponding complex modulus, E', where 

2nd the fiexurzl damping is siven by 

- E?' 

t2n 8, = -=; = q. c 

I 

The complex frequency quation is given by 

pA1.w: A:=---- €'I, ' 

where A. is given by the boundary condition. Since 

E' is complex, the frequency values are also complex and can be wri~ten as 

w,=wW:+iw:. (6) 

Separating the real and imapinary pans of eqn (5) 

results in 

6 

tan = z, 

(7) 

where 6 is rhe logarjtiunjc decrement obtained 

from 2 free vibration decay cxpcrhcnr. 

Th: uivcial COnStiKUti\Y equation for marnx 

cracking of clastic-composites under isothemal 

conditions is modelled by[]<, 151. 

u, = EE~ -la,. (1 0) 

where a, is an internal state variable representing 

mauk cracking in each ply. The changes in damping and stiliness due to microstructural damage 

are hisror). dcpend=nt and spatially variable. For 2 

futed damage stare rne dynamic prop- -des ar: 

slowly varying functions of time so that the composite material behaves like a linear viscoclu~c 

body. Undpr sinusoidal loading, experimental 

resulu[lO) indicate that the principle of superposition is valid for a damage state that is ked and 

subjected to small amplitude vibrauons. Based on 

these obsemations it is proposed that eqn (10) czn 

be modified for sinusoidal loading at fixed damage 

state IO 

(3 

where 
Prediction of darnage dependent damping in kmirukd compositu 

I 

I 

I 

I 

~I 

I 

~I 

I 

I 

11s 

evidence that damping increases with microsuucturd damage md the resonant frequencies 

6, = U,A ed. decrease. Equation (18) can therefore be written as 

Further insight into the effects of fixed damage 

on damping can be gained by studying the energy 

dissipated in a cycle of sinusoidal motion. Therefore, Appluarion of ~hc model IO crossply laminaus 

The moment~urvaturc relationship for a viscoelastic Euler-Bernoulli &am vibrating sinusoidally with frcquency w can & wn'nen as 

In order to obtain a meaningful expression for 

energy dissipation the real part of 6- and i, 

should be used in the evaluauon of (13). Thus, 

For plies of uniform thickness the complex flexural 

modulus is given in terms of the complex ply 

moduli and stacking geometry by[16]: 

E;(3j2-3j+l)] for even h' 

Real(6,) = Real[(E'- i-r, eu 

+(I'- iI?a,, ei'] 

for odd A'. 

= [€ sin wr - ,?7 cos wr] e,, 

In this work symmemc laminated beams composed 

solely of longitudinal and nansverse plies Will be 

considered. For this case eqn (22) reduces 10 

Using (15) and (14) in (13) the energy dissipated (23) 

per cycle is given by 

where D and b are functions of stacking geomeny. 

The fiexural damping is given by 

Note that for a,, =O (no damage) the above 

expression reduces to the case of energy dissipation 

in a linear viscoelrstic body with no damage. 

The main energy stored ourins the peak displacement is In the presence of transverse maaix cracks in the 

90' plies, it is assumed here that E; and E; will be 

aff ected by damage 2nd the values of E; and E; 

remain unchanged since no cracking occurs in the 

0" plies. 

. Comparing eqns'(20) and (24), the loss factor in 

the presence of transverse matrix cracks is thus 

given by 

a€; i CAE; + bE; (25) ' = a€; + CAE; t bET' 

The loss factor is 

Furthermore. it will be assumed that 

where CAE; and CAE; are functions of the total 

surface area of manix cracks and the location of 

the cracks. LE; 2nd A€> as functions of crack 

damage can be extracted from a single simplified 

stacking sequence. This infomarion may then be 

utilized in (25) to predict the damping as 2 funcuon 

of damage for difierent stacking sequences. 

where AE" is the increve in loss m@iulm and 4E' 

is the decrease in storage modulus due to dzmage. 

This assumption is made based on the experimental 

CAS 30:1/2-i 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

116 S. KALYANSUNDARAM BMI D. H. ALLEN 

DISCUSSION OF RESULTS 

Expcrimental rerul~s[l I] obtained from crossply 

laminates with different stacking sequences will be 

used to verify the validity of the model developed 

in the previous section. By testing laminates built 

from 0 or 90" plies, the values of EL and E; can 

be obtained from resonant frequency measurements through eqn (8). Using the damping 

measurements, the values of E; and FT can be 

obtained from eqn (4). These values can be used to 

predict the damping of any crossply laminates with 

various combinations of 0 and 90" plies. Figure 1 is 

a comparison of experimental and theoretical 

damping values for difl went laminates. For 

pol~mer matrix composites the viscoelasticity of 

the matrix is the major source of energy dissipation. Hence, the material damping of a crossply 

laminated beam is dominated by the location and 

the number of 90" plies. The predicted and 

experimental damping values indicate that the 

Aexural damping increases with distance from the 

midplane of the beam. 

For predicting the damage dependent damping 

for difiercnt stacking sequences, the increase in 

loss modulus as a function of m2k crack damage 

can be constructed by experimentally measuring 

the damping increase in 2 [0/90/0], laminate. The 

change in loss factor as a function of transverse 

matrix crack surface area is given experimentally 

by 

A~(J) = 0.0002(~) - 1.710s X IO-'?. (26) 

Foi this laminate the darnping increased by 3570, 

while the reduction in rhe resonant frequency uas 

about 0.07y0. Lising eqn (35) for 2 [0/90/0], 

laminate, the change in rhc loss modulus as 2 function of surface area of matrix cracks is given by 

The cxpcrimental change in the storage modulus is 

negligible. 

The experimental and theoretical predictions of 

damping of a [90/0/90], laminate at two diff erent 

damage states are illustrated in Fig. 3. The descriptions of rhc damage states are given in Table 

1. For this laminate there are more transverse 

matrix cracks at the outer 90" plies than the 90' 

plies near the midplane. The damping increases by 

nearly 100%. The agreement between theoretical 

and experimental results is good. The reduction in 

the first mode resonant frequency (an indication of 

stiff ness loss) is only about 770. This trend clearly 

indicates that damping changes are more sensitiw 

to the microstructuraliy induced damping changes 

than the suff ness loss. Figure 3 depicu the eff ect of 

the matrix damage on 2 [0/90/0/90>, laminate. The 

corresponding damage slates arc given in Table 2. 

The theoretical model predicts the increzse in 

damping due to transverse matrix cracks IO be 2 

function of both roral ma^ crack area and the 

location of the 90" ply in the laminate. It is interesting to note that for approximately the same 

amount of rota1 rnauix. crack area the dampinf 

chanze in 2 [90/0/90], laminate is s:\'eial times 

hisher than the change 'in the [0!90/0!90], 

laminate. Tnis is due to the fact that the matrix 

crack damage in 90" plies faitner away from the 

midplane of the beam uill lead IO 2 lzrfer increase 

in damping than tne 90" plies n~ar tne midplane. 

This uend is captured by borh the theoretical and 

... 

_. 

I 

j-. ., 

I 

j 

! 

- 

Fi:. 1. DamFing values for various laminates. Fig. 2. Danipc ocpcna-n; ozmjing vaIucs for [?O/O!?OJ, 

izminzre. 

ORIGINAL PAGE IS 

OF POOR QUALITY 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

~ 

Prediction of damage depuxknt damping in laminaid compitcr 117 

D~N~c SL~IC 1 Dmgr Slate I1 

' Fig. 3. Damaze dependent damping values for [O/9OJ2, 

laminate. 

D~rvpr Scale I 

Fig. 4. Damage dependent damping values 

[90/0/902], laminate. 

- 

for 

experimental results. Figure 4 illustrates the effect 

of matrix cracks on a [90/0/90,], laminate (see 

Table 3). The agreement berween the theoretical 

and experimental damping values for the damage 

states is good. However, the theoretical model 

tends to overpredict the increase in damping from 

undamaged damping valucs as compared to the 

experimental results. 

CONCLUSlOKS 

A constitutive model has been developed for 

including the efiect of microstructural damage on 

Table 1. Description of darnage state for 

[90/0/90]. alrninate 

Stacking Damage state I Damage state 11 

gcomey (crac krlm.) (crac kslin.) 

90 1.9 21.3 

0 

90 0 2.9 

90 0 2.9 

0 

90 1.9 30.2 

- - 

- - 

Table 2. Description of damage state for 

[0/9Oj2, laminate -~ ~ 

Sucking Damage sute I Damage state 11 

gcometq (crackslin.) (crackslin.) 

90 5.0 10.3 

90 6.3 15.0 

90 6.3 15.0 

90 3.1 6.1 

0 - 

0 - 

- 

- 

0 - 

0 - 

- 

- 

Table 2. Description of damage state for 

[90/0/901], laminare 

Stacking Damage state I Damage state I1 .- eeomctr). (crackslin.) (crac kslin.) 

90 

0 

90 

90 

90 

10.9 

.3.9 

3.9 

3.9 

- 

00 3.9 2i.2 

90 11.9 26.7 

0 - - 

This model is valid for a fmed damage sfate and 

small amplitude vibrations. Under these usumptions tne composite material behaves like 2 quasilinear viscoelzstic body. The vaiidiry of the 

theoretical formulation has been verified for 

crossply graphite-poxy laminates with a variety of 

sacking sequences. Damage dependent material 

constants obtained from [0/90/0], laminate have 

been used to predict the increase in damping in 

[90/0/90],, [0/90/0/90], and [90/0/902], laminates 

for different damage states. Damping is found to 

be more sensitive to microstrucruial darnage than 

the stiff ness loss. Thus, damping holds some promise for further studying the damage development in 

composite structural components. 

kcknowlcdgcmcr.fi-Thc authors wish to acknou*lcd_ec 

rhe finaniial SUDDOTi urovided bv NASA Johnson SDacc 1' . the dynamic response of laminated composits. . Cente: under granr number NAG 9-140. 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

118 S. KALYANASUNDARL~* ud D. H. ALLW 

RJ3ZRENCts 

1. K. L. Reifsnider and K. Jmiron, Fracture of fatigueloaded composite laminates. hl. 1. Fatigue 4, 187- 

197 (1982). ‘ 

2. H. L Highsmith, W. W. Slinchcomb and K. L. Reifsnider, Siiffness reduction resulting from trrnfvet% 

cracking in fiber reinforced composite laminrtes. 

Virginia Polylechnic Institute and State University. 

Blacksburg. VA. \’PI-E-81.33 (1962). 

3. J. E. Masters snd K. L. Rcifsnidcr. An investigation 

of cumulative damage development in quasi-irotropic grrphitclcpoxy. In Damage in Cornprtv 

Mavnal (Edited by K. L. Reifrnider). pp. 40-62. 

ASTM STP775. Amcricrn Society for Testing and 

Materials. Philadelphia. PA (1982). 

4. R. G. Norvcll. An investigation of damage accumulation in graphitelepoxy laminates. 7cxu AdrM 

University thesis (1985). 

5. S. Kalyanasundaram. J. D. LUG M‘. E. Haisler and 

D. H. Allen. EBcct of degradation of mrlcrirl properties on the dynamic response of large space SVUCtures. 1. 5pcccrafr Rockcrc 23. 297-302 (1966). 

6. A. B. Schulu and D. h’. M’awick, Vibration response: A non-desmcuve test lor fatigue crack 

damage in filament-reinlorced composites. J. Cornp. 

Maicr. 5, 39C404 (1971). 

7. A. T. DiBenedrto. J. V. Gavchel. R. L. Thomas and 

J. N’. Barlow. Nondestmctive determination of latiguc crack damage in composites using vibrauon 

mu. J. MaicriaLr 7. 31 1-215 (1972). 

8. R. D. Adams. J. E. Fittcrof;. h’. L. Hancox and U’. N. 

Reynolds, E€ect o! shea: darnage on the rorsional 

behavior of carbon fibre rcinlorced plastics. J. 

Comp. Maw. 5, 66-75 (1973). 

9. R chudrr. A. K. Mdlik ud R. Rabbk-n. 

Damping u a measure of damarc in composites. 1. 

7cch. &unci& ASCE 108, loblll (1962). 

IO. R. Plunken. Damping in fiber reinforced h~nin8ted 

compositu 81 high curin. 1. COT. Maw. SI&. 14, 

109-117 (1980). 

11. S. A. Smith. D. H. Allen and C. E. Hams. An 

expenmental investigrtion of damagc-depcndcnl 

malcrirl damping in Iaminrtcd composites. An Procecdinp AIAA 29th %UCNICJ. Snucwa/ Dynamiu 

and Maleria& Conlrrcncc, pp. 1501-1510. American 

Institute of Aeronautics (1966). 

12. D. H. Allen. C. E. Hams and A. L. Highsmith. Prediction rnd expenmental obscn@rtion of damage 

dependent damping in laminated composite beams. 

In Tht Rok o/ Damping in Vibration and Kobe ConJrO/ (Edited by L. Rogers and J. C. Simonis). DEVol. 5, pp. 33-164. American Society of Mechanical Engineers (1967). 

13. y. Weitsman, A continuum damage model for viscoelastic materials. Mechanics and Materials Center. 

Texas AkM University. hfM 576247-17 (1967). 

14. D. H. Allen. C. E. Hams and S. E. Groves, A thcrmomcchanical constitutive theory for clastic composites with distributed damage - Pan 1: theoretical 

development. Inr. J. Soli& Smrrr. 23, 1301-1316 

(1967). 

15. D. H. Allen. C. E. Hams and S. E. Groves, k thcrmomechanical constituti\*c theor\. for elasuc composites with distributed damage - Pan 11: appiication IO matrix cracking ir. iaminated compositcs. 

Inf. J. Solid5 h~r. 23. 13!$-1536 (1967). 

16. R. F. Gibson and R. Plunkert. Dynamic mechancial 

behavior of fi‘kr-reinforced composites: measurement and analysis. J. Cornp. Mow. 10, 524-341 

(1976). 

. 

ORIGINAL PAGE IS 

OF POOR QUALITY