NASA Technical Memorandum 102567 ...............

Computational Simulation of Damping

in Composite Structures .......

D.A. Saravanos and C.C. Chami_ .....................

Lewis Research Center .......

Cleveland, Ohio

Prepared for the

2nd National Congress in Mechanics

Athens, Greece, June 29--July 1, 1989

NASA

_(NA_A-T_-107567) COHPUTATIgNAL SIMULATION

OF OA_PTN G IN CnM°0 qlTE STFJICTURES (NASA)

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COMPUTATIONAL SIMULATION OF DAMPING IN COMPOSITE STRUCTURES

D.A. Saravanos* and C.C. Chamis**

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135

ABSTRACT

A computational methodology is developed for the prediction of passive damping

in composite structures. The method involves multiple levels of damping modelIing by

integrating micromechanics, laminate, and structural damping theories. The effects of

temperature and moisture on structural damping are included. The simulation of damping

in the structural level is accomplished with £nite-element discretization. Applications

are performed on graphiteepoxy composite beams, plates, and shells to illustrate the

methodology. Additional parametric studies demonstrate the variation of structural modal

damping with respect to ply angles, !_ber volume ratio, and temperature.

NOMENCLATURE

A

lAd]

[A_]

[B]

[C]

[C,]

[D,]

E

ft

F

G

Area.

3 x 3 extensional laminate damping matrix.

3 x 3 extensional laminate stiffness matrix.

Strain shape function matrix.

Global damping matrix.

3 x 3 coupling laminate damping matrix.

3 × 3 coupling laminate stiffness matrix.

3 x 3 flexurM laminate damping matrix.

3 × 3 flexurM laminate stiffness matrix.

Normal modulus.

Force.

Discretized force.

Shear modulus.

*National Research Council - NASA Research Associate at Lewis Research Center.

**Senior Research Scientist.

tBold characters indicate vectors.
k

[K]

m

[/]

[u]

P

t

T

V

U

U_Uy_'az

W_ AW

W,_W

E

6

/J

P

O"

Cx,¢y,¢=

¢

o)

Volume ratio.

Global stiffness matrix.

Percent of moisture by weight.

Global mass matrix.

Displacement shape function matrix.

Property.

Time.

Temperature.

Volume.

Displacement.

Deflections along x,y,z structural axes.

Stored and dissipated specific strain energies.

Stored and dissipated strain energies.

Engineering strain.

Damping coefficient matrix.

Fiber orientation angle.

Curvature.

Poisson's ratio.

Mass density.

Stress.

Rotations about x,y,z structural axes.

Specific damping capacity.

Angular velocity.

Subscripts

a

b

C

d

e

f

Dynamic amplitude.

Field force.

Ply (off-axis).

Damping.

Finite element.

Fiber.

2
gd Glass transition, dry.

gw Glass transition, wet.

i Node.

l Ply (on-axis).

/) Laminate.

m Matrix.

M Mechanical.

n n-th vibration mode.

o Reference.

t Surface traction.

Superscripts

o Laminate mid-plane.

Direction

11 Normal longitudinal.

22 Normal in-plane transverse.

12 Shear in-plane.

n Normal.

s Shear.

1. INTRODUCTION

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

recognized. Passive damping is a significant dynamic parameter for vibration and sound

control, fatigue endurance, and impact resistance. However, present light-weight and highperformance requirements imposed on most structural applications restrict the use of many

traditional non-structural damping sources. It is desirable, therefore, to use materials

which combine high damping in addition to other favorable mechanical properties. Fiber

reinforced polymer- matrix composites are known to exhibit material damping one or two

orders higher than most common metals as a result of the high damping of polymer matrices and the material heterogeneity. Composites are already preferred in many structural
applications due to their high specific Stiffness and strength, hence, the option of passive

structurM damping is an added advantage to these advanced materials. Recent work on

the damping of unidirectional composites [1] and composite laminates [2] has shown that

composite damping depends on an array of micromechanics and laminate parameters, including constituent material properties, fiber volume ratios, ply angles, ply thicknesses,

ply stacking sequence, temperature, moisture, and existing damage. Composite damping

is also anisotropic, but exhibits an opposite anisotropy trend than stiffness and strength,

being minimum in the direction of the fibers and maximum in the transverse direction and

in shear. It is apparent, therefore, that in order to realize significant structural benefits

from the damping of composite materials, integrated structural mechanics theories should

be formulated correlating the damping of composite structures to parameters of the basic constituent materials, laminate configuration, hygro-thermal conditions, and structural

geometry.

Unified mechanics theories are available for many elastic properties [3] required for

the prediction of the undamped dynamic response of composite structures. However, it

seems that an integrated methodology for predicting the damping of composite structures,

and their damped dynamic response, needs yet to be developed. Reported research on

the subject is mostly limited to the damping of unidirectional composites and laminates.

Analytical and experimental studies on the damping of continuous unidirectional fiber

composites have been reported [4-6]. These studies mostly assumed nondissipative fibers,

were restricted to longitudinal and in-plane shear damping, and overlooked the anisotropic

fiber damping properties. Theoretical and experimental work for the damping in discontinuous fiber composites is presented in refs. 7-9. The damping of off-axis composites and

symmetric laminates has been investigated [7,9-12]. Work on the structural damping of

symmetric composite plates has been also reported [13,14].

In this paper, an integrated finite-element based method is developed for simulating

the damping of composite structures. In contrast to the limitations of other methods

[13,14], the present methodology is applicable to both plate and shell-type structures and

4
general laminate configurations. It includes damping due to flexure (bending and torsion),

in-plane extension, and extension-flexure coupling. Furthermore, the method includes

the effects of temperature, moisture, micro-damage_ and interlaminar matrix layers on

structural damping. A specialty triangular plate element has been developed for the direct

formulation of finite element damping matrices based on micromechanics and laminate

damping theories. An additional feature of the method is the capability to calculate the

modal damping capacities of composite structures. Applications of the method are focused

on composite beams, plates, and shells. In this manner, the effect of structural geometry

on the structural composite damping is illustrated. Moreover, parametric studies illustrate

the effect of fiber volume ratio (FVR), ply angles, and operating temperature on structural

composite damping.

2. METHOD

In this section an integrated theory for simulating structural composite damping is

briefly described. The on-axis damping of each composite material (damping along the

material axes) is synthesized from the constituent fiber/matrix properties and fiber volume ratios (FVR's). Then, the damping capacities of off-axis composites are calculated

and, subsequently, the local laminate damping capacity is formulated. Finite element

discretization is utilized for the assembly of the global damping matrix and/or the calculation of modal structural damping. Analogous approaches are utilized for other mechanical

properties [3].

Damped Dynamic Response

Application of the principle of virtual work to an elastic structure under dynamic

loading requires that the work of discrete forces, body forces_ and surface tractions during

an infinitesimal virtual displacement should be equal to the sum of strain energy and

dissipated strain energy variations,

5
where, gui'rFi is the virtual work of discrete forces, 6uTfb the virtual work of non-inertial

body forces per unit volume, guT(--pfi) the virtual work of inertial body forces per unit

volume, 6uWf't the virtual work of surface tractions per unit area, _To the variation of

specific strain energy, and 6(Aw) the dissipated specific strain energy during the virtual

motion.

The virtual dissipated strain energy in eq. (1) depends on the viscoelastic behavior

of the material. Assuming a Voigt type of solid, the virtual dissipated strain energy is

express as,

s(Aw) = Jv 6 T[<] av (2)

By combining eqs. (1-2) and applying a finite element discretization scheme a = [N]u

and _ = [B]u, eq. (1) results in a system of discrete dynamic equations:

[M]ii + [C]fi + [K]u = Fi + Fb + Ft (3)

(4.1)

Aw= f (5)

/,

[M] =/v [N]T p[N]dV

[C] =/v[B]T[(][B]dV (4.2)

[K] =/v[B]T[EI[B]dV (4.3)

where, [N] and [B] are the displacement and strain shape functions respectively. In the

previous equations, [E] and [(] represent the local stiffness and damping coefficient matrices

of the material respectively. In the case of plate/shell composite structures, [E] and [(]

represent the local laminate stiffness and damping matrices [EL] and [_L] respectively.

Laminate Damping

Based on eq. (2), integration over a complete vibration cycle provides the specific

dissipated strain energy per cycle Aw,
For sinusoida/vibration, e = easin(wt), the previous equation yields,

Aw= (6)

In the case of composite laminates, the generalized strain eL represents the midplane strain

and curvature vector,

eL,a = {e°,k} (7)

and in accordance with eq. (6) the dissipated strain energy per cycle per unit area is:

AwL= (s)

As shown in ref. 2, The dissipated strain energy of a laminate is also expressed in

terms of the damping matrices [Ad], [Cd], and [Da].

AWL= _eL,a 1 T [[Ad][Ca] [Dd][Ca]]_L,a (9)

In the previous equation, the extensionai damping matrix [Ad] is associated with extensional deformations, the flexural damping matrix [Dd] is associated with bending and

torsional deformations, and the extension-flexure coupling damping matrix [Cd] is associated with coupled extension-flexure deformations. All three matrices in eq. (9) are fully

populated. The damping matrices depend on the off- axis SDC's of the individual plies

and the laminate configuration. They also include the interlaminar damping contributions

of the interply matrix layer, which have shown to be a significant additional source of

damping in composite laminates [2]. The damping coefficient matrix [eLi is expressed in

terms of the previous damping matrices by combining eqs. (8) and (9),

Ref. 1 describes the micromechanics theory for the damping of composite plies. The

off-axis composite damping is related to the on-axis elastic and dissipative properties and

fiber orientation. The off-axis damping matrix [¢¢] is fully populated due to in-plane

coupling between extension and shear. The on-axis SDC's depend on the hysteretic damping and the elastic properties of the matrix and fibers, the fiber volume ratio, and the

interfacia/friction damping from broken/debonded fibers.

7
Hygrothermal Effect

Temperature and moisture primarily affect the properties of the polymer matrix including damping, as a result, many properties of composites are sensitive to hygrothermal

variations. In this manner, temperature and moisture may have a definite effect on the

response of composite structures. Previous studies [15] have shown that the hygro-thermal

effect on most mechanical properties of the matrix can be expressed as,

PM Tg, -T o

- [Tg,_ - To ] .s

(10)

Based on the fact that for temperature ranges below the glass transition the matrix damping is increasing with temperature, an inverted form of equation (10) has been proposed

for the hygro-thermal effect on the matrix damping [1]:

To] (11)

The wet glass transition temperature is [15],

Tg_, = Tg,l(0.005rn _ - 0.1mz + 1) (12)

The exponent q in eq. (11) can be correlated to experimental damping data of each

individual polymer matrix. In the present study, the value q=0.5 is assumed. The effects

of temperature and moisture on structural damping axe simulated based on the previously

described micromechanics, laminate, and structural damping theories. A similar approach

is used for other mechanical properties.

Structural Modal Damping

Since the elasto-dynamic response of any structure, undergoing small deflections, is

a linear combination of individual vibration modes, structural damping associated with

each mode would provide a natural and meaningful measure regarding the overall damping

capacity of the composite structure. In the remaining sections, the global damping capacity

of the structure for a particular mode will be identified as structural modal damping.
Thus, an array of modal damping factors is generated,eachonerepresentingthe damping

capacity of the structure vibrating solely at the respectivemode shape. The modal specific

damping capacity (SDC) associatedwith the n-th vibration mode ¢,_is:

¢,, = L, w,,av

fv,,,,,d v (is)

where Awn and w_ represent the dissipated and maximum specific modal strain energies

per cycle. For the case of plate/shell composite structures the structural modal damping

becomes,

¢_ fA AWL,_dA (14)

"- fA wLndA

where Awn,_ and WL,_ are respectively the dissipated and maximum laminate modal strain

energies of the n-th mode, as defined in eq. (9).

The integrations in eq. (14) are performed numerically using finite element discretization. Such a finite element scheme has been developed for a triangular plate element with

3 nodes, 6 degrees of freedom per node ul = {u_, uv, uz, ¢_, ev, ¢_}, linear shape functions

for membrane deflections, and cubic shape functions for flexure.

Integration is first performed over the area of the triangular finite element. The

damping and maximum strain energies per cycle, over the area of the triangular element

Ae would be respectively:

1 /A Ue,nT[Be]T[ [Ad]

1

14zem = 2 /A Ue'nT[Be]T [ [A'][Cs]

[cd]

[Od]] [B_]ue,ndA,

[C,]] [B_]ue,=dA_ [D.]

(15)

(16)

where [A,], [C,], and [D s], are the extensional, coupling, and flexural laminate stiffness

matrices. [B_] is the element strain shape function matrix, and Ue.n is the nodal displacement vector of the respective mode. Both integrals are calculated numerically using

three-point quadrature. The modal SDC of the n-th mode ¢,_ is assembled according to

eq. (14):

Nil

E (aw,,,,)j

¢_ = j=l

N,

j=l

(17)
For lightly damped systems with nonclustered natural frequencies, as may be the case

of most composite structures, use of the undamped modes of eq. (3) will provide sufficient

accuracy. In this case, the direct solution of the damped dynamic system (3) may be

substituted by a less computationally intensive scheme of modal superposition.

3. APPLICATIONS AND DISCUSSION

Modal damping capacities of three different composite structures were simulated to

illustrate the present methodology. The three structures were: (1) a composite beam 16in

long, 2in wide, 0.2in thick (in some cases 0.08in thick); (2) a square plate 16in long, 16in

wide, 0.2in thick; and (3) a cylindrical shell panel 16in long, 16in wide, 10in in radius, and

0.2in thick. These three typical structural components were selected in order to demonstrate the effect of structural configuration on the modal SDC's. The structural complexity

increases from Structure 1 (beam) to Structure 3 (shell). All structures were assumed fabricated from the same composite system of high-modulus surface treated graphite (HM-S)

fiber in an epoxy matrix. The mechanical properties of this composite system are shown

in Table 1. The composite system was selected to match the 0.50 FVR HM-S/DX210

graphite/epoxy composite for which experimental damping data values are available [16].

The damping properties of the fiber and matrix for this composite system were backcalculated from measured on-axis SDC's based on the micromechanics damping theory in

ref. 1. The dry glass transition temperature of the matrix was taken to be 215°C (420°F).

Each structure was modeled using a finite element mesh of 55 nodes and 80 triangular

plate elements. Undamped mode shapes were used in the formation of structural modal

SDC's

The SDC of the first mode of the composite beam in free-free support with a symmetric

(8/90+0/45+0/-45+8)s laminate configuration is shown in Fig. 1 as a function of the outer

ply fiber orientation angle 0 with respect to azdal beam direction. Each ply was 0.010in

thick. The predictions correlate very well with measured damping values reported in ref.

16. The measurements were performed on composite beams vibrating at their first mode

(first bending) in a nearly free-free configuration. The experiments were conducted in

10
order to measurethe flexural SDC of the composite laminate, however, the measured

damping wasactually structural modal damping. The predicted laminate SDC basedon

the laminate damping theory reported in ref. 2, is also plotted in Fig. 1. The predicted

values for both structural and local laminate damping are very close, illustrating that

testing of composite beamsin free- free support will provide good estimations of passive

laminate damping.

Additional parametric studies were conducted regarding the effect of material and

environmental parameterson structural damping. Due to the spacehmitation, the studies

wererestricted on the effect of FVR, fiber orientation angle,and operating temperature T

on the modal SDC's of the previously described structures in cantilever support (Fig. 2).

The laminate configuration in the latter cases was a symmetric angle-ply ((-l-0)s)s with 20

plies through the thickness, each 0.010in thick. In all parametric studies, one parameter

was varying and the remaining two were held fixed at the following nominal values: 0.50

FVR, +45 ° ply angles, and room temperature (21.1°C (700F)).

Cantilever Beam

The first four modal SDC's and natural frequencies of the 0.50 FVR HM- S/Epoxy

cantilever beam at room temperature are plotted in Fig. 3. The symbols indicate the

simulation points. As seen in Fig. 3, only the 1st modal SDC exhibits uniform variation

with 0. Mode 1 was always the 1st bending mode of the beam. The SDC is rapidly

increasing with the ply angle and follows the trend of the flexural laminate SDC [2].

However, the SDC's of the three higher modes vary irregularly as the ply angles increase

from 0 to 90 degrees. The variation of the anisotropy of the composite laminate and the

dependence of laminate properties on the fiber orientation angle, have caused the higher

modes to switch shape between out-of-plane bending, torsion, and in-plane bending/shear.

For example, mode 2 at 0 ° is the 1st torsional mode, and mode 3 is the 2rid bending.

Near 15 ° both modes have equal frequencies and switch shapes. Similar switchings in

modal shapes occur between the 3rd and 4th mode. Mode 3 at 0 ° is the 2nd out-of-plane

bending, then subsequently switches to the 1st torsion, to the 1st in-plane bending, to the

11
3rd out-of- plane bending, and finally to the 1st in-plane bending. Mode 4 at 0 ° is the 1st

in-plane bending, and then switches to the 1st torsion, the 3rd bending, the 1st in-plane

bending, and the 3rd bending respectively.

As seen in Fig. 3, at double natural frequencies the respective modal SDC's change

rapidly. This happens because the modes switch shapes, therefore, the strains and stresses

associated with modal SDC's also change near the double natural frequencies resulting in

the observed damping variations. The out-of-plane 1st, 2nd, and 3rd bending modes follow

similar trends to the laminate flexural SDC, but they have different values for ply angles

between 15 ° and 70 ° due to transverse bending in the modes.

Fig. 4 presents the variation of modal SDC's and natural frequencies of a (4-45 °)

cross-ply laminate at room temperature with respect to the FVR. The modal SDC's vary

more uniformly with the FVR than with the ply angle. A switching occurs between modes

3 and 4. This reduction of modal SDC's as the FVR increases was expected [1].

The temperature effect on the modal SDC's and natural frequencies of the 0.50 FVR

(4-45 ° ) cantilever beam is illustrated in Fig. 5. The temperature significantly affects the

modal SDC's but has small effect on the natural frequencies.

Cantilever Plate

Fig. 6 shows the variation of modal SDC's and natural frequencies with respect to

the ply angle. Mode 1 is the 1st out-of-plane bending along the x-axis coupled with some

bending in the y-direction, and mode 2 is mostly the 1st torsional modal shape. Mode 3

is initially the 1st out-of-plane bending in the y-directlon, then as the ply angle increases

is gradually coupled with the 2nd out-of-plane bending mode in x-direction, and near 90 °

is mostly the 2nd bending in x-direction. The 4th mode is mostly the 2nd torsional shape

near 0 °, switches gradually to a coupled out-of-plane bending mode in x and y-directions,

and then switches to the 2nd torsional shape. As seen in Fig. 6, the modal SDC's follow

more or less continuous variations with the ply angle. Again, the modal SDC's follow the

opposite trend of the natural frequencies. In contrast to the beam, the plate is a twodimensional structure and its modes are mostly coupled changing shapes gradually. The

12
only exemption is mode 4 near 70% The coupled mode shapes have resulted in different

variations of the modal SDC's than in the previous case of the beam. The modal SDC's

of the plate do not follow closely the trends of laminate damping in bending and torsion.

Furthermore, the results illustrate the importance of the aspect ratio on modal damping.

The curves in Figs. 3 and 6 suggest that specimens of low aspect ratios may yield lower

experimental damping values for angle-ply composites and laminates. The same figures

also demonstrate that measured modal damping values depend on structure type and

should be properly interpreted.

Fig. 7 presents the variation of modal SDC's and natural frequencies of the +45 ° plate

with respect to the FVlZ. The modal SDC's of the plate decrease faster than the respective

SDC's of the beam in Fig. 3, and resemble more the variation of the longitudinal on-axis

damping of unidirectional composites. The fact that the structural damping of the plate

appears more fiber controlled than in the case of the beam has been attributed to the

two-dimensional nature of the plate.

The effect of temperature on the modal SDC's and natural frequencies is shown in

Fig. 8. The modal SDC's are more sensitive to temperature variations than the natural

frequencies. Both SDC's and natural frequencies of the plate are less sensitive to temperature than the SDC's and natural frequencies of the beam. This is in agreement with the

previous observation that the modal SDC's of the plate appear more fiber controlled than

the modal SDC's of the beam.

Cylindrical Shell

Compared to the previous beam and plate, the cylindrical shell panel in Fig. 2c

incorporates additional structural coupling between flexural and membrane deflections.

As a result, the cylindrical composite shell has the more complicated mode shapes of the

three structures considered in this study. Fig. 9 shows the variation of modal SDC's and

natural frequencies vs. the ply angle. All modal SDC's have high values at 0 ° and decrease

continuously as the fiber angle is increasing. This type of variation is substantially different

than the SDC's of the beam and plate. The structural membrane-flexure coupling and the

13
coupled mode shapesare in part responsiblefor this type of behavior.

At 0 ° , mode 1 is an opening mode, then is coupled with the 1st torsional shape as

a result of increased material coupling between flexure and torsion, and for angles greater

than 50 ° becomes mostly the 1st torsional modal shape. Mode 2 follows the opposite

pattern. Interestingly, the SDC of the torsional modal shape is not increasing for ply

angles higher than 45 degrees and follows the SDC of the opening mode. Modes 3 and 4

follow almost a similar pattern. Near 00 they are respectively the 2nd opening and 2nd

torsional modal shapes, then they appear coupled and switch shapes near 15 ° , and switch

again near 55 °. In general, the modal SDC's of the shell follow different variations with

the ply angle than the previous two plane composite structures. The results illustrate

the significant effect which curvature may have on the structural damping and dynamic

response.

Typical variations of modal SDC's and natural frequencies with respect to the FVR

are shown in Fig. 10. All modal SDC's decrease rapidly at low FVR's and then decrease

slowly. For FVR's of practical interest, the modal SDC's show small sensitivity to FVR.

The temperature effect on modal SDC's and natural frequencies is shown in Fig. 11.

As in the case of the plate, the modal SDC's increase with temperature. The natural

frequencies decrease slowly as the temperature increases and remain mostly insensitive to

the temperature variation.

4. CONCLUSIONS

An integrated methodology was developed for the computational simulation of structural damping in beam/plate/shell type composite structures. The method incorporates

micromechanics and laminate theories for composite damping, and can model the effect of

constituent material parameters, temperature, moisture, general laminate configurations,

and structural geometry. The structural damping simulation is based on finite element

analysis and a specialty finite element was developed for this purpose. The method has

been encoded into an in-house structural-analysis research computer code.

14
Correlations with limited available experimental data, illustrated the accuracy of the

method. The modal SDC's of a composite beam, a composite plate, and a composite shell

panel were simulated. Parametric studies illustrated the sensitivity of modal structural

damping to the fiber volume ratio, ply angles, temperature, and structural shape. The

obtained results lead to the following conclusions:

i. The structural damping has shown significant dependence on structural configuration.

Each modal SDC has also shown great dependence on the shape of the respective mode.

The damping of beam-type structural components is most sensitive to variations of

fiber volume ratio, fiber orientation, and temperature, while the damping of shell-type

components the least.

ii. In case of simple composite structural components (beams), the 1st modal SDC follows the trend of the ftexural laminate SDC, but the SDC's of the higher modes

have shown a pattern of rapid variations near double natural frequencies. In more

complicated composite structures (plates and shells), the pattern of rapid switchings

was less predominant, however, unpredictability in modal SDC's was induced as a

result of coupled mode shapes, l_Ience, structural analysis is recommended for reliable

predictions of structural damping and proper measured data interpretation.

iii. The aspect ratio has a significant effect on the modal SDC's of composite plane structures (beams or plates). The effect was more significant for higher modes. Geometric

curvature has also a significant effect on composite structural damping. The results

illustrate that the aspect ratio and warping of specimens may significantly affect the

accuracy of experimental damping measurements in off-axis composite laminates, especially in higher modes.

iv. PIy angles were found to be the most important parameters affecting the modal damping of composite structures. Significant alterations in the dynamic response may occur

by tailoring the ply angles. The effect of fiber volume ratio on modal SDC's was less

important.

v. The temperature had also a significant effect on structural modal damping. In all

15
structural cases, structural damping was more sensitive than the natural frequencies

to temperature variations.

Overall, the application examples demonstrated the versatility of the current method

in the dynamic analysis of composite structures. Proper tailoring of the basic composite

materials, lamination, and structural shape may significantly improve the damped dynamic

response of composite structures. The results also suggest that the tailoring of composite

structures for improved damped dynamic performance would be a complicated task and

may be best accomplished by combining the methodology described herein with formal

optimization methods.

ACKNOWLEDGEMENT

This work was performed while the author held a National Research Council - NASA

Research Associateship.

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16
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17
Society for Testing and Materials, 160-192 (1978).

16. Ni R. G. and Adams R. D. "The Damping and Dynamic Moduli of Symmetric Laminated Composite Beams - Theoretical and Experimental Results," Journal of Comoosite Materials, 18:104-121 (1984).

Table 1. Mechanical properties of HM-S/epoxy system.

Epoxy HM-S Graphite 50o HM-S/Epoxy(ref. [16])

E,,_= 0.500 Mpsi

(3.45 GPa)

G,_ = 0.185 Mpsi

(1.27 GPa)

¢,,_ = 10.30 %

"¢,,_s = 11.75 %

E¢11 = 55.0 Mpsi

(379.3 GPa)

E:22 = 0.9 Mpsi

( 6.2 GPa)

G f12 = 1.1 Mpsi

( 7.6 GPa)

vii2 = 0.20

¢:11 = 0.4 %

¢:._2 = 0.4 %

¢:12 = 0.4 %

¢al = 0.45 %

¢z22 = 4.22 %

¢a2 = 7.05 %

18
c=.

d

3,33

2.5(]

1.67

•83)

u FREE-FREE BEAM

0 MEASUREMENT [I(;]

..... LA/IINATETHEORY [2]

of%,

! X

I I I _ _I I

30 60 90 120 150 180

O, DEG

FIGURE I. - FIRST MODAL SDC OF THE O.SO FVR (O/90+B/l15+

O/-LIS+B)s HM-S/EPOXY BEAM IN FREE-FREE SUPPORT.

IG

(a) CANTILEVER BEAM.

Z

Y

___.._" 16 -

(b) CANTILEVER PLATE.

Z

(c) CANTILEVER SHELL.

FIGURE 2. - THE THREE COMPOSITES STRUCTURES.

19
-3-

6

5

4

3

2

I

0

MODE

I

[-- '--'0-- 2 _ 4')

+...,f

, ,_ _ -_. _._._

- _ J

I I I I I I

I ,.,_'_._. ,/_ \

500_._- _,

200,J -_%% %_

%% -+_<++,_

100 - o_:_ _:}o_:} _3o Doc:__oo_o: ]

15 30 45 GO 75 90

0, DEG

FIGURE 3. EFFECT OF PLY ANGLE0 ON TIE MODALPROPERTIES

OF THE I'_-S/EPOXY BEAM. (IL'VR= 0.50, T = 70 OF).

=_

I,L

l_ODE

12 -- ----o--- 1

--o-- 2

---¢,--- 3

I 1 I I I I

++50--

75

I

0 .117 .233 .350 .467 .583 .700

FIBER VOLUMERATIO

FIGURE _1. - EFFECT OF F"VRON THE MODALPROPERTIESOF THE

I_I-S/EPOXY EAR. (e = ±45°, T = 70 OF).

20
15--

12

6

MODE

I

_ 2

.... 4 _

I I [ I I ]

420 --

350 --

• 280 --

210-

LL

lqO --

70 --

I I I I_, I I

20 40 60 80 100 120

T - To, DEG

FIGURE 5. - EFFECT OF TEMPERATUREON THE MODALPROPERTIES

OF THE HM-S/EPOXY BEAM. (FVR = 0.50, e = ±q50).

7.5 _" RODE

----o---- 1

I --n-- 2 P

3.0

^_ _

°Ts

270 f--- . ..,_._-A-_-z_

; 1801--

135

45

15 30 45 60 75 90

0, DEG

FIGURE 6. - EFFECT OF PLY ANGLE 0 ON THE MODAL PROPERTIES

OF THE HM-S/EPOXY PLATE. (FVR = 0.50, T = 70 OF).

21
270 --

225 --

• 180

155

u_

45

m j.S_,oS_; J

-- Is SSS /"/'/ _

/

------ I 1 I I I I

.117 .233 .350 ,h67 .583 .700

FIBER VOLURERATIO

FIGURE 7. - EFFECT OF F-VR ON THE MODALPROPERTES OF THE

HR-S/EPOXY PLATE. (0 = _:_5°, T = 70 °F),

240 --

20O

160

z

12o

I.L

_ 8o

_0

B

m

I 1 1 1 I I

0 20 40 60 80 100 120

T - To, DES

FIGURE 8. - EFFECT OF TEMPERATUREON TIlE MODALPROPERTIES

OF THE HR-S/EPOXY PLATE. (FVR = 0.50, 0 = :L450),

22
.7

!

• 320I-- s s.s_"

0 15 50 q5

e, z>e_ FIGORE 10, - EFFECT OF FVU ON THE _ I_OPERT[ES OF "THE

F|GLIRE 9. - EFFECT OF PLY ANGLE0 ON THE RO1)A[.pROPERTIES HIR-S,/1F..POXYSHELL, (0 - :I:elS°, T = 70 OF).

OF THE HR-S/EPOX'YSHELL. (FVR = 0.50, T = 70 OF),

23
1.8

1.5

1.2

x_ .90

c

.50

•30

450

375

300

,T .

225

d

150

75

0

_ 2

1 ) I I I I

_ _ i_i_ _ _ _ .._im -

I I I I I I

20 _0 GO 80 100 120

T - T 0, DEG

FIGURE ll. - EFFECT" OF TEMPERATURE ON THE PA)DAL PROPERTIES

OF THE HIq-SAEPOXY SHELL. (FVR = 0.50, 0 = i450).

24
Natienal Aeronautics and Report Documentation Page

Space Administration

1, Report No. 2. Government Accession No. 3. Recipient's Catalog No.

" NASA TM-102567

4. Title and Subtitle 5. Report Date

Computational Simulation of Damping in Composite Structures

7. Author(s)

D.A. Saravanos and C.C. Chamis

9.

12.

Performing Organization Name and Address

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135-3191

Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, D.C. 20546-0001

6. Performing Organization Code

8. Performing Organization Report No.

E-5143

10. Work Unit No.

505-63-1B

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

15 Supplementary Notes

Prepared for the 2nd National Congress in Mechanics, Athens, Greece, June 29--July 1, 1989. D.A. Saravanos,

National Research Council--NASA Research Associate at Lewis Research Center. C.C. Chamis, NASA Lewis

Research Center.

16. Abstract

A computational methodology is developed for the prediction of passive damping in composite structures. The

method involves multiple levels of damping modeling by integrating micromechanics, laminate, and structural

damping theories. The effects of temperature and moisture on structural damping are included. The simulation of

damping in the structural level is accomplished with finite-element discretization. Applications are performed on

graphite/epoxy comlx_sitc bearns, plates, and shells to illustrate the methodology. Additional parametric studies demonstrate the variation of structural modal damping with respect to ply angles, fiber volume ratio, and temperature.

i 17. Key Words (Suggested by Author(s))

Properties; Damping; Composites; Composite structures;

Fiber volume; Ply angles; Temperature; Beams; Plates;

Shells

18. Distribution Statement

Unclassified - Unlimited

Subject Category 39

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

Unclassified Unclassified 26

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