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Thermoviscoplastic Nonlinear Constitutive 
Relationships for Structural Analysis 
of High Temperature Metal 
Matrix Composites 



tHASA-Tfl*8729i) IHEEJSCVISCCEIASIIC 

NONLINEAB CONSTITUTIVE RELAltCNSHlPS FOE 

S1RUC20KAL ANALiSlS CF HIGH liKEEEAIUHB 

HETAL HATEIX CCKECSITES {NASA) 25 p 

HC A02/BF A01 CSCL 11D G3/2H 



HQ6-24756 



Unclas 
42913 



Christos C. Chamis and Dale A. Hopkins 
Lewis Research Center 
Cleveland, Ohio 




Presented at the 

First Symposium on Testing Technology of Metal Matrix Composites 
sponsored by the American Society for Testing and Materials 
Nashville, Tennessee, November 18-20, 1985 



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CONTENTS 



Page 

ABS1RACT ! 

INTRODUCTION ! 

EQUATION FORM/FEATURES 2 

APPLICATION . . . 4 

POSSIBLE EXTENSIONS/LIMITATIONS 7 

CONCLUSIONS/REFERENCES 8 



'"" " : "' ' ■ " ""■ y\ 

THERMOVISCOPLASTIC NONLINEAR CONSTITUTIVE RELATIONSHIPS 
FOR STRUCTURAL ANALYSIS OF HIGH TEMPERATURE METAL MATRIX COMPOSITES 

ChHstos C. Chamls* and DaTe A. Hopkins** 
National Aeronautics and Space Administration ^ 

Lewis Research Center 
Cleveland, Ohio 44135 

SUMMARY 

A set of thermovlscoplastlc nonlinear constitutive relationships (TVP-NCR) T 

1s presented. This set 1s unique and has been developed mainly for application ,] 

to high- temperature metal-matrix composites (HT-MMC) &nd is applicable to ther- i 

mal and mechanical properties. Formulation of the TVP-NCR 1s based at the .< 

mkromechanlcs level. The TVP-NCR are of simple form and readily Integrated 

into nonlinear composite structural analysis. Results show that this unique 4 

§ set of TVP-NCR Is computationally effective. It provides a direct means for ; 

£ predicting complex materials behavior at all levels of the composite Simula- j 

,i tlon; that 1s, from the constituent materials, through the several levels of : *| 

composite mechanics, and up to the global response of complex HT-MMC structural \ -, 

components. \ 

INTRODUCTION 

High temperature metal matrix composites (HT-MMC) are emerging as mate- 
rials with potentially high payoffs In structural applications. Realization 
of these payoffs depends on the parallel and synergistic development of (1) a 
technology base for fabricating HT-MMC structures and components, (2) experi- 
mental techniques for measuring their thermal and mechanical characteristics, 
and (3) computational methodologies for predicting their nonlinear thermovlsco- 
plastlc (TVP) behavior 1n complex service environments. In fact, development 
of computational methodologies should precede the other two because the struc- 
tural integrity and durability of HT-MMC can be numerically assessed and the 
potential payoff for the specific application can be closely estimated. In 
this way, 1t 1s possible to minimize the costly and time consuming experimental 
effort that would otherwise be required m the absence of a predictive 
capability. 

Reteht research at NASA Lewis 1s directed towards the development of a 
computational capability to predict the nonlinear TVP behavior of HT-MMC. This 
capability 1s schematically depicted 1n figure 1. As can be seen in this 
figure the capability consists of several computational modules encompassing 
the material TVP behavior (bottom), composite mechanics (sides), and the finite 
element analysis of structural components (top). The TVP computational module 
consists of mathematical models which formally and explicitly relate the 
dependence of the constituent material properties on 

*Sen1or Research Engineer, Aerospace Structures/Composites. \ 

**Aerospace Structures Engineer. \ 



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i^twE'Jlnli IV St T S ' ? nd (3) t1me - These wth«nat1cal models are col- 
NCftJ !L"i !?M her T^° plast1c nonl1n "r constitutive relation ships (TVP. 
NCR). The objective of this report Is to present the ivp url aIIIIITJ* V 
part of the computational capability for HT-m? structures ° P ' d " * 

t1onIlVffec!?^« e JH SCr1bed / e,re , deve1oped w1th an em P h "1s on computa- 
tional effectiveness and are unique m their following features- m SJIrir 

form - they are applicable to all constituent mateMal proper? es the£!l 
SKli?!' 1nclud1ft 9 strength), (2) evolutionary - they a?e e«ly extend/ 
modified to accommodate additional effects such as strain rat*^ m!fl? IS? 

tL rad Mj°: n il?L 1s T rph1c v?? y have >^ «^ ^a d t : < f _ 

This unique set of TVP-NCR consists Of products of terms with unknnur, 
exponents The exponents are determined fo° the specific material anS tine of 
tTp NcTeoualtJn 6 !^ 'V!*^'- th I" Product terms aVFr^ulr.ft.'XJ ° 
Ind"?Sf XI" E ac " unt for: 0> temperature dependence, (2) stress level 
T.t i?L Jh?" Pat J: Three ex P dn ** ts "*ed to be determined to complete y 
i5l!^J h1$ equa t 10n ' These "P^nts are determined from available expeM 
mental data or estimated from anticipated behavior of the particular prodScT 

The computational effectiveness of this unique set of TVP nch 1< ftu , lliafo , 
using the computational capability depicted schemat cal?y [f^ll i he 

St Sc Ihd l2Trl\° f d tUrb1 " 6 bUde ' made from f1ber relnfo ?ed superanoy 
HT-MMC and subject to representative mission loading conditions 1s determined 
The effectiveness of the TVP-NCR to represent the physlca behavior of th! 

7&*llill* T PUtat1 ° nally assessed b * Perturbing the exponents and comparing 
the structural response of the blade to the unperturbed case. comparing 



i 

i i 



EQUATION FORM/FEATURES 
Is aI h follIX! C f ° rm S6leCted f ° r the TVP " NCR f ° r the co " st1 tuents of HT-MMC 
(1) Mechanical property (moduli, strength) P M 



Mo 



>" 1o J l> 



a 



m 



i 



F ' °o 



lA 



(1) 



(2> ^a r c1t Y rP Perty (expaf, ' 1ort coeff ^lents, thermal conductivity, heat 



P, 



10 T" J [ Sf 



- 



m 



Sr - 

V- % 



(2) 



V 



n 



W-. 



v^: r 






::*£'■■ 



**■ 



The notation used in equations (1) and (2) 1s as follows: 

PHj denotes the current property of Interest 

P 1s the corresponding property at reference conditions 

Tm Is the melting temperature 

T 1s the current temperature 

1 1s the reference temperature at which P is determined 

Sp 1s the fracture stress determined at T conditions 

o the reference stress at which P 1s determined 

Sf Is an appropriately selected stress rate, for example, the stress 
rate at which penetration occurs during Impact 

o c Is the stress rate at which P 1s determined, and 

<j 1s the current stress rate 

The exponents n, m, and l are empirical parameters which are determined 
from available experimental data or estimated from the anticipated behavior 
of the particular product term. 

The first term on the right side of equation (1) represents the tempera- 
ture dependence, the second represents the stress dependence, and the third 
represents the rate dependence or the time dependence, In part. The other 
part of the time dependence 1s through the direct time Integration as will 
be described later. Equations (1) and (2) describe, then, material behavior 
1n the temperature- stress- time space. 

Each term on the right side of equations (1) and (2) describes a mono 
tonic functional dependence of P/P from some Initial property value to 
a terminal or ultimate material state. The specific shape of the function 
depends on the exponent as 1s shown In figure 2 for a fixed exponent and In 
figure 3 for a fixed T/T F ratio. By judicious selection of the exponent 
and the initial and terminal values, a variety of functional dependences can 
be simulated using equations (1) and (2). 

lhe form of equations (1) and (2) makes it convenient (provides direct 
feedback) to select the various parameters so that the functional dependence 
described Is consistent with the physical considerations. For example, 1t 
1s well known that the melting temperature (1 H ) Is a fundamental parameter 
1n metals and that the mechanical properties are "zero" at T(y|. Also, the 
stress at fracture for some reference condition 1s readily determined by 
simple experiments. In addition, the ultimate value of the stress rate may 
be determined from high velocity impact penetration tests. It can be seen 
from figure 2 that the P/P Increases/decreases very rapidly as the melt 
Ing temperature (or any terminal value) 1s approached. Furthermore, the form 
of the equations make It convenient to evaluate the exponents from available 
data since each term 1s "Isolated" from the others, that 1s, the other terms 



V 



can be taken at reference conditions and will be unity. The equations are 
computationally effective since each term requires simple substitution of 
values and one exponentiation. 

The form of the TVP-NCR selected has all the desirable features men- 
tioned 1n the introduction. These features can be conveniently summarized 
Into three groups: (1J physical, (2) fundamental, and (3) computational, as 
follows: 

(1) Physical - The constitutive relationships describe dependence on: tem- 
perature, time, stress, stress rate, and complete property degradation 
as the ultimate value 1s approached. 

(2) Fundamental - the constitutive relationships are: 

generic - they are applicable to all constituent material proper- 
ties (fig. 4) 

evolutionary - they are easily extended to Include additional 
dependence, for example cyclic (mechanical, thermal) 

isomorphic - they have the same form for all the properties 

unified - they are fully coupled from the initial to the terminal 
material state 

universal - they are equally applicable to any three constituents 
(fibers, matrix, interphase) 

nondimenslonal - they are normallzable with respect to reference 
and ultimate values. 

(3) Computational - the constitutive relationships are: ' 

computationally effective - they only require simple substitution * 

and exponentiation L< 

easily Integrated into nonlinear composite mechanics and struc- 
tural analysis codes - they can be fully Integrated using only a 
few programming statements 

i 
APPLICATION V 

The 1VP-NCR were Integrated into a special-purpose computer code for 
structural analysis of turbine blades fnade from HT-MMC (ref. 1). This code 
1s depicted schematically in figure 1. The TVP-NCR describe the constituent 
material properties 1n the material space as shown at the bottom of the 
figure. Notfe that the cumulative time* temperature* <»rid stress at the cur- 
rent state are tracked through the Integrated computational process as shown 
1n the figure. Once the current properties for the constituent materials 
have been determined, they are used In the various levels of composite mech- 
anics to generate the quantities required for the global structural analysis 
(ref. 1). 



1 



M 



MMMWMMMBPMi 




volume ratio of the fih<>r ui*c n * T k i * <uer/superai ioy hi~mmc. The 
»uL n ^ r % t :;^ Pendent PWrtlt.) are estimated or deduced fro"ot*er 



p >dt^;f t f ;=it t^?Mr r£ 9" 

If he S»L°? ^,'? C ?^ r t???"" "^"P""" to the outermost ply (Ply no'li 
at the nodal point Identified by the arrow 1n fitmre * (*«** ;« IX i ' 

of thJ JJJJl!J;.i he secondset was Performed to determine the sensitivity 
lift rt ructura ^ r ^Ponse to arbitrary perturbations of the 1VP-NCR exoon 
tlowl I?LL e t?nn *!* of * nal * s ? s ^onstrates the Importance of compu a 
£X . ? 2*!°" s1nce 1t P rov1d « an assessment of the accuracy of the 
data required to experimentally determine the parameters In the W-NCR 

f»hi 7 J e / esu J ts obtained from both structural analyse- are summarized in 
fo? thl wJ he 9 "J? 1 V f MableS (cru1se and ^sldual state) aTn"abe II 

cases ? ) I*?/ \ a V e L {C : uU V tate only) - The resu1t * are grouped 1n5o 
cases 1-7. case 1 1s the baseline case and has the "best" values for Ihl 

'Z'Ztr^ th ? TVP ' NCR - Cases 2 ~ 7 constitute the sen 1t1v ty ana y es 

the£b^M 

obtain data for these exponents which 1s accurate to within 15 percent 
especially 1n these high temperature ranges. percent 

/*. k , lh M 9 ? test change (16 Percent) 1n the frequency occurs 1n cat* 3 
(table I) where the perturbations in the exponents decrease ?he mechanLl 
properties and Increase the thermal properties of the matrix inT^*!<\ 
change (15 percent) 1h the ply stress occu n case 6 ?tab?e , u * 
perturbations in the exponents (1) Increase the "ber m a c 'p? p° t e 

prope e'fof h°e 6 f?LJ h h,ft r1X ' iM * 1°^^ < 2) decrease th/SSSl 
properties of the fiber but increase those of the matrix The residual <<■;.** 

Ply stresses for all case^re negligible for all pJactUal purposes '* 






The Integrated aralysls generates properties at all levels of the com- 
posite behavior simulation. Behavior of the longitudinal (fiber direction) 
modulus (En) of the constituents and ply are shown graphically 1n 
figures 7(a) and (b) for cases 1 and 6 throughout the mission duration. The 
modulus decreases Initially during the start-up and climb part of the mis- 
sion. The modulus levels off during the steady state (cruise) part of the 
mission. Finally, the modulus Increases during the landing and engine cut- 
off part of the mission. The significant point 1s that the TVP-NCR appear 
to represent the material behavior as would be Intuitively anticipated for 
this type of flight mission. 

The corresponding behavior for the transverse (perpendicular to fiber) 
modulus (E 2 2) 1s shown 1n figures 8(a) and (b) and for the 1n-p1ane shear 
modulus (G12) in figures 9(a) and (b). The behavior of these moduli 1s 
about the same as that for the longitudinal modulus (En). 

The behavior of the longitudinal thermal expansion coefficient (an) 
throughout the mission 1s shown In figures 10(a) and (b) for cases 1 and 6. 
This coefficient Initially Increases rapidly (during climb), levels off dur- 
ing cruise, and slowly decreases to about Its Initial value during landing. 
This type of behavior 1s to be expected since the thermal expansion coeffi- 
cients increase with increasing temperature. Note that even though an 
for the matrix for case 6 is about 30 percent smaller than for case 1, the 
coefficients for the ply and the interphase are about the same. This illu- 
strates, 1n part, the restraining Influence of the fibers in the HT-MMC 
behavior and in addition the Importance of having TVP-NCR defined at the 
mlcromechanics level. The corresponding behavior for the transverse thermal 
expansion coefficient (a 2 2) is shown in figures 11(a) \nd (b). The 
behavior of a22 is similar to that of an • 

The behavior of the three lowest natural frequencies of the airfoil 
throughout the flight mission is shown in figures 12(a) and (b) for cases 1 
and 6. Note that each frequency behaves somewhat differently. This is 
expected since each frequency is influenced differently by the centrifugal 
force. The second and third frequencies decrease during climb, increase 
during the early part of cruise, remain constant during the major portion of 
the cruise, and increase gradually to about their initial value during land- 
ing and engine cut-off. On the other hand* the first frequency increases 
sharply during climb (due to centrifugal force stiffening), levels off during 
cruise, increas&S sMghtly during landing (cooling but speed retained) and 
gradually decreases to approximately its initial value (zero- speed) . The 
coupled behavior of these three frequencies throughout the flight mission 
further demonstrates the computational effectiveness of the TVP-NCR to 
represent the physics of the HT-MMC from the constituent materials level to 
the component global structural response. 

The longitudinal stress (on) behavior throughout the mission in the 
constituents and 1h the ply is shown in figures 13(a) and (b) for cases 1 and 
6. The stress 1h the fiber Increases very rapidly during climb, decreases 
gradually during cruise, decreases rapidly during landing, and decreases 
gradually to a small residual compressive stress at engine cut off. Ihe ply 
stress exhibits the same behavior as the fiber stress but 1s much lower 1n 
magnitude. The stress in the matrix increases (compressively) very rapidly 



1 



W*i * 



•P"^ 



mmmmm 



during climb, remains compressive during cruise and decreases gradually to a 
residual tensile value during landing and engine cut-off. 

The corresponding behavior for the transverse stress (o 2 2) 1s shown 1n 
figures 14(a) and (b) and that for the IntralarMnar shear stress (o 12 ) 1n 
figures 15(a) and (b). Note there are three different regions (A f B, and C) 
for the matrix and two different regions (B and C) for the Interphase 1n 
which c 2 2 ahd 12 are computed. These regions correspond to the 1ntra- 
lamlftar regions shown 1n figure 4. The Interesting points to note are: (1) 
the matrix in the different regions 1s subjected to both tensile and compres- 
sive transverse stresses which can be of substantial magnitude, (2) the 
interphase can be subjected to relatively high transverse tensile stresses 
which may cause Interfaclal damage, (3) the fiber 1s subjected to very high 
transverse tensile stresses which could cause fiber splitting, and (4) the 
transverse ply stress 1s relatively small compared to the stress distribution 
1n the constituents. 



Collectively, the 
effectiveness of the 1 
constituents at the ml 
anks) level. Determl 
possible because the 1 
and the formulation 1s 
gradual decrease Indie 
the material thermovls 
creep. 



local stress behavior demonstrates the computational 
VP-NCR to predict the Instantaneous behavior of the 
cromechanlcs level as well as at the ply (macromech- 
natlon of the stress behavior 1n the constituents 1s 
VP-NiJR are referred to the constituent material space 

based at the composite mlcromechanlcs level. Ihe 
ated for all the stresses during cruise 1s caused by 
coplastlc behavior and may be thought of as a form of 



POSSIBLE EXUNSiONS/i. IMITATIONS 

The 1VP-NCR can be extended to Include thermal cycle and mechanical 
cycle effects, diffusion, other material degradation effects, as well as 
time directly. Though terms for these factors are easily added since they 
will be of similar form, 1t 1s not necessarily clear which of these will 
contribute to Independent material behavior. 

In the direct time Integration analysis, the effects of temperature are 
directly accounted for. Any ratchettlng, for example, wilt be part of the 
residual displacements. Also, residual thermal stresses and other stresses 
are known and constitute a part of the cumulative stress history. On the 
other hand, vibratory stress effects are not accounted for 1n the direct 
time Integration of the structural analysis. Though these effects can be 
accounted for through the stress rate, vibratory stress effects may Indeed 
contribute to Independent behavior. Diffusion or any other material degrada 
tlon can be Incorporated once the type of degradation has been defined. 
Other extensions will become self evident as HT-MMC start being extensively 
applied 1n environments where limited o no property data are available. 

Some limitations of the 1VP-NCR described herein are that they: (1) 
must be used at the current Instant of time, (2) must be used with a direct 
time Integration nonlinear composite structural analysis, (3) cannot be 
verified experimentally at all levels of the composite mechanics analysis 
and at the very high temperatures, and (4) do not Incorporate Initial tangent 
unloading or possible shakedown 1n the classical plasticity sense. Whether 
(3) and (4) are serious limitations 1s yet to be determined. At this stage 



I 

^.-^< 




of the development 1t 1s prudent to say that these TVP-NCR must be used 
judiciously in design studies relying mainly on sensitivity analyses and 
other Judgment factors that are appropriate for the specific case 



CONCLUSIONS 

/tvp Uppw!! 6 kJVI thermov *scoplast1c nonlinear constitutive relationships 
(TVP-NCR) for high-temperature metal-matrix composites (HT-MMC) has been 

cabtlT.n^ 5 prese i; te ?- n™ 5 S6t of 7VP " NCR 1s of ^P le form? Is appli- 
cable to all thermomechanlcal properties, 1s fully coupled, 1s readily 1nte- 

SfeSilI! !nnnr!Mi^!!; P0S H te struc * u ^l analyses , and 1s computationally 
effective. Applicability and computational efficiency were demonstrated 
through an application to a HT-MMC turbine blade structural anal? Is Sen- 
sitivity analyses Indicated that substantial perturbations 1n the TVP-NCR 
exponents have rather minimal effect on the global and local response of the 
structure These TVP-NCR make it possible to trace the history of Hl-MMC 

mirrn^h]n5 0ft, T er, J S K f !: om fabr ] cat1on th ™"9h service and from the composite 
micromechanlcs to global composite structural response. The TVP-NCR are 
suitable for preliminary designs and parametric studies. They should be 
tal lj Tr flT d 1n 6 J eSi9n a PP l1cat1 <™ since they have not been experlmen- 



REFERENCE 

1. Wins, D.A., "Nonlinear Analysis of H1gh-7emperature Multl layered 
Fiber Composite Structures," NASA TM-83754, National Aeronautics and 
Space Administration, Washington, D.C., 1984. 



■>, 



7 



TABLE I. - EXPONENT PERTURBATION EFFECTS ON STRUCTURAL RESPONSE 



Case 


Constituent 


Property 


Change 


Exponent 


Structural response 


Percent 
freq. 




Cruise conditions 


Residual 


Cruise/ 
res. 


N 


M 


L 


Displ. 


Unt't 


Freq. 


Displ. 


Unt't 


Freq. 


1 


Fiber 
Matrix 


F 

a 


R 

R 
R 

R 


0.3 
.6 
.8 
.2 


0.6 

.1 

2.8 

.1 


0.1 
.05 
.1 
.05 


0.015 


-0.22 


3650 


0.00016 


-0.04 


4380 


0/0 


2 


Fiber 
Matrix 




R 
R 
I 
D 


0.3 
.6 
.4 
.1 


0.6 

.1 
2.0 

.05 


0.1 
.05 
.1 
.05 


0.015 


-0.16 


3840 


0.00016 


-0.03 


4550 


+S/+4 


3 


Fiber 
Matrix 


Pt 


R 
R 

D 
I 


0.3 
.6 

1.2 
.6 


0.6 

.1 
4.0 
0.2 


0.1 

.05 
0.1 
0.05 


0.017 


-0.31 


4230 


0.00012 


-0.05 


4650 


+16/+6 


4 


Fiber 
Matrix 


ft 


I 

R 
R 


0.1 
.2 
.8 
.2 


0.2 

.05 
2.8 

.1 


0.1 
.05 
.1 
.05 


0.014 


-0.20 


3470 


0.00010 


-0.04 


4370 


-5/~ 


5 


Fiber 
Matrix 


% 
% 



I 
R 
R 


0.5 

1.0 

.8 

.2 


1.0 

.2 

2.8 

.1 


0.1 
.05 
.1 
.05 


0.016 


-0.23 


4050 


0.00020 


-0.02 


4540 


+11/+4 


6 


Fiber 
Matrix 


? 
* 


I 

D 

I 


0.1 
.2 

1.2 
.6 


0.2 

.05 
4.0 

.2 


0.1 
.05 
.1 
.05 


0.015 


-0.29 


3660 


0.00021 


-0.06 


4580 


~/ + 6 


7 


Fiber 
Matrix 


ft 


D 
I 
I 



0.5 

1.0 

.4 

.1 


1.0 

.2 
2.0 

.05 


0.1 
.05 
.1 
.05 


0.0J !: > 


-0.17 


4040 


0.00018 


-0.02 


4350 


+11/+6 



P M - Mech. Prop. 

Pj - Thermal Prop. 

R - Reference 

I - Increase 

D - Decrease 

Units: Displ. - In: Unt't - Deg: Freq., Hz 



« t 

i i 



.i-ri 



IV 



\ / 



'J 






"»*■ 



TABLE II. - EXPONENT PERTURBATION EFFECTS ON PLY STRESSES 



B > 



Case 


Constituent 


Property 


Chanqe 


Exponent. 


Ply stresses 


Percent 
tot n 


Cruise conditions 


Residual 












N 


M 


L 


atii 


°*22 


o*12 


a m 


ot 2 2 


**12 


Cruise/ 
res. 


1 


Fiber 


i* 


R 


0.3 


0.6 


0.1 




















P T 


R 


.6 


.1 


.05 


29.2 


8.4 


-13.7 


0.4 


-0.6 


• 







Matrix 


s? 


R 
R 


.8 
.2 


2.8 

.1 


.1 
.05 
















2 


Fiber 


E* 


R 


0.3 


0.6 


0.1 




















P T 


R 


.6 


.1 


.05 


27.6 


10.2 


-13.6 


0.5 


-0.4 


~ 


-5 




Matrix 


PM 

pt 


I 



.4 
.1 


2.0 
.05 


.1 
.05 
















3 


Fiber 


&* 


R 


0.3 


0.6 


0.1 




















P T 


R 


.6 


.1 


.05 


32.1 


6.0 


-13.7 


0.5 


-0.7 


*. 


+10 




Matrix 


PM 

pt 




I 


1.2 
0.6 


4.0 
0.2 


0.1 
0.05 
















4 


Fiber 


ft 


I 


0.1 


0.2 


0.1 




















p T 





.2 


.05 


.05 


30.8 


. 7.1 


-13.9 


1.9 


-1.8 


~ 


+5 




Matrix 


f? 


R 
R 


.8 
.2 


2.8 

.1 


.1 
.05 
















5 


Fiber 


S* 


D 


0.5 


1.0 


0.1 




















P T 


I 


1.0 


.2 


.05 


33.2 


10.4 


-16.1 


5.2 


-1.4 


-2.8 


+14 




Matrix 


Pt 


R 
R 


.8 
.2 


2.8 

.1 


.1 
.05 
















6 


Fiber 


E* 


I 


0.1 


0.2 


0.1 




















P T 


D 


.2 


.05 


.05 


33.7 


4.5 


-13.7 


1.4 


-1.4 


-0.2 


+15 




Matrix 


p M 

pt 




I 


1.2 
.6 


4,0 
.2 


.1 
.05 
















7 


Fiber 


5« 





0.5 


1.0 


0.1 




















Pt 


I 


1.0 


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26.7 


11.1 


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- 


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-9 




Matrix 


o PM 

p t 


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D 


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P M - Mech. Prop 
Pj - Thermal Prop 
R - Reference 
I - Increase 
D - Decrease 




w w y y , « ! w. 



- , w^" 



-V 4 






TO 

GLOBAL 
STRUCTURAL 
ANALYSIS 



LAMINATE 




f LAMINATE 
THEORY 



COMPOSITE 

MICROMECHANICS V 
THEORY 



^ 



COMPONENT 



CONSTITUENTS 




V 



FROM 
GLOBAL 
STRUCTURAL 
ANALYSIS 




LAMINA"' ( 
THEORY { 



NONLINEAR 
MATERIAL n 
MODEL 



LAMINATE 



PLY 



> COMPOSITE 

* MICROMECHANICS 
>' THEORY 



MATERIAL PROPERTIES 
P * P(0, T. t) 



Fig. 1. "Integrated nonlinear composite structural analysis. 




Fig ; , 2 ;. * Thermoviscoplastic nonlinear constitutive 
relationships - typical behavior for a given 
exponent. y 



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^^ 




-2 



-1 1 

EXPONENT, n 



3 



Fig. 3. -Thermoviscoplastic nonlinear constitutive 
relationships typical behavior for given T/T F 
ratio. 



SUBREGIONS OF 
INTRALAMINAR 
NONUNIFORMS -^ 




MATRIX 

INTERPHASE 

FIBER 



**2 



Fig. 4. -Typical interconstituent regions for composite micromechanics. 



J, 






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v f ■ 0. 50 
\ g * 0.01 in. 
9 • [±45L 




Fig. 5. - Finiie-element model for hollow turbine blade 
airfoil of high temperature metal matrix composite. 




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120 



80 



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6000 

4000 
2000 



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8000 
6000 

4000 
2000 



20 



40 



60 



i 



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60 



Fig. 6. -Flight mission profile for turbine blade structural analysis. Node6 ; 



Ply 4. 



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»i i i «a ~w* 



60 



40 



20 



i o 

o 
o 

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Fig. 7. - Longitudinal modulus behavior during flight mission. Node 6* 
ply 4. 



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Fig. 8. - Transverse modulus behavior during flight mission. Node 6- 
ply 4. * 



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M 


MATRIX 










D 


INTERPHASE 










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2 40 


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Fi L?' I ln ! raIaminar she * r modulus behavior during flight mission 
Node 6; ply 4. 



s^aBCTEsrsrarsmEEiS 



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z 
o 

z 



0£ 



s 



4- 



£ 2 



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D INTERPHASE 

L PLY 

CASE 1 




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40 



60 



7 




J L^J L 



40 



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6 
4h 



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20 



40 



60 



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Fig. 10. - Longitudinal thermal expansion coefficient behavior during flight 
mission. Node 6; ply 4. 



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6 
4 
2 



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2 



J L 



20 



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60 




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TIME, t, min 



40 



60 



Fig. 11. -Transverse thermal expansion coefficient behavior during flight 
mission. 







CASEl 



- 1 1st MODE 

2 2nd MODE 

3 3rd MODE 



J 1 

60 




CASE 6 



20 



CASE 6 



J L 



40 



(a) 
60 




— L ^p — i — ' ' i 

2 40 60 

TIME, t, min 

Fig. 12. -Turbine airfoil frequencies behavior during flight mission. 




V 



im&trgm 



MMtfMi 



111* Iri 




fe ) 






120 n 



F FIBER 

M MATRIX 

D INTERPHASE 

L PLY 



CASE 6 



4/> 

UJ 



Z 

o 




CASE 6 




60 

TIME, t, min 

Fig. 13. - Longitudinal stress variation during flight mission. Node 6; 
Ply 4. 



i 

ill 



i 




g gg^a-i* 



^*w 



1 FIDER 

2 MATRIX (A) 

3 MATRIX (B) 

4 MATRIX (C) 

5 INTERPHASE <B) 

6 INTERPHASE (C) 

7 PLY 



D 



UJ 



200 


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6^ 


W 


100 


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> 


4-i 






\ 







r- 






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1 1 


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I I 


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TIME, t, min 
Fig. 14. -Transverse stress variation during flight mission. Node 6; ply 4. 



! * 



•zvmzzr 



LMUL JL.-l*ll-JHLh-: 



1 FIBER 

2 MATRIX (A) 

3 MATRIX (B) 

4 MATRIX (C) 

5 INTERPHASE (B) 

6 INTERPHASE (C) 

7 PLY 




60 

TIME, t, min 

Fig. 15. - Intralaminar shear stress variation during flight mission. 
Node 6; ply 4. 




1. Report No. 

NASA TM-87291 



2. Government Accession No. 



3. Recipient's Catalog No. 



4. Title and Subtitle 



5. Report Date 



Thernrtovlscoplastlc Nonlinear Constitutive Relation- 
ships for Structural Analysis of High Temperature 
Metal Matrix Composites 



6. Performing Organization Code 

505-63-11 



7. Authors) 

Chrlstos C. Chamls and Oale A. Hopkins 



6. Performing Organization Report No. 

E-2998 



10. Work Unit No. 



9. Performing Organization Name and Address 

National Aeronautics and Space Administration 
Lewis Research Center 
Cleveland, Ohio 44135 



11. Contract or Grant No. 



12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 
Washington* O.C. 20546 



13. Type of Report and Period Covered 

Technical Memorandum 



14. Sponsoring Agency Code 



15. Supplementary Notes 

Presented at the First Symposium on Testing Technology of Metal Matrix 
Composites, sponsored by the American Society for Testing and Materials, 
Nashville, Tennessee, November 18-20, 1985. 



16. Abstract 

A set of thermov1scoplast1c nonlinear constitutive relationships (1VP-NCR) 1s 
presented. This set Is unique and has been developed mainly for application to 
high- temperature metal-matrix composites (HT-MMC) and Is applicable to thermal 
and mechanical properties. Formulation of the TVP-NCR 1s based at the micro- 
mechanics level. The TVP-NCR are of simple form and readily Integrated Into non- 
linear composite structural analysis. Results show that this unique set of 
TVP-NCR 1s computationally effective. It provides a direct means for predicting 
complex materials behavior at all levels of the composite simulation; that 1s, 
from the constituent materials, through the several levels of composite mech- 
anics, and up to the global response of complex HT-MMC structural components. 



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

Fiber composites; Metal matrix; Thermovisco- 
plastic behavior; Composite mechanics; Structural 
analysis; Stress analysis; Turbine blades; Thermal 
properties; Mechanical properties; Micro 
stresses; Vibration frequencies. 



19. Security Classif. (of this report) 

Unclassified 



18. Distribution Statement 

Unclassified - unlimited 
STAR Category 24 



20. Security Classif. (of this page) 

Unclatss1f1jed- 



21. No. of pages 



22. Price* 



\-_-4 



*For sale by the National Technical Information Service. Springfield, Virginia 22161