Probabilistic Modeling Of High-Temperature Material PropertiesOf A 5-Harness 0/90 Sylramic Fiber/CVI-SiC/MI-SiC Woven Composite
Abstract
An integrated probabilistic approach was developedand used to determine thermal and mechanicalproperties and their probabilistic distributions of a 5-harness 0/90 Sylramic fiber/CVI-SiC/MI-SiC wovenceramic matrix composite (CMC) at hightemperatures. The purpose of this study was todevelop quantitative probabilistic information on thisCMC to help complete the evaluation for its potentialapplication for HSCT combustor. This study includedquantifying the influences of uncertainties inherent inprimitive variables on selected response variables ofthe CMC at 2200 F. The quantitative information ispresented in the form of cumulative densityfunctions(CDFs), Probability Density Functions(PDFs) and sensitivity analyses. Results indicate thatthe scatters in all the response variables . The scattersin the response variables were reduced by 30-50%when the uncertainties in primitive variables, whichshowed the most influence, were reduced by 50%.
Introduction
The Enabling Propulsion Material (EPM) Projectteam of NASA Lewis Research Center (LeRC) isevaluating the Sylramic fiber/CVI-SiC/MI-SiCwoven composite (CMC) for its potentialapplications for Propulsion System components inthe High Speed Civil Transport (HSCT).
The 0/90 woven composite is constructed with twosets of mutually orthogonal sets of fiber towsinterlaced with each other, to form a layer. The layeris then coated with BN by means of chemical vaporinfiltration (CVI). The CVI process is also used todeposit SiC into the fiber tow (CVI-SiC). This CVI-SiC matrix fills up the fiber tow area and generallywould have a thin coating around the fiber tow.Melt-infiltrated process is then used to deposit theSiC between the fiber tows(MI-SiC). A typicalsection of a 5-harness composite is shown in Figure1.
Propulsion System components in the HSCT arerequired to have an assured life of several thousand
hours. The process of reliability estimations for thesecomponents is quite complex and requires aknowledge of uncertainties that occur in variousscales. Unlike conventional materials, the propertiesof the CMC display considerable scatter because ofthe uncertainties involved at two levels. They are (1)the constituent level (fiber, matrix, and interphase)properties and (2) the fabrication level. All theseuncertainties cause scatters at the laminate level. Thislevel includes fiber volume fraction, interphasethickness and properties, matrix void volume fractionand geometrical parameters of the laminate - such asfiber tow spacing and the fiber count per tow.
It is necessary to quantify the scatters in the responsevariables taking into account the inherentuncertainties in the variables at levels (1) & (2), alsocalled, primitive variables in order to be able toassure the required reliability of structuralcomponents.
Fabrication related variables consist of boron nitride(BN) coating thickness, fiber tow spacing, fiber andvoid volume fractions. Material variables consist ofmoduli, thermal conductivities, and thermalexpansion coefficients of Sylramic fiber, CVI-SiCand Ml-SiC matrices, and BN coating. Responsevariables are those that characterize the compositebehavior such as the in-plane modulus, through-thickness thermal expansion coefficient and thermalconductivity.
In the current practice of deterministic approaches,uncertainties are usually accounted for by the safetyfactors. This could often result in a design withunknown risk of failure. Consequently, it impedesthe application potential of CMCs for aerospacecomponents.The primary objective of this work is to determinethe properties of the CMC and their probabilisticdistributions, accounting for the uncertainties in theconstituent properties and fabrication relatedvariables. Such information could then be used todesign the structural component to meet thenecessary requirements of life.
1. to estimate the composite properties and theirprobabilistic distributions of a 5-harness 0/90Sylramic fiber/CVI-SiC/MI-SiC woven ceramicmatrix composite (CMC), accounting for theuncertainties in its constituent properties andfabrication related variables.
2. to quantify the influence of uncertainties in thematerial and fabrication relatedvariables on the overall thermal and mechanicalproperties of the CMC and identify themost influential ones
3. to develop a composition of the composite whichwill meet practically all the required specifications bysetting tighter tolerances on the most influentialvariables
Variable of the Study
Primitive: It includes constituent propertyvariables and fabrication related variables. Theconstituent property variables include fiber Young'smodulus, thermal conductivity, and coefficient ofthermal expansion of fiber, Young's modulus thermalconductivity and coefficient of thermal expansion ofmatrix, Young's modulus, thermal conductivity andcoefficient of thermal expansion of coating. Thefabrication related variables are coating thickness,fiber tow spacing, fiber volume fraction, and voidvolume fraction.
Response: These are in-plane modulus, throughthickness thermal expansion coefficient and thermalconductivity of laminate
Analysis Approach
The approach taken in this effort was to combineceramic matrix composite analysis embedded in thecomputer code CEMCAN (Ceramic MatrixComposite Analyzer) [References 1-2] and fastprobability integration techniques (FPI) available hiNESSUS (Numerical Evaluation of StochasticStructures Under Stress) [Reference 3]. A schematicof the integrated approach is shown in Fig. 2.
The role of CEMCAN was to provide the functional
relationships (micromechanics and macromechanics)that tie the constituent property values to anequivalent composite behavior.
The role of FPI was to perform probabilistic analysisutilizing the properties generated by CEMCAN. Inaddition, FPI was used to perform sensitivity analysisto rank primitive variables hi order of their influenceon a specific response variable.
The primitive variables are assumed to beindependent and have normal distributions . Theprimitive variables are perturbed with hi the theirassumed distributions. Response variables ,_ areestimated for each set of values of primitivevariables. Based on all the values thus generated ,distributions of the response variables are generated.
FPI is also used to generate probability densityfunctions (PDF) and cumulative density functions(CDF) for ply/laminate properties of the CMC. ThePDF gives a relationship between a range of valuesof a property (response variable) and the probabilityof its occurrence. The CDF gives a relationshipbetween a value up to certain magnitude of aproperty and the probability of its occurrence. Inaddition, the response variable probabilitysensitivities to inherent scatter in primitive variablesare obtained from FPI.
Simulation Technique and Stepwise Procedure
There are a number of approaches available forobtaining probabilistic response for a given set ofindependent primitive variables. The Monte-Carlosimulation technique is one such fairly commontechnique to obtain a response and its densityfunctions.
In the Monte Carlo technique, random values of theinput variables are selected from their distributions tocompute the value of the response variabledeterministically. This is repeated usually severalhundreds times in order to be able to build theresponse probabilistic density distributions.Although this technique is a simple, very accurateand easy approach, it requires enormous amount ofresources to generate reasonably accurate CDFs ofresponse variables. This was the reason that NASALeRC took initiative to develop Fast ProbabilityIntegrator (FPI) which approximates Monte CarloTechnique fairly accurately and requires orders of
magnitude less number of computational runs. Thiscode was developed to solve a large class ofengineering problems.
A 4 step procedure to characterize the behavior of theselected CMC using integrated approach is asfollows:
1. Select the response variable(s), andcorresponding primitive variables and theirprobabilistic distributions. For example,for composite longitudinal modulus(response variable), the primitive variableswould be the fiber modulus, matrixmodulus, fiber volume ratio, etc.
2. Run CEMCAN for the selected set of values(means & standard deviations) of primitivevariables to get material properties using themicro-macro-mechanics.
3. The whole process is repeated severalhundred times to generate a table of materialproperties ( response variables) values thatcorrespond to perturbed values of theprimitive variables.
4. The FPI analysis is then run making use ofthe previously generated table tocompute the CDF and the correspondingsensitivities of the response. The primitivevariables are then ranked by FPI in order oftheir influences
Sensitivity values could be + or — in nature. Apositive value of sensitivity indicates that themagnitude or scatter of a given response variable willincrease with an increase in the magnitude or scatterof that particular primitive variable. Increase in themagnitude or scatter of the response variable is notdeterministic. Variable with the highest absolutesensitivity value is defined to be the most influentialvariable. Variable with next lower absolutesensitivity value is second most influential variableand so on.
The sensitivity information thus obtained from FPI isvery useful from the design point of view. Forexample, reliability in design can be improved whenuncertainties in the most influential variables are
controlled. Those primitive variables which do nothave significant influences deterministically couldnevertheless have strong influences on the response
scatters if these primitive variables have largescatters. Weak physical variables with largeuncertainties may have probabilistic sensitivityfactors more important than strong physical variableswith small standard deviations. Variables with noscatter (deterministic) will obviously result hi zerovalues for the sensitivity implying the responsescatter is unaffected by such variables.
Results and Discussions
Mean values of the primitive variables are shown hiTable I. As mentioned before, the scatters in thecomposite properties occur at various levels both dueto inherent uncertainties hi the material properties aswell as hi fabrication processes. They may occur atthe constituent (fiber, matrix or interphase) level, atthe ply level (fiber volume ratio, void volume ratio,thickness of interfacial region etc.) and/or at thelaminate level (ply angle, lay-up etc.).
The scatter in this study refers to - and + threestandard deviations from the mean value. This rangewas particularly selected to generate informationwhich is directly useful for the designers.
However, hi this study, uncertainties in theconstituent material properties and fabrication relatedvariables are considered to quantify the scatters hisome of the laminate level properties. Specifically,fiber modulus, matrix modulus, coating modulus,thickness of the BN coating, coefficients of thethermal expansion of the fiber, matrix and theinterphase, thermal conductivities of the fiber, areconsidered random, while other parameters havebeen assumed to be deterministic.
In-plane and through-the-thickness Young's modulusare chosen as response variables to characterize theCMC's mechanical behavior, and coefficients ofthermal expansion and thermal conductivity tocharacterize thermal behavior.
The probabilistic and cumulative density functions ofin-plane Young's modulus, along with its sensitivityto the various primitive variables are shown hiFigures 4-6. The computed mean value of in-planemodulus for this particular composite is 33.33 Msi.,with a scatter range of 29.54 to 37.13 Msi. Asexpected, the three most influential variables for in-plane Young's modulus are fiber and CVI-SiCmoduli, and fiber tow spacing (Figure 5). In
controlling the scatter in the in-plane modulus, thebiggest pay-off will result from controlling the scatterin the fiber and CVI-SiC moduli, and fiber towspacing. Other primitive variables such as modulusof the BN coating and void volume fraction haveminimal influence.
These results imply that tightening the tolerances ofCVI-SiC moduli and fiber tow spacing will resultin lower scatter in the CMC in-plane modulus.Tightening tolerances in the remaining primitivevariables will result in a zero to marginal reduction inthe scatter. To estimate the reduction in the scatter,the PDF of the modulus was regenerated with theuncertainties of the three most influential variables,i.e. fiber/CVI-SiC/MI-SiC moduli, reduced by 50%.It shows that the scatter range of the in-planemodulus is reduced by 29%. The results are shown inFigure 7.
The CDFs, PDFs and sensitivity analysis are shownin Figures 8-10 respectively for through-the-thickness Young's modulus. The sensitivity analysisresults show, with the exception of fiber and BNmoduli, and void volume fraction, that all theprimitive variables considered appear to significantlyinfluence the scatter in thru-thickness modulus. Byreducing the standard deviations of the three mostinfluential variables (fiber volume fraction, BNthickness, and fiber tow spacing) by 50%, the scatterrange of the thru-thickness ply modulus is reduced by35%, as shown in Figure 11.
The CDFs, PDFs and sensitivity analysis for in-planeand through-the-thickness thermal conductivities areshown in Figures 12 -17 respectively. The predictedscatter for the in-plane thermal conductivity is 8.82 —12.29 Btu/hr-ft-F and for through-the-thickness is7.16 - 10.49 Btu/hr-ft-F. Fiber thermal conductivityis the most influential variable for in-plane compositethermal conductivity while the BN coating thermalconductivity is for thru-thickness thermalconductivity. Most of the remaining primitivevariables appear to significantly influence the scatterin the through-the-thickness thermal conductivityresponse. Fiber tow thermal conductivity (i.e. fiberand CVI-SiC conductivities) has the most influenceon the scatter of the in-plane thermal conductivity.Void volume fraction has only minimal influences onboth the in-plane and thru-thickness thermalconductivities. By reducing the uncertainties in thethree most influential variables by 50%, the scatterrange of the in-plane and thru-thickness ply thermal
conductivities are reduced by 38% and 29%,respectively. They are shown in Figures 18 and 19.
The CDFs, PDFs and sensitivity analysis for in-planeand through-the-thickness coefficients of thermalexpansion are shown in Figures 20 through 25. Thepredicted mean values of in-plane and through-the-thickness coefficients of thermal expansion areessentially the same -3.37 and 3.29 ppm/F,respectively. The scatter ranges for both in-plane andthru-thickness expansion coefficients are also quitesimilar - 2.78 to 3.97 and 2.66 to 3.92 ppm/F,respectively. For the in-plane thermal expansioncoefficient, the most influential variables are thethermal expansion coefficients of the fiber and CVI-SiC. The coefficient of thermal expansion of the MI-SiC matrix has a moderate effect. The remainingvariables essentially have insignificant influences.This behavior is expected as the in-plane thermalexpansion of composite is essentially controlled bythe fiber tows.
For the through-the-thickness thermal expansioncoefficient, the sensitivity plot (Figure 25) showsthat the dominant variables are the thermal expansioncoefficients of CVI-SiC and Ml-SiC matrices. Theseresults are to some extent intuitively obvious. Thebehavior in the thru-thickness direction is essentiallymatrix dominated and therefore the matrix propertieshave greater effect on the response scatter. Byreducing the standard deviations of the three mostinfluential variables by 50%, the scatter range of thein-plane and thru-thickness ply thermal expansioncoefficients are reduced by 50% and 49%,respectively. They are shown in Figures 26 and 27.
Summary
An integrated probabilistic approach was developedand used to determine thermal and mechanicalproperties and their probabilistic distributions of aselected CMC.
Influences of uncertainties inherent in primitivevariables on selected response variables of theselected CMC at 2200 F were quantified. Primitivevariables included material and fabrication relatedvariables; material variables are moduli, thermalconductivities and thermal expansion coefficients ofsylramic fiber, CVI-SiC and Ml-SiC matrices andBN coating, and Fabrication related variables are BNcoating thickness, fiber tow spacing, fiber and voidvolume fractions.
Cumulative density and probability density functionshave been developed for the response variableswhich include Young's modulus, coefficient ofthermal expansion and thermal conductivites. Also,sensitivity analyses were performed to identify theprimitive variables which have the most influence onthe response variables. This information is directlyuseful for the designers.
The results indicate that the scatter in the responsevariables were reduced by 30-50% when theuncertainties in the most influential primitivevariables were reduced by 50%.
Structural Reliability and Reliability SensitivityAnalysis. AIAA Journal, Vol. 32, No. 8, Aug. 1994,pp.1717-1723. Also presented at the 34th SDM
Conference, 1993.
4. Murthy, P.L.N.; Chamis, C.C.; and Mital, S.K.:Computational Simulation of ContinuousFiber-Reinforced Ceramic Matrix CompositesBehavior. NASATP-3602,1996.
Future Work
quantify scatters of life predictions of agiven a life prediction model incorporatinguncertainties in the primitive variables
quantify the influence of the remainingvariables on the response variables
incorporate input from manufacturers oncontrolling the primitive variables andoptimize the scatter in the response variables
Conclusions
A simple probabilistic approach has beenestablished using the existing codes toconduct such pilot or detailed studies
This approach has been used to demonstratehow to quantify and control scatters in theresponse variables
References
1. Mital, S.K.; Murthy, P.L.N.; CEMCAN - CeramicMatrix Composites Analyzer User s Guide - Version2.0, NASA TM 107187, April 1996.
2. Mital, S.K.; Murthy, P.L.N. and Chamis, C.C.:Micromechanics for Ceramic Matrix Composites viaFiber Substructuring, Journal of CompositeMaterials, Vol. 29, No. 5, 1995, pp. 614-633
