Determination of Material Properties for ANSYS Progressive Damage Analysis of Laminated ...

28 Pages·2017·0.53 MB·English

Determination of Material Properties for ANSYS Progressive Damage
Analysis of Laminated Composites
EverJ.Barberoa,MehdiShahbazib,c
aCorrespondingauthor.WestVirginiaUniversity,Morgantown,WV,USA
bGraduateresearchassistant.WestVirginiaUniversity,Morgantown,WV,USA
cFinalpublicationavailableathttps:/doi.org/10.1016/j.compstruct.2017.05.074
Abstract
A method is presented to calculate the material parameters required by the progressive damage analysis
(PDA)material-modelinANSYS(cid:13)R. TheproposedmethodisbasedonfittingresultscalculatedwithPDA
to experimental data by using Design of Experiments and Direct Optimization. The method uses experi-
mentalmodulus-reductionvs. straindataforonlytwolaminatestofitalltheparametersrequiredbyPDA
transverse/sheardamagemode. Fittedparametersarethenusedtopredictandcomparewiththeexperimen-
talresponseofotherlaminates.MeshsensitivityofPDAisstudiedbyperformingp-andh-meshrefinement.
PDAcanthenbeusedtopredicttheresponsewithdamageevolutionofanylayup.
Keywords: ANSYS;Matrixdamage;Parameteridentification;Damagemodel;Damageparameters;
Insitu;Fracture.
1. Introduction
Damage initiation and propagation in composite materials are of particular importance for the design,
production,certification,andmonitoringofanincreasinglylargevarietyofstructures. Whilemostmodels
areimplementedonshell-typeformulations,someareimplementedonsolid-typefiniteelementformulations
[1].
One approach is to predict the response of each lamina using a meso-model that couples the behavior
ofeachlaminawithadjacentinterlaminarlayers[2]. Anintralaminardamagemodeltakesintoaccountthe
effects of transverse matrix crack and the interlaminar layer takes into account delaminations. Parameter
identificationandvalidationwithopen-holetensiletestsonquasi-isotropiclaminatesisdescribedin[3].
Another approach is to use serial-parallel mixing theory [4, 5, 6], or multi-continuum theory (MCT)
[7],tosimulatetheonsetandpropagationoftransversematrixcrackingandotherfailuremechanisms. The
composite response is obtained from the constitutive behavior of its constituents, each one of them simu-
lated with its own constitutive law. In this way, it is possible to use any given non-linear material model,
suchasdamageorplasticity,tocharacterizetheconstituents. Theaccuracyofdamageonsetandevolution
predictionsdependsontheaccuracyoftheconstituentmodelsused.
Most of the models developed specifically to address the phenomenon of transverse matrix cracking
are meso-models, meaning that each lamina is considered to be a homogeneous, orthotropic material [8].
Somemodels,suchasProgressiveDamageAnalysis(PDA)usestrengthpropertiestopredictdamageonset,
and fracture mechanics properties to predict damage evolution [9, 10]. Others, such as Discrete Damage
Mechanics(DDM)[11]useonlyfracturemechanicspropertiestopredictbothdamageinitiationanddamage
evolution.
PreprintsubmittedtoCompositeStructures176:768–779(2017).DOI:10.1016/j.compstruct.2017.05.074 March27,2018
CompositeStructures176:768–779(2017) DOI:10.1016/j.compstruct.2017.05.074 2
Regardless of how initiation is predicted, evolution is commonly predicted by establishing a relation
betweentheavailablestrainenergyinthematerialandthedensityofmatrixcracks[12,13,14,15,16]. For
example, [17] uses Finite Fracture Mechanics (FFM) to obtain the energy release rate required to double
thecrackdensity,bypropagationofanewcrackbetweentwoexistingcracks,orbycontinuousvariationof
crackdensity[18]. Similarly,theequivalentconstrainedmodel(ECM)[19,20],definesalawthatprovides
theevolutionofstiffnessasafunctionofmatrixcrackdensity.
Regardingthesolutionofthestressandstrainfieldinthevicinityofacrack,whichisnecessarytocalcu-
latethestrainenergyreleaserate(ERR),mostmodelsapproximatetheproblemasbeingperiodic. However,
numerical models such as PDA are not constrained by this assumption. While most models calculate the
ERR using the equations of elasticity, with various degrees of approximation, others use Crack Opening
Displacementapproximations[14,16,21].
Someoftheabovementionedformulationsprovideanalyticalexpressionsthatcanbeusedtoobtainthe
mechanicalresponseforsimplegeometryandloadconfigurations. However,itisoftennecessarytoinclude
the constitutive model into Finite Element Analysis (FEA) software. Some models have been included in
commercialFEAsoftware[9,10,22]. OthersareavailableaspluginsforexistingFEAsoftware[7,23],or
asuserprogrammablefeatures,includingUMAT,UGENS,UserMAT[24],andVUMAT[1].
ANSYSMechanicalprovidesprogressivedamageanalysis(PDA)startingwithrelease15. Furthermore,
ANSYS Workbench allows optimization of any set of variables to any user defined objective defined in a
Mechanical APDL (MAPDL) model by importing the APDL script into Workbench and using Design of
Experiments (DoE) and Direct Optimization (DO). Since PDA is not implemented in the graphical user
interface(GUI)ofANSYSWorkbench,theusermustuseAPDLcommandstodefinethedamageinitiation
criterion,damageevolutionlaw,andmaterialparameters,asdescribedinthismanuscript.
Althoughelasticmoduliareavailableformanycompositematerialsystems,thesameisnottrueforthe
additionalmaterialpropertiesrequiredbyPDA.However,laminatemodulusandPoisson’sratiodegradation
oflaminatedcompositesasafunctionofappliedstrainareavailableforseveralmaterialsystems,forexample
[21,25,26]. ThisstudyshowshowtouseavailabledatatoinferthematerialparametersrequiredbyPDA.
Specifically, the main purpose of this study is to find in situ values [27] of transverse tensile strength F ,
2t
in-planeshearstrength F ,andenergydissipationperunitareaG (transversetensile/shearmode)forthe
6 c
materialsystem(compositelamina)whichcanbeusedinPDAtopredictdamageinitiationandevolutionof
laminatedcompositestructuresbuiltwiththesamematerialsystembutdifferentlaminatestackingsequence
(LSS),geometry,loads,andboundaryconditions. Experimentaldatafrom[21,25]isusedforillustrationin
thisstudy.
The proposed method requires experimental data in the form of laminate extensional modulus E re-
x
duction(orlaminateshearmodulusG reduction,orlaminatePoisson’sratioν reduction)vs. damagefor
xy xy
two laminates. One laminate should have a significant component of mode I (opening) crack propagation
andtheotherasignificantcomponentofmodeII(shear)crackpropagation,suchasforexample[0/90] and
S
[0/±θ] ,respectively. Extensionalmodulusreductiontestsaretheeasiesttoperformbuttheusercoulduse
S
laminateshearorPoisson’sratioreductionaswell.
ThestatedobjectiveisachievedbyminimizingtheerrorbetweenPDApredictionsandavailableexperi-
mentaldata.OncetheinputparametersF ,F ,G arefound,theaccuracyofPDApredictionsischeckedby
2t 6 c
comparingthosepredictionswithexperimentaldataforotherlaminatesthathasnotbeenusedtofittheinput
parameters. Infact, experimentaldataforonlytwolaminatesarerequiredtofittheparameters. Theinput
parameters are fitted using an specific mesh (one element) and type of element (SHELL 181). However,
severaltypesofelementsandvariablemeshrefinementarecustomarilyusedfortheanalysisofacomplex
structure. Therefore, sensitivity of the PDA predictions with respect to mesh refinement and element type
(quadraticvs. linear)isassessedinthisworkbyperformingbothp-andh-refinement.
CompositeStructures176:768–779(2017) DOI:10.1016/j.compstruct.2017.05.074 3
2. ProgressiveDamageAnalysis
Toperformprogressivedamageanalysisofcompositematerials,theuserneedstoprovidelinearelastic
orthotropicmaterialpropertiesandtwomaterialmodels: damageinitiationanddamageevolutionlaw.
2.1. DamageInitiationCriteria
WithdamageinitiationcriteriatheusercandefinehowPDAcalculatestheonset(initiation)ofmaterial
damage. TheavailableinitiationcriteriainANSYSareMaximumStrain,MaximumStress,Hashin,Puck,
LaRC03,andLaRC04. Besides,theusercandefineuptonineadditionalcriteriaasuserdefinedinitiation
criteria, but only the Hashin criterion works with PDA. The remaining only work with instant stiffness
reduction(MPDG).Thelaterissimilartoply-dropoffbuttheamountofstiffnessdropcanbespecifiedin
therange0–100%oftheundamagedstiffness.WithMPDG,theusercanchoosefailurecriteriaamongthose
mentionedforeachofthedamagemodes,whichareassumedtobeuncoupled.
Using the Hashin initiation criteria [28], PDA incorporates the following four modes of damage initi-
ation: fiber tension, fiber compression, matrix tension, and matrix compression, which are represented by
damageinitiationindexesI ,I ,I andI thatindicatewhetheradamageinitiationcriterioninadamage
ft fc mt mc
modehasbeensatisfiedornot. Damageinitiationoccurswhenanyoftheindexesexceeds1.0.
Damage initiation indexes are unfortunately called “failure” indexes in the literature, despite the fact
that,foraproperlydesignedlaminate,usuallynocatastrophic“failure”occursatthatpointbutratheronlya
smallamountofdamageappears[29,30,31]. Thefourdamageinitiationindexesaredescribednext.
2.1.1. Fibertension(σ ≥0)
11
Fibertensionisamisnomersometimesusedintheliterature,sincethismodeactuallyrepresentslongitu-
dinaltensionofthecompositelamina,notthefiber. Thecorrespondingdamageinitiationindexiscomputed
asfollows
(cid:18)σ (cid:19)2 (cid:18)σ (cid:19)2
It = 11 +α 12 (1)
f F F
1t 6
whereαdeterminesthecontributionofthein-planeshearstresstothiscriterion.
2.1.2. FiberCompression(σ <0)
11
Thedamageinitiationindexforlongitudinalcompressionofthecompositelaminaiscomputedasfol-
lows
(cid:18)σ (cid:19)2
Ic = 11 (2)
f F
1c
2.1.3. Matrixtension(σ ≥0)
22
Thisisalsomisnomer,sincethismodeactuallyrepresentstransversetensionandin-planeshearofthe
compositelamina,notthematrix. Theconfusionisduetothefactthatthisisamatrix-dominatedmode,but
still the criterion is applied at the meso-scale; that is at the lamina level, not at the micro-scale where the
fiberandmatrixwouldbeanalyzedseparately. Furthermore,theparametersinvolved(F ,F )arethoseof
2t 6
alamina,notoffiberandmatrixseparately,andtheresultingindexappliestothelamina,nottothematrix
(cid:18)σ (cid:19)2 (cid:18)σ (cid:19)2
It = 22 + 12 (3)
m F F
2t 6
CompositeStructures176:768–779(2017) DOI:10.1016/j.compstruct.2017.05.074 4
2.1.4. Matrixcompression(σ <0)
22
Thedamageinitiationindexfortransversecompressionofthecompositelaminaiscomputedasfollows
(cid:18)σ (cid:19)2 (cid:20)(cid:18)F (cid:19)2 (cid:21)σ (cid:18)σ (cid:19)2
Ic = 22 + 2c −1 22 + 12 (4)
m 2F 2F F F
4 4 2c 6
whereσ arethecomponentsofthestresstensor; F and F arethetensileandcompressivestrengthsof
ij 1t 1c
a lamina in the longitudinal (fiber) direction; F and F are the tensile and compressive strengths in the
2t 2c
transversedirection;F andF arethein-planeandintralaminarshearstrengths.
6 4
In this work, to obtain the model proposed by Hashin and Rotem [28], in (1) we set α = 0 and in (4)
F =1/2F . Alsonotethatinsitustrengthshouldbeusedforallmatrix-dominatedmodes[32].
4 2c
The APDL command for Hashin initiation criteria for all damage modes is TB, DMGI, as it is shown
below:
! Damage detection using failure criteria
TB, DMGI, 1, 1, 4, FCRT
TBTEMP,0
! 4 is the value for selecting Hashin criteria,
! which is here selected for all four failure modes
TBDATA,1,4,4,4,4
2.2. MaterialStrengthLimits
Toevaluatethedamageinitiationcriteria,theuserdefinesthemaximumstressesorstrainsthatamaterial
cantoleratebeforedamageoccurs. Requiredinputsdependonthechosencriteriainthedamageinitiation
part. Forinstance,forHashincriteriatheuserneedstodefineinsitutensileandcompressionstrengthin1,
2,and3laminaorientations(calledX,Y,andZdirectionsinANSYS),andtheshearstrengthin12,13,and
23laminaplanes(calledXY,XZ,andYZinANSYS).
Sincefiberdominatedstrengthvaluesareatleastoneorderofmagnitude(10X)largerthanmatrixdomi-
natedstrengths,matrixmodesoccurmuchearlierinthelifeofthestructurethanfibermodes. Forproperly
designedlaminates(e.g.,accordingtothe10%ruleandotherindustry-acceptedpractices[33])fibermodes
(§2.1.1, §2.1.2) do not occur until nearly the end of the life. Furthermore, transverse matrix compression
(§2.1.4) does not result in progressive damage but rather leads to sudden failure according to the Mohr-
Coulomb criterion [34]. Therefore, this study focuses on matrix tension and shear modes (§2.1.3), which
areknowntoleadtosubstantialprogressivedamage[25,21].
InitialvaluesofinsitutransversetensilestrengthF ,insituin-planeshearstrengthF ,andintralaminar
2t 6
shearstrengthF (calledXT,XY,andXZinANSYS)shouldbedefinedintheAPDLscript. Thecommand
4
formaterialstrengthlimitisTB,FCLI,asshownbelow:
! Material Strengths
TB,FCLI,1,1,6
TBTEMP,0
! Failure Stress, Fiber Tension
TBDATA,1, F
1t
! Failure Stress, Fiber Compression
TBDATA,2,F
1c
! Toughness Stress, Matrix Tension
TBDATA,3,F
2t
CompositeStructures176:768–779(2017) DOI:10.1016/j.compstruct.2017.05.074 5
! Failure Stress, Matrix Compression
TBDATA,4,F
2c
! Failure Stress, XY Shear
TBDATA,7,F
6
! Failure Stress, YZ Shear
TBDATA,8,F
4
As it was previously stated, the damage initiation part of PDA requires six material parameters listed
above. Twoofthose,F andF ,areinsituvalues. Insituvaluescanbecalculatedusingequationsthatin-
2t 6
volvethenominalstrengthsoftheunidirectionallaminaandthecorrespondingcriticalenergyreleaserates
(ERRs)G ,G inmodesIandII[34],orthetransitionthicknessofthematerial[33,§7.2.4],[35]. Howe-
Ic IIc
ver, thesecriticalERRsandtransitionthicknessesarenotusuallyavailableintheliterature. Experimental
methodstodetermineinsituvaluesarenotavailableeither.Byfocusingononedamagemode,namelytrans-
versetension/sheardamage(§2.1.3),itisshowninthisworkthatthematerialparametersrequiredbyPDA
(F ,F ,andG ) can beobtained by fitting PDA modelresults to suitable experimental data. Thus, critical
2t 6 c
ERRs, transition thicknesses, and in situ strength values, which are not available in the literature and are
verydifficulttodetermineexperimentally,canbesimplyobtainedwiththemethodologyproposedherein.
2.3. DamageEvolution
After satisfying the selected initiation criteria, further loading will degrade the material. The damage
evolutionlawdetermineshowthematerialdegrades. InANSYS,therearetwooptionsfordamageevolu-
tion: instant stiffness reduction (MPDG) and continuum damage mechanics (PDA). Since instant stiffness
reduction,whichissuddenlyappliedwhenthecriterionissatisfied,doesnotprovideanyinformationabout
damageevolution,thisstudyusesthePDAmethodfordamageevolution.
PDA damage evolution requires eight parameters: four values of energy dissipated per unit area (G ,
c
Figure1)andfourviscositydampingcoefficients(η)fortensionandcompressioninbothfiberandmatrix
dominatedmodes. Theenergydissipatedperunitareaisdefinedas:
Figure1:Equivalentstressσevs.equivalentdisplacementue.
(cid:90) uu
e
G = σ du (5)
e e
0
CompositeStructures176:768–779(2017) DOI:10.1016/j.compstruct.2017.05.074 6
where:
σ =istheequivalentstress.Forsimpleuniaxialstressstate,theequivalentstressistheactualstress.For
e
complexstressstate,theequivalentstressesandstrainsarecalculatedbasedonHashin’sdamageinitiation
criteria.
u =istheequivalentdisplacement. Forsimpleuniaxialstateofstressandstrain,u = (cid:15) L ,with(cid:15) is
e e e c e
theuniaxialstrainandL isthelengthoftheelementinthestressdirection. Forcomplexstatesofstrain,(cid:15)
c e
istheequivalentstrainandL isthecharacteristiclengthoftheelement.
c
uu =istheultimateequivalentdisplacement,wheretotalmaterialstiffnessislostforthespecificmode.
e
L isthecharacteristiclength. For2Delements(PLANEandSHELL),itisinternallydefinedinterms
c
oftheareaA oftheelement,asfollows[36]:
e
(cid:112)
Lc=1.12 A ;forsquareelements
e
(cid:112)
Lc=1.52 A ;fortriangularelements (6)
e
Viscous damping coefficients η are also specified respectively for each of the damage modes. For a
specificdamagemode,thedamageevolutionisinternallyregularizedasfollows:
η ∆t
dt(cid:48)+∆t = η+∆t dt(cid:48)+ η+∆t dt+∆t (7)
where:
d(cid:48) =Regularizeddamagevariableatcurrenttime.
t+∆t
dt+∆t isusedformaterialdegradation
d(cid:48) =Regularizeddamagevariableatprevioustime.
t
dt+∆t =Un-regularizedcurrentdamagevariable
Thedampingcoefficientsmayneedtobeadjustedifconvergenceproblemsareencountered,butallthe
casesinthisworkexecutedflawlesslybyjustusingthedefaultvaluesthatarebuiltintoANSYS.
ThecommandfordefiningdamageevolutioninAPDLisTB,DMGE,asshownbelow.
! Damage Evolution with CDM Method
TB,DMGE,1,1,8,CDM
TBTEMP,0
! Fracture Toughness, Fiber Tensile
TBDATA,1,Gft
c
! Viscosity Damping Coefficient, Fiber Tensile
TBDATA,2, ηft
! Fracture Toughness, Fiber Compressive
TBDATA,3,Gfc1E6
c
! Viscosity Damping Coefficient, Fiber Compressive
TBDATA,4, ηfc
! Fracture Toughness, Matrix Tensile
TBDATA,5,G
c
! Viscosity Damping Coefficient, Matrix Tensile
TBDATA,6, ηmt
! Fracture Toughness, Matrix Compressive
CompositeStructures176:768–779(2017) DOI:10.1016/j.compstruct.2017.05.074 7
TBDATA,7,Gmc
c
! Viscosity Damping Coefficient, Matrix Compressive
TBDATA,8, ηmc
3. Methodology
InthissectionweproposeamethodologyusingDesignofExperiments(DoE)andDesignOptimization
(DO) to adjust the values of F , F , andG , so that the PDA prediction closely approximates the experi-
2t 6 c
mentaldata.
FirstweuseDoEtoidentifythelaminatesthataremostsensitivetoeachparameter. Thefocusatthis
pointistoidentifytheminimumnumberofexperimentsthatareneededtoadjusttheparameters.Inthisway,
additional experiments conducted with different laminate stacking sequences (LSS) are not used to adjust
parametersbuttoassesthequalityofthepredictions.
In principle, the DoE technique is used to find the location of sampling points in a way that the space
of random input parameters X = {F ,F ,G } is explored in the most efficient way and that the output
2t 6 c
functionDcanbeobtainedwiththeminimumnumberofsamplingpoints. Inthisstudytheoutputfunction
(also called objective function) is the error between the predictions and the experimental data. Given N
experimentalvaluesoflaminatemodulusE((cid:15)),theerrorisdefinedas
i
1 (cid:118)(cid:117)(cid:116)(cid:88)N (cid:32)E(cid:12)(cid:12)ANSYS−PDA E(cid:12)(cid:12)Experimental(cid:33)2
D= (cid:12)(cid:12) − (cid:12)(cid:12) (8)
N i=1 E(cid:101)(cid:12)(cid:15)=(cid:15)i E(cid:101)(cid:12)(cid:15)=(cid:15)i
where (cid:15) is the strain applied to the laminate, and i = 1...N. Furthermore, E and E(cid:101)are laminate elastic
modulus for damaged and undamaged laminate, respectively, and N is the number of experimental data
pointsavailable.
DoEtoolscanbefoundinANSYSWorkbenchintheDesignExplorer(DE)module,whichincludesalso
DirectOptimization(DO),ParametersCorrelation,ResponseSurface(RS),ResponseSurfaceOptimization
(RSO),andSixSigmaAnalysis.
Let’s denote the input parameters by the array X = {F ,F ,G }. In this case, the output function
2t 6 c
D = D(X) can be calculated by evaluation of eq. (8) through execution of the finite element analysis
(FEA)codeforN valuesofstrain. EachFEAanalysisiscontrolledbytheAPDLscript,whichcallsforthe
evaluationofthenon-linearresponseofthedamaginglaminateforeachvalueofstrain,withparametersX.
Ifthemeshisrefined,theseevaluationscouldbecomputationallyintensive.
Analternativetodirectevaluationoftheoutputfunctionistoapproximateitwithamultivariatequadratic
polynomial. Theapproximationiscalledresponsesurface(RS).Itcanbeconstructedwithonlyfewactual
evaluationsoftheoutputbychoosingasmallnumberofsamplingpointsfortheinput. Thesamplingpoints
arechosenusingDesignofExperiments(DoE)theory. Thenumberofevaluationsneededtoconstructthe
responsesurface(RS)andfordirectoptimization(DO)areshowninTable1.
TheshapeoftheresponsesurfacecanbeinferredbyobservingthevariationoftheoutputDasafunction
ofoneinputatatime. ThisisshowninFigures2,3,and4,forlaminate#1(Table2). Theabscissaspans
eachoftheinputsandtheordinatemeasurestheerror(8)betweenpredictedandexperimentaldata.
ByobservingtheRSwecanunderstandwhichparameters(inputs)havemosteffectontheerror(output).
FromFigure4itisclearthattheerrorisnotsensitiveto F forlaminate#1. Thisisbecausenolaminain
6
laminate#1issubjecttoshear.Therefore,itisdecidedtousetheexperimentaldataoflaminate#1toevaluate
CompositeStructures176:768–779(2017) DOI:10.1016/j.compstruct.2017.05.074 8
Table1: NumberofFEAevaluationsused(a)toconstructtheresponsesurface(RS)and(b)toadjusttheinputparametersbydirect
optimization(DO).
#ofevaluations
#ofinputs Inputs RS DO(screening)
1 F 5 100
6
2 F ,G 9 100
6 c
3 F ,F ,G 15 100
2t 6 c
Figure2:Responsesurfacechart.Error(D)vs.F2t.
Figure3:Responsesurfacechart.Error(D)vs.Gc.
CompositeStructures176:768–779(2017) DOI:10.1016/j.compstruct.2017.05.074 9
Figure4:Responsesurfacechart.Error(D)vs.F6.
Table2:Laminatestackingsequenceforalllaminatesforwhichexperimentaldataisavailable[21,25].
Laminate# LSS
1 [0 /90 ]
2 4 S
2 [±15/90 ]
4 S
3 [±30/90 ]
4 S
4 [±40/90 ]
4 S
5 [0/90 /0 ]
8 1/2 S
6 [0/±70 /0 ]
4 1/2 S
7 [0/±55 /0 ]
4 1/2 S
8 [0/±40 /0 ]
4 1/2 S
9 [0/±25 /0 ]
4 1/2 S
CompositeStructures176:768–779(2017) DOI:10.1016/j.compstruct.2017.05.074 10
Table3:SensitivityS oftheoutput(error)toeachinput(parameter).
Inputrange ErrorD Sensitivity Laminate
Input min max min(D) max(D) ave(D) S #
F 72 88 0.0022 0.0041 0.00297 0.64 #1
2t
G 22.5 27.5 0.0031 0.0020 0.0026 -0.42 #8
c
F 50 88 0.076 0.102 0.093 -0.28 #8
6
onlytwoinputparameters(F andG ). Thismakestheoptimizationalgorithmtorunfaster. Thechartsin
2t c
Figures2–4aredrawnfortheinputrangesgiveninTable3.
The RS is a multivariate quadratic polynomial that approximates the actual output function D(X) as a
functionoftheinputvariables. Inthisstudy,D = f(X). ThesensitivityS oftheoutputtoinputX inthe
RS i
userspecifiedinterval[X ,X ]iscalculatedas
min max
max(D)−min(D)
S = (9)
average(D)
andtabulatedinTable3. Notethatthesensitivitycanbecalculatedfromtheerrorevaluateddirectlyfrom
FEAanalysis(DO)orfromResponseSurface(RS).Thelaterismuchmoreexpedientthantheformer,asit
canbeseeninTable1,consideringthenumberofevaluationsneeded.
The input parameters can be adjusted with any mesh and any type of elements that represent the gage
section of the specimen, or a single element to represent a single material point of the specimen. For
expediency,asinglelinearelement(SHELL181)isusedinthisstudy.
3.1. APDL
ANSYSParametricDesignLanguage(APDL)[24]mustbeusedinordertopredictdamagewithPDA
[36]. In this study, we need three input parameters: F ,F ,G and one output parameter: D. In APDL,
2t 6 c
parametersaredefineddynamically,justbygivingthemanameandvalue,asfollows:
/TITLE, Laminate #1: [0-2/90-4]s
/UNITS,MPA ! Units are in mm, MPa, and Newtons
/PREP7
F2t = 80.0 ! MPa
F6 = 48.0 ! MPa
Gc = 25.0 ! kJ/m^2
Next, APDL commands are used to specify the element type, the laminate stacking sequence (LSS)
with SECDATA command (Table 2), the elastic properties with MP command (Table 4), and the material
strengths with TB command (Table 4). The material strengths values given above for F ,F , are simply
2t 6
initial(guess)valuesfortheoptimization.
ET,1,SHELL181 ! Element type
! LAMINATE STACKING SEQUENCE
SECTYPE,1,SHELL,,#1 ! SECTYPE, SEID, TYPE, SUBTYPE, NAME
SECDATA,0.288,1,0 ! thickness, material, angle
SECDATA,1.152,1,90
SECDATA,0.288,1,0
! ORTHOTROPIC MATERIAL PROPERTIES