Fatigue Prediction for Composite Materials and Structures

Transcription

1 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures Oar ALOMON*, Fernando RATELLINI, ergio OLLER and Eugenio OÑATE CIMNE (International Center or Nuerical Methods in Engineering) Building C-1, Capus Nord UC -C/ Gran Capitán s/n Barcelona AIN ABTRACT * saloon@cine.upc.edu The objective o this paper is to present a new coputational ethodology or predicting the durability o structures ade o coposite aterials based on epoxy atrix with long carbon ibres. To analyse the behaviour o coposite aterials an Enhanced erial-arallel constitutive odel (E odel) is developed assuing coponents behave as parallel aterials in the ibres alignent direction and as serial aterials in orthogonal directions. It allows equal coponent strains in the ibre direction and equal stresses in the transverse directions. The E odel brings answers on the non-linear behaviour o coposites, where Classical Micro-echanics Forulas are restricted to their linear elastic part. Constitutive tensors o the coposite aterials are obtained ro echanical properties and internal variables o their coponents within a continuu raewor. Anisotropy and dierent constitutive odels (plasticity, daage and atigue) or each phase are considered. The atigue prediction odel is based in a continuu echanic stress lie approach or each o the coponent aterials. -N curves are proposed or each phase. A cuulative atigue daage index is used to update the echanical properties o the coponents and with these properties update layers properties and copute residual stiness o the lainate. Constitutive odels or the coposite aterials have been ipleented into a general purpose inite eleent code (COMET) and a user riendly interace or coposite aterial data input in a pre-post processing odule (GiD) has been developed. The ethodology is validated using experiental data ro tests on CFRR coposite aterial saples. 1.0 INTRODUCTION Fatigue is deined as "the process o peranent, progressive and localized structural change which occurs to a aterial point subjected to strains and stresses o variable aplitudes which produces cracs leading to total ailure ater a certain nuber o cycles" and it is the ain cause o ailure o achine parts in service, in echaniss and structural eleents in aeronautics, naval and autootive industries. Fatigue ailure can occurs under load conditions well below the strength liit o the aterial. Typically, a progressive loss o strength occurs depending on the nuber o stress/strain cycles, reversion index, load aplitude, etc. This loss o strength induce the aterial to inelastic behaviour, which ay be interpreted as icro-cracing ollowed by crac coalescence leading to the inal collapse o structural parts. aloon, O.; Rastellini, F.; Oller,.; Oñate, E. (2005) Fatigue rediction or Coposite Materials and tructures. In Evaluation, Control and revention o High Cycle Fatigue in Gas Turbine Engines or Land, ea and Air Vehicles (pp ). Meeting roceedings RTO-M-AVT-121, aper 31. Neuilly-sur-eine, France: RTO. Available ro: RTO-M-AVT UNCLAIFIED/UNLIMITED

2 Report Docuentation age For Approved OMB No ublic reporting burden or the collection o inoration is estiated to average 1 hour per response, including the tie or reviewing instructions, searching existing data sources, gathering and aintaining the data needed, and copleting and reviewing the collection o inoration. end coents regarding this burden estiate or any other aspect o this collection o inoration, including suggestions or reducing this burden, to Washington Headquarters ervices, Directorate or Inoration Operations and Reports, 1215 Jeerson Davis Highway, uite 1204, Arlington VA Respondents should be aware that notwithstanding any other provision o law, no person shall be subject to a penalty or ailing to coply with a collection o inoration i it does not display a currently valid OMB control nuber. 1. REORT DATE 01 OCT REORT TYE N/A 3. DATE COVERED - 4. TITLE AND UBTITLE Fatigue rediction or Coposite Materials and tructures 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. ROGRAM ELEMENT NUMBER 6. AUTHOR() 5d. ROJECT NUMBER 5e. TAK NUMBER 5. WORK UNIT NUMBER 7. ERFORMING ORGANIZATION NAME() AND ADDRE(E) CIMNE (International Center or Nuerical Methods in Engineering) Building C-1, Capus Nord UC -C/ Gran Capitán s/n Barcelona AIN 8. ERFORMING ORGANIZATION REORT NUMBER 9. ONORING/MONITORING AGENCY NAME() AND ADDRE(E) 10. ONOR/MONITOR ACRONYM() 12. DITRIBUTION/AVAILABILITY TATEMENT Approved or public release, distribution unliited 11. ONOR/MONITOR REORT NUMBER() 13. ULEMENTARY NOTE ee also ADM202115, RTO-M-AVT-121. Evaluation, Control and revention o High Cycle Fatigue in Gas Turbine Engines or Land, ea and Air Vehicles., The original docuent contains color iages. 14. ABTRACT 15. UBJECT TERM 16. ECURITY CLAIFICATION OF: 17. LIMITATION OF ABTRACT UU a. REORT unclassiied b. ABTRACT unclassiied c. THI AGE unclassiied 18. NUMBER OF AGE 22 19a. NAME OF REONIBLE ERON tandard For 298 (Rev. 8-98) rescribed by ANI td Z39-18

3 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures This study is based on the hypothesis that atigue daage is essentially o the sae nature as echanical daage and can be described via an internal variable allowing the adequate treatent o the accuulation and localization o dislocations, thereore the theoretical structure o continuu echanics (such as plasticity and daage) is suitable or the study o non linear atigue probles. The inelastic theories o plasticity and/or daage solve the proble o aterial behaviour beyond the elastic range and both theories allow the study o the change in strength that a aterial point suers by inelastic eects, however they are not sensitive to cyclic load eects. In this wor the standard inelastic theories are odiied to account or atigue eect coupled with non-atigue behaviour [1]. Concerning coposite aterials, in this wor, a nuerical odel to assess the constitutive non-linear behaviour o a unidirectional laina is conceived [2]. It is based on the appropriate anageent o the constitutive odels o coponent phases within a continuu raewor. With such ai, a strategy is developed to decouple phases that ay also consider possible inelastic strains and/or elastic degradation o the stiness based on previous developent o the authors [3], [4]. Input data are the echanical properties o each coponent (i.e.: ibres and atrix in case o ibre-reinorced lainates) and its distribution inside the coposite. This ethodology o ixing coponents with the strategy or phases decoupling allow each coponent to retain, unchanged, its original constitutive law (isotropic or anisotropic, linear or non-linear) while conditioning the coposite global response. Dierent constitutive odels (plasticity, daage, atigue, creep, etc.) or each phase ay be taen into account. The coposite odel (or a single laina) is cobined with classical laination theory to describe lainates consisting o unidirectional continuously reinorced layers. Its relative siplicity and the resulting nuerical eiciency ae this approach well suited or ipleentation as a aterial odel in Finite Eleent progras or studying the elastoplastic response o structures or coponents ade o continuously reinorced or lainated coposites. The previous entioned atigue odel is applied to each coponent in order to obtain the durability o the whole coposite lainate. 2.0 CONTINUUM MECHANICAL MODEL FOR FATIGUE ANALYI It is assued that each point o the solid ollows a daage-elasto-plastic constitutive law with the stress () evolution depending on the ree elastic strain variable (E e ) and a set o internal plastic and daage variables q={α p,d}={e p, κ p,d=κ d }, were E p and κ ini (κ=κ p +κ d ) 1 represent the plastic part o the strain and a unit noralized dissipation coposed by the plastic plus daage parts, respectively. The ree energy or isotheral, isentropic and adiabatic processes and or sall elastic strains and large plastic strains is written in the reerence coniguration, accepting the additivity o its elastic Ψ e p and plastic Ψ parts, as e o e p p [ E C E ] + Ψ ( α ) e e p p 1 Ψ =Ψ ( E, d ) + Ψ ( α i ) = (1 d ) o l l i (1) 2 where E e is elastic Green strain tensor, 0 d is the aterial density, d = κ is the internal echanical daage ini d ini ini variable or the daage processes d ( d = κ ) 1 with the initial value d κ provided by the deect 0 level, and C l the initial constitutive tensor. The stress tensor in the reerence coniguration can be expressed as 31-2 RTO-M-AVT-121 UNCLAIFIED/UNLIMITED

4 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures = o Ψ E e = ( 1 d) C o l E e l (2) For the plastic behaviour, the general ors o the yield F and potential G plastic unctions tae into account the inluence o the current stress state, the internal plastic variables, and other variables such as the nuber o cycles N: F( G(, κ ) = ) = g( ( ) K( ) = constant., κ, N) (3) where ) and g ) are the uniaxial equivalent stress unctions, K(, κ, N) is the strength threshold ( ( (see Figure 1). All the internal variables at current tie t are obtained by eans o an integration process α p i t = α p p p & dt, starting ro its evolution law (, p α = λ& H α ), where λ is the plastic consistent actor. 0 i i i l K(,κ,N) κ 0 κ = κ p +κ d 1 Figure 1: Uniaxial strength threshold or a sybolic aterial. The daage unction is deined as _ ( G D ( _, κ ) = ( ) K(, κ, N) (4) where ) is the uniaxial equivalent stress unction in the undaaged space, K(, κ, N) is the sae d t strength threshold as in (3) and κ = d = d& dt the daage internal variable with an evolution deined as 0 D d = µ & H ( l, d), where µ is the consistency daage actor. In both, (3) and (4), the noralized dissipation p d p d ax p d ax is deined as κ = κ + κ = ( Ξ + Ξ )/ Ξ, where Ξ, Ξ, Ξ are the Clausius-Duhe dissipation or the current plastic, daage process and its axiu capacity o the solid dissipation at each point, respectively. The eect o the nuber o cycles on the plastic and/or daage consistency conditions ( F & D = 0, G & = 0) is introduced as ollows, RTO-M-AVT UNCLAIFIED/UNLIMITED

5 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures ( ) K(, κ). red ( N, ed, R) = 0 (5) K(, R, N) ( ) K(, κ). red ( N, ed, R) = 0 (6) K(, R, N) where 0 red 1 represent the unit noralized reduction part o the strength threshold K plastic and/or daage strength evolution by load cyclic eect. -N Curves tress-nº o cycles (-N) curves are experientally obtained by subjecting identical sooth speciens to cyclic haronic stresses and establishing their lie span easured in nuber o cycles. The curves depend on the level o the axiu applied stress and on the ratio between the lowest and the highest stresses ( R = in / ax ). In previous wors ([1],[5],[6]) an exponential unction to approach steel and aluiniu experiental -N curves was proposed. This unction depends on and is capable o dealing with any value o the ratio between iniu and axiu stress. However, it is a bit diicult to adjust its paraeters to obtain a good approxiation to experiental curves, which are usually not deined over the whole lie-span o the aterial. Instead o using such an exponential unction, here the experiental data is introduced by points in a table-lie ode. Usually, -N curves are obtained or a ully reversed stress state ( R = in / ax = 1) by rotating bending atigue tests. This zero ean stress is however not typical o real industrial coponents woring under cyclic loads. Based on the actual value o the R ratio and a basic value o the endurance stress e (or R = -1) the odel proposed postulates a threshold stress th. The eaning o th is that o an endurance stress liit or a given value o R = in / ax ; i the actual value o R is R= -1 then, th = e. th th = e = e + ( + ( u u e e ) ( R) ) ( / R) THR1 THR2 abs( R) 1 abs( R) 1 (7) THR1 and THR2 are aterial paraeters that need to be adjusted according to experiental tests. Cyclic trength Reduction Function The -N curves proposed in the previous section are atigue lie estiators or a aterial point with a ixed axiu stress and a given ratio R. I, ater a nuber o cycles lower than the cycles to ailure, the constant aplitude cyclic loads giving that axiu stress ax (and ratio R) are reoved, soe change in u is expected due to accuulation o atigue cycles. In order to describe the variation o u the ollowing unction is proposed: red( R, Ncycles) = exp( B0 (log10( Ncycles)) B0 = log10( ax / ) /(log10( N )) u F BETAF BETAF ) (8) 31-4 RTO-M-AVT-121 UNCLAIFIED/UNLIMITED

6 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures BETAF is a aterial paraeter and Tie Advancing trategy N F the nuber o cycles to ailure. An advantage o the ethodology presented consists in the way the loading is applied. In a echanical proble each load is applied in two intervals, in the ollowing order (see Figure 2), -Tracing load, (described by ai periods on Figure 2). It is used to obtain the stress ratio R = in / ax at each integration point, ollowing the load path during several cycles until the R relationship tends to a constant value. This occurs when the ollowing nor is satisied, i+ 1 G R R η = + R G i 1 G i G 0 (9) where R i G i ax G = is coputed at each Gauss interpolation point or the load increent i. in / F N a1 b1 a2 b1 a3 b3 Interval 1 Interval 2 Interval 3 Figure 2: cheatic tie advancing loads representation. ai describes the tracing load doain on the i interval. bi describes the enveloping load doain on the sae i interval. -Enveloping load, (described by bi periods on Figure 2). Ater the irst tracing load interval (ai), the nuber o cycles N is increased while eeping constant the axiu applied load (thic line in Figure 2) and the stress ratio R. In this new load interval, the variable is not the load level (ept constant) but the nuber o cycles. This two-stages strategy allows a very ast advance in the tie loading. A new interval with the two stages explained should be added or each change in the loading level. RTO-M-AVT UNCLAIFIED/UNLIMITED

7 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures 3.0 CONTITUTIVE MODEL FOR FIBRE REINFORCED LAMINA In coposite aterials each coponent aes the coposite behaviour dependent on its own constitutive law according to its voluetric participation. In addition, the coponents orphological distribution inside the coposite becoes essential, giving better properties to the coposite aterial than its coponent parts. Classical Mixing Theory (CMT), also nown as Rule o Mixtures (ROM), taes into account the voluetric contribution o coponents but not its orphological distribution. Thereore, CMT can be only useul to obtain basic properties o coposite aterials assuing parallel behaviour (coponents with sae strains in all directions). This hypothesis is a strong liitation or the use o CMT to predict the behaviour o ost coposites and consequently odiications to this theory are always required. In the late 1970s, a siple icroechanically-based constitutive relation or coposite aterials has been developed [8]. This odel consists o aligned continuous ibres ebedded in an elastoplastic atrix. Approxiate stress and strain ields are prescribed or atrix and ibres, strain coupling in the axial direction and stress coupling or all other stress and strain coponents [9]. The odel account or load sharing (ibresatrix) in the axial direction, but ibres do not interact with the deoration o the atrix in any other direction, leading to a response that underestiate the transverse and shear stiness. To overcoe these liitations, a general nuerical procedure or non-linear constitutive behavior siulation o the unidirectional lainas is developed. This odel is able to capture the siultaneous serial and parallel behaviours, which taes place inside the coposite, independently o loading directions. It is based on the appropriate anageent o the constitutive odels o coponent phases within a continuu raewor. With such an ai, a strategy is developed to decouple phases that ay also contribute possible inelastic strains and/or elastic degradation o the stiness. Input data are the echanical properties o each coponent (i.e.: ibres and atrix in case o ibre-reinorced lainates) and its distribution inside the coposite. This ethodology o ixing coponents with the strategy or phase decoupling allows each coponent to retain, unchanged, its original constitutive law (isotropic or anisotropic, linear or non-linear) while conditioning the coposite global response. Dierent constitutive odels (plasticity, daage, atigue, creep, etc.) or each phase ay be taen into account. 3.1 Deinition o erial and arallel arts o train/tress Tensors The odel considers that in a certain direction (or directions) the copounding aterials behave in parallel, while their behaviour is serial in the reaining directions. For this reason, it is necessary to deine and split the serial and parallel parts o strain and stress tensors. Dierent types o serial-parallel ixtures ay be considered: ro the pure parallel (sae strain ield or all phases) or the pure serial (sae stress ield or all phases), up to the cases o parallel behaviour in one direction (ibre reinorced atrices) or in a plane (biphasic lainates). This wor is ocus on the unidirectional ibres case. Unidirectional Fibres Case Being e 1, the director versor that deterines the parallel behaviour (ibre direction), and being e 2 and e 3 the director tensors, orthogonal to e 1, that deterines the directions where serial behaviour taes place, the parallel projector tensor N ay be deined as ollows: N = e1 e1 (10) 31-6 RTO-M-AVT-121 UNCLAIFIED/UNLIMITED

8 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures It is a second order tensor that gives the parallel (to ibres) projection o a generic vector v : v = N v (11) It is now deined the 4 th -ordered parallel projector tensor : = N N (12) The serial projector tensor is evaluated as its copleent: = I (13) Both tensors allow inding the parallel part o the strain tensor, ε : ε = : ε (14) and its serial part, ε : ε = : ε (15) By this ean, the strain state is split into its parallel and serial parts: ε = ε + ε (16) The stress state ay be split analogously: σ = σ + σ (17) where: σ σ = = : σ : σ (18) (19) 3.2 Hypothesis or the Nuerical Modelling The present nuerical odel orulation is based on the ollowing hypothesis: 1) Coponent aterials have the sae strain in parallel (ibre) direction, 2) and the sae stress in serial directions. 3) Coposite aterial response is in direct relation with the volue ractions o copounding aterials. RTO-M-AVT UNCLAIFIED/UNLIMITED

9 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures 4) Hoogeneous distribution o phases is considered. 5) erect bonding between coponents is also considered. 3.3 Equations that Govern the erial-arallel Behaviour Constitutive Equations o Copounding Materials A general case is considered, with aterials that ay undergo degradation not only because o plastic strains growth but also because o deterioration or elastic daage in their stiness: (,, ) (,, ) σ = ε ε C p d σ = ε ε C p d As an exaple, the particular case o additive plasticity is shown: σ = C ε ε : ( ) : ( ) d p σ = C ε ε d p Constitutive equations o copounding aterials ay be rewritten taing into account the serial and parallel split o strain and stress tensors eqs. (16) and (17) : Equilibriu and Copatibility Equations c c d c d c c p σ C C ε ε : c = c d c d c c p σ C C ε ε c d c d C = : C : c d c d C = : C : where: with: c=, c d c d C = : C : c d c d C = : C : The equations that deine the stress equilibriu and establish the strain copatibility between coponents arise ro the analysis o the previously orulated hypotheses: arallel behaviour: (20) (21) ε = ε = ε σ = σ + σ (22) (23) 31-8 RTO-M-AVT-121 UNCLAIFIED/UNLIMITED

10 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures erial behaviour: σ = σ = σ ε = ε + ε (24) (25) where and are volue raction coeicients, ulilling: + = Exposition o the roble Given the coposite strain state ε at the present tie ( t+ t ), and nown the coponents internal variables o the previous step (t ), ind the strain state o the coponents and also the corresponding stress states at the present tie that ulil the equilibriu, copatibility and constitutive equations described in the previous section. The ollowing chart point up the nown and unnown variables o the proble: t+ t Known variables: [ ] ε, t α, t α ree variable. internal variables Unnown variables: t+ t ε, t + t ε, dependent variables + t σ, t + t t+ t σ, [ σ ], t+ t α, t + t α. updated internal variables t Variables α and α group all the set o internal variables corresponding to the coponents, lie or exaple internal variables o daage and/or plasticity that deine the aterials state: { } { } α : = ε, C p d α : = ε, C p d 3.5 Algorith or Copounding Behaviours It is now introduced the algorith or the constitutive odel o a coposite aterial ade up o behaviours o dierent aterials (e.g.: ibre and atrix). It will be seen that the odel that sets out constitutes, in act, a anager o non-linear constitutive odels, which cobines the hypotheses enunciated or coposites with general aterial odels (siple or coplex ones). Each increase o load ay cause evolution o the internal variables o the coponent aterials, producing an stress disequilibriu incopatible with the second hypothesis (see section 3.2). This stress disequilibriu constitutes the residue to iniize: RTO-M-AVT UNCLAIFIED/UNLIMITED

11 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures σ = σ σ (26) It is chosen as unnown the total serial strain o one o the aterials in this case the one ro the atrix ε ) since the strain o the other aterial depends on the irst one: ( 1 ( ) = ( ) ( ) ( ) ε ε ε ε (27) ( ) σ ε = σ ε σ ε ε (28) The derivative o the objective unction (residue) with respect to the unnown: [ ] σ = σ σ σ σ ε = : ε ε ε ε ε gives the expression o the Jacobian: [ σ ] σ σ ε ε ε ε = ε ε ε J = = - : ( ) T T = C - C : - I T T C + C = that it is used to update the unnown at each local iterative step o the Newton-Raphson ethod: (29) (30) [ ] + 1 ε 1 : = ε σ Note that, ro now on, when [ ] t t is shown, it should be understood [ ] J (31) +. tep 1. Initial Approxiation. The initial approxiation o the unnown can be established when considering that the strain increent aintains the tangent evolution o the previous step: First, the constitutive odels o each aterial are evaluated to deterine their constitutive tangent tensors corresponding to the previous step. t t t t ε ; α ε ; α Matrix Fibre Model Model t T t T C C RTO-M-AVT-121 UNCLAIFIED/UNLIMITED

12 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures Then, the increent o the unnown is deterined supposing that the total strain increent is distributed aong constituent aterials according to their previous step tangent stiness: [ ] = + t+ t t [ ε ] = [ ε] [ ε ] t [ ] + t t [ ] ε ε ε t t t ε ε ε Finally, the initial value o the unnown = 0 ε : T : + T T = A C ε C C : ε [ ] ( ) [ ] T T where: ( ) 1 A= C + C ε is established. t ε = ε + ε = 0 0 tep 2. Residue Evaluation. The total strain tensors o the coponents ( and ) value o ε. ε ε are deterined as a unction o the updated t+ t ε = ε + ε where ε = ε = [ ε ] 1 t+ t ε = ε + ε where ε = [ ε] ε The constitutive odels o each aterial are checed to deterine their updated internal variables and stress states. t t ε ; α ε ; α Matrix Model Fibre Model σ ; α σ ; α Thereater, the residue is evaluated to ind out i the stress equilibriu (i.e., the convergence o the odel) has been achieved. RTO-M-AVT UNCLAIFIED/UNLIMITED

13 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures [ ] = where: = : σ σ σ σ σ σ = : σ tep 3. Convergence Checing. In order to choose the tolerance, the order o agnitude o the serial stresses ust be considered as a reerence. I the stresses o the previous step are dierent to zero, it is taen the iniu between the; on the contrary, the linearized stresses are taen as a reerence. = t t { σ σ } { t T t T C [ ε ] C [ ε ] } re 1 in, re 2= in :, : The tolerance is chosen based on the reerence value: I re 1 > 0 then: reer = re 1 else: reer = re 2 toler = reer 10 I the nor o the residue is greater than a tolerance, then to go to step 4 correction o the unnown, otherwise go to step 5 update o variables. [ σ ] 4 I > toler then: goto tep 4. else: goto tep 5. tep 4. Correction o the Unnown. Evaluation o tangent constitutive tensors o both aterials. The constitutive odels o each aterial are evaluated to deterine their tangent constitutive tensors. ε ; α ε ; α Matrix Fibre Model Model T T C C RTO-M-AVT-121 UNCLAIFIED/UNLIMITED

14 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures Calculation o the Jacobian. T T T T J = C + where: = : : C C C T T C = : C : Update o the unnown. [ ] 1 := J : ε ε σ := + 1 Go to step 2 residue evaluation. tep 5. Update o Variables. Once obtained the convergence o the constitutive odel, all the variables o the odel ust be updated: t+ t t+ t t+ t ε = ε ; α = α ; σ = σ ε = ε ; α = α ; σ = σ t+ t t+ t t+ t tep 6. Update o the Coposite tress tate. The stress tensor o the coposite aterial is calculated taing into account equations (23), (24) and (17): [ ] σ = σ + σ + σ t+ t t+ t t+ t t+ t The presented algorith anages to copound o behaviours o aterials ulilling the hypotheses o the series-parallel proble. Note that the sae algorith ay also deal with the extree cases o pure serial or pure parallel behaviours. 3.6 Tangent Constitutive Tensor or the Coposite Once obtained the local convergence o the coposite odel (anager o constitutive odels), it is necessary to ind out the coposite tangent constitutive tensor that will allow to achieve the convergence o the global proble. In order to get this tangent tensor it is necessary to dierentiate the coposite stress tensor resulted or the solution o the linearized syste o governing equations. Note that in the linearization it is used the tangent constitutive tensors o the coponent phases C T and T C (provided by each siple odel) which are based on the updated strain states and internal variables corresponding to each coponent. + + ( ε, α ) C = g (32) T t t t t RTO-M-AVT UNCLAIFIED/UNLIMITED

15 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures + + ( ε, α ) C = g (33) T t t t t The expression o the algorithic tangent tensor is obtained dierentiating the coposite stress tensor with respect to strain tensor in a way consistent with the integration algorith: where: C σ σ = = = = ε ε T T T σ σ ε ε C C T T ε ε σ σ C C ( + ) + ( - ) : :( - ) ( : : + : : ) ( : : + : : ) ½ ( : : + : : ) C = C C C C C C C T T T T T T T C = C A C C A C T T T T T C = C A C C A C T T T T T C = C A C C A C T T T T T (34) T T with: ( ) 1 A= C + C It can be clearly observed that the expression o the coposite tangent tensor when the tangent tensors o coponents ( C T and C T ) are also syetrical. T C aintains its syetry 3.7 Enrichent o the Transversal Behaviour The assuption o equal stress in orthogonal directions to the ibre (pure serial behaviour in transverse directions), constitute a lower bound or the transverse stiness o the coposite. Experiental results conir this act (see next igure). For this reason, an enrichent in the proposed nuerical odel is required to be able to predict the transversal behaviour ore accurately. That is, the Basic odel is upgraded into the Enriched erial-arallel odel (E odel). In order not to resort to a ore coplex odel, an iproveent is perored that aintains the structure o the previously developed algorith: the behaviour o the ost deorable aterial (atrix) is stiened by eans o a gaa paraeter. The structure o the new algorith is siilar to the previous one, being its only T C instead o ε, σ, C T, whose expressions are: dierence the use o ( ε ), ( σ ), ( ) 1 ( ) : ε = ( σ ) = : σ ( ) = : : K ε (35) K (36) C K C K (37) T T RTO-M-AVT-121 UNCLAIFIED/UNLIMITED

16 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures where: K = : I: + γ : I: By eans o icro-echanical considerations it is possible to tae values o the gaa paraeter as a unction o the ibre voluetric raction ( ) ( R E/ E) and the ratio between ibre and atrix Young odules = (see Figure 3). For exaple: γ = 1.31 or R = 20 and = Gaa R=60 R=40 R=20 R=10 R=5 R= Voluetric raction o ibers Figure 3: Estiative values o gaa as a unction o the ibre voluetric raction or dierent stiness ratios. In the ollowing, the predictive capacity o pure erial-arallel () odel and enriched odel or the transversal stiness o a glass-epoxy laina with E F /E M =21.19, ν F =0.22, ν M =0.38 (stiness ratio and oisson ratios) is copared against experiental data and against the estiation given by broadly used sei-epirical orulas. An hexahedral coposite eleent is subjected to pure transversal load, at dierent ibre volue ractions. In Figure 4, the adiensional curves E 2 /E vs. resulting ro nuerical siulations are given together with experiental values taen ro Barbero [10] (pag. 72). The curves resulting ro perect inverse R.O.M. and ro Halpin-Tsai equation are also given in the sae igure. RTO-M-AVT UNCLAIFIED/UNLIMITED

17 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures E 2/E M Experients inverse R.O.M. Halpin-Tsai odel Enriched odel V Figure 4: Relative transversal stiness o a glass-epoxy laina as a unction o ibre volue raction. Nuerical results copared against experiental data and against estiations given by broadly used sei-epirical orulas. The transversal stiness obtained by pure odel results to be slightly greater than the one given by inverse ROM, due to oisson eects. O course, when oisson ratios are set equal to zero a perect inverse ROM curve is recovered. The graph shows that siple odel, as well as inverse ROM, underestiates the experiental values. On the other side enriched odel, with an appropriate gaa, obtains an approxiation to experiental data as good as the one given by Halpin-Tsai equation. 4.0 CALIBRATION AND VALIDATION OF THE MODEL The nuerical odel proposed to siulate the behaviour o coposite aterials is based on the appropriate anageent o the constitutive odels o the dierent coponent aterials considering the orphology and distribution o ibres within the atrix [2]. oe echanical paraeters, such as the transverse Young s odulus or anisotropic ibres (e.g.: carbon ibres), are ipossible to obtain by eans o direct easureents. Thereore, it ust be deduced indirectly by eans o calibration o this paraeter to it the stiness o the laina, or by eans o a orula that considers the transverse Young odulus o the coposite, the Young odulus o the atrix and the voluetric participation o the ibre [10]. Once these paraeters are deterined they reain lined to that coponent aterial since they are its attribute; they are independent o ibres direction, their voluetric participation, the loads or the ind o proble. In order to study the atigue phenoenon in coposed aterials, it is necessary to calibrate the paraeters that deterine the lie curves -N o the atrix and o the ibre, with the tests done on lainas o coposite. Then, the tests on lainates and structures will allow veriying i the calibration has been ade successully RTO-M-AVT-121 UNCLAIFIED/UNLIMITED

18 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures As calibration ethodology or the atigue phenoenon, the ollowing tips set out: calibrate the lie curve -N o the atrix with atigue tests done on lainas with ibres at 90 degrees; calibrate the lie curve -N o the ibre with atigue tests done on lainas with ibres at 0 degrees; veriy that the calibrations respond to the behaviour o lainas at 90 degrees; ailing that, return to the irst step; veriy with the response o the atigue tests done on [+45/-45]sy lainates; ailing that, it will be necessary to include an interphase "aterial" and to calibrate it (currently the odel does not include the interace since perect adhesion between ibre and atrix is assued, but this one could be taen into account by eans o slight changes in the strain copatibility equations); veriy the values with other lainates and/or structures. The odel is validated with experiental testing done at Institute or tatic and Dynaics (ID) o University o tuttgart on CFRR aterial saples. Carbon ibre here is Tenax HTA 5241 and the resin used is Ruetapox VE4434. Values or echanic properties o coponents have been given by anuacturers and they are listed in Table 1. Table 1: Materials echanical properties given by anuacturers Coponents roperties Unit Fibres values Resin values Density ( c ) g/ Tensile strength Ma 3950 Tensile strain % 1.5 Young s odulus parallel Ga Young s odulus perpendicular Ga 28 4 hear odulus Ga oisson s ratio (ν xy ) Mass (M c ) g Voluetric content ratio ( c ) The lay-up has been chosen to get a syetric lainate coniguration. 3-layer specien [0 90 0] [ ] [0 90 0] 4-layer specien [ ] [ ] [ ] [ ] 5-layer specien [ ] [ ] [ ] [ ] [ ] The thicness o the lainas is chosen in such a way that the nubers o ibres are the sae in both directions, inner layers twice thic as the outer ones. tatic testing: For each type o specien (3, 4, 5-layers) ive orce controlled tensile test (European tandard EN 2561) have been carried out to ind out the Young s odulus, the oisson s ratio and the ultiate strength (stress at load to racture). Figures 5 and 6 show the coparison between experiental results and nuerical siulations or the 3-layer speciens at 0º and 45º loading directions. Looing at the brea pattern o the speciens it can be seen that RTO-M-AVT UNCLAIFIED/UNLIMITED

19 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures ailure is caused by racture o ibres in the 0º loading direction (quasi-linear stress-strain curve) and due to shear ailure o the atrix in the 45 case (strong non-linear stress-strain curve). Both situations are accurately reproduced by the nuerical siulations. Figure 5: Tensile trength Test - 3-layer specien [0 90 0] [ ] [0 90 0]. Dynaic testing: Tension cyclic loading with a ratio in / ax = 0.1 and axiu stress depending on the thicness o the saples have been used or durability tests. For 3-layered speciens, load levels o 440, 480 and 500 Ma are applied up to ailure. The daage progress within the inspected speciens are observed in dierent ways: tiness onitoring (stiness decrease is a concoitant o daage progress), hysteresis easureent (dissipated energy equivalent to area o hysteresis loop and daage progress) and heat radiation. In Figure 7 (let side) resulting stiness plots over the nuber o load-cycles or dierent loadlevels are suarized. On the right side o Fig. 7 the corresponding results o nuerical siulations are presented. Good agreeent is ound or the 440 Ma load case, acceptable or the 480 Ma case and it is diicult to assert about the 500 Ma loading case. Looing at the experiental results or the three dierent load-levels a signiicantly increasing variation o the nuber o load-cycles to ailure can be observed RTO-M-AVT-121 UNCLAIFIED/UNLIMITED

20 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures Figure 6: Tensile trength Test - 3-layer specien [ ] [ ] [ ]. 6.00E E+02 tress [N/2] 4.00E E E N/2 480 N/2 500 N/2 1.00E E+00 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E E E+04 tiness 1.7E E E N/2 480 N/2 500 N/2 1.4E+04 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 load cycles Figure 7: Fatigue behaviour o 3-layer speciens. Experiental results on the let, nuerical siulations on the right. RTO-M-AVT UNCLAIFIED/UNLIMITED

21 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures 5.0 CONCLUION The erial-arallel () odel is cobined with classical laination theory to describe lainates consisting o unidirectional continuously reinorced layers. Its relative siplicity and the resulting nuerical eiciency ae the approach well suited or ipleentation as a aterial odel in Finite Eleent progras or studying the elastoplastic response o structures or coponents ade o continuously reinorced or lainated coposites. In addition, it requires relatively sall coputational resources when ipleented into a structural FE code. Its initial drawbac o underestiation o the transverse and shear stiness, is then iproved upon with the Enriched odel. A atigue odel based on continuu echanics accounting or elasto-plastic-daage constitutive equations, previously developed by the authors, has been applied to the coponents o CFRR aterial in order to obtain the durability o the whole coposite lainate. Coparison between experiental and nuerical testing carried out on aterial saples allow saying the ethodology presented here is very proising or durability analysis o coposite aterials and structures. ACKNOWLEDGMENT This wor has been partially unded by European Coission under the GROWTH roject GRD , COMA, Contract G5RD-CT and by the panish governent through FU grant or F. Rastellini. This support as well as the experiental results obtained by Dr. C. Hoann at Institute or tatic and Dynaics (ID), University o tuttgart is grateully acnowledged. We are indebted to R. erpieri or his help on the nuerical odel validations. REFERENCE [1] Oller, ; aloon, O.; Oñate, E. (2005). A continuu echanics odel or echanical atigue analysis. Coputational Materials cience 32, [2] Rastellini, F.; Oller,. (2004). Modelado nuérico de no linealidad constitutiva en lainados copuestos - Teoría de ezclas. Métodos Coputacionais e Engenharia. Lisboa (ortugal): AMTAC. [3] Rastellini, F.; Oller,.; aloon, O.; Oñate, E. (2003). Teoría de Mezclas erie-aralelo Avanzada para el Análisis de Materiales Copuestos V Congreso de la Asociacion Española de Materiales Copuestos, AEMAC, 2003, pág IBN A. Mirevete & J. Cuartero (Eds) [4] Rastellini, F.; Oller,.; aloon, O.; Oñate, E. (2003). Advanced serial-parallel ixing theory or coposite aterials analysis. Continuu basis and inite eleent applications, COMLA 2003, roceeding (CD) o the VII International Conerence on Coputational lasticity IBN: CIMNE, Barcelona. [5] aloon, O.; Oller,.; Oñate, E. (2002) Industrial Application o Fatigue Daage Analysis and FEM, NAFEM-FENET Technology Worshops. Durability and Lie Extension. Zurich(Geroldswill), witzerland. [6] aloon, O.; Oller,.; Oñate, E. (2002) Fatigue Daage Modelling and Finite Eleents Analysis Methodology: Continuu Basis and Applications, FATIGUE 2002, roceeding o the Eighth International Fatigue Congress, Ed. A.F.Blo, tochol, weden, pag , Vol.4/ RTO-M-AVT-121 UNCLAIFIED/UNLIMITED

22 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures [7] aloon, O.; Oller,.; Oñate, E. (2002) Fatigue Analysis o Materials and tructures using a Continuu Daage Model, International Journal o Foring rocesses, Vol. 5, Nº 2-3-4, gs , Ed. HERME cience ublications, France [8] Y.A. Bahei-el-Din. lastic Analysis o Metal Matrix Coposite Lainates, Doctoral Thesis, Due University, Durha, NC, [9] G.J. Dvora, Y.A. Bahei-el-Din. lasticity analysis o ibrous coposites, J. Appl. Mech. 49, , [10] Barbero, E. J. (1998). Introduction to coposite aterials design. London: Taylor & Francis. RTO-M-AVT UNCLAIFIED/UNLIMITED

23 UNCLAIFIED/UNLIMITED Fatigue rediction or Coposite Materials and tructures YMOIA DICUION AER NO: 31 Author s nae: O. aloon Discussor s nae: J. Calcaterra Question: In the continuu atigue orulation, you assued a atigue endurance liit stress- Are there data to validate this assuption or your aterial? Answer: For coposite aterials experiental data corresponding to coponents ust be obtained ro tests on the inal aterial (coposite) using special conigurations and nown data or one o the coposites. This is not an easy tas but can be done. Anyway, a ixed stress endurance liit is not a requireent or the odel. I stress continues decreasing as a unction o nuber o cycles, this inoration is the iportant. There is no need or a constant endurance liit RTO-M-AVT-121 UNCLAIFIED/UNLIMITED