Fatigue Life Prediction in Composite Materials

Transcription

1 Autodesk Helius PFA 2017 Fatigue Lie Prediction in Coposite Materials Predicting atigue lie in coposites is challenging because even siple load states lead to coplex behavior in the individual iber and atrix constituents. Furtherore, the eects o requency and load history can only be dealt with adequately by using a physics-based approach. Autodesk Helius PFA addresses this challenges by cobining ulticontinuu technology with the kinetic theory o racture to introduce an eective technology or progressive daage odeling o atigue loading or both unidirectional and woven coposites.

2 Contents Introduction...4 The Proble...5 Previous Options...6 The Solution...7 Theoretical Developent: Fatigue Failure in Polyers...8 Multicontinuu Theory or Unidirectional Coposites Extracting Paraeters or Use with Kinetic Theory Technology Deonstration Ipleentation Suary Reerences... 14

3 Table o Figures Figure 1. Typical three diensional ailure suraces or static and atigue ailure in a carbon epoxy laina... 2 Figure 2. An exaple o siple cyclic loading and its saw-tooth approxiation Figure 3. O-axis coposite tension-tension atigue lie predictions at roo teperature (dashed lines) copared with experiental data (arkers) ro Kawai et al. [9]

4 Introduction It is no news that the use o coposite aterials is on the rise. The extent o their use is increasing in any engineering applications ro aerospace to inrastructure to alternative energy. Nearly 20 years ago, the Boeing 777 was introduced with coposites coprising roughly 10% o the structural weight [1]. This included the cabin loor and uch o the tail section. Recently, the state-o-the-art Boeing 787 Drealiner aircrat was rolled out, which utilizes advanced coposites in 50% o the aircrat. This includes 100% o the skin, entire sections o the uselage, the tail, and the wing box and wing skins [2, 3]. General Motors recently indicated part o their strategy or uture vehicle design, production, and globalization includes an expanded use o carbon iber [4]. And, wind energy new capacity installation is expected to continue growth at an average rate o 13.8% per year over the next ive years. The resultant need or new turbines is expected to increase the industry deand or coposites at an annual growth rate o 18% to 1,243 illion lbs by 2013 [5]. With the increasing use o coposite aterials coes an increasing need to understand their behavior and design lie. Many o these applications include conditions that include repetitive loading cycles, thus deanding the ability to understand and evaluate atigue in coposite structures. Yet this understanding has reained a challenge or the design engineer. The need or the ability to apply physics-based theories and principles to coposite aterials has been widely recognized. Eorts to develop this capability or atigue analysis o large structures have been hindered by excess coputational tie and the inability to separate dierences in physical behavior exhibited by each constituent o the coposite. In this paper, a new approach or atigue lie prediction in coposites is introduced. The stresses and strains o the coposite constituents are calculated using ulticontinuu theory (MCT), which requires negligible additional coputation tie beyond that o standard inite eleent analysis o a hoogenized aterial. The constituent stresses are used in cobination with kinetic theory to predict atigue lie in large-scale coposite structures under a variety o coplex load states. This ethod dovetails readily with existing inite eleent analysis tools and requires inial coposite atigue characterization. Thus, it provides the needed bridge between physics and large-scale structural analysis.

5 The Proble Fiber reinorced polyer coposites have becoe an attractive replaceent or heavier etals due to their superior atigue and corrosion properties. Although they ay be less susceptible to atigue ailure than etals, atigue can still occur. This is especially true when environental actors such as teperature and huidity becoe signiicant. In general, the lack o understanding about coposite structural atigue is a signiicant setback or any industries. Fatigue ailures represent the greatest uncertainty with regard to the long-ter service lietie o the ajor structural coponents o a wind turbine, oten leading the designer to either add excess weight to copensate or to inadequately size a blade resulting in racture both costly errors liiting the cost advantage o wind energy [6]. During the 2008 Society o Plastics Engineers Autootive Coposites Conerence and Exhibition, the need or advanced predictive engineering techniques was discussed as a roadblock to a ore widespread use o coposites in autootive design [4]. A recent European study also concluded that or coposites to take hold in autootive structural design, there will need to be continued developent o ailure theories, daage odeling, and atigue lie prediction [7]. For the design engineer or structural analyst, there are signiicant challenges to predicting atigue behavior in coposites: Liited Fatigue Characterization Data Coupon-level coposite atigue test data is an unreliable source or design because it doesn t capture the range o paraeters that are oten observed in real structures. Fatigue characterization is typically perored under sipliied loading conditions and generally consists o uniaxial, constant aplitude tensile atigue loading. However, any applications are subjected to ulitaxial, variable aplitude loads that involve dierent environents. In addition, atigue ailure depends on the speciic layup and geoetry o the structure thus, characterization data is not likely to be applicable to structural design. Physics-based Theories Cannot Be Easily Applied at Coposite-Level In a large structural analysis, typically only hoogenized coposite stresses are coputed. The proble with utilizing these stresses to predict atigue behavior is that they do not represent the stresses in the individual constituent aterials. Fiber & atrix stresses drive coposite atigue behavior. As such, an accurate deterination o coposite atigue requires access to the constituent properties. Historically, no coercial solution has existed which can reliably extract this inoration without exotic aterial data or substantial coputational cost. To eet the needs o coposite structural evaluation, an eective solution to accurately odel coposite atigue should: Account or aterial behavior at the atoic or deect level so that environental eects (teperature, oisture, cheical exposure, etc.) can be incorporated Consider atigue daage to a particular constituent aterial within the coposite, not daage to the hoogenized coposite Require a inial input data set or characterization o atigue behavior Apply to ultiple types o loading and load histories Apply to any coposite lainate layup Be eicient enough to use or a routine structural analysis 5

6 Previous Options Coposite atigue has been studied or decades and any o the proposed odeling approaches satisy soe o the above requireents. But currently a odeling approach does not exist that satisies all o the. A range o approaches have considered several dierent types o atigue loading: constant aplitude, block loading, and spectral loading. Yet no approach has been satisactorily applied to the entire range o probles that ust be addressed. Early studies in predicting coposite atigue, such as those o Hashin and Rote [8], relied on acroscopic coposite strengths. This approach requires a inial data set, but satisies none o the other requireents. More sophisticated acroscopic approaches, such as calculating residual strength or residual stiness, have also been used, but they require uch ore experiental data and still apply only to a particular lainate. Microstructural odeling o a coposite can yield high-idelity constituent stresses and strains but is too coputationally intensive or large-scale structural analyses. Moreover, nearly all o these techniques require a large aount o experiental data to characterize the aterial. Most predictive theories pertain to a speciic load history at a speciic teperature and are not easily generalized to capture ultiaxial load states, variable aplitude or spectral loading, teperature changes, or environental eects. In addition, any general solution ust be able to be ipleented into design codes, such as inite eleent analysis sotware, with coputational eiciency. This goal is unlikely to be achieved as long as coposite atigue predictions are based on purely epirical relationships. Physics-based concepts ust be applied to the coposite in order to use a inial aount o characterization data to predict the atigue lie o a coposite under a variety o loading and environental conditions. But applying physics-based theories requires knowledge o constituent-level behavior. Because the ajority o coposite atigue analyses use hoogenized coposite stresses/strains rather than constituent stresses/strains, direct application o relevant aterial physics is nearly ipossible.

7 The Solution The challenge in obtaining constituent aterial stresses in large-scale structural inite eleent analyses has been reedied by Autodesk Helius PFA, which uses ulticontinuu theory to eiciently extract average constituent stresses and strains ro coposite stress and strain. Using the Helius PFA s raework, we can now apply the appropriate physics - in this case polyer kinetics - to the constituent aterials individually in order to predict coposite atigue response. The atigue solution ipleentation uses constituent stresses calculated using Helius PFA and then applies the wellestablished kinetic theory o racture to produce a physics-based ethod or predicting atigue lie in coposite aterials. One crucial observation regarding the character o coposite atigue ailure is that, with the exception o nearly perect axial loading, atigue ailure is a atrix-doinated event [9, 10, 11]. This eans that the physical behavior o polyers is the physics-based ingredient needed to predict atigue. As such, the kinetic theory o racture, a proven physics-based theory or polyer atigue lie, is applied to the atrix stress in the coposite to predict the reduction in strength based on the nuber o cycles. In contrast, the iber ailure planes reain nearly unchanged as their strength is unaected by cyclic loading. Figure 1 shows a representation o a static and atigue ailure envelope or a unidirectional graphite coposite. Fiber Failure Plane Figure 1. Typical three diensional ailure suraces or static and atigue ailure in a carbon epoxy laina Fatigue Matrix Failure Surace Static Matrix Failure Surace 7

8 Theoretical Developent: Fatigue Failure in Polyers The relationship between polyer kinetics and echanical behavior was developed ore than ive decades ago by Zhurkov [12, 13] and Colean [14, 15] in parallel eorts. Zhurkov used experiental observations to show that the conceiving o strength in ters o olecular kinetics was well-ounded. Most iportantly, Zhurkov showed that the bond rupture rate deterines the racture strength o a polyer and the tie to ailure under a creep load, where the bond rupture rate Kb under a tensile load σ has the or KK bb = νν 0 eeeeee UU γγγγ. (1) kkkk In Equation (1), U is an activation energy that is closely related to bond energy, γ is an activation volue, and k is the Boltzann constant. (Note: this is the Eyring equation, as discussed by Ward [17].) Approaching the proble ro a statistical echanics approach, Colean developed a siilar equation and noted that it could also be used to predict polyer atigue lie [15]. The value ν0 is the oscillation requency o the ato, which should be proportional to kt/h, where h is Planck s constant; at roo teperature kt/h = s -1. Zhurkov [12] reports a value o s -1 or this ter, while Colean [14] reports a value o s -1. As a irst-order approxiation, we siply use kt/h, such that Equation (1) becoes KK bb = kkkk eeeeee UU bb γγ bb σσ. (2) h kkkk Siilarly, or bond healing we can write KK h = kkkk eeeeee UU h γγ h σσ. (3) h kkkk Hansen and Baker-Jarvis [16] cobined these earlier works to develop a rate-dependent kinetic theory o racture or polyers, which successully predicted the strength o polyers subjected to a wide range o stress rates. In their orulation, they introduced a dierential equation or the evolution o a daage variable n with tie t, where the evolution o the daage variable is directly related to the bond rupture rate as: dddd dddd = (nn 0 nn) λλ (KK bb KK h ), nn(0) = nn ii (4) where nn 0 is a paraeter deterined by enorcing the condition 1 dddd = 1 (5) 0 (nn 0 nn) λλ

9 and nn ii is the value o daage at the beginning o the atigue loading. The distinction between this orulation and that o Hansen and Baker-Jarvis is that their orulation assued λ=1. This value is the deault value used in our current ipleentation 1. For the case o λ=1, nn 0 = ee ee 1. Assuing a value o 1.0 or λ, and cobining Equations (2), (3), and (4) gives the starting equation or deterining the atigue lie o a polyer. dddd = (nn dddd 0 nn) kkkk eeeeee UU bb γγ bb σσ eeeeee (tt) eeeeee UU h, nn(0) = nn h kkkk kkkk ii (6) Note that the ter containing the activation volue, γ, or healing in Equation (3) has been assued zero. Healing generally only occurs at low stress values, and thus the contribution o the stress to healing at this level is negligible. Solving Equation (6) yields the evolution o the daage paraeter with tie, which can be written as 2 : nn(tt) = nn 0 (nn 0 nn ii )eexxxx kkkk tt eeeeee UU bb γγ bb σσ eeeeee (ττ) eeeeee UU h h 0 kkkk kkkk dddd (7) The atigue solution uses Equation (7) with calibrated values o UU bb, γγ bb, and UU h to deterine the tie at which the daage paraeter becoes unity (n=1). Equation (7) ust be nuerically integrated over a load history to identiy the nuber o cycles to ultiate ailure. Mean Stress Eects Equation (7) can provide the atigue lie or a polyer under a deined load history. When applied to varying load aplitudes and requencies, however, Equation (7) cannot capture the ean stress eect appropriately. The ean stress eect is generally deined as the eect o varying requency, load aplitude, or ean stress on the atigue lie o a aterial, as shown in Figure 2. 1 For a single linear elastic analysis, i.e. no progressive ailure, the choice o λ is irrelevant. All solutions o λ will yield the sae nuber o cycles to ailure. However, this actor could be critical or block loading and progressive ailure siulation. 2 Note that we have assued that the orulation in Equation (7) assues that U and T are constant during the atigue process. 9

10 Figure 2. Stiness reduction o woven abric coposites subjected to axiu atigue stress o 80 MPa and a stress ratio R o While there ay be any undaental physics which contribute to the ean stress eect in a polyer, one o the ain contributors is through to be hysteresis heating, as shown in Figure 3. Figure 3. Hysteresis heating during atigue tests in the bias direction. Woven abric coposite was subjected to axiu atigue tress o 80 MPa and a stress ratio R o We can then assue that the hysteresis heating contributes to a rise in teperature. This iners that the teperature in Equation (7) is a unction o the requency and stress aplitude, which would give nn(tt) = nn 0 (nn 0 nn ii )eeeeee kktt tt eeeeee UU bb γγ bb σσ eeeeee (ττ) eeeeee UU h h 0 kktt kktt dddd (8)

11 The new paraeter, T*, in Equation (8) is a odiied teperature. We can write the change in teperature as an energy equation; the current or o which is = TT ii TT ψψ ΔΔσσ eeeeee,ii 2 ee kkkktt ii + tt + ψψ ΔΔσσ eeeeee,ii TT ii+1 ΔΔΔΔ2 ii ΔΔΔΔ ii 2 kk 2 (8) The above equation can be sued over the entire load history to identiy the change in teperature as the nuber o cycles progresses. In the above, ψψ is a constant o proportionality, Δσσ eeeeee is the agnitude o the eective stress change over a tie period Δtt, and n is the nuber o dierent stress ranges in a cycle. Multicontinuu Theory or Unidirectional Coposites In order to apply kinetic theory to coposite structures, the stress in the iber and atrix constituents ust be deterined. Multicontinuu theory, as developed or two-constituent coposite aterials by Garnich and Hansen [19, 20, ] and now coercially ipleented in Helius PFA, provides an elegant and coputationally eicient ethod to deterine volue-averaged stresses o the atrix and iber. In this report, we are interested in atrix-doinated atigue ailure, so the average atrix stress is the value o iportance. The exact value o average stress in the atrix σ can be written as σσ = QQ σσ cc ψ (ΔΔΔΔ) (8) where Q ψ = C = C c φ A = φ a = C η φ C c 1 { Cc ( φ I + φ A) } 1 φ [( C C )( φ I + φ A) ] a + η ( φ I + φ A) 1 { η } 1 ( C C ) ( C C ) c c c η φ C η c (9) In Equations (8) and (9), σc is the six-coponent coposite stress vector; T is the teperature change ro the stress-ree state; Ci (i = c,, ) are the reduced stiness atrices or coposite, iber, and atrix, respectively; φ and φ are the iber and atrix volue ractions, respectively; and ηi (i = c,, ) are the theral expansion coeicients o the coposite, iber, and atrix, written as six-coponent strain vectors where the shear coponents are zero. Extracting Paraeters or Use with Kinetic Theory Ater deterining the static in situ properties, the ollowing aterial atigue characterization process ust be carried out or each identiiable atigue ailure ode within the coposite laina. The process is identical or each, so it is only described once. The aterial atigue characterization process involves an initial step ollowed by an iterative step. The ollowing processes deine the initial step: a) Read in S-N calibration curves and their corresponding paraeters, R,, and T, and aterial properties b) Rotate the stress S into the principal coordinate syste o the coposite to get a coposite stress tensor σc c) Extract the atrix stress tensor σ ro the coposite stress tensor using MCT 11

12 d) Map the atrix stress tensor to an eective stress depending on the icrostructure and whether the loading is on-axis or oaxis. e) Use the value o R to calculate σe,in = Rσe,ax. ) At the conclusion o this initial step, we have three vectors, σe,ax,σe,in and N, o length i, where i is the total nuber o data points in the S-N curve(s). The iterative step o the aterial atigue characterization involves adjusting U, γ, and Ψ to iniize the error between the log o the cycles to ailure and the log o the predicted nuber o cycles to ailure. Accoplishing this task requires an assuption about the load history. We assue that the load history has a saw-toothed shape, as shown by the red curve in Figure 4. As with the static in situ properties, the gradient ethod o steepest descent is used to optiize the atigue paraeters. Figure 4. An exaple o siple cyclic loading and its saw-tooth approxiation. As a irst-order approxiation, we assue that the stiness properties do not degrade with increasing n, as has been observed in soe experiental work on coposites [18]. Using these assuptions and solving Equation (7) gives the nuber o cycles to ailure N.

13 Technology Deonstration In the ollowing subsection, we apply the atigue odel to o-axis atigue o unidirectional T800H/2500EP carbon/epoxy coposites. We copare our results with experiental data reported by Kawai et al. [9] at roo teperature and show that our odel can be used or prediction o atigue ailure. Only two additional paraeters beyond static ailure data are needed: activation energy U and activation volue γ. Using o-axis roo teperature atigue data reported by Kawai et al. [9] or T800H/2500EP, we deterined a best it o U = kj/ol and γ = kj/mpa-ol. The results o these odel predictions are shown in the graph below where the arkers depict experiental data and the dashed lines indicate odel predictions. Note that these paraeters (U and γ) are deterined ro a typical S-N curve. Figure 5. O-axis coposite tension-tension atigue lie predictions at roo teperature (dashed lines) copared with experiental data (arkers) ro Kawai et al. [9]. Ipleentation The use o ulticontinuu theory to extract the atrix stresses allows the kinetic theory to be utilized in large-scale analyses. In a conventional inite eleent analysis, ulticontinuu theory as used by Helius PFA, adds only about 3% to the coputation tie required or a large-scale structural analysis. Given the siple nature o the odel proposed, we expect a siilar coputational burden ipleenting this atigue lie prediction odel into existing inite eleent sotware. Together with Helius PFA, the atigue eature perits rapid, robust atigue lie analysis o large-scale structures. 13

14 Suary In the atigue solution, kinetic theory is applied to atrix stresses and is used to predict coposite atigue lie under any loading condition, including ultiaxial load states and varying load histories. Results deonstrate that, using ulticontinuu theory, kinetics can be applied to coposite aterials at the constituent level to predict atigue lie ro undaental physical properties o the polyer atrix. For urther details regarding the atigue eature o Helius PFA, reer to Section 11 o the Helius PFA Theory Manual. Reerences 1. West, K. "Boeing 777 Lights up with Plastics." Seattle Post-Intelligencer. 1 June May 2009 < 2. H. Baldwin. 787 Drealiner Unatched Econoics, Perorance and Passenger Appeal. Aviation Week & Space Technology Market Suppleent. 14 March J. Rakow and A. Pettinger. Failure Analysis o Coposite Structures in Aircrat Accidents. Proceedings o the ISASI 2006 Annual Air Saety Seinar. Septeber p M. LeGault SPE Autootive Coposites Conerence and Exhibition. Coposites Technology. 1 Deceber May 2009 < articles/2008-spe-autootive-coposites-conerence-apexhibition.aspx> 5. Future Material Needs or the Global Wind Energy Market, March May < 6. Coission on Engineering and Technical Systes. Assessent o Research Needs or Wind Turbine Rotor Materials Technology. Washington, D.C.: National Acadey Press, J. Carruthers, E. Mangino and G. Pitarresi. The Future Use o Structural Coposite Materials in the Autootive Industry International Journal o Vehicle Design 2007 Vol. 44, No. ¾ pp Z. Hashin and A. Rote, A atigue ailure criterion or iber reinorced aterials, Journal o Coposite Materials, vol. 7, 1973, p M. Kawai, S. Yajia, A. Hachinohe, and Y. Takano, O-axis atigue behavior o unidirectional carbon iberreinorced coposites at roo and high teperatures, Journal o Coposite Materials, vol. 35, 2001, pp J. Awerbuch and H. Hahn, O-axis atigue o graphite/epoxy coposite, Fatigue o Fibrous Coposite Materials, San Francisco: ASTM, 1981, pp J. Peterann and A. Plutree, A uniied atigue ailure criterion or unidirectional lainates, Coposites: Part A, vol. 32, 2001, pp S. Zhurkov, Kinetic Concept o the Strength o Solids, International Journal o Fracture, vol. 1, 1965, pp S. Zhurkov and V. Kuksenko, The icroechanics o polyer racture, International Journal o Fracture, vol. 11, 1975, pp B.D. Colean, Tie Dependence o Mechanical Breakdown in Bundles o Fibers. I. Constant Total Load, Journal o applied physics, vol. 28, 1957, pp B. Colean and D. Marquardt, Tie dependence o echanical breakdown in bundles o ibers. II. The iniinite ideal bundle under linearly increasing loads, Journal o Applied Physics, vol. 28, 1957, pp A. Hansen and J. Baker-Jarvis, A rate dependent kinetic theory o racture or polyers, International Journal o Fracture, vol. 44, 1990, pp I. Ward, Mechanical Properties o Solid Polyers, New York: John Wiley & Sons, H. El Kadi and F. Ellyin, Eect o stress ratio on the atigue o unidirectional glass ibre/epoxy coposite lainae, Coposites, vol. 25, 1994, pp M.R. Garnich and A.C. Hansen, A Multicontinuu Theory or Theral-Elastic Finite Eleent Analysis o Coposite Materials, Journal o Coposite Materials, vol. 31, Jan. 1997, pp M.R. Garnich and A.C. Hansen, A Multicontinuu Approach to Structural Analysis o Linear Viscoelastic Coposite Materials, J Appl Mech, vol. 64, Dec. 1997, pp K. Takeura and T. Fuji, "Fatigue Daage and Fracture o Carbon Fabric/Epoxy Coposites under Tension- Tension Loading," JSME International Journal, pp , Autodesk and the Autodesk logo are registered tradearks or tradearks o Autodesk, Inc., and/or its subsidiaries and/or ailiates in the USA and/or other countries. All other brand naes, product naes, or tradearks belong to their respective holders. Autodesk reserves the right to alter product and services oerings, and speciications and pricing at any tie without notice, and is not responsible or typographical or graphical errors that ay appear in this docuent Autodesk, Inc. All rights reserved.