MAE: An Integrated Design Tool for Failure and Life Prediction of Composites

Transcription

1 MAE: An Integrated Design Tool or Failure and Lie Prediction o Composites SANGWOOK SIHN* AND JIN WOO PARK University o Dayton Research Institute, 300 College Park Dayton, OH , USA ABSTRACT: An advanced strength and lie prediction tool, MAE, was developed or the analysis and design o composite structures and components. The MAE integrates three theories and methods: micromechanics o ailure (MMF), an accelerated testing method (ATM), and an evolution o damage (EOD). The MAE can serve as a useul tool to predict damage initiation, progression and lie under various durability loading and environmental conditions. It can handle the inhomogeneous complex geometries and structural components such as open-hole, illed-hole, bonded/bolted joint, stiened panel, textile, etc. Thereore, it can help select not only the material and laminates but also optimal geometrical conigurations. The MAE modules were implemented and integrated with a commercial inite element sotware, Abaqus, or better reliability and maintainability. Several examples o strength and lie predictions o open-hole and double-edge notched specimens demonstrated the capability o the MAE as an advanced tool or the composite durability design. KEY WORDS: creep, atigue, durability, composites, inite element. INTRODUCTION FIBER-REINFORCED LAMINATED COMPOSITES have been used in many structural applications such as airplanes, ships and sporting goods because o their superior speciic properties compared with metal materials. The composite structures can be damaged under mechanical and thermal loadings. Typical damage behaviors in laminated composites are transverse microcracking, iber-splitting, iber-breakage and delamination. The transverse microcracking through the thickness o the ply occurs as the irst-ply ailure and delamination damage ollows. The iber breakage usually occurs at the last stage o the ailure. However, a catastrophic ailure can occur only with the microcracking and delamination damage without iber breakage. The ailure behavior in laminated composites is usually complicated and highly dependent on the properties o the *Author to whom correspondence should be addressed. sangwook@stanordalumni.org Figures 1 19 appear in color online: Journal o COMPOSITE MATERIALS, Vol. 42, No. 18/ /08/ $10.00/0 DOI: / ß SAGE Publications 2008 Los Angeles, London, New Delhi and Singapore

2 1968 S. SIHN AND J. W. PARK constituent materials, iber orientation, stacking sequence, nature o loading, environment conditions, etc. When there exist abnormalities such as holes or notches in the composite structures and components, the mechanical and thermal stresses are concentrated in local spots causing initiation o the damage. With an increase o load amplitudes, the damage grows and progresses urther with more microcracks, longer splitting and larger delaminations until ultimate ailure. Due to the viscoelastic nature o the material properties, damage can grow even without an increase o load, as the composite properties can decrease with the passage o time as well as changes in the environmental conditions such as high temperature and moisture content. For example, composites subject to constant creep loading or constant amplitude o atigue cyclic loading could ail, although the amplitude o load is lower than the critical static strength. Thereore, the analysis and design o composite structures and components must consider the durability o composites under long-term loading and environmental conditions. As stated earlier, the damage phenomena in composites are complicated. Unlike metal cases, the damage does not ollow a dominant crack, posing much diiculty in predicting the ailure and lie o composites with Paris law type methods. Furthermore, because o the viscoelastic nature, the material properties o composites are subject to change under various environmental conditions including temperature and humidity. Thereore, it is essential to develop analytic tools to predict such complex damage behaviors in the composites in order to insert them into the real structures in aerospace, automotive, naval and wind energy applications. The objective o the present study is to develop a damage and lie prediction tool or the analysis and design o the composite structures and components subject to various loading and environmental conditions. We have made an eort to integrate three theories and methods: micromechanics o ailure (MMF), an accelerated testing method (ATM) and an evolution o damage (EOD), which are key theories and methods or the strength and lie prediction o composite structures. Without going into great detail, key eatures o the three theories and methods will be highlighted and summarized in the ollowing subsections. Micromechanics o Failure (MMF) Conventional damage prediction o composites has been conducted on a macromechanical level with homogenized stiness and strength properties. The properties o iber and matrix constituents in the composite ply are smeared in the homogenized properties. A structural analysis then calculates macrostress components o the composite ply with homogenized properties. The macrostress components are compared with the macro-level strength properties or the prediction o damage initiation. Many ailure criteria are available to determine whether the composite ply ails or not. The homogenization process in this scheme introduces interaction parameters, which can only be determined rom combined stress tests where both normal stress components are nonzero. Because o the diiculty in perorming the combined-stress test, the interaction term is assumed empirically [1]. Moreover, there exists an ambiguity o how to treat the homogenized properties or degraded or damaged plies. Oten empirically-determined degradation actors are assumed in this case. Christensen [2,3] derived a general 3-D theory o ailure or broad classes o homogeneous isotropic elastic materials rom very ductile materials to very brittle materials including

3 MAE: An Integrated Design Tool or Failure and Lie Prediction o Composites 1969 polymer, metal, and even geological materials (rocks). The ailure theory was urther developed to consider composites having two constituents, iber and matrix, as well as their interaces through analytical and numerical micromechanical calculation [4,5]. Contrary to the macromechanical damage prediction, the MMF conducts the damage assessment at the micromechanics level with the iber and matrix constituents. Microstress components o the iber and matrix are calculated rom the macrostress components o the ply, and then compared with micro-level strengths o the constituents. The micro-level stiness and strength properties are homogeneous within the iber or matrix constituents. Thereore, no homogenization o the properties is necessary, and thus neither is the interaction parameter. Furthermore, it is more straightorward to handle the degradation and damage o the constituents than those o the homogenized plies. The degradation or damage actors o the constituents can experimentally be determined without assumption, as will be stated later in this paper. Thereore, more accurate ailure prediction is possible with the MMF. Furthermore, through the micromechanical calculation, stiness and strength properties o the homogenenized plies can be predicted rom those o the constituents. Thereore, any changes in the constituents material properties during long-term loading and environmental conditions can easily be incorporated through the micromechanical calculation. The versatility o varying the constituents properties subject to the loading and environmental conditions provides an opportunity to link the MMF with lie prediction capability using master curves, which will be explained in the next subsection. Accelerated Testing Method (ATM) The ATM is a way o rapidly generating a long-term durability database with accelerated testing conducted at elevated temperatures. The ATM is based on a time temperature superposition principle (TTSP) o viscoelastic materials. The time temperature equivalency o the TTSP has been a well-established phenomenon or non-destructive properties, such as creep compliance and storage modulus, o many viscoelastic materials, and can be applied to predict long-term behavior o the non-destructive properties o viscoelastic materials rom relatively short-term tests at several elevated temperatures [6]. Since Miyano s pioneering work, there have been many successul attempts to extend the TTSP to destructive properties such as static, creep and atigue strengths [7 13]. The ATM has been developed based on experimental observation that the composite plies and laminates ollow the time- and temperature-dependencies o the viscoelastic matrix material. The strength o the viscoelastic material subject to loadings at various temperatures and rates o loading can be measured and ormulated as a unction o the temperature and the time to ailure. Using the TTSP, modulus and strength data, measured at several elevated temperatures or short-term loading, are shited to long-term loading range at a reerence temperature, and thus superposed to orm the master curves o modulus and strength at the reerence temperature, respectively. By using the master curves, the time and the temperature are interchangeable, meaning acceleration in time can be achieved by raising the temperature. Once the master curves are generated with the accelerated testing, they can be used to obtain the long-term modulus and strength data under various loading and environmental conditions, such as stress levels o creep and atigue loadings, a requency and a stress ratio o atigue loading, temperature, moisture, etc. The property reduction due to the

4 1970 S. SIHN AND J. W. PARK degradation and damage o the constituents can be determined experimentally by the master curves, not relying on the assumed property-knockdown actors. Thereore, with the database ormulated as the master curves, more reliable predictions or lie and residual strength o composite structures and components are possible under arbitrary creep and atigue loadings. Compared to other existing methods [14], the ATM is a very versatile method that can handle a wide spectrum o loading and environmental conditions. Furthermore, based on crack kinetics o the viscoelastic materials [15,16], the master curves can be ormulated with a ew it constants, taking probabilistic aspects into account by considering a Weibull distribution o the scatter o the data. Evolution o Damage (EOD) The composite laminates consist o several iber orientations and stacking sequences. Thereore, upon ailure, not only the initiation o the ailure but also the propagation behavior o damage signiicantly aect the perormance o the composites. Thereore, the damage progression beyond the initial ailure needs to be considered or more advanced composite analysis and design. The EOD considers not only the initiation o the damage, but also the damage progression beyond the irst-ply ailure. In conjunction with a inite element (FE) method, the damage initiation and propagation can be simulated either by a conventional material property degradation method (MPDM) [17 19] or a novel element ailure method (EFM) [20 24]. The MPDM simulates the damage by degrading the material property o the damaged elements, while the EFM achieves this by modiying nodal orces o the damaged elements. With the help o numerical FE analysis, we can handle various cases with stress concentration observed in notched open-hole, illed-hole, slotted-hole, etc. Furthermore, we can handle complex geometries such as bonded and bolted joints, stiened panels, sandwich construction and textile composites. Integration o MMF and ATM Out o the three theories and methods stated above, the MMF and the ATM were combined and integrated into a computer program, Super Mic-Mac (SMM) [25,26]. The SMM has been used to predict lie under creep and atigue loadings, and was demonstrated to generate conventional S N curves and to predict residual strengths. The prediction compared well with experimental data measured with various laminate lay-ups o unidirectional plies [0] and [90], cross ply [45] and quasi-isotropic under tensile tensile and tensile compressive atigue loading [27]. These are all unnotched smooth specimens. The SMM is a robust and powerul preliminary design tool or material and laminate selection using its strength and lie prediction capability or the composites. It is based on an MS Excel spreadsheet, making it simple to use and robust. Optimization can be accomplished easily through its built-in parametric-study modules. The SMM can handle irst- and last-ply ailures using a simple progressive damage scheme. I the damage is predicted by the MMF criteria, it will degrade the material properties o the iber and/or matrix constituents depending on the damage mode. I the ailure occurs in the matrix, the matrix properties will be degraded and the micromechanical calculation will update the corresponding ply properties. I iber ailure occurs, both matrix and the iber properties

5 MAE: An Integrated Design Tool or Failure and Lie Prediction o Composites 1971 will be degraded. Note that the SMM uses an assumption o homogeneous damage, meaning that the entire ply layer damages upon ailure. Because o the simpliied homogenous damage scheme, it is sometimes diicult to make an accurate prediction o the strength and lie when the stresses are inhomogeneous and thus the damage occurs in local regions at concentrated stresses near holes, stieners, joints, etc. In these cases, the damage progression becomes important but is too complicated to be handled with the simpliied progressive damage scheme. Thereore, the SMM is adequate or the cases with the homogeneous stress and strain state such as unnotched smooth unitape composites. I the composites possess inhomogeneity, such as notches, causing the stress concentration, the SMM may not be able to predict the accurate lie and strength. In this case, more sophisticated EOD schemes based on the FE method need to be used to handle the complicated damage cases. Furthermore, the SMM uses the master curves to obtain the stiness and the strength properties at a given time. For example, i the composite structure is in operation or 10 years, this load duration is plugged into the master curves to obtain the reduced stiness and strength properties. Thereore, the history o loads and damage accumulated during the 10 years do not aect the property reduction. The eect o load sequence, load history and damage history during the 10 years are not considered in the SMM. Integration o MMF and EOD The MMF and the EOD can be integrated via the MPDM and the EFM to predict the damage initiation and progression, as stated earlier. The MPDM and the EFM programs are sophisticated analysis tools or strength prediction o composites. Since they are accompanied by the numerical FE analysis, they can handle inhomogeneous stress and strain states such as notched holes, joints, textile composites, etc. The loading can be either static or dynamic loading or impact analysis. The damage progression is predicted element by element, so the detailed progression can be predicted beyond the irst-ply ailure up to the last-ply ailure. However, the integration o MMF and EOD is not enough to make the durability prediction or the lie o the composite structures under various conditions o long-term creep and atigue loadings. For example, the combination o MMF and EOD will not predict any damage i a constant creep stress is applied below a critical stress level. However, damage would occur in reality because o the relaxation o the modulus and the degradation o the strength over time. Such updates on the properties and damage status o the elements are necessary to enable the lie prediction analysis. Thereore, the ATM using the master curves needs to be incorporated and linked with these programs. Integration o MMF, ATM and EOD The present article introduces a new eort to integrate all three components, MMF, ATM and EOD, to predict the damage progression o complicated composite components and structures under long-term creep and atigue loadings. The integration was implemented into a computer program, MAE, which takes the three irst letters rom MMF, ATM and EOD. The MAE is an advanced tool or strength and lie prediction o composites. It is a design and analysis tool capable o damage and lie prediction beyond the

6 1972 S. SIHN AND J. W. PARK irst-ply ailure to detailed damage progression up to ultimate ailure. It can handle inhomogeneous complex geometries and structural components such as open-hole, illedhole, bonded/bolted joint, stiened panel, textile, etc. Thereore, it can help select not only the material and laminates, but also optimal geometrical conigurations. We also have made an eort to implement the MAE in conjunction with commercial FE sotware. In this way, the FE analysis will be more reliable and maintainable. The integration method o the three components into the MAE will be explained in the next section, ollowed by the demonstration o several examples o the strength and lie predictions using the MAE. IMPLEMENTATION METHODS FOR STRENGTH AND LIFE PREDICTION The integration o the three methods, MMF, ATM and EOD, into the MAE or longterm durability prediction o complex structures was achieved with implementation methods described hereater. Suppose a composite structure is under a history o creep and atigue loadings, as shown in Figure 1. Dierent parts (or elements in the FE sense) o the structure, marked by rectangles in the composite structure in Figure 1, will undergo loads with dierent magnitudes yet with a similar loading type. We can subdivide the time scale into subtime segments, t i, and conduct the FE analysis at each time step in a quasi-static way. Property change and damage prediction with the increase o load duration are accounted or by the master curves o stiness and strength, respectively, as will be explained in detail below. Note that or less critically loaded sections, large time segments can be used in order to expedite the calculation, while or highly loaded sections, small time segments can be used to maintain the accuracy o the calculation. As the duration o loading increases, the material properties change with time due to the viscoelasticity o the material. The change o the viscoelastic stiness properties are accounted or with the stiness master curves o either the creep compliance or the storage modulus as the duration o loading increases. The constituent iber and matrix properties are updated at every time step based on the stiness master curve. At a given time t i,we can obtain the reduced stiness properties easily by reading the master curves, as depicted in Figure 2. With the updated properties, the FE analysis is conducted quasi-statically at each time step. Note that the cyclic atigue loading can be treated in the same manner as the creep loading. The input o the creep loading is the applied constant stress level, while the input o the atigue loading is the maximum and minimum stress levels o the cyclic loading. The cyclic eect will be handled by the atigue strength master curves via parameters such as a atigue requency and a stress ratio. Composite structure t i Creep Fatigue Fatigue Creep Fatigue Time t 1 t t i Figure 1. A schematic o the subtime segments o a spectrum o creep and atigue loadings with the increase o time.

7 MAE: An Integrated Design Tool or Failure and Lie Prediction o Composites 1973 Figure 3 shows a detailed low diagram explaining how the MAE computes and updates the ply properties at time t i using the ATM and the micromechanical calculation. When the loading duration increases to time t i with a time segment t i, the corresponding stiness o the matrix constituent (E m ) at time t i can be read rom the stiness master curve. Since the iber constituent is nearly elastic, the iber properties are assumed to be unchanged with time. The same assumption was made or the Poisson s ratio and the coeicients o thermal expansion o iber and matrix constituents. With the stinesses o iber (E ) and matrix (E m ), iber volume raction (v ) and environment conditions such as temperature (T) and moisture content (c) as variables, we can obtain the ply properties by the micromechanical (MM) calculation. This MM calculation can be conducted either analytically using an octagonal unit cell model or numerically using the FE analysis [4,28,29]. In the latter case, the FE calculation can be conducted with various combinations o iber matrix modulus ratios (E /E m ) and the iber volume ractions, and can be tabulated as a database in advance. The actual MM calculation during the MAE procedure can thus be achieved eiciently by a simple and robust table lookup, Stiness master curve Reduced stiness t 1... t i Log t Figure 2. Reading o stiness master curve rom a given time t i. Increase time step: t i = t i 1 + t i Obtain constituent properties E, E m,... at time t i ATM E m (t i ) Storage modulus E (t,t)(gpa) Creep compliance or storage modulus master curve Log t (min) Log t (min) T 0 = 25 C t 0 = 1min Storage modulus E (t,t 0 )(GPa) t i MM: E, v v, T, c E m (t i ) Obtain ply properties at t i (E L (t i ), E T (t i ), G LT (t i ), G TT (t i ),...) Figure 3. A low diagram to show how to obtain ply properties at time t i using ATM and MM.

8 1974 S. SIHN AND J. W. PARK Strength master curve Element stress t Log t Figure 4. Reading o strength master curve rom a given time t i. so that the homogenized ply properties at time t i can be calculated rom the constituent properties easily and instantly. These constituents and ply properties are updated at every time step and used in the FE calculation o the time step. Using the updated properties at every time step, the FE analysis is conducted to calculate stress and strain components on all elements. Severity o damage in the elements is then evaluated using the strength master curves (either creep or atigue strength master curves depending on the loading type). The elements can be partially damaged or completely damaged based on the damage accumulation on the elements. Note that the interpretation o the strength master curves is dierent rom that o the stiness master curves. The stiness master curve relates the changes o the material properties with respect to time. Thereore, it can be considered that the stiness reduces with time by tracing the stiness master curve. However, the strength master curve is just a collection o ailure lives subject to various stress levels. For example, the creep strength master curve was obtained by applying loadings maintained at several constant stress levels and measuring the time at ailure. Thereore, it is not guaranteed that these strengths will ollow the master curve with time, and is not appropriate to obtain the reduced strength corresponding to time t i regardless o the stress history applied to the elements. The correct way o interpreting and using the strength master curve is to obtain the time to ailure, t, corresponding to the stress components evaluated at the inite elements, as Figure 4 shows. When the applied loading type is creep, the elemental stress component corresponds to the creep stress level and is plugged into the creep strength master curve. When the applied loading type is atigue, the elemental stress component corresponds to the maximum stress level o the cyclic loading and is plugged into the atigue strength master curve or a given stress ratio. We can assume that the element subject to the applied stress level damages ractionally during the time segment, t i. By considering the time to complete ailure t subject to the same stress level, a damage raction (D i ) o the element during t i can be determined with a time ratio, t i =t. I the time segment o load duration were equal to t, the damage raction would become 1, which means complete ailure. Thereore, the range o the damage raction, D i, is rom zero or an undamaged state to unity or a completely damaged one. I the damage raction is less than 1, it is considered as a partial element ailure. In the partially damaged case, the damage raction is accumulated with the one obtained in the previous time steps. For simplicity, a linear cumulative damage (LCD) law was adopted in the present study. The key step in the MAE program is to calculate the partial damage ractions at each time increment and store them or iber and matrix constituents o all the inite elements until complete element ailure occurs. Figure 5 shows a detailed low diagram explaining how the MAE computes and updates the elemental damage ractions at time t i using

9 MAE: An Integrated Design Tool or Failure and Lie Prediction o Composites 1975 Increase time step: t i = t i 1 +Dt i s s Creep or atigue strength master curve FE analysis at time t i s m(e) Obtain macro stress/strain: MMM t (e) Log t For each element (e), get micro stress/strain: σ m(e), ε m(e) Damage : D (e) (t i ) = D (e) (t i 1 )+ raction Dt i t (e) Figure 5. A low diagram to show how to obtain damage raction at time t i using ATM and MMM. the ATM and the micromechanical calculation. For the time segment t i, the FE analysis is conducted with the updated ply properties at t i as obtained in Figure 3. The FE analysis will calculate the stress and strain components o all the elements. These components are macro-level stresses and strains o the ply in the homogenized elements. The micromechanical modiication (MMM) through either the analytical model or the numerical FE model is then used to ampliy the macrostresses into the microstress components in iber and matrix constituents o every inite element. These microstresses are used to calculate two stress invariants, vm and I 1. The microstress invariants are plugged into the strength master curves o ATM to obtain the time to ailure or the element t ðeþ. The procedures implemented in the MAE program to obtain the micro-level strength master curves rom the macro-level ones and to obtain the time to ailure t ðeþ rom the microstress invariants, vm and I 1, are detailed in the Appendix. As stated earlier, either creep or atigue strength master curves are used depending on the applied loading type. Note that while the creep strength master curve is represented with a single curve, the atigue strength master curves are ormulated with several curves corresponding to several requencies or number o cycles. Note also that the creep strength can be considered as a special case o atigue strength with the stress ratio R ¼ 1. Thereore, with the atigue strength master curves drawn or R ¼ 0, the t ðeþ at any R can be interpolated with two values o t ðeþ rom R ¼ 0 and R ¼ 1. The t ðeþ would be the time to ailure i the elements are subject to the stresses rom beginning to ailure. However, since the stresses are applied during the time segment t i, the damage raction during this time segment is calculated as t i =t ðeþ. This damage raction at the current time step can then be accumulated linearly with the one obtained in the previous time step in such a way that D ðeþ ðt i Þ¼D ðeþ ðt i 1 Þþ t i t ðeþ, ð1þ where D ðeþ ðt i 1 Þ and D ðeþ ðt i Þ are the damage ractions at the previous and the current time steps, respectively. Using the LCD law, we can judge whether the elements are sae or have ailed at this time step. I the cumulative damage raction is less than 1, the element is

10 1976 S. SIHN AND J. W. PARK considered as being still sae. I the cumulative damage raction becomes greater than 1, the element is considered as having ailed. Since the damage ractions are accumulated or each constituent o the elements, the mode o ailure can easily be identiied in the orm o either iber or matrix ailure. I the matrix damage occurs in the element, only the properties o the matrix degrade, while in the case o iber damage, all the properties o both iber and matrix degrade. The damaged elements are then accounted or in the subsequent FE analyses by either the property degradation approach o the MPDM or the nodal orce modiication o the EFM. Figure 6 summarizes an overall low chart or the MAE procedure. The program starts with an initial time t ¼ 0 and a time segment t i, so the irst time is set to t i. At this time, the constituent properties o iber and matrix are obtained rom the stiness master curve. The homogenized ply properties are then calculated using the micromechanical (MM) calculation. With the ply properties input, the FE analysis is conducted at the macro (ply) level. The post-processing o the FE analysis yields the macrostresses and macrostrains at time t i. Using the micromechanical modiication (MMM), we can obtain the microstresses and microstrains in the iber and the matrix. The stress invariants are calculated rom the microstresses. Using the MMF and its corresponding micro-level strength master curves rom the ATM, the elements microstress invariants are plugged into the strength master curves to obtain the time to ailure t. The damage raction is calculated and accumulated linearly or each constituent o all elements based on the LCD. When the accumulated damage raction is less than 1, the next time segment is adjusted. I the damage raction is greater than 1, the corresponding elements are considered as having completely damaged, triggering the EOD scheme to begin. We can degrade iber and/or matrix properties depending on the damage mode i the MPDM is used, or can modiy the nodal orces on the ailed elements or the EFM. Start Initialize time, time step (t=0, t i ) Time increment t i = t i 1 + t i For each element (e), Get micro-stress/strain σ m(e),ε m(e) (e) Obtain t o each element by ATM master curves lookup MMF ATM Element loop ATM MM Obtain constituent properties E,E m, at time t i MMMM Compute cumulative damage (e) (e) t D ti = D ti 1 + i (e) t LCD Obtain ply properties E 1,E 2, at time t i (e) D ti >1? 1 Yes EOD: Degrade property(mpdm)/ modiy nodal orce(efm) FE analysis And recover macro-response at time t i : σ (i), ε (i) No Adjust time segments, t i No End condition met? Yes Stop Figure 6. A low diagram or implementation o MAE.

11 MAE: An Integrated Design Tool or Failure and Lie Prediction o Composites 1977 In the current MAE program, we have implemented the MPDM only. For the MPDM, we assumed that the transverse Young s modulus and longitudinal and transverse shear moduli retained 1% o the intact moduli subject to the matrix damage, while all the moduli including the longitudinal Young s modulus retained 1% o the intact moduli subject to the iber and ull damage. The Poisson s ratios were arguably assumed to be unchanged subject to the damage. The sensitivity o the degradation actors and the eect o the Poisson s ratio are beyond the scope o this paper, but worth consideration in uture study. Ater the EOD scheme takes necessary actions on all the damaged elements, the time segment is adjusted or the next iteration. I any preset end condition is satisied, the program stops. Otherwise, the procedure will resume with the next time increment step, as Figure 6 shows. The entire procedure has been implemented into the commercial FE sotware, Abaqus, using the Python scripting eature o Abaqus, so that the FE calculation becomes reliable and maintainable. EXAMPLE CASES Several example cases will be shown to demonstrate the validity, capability, and usage o the MAE prediction. First, the validity o the multiple time step scheme will be studied by reproducing creep strength master curves. The second example is the damage prediction o an open-hole tension specimen under creep loading. It will be ollowed by a comparison o the present MAE method using the multiple time step scheme with a simplistic method without time stepping. The next example is the usage o the MAE or a study on the eect o loading sequence. The last example case is the damage progression o a double-edge notched specimen under creep loading. REPRODUCTION OF CREEP STRENGTH MASTER CURVES The irst example is the reproduction o the creep strength master curves using the multiple time step scheme. We have modeled unnotched smooth specimens with [0] and [90] lay-ups to simulate the experimental procedures o generating the creep strength master curves. Four modes o macro ailure were considered or the creep strength under longitudinal tension (X), longitudinal compression (X 0 ), transverse tension (Y), and transverse compression (Y 0 ). X and X 0 were obtained rom [0] specimens, and Y and Y 0 were rom [90] specimens. The material system used in this and the subsequent calculations were TR30S/epoxy consisting o a carbon iber and epoxy matrix. The stiness and strength properties and the master curves or the TR30S/epoxy were obtained rom experimental measurement by Miyano [30]. Figure 7 shows the our creep strength master curves. The creep strengths are normalized by their corresponding static strengths. The time scales are plotted up to 10 3 years o load duration. The dots are the MAE predictions using the multiple time step scheme. The good agreement or all our ailure modes shown here signiies that the MAE successully reproduced the creep strength master curves within this time period. Note that the X master curve is nearly horizontal during 100 years, which indicates a nearly elastic behavior, and that it is extremely hard or the ibers to ail by aging. Thereore, the present assumption o the elastic iber properties was appropriate.

12 1978 S. SIHN AND J. W. PARK S c /S c0 0.4 X creep master curve 0.2 X creep master curve Y creep master curve Y creep master curve MAE Time (year) Figure 7. Normalized master curves o creep strengths, X, X 0, Y, and Y 0 against duration o the creep loadings and their predictions by the present MAE. OPEN-HOLE TENSION UNDER CREEP LOADING The second example is a simulation o the damage progression in an open-hole tension specimen under constant creep loading. The laminate lay-up is [0/90] s. The FE model consists o 36,544 3-D solid elements and 55,812 nodes, and the total number o degrees o reedom is 167,436. The specimen is ixed at one side and loaded at the other side, as shown in Figure 8. Note that the FE meshes are aligned with the [0] and [90] lay-ups. The mesh sensitivity, such as the eect o unaligned meshes, was not considered in the present study but is worthwhile being treated in a uture study. Figure 9 shows the increase o the number o damaged elements and the maximum displacement subject to the increase o the load duration. The maximum displacement is calculated at the loaded side o the grip. It is observed that the damage grows signiicantly ater one year o creep loading duration. The maximum displacement at the loaded grip increases due to the element damage as well as the decrease o the stiness with time. The damage proiles at the upper side o Figure 9(b) indicate the splitting damage initiates near the hole edge and propagates in the loading direction as a dominant damage mode. Figure 10 shows the damage progression o the [0/90] s laminate under the creep loading ater 3, 10, 30 and 50 years o load durations. The damage proiles in both [90] and [0] layers are plotted. Dierent colors represent dierent damage modes o iber and matrix constituents. The splitting damage o the specimen grows in size with the increase o load duration subject to the creep load o constant magnitude. Note that typical microcracking damage near the ree edges at the sides o the specimens was not predicted in this MAE simulation since our interest ocused more on the hole edge rather than the ree edges, so the FE meshes were not reined enough to capture the microcracking damage starting at the ree edges.

13 MAE: An Integrated Design Tool or Failure and Lie Prediction o Composites 1979 Figure 8. An FE model or an open-hole tension specimen under creep loading. (b) (a) No. o damaged elements Time (year) Max. displacement at grip (mm) Time (year) Figure 9. (a) Number o damaged elements and (b) maximum displacement at loading grip against increase o load duration subject to a constant creep load on an open-hole tension specimen. 3 years 10 years 30 years 50 years [90] [0] Figure 10. Prediction o damage progression o a [0/90] s laminate under constant creep loading ater 3, 10, 30 and 50 years o load durations. MAE VERSUS SIMPLE METHODS As stated earlier, the MAE method uses the multiple time step scheme with the degraded stinesses and the damage ractions o the constituents updated at each time step.

14 1980 S. SIHN AND J. W. PARK Thereore, the damage proiles at earlier time steps aect the damage progression at the current and uture time steps. In contrast to the MAE method, we can take a simplistic approach without considering multiple time steps. In the simplistic approach, not only the degraded stinesses but also the degraded strengths are obtained rom the master curves directly or a given time, as Figure 11 shows. As stated earlier, the ormer is a valid way o reading the stiness master curves, while the latter may not be valid. I the reduction in stiness and strength are moderate, we expect that both methods would yield similar results. We used the open-hole tension specimen under the creep load to predict the damage progression near the hole edge with the increase o load duration, and compared two predictions made by the MAE method with the multiple time step scheme and the simplistic approach without the time step scheme. In the simplistic method, we made the prediction using the reduced stiness and strength at the applied time, so that the damage proiles at the earlier time steps do not aect the later damage progression. Figure 12 shows the damage progression in the [90] layer o the [0/90] s open-hole specimen under the creep loading ater 10, 30, and 50 years o load durations. As the igure shows, the MAE and the simplistic methods result in completely dierent damage patterns even at Stiness/strength master curve Reduced strength/stiness Applied time Log t Figure 11. A simplistic approach to obtain reduced stiness and strength properties rom the master curves at a given time. 10 years 30 years 50 years (a) MAE (b) Simple Figure 12. Prediction o damage progression o a [0/90] s laminate under constant creep loading ater 10, 30 and 50 years o load durations using (a) the MAE method and (b) the simplistic method.

15 MAE: An Integrated Design Tool or Failure and Lie Prediction o Composites years o loading. While the MAE predicts the growth o the splitting damage, the simplistic method results in the growth o transverse microcracking damage. In general, because o the multiple time step scheme, the MAE takes a longer calculation time than the simplistic method. However, in terms o accuracy in ailure and lie prediction, it is essential to check the results with the MAE and not to rely only on those rom the simplistic approach. Otherwise, the simplistic approach might mislead the structure and component design to a wrong direction. Here, the MAE can serve as a useul and more sophisticated tool to be complementary to simplistic, yet robust, tools such as the SMM. EFFECT OF LOADING SEQUENCE As seen in the previous example, the damage proiles at the earlier time steps can aect the damage progression at the current and uture time steps. Thereore, we can expect that the load sequence o the creep or the atigue loading can also aect the damage behavior. As a simple example, we considered two dierent sequences o creep loading: low high and high low. For a given average stress level,, load sequence 1 is the case o low high sequence, where a 75% level o the average stress was applied during the irst hal o a load duration, and then another 150% level was applied during the later hal. Load sequence 2 is the case o high low, where the 150% level was applied irst, and then the 75% level was applied later. Figure 13 shows the two dierent load sequences. Since the durations o the two applied stress levels are the same, the damage ractions based on the LCD law at the end o the each load sequence are expected be the same in both cases. Hence, without considering the damage history, the prediction will be exactly the same, irrespective o the sequences o loading. Figure 14 shows the increase o the damaged elements with the increase o load duration under two dierent sequences o the creep loads. It can be seen that the low high load sequence results in more damaged elements than the high-low case ater 30 years. It can be speculated that this is due to a act that the material properties at the ormer period are higher than those at the latter period. As a result, the damage sustained rom the high load at the ormer period has less eect on the integrity o the structure than that at the latter period. Figure 15 shows the comparison o the damage proiles under two sequences o loading. Load sequence 1 with low high sequence yields more damaged elements with longer splitting damage. It is worthwhile noting that although the LCD law was used, the MAE is capable o yielding the history-dependent damage progression, and thus capturing the dierences in the damage pattern subject to the dierent load sequences. (a) 1.5σ σ 0.75σ (b) 1.5σ σ 0.75σ 0.5t t 0.5t t Figure 13. Two dierent sequences o creep loading: (a) load sequence 1 (low high) and (b) load sequence 2 (high low).

16 1982 S. SIHN AND J. W. PARK Load sequence 1 (Low-High) Load sequence 2 (High-Low) No. damaged elements Time (year) Figure 14. Number o damaged elements against increase o load duration subject to two dierent sequences o constant creep loads on an open-hole [0/90] s tension specimen. (a) Load sequence 1 (low high) (b) Load sequence 2 (high low) (90) (0) Figure 15. Prediction o damage progression o a [0/90] s open-hole tension specimen ater 30 years o load duration under (a) low high and (b) high low load sequences. DOUBLE-EDGE NOTCHED SPECIMEN UNDER TENSILE CREEP LOADING The last example o MAE prediction is the double-edge notched specimen under tensile creep loading. Figure 16 shows the geometry and the meshes o the model. By taking advantage o symmetry, we modeled an upper hal portion only and applied a symmetric boundary condition along a symmetry line. One side o the specimen was clamped, and the other side was subject to a creep load at a constant stress level. The FE meshes were reined toward the notch tip. The laminate lay-up was [0/90] s. Figure 17 shows the damage proiles in [0] and [90] lay-ups near the notch tip ater 1, 3 and 25 years o the creep load duration, and the increase o the number o damaged elements

17 MAE: An Integrated Design Tool or Failure and Lie Prediction o Composites 1983 Clamped BC Y Z X Symmetric BC Surace traction (150 MPa) Figure 16. Geometry and FE meshes o a double-edge notched specimen under tensile creep loading. (0) (90) 1 year 3 year 25 year No. damaged elements No. damaged elements E+00 1.E+01 1.E+02 1.E+03 1.E E E E E E+06 Time (minutes) 1.E+05 1.E+06 1.E+07 1.E+08 Log t (minutes) 1.0E E E+07 Figure 17. Prediction o damage progression o a [0/90] s double-edge notched specimen under tensile creep loading ater 1, 3 and 25 years o load durations using the MAE. with the increase o load duration. An experimental observation made by Beaumont [31] with X-ray images o the double-edge notched specimens taken at various load levels indicates similar damage proiles with the splitting damage at the notch tips. Note that the X-ray images were taken with the specimen under static loading, while the present prediction was made or the creep specimen. Although it was not possible to make a direct comparison due to the diiculty in conducting the creep tests or the duration o 1 25 years, the resemblance o the damage pattern indicates that the MAE can be a useul tool to capture the damage progression in the notched composites having the stress concentrations.

18 1984 S. SIHN AND J. W. PARK SUMMARY AND CONCLUSIONS An advanced strength and lie prediction tool, MAE, was developed or the analysis and design o composite structures and components. The MAE, based on numerical FE analysis, treats the creep and atigue loading as multiple time steps o the quasi-static loading. The property degradation and the damage due to the load duration are handled by the modulus and strength master curves rom the ATM, respectively. The damage ractions are calculated by the ailure criteria rom the MMF. The damage evolution beyond the initiation o the damage is accounted or by the EOD scheme. The MAE can handle the long-term creep and atigue strengths o various materials, laminate lay-ups, complex shapes such as open hole, illed hole, bonded and bolted joints, stiened panels, textile composites, etc., subject to combined loadings and hot wet environmental conditions at various requencies and stress ratios. Thereore, the MAE can serve as a useul tool to predict damage initiation, progression and lie under various durability loading and environmental conditions. The MAE also can serve as a useul design tool in selecting not only the material and laminates, but also optimal geometrical conigurations, and help reduce the case-by-case time- and cost-consuming (sometimes impossible) testing during the preliminary design stage. The MAE modules were implemented and integrated with the commercial FE sotware, Abaqus, or better reliability and maintainability. Note that the MAE is not meant to replace or eliminate the preliminary design tool, SMM. Rather, it serves as a complementary tool to the SMM. By combining the robustness o SMM and the versatility o MAE, the composite design will be more complete and reasonable. Several examples were presented to demonstrate how the MAE prediction can be utilized in the design o the composite structures. The creep strengths o the unidirectional [0] and [90] laminates were predicted well with MAE with the multiple time step scheme at several applied stress levels. The damage initiation and progression o the cross-ply laminate [0/90] s with the open hole were predicted by the MAE or several decades o tensile creep loading. The MAE also demonstrated distinct damage patterns under two dierent sequences o creep loads on the open-hole tension [0/90] s laminates. The low high load sequence causes more damage to the open-hole specimen than the high low one. The damage progression predicted by the MAE near the double-edge notched tip was indirectly compared well with the experimental observation. APPENDIX Construction o Micro-level Strength Master Curves or Constituents by Back-Calculation Composites properties are usually measured in the ply level, and thereore, the microlevel properties may not be available in many cases. Furthermore, it is sometimes diicult to directly measure the strengths o the constituents, especially at elevated temperatures, because o the viscoelastic nature o the material. For example, neat resin materials would not yield or break under compressive loading at high temperatures. For the MMF, however, it is necessary to generate the micro-level strength master curves or the iber and matrix constituents. In the present study, we used the micromechanical calculation to back-calculate the microstrength master curves or iber and matrix constituents rom the ply-level macrostrength master curves. The back-calculation procedure or the

19 MAE: An Integrated Design Tool or Failure and Lie Prediction o Composites 1985 microstrength master curves o matrix material will be briely summarized here. A similar procedure can be derived to produce those o the iber material. For a given time to ailure, t tensile and compressive macrostrengths, Y and Y 0, are obtained rom their corresponding macrostrength master curves. By the micromechanical modiication (MMM), microstresses can be obtained rom the macrostresses corresponding to the macrostrengths. The microstresses are evaluated at several locations in the matrix constituent, and yield two microstress invariants, ðiþ I 1 and ðiþ vm, at each point (i). Tensile and compressive microstrengths, T m and C m, o matrix material at time t can then be determined by iterating through the microstress invariants or all points and inding the lowest pair o strengths that satisy the MMF criteria [2,4]: vm þ ð C m T m Þ ðiþ I 1 C m T m 50: ð2þ ðiþ2 The above procedure repeats or dierent values o t. These discrete values o T m and C m microstrengths or all t are then interpolated to orm their microstrength master curves o the matrix constituent. In the present study, a cubic spline was employed or the interpolation task. Figure 18 summarizes the above procedure o obtaining the microstrength master curves o matrix constituent rom the macrostrength master curves. Calculation o Time-to-Failure rom Microstress Invariants using Constituent Master Curves The MAE calculation requires the determination o the time to ailure or each element, by which the damage ractions o the element is accumulated. The macrostress components o each element obtained rom the quasi-static FE analysis are ampliied by the MMM calculation to yield the microstresses and their invariants, ðiþ I 1 and ðiþ vm, Input: Macrostrength master curves (Y and Y ) as unction o t Output: Constituent microstrength master curves (T m and C m ) as unction o t For each t {t,1,t,2,t,3,...,t,n } For given t, obtain σ Y (t ) = [0,Y(t ),0,0,0,0] T macrostresses rom macrostrength σ Y (t ) = [0, Y (t ),0,0,0,0] T master curves. Obtain microstress invariants by micromechanical modiication (MMM). Find T m and C m that yield the lowest value o a unction orm or MMF criteria rom microstress invariants at all points. (i)2 (i) σ nm + (C m T m )σ I1 C m T m < 0 Interpolate discrete (T m, C m ) using c-splines. (i) (i) σ I1 (t ) and σ nm (t ) (i = 1,2,3,...,n p ) Macrostrength master curves {T m (t, 1 ), T m (t, 2 ),...,T m (t, n )} {C m (t, 1 ), C m (t, 2 ),...,C m (t, n )} Y Y(t ) Y Y (t ) MMM t t Time Time (i) Figure 18. A low diagram to show how to construct microstrength master curves o matrix constituent rom macrostrength master curves.

20 1986 S. SIHN AND J. W. PARK at several locations in the constituents. The time to ailure t ðiþ at each point (i) can then be determined by solving the MMF equation: h i F t ðiþ ¼ ðiþ2 vm tðiþ þ C m t ðiþ T m t ðiþ t ðiþ C m t ðiþ T m t ðiþ ¼ 0, ð3þ where T m ðt ðiþ Þ and C m ðt ðiþ Þ are obtained rom the microstrength master curves o the matrix constituent at time t ðiþ. Since the t ðiþ cannot be obtained explicitly rom Equation (3), iterative root-inding methods, such as secant and Brent s methods [32], are used to solve or the t ðiþ. A minimum value among the t ðiþ computed or every predetermined point in the micromechanics model is taken as the time to ailure o the element. This procedure repeats or all the inite elements. Thereore, this process can become computationally expensive with the increment o local points and the reinement o FE meshes. It should be noted that during the time period t ðiþ, the macrostresses in the element kept constant in accordance with the meaning o the macrostrength master curves. However, the microstresses are not constant but deviate rom their initial states because o the change in the constituents moduli over time, especially the matrix properties. To account or the variations o microstresses, the above procedure must repeat until t ðiþ converges, where each subsequent process utilizes the microstresses and their invariants recomputed with the degraded moduli rom previously obtained time to ailure. In this calibration step, only the critical location in the micromechanical model is considered. This additional procedure improves the accuracy o t ðiþ with the penalty o computational cost. To alleviate the computational burden, the microstresses as well as the macrostresses were assumed to be constant with respect to time in the present study, so that ðiþ I 1 ðt i Þ ðiþ I 1 ðt ðiþ Þ and ðiþ vm ðt iþ ðiþ vm ðtðiþ Þ, where t i is the current time at the MAE time steps. Not reported here, a sensitivity study showed that the assumption o the constant microstresses has negligible eect on the estimated ailure lie within a reasonable time period (e.g., 100 years) due to the multiple time stepping algorithm o the MAE. Figure 19 summarizes the above procedure o obtaining the time to ailure o the elements. ðiþ I 1 σ (e) Macrostresses in an element By micromechanical modiication (MMM), obtain microstresses and their invariants (i) σ (e) (i = 1,2,3,...,n p ) (i) (i) σ νm, σ l1 Find root o the equation (solve or t ) using root-inding methods (i.e. Secant, Brent s Method, etc.) T m MMM (i) Microstrength master curves (i)2 σ νm (t (i) )+[C m (t (i) ) T m (t (i) )] σ (i) l1 (t (i) ) T m (t (i) ) Repeat or next element C m (t (i) ) T m (t (i) )=0 Microstresses are assumed to be constant C m C m (t (i) ) t (i) Time Find time to ailure o element t (i) Time t = min (t (1),t (1),t (1),...,t (n p ) ) Figure 19. A low diagram to show how to obtain time to ailure o elements using MMF and ATM.