Computational tools for the fatigue life modeling and prediction of composite materials and structures

Computational tools for the fatigue life modeling and prediction of composite materials and structures

18

Anastasios P. Vassilopoulosa, Julia Maier b, Gerald Pinter b, Christian Gaierc a Ecole Polytechnique F
ed
erale de Lausanne (EPFL), Composite Construction Laboratory (CCLab), Lausanne, Switzerland, bMaterials Science and Testing of Polymers, Montanuniversitaet Leoben, Leoben, Austria, cMagna Powertrain, Engineering Center Steyr GmbH&CoKG, St. Valentin, Austria

18.1

Introduction

As it is clearly described in the literature and other chapters of this volume, fatigue of composites has been recognized as a critical phenomenon activating damage mechanisms that affect the durability of composites and structures made of them, since very early, for example, Ref. [1], Chapter 1. Significant experimental programs have been established and a wide range of experimental investigations have been conducted in order to investigate the fatigue behavior of several types of composite materials, identify the mechanisms that cause failure, and eventually quantify their effects on the structural integrity of the examined components [2–5]. In parallel to the experimental investigations, an abundance of models have been established in order to simulate the exhibited behavior; actually to fit the material experimental behavior and provide mathematical representation of it in order to allow interpolation between the experimentally defined ranges. Typical examples of such models are the S-N curve algorithms that have been introduced, already since 1910 by Basquin [6, 7] or those models trying to simulate the evolution of the fatigue stiffness with loading cycles, both under constant amplitude fatigue loading patterns. In a second stage, more sophisticated models were introduced for the prediction of the material behavior under “unseen” loading cases, see, for example, Refs. [8–10] for fatigue failure criteria, or Refs. [11–13] for constant life diagrams (CLDs). Such theories are described as empirical, or phenomenological in the literature [8, 14, 15]. In contrast to the phenomenological modeling approaches, models based on the measurement and the evolution assessment of actual damage mechanisms into the material have been introduced and used in progressive damage modeling algorithms. Appropriate damage evolution laws have to be derived and calibrated, by using experimental evidence, before such models are able to provide any type of information regarding the material’s performance under cyclic loading. Models have been Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00018-8 © 2020 Elsevier Ltd. All rights reserved.

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Fatigue Life Prediction of Composites and Composite Structures

developed at different scales, based either on fracture mechanics [16–19] and/or the estimation of S-N curves at the laminate, for example, Refs. [4, 17], the lamina, or the fiber/matrix levels [10, 20, 21] in order to fulfill this requirement. Application of the progressive damage models is iterative; once the failure criterion (at any selected scale level) is satisfied a number of properties of the failed elements are degraded followed by the calculation of the new stress field (stress redistribution). The procedure is repeated until the failure condition is fulfilled. Depending on the method, the failure condition could be the final failure of the laminate, when a specific number of cycles is reached, or when a certain value of the damage metric is achieved, for example, 10% stiffness degradation (Chapter 15, Ref. [22]), or a predefined crack length [23]. In some cases, combinations of the aforementioned conditions are utilized in order to calibrate the model parameters and aim for more reliable fatigue behavior predictions, for example, Ref. [24]. Initially, such theories were developed mainly based on the constant amplitude and block loading fatigue data, and their application was limited to the simulation of constant amplitude fatigue behavior for several composite systems, since variable amplitude (VA) data were very limited to allow validation. However, over the last three decades, more fatigue data have been produced in laboratories concerning fiberreinforced composite materials in order to examine their behavior under realistic loading situations, including VA fatigue loading and complex environmental conditions, for example, Refs. [2–4, 25–30]. In parallel to the experimental work, theoretical models are developed for the simulation of the fatigue behavior of the examined materials under different thermomechanical loading conditions and the prediction of material fatigue life under complex stress states that may arise during the operation of a structure in the open air [8, 29, 31–33]. A fatigue life prediction methodology is usually based on the development of empirical relationships between the applied loads and the fatigue lifetime of the examined materials. Such relationships can be developed at different scales depending on the followed approach. The implementation of a numerical procedure for fatigue analysis consists of a number of distinct calculation modules, related to life prediction. Some of these are purely conjectural or of a semiempirical nature, for example, the failure criteria, presented in Refs. [17, 32], while others rely heavily on experimental data, for example, S-N curves and CLDs [7, 11, 34]. In cases of composite laminates under uniaxial loading, leading to uniform axial stress fields, the situation can be substantially simplified since almost all relevant procedures could be implemented by experiment. On the other hand, for complex stress states, the laminated material is considered as being a homogeneous orthotropic medium and its experimental characterization, that is, static and fatigue strength, is performed for both material principal directions and in-plane shear [35]. This laminate approach (also cited as the building block testing approach [36]) is a straightforward process for predicting fatigue strength under plane stress conditions, avoiding the consideration of damage modeling and interaction effects between the plies and stress redistribution, and can be reliably used when limited stacking sequence variations are present in a structural element. The building block approach, used together with analysis, is considered essential to the qualification/certification of

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composite structures due to their sensitivity to out-of-plane loads, the multiplicity of composite failure modes and the lack of standard analytical methods [36]. The approach was implemented by Philippidis and Vassilopoulos [3, 32, 37, 38] for a glass/polyester multidirectional laminate of [0/
45] stacking sequence and was shown to yield satisfactory predictions for fatigue strength predictions under complex stress conditions for both constant amplitude and VA loading. A straightforward algorithm must be followed comprising steps dealing with the analysis of the load to determine the developed stress fields, the interpretation of the fatigue data of the examined laminates to derive the S-N curves and the corresponding CLDs, the fatigue failure criteria of the calculation of the design allowables, and finally the damage accumulation based on the selected damage rule. In large composite structures, consisting of numerous different materials and laminate configurations, a lamina-to-laminate approach seems more appropriate, although requiring the development of additional calculation modules able to take into account the implications in local stress fields, stress redistribution in neighboring plies, and finally, how damage propagates as a function of loading cycles, for example, Refs. [17, 35, 39]. According to this, more refined approach, the material properties of basic building plies need to be experimentally derived. The properties of any new laminate configuration, consisting of the basic building plies, are then estimated based on the existing theoretical procedures. The failure analysis is based on a progressive damage modeling, considering failed layers and stress redistributions in the laminate according to the applied load history. The analysis of structures based on the fatigue experimental data from substructural elements leads inevitably to lower scales where, for example, the fatigue behavior of a laminate is estimated (predicted) when information about the fatigue behavior at the level of the layer (as described in the previous paragraph) or of the fiber and the matrix is available. This approach goes one level lower in scale, when compared to the “lamina-to-laminate approach” described above, reaching the microscale level. Structural laminate properties are broken down to the ply level and eventually to the constituent material microscale level. At this basic level, the model taking microscopic defects into account, calculates the stress fields developed in both the fibers and matrix, and using appropriate failure criteria defines local failure, and depending on the method used is capable of identifying the failure mode as well. Then, based on a combination of micro-mechanics and laminate theories, the model is rebuilt up to the laminate in order to provide as output an estimation of the laminate lifetime, for example, Ref. [24]. The objective of the aforementioned theoretical approaches is to replace, in a way, the need for excessive experimental campaigns and develop virtual testing environments, which, after validation can be used to satisfy a multitude of tasks. Validated analytical/numerical procedures can be used, in between others, for: l

l

The simulation of the material’s response under selected loading patterns, in terms of S-N curves, stiffness degradation models, and CLDs. The lifetime estimation under unseen loading patterns, including irregular realistic spectra, through the use of appropriate fatigue failure criteria, and damage rules.

638 l

l

Fatigue Life Prediction of Composites and Composite Structures

The simulation of the damage progression, identifying damage modes, and predicting eventual material failure. Material selection/optimization, especially through bottom-up micro-mechanical multiscale approaches able to assist in selecting the appropriate mix of materials, as well as joining techniques.

Virtual testing serves for reducing physical prototypes and complicated experimental campaigns. Nevertheless, the existence of validated fundamental modeling and analysis methods does not mean that testing is not necessary. It rather implies that less validation against actual data will be required at complex and large-scale structural levels [20]. Development of numerical and analytical methods for life modeling and prediction of composite materials, structural elements, and complete structures goes hand in hand with experimental campaigns. Valid experimental data are always necessary in order to validate the developed theoretical methodologies in order to be able to be used for virtual testing of different loading and material configuration scenarios. The drawback of virtual testing is the lack of confidence due to the lack of validation, as well as the lack of certification processes for virtual testing environments [20], especially given the scarcity of structural failure data against which to benchmark. Empirical, phenomenological methods and especially progressive damage methods have not yet received enough verification and validation to be used for structural analysis. In a comparison of composite damage computational tools performed in 2017 [40], it was revealed that on average blind predictions of the behavior of notched laminates under constant amplitude fatigue at R ¼ 0.1, from all models, differed by 42% from the available experimental data. This error was reduced to around 18% only when a series of model parameters was recalibrated. Neither compression nor spectrum cycling loading cases were investigated in this study. This uncertainty appeals for additional efforts for the improvement of the accuracy of fatigue life modeling and prediction procedures in order to have such tools accepted for fatigue structural designs. It is obvious that independent of the adopted approach for the life estimation, implementation of the necessary steps into a computer program is indispensable. Software products have been developed to assist the research community in design processes using methods that are based either on the micromechanics modeling of fatigue life, for example, Autodesk Helius PFA [21] or on macromechanical approaches, such as the phenomenological laminate approach, for example, CCfatigue [33]. This chapter presents some of the existing computational tools for fatigue life modeling/prediction of composite materials and structures and discusses their potential to be used as reliable virtual testing tools in the future.

18.2

Engineering software for fatigue life modeling/ prediction

Computational tools for the modeling of the fatigue behavior and the lifetime prediction of composite materials have been established during the last decades. Such computer programs are either in-house developed software packages aiming to cover

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specific demands of research institutes and/or industries, or modules in commercial software packages aiming to cover a wider range of applications. In between these two, there are also tools that although have been developed with a purely research outlook, became mature enough after been validated against a series of experimental data and are in a pre-commercial development phase. In composites, the engineers, simultaneously to the structural design, have the ability to design the material itself. Therefore, the design process has to go through an optimization procedure, in which such analysis tools can play a significant role by allowing engineers and manufacturers to try several potential designs and focus on the optimal solution (or at least limited number of solutions) instead of undergoing physical testing of too many options. Computer simulation tools should be able to define where material is needed, and where is too much. They should either estimate fatigue life, or define critical load cases in order to refine the design and designate and validate accelerated fatigue testing procedures. Plenty of software frameworks able to perform in some extend fatigue life predictions are available. A family of tools is based on a multiscale progressive damage analysis approach, for example, GENOA (http://alphastarcorp.com/genoa/), the Multiscale Designer (MDS-C) (https://altairhyperworks.com/product/Multiscale-Designer), or the Fe-safe/composites (http://www.digitaleng.news/pics/pdfs/fe-safe_Composites.pdf) that uses a multi-continuum theory developed by Helius:Fatigue. (All web sites accessed on January 2019.) The idea of such tools is the development of constitutive models for the description of fatigue damage at the microscale level and then, in order to analyze largescale components through multiscale decomposition procedures. Models of this family can be accurate enough when appropriate experimental data are available in order to fit all model parameters from the micro to the macroscale. After successful parameter calibration procedures, those models can, quite accurately, predict the right type, amount and location of damage as a function of the fatigue cycles [40]. Nevertheless, when the available experimental data for calibration are limited, model predictions can be significantly different from experimental results. It is well documented today that the accuracy of the fatigue progressive damage models is still low for predicting the response of composite laminates, while their performance has yet to be validated for other loading cases, such as under compression and spectrum loading profiles [40]. A less data demanding family of models rely on macro mechanical/phenomenological approaches, for example, FOCUS6 (https://wmc.eu/focus6.php), CCfatigue, finite element fatigue (FEMFAT) (see later in this chapter). These theories are based on constitutive models for the material description at the lamina (e.g., FEMFAT), the laminate, or the structural component level (e.g., CCfatigue). The former case implements a lamina-tolaminate approach in order to use the experimental information at the lamina level in order, through the appropriate modeling, to simulate the fatigue behavior of laminates made of the same laminae. The theories of the latter case are more straightforward using the so-called “building block” approach, and can be used for design verification and life assessment of materials and structures under any random spectrum, given that enough constant amplitude fatigue data are available. Appropriate failure criteria are employed by these theories in order to take into account the effect of complex, multiaxial fatigue stress states on the fatigue life of the examined material, or structural component.

640

18.3

Fatigue Life Prediction of Composites and Composite Structures

FEMFAT laminate approach

For metallic materials, theories based on the fatigue strength, usually represented by stress vs number of cycles curves (S-N curves), are widely spread and have been implemented successfully in software tools for fatigue life prediction. One software tool is the FEMFAT developed by Magna Powertrain Engineering Center Steyr GmbH & Co KG (St. Valentin, Austria) [41]. In contrast to the studies published for composite materials so far, a very comprehensive, engineering approach is used. The real part geometry, quasi-static, and fatigue material data reflecting effects on the material behavior, the applied load-time history caused by the application, and local stresses calculated by finite element (FE) analysis are taken into account. For each node of the FE mesh, local S-N curves are predicted [42, 43]. Critical damages are calculated according to the critical plane concept [44–46]. Thereby, damage accumulation is performed for all planes at defined angles, at each node. The plane, in which the calculated damage reaches a maximum, is considered as critical. The equivalent stresses occurring in the critical planes are classified by rainflow-counting. Subsequently, damages are calculated based on the local S-N curves and accumulated to the total damage sum. This software tool has been successfully adapted for fatigue life prediction of orthotropic materials [47]. For injection molded short-fiberreinforced plastics, anisotropic material behavior and effects caused by the injection molding process can already be taken into account. The functionality of simulation chains from injection molding simulation to lifetime prediction has been presented and validated in different studies [48–51].

18.3.1 Fatigue life prediction method for laminates To meet the fatigue characteristics of continuously fiber-reinforced composites, the fatigue solver FEMFAT has been extended with a new module for lifetime estimation of laminates. Within this software tool for laminates, standard methods for the assessment of metallic parts based on the S-N curves have been adapted for laminates. In order to take the characteristic damage modes of composite materials into account, the three failure modes; fiber failure (FF), interfiber failure (IFF), and, optionally, delamination according to Puck are included in the software. For each ply of the laminate, the lifetime prediction is performed. For the assessment of FF, the stress history of the normal stress σ 1 longitudinal to the fiber orientation is calculated by linear superimposition of in general multiaxial load channels. A rainflow counting algorithm is applied to obtain an amplitude-meanrainflow-matrix of closed load cycles. Subsequently, the partial damages are analyzed by using experimentally measured material S-N curves and are linearly accumulated according to Palmgren/Miner [52, 53]. For the IFF modes, the same procedure is performed for the normal stress σ 2 transverse to the fiber orientation and for the in-plane shear stress τ21 and the respective material S-N curves in the fatigue life software. To apply Puck’s criterion also combinations of σ 2 and τ21 have to be considered. Nevertheless, for nonproportional loading, the stress vector spanned by σ 2 and τ21 may

Computational tools for the fatigue life modeling and prediction

641

change its direction with respect to time. It is difficult to apply a rainflow counting procedure in such a case. To solve this problem, a simplified version of the so-called “Critical Plane-Critical Component” approach was developed [40, 46, 54]. The stress vector is projected onto several fixed directions with the given unit ! vector v i : !

!

σ eqv ðtÞ ¼σ ðtÞ  ν i ¼ σ 2 ðtÞ cos φi + τ21 ðtÞsin φi , i ¼ 1, …, N

(18.1)

For each orientation direction, rainflow counting and damage analysis of the resulting equivalent stress can be performed without any restrictions. The direction, which delivers the maximum damage, is assumed critical for fatigue failure. It can be mathematically interpreted as the critical component of the stress vector and it defines the type of failure mode A, B, or C. To assure that each mode is covered by at least one direction, an angle of 30 degree is used between directions as default, leading to six directions in the σ 2-τ21-plane as illustrated in Fig. 18.1. For the damage analysis of the intermediate directions S-N curves are used, which are obtained by interpolation between the S-N curves for σ 2 and τ21. To consider the influence of the mean stress, CLDs are constructed from quasistatic and cyclic material parameters such as ultimate tensile strength, ultimate compressive strength, ultimate shear strength, alternating and pulsating fatigue limit for a given number of cycles as, for example, 5  106 [52, 53]. Due to the different strengths under tensile and compressive loads of composite materials caused by the anisotropic macroscopic material behavior, usually highly asymmetric CLDs are obtained for normal stresses, although for shear stresses, the CLD is symmetric. For intermediate directions, which cannot all be covered by experiments, the CLDs are interpolated with a smooth transition between tensile-compressive loading at an angle of φ ¼ 0 degree and shear loading at φ ¼ 90 degree, resulting in a constant life surface as illustrated in Fig. 18.2. The static limitations of the constant life surface at the left-hand

Modus/mode B Modus/mode C

t21

c

Modus/mode A PA⊥⎥⎥

c P ⊥⎥⎥ C

A

t 21c= R ⊥⎥⎥ 1+2 P⊥⎥⎥ – R⊥⊥ /R⊥⎥⎥

RA⊥⊥

0

s2

Fig. 18.1 Fatigue assessment of different load directions in the laminate plane.

642

Fatigue Life Prediction of Composites and Composite Structures Modus/mode B Modus/mode C

t21

c

Modus/mode A l p⊥⎥⎥

c p⊥⎥⎥

t21c = R⊥⎥⎥

c A 1 + 2p • R /R ⊥⎥⎥

⊥⊥

⊥⎥⎥

j s2

s

fl ⊥

ngle

j [de

g]

0°

100

ra Pola

Endurance limit [Mpa]

A R⊥⊥

0

t fl

90°

0 0

–200

90

Mean stress [Mpa]

Fig. 18.2 Constant life surface for IFF according to Refs. [55–57] with an interpolated constant life line (red) for a given angle φ of the considered loading direction.

side (green line in compressive domain) and right-hand side (yellow line in the tensile domain) are determined by Puck’s criterion in the software routine. The intralaminar stresses σ 1, σ 2, and τ21, which are acting in the plies, can be analyzed by the finite element method (FEM) with shell elements, whereas for interlaminar stresses σ 3, τ32, and τ31, which are acting between plies and causing delamination, solid elements are needed, which is even more expensive. For the fatigue assessment of interlaminar stresses, the same procedure is used as for intralaminar stresses with different S-N curves and CLDs. In Fig. 18.1, σ 2 is replaced by σ 3 and τ21 by τ32 and τ31, respectively. Nevertheless, it is difficult to measure S-N curves for interlaminar stresses. Interlaminar shear stresses can be generated by a three-point bending load applied to specimens [58]. According to Ref. [59] a high load in fiber direction reduces the strength for IFF by the following elliptical weakening factor with the two material parameters s and a: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   σ 1 =Rk  s 2 ηw ¼ 1  σ 1 =Rk > s: a

(18.2)

For cyclic loading, the weakening factor may change during a load cycle because σ 1 is generally a function of time. Therefore, the IFF strength is not unique for the damage

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643

Fig. 18.3 Fatigue analysis procedure for a laminate structure.

analysis of the cycle. To overcome this theoretical problem, the stress components σ 2 and τ21 will be increased by the time-dependent weakening factor instead (σ0 2, σ0 21) and assessed with the original non-reduced IFF strength [58]: σ 02 ðtÞ ¼

σ 2 ðt Þ ηW ðtÞ

τ021 ðtÞ ¼

τ21 ðtÞ ηW ðtÞ

(18.3)

The fatigue analysis procedure for a laminate component is exemplified in Fig. 18.3. Up to six nested loops are required in order to cover the different scale levels of whole structures. All laminate elements resulting from the FE analysis, the nodes at the elements and the individual plies need to be considered. Furthermore, fatigue analyses have to be performed for each stress component at top and bottom side of each ply. To take the anisotropic material behavior into account, the different stress components σ 1, σ 2, and τ21 are included. To enable the consideration of various, realistic time ranges, rainflow counting is applied. For acceleration of the procedure appropriate filters have been implemented to select only highly stressed plies for the fatigue analysis. In addition, a parallel analysis on several CPUs is possible. For this case, the component is automatically divided into several parts, which are distributed to the CPUs for parallel computing. After finishing, the results are automatically merged together.

18.3.2 Experimental work Unidirectional (UD) laminae made of carbon fibers and epoxy resin were tested at angles of 0, 45, and 90 degree. Layups consisting of
45 degree and [0°/+45°/45°/90°]S layers were also investigated and both layups were symmetric referring to the middle plane. The fiber volume content of all produced specimens was 55% (measured by thermogravimetric analysis as published in Ref. [60]). UD 0 degree plies consisted of four layers; all other specimens were made of eight layers.

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Fatigue Life Prediction of Composites and Composite Structures

The specimens’ geometry used in quasi-static tensile and tension-tension fatigue tests was 200  10  1 mm (length width  thickness) for UD specimens in fiber direction and 200  20  2 mm for all other specimens. For quasi-static compression and tension-compression fatigue tests, specimens’ geometry was 110  10  2 mm (UD 0 degree) and 110  20  2 mm, respectively. Aluminum tabs (length: 50 mm, thickness: 1 mm) were glued on both sides of all specimens. Quasi-static and fatigue tests were performed on a servo-hydraulic test machine equipped with a load frame and load cell for 100 kN by MTS Systems Corporations (Minnesota, USA) at room temperature. Hydraulic wedge pressure of 5 MPa was chosen in order to prevent slipping without damaging the specimens. Good adhesion between aluminum tabs and carbon/epoxy specimens was assured in preliminary tests. Gauge length was 100 mm for tensile and 10mm for compression loads. Quasi-static tension and compression tests were performed with a test speed of 0.5 mm/min until failure. During quasi-static tests, a digital image correlation (DIC) system by GOM (Braunschweig, Deutschland) was used for strain measurement in and transverse to fiber direction in order to calculate Poisson’s ratios. Tensile moduli were calculated according to Ref. [61]. Compressive moduli were evaluated between 0.001 and 0.003 absolute strain according to Ref. [55]. Shear moduli were evaluated as in-plane shear response in tensile tests with
45 degree specimens [56]. Tension-tension fatigue tests were performed with the R-value (¼minimum force/maximum force) of 0.1, tension-compression fatigue tests with R ¼  1 until total failure. For the creation of S-N curves, specimens were tested at four different stress levels. Maximum cyclic stresses between approximately 80% and 65% of the ultimate tensile strengths were chosen as load levels for UD 0 degree and between 60% and 35% for the tested off-axis specimens in tensiontension fatigue tests. In tension-compression fatigue tests, the maximum cyclic tensile stresses applied were between approximately 25% and 15% of the ultimate tensile strengths for UD 0 degree specimens and between approximately 65% and 25% for off-axis specimens [62]. A minimum of three specimens were tested on each stress level. Specimens tested on the lowest stress level, which did not fail, were manually stopped after approximately 2  106 cycles. Specimens’ temperatures were monitored by infrared sensors in all fatigue tests. Test frequencies were chosen between 2 and 10 Hz depending on the load amplitude and the specimens’ tendency for hysteretic heating in order to limit the temperature increase to a maximum of 5°C and consequently minimize the influence on the materials’ properties [63]. Results of fatigue tests were evaluated statistically to calculate the slope of the S-N curves k, scatter with Ts and the nominal stress amplitude after 5  106 cycles σ a,5106 according to Ref. [64].

18.3.3 Experimental results 18.3.3.1 Test results for quasi-static loading The results of quasi-static tensile and compressive tests are presented in the way they were used as input parameters for the software tools. Material parameters in fiber direction are summarized in Table 18.1, mechanical properties transverse to fiber direction are presented in Table 18.2, and shear properties evaluated with

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Table 18.1 Quasi-static input parameters for FE analysis and fatigue-life prediction evaluated in quasi-static tension and compression tests in fiber direction Young’s modulus in fiber direction [GPa] Elastic Poisson’s ratio [] Elongation at rupture [%] Ultimate tensile strength [MPa] Ultimate compressive strength [MPa]

107.0 0.34 1.35 1550 549

Table 18.2 Quasi-static input parameters for FE analysis and fatigue-life prediction evaluated in quasi-static tension and compression tests transverse to fiber direction Young’s modulus transverse to fiber direction [GPa] Ultimate tensile strength [MPa] Ultimate compressive strength [MPa]

5.5 33 89

Table 18.3 Quasi-static shear input parameters for FE analysis and fatigue-life prediction evaluated in quasi-static tension tests with
45 degree specimens In-plane shear modulus G12 [GPa] In-plane shear strength [MPa]

3.3 74


45 degree specimens in Table 18.3. The shear moduli G13 and G23 necessary for the FE analysis could not be measured experimentally. Based on the assumption of transversally isotropic material behavior of UD plies, the shear modulus G13 was assumed to be equal to the measured shear modulus G12 [59]. The third shear modulus G23 was set as 2.0 GPa based on the relation between the three shear moduli in calculated material data for UD carbon/epoxy plies from literature [59]. In order to calculate the values of the shear moduli for the material used herein, micromechanical modeling based on the measured material parameters could be used [65].

18.3.4 Results of fatigue tests Fatigue data of UD 0 degree, UD 90 degree, and
45 degree were used as input parameters for the fatigue life software. The experimentally measured S-N curves for R ¼ 0.1 and R ¼  1 at room temperature are illustrated in terms of nominal stress amplitudes σ a vs number of cycles in Fig. 18.4. For statistical evaluation of the measured fatigue data, a linear model according to Eq. (18.4) was used. The parameters of the linear model were calculated as maximum likelihood estimators according to the ASTM E739 [64], assuming that: all fatigue life data pertain to a random sample and are independent, there are no run-outs, the S-N curve can be described by a linear model, a two-parameter log-normal distribution describes the fatigue life and the variance is constant. The maximum likelihood estimators A^ and B^ were calculated

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Fatigue Life Prediction of Composites and Composite Structures

Fig. 18.4 S-N curves of 0, 90, and
45 degree coupons tested at stress ratios R ¼ 0.1 and R ¼  1 at room temperature until failure, which were used as input parameters for the fatigue software.

according to Eqs. (18.5) and (18.6). X and Y describe the average values of the log Si and the log Ni, respectively, taking the total sample size m into account [64]. For further calculation of arbitrary data points in the S-N diagrams, Eq. (18.7) was used where σ 1 > σ 2 and N1 < N2 and k represents the statistically evaluated slope of the S-N curve. The values for the slope k, scatter width Ts and nominal stress amplitude after 1 cycle σ a,0 and after 5  106 cycles σ a,5106 of the S-N curves are summarized in detail in Table 18.4. log N ¼ A + Blog S

(18.4)

^ A^ ¼ Y  BX

(18.5)

Xm  B^ ¼

   X  X ∗ Y  Y i i i¼1 Xm   Xi  X 2 i¼1

 σ1 ¼ σ2

N1 N2

(18.6)

 1 k

(18.7)

Tension-tension fatigue amplitudes (R ¼ 0.1) were higher for specimens in fiber direction than alternating amplitudes (R ¼  1) as a result of the quasi-static ultimate compressive strength being approximately 30% of the ultimate tensile strength

Computational tools for the fatigue life modeling and prediction

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Table 18.4 Input S-N curves for the fatigue-life prediction Specimen

R [2]

k [2]

Ts [2]

σ a,0 [MPa]

σ a,5×106 [MPa]

UD 0 degree UD 90 degree
45 degree UD 0 degree UD 90 degree
45 degree

0.1 0.1 0.1 1 1 1

24.9 11.1 17.0 13.4 8.0 13.5

1/1.14 1/1.15 1/1.13 1/1.41 1/1.58 1/1.45

808.0 19.2 73.2 782.5 29.0 136.3

435.2 4.8 29.5 248.4 4.2 43.4

Slope k, scatter width Ts, the nominal stress amplitude after 1 cycle σ a,0 and the nominal stress amplitude after 5*106 cycles σ a,5*106 for R ¼ 0.1 and R ¼ 1 evaluated from experimental fatigue tests.

(Table 18.1). On the contrary, alternating amplitudes were higher than tension-tension amplitudes for
45 degree specimens. For specimens tested transverse to fiber direction, the mean stress did not significantly influence the fatigue strengths. For the implementation in the fatigue life prediction software, the respective maximum cyclic stresses for R ¼ 0 and R ¼  1 after 5  106 cycles and the slopes of S-N curves were required. Therefore, fatigue data measured at R ¼ 0.1 and the mean stress effect were used to calculate pulsating tension fatigue strength for R ¼ 0 of 928.2 MPa (in fiber direction) and 9.95 MPa (transverse to fiber direction). Shear fatigue strength after 5*106 required by the fatigue software was calculated by dividing the fatigue strengths of
45 degree specimen by two [56]. Consequently, pulsating shear fatigue strength for R ¼ 0 was 30.8 MPa. The alternating fatigue strengths were equal to the stress amplitudes σ a,5106 measured at R ¼  1 in Table 18.4. For verification of the described fatigue analysis method, fatigue tests with pulsating loading (R ¼ 0.1) of two additional laminate configurations for UD 45 degree and for a multilayer composite [0°/+45°/45°/90°]S were performed. Test fatigue results of the two layups used for validation are illustrated in comparison to the maximum cyclic stresses of UD 0 degree, UD 90 degree, and
45 degree in Fig. 18.5. The S-N curves measured in tension-tension fatigue tests are illustrated in terms of maximum cyclic stress vs number of cycles in order to ease the comparison with the respective quasi-static tensile strengths. Slope k, scatter width Ts, and the calculated nominal stress amplitudes after 5  106 cycles σ a,5106 of the S-N curves can be found in Table 18.5.

18.3.5 Simulation results 18.3.5.1 FE analysis For the FE analysis of deformation and stresses with the FE solver ABAQUS, the specimens were modeled with linear quadrilateral shell elements. In the ABAQUS input file, the thickness and material of each ply and also the orientation of the fibers are defined in a shell section with the attribute COMPOSITE. The used stiffness parameters are shown in Tables 18.1–18.3. To obtain correct clamping conditions,

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Fatigue Life Prediction of Composites and Composite Structures

Fig. 18.5 S-N curves of 0, 45, 90,
45 degree, and [0°/+45°/45°/90°]S coupons tested at stress ratio R ¼ 0.1 until failure [66]. Table 18.5 Verification S-N curves for the fatigue-life prediction Specimen

R [2]

k [2]

Ts [2]

σ a,0 [MPa]

σ a,5*106 [MPa]

UD 45 degree Multilayer

0.1 0.1

9.2 13.0

1/1.12 1/1.21

56.7 360.3

10.6 109.8

Slope k, scatter width Ts, the nominal stress amplitude after 1 cycle σ a,0 and the nominal stress amplitude after 5*106 cycles σ a,5*106 for R ¼ 0.1 evaluated from experimental fatigue tests.

also the aluminum tabs were modeled with shell elements and connected to the laminate with tie contacts. For modeling of the aluminum tabs, tensile modulus of 70 GPa and Poisson’s ratio ν ¼ 0.34 were assumed. The FE mesh of the UD 0 degree specimens’ geometry consists of 2000 quadrilateral shell elements for the carbon fiberreinforced laminate and additional 2000 quadrilateral shell elements for the aluminum tabs. For UD 45 degree and UD 90 degree specimens, the number of elements is doubled according to the double width of 20 mm, leading to a total number of 8000 elements.

18.3.5.2 Fatigue life prediction For fatigue lifetime prediction with FEMFAT, an input material dataset was generated as a first step. Quasi-static material properties as presented in Tables 18.1–18.3 were used. Furthermore, the fatigue strengths for alternating and pulsating loading in and

Computational tools for the fatigue life modeling and prediction

649

–∞

=0

=

7 5

8

10 9

1

1000

= –∞

60

=0

R

R = –1

0

R

0

4

R

R

500

R = –1

transverse to the fiber orientation and for shear loading were implemented. Based on those data, CLDs were constructed. In contrast to the behavior known from metallic materials, the tensile mean stresses had a positive effect on the fatigue life for loading in fiber direction which corresponded to the high quasi-static material properties in fiber direction (Table 18.1). As a result of the different quasi-static strengths under tensile and compressive loads, the CLDs for loading longitudinal and transversal to fiber direction appeared asymmetric. In contrast to that the CLD for shear loading appeared symmetric (Fig. 18.6). Subsequently, fatigue life predictions were performed for the UD 0 degree, UD 90 degree, and
45 degree specimen in order to check the validity of the input data. The check was assessed successfully if the fatigue simulation produced the same S-N curves as the ones measured experimentally. A good fit with experimental test results could be achieved for UD 0 degree and UD 90 degree specimens as illustrated with red lines drawn in comparison to the experimental results (Fig. 18.7 and Table 18.6). For the two implemented load cases, tension-tension and tension-compression, the produced input data in and transverse to fiber direction fitted the experimental fatigue strengths very well. For
45 degree, the damage distribution along the specimens calculated by the software was not as homogeneous as for UD 0 degree and UD 90 degree specimens. The disturbed damage distribution in the area of the clamping of the specimen is illustrated in Fig. 18.8. This effect might be caused by the two different

0

30

60

R –∞

R

=

=0

0

R = –1

–60

0 –60

0

Fig. 18.6 Constant life diagram for loading in fiber direction (top), shear CLD (middle), and CLD for loading transverse to the fiber orientation (bottom).

Fatigue Life Prediction of Composites and Composite Structures

Nominal stress amplitude sa (log.) [MPa]

650

Fatigue tests CFRP 55% fibre volume content j = 2–10Hz, R 0.1, R-1, RT

UD 0° 55% R0.1 ±45° R0.1 UD 90° 55% R0.1

UD 0° 55% R-1 ±45° R-1 UD 90° 55% R-1

1000

R0.1

R-1

100

Result at middle position for R-1

Result near clamping position for R0.1 Result at middle position for R0.1 R0.1

10

R-1

1 102

103 104 105 Number of cycles N (log.) [cycles]

106

107

Fig. 18.7 Validation of simulated input parameters (red lines) with experimental input parameters. Table 18.6 Simulation results for UD 0 degree, UD 90 degree and
45 degree Specimen

R [2]

k [2]

σ a,0 [MPa]

σ a,5*106 [MPa]

UD 0 degree UD 90 degree
45 degree middle
45 degree clamping UD 0 degree UD 90 degree
45 degree

0.1 0.1 0.1 0.1 1 1 1

13.4 8.0 13.5 13.5 13.4 8.0 13.5

1311.5 34.4 91.2 80.9 745.5 28.2 136.1

414.8 5.0 29.1 25.8 235.8 4.1 43.4

directions of layers within the specimen, +45 degree and 45 degree, resulting in a stress introduction into the specimen different from uniaxial specimen. However, the obtained damage distribution reflected the layup of the laminates and corresponded to the failure locations monitored in the tested coupons. To address this effect, two different input parameter sets were tested, one corresponding to the area of clamping and one to a position in the middle of the specimen. Consequently, the correlation between calculation and experimental results was best in the undisturbed middle region of the specimen which was used as input for subsequent fatigue life prediction (Fig. 18.7). For
45 degree, the results depended on the position as shown in Fig. 18.8. Based on the input parameter sets validated for R ¼ 0.1 and R ¼  1, the fatigue life of the UD 45 degree specimens, in which all layers were aligned at the same angle of 45 degree, and of a multiaxial layup with the stacking sequence [0°/+45°/45°/90°]S

Computational tools for the fatigue life modeling and prediction

651

Fig. 18.8 Comparison of simulated damage distribution with the experimental failure of a
45 degree coupon at stress ratio R ¼ 0.1.

Fig. 18.9 Comparison of simulated damage distribution with the experimental failure of a UD 45 degree coupon at stress ratio R ¼ 0.1. The pink area is the area with highest damage.

were predicted subsequently. Due to the UD layup of the UD 45 degree specimen and the fixed clamping, asymmetric stress distribution resulting in inhomogeneous damage distribution was obtained (Fig. 18.9). For this case, not the shear stress was responsible for fatigue failure as supposed by intuition, but the normal stress transverse to the fiber direction for both investigated locations at the middle of the specimen and at the

Fatigue Life Prediction of Composites and Composite Structures

Nominal maximum stress σmax (log.) [N/mm2]

652

10,000

Fatigue tests CFRP 55% fibre volume content j = 3–10Hz, R 0.1, RT

UD 0° UD 45° UD 90° ±45° [0/45/–45/90/symm.]

1000

Ply-45°

Inside ply 90°

Ply-45°

100 Result at middle position Result at clamping

10 102

103

104 105 106 Number of cycles N (log.) [cycles]

107

Fig. 18.10 Comparison of simulation results (red lines) with test results for UD 45 degree and multilayered composite [0°/+45°/45°/90°]S.

clamping position. Consequently, the best correlation with experimental results was again obtained for the middle position as presented in Fig. 18.10 and Table 18.7. It can be assumed, that the stresses at the clamping position were too high because of singularity effects at the edges, where aluminum tabs were connected to the laminate. Only a very detailed modeling of notch radii with finite solid elements may increase the accuracy at such positions, which nevertheless would be too much effort in daily engineering practice. For the multilayered composite, the predictive situation was quite different. Due to the functional principle of analyzing the laminate ply by ply, interactions between the plies influencing in the total behavior of the entire laminate were not taken into account. Consequently, the fatigue life prediction assuming that the 90 degree ply at the middle plane of the [0°/+45°/45°/90°]S specimen was critical for failure which resulted in predictions underestimating the test result for total failure (red line in

Table 18.7 Simulation results for UD 45 degree and multi-layer composite [0°/+45°/45°/ 90°]S Specimen

R [2]

k [2]

σ a,0 [MPa]

σ 6a,5*10 [MPa]

UD 45 degree middle UD 45 degree clamping Multilayer 90 degree ply Multilayer 45 degree ply Multilayer +45 degree ply

0.1 0.1 0.1 0.1 0.1

8.0 8.0 8.0 8.0 8.0

70.1 53.6 266.1 422.9 507.5

10.2 7.8 38.7 61.5 73.8

Computational tools for the fatigue life modeling and prediction

653

Fig. 18.11 Experimental failure of [0°/+45°/45°/90]S carbon/epoxy coupon after fatigue test at stress ratio R ¼ 0.1.

Fig. 18.10 and Table 18.7). If continuing the fatigue life calculation after the assumed failure of the 90 degree plies,
45 and 0 degree would still carry load until the neighboring plies with 45 degree orientation and subsequently the plies with +45 degree orientation would fail, schematically drawn in Fig. 18.10. At last, only the 0 degree plies would carry the load according to the simulation. Therefore, the failure criterion for the analysis was different from test: while the initial crack in the weakest ply defined the failure for analysis, the experimental tests were performed until fracture of the whole compound (Fig. 18.11). However, it is difficult to detect the initial crack during the fatigue test in practice, because cracks may start somewhere inside the compound and can usually not be found with surface investigations [66]. For the simulation of the total fracture, stiffness degradation will have to be considered in an iterative way.

18.3.6 Example: Multifunctional truck cross member As example for a component, a truck cross member is presented made of a combination of continuous fiber-reinforced plastic and steel. Beside the function as an important structure of the truck main frame, the part represents a liquid/compressed natural gas (CNG) reservoir at the same time. The biggest challenge from a design point of view was the fixing within the truck main frame structure with steel brackets (Fig. 18.12). The cross member was made up of several functional layers. The inner layer of polyethylene high density (PEHD) makes the vessel airtight. During production, this liner is used as a core for the 90 degree winding and for the braiding process of the carbon fiber-reinforced plastic (CFRP) outer layers. The 90 degree winding carries the main pressure load from inside and the braided layers take over the longitudinal loads from the steel calottes and the forces from the cross member function. For the ABAQUS analysis, a simplified composite shell model was used consisting of five UD plies, which represents the axial yarn and the braider yarns [67].

654

Fatigue Life Prediction of Composites and Composite Structures

Fig. 18.12 Truck cross member with steel brackets.

For the fatigue life assessment with FEMFAT laminate, two load cases were examined and compared with component tests: a sinusoidal bending and a sinusoidal torsion load pulsating between zero and maximum load (stress ratio R ¼ 0). The simulations gave a good agreement with regard to failure location, however, the predicted lifetime was much too low. The reason, of course, is that FEMFAT identifies failure based on the first failure in the 90 degree plies, what is not conform with the failure criterion in the component test, what is already a quite large delamination after detection. For every ply, the critical stress component was the normal stress σ 2 perpendicular to the fiber orientation. For both load cases, the critical ply was on the outside [58]. The results for torsion are shown in Fig. 18.13.

Fig. 18.13 Component testing with torsion load: observed crack (A) and simulated damage distribution for the most critical ply (B).

Computational tools for the fatigue life modeling and prediction

655

18.3.7 Summary A fatigue life prediction method for laminates under random multiaxial loads, based on the S-N curves derived at the lamina level has been presented in this section. For the modeling, quasi-static and fatigue data of a UD carbon/epoxy laminate were measured longitudinal and transverse to the fiber directions under tensile and compressive loads and also for shear loading to characterize the anisotropic fatigue behavior of the material. The obtained experimental data were used as input parameters for the fatigue solver FEMFAT laminate which enables the assessment of fiber and interfiber fracture with ABAQUS composite shell elements, and recently delamination with ABAQUS composite solid elements, even for nonproportional loading. Fatigue life of a UD 45 degree laminate and a multilayer composite were calculated and validated with experimental results. As example for a structural component, a truck cross member was presented with test and simulation results. As a first step, the cyclic input material data were assessed in order to check the validity of the calculations. The simulated input parameters correlated well to the experimental inputs in and transverse to fiber direction and also for
45 degree orientation. However, clamping positions should be excluded from fatigue assessment, because finite shell element models could not represent the physical conditions adequately. Instead, solid elements can be used; however, they necessitate expensive numerical procedures and the experiences to be gained. The fatigue life of the UD 45 degree could be predicted very accurately. For the fatigue life calculation of the multilayered composites with the stacking sequence [0°/+45°/45°/90°]S, the fatigue software assumed failure as soon as the weakest 90 degree ply became critical (similar to the weakest link concept), whereas tests were performed until total failure which lead to a significant difference between simulation and test. However, it could be shown that considerations beyond this point can lead to improved predictions. For the accurate software-based prediction of total failure of the complete composite structure, additional research and development activities will be necessary. Especially stiffness degradation caused by fatigue-induced damage mechanisms must be considered in an iterative way [57, 68]. Stress and fatigue analyses could be conducted with iteratively adapted stiffness parameters. Mathematical models will have to be developed to describe the reduction of stiffness parameters in dependence of the damage evolution. In addition, delamination will have to be considered by taking into account stress components perpendicular to the laminate’s plane. However, only finite solid elements are able to deliver these stress components with sufficient accuracy.

18.3.8 Nomenclature Definitions according to [69]: R k t, R k c … tensile and compressive strength of UD lamina parallel to fiber direction. R ? t, R ? c … tensile and compressive strength of UD lamina transverse to fiber direction. Rk? … in-plane shear strength of UD lamina. RA?? … fracture resistance of an action-plane action parallel to the fiber direction against its fracture due to τ?? stressing acting on it.

656

Fatigue Life Prediction of Composites and Composite Structures

p k t, p k c … inclination of (τ21, σ 2)-fracture curve at σ 2 ¼ 0. t for the range σ 2 > 0 (tension). c for the range σ 2 < 0 (compression).

18.4

Description of CCfatigue and case studies

A classical fatigue life prediction methodology based on the building block approach is presented in this section. The computer software in which this procedure has been implemented allows the estimation of the fatigue lifetime of UD or multidirectional laminates and structural components under multiaxial stress states of VA, when basic quasi-static and fatigue data are available. The procedure is capable of performing different tasks in an articulated method aiming the life prediction of each examined material under given loading patterns. The steps for the procedure include routines for cycle counting, the derivation of S-N curves, the establishment of appropriate CLDs, the estimation of the fatigue failure under multiaxial fatigue stress states, and finally the calculation of the failure index by using the linear Miner damage summation rule. A wide range of solvers has been implementing for each step of the procedure, allowing the benchmarking of selected methods and the selection of the most appropriate for each subproblem of the life prediction methodology. The efficiency of the CCfatigue software has been already validated in previous works (Chapter 11, Refs. [8, 33]) by comparisons of the theoretical predictions to available experimental data from two material systems. A third dataset for which data from quasi-static, constant amplitude, as well as VA loading patterns are available [25] is used in this chapter for the demonstration of the software and the reassessment of its predicting ability. Data from previously used datasets [8, 25] are used for the demonstration of the CCfatigue software for the fatigue life prediction under multiaxial stress states.

18.4.1 Datasets description The first experimental program used in this chapter for the demonstration of the CCfatigue software refers to the fatigue behavior of a material fabricated by a filament winding technique at
30 degree [25]. Specimens were cut from a GFRP (E-glass fibers with Epon826 epoxy resin) cylinder with a large enough diameter to supply samples that could be considered as flat. Results from quasi-static as well as fatigue loading patterns are provided. Two sets of constant amplitude fatigue data, at tensiontension, R ¼ 0.1, and reversed fatigue, R ¼  1, can be retrieved from Ref. [25]. In addition, two sets of VA fatigue results are also available; one referring to experiments under the WInd SPEctrum Reference (WISPER) spectrum [70], and a second one referring to fatigue data obtained after applying the WISPERX spectrum. The development of the WISPER standardized irregular spectrum was supervised by the International Energy Association (IEA) and a number of European industrial partners and research institutes active in the wind energy domain. The spectrum was based on the measurements of bending moment loadings at different sites and for

Computational tools for the fatigue life modeling and prediction

657

different types of wind turbine rotor blades. A total number of nine different wind turbines with rotor diameters of between 11.7 and 100 m and rotor blades made of different materials such as metals, GFRP, wood, and epoxy resins were considered. The WISPER standardized spectrum consists of 132,711 loading cycles, a high percentage of which are of low amplitude. Therefore, a long period is needed for the failure of a specimen under this spectrum and it is for this reason that the WISPERX spectrum was derived. WISPERX is a short version of WISPER containing approximately 1/10th of the WISPER loading cycles, while theoretically producing the same damage as its parent spectrum. The WISPERX spectrum comprises 12,831 loading cycles since all cycles with a range of below level 17 of WISPER have been removed. All data are summarized in Table 18.8, and graphically presented in Fig. 18.14 (constant amplitude fatigue) and in Fig. 18.15 (VA under WISPER and WISPERX).

18.4.2 Multidirectional laminate [0/(
45)2/0]T Another dataset is used for the demonstration of the software ability to provide fatigue life predictions under multiaxial stress states and compare relevant theoretical models. The dataset is taken from Ref. [8] and consists of a series of S-N fatigue data from specimens cut from a multidirectional laminate at different angles. Experiments at different R-ratios were carried out to cover several cases between compressioncompression, tension-tension, and reversed fatigue loading. The material system was E-glass/polyester, with E-glass fibers being supplied by Ahlstrom Glassfibre and the polyester resin, Chempol 80 THIX, by Interchem. A systematic experimental investigation was undertaken, consisting of static and fatigue tests on straight edge specimens cut at various directions from a multidirectional laminate. The stacking sequence of the E-glass/polyester plate consisted of four layers, 2 UD, UD lamina of 100% aligned warp fibers, with a weight of 700 g/m2 as outer layers and two stitched laminae with fibers along
45 degree directions, of 450 g/m2, 225 g/m2 in each off-axis angle. Considering the UD layer fibers as being along the 0 degree direction, the layup can be encoded as [0/(
45)2/0]T. Specimens were cut using a diamond wheel at 0 degree, on-axis, and 15, 30, 45, 60, 75, and 90 degree off-axis directions. The fatigue results are presented in Figs. 18.16–18.19. Details about the experimental program and the data analysis can be found in Ref. [8] In addition to the constant amplitude fatigue experiments, 30 specimens cut at 0, 30, and 60 degree were experimentally investigated under a modified version of the WISPEX spectrum, denoted MWX in Ref. [8]. MWX is a shifted version of the original to produce only tensile loads. The lower positive stress level is the first level of the original WISPERX time series, in which level 25 is considered as zero stress level. The same number of cycles as WISPERX, that is, 12,831, is maintained. In all, 15 specimens at 0 degree were loaded at three different maximum stress levels, 10 specimens cut at 30 degree also at three different maximum stress levels, and five specimens cut at 60 degree at two different stress levels. The experimental results are presented in Fig. 18.20. The off-axis VA fatigue data will be used in the following paragraphs for the demonstration of the CCfatigue software ability to predict fatigue life of composite materials under multiaxial stress states of VA.

Table 18.8 Quasi static and fatigue data—FFA material CA, R 50

CA, R 51

WISPER

WISPERX

UTS (MPa)

σ max (MPa)

N

σ max (MPa)

N

σ max (MPa)

N

Spectrum passes

σ max (MPa)

N

spectrum passes

275.6 273.3

112.5 105.0 84.4 84.4 84.2 79.0 79.0 67.4 56.6 56.4 56.1 52.6 52.6 52.6 50.7 50.6 50.5 45.0 42.1 42.1 42.1 39.2

400 110 1000 700 1300 940 400 4900 32,000 29,900 80,300 2354 2095 1700 92,000 327,900 359,300 660,400 245,000 230,000 14,200 4,175,800

142.1 141.2 140.9 140.6 114.3 112.9 112.9 112.9 112.7 105.3 105.3 105.3 101.4 97.1 92.1 92.1 92.1 90.3 90.3 90.2 84.7 84.7

445 4200 2640 4600 89,800 46,300 35,100 31,300 6200 3940 3810 2000 125,770 768,700 198,000 112,000 46,700 2,685,200 1,841,500 227,009 602,100 567,900

217.1 217.1 194.3 194.3 194.3 194.3 194.3 194.3 194.3 194.3 194.3 182.9 171.4 171.4 171.4 171.4 171.4 171.4 171.4 171.4 171.4 160.0

232,000 194,000 871,000 761,000 725,000 725,000 675,000 536,000 437,000 327,000 149,000 3,645,000 3,380,000 2,981,000 2,319,000 1,920,000 1,671,000 1,471,000 1,390,000 1,123,000 991,000 2,531,000

1.75 1.46 6.56 5.73 5.46 5.46 5.09 4.04 3.29 2.46 1.12 27.47 25.47 22.46 17.47 14.47 12.59 11.08 10.47 8.46 7.47 19.07

171.4 171.4 171.4 171.4 171.4 171.4 171.4 171.4 171.4 148.6 148.6 148.6 148.6 148.6 148.6

438,194 289,044 258,928 135,061 122,230 77,565 70,900 66,275 33,282 986,119 598,577 422,455 263,450 221,448 160,724

34.15 22.53 20.18 10.53 9.53 6.05 5.53 5.17 2.59 76.85 46.65 32.92 20.53 17.26 12.53

39.2 39.2 36.8 36.8 36.8 36.8 36.8

4,153,400 3,975,500 2,530,000 2,200,000 1,540,000 443,000 280,000

84.5 79.0 79.0 79.0 79.0 78.7 78.7 68.0 65.8 65.8

779,200 391,000 111,000 93,600 61,000 4,286,000 136,300 11,254,000 1,310,000 596,000

148.6 148.6 148.6 148.6 148.6 137.1 137.1 137.1

11,475,000 10,281,000 8,157,000 4,344,000 3,909,000 11,398,000 9,143,000 8,450,000

86.47 77.47 61.46 32.73 29.45 85.89 68.89 63.67

660

Fatigue Life Prediction of Composites and Composite Structures

Fig. 18.14 Tensile strength and constant amplitude fatigue data—FFA material [25].

Fig. 18.15 Variable amplitude fatigue data—FFA material [25].

18.4.3 CCfatigue software application As mentioned in the previous paragraphs of this chapter, CCfatigue follows the building block approach and a classical fatigue life prediction procedure that comprises a number of sequentially executed modules. The process is based on an articulated algorithm consisting of a pre-processor for handling the input date, four to five basic steps

Computational tools for the fatigue life modeling and prediction

661

Fig. 18.16 Fatigue data and Power S-N curves under compression-compression loading, R ¼ 10 [8].

Fig. 18.17 Fatigue data and power S-N curves under reversed loading, R ¼ 1 [8].

(according to the needs), and a post-processor to produce (numerically and graphically) the output, as presented in Fig. 18.21: l

l

Cycle counting, to convert VA time series into blocks of certain numbers of cycles corresponding to constant amplitude and mean values. Interpretation of fatigue data to determine and apply the appropriate S-N formulation for the examined material.

662

Fatigue Life Prediction of Composites and Composite Structures

Fig. 18.18 Fatigue data and power S-N curves under tension-tension loading, R ¼ 0.1 [8].

Fig. 18.19 Fatigue data and power S-N curves under tension-tension loading, R ¼ 0.5 [8]. l

l

l

Selection of the appropriate formulation to take the mean stress effect on the fatigue life of the examined material into account. Use of the appropriate fatigue failure criterion to calculate the allowable number of cycles for each loading block that results after cycle counting, when multidirectional stress states develop. Calculate the sum of the partial damage caused to the material by each of the applied loading blocks estimated using the cycle counting method.

Computational tools for the fatigue life modeling and prediction

663

Fig. 18.20 Fatigue data under MWX spectrum loading [8].

Modeling constant amplitude fatigue behavior involves the determination of the S-N curves (plot of cyclic stress vs life), typically by grouping data at a single R-value. Interpretation of the fatigue behavior for the assessment of the mean stress effect results in the construction of the CLD. These two processes can be treated as separate steps, but are related in the sense that the CLD is constructed from the available S-N curves, or S-N curve fatigue data, while new S-N curves can be estimated from the CLD. In case of multiaxial stress states, for example, when laminates are loaded off-axis, it is necessary to derive CLDs for each off-axis angle, namely for each combination of the stress tensor components. An experimental campaign for a task like that is extremely laborious, only to cover a limited amount of selected cases, thus, constituting the implementation of fatigue failure criteria indispensable. Detailed information regarding the steps that concern the cycle counting methods, the interpretation of the fatigue data, CLDs, fatigue failure criteria, and damage accumulation rules, can be found in Ref. [8]. The CCfatigue software accommodates all the necessary modules to address all the aforementioned steps of a life prediction methodology through a sequential articulated procedure, as well as to perform individual calculations for the material behavior modeling. The welcome screen of the current version of the CCfatigue software is presented in Fig. 18.22. The five successive steps are shown at the left side with tabs. The user can run each step independently of the others, select the desired solver from those available in the software library, and obtain the solution. Results from different methods can easily be created for comparisons. The data structure of the output file for each step has been designed in such a way that it can be used as the input file for the following step. By adopting a modular infrastructure, each component of the software can be updated without any substantial amount of modifications to the existing software interface. New routines can be easily implemented in the software library, increasing the number

664

Fatigue Life Prediction of Composites and Composite Structures

Variable amplitude spectrum

Static strengths & constant amplitude experimental fatigue data

Input

Processing

S-N Curve Interpretation of the CA fatigue data

Cycle Counting Convert VA load time series to CA blocks

Fatigue Failure Multiaxial stress states

Yes

Multiaxial fatigue failure criteria

No CLD Assess the effect of stress-ratio on the fatigue life

Design Allowables Partial damage coefficients

Damage Summation Miner or non-linear damage coefficient

Damage Index – life assessment

Output

Fig. 18.21 Flowchart of the CCfatigue software.

of solvers for each step and allowing the benchmarking of different methods established by different research groups, and strengthening collaborations. The application of the software for the life prediction of the selected materials and the demonstration of the modules for each step are presented in the following sections.

Computational tools for the fatigue life modeling and prediction

665

Fig. 18.22 CCfatigue software welcome screen.

18.4.4 Variable loading fatigue lifetime prediction The irregular loading patterns under which the material performance has to be predicted are traditionally subjected to a cycle counting analysis in order to count the occurrences of cycles of various characteristics (amplitude, mean value, etc.) in the spectrum. There are several methods for doing this analysis; four of the most commonly used are implemented in the CCfatigue software framework. The cycle counting analysis results to a matrix, containing groups of cycles with certain range and mean levels. A cumulative percentage of spectrum cycles (showing the percentage of the measured cycles with ranges more than given values) in the WISPERX spectrum is presented in Fig. 18.23. An alternative representation for the cycle counting results is shown in Fig. 18.24, where the numbers of cycles (the size of the bubbles) for certain range and mean stress values can be depicted by this bubble chart. As expected, see Fig. 18.24, >90% of the counted cycles have a stress range <20 MPa, although possessing different mean cyclic stress values and therefore, “belong” to fatigue data with different R-ratios. The material constant amplitude fatigue behavior should be also modeled after analyzing the loading pattern. In this case, the experimental fatigue data are analyzed in order to derive appropriate S-N curves. Several S-N models exist in the literature for the modeling of the CA fatigue life of composite materials [7]. Simple linear

666

Fatigue Life Prediction of Composites and Composite Structures

Fig. 18.23 Cycle counting of the WISPERX spectrum with different methods.

smean (MPa)

50

40

30 15

20

25

30

Δs (MPa)

Fig. 18.24 Bubble chart showing blocks of cycles with given range and mean values.

Computational tools for the fatigue life modeling and prediction

667

regression techniques (also introduced in ASTM standards—Ref. [71]) as well as more sophisticated methods for the statistical interpretation of composite’s fatigue data [72, 73] have been implemented in the CCfatigue software. Application of these methods to the available experimental data allows the derivation of the S-N curves, as shown, for example, in Fig. 18.25, for the specimens examined under tension-tension constant amplitude fatigue at an R-ratio of 0.1. As shown in Fig. 18.25, selection of different S-N curve model can lead to significant differences in fatigue life prediction, especially when the models are used to estimate fatigue lives at the low or at the high-cycle fatigue regimes. It is also obvious that extrapolation of the models, outside the regions where experimental fatigue data exist should be performed with caution. For the case shown in Fig. 18.25, for example, extrapolation of the exponential S-N curve [called also Lin-Lin S-N curve, expressed by the equation: σ max ¼ A + B log(N)] after 109 cycles can “predict” material failure at zero stress levels. For the material dataset examined here, “Whitney” model seems to be less conservative when extrapolating predictions at the high-cycle fatigue region for this material.

Fig. 18.25 Example of output plot for S-N curves by CCfatigue software.

668

Fatigue Life Prediction of Composites and Composite Structures

Predictive tools are necessary in order to estimate the fatigue life of the examined material under different S-N curves, for which experimental data are not available. CLDs reflect the combined effect of mean stress and material anisotropy on the fatigue life of the examined composite material. Furthermore, they offer a predictive tool for the estimation of the fatigue life of the material under loading patterns for which no experimental data exist. The main parameters that define a CLD are the mean cyclic stress, σ m, the cyclic stress amplitude, σ a, and the R-ratio. A large number of methods have been programmed in CCfatigue software for this task. The detailed theoretical background of all used methods together with comparisons to a wide range of experimental data can be found in Refs. [8, 11]. The CCfatigue set-up & Run tab for the estimation of the CLDs is presented in Fig. 18.26, with the output window, presenting the piecewise linear CLD for the examined material, shown in the embedded screen shot. Different CLD models can be derived, while different S-N curve models can be used as the input for each one of these models. It is obvious that the selection of an accurate CLD formulation is essential for the overall accuracy of a fatigue life prediction methodology. As shown, the “wrong” choice can produce very conservative or very optimistic S-N curves, which is directly reflected in the corresponding life assessment.

Fig. 18.26 Set-up & Run tab for constant life diagrams—the available methods are shown in the drop down menu—Output in the embedded screen shot.

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Fig. 18.27 CCfatigue software interface for “set-up & Run” the damage accumulation module.

Damage summation is performed during the last step of the CCfatigue software framework—a typical “set-up & Run” tab is presented in Fig. 18.27. The combination of the solvers for the solution of each step of the articulated methodology can be selected in this interface. A fatigue damage index is the output of this module, corresponding to the damage accumulated in the material after application of the selected fatigue spectrum. The linear Miner rule has been implemented in the current version of the software. The output is saved in a text file and can be graphically visualized in terms of spectrum passes vs maximum stress level, as presented in Fig. 18.28 for the WISPER spectrum and in Fig. 18.29 for the WISPERX spectrum. As shown in both Figs. 18.28 and 18.29, the selected methods for solving each step of the methodology were able to provide quite accurate lifetime predictions for this material. The selection of the power S-N curve together with the other chosen methods provides accurate results for the WISPERX spectrum (see Fig. 18.29) while is conservative when predicting the WISPER fatigue lifetime at high stress levels, as presented in Fig. 18.28. Nevertheless, when the S-N curve based on the method of Whitney [73] is used, the predictions for the WISPER spectrum are also as accurate as those for the

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Fatigue Life Prediction of Composites and Composite Structures

Fig. 18.28 Theoretical prediction vs experimental data; WISPER spectrum.

Fig. 18.29 Theoretical prediction vs experimental data; WISPERX spectrum.

WISPERX spectrum. This is because WISPER contains a large number of cycles with low stress level (which have been removed in WISPERX) and if one observes Fig. 18.25, where the S-N curves are compared, all methods, including the so-called power provides S-N curves that are conservative at the high-cycle fatigue region, except the “Whitney” method.

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18.4.5 Multiaxial fatigue lifetime prediction As mentioned above, the CCfatigue software can also be used for the fatigue analysis of composite laminates under multiaxial (irregular) stress states. In such situations, fatigue failure criteria should be derived and implemented in the software. Implementation of multiaxial fatigue failure criteria requests additional material data to be experimentally derived (usually enough fatigue data for the S-N curves along the principal material axes and for shear [8]). When such datasets are available, the multiaxial fatigue failure criteria are capable (at a certain degree of accuracy) to predict fatigue behavior under any off-axis angle or under biaxial stress states as described in Ref. [8]. CCfatigue software implemented a large number of fatigue failure theories that can be easily applied for this task. A detailed description of the theoretical background and a demonstration of their application for the fatigue life prediction of a wide range of materials under different conditions can be found in Ref. [8]. The selection of the failure theory, and the appropriate input is defined in the “Fatigue Failure” tab of CCfatigue. The set-up & Run tab for the multiaxial fatigue failure analysis is presented in Fig. 18.30. Six methods have been programmed in the current version of the software; different input is requested for the application of each one of them as described in Ref. [8]. The accuracy of these methods has been assessed

Fig. 18.30 Set-up & Run tab for the multiaxial fatigue failure analysis.

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Fatigue Life Prediction of Composites and Composite Structures

in Ref. [8] by comparisons to the wide range of off-axis and biaxial fatigue experimental data. The method implementing the failure tensor polynomial in fatigue (FTPF) [8, 37] is used here for the demonstration. According to this model, the S-N curve at any off-axis angle, θ, and a given R-ratio, can be expressed as a function of those at the longitudinal (X), the transverse (Y) directions, and the one for shear (S). vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1  σx ¼ u u 4 4 2 t cos θ sin θ cos θ sin 2 θ cos 2 θ sin 2 θ + +  X2 Y2 XY S2

(18.8)

For the case of the [0/(
45)2/0]T laminate investigated in this case study, it has been shown in Ref. [37] that the S-N curve for shear can be safely established by the S-N curve for 45 degree off-axis. Nevertheless, Eq. (18.8) can be used as well for the calculation of the S-N curve under shear, when it is not possible to derive it experimentally. For this, an off-axis S-N curve (σ x) must be used as the reference curve and Eq. (18.8) must thus be solved for S. For the prediction of the constant amplitude fatigue lifetime of a composite laminate along different off-axis directions (therefore, at different multiaxial stress states) the FTPF method requires data for the constant amplitude fatigue life of the same material along the principal directions and shear, as shown in Fig. 18.30 under “fatigue data.” The results of the application of FTPF for the prediction of the off-axis fatigue behavior of the [0/(
45)2/0]T laminate under different stress ratios is presented in Figs. 18.31–18.33.

Fig. 18.31 Off-axis fatigue life prediction under constant amplitude, R ¼ 10.

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Fig. 18.32 Off-axis fatigue life prediction under constant amplitude, R ¼ 1.

As shown, the FTPF fatigue failure theory, in all cases but one (R ¼ 0.1, 15 degree off-axis, Fig. 18.33) produced accurate predictions of the fatigue life of the examined material. Other methods can also be used for the estimation of the off-axis fatigue life of the multidirectional laminates. For example, the predictions of the method proposed by Sims and Brogdon (SB) are also presented in Figs. 18.31–18.33 as well for

Fig. 18.33 Off-axis fatigue life prediction under constant amplitude, R ¼ 0.1.

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comparisons. More details about the theoretical background of this method and others that are already implemented in the CCfatigue software can be found in Ref. [8]. The same theoretical model(s) can be used for the estimation of the fatigue life of the examined laminate under VA loading. In this case, the FTPF is used in combination to appropriate CLD formulations in order to derive S-N curve formulations for the desired off-axis angle under each R-ratio resulting from the cycle counting analysis of the VA spectrum. This procedure resembles to the derivation of the S-N curves and eventually the CLDs for each desirable off-axis angle, through the application of a fatigue failure theory (e.g., FTPF) by using experimental data for the principal directions and shear. Application of this procedure to the multidirectional laminate examined in this section derives the VA fatigue life prediction results presented in Fig. 18.34 for the offaxis angles of 30 and 60 degree. As can be seen, the prediction for the 60 degree off-axis specimens is accurate for all the examined range of stresses. For the 30 degree off-axis specimens, the theoretical predictions are conservative for high stresses, while they become nonconservative for low stresses. The predictions at high stress levels can be improved (see Fig. 18.34) if the quasi-static strength data will be used, together with the constant amplitude fatigue data, for the derivation of the S-N curves, by employing, for example, the Sendeckyj method as described in Ref. [72] instead of the power law curve. The individual estimated Miner coefficient for the different maximum stress levels for both off-axis angle specimens are given in Table 18.9. Miner values <1 mean conservative predictions, while higher than one denotes that the model predicts longer fatigue life than what the specimen actually sustains during the experiment. Average values of 0.711 for the 30 degree and 1.243, for the 60 degree off-axis specimens, prove the potential of the selected methods, used by means of the CCfatigue software, to reliably predict the fatigue life of the examined material.

Fig. 18.34 Off-axis fatigue life prediction under MWISPERX spectrum.

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Table 18.9 Experimental data and theoretical predictions for 30 and 60 degree off-axis specimens under the MWX spectrum σ max (MPa)

Spectrum passes

Predicted spectrum passes

Miner coefficient

354.60 229.81 354.60 52.18 97.87 149.73 97.87 0.89 0.40 0.73

0.22 0.36 0.32 0.30 0.27 0.26 0.77 1.42 1.29 1.91

4.11 6.27 49.37 24.08 58.30

0.80 1.80 1.57 1.22 0.83

Average

30 degree off-axis 79.4 81.55 79.24 90.1 86.64 83.82 86.06 114.42 119.35 115.37

78.87 81.83 113.23 15.74 26.47 38.65 75.25 1.26 0.51 1.39

0.711

60 degree off axis 62.71 60.98 53.18 55.86 52.67

3.27 11.3 77.36 29.47 48.11

1.243

The derived average Miner values are acceptable, especially when compared to those presented in the past for composite materials, for example, Ref. [74, 75]. Nevertheless, individual values for specific stress levels are far from accurate predicting ca. 2 times less (1.91) and up to ca. 5 times more (0.22) lifetime compared to the experiments. The prediction accuracy can be improved by recalibrating the models, for example, by performing nonlinear damage summation, as it was discussed in Ref. [8] or by using constant amplitude fatigue data at the off-axis directions to improve the S-N curve predictions through the selected fatigue failure theory. However, the major drawback in using such nonlinear life prediction scheme is the requirement of small number of experimental data under each irregular spectrum of interest and it is, therefore, not practical for applications involving a large number of different load cases.

18.4.6 Summary The accuracy of the lifetime prediction of composite materials under VA loading is based on the accuracy of a series of processes before reaching the final goal. According to the building block approach, material property evaluation follows a direct characterization methodology; the laminated plate (the building block) is experimentally investigated in its symmetry directions and for shear and the respective fatigue strength parameters are derived. The proposed procedure is, therefore, suitable for design verification, that is, the life prediction of a laminate or a structural

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Fatigue Life Prediction of Composites and Composite Structures

component designed for ultimate (static) or stability critical loads. If fatigue design optimization is required, a “ply-to-laminate” prediction methodology and associated ply characterization techniques should be adopted instead. The application of the CCfatigue software for the prediction of the off-axis fatigue behavior of a multidirectional laminate, as well as for the lifetime prediction of angle ply and multidirectional laminates under VA loading has been demonstrated in this section. The predictive accuracy of the software has been validated by comparisons to a series of experimental results from the literature. It has been proven that when appropriate experimental data are available, the software is able to provide accurate predictions of the material fatigue life under “unseen” VA spectra, by performing a blind fatigue analysis. An asset of the software is the possibility it offers to provide output at the intermediate stages of the life prediction procedure, assisting data analysis procedures, allowing the performance of rigorous calculations, comparisons of models for the interpretation of the fatigue data and their statistical analysis (e.g., S-N curves, CLDs), etc., in order to support model benchmarking procedures.

18.5

Conclusions and outlook

Several commercial and academic codes have been presented during the last two decades for the assessment of fatigue life of composite materials and structures. The trend during the recent years is to develop multiscale progressive damage tools able to estimate fatigue life, but as well as simulate fatigue damage, recognize damage mechanisms, and quantify their effect on the fatigue life. Reliable tools, validated against a series of experimental data for a variety of composite materials and structures would be valuable tools on the hands of researchers and design engineers. Accurate simulations can reduce drastically the amount of physical testing at any level. Understanding of the material behavior through virtual testing can significantly assist optimization procedures, as the theoretical models can provide life estimations for different loading profiles, and designate where material is needed or where material could be removed in a structural element under given fatigue loads. An abundance of formulations exist today, others more, others less evaluated. Nevertheless, experience showed that a lot of efforts have to be allocated in the topic, before one, or some of the existing solutions become well-spread commercial tools, easily accessible by composite engineers in different engineering domains. Additional databases containing data for benchmarking are necessary in order to allow the developed virtual testing tools to gain confidence.

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