# Fatigue Analysis of Continuously Carbon Fiber Reinforced Laminates.

INTRODUCTIONContinuously fiber reinforced composites have been increasingly applied in various applications during the last decades due to their usability in lightweight constructions and outstanding mechanical properties. To assure both weight reduction and safety, material utilization has to be optimized by producing load-tailored and individually designed composite parts. Assessing expected stresses and possibly occurring failure is of tremendous importance to achieve these objectives and can be realized by conducting mechanical tests and using predictive theories. In contrast to most metallic materials, highly anisotropic material behavior has to be taken into account influencing not only the mechanical properties but also damage mechanisms.

Among different theories describing and predicting composite failure, Puck [1, 2, 3] characterized failure under quasi-static loads with a stress-based criterion considering five different failure modes. Due to the differentiation between fiber and matrix dominating failure mechanisms, Puck's criterion can consider physically motivated material behavior rather than criteria by e.g. Hashin or Tsai-Hill [4, 5]. Puck's five failure modes are defined as two modes for fiber failure (FF) under tension and compression load and three modes describing inter-fiber failure mechanisms (IFF). The IFF modes distinguish between matrix failure due to tension (Mode A), compression (Mode B) or combined compression and shear load (Mode C) [3] (Figure 1). Consequently, Puck's criterion allows not only the prediction of critical stresses but of the expected failure mode as well. Failure criteria are often visualized by fracture surfaces or fracture bodies. Puck's criterion and Tsai-, Hill- or Wu-like global stress criteria for the ([sigma]1, [sigma]2, [tau]21)-space are compared in a schematic way in Figure 2 [6]. The definition of the parameters can be found in the section "Nomenclature" at the end of the paper.

If composite structures are exposed to cyclic loads such as mechanical loads or temperature changes, which is a very likely scenario during their operation time, the prediction of the material behavior becomes even more complex due to the occurring damage mechanisms. Common damage mechanisms in composite materials are matrix cracking, fiber matrix debonding, delamination or fiber fracture. These damage mechanisms may change, progress or interact during fatigue-life and decrease the mechanical properties such as stiffness and strength [7, 8, 9]. Additionally, fatigue-induced damage mechanisms depend not only on the direction of fibers in relation to the applied load, but also on the amplitude and frequency of the cyclic load [10], and on the mechanical mean stress. Due to the described material behavior, accurate fatigue-life prediction tools considering the variety of aspects of continuously fiber-reinforced materials are still in their early stages. Various studies regarding this issue have been published in the last decades pursuing different approaches e.g. [11, 12, 13, 14].

For metallic materials, theories based on fatigue strength, usually represented by stress versus 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 Finite Element Fatigue (FEMFAT) developed by Magna Powertrain Engineering Center Steyr GmbH & Co KG (St. Valentin, Austria) [15]. 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 finite element mesh, local S-N curves are predicted [16, 17]. Critical damages are calculated according to the critical plane concept [18, 19,20]. 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 [21]. For injection molded short fiber reinforced 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 [22, 23,24, 25].

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 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 FF, 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 [[sigma].sub.1] longitudinal to the fiber orientation is calculated by linear superimposition of in general multiaxial load channels (Figure 3 and 4). A rainflow counting algorithm is applied to obtain an amplitude-mean-rainflow-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 [26, 27]. For the IFF modes illustrated in Figure 1, the same procedure is performed for the normal stress [[sigma].sub.2] transverse to the fiber orientation and for the in-plane shear stress [[tau].sub.21] and the respective material S-N curves in the fatigue-life software. The material S-N curve for shear can be measured with specimens consisting of [+ or -]45[degrees] lay-ups, see Figure 4.

To apply Puck's criterion, according to Figure 1, also combinations of [[sigma].sub.2] and [[tau].sub.21] have to be considered [1, 2, 3]. Nevertheless, for non-proportional loading the stress vector spanned by [[sigma].sub.2] and [[tau].sub.21] may change its direction with respect to time (Figure 5 top). 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 [19, 20, 28]. The stress vector is projected onto several fixed directions with the given unit vector [[??].sub.i] (1).

[[sigma].sub.eqv] = [??](t)*[[??].sub.i] = [[sigma].sub.2](t)cos[[phi].sub.i]

i = 1,..., N (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[degrees] is used between directions as default, leading to six directions in the [[sigma].sub.2]-[[tau].sub.21]-plane as illustrated in Figure 5 bottom. For the damage analysis of the intermediate directions S-N curves are used, which are obtained by interpolation between the S-N curves for [[sigma].sub.2] and [[TAU].sub.21].

To consider the influence of the mean stress, Haigh-diagrams are constructed from quasi-static 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 e.g. 5*[10.sup.6] [26, 27]. Due to the different strengths under tensile and compressive loads of composite materials caused by the anisotropic macroscopic material behavior, usually highly asymmetric Haigh-diagrams are obtained. For shear loading, the Haigh-diagram must be symmetric. For intermediate directions, which cannot all be covered by experimental tests, the Haigh-diagrams are interpolated with a smooth transition between tensile-compressive loading at an angle of [phi] = 0[degrees] and shear loading at [phi] = 90[degrees], resulting in a Haigh-surface as illustrated in Figure 6. The static limitations of the Haigh-surface at the left hand side (green line in compressive domain) and right hand side (yellow line in the tensile domain) are determined by Puck's criterion.

The intra-laminar stresses [[sigma].sub.1] [[sigma].sub.2] and [[tau].sub.21], which are acting in the plies, can be analyzed by the finite element method (FEM) with shell elements, whereas for inter-laminar stresses [[sigma].sub.3], [[tau].sub.32] and [[tau].sub.31], which are acting between plies and causing delamination, solid elements are needed, which is much more expensive. For the fatigue assessment of inter-laminar stresses, the same procedure is used as for intra-laminar stresses with different S-N curves and Haigh-diagrams. In Figure 5, [[sigma].sub.2] is just replaced by [[sigma].sub.3] and [[tau].sub.21] by [[tau].sub.32] and [[tau].sub.31], respectively.

Nevertheless, it is difficult to measure S-N curves for imterlaminar stresses. Interlaminar shear stresses can be generated by a three-point bending load applied to specimens as shown in Figure 7.

Acc. [36] (see also Figure 2 "Puck's fracture cigar"), 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:

[[eta].sub.w] = [square root of 1- [([[sigma].sub.1]/[R.sub.[parallel]] - s/a).sup.2]]/a [[sigma].sub.1][R.sub.[parallel]]/>s (2)

For cyclic loading, the weakening factor may change during a load cycle because [[sigma].sub.1] is generally a function of time. Therefore, the IFF strength is not unique for the damage analysis of the cycle. To overcome this theoretical problem, the stress components [[sigma].sub.2] and [[tau].sub.21] will be increased by the time dependent weakening factor instead, and assessed with the original non-reduced IFF strength:

[[sigma].sub.2] (t) = [[sigma].sub.2](t)/[[eta].sub.W](t) [[tau].sub.21] = [[tau].sub.21](t)/[[eta].sub.W](t)(3)

The fatigue analysis procedure for a laminate component is exemplified in Figure 8. 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 [[sigma].sub.1] [[sigma].sub.2] and [[tau].sub.21] 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.

EXPERIMENTAL WORK

Unidirectional (UD) lamina made of carbon fibers and epoxy resin were tested at angles of 0[degrees], 45[degrees] and 90[degrees]. Lay-ups consisting of [+ or -]45[degrees] and [07+457-457907symm.] layers were also investigated and both lay-ups were symmetric referring to the middle plane. The required carbon/epoxy plates were produced in a pressure-driven vacuum assisted resin transfer molding process. The resin - hardener mixture was injected with a pressure of 6 bar into the cavity and all plates were cured at 80 [degrees]C for 5 hours. Subsequently, specimens were milled from the plates with diamond blades. The fiber volume content of all produced specimens was 55 % (measured by thermogravimetric analysis as published in [29]). UD 0[degrees] plies consisted of four layers; all other specimens were made of eight layers. The specimens' geometry used in quasi-static tensile and tension-tension fatigue tests was

200x10x1 mm (length x width x thickness) for UD specimens in fiber direction and 200x20x2 mm for all other specimens. For quasi-static compression and tension-compression fatigue tests, specimens' geometry was 110x10x2 mm (UD 0[degrees]) and 110x20x2 mm, respectively. Aluminum tabs (length: 50 mm, thickness: 1 mm) were glued on both sides of all specimens. The schematic dimensions for UD 0[degrees] are illustrated for tension (Figure 9) and compression tests (Figure 10).

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 10 mm 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 [30], Compressive moduli were evaluated between 0.001 and 0.003 absolute strain according to [31]. Shear moduli were evaluated as in-plane shear response in tensile tests with [+ or -]45[degrees] specimens [32]. Tension-tension fatigue tests were performed with the R-value (= minimum force /maximum force) of 0.1, tension-compression fatigue tests with R = -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[degrees] and between 60 % and 35 % for the tested off-axis specimens in tension-tension 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[degrees] specimens and between approximately 65 % and 25 % for off-axis specimens [33]. 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*[10.sup.6] 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 [degrees]C and consequently minimize the influence on the materials' properties [34]. 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*[10.sup.6] cycles [[sigma].sub.a,5*10.sup.6] according to [35].

EXPERIMENTAL RESULTS

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 1, mechanical properties transverse to fiber direction are presented in Table 2 and shear properties evaluated with [+ or -]45[degrees] specimens in Table 3. The shear moduli [G.sub.13] and [G.sub.23] necessary for the finite element analysis could not be measured experimentally. Based on the assumption of transversally isotropic material behavior of unidirectional plies, the shear modulus [G.sub.13] was assumed to be equal to the measured shear modulus [G.sub.12] [36]. The third shear modulus [G.sub.23] was set as 2.0 GPa based on the relation between the three shear moduli in calculated material data for unidirectional carbon/epoxy plies from literature [36]. In order to calculate the values of the shear moduli for the material used herein, micromechanical modelling based on the measured material parameters could be used [37].

Results of Fatigue Tests

Fatigue data of UD 0[degrees], UD 90[degrees] and [+ or -]45[degrees] were used as input parameters for the fatigue-life software. The experimentally measured S-N curves for R = 0.1 and R = -1 at room temperature are illustrated in terms of nominal stress amplitudes [[sigma].sub.a] versus number of cycles in Figure 11. For curve fitting, a maximum likelihood estimator according ASTM 739 was applied [35], 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 slope k, scatter width [T.sub.s] and nominal stress amplitude after 5*[10.sup.6] cycles [[sigma].sub.a,5*10.sup.6] of the S-N curves are summarized in detail in Table 4. Pulsating fatigue amplitudes (R = 0.1) were higher for specimens in fiber direction than alternating amplitudes (R = -1) as a result of the quasi-static ultimate compressive strength being approximately 30 % of the ultimate tensile strength (Table 1). On the contrary, alternating amplitudes were higher than pulsating amplitudes for [+ or -]45[degrees] 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 = 0 and R = -1 after 5*[10.sup.6] cycles and the slopes of S-N curves were required. Therefore, fatigue data measured at R = 0.1 and the mean stress effect were used to calculate pulsating tension fatigue strength for R = 0 of 928.2 MPa (in fiber direction) and 9.95 MPa (transverse to fiber direction). Shear fatigue strength after 5*[10.sup.6] required by the fatigue software were calculated by dividing the fatigue strengths of [+ or -]45[degrees] specimen by two [32]. Consequently, pulsating shear fatigue strength for R = 0 was 30.8 MPa. The alternating fatigue strengths were equal to the stress amplitudes [[sigma].sub.a,5*10.sup.6] measured at R = -1 in Table 4.

For verification of the described fatigue analysis method, fatigue tests with pulsating loading (R = 0.1) of two additional laminate configurations for UD 45[degrees] and for a multi-layer composite [0[degrees]/+45[degrees]/-45[degrees]/90[degrees]/symm.] were performed. Test fatigue results of the two lay-ups used for validation are illustrated in comparison to the maximum cyclic stresses of UD 0[degrees], UD 90[degrees] and [+ or -]45[degrees] in Figure 12. The S-N curves measured in tension-tension fatigue tests are illustrated in terms of maximum cyclic stress versus number of cycles in order to ease the comparison with the respective quasi-static tensile strengths. Slope k, scatter width [T.sub.s] and the calculated nominal stress amplitudes after 5*[10.sup.6] cycles [[sigma].sub.a,5*10.sup.6] of the S-N curves can be found in Table 5.

SIMULATION RESULTS

FE Analysis

For the finite element analysis of deformation and stresses with the FE solver ABAQUS, the specimens were modelled 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 1 to 3. To obtain correct clamping conditions, also the aluminum tabs were modelled with shell elements and connected to the laminate with tie contacts. For modelling of the aluminum tabs, tensile modulus of 70 GPa and Poisson's ratio v = 0.34 were assumed. Figure 13 shows the finite element mesh of the UD 0[degrees] specimens' geometry used for quasi-static tensile and tension-tension fatigue tests. It consists of 2000 quadrilateral shell elements for the carbon fiber reinforced laminate and additional 2000 quadrilateral shell elements for the aluminum tabs. For UD 45 [degrees] and UD 90 [degrees] specimens, the number of elements is doubled according to the double width of 20 mm, leading to a total number of 8000 elements (Figure 14).

Fatigue Life Prediction

For fatigue-life prediction with FEMFAT an input material dataset was generated as a first step. Quasi-static material properties as presented in Tables 1 to 3 were used. Furthermore, the fatigue strengths for alternating and pulsating loading in and transverse to the fiber orientation and for shear loading were implemented. Based on those data, Haigh-diagrams 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 1). As a result of the different quasi-static strengths under tensile and compressive loads, the Haigh diagrams for loading longitudinal and transversal to fiber direction appeared asymmetric. In contrast to that the Haigh-diagram for shear loading appeared symmetric (Figure 15).

Subsequently, fatigue-life predictions were performed for the UD 0[degrees], UD 90[degrees] and [+ or -]45[degrees] 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[degrees] and UD 90[degrees] specimens as illustrated with red lines drawn in comparison to the experimental results (Figure 16 and Table 6). For the two implemented load cases, tension-tension and tensioncompression, the produced input data in and transverse to fiber direction fitted the experimental fatigue strengths very well. For [+ or -]45[degrees], the damage distribution along the specimens calculated by the software was not as homogeneous as for UD 0[degrees] and UD 90[degrees] specimens. The disturbed damage distribution in the area of the clamping of the specimen is illustrated in Figure 17. This effect might be caused by the two different directions of layers within the specimen, +45[degrees] and -45[degrees], resulting in a stress introduction into the specimen different from uniaxial specimen. However, the obtained damage distribution reflected the lay-up 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 (Figure 16).

Based on the input parameter sets validated for R = 0.1 and R = -1, the fatigue-life of the UD 45[degrees] specimens, in which all layers were aligned at the same angle of 45[degrees], and of a multiaxial lay-up with the stacking sequence [0[degrees]/+45[degrees]/-45[degrees]/90[degrees]/symm.] were predicted subsequently. Due to the unidirectional lay-up of the UD 45[degrees] specimen and the fixed clamping, asymmetric stress distribution resulting in inhomogeneous damage distribution were obtained (Figure 18). 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 clamping position. Consequently, the best correlation with experimental results was again obtained for the middle position as presented in Figure 19 and Table 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 modelling 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 multi-layered 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[degrees] ply at the middle plane of the [0[degrees]/+45[degrees]/-45[degrees]/90[degrees]/symm.] specimen was critical for failure which resulted in predictions underestimating the test result for total failure (red line in Figure 19 and Table 7). If continuing the fatigue-life calculation after the assumed failure of the 90[degrees] plies, [+ or -]45[degrees] and 0[degrees] would still carry load until the neighboring plies with -45[degrees] orientation and subsequently the plies with +45[degrees] orientation would fail, schematically drawn in Figure 19. At last, only the 0[degrees] 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 (Figure 20). 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 [38]. For the simulation of the total fracture, stiffness degradation will have to be considered in an iterative way.

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 (Figure 21). Beside the function as an important structure of the truck main frame, the part represents a liquid/CNG reservoir at the same time. The management of design issues, the assessment regarding stiffness and strength was the major focus of the project [41].

By means of FE analysis, performed with ABAQUS, the structure was optimized regarding manufacturing possibilities in combination with the other essential properties like stiffness and strength. Many activities were started to find an appropriate combination of the plastic part and the steel brackets, which enable the fixing within the truck main frame structure. Due to the fact that the function integrated structure is much stiffer (resulting from the bigger size and cylindrical shape) than the original cross member, the specific decoupling of the truck main frame and the new developed module was challenging. After solving the main design problems, the development team focused on the optimized layout of the plies with respect to stiffness and durability. Finally, the verification of the analysis results was done regarding stiffness and fatigue results at a frame test rig.

The cross member consisted of several layers: The liner is the most inner part. Its function is to make the vessel airtight. The material is high-density polyethylene (HDPE). During the production the outline of this volume has an additional function. It is used as core for the 90[degrees] winding and for the braiding process. The 90[degrees] winding carries the main pressure load which acts at the inside. The operation pressure for a compressed natural gas (CNG) tank is 200 bar. The burst pressure is 450 bar. The material is a CFRP (carbon fiber reinforced plastic). The stress and fatigue assessment was focused on the outside layer, which is a braiding also made of CFRP. The longitudinal force from the two steel calottes and the forces from the cross member function are taken over by this layer. For the ABAQUS analysis a simplified composite shell model was used consisting of five unidirectional plies, which represents the axial yarn and the braider yarns (Figure 22). The waviness was not considered.

Some prototypes have been produced for fatigue testing. Two load cases have been considered: A sinusoidal bending and a sinusoidal torsion load pulsating between zero and maximum load (stress ratio R=0). The fatigue analysis was performed with FEMFAT laminate based on ABAQUS composite shell element stress results. The locations of the observed cracks were different for the two load cases. The fatigue simulation predicted the locations for both load cases correctly, whereas the predicted lifetime was too conservative. The main reason is, that FEMFAT predicts the lifetime until the first failure in the 90-degree plies, which occurs much earlier and invisible inside the CFRP than the observed crack, which is already a quite large delamination after detection. For every ply the critical stress component was the normal stress [[sigma].sub.2] perpendicular to the fiber orientation. For both load cases the critical ply was on the outside. The results for bending are shown in Figure 23.

SUMMARY/CONCLUSIONS

In this work, a fatigue life prediction method for laminates based on S-N curves is presented, which is applicable even for general random-like and multi-axial loads. Further, quasi-static and fatigue data of a carbon/epoxy laminate were measured longitudinal and transverse to the fiber orientation 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 inter fiber fracture with ABAQUS composite shell elements, and recently delamination with ABAQUS composite solid elements, even for non-proportional loading. Fatigue-life of a unidirectional 45[degrees] laminate and a multilayer composite were calculated and validated with experimental results. As example for a real life 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 [+ or -]45[degrees] 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, but which is expensive and experiences still must be gained. The fatigue-life of the UD 45[degrees] could be predicted very accurately. For the fatigue-life calculation of the multi-layered composites with the stacking sequence [0[degrees]/+45[degrees]/-45[degrees]/90[degrees]/symm.], the fatigue software assumed failure as soon as the weakest 90[degrees] ply became critical (similar to the weakest link concept), whereas tests were performed until total failure which lead to a large gap 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 [39, 40]. 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.

REFERENCES

[1.] Puck A. Festigkeitsanalyse von Faser-Matrix-Laminaten: Modelle fur die Praxis. Munchen, Wien: Hanser; 1996.

[2.] Puck A, Schurmann H. Failure analysis of FRP laminates by means of physically based phenomenological models. Composites Science and Technology 1998(58):1045-67.

[3.] Puck A, Schurmann H. Failure analysis of FRP laminates by means of physically based phenomenological models. Composite Science and Technology 2002(62):1633-62.

[4.] Hashin Z. Finite thermoelastic fracture criterion with application to laminate cracking analysis. J.Mech.Phys.Solids 1996(Vol.44 No.7):1129-45.

[5.] Tsai S, Wu E. A General Theory of Strength for Anisotropic Materials. Journal of Composite Materials 1971(Vol. 5):58-80.

[6.] Deuschle H. 3D failure analysis of UD fibre reinforced composites: Puck's theory within FEA. Dissertation. Stuttgart; 2010.

[7.] Talreja R. Fatigue of Composite Materials. Pennsylvania, U.S.A: Technomic Publishing Inc; 1987.

[8.] Talreja R, Singh CV. Damage and failure of composite materials. Cambridge, New York: Cambridge University Press; 2012.

[9.] Reifsnider KL (ed.). Fatigue of composite materials. Amsterdam, New York: Elsevier; 1991.

[10.] Eftekhari M. and Fatemi A., "On the Strengthening Effect of Increasing Cycling Frequency on Fatigue Behavior of Some Polymers and Their Composites: Experiments and Modeling", Int. J. of Fatigue, 2016;87:153-166.

[11.] Kawai M. A phenomenological model for off-axis fatigue behavior of unidirectional polymer matrix composites under different stress ratios. Composites Part A: Applied Science and Manufacturing 2004;35(7-8):955-63.

[12.] Ladeveze P, Lubineau G. A computational mesodamage model for life prediction for laminates. In: Harris B, editor. Fatigue in composites: Science and technology of the fatigue response of fibre-reinforced plastics. Cambridge: Woodhead; CRC Press; 2003, p. 432-41.

[13.] Lubineau G, Ladeveze P, Violeau D. Durability of CFRP laminates under thermomechanical loading: A micro-meso damage model. Composites Science and Technology 2006;66(7-8):983-92.

[14.] Philippidis TP, Vassilopoulos AP. Life prediction methodology for GFRP laminates under spectrum loading. Composites Part A: Applied Science and Manufacturing 2004;35(6):657-66.

[15.] Magna Engineering Center Steyr. FEMFAT software: advanced engineering services by Magna Powertrain; Available from: http://www.femfat.com/.

[16.] Eichlseder W. Fatigue analysis by local stress concept. Computers & Structures 2002(80):2109-13.

[17.] Unger B, Dannbauer H, Eichlseder W. Virtual Fatigue Optimisation of Automotive Structures. Engineering Integrity 2004;15(1):6-13.

[18.] Gaier C, Dannbauer H. A Multiaxial Fatigue Analysis Method for Ductile, Semi-Ductile, and Brittle Materials: Arabian Journal for Science & Engineering, Volume 33, Number 1B, April 2008, pp. 223-235.

[19.] Gaier C, Pramhas G, Steiner W. An Extended Critical Plane Criterion for General Load Situations. In: Blom AF, editor. Proceedings of the 8th International Fatigue Conference June 3-7 2002 Stockholm Sweden //Heath, West Midlands, UK: Engineering Materials Advisory Services; 2002, pp. 259-66.

[20.] Gaier C, Hofwimmer K, Dannbauer H. Accurate and Efficient Methods for Multiaxial Fatigue Analysis, Proceedings NAFEMS World Congress, 9th-12th June 2013, Salzburg.

[21.] Gaier C, Unger B, Dannbauer H. Multiaxial fatigue analysis of orthotopic materials. Rev. Metall. 2010;107(9):369-75."

[22.] Guster C, Pinter G, Mosenbacher A, Eichlseder W. Evaluation of a Simulation Process for Fatigue Life Calculation of Short Fibre Reinforced Plastic Components. Procedia Engineering 2011;10:2104-9.

[23.] Mosenbacher A, Brunbauer J, Pichler PF, Guster C, Pinter G. Modelling and validation of fatigue life calculation for short fibre reinforced injection moulded parts. In: ECCM16, editor. 16th European Conference on Composite Materials; 2014.

[24.] Gaier C. Investigations on a statistical measure of the nonproportionality of stresses. International Journal of Fatigue 2004;26(4):331-7.

[25.] Gaier C, Fleischer H, Guster C, Pinter G. Influence of Fiber Orientation, Temperature and Moisture on the Fatigue Behavior of Injection Molded Fiber Reinforced Thermoplastics // Einfluss von Faserorientierung, Temperatur und Feuchtigkeit auf das Ermudungsverhalten von kurzfaserverstarkten Thermoplasten. MP 2010;52(7-8):534-42.

[26.] Radaj D. Ermudungsfestigkeit: Grundlagen fur Leichtbau, Maschinenund Stahlbau. 3rd ed. Berlin [u.a.]: Springer; 2007.

[27.] Haibach E. Betriebsfestigkeit: Verfahren und Daten zur Bauteilberechnung. Berlin: Springer; 2006.

[28.] Orth M, Butz M, Gaier C. Entwicklung einer Methode zur Betriebsfestigkeitsanalyse von Strukturbauteilen aus CFK unter Zuhilfenahme verschiedener statischer Versagenskriterien // Development of a method for the durability assessment of structural parts made from carbon fiber reinforced plastics (CFRP) based on various static failure criteria. In: DVM, editor. 40. Tagung DVM Arbeitskreis Betriebsfestigkeit; 2013.

[29.] Brunbauer J, Stadler H, Pinter G. Mechanical properties, fatigue damage and microstructure of carbon/epoxy laminates depending on fibre volume content. International Journal of Fatigue 2015;70:85-92.

[30.] ASTM International. Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials (ASTM D3039/D3039M-00). West Conshohocken, PA, United States: ASTM International; 2000.

[31.] ASTM International. Standard Test Method for Compressive Properties of Polymer Matrix Composite Materials with Unsupported Gage Section by Shear Loading (ASTM D3410/D3410M-03). West Conshohocken, PA, United States: ASTM International; 2008.

[32.] ASTM International. Standard Test Method for In-Plane Shear Response of Polymer Matrix Composite Materials by Tensile Test of a [+ or -]45[degrees] Laminate (ASTM D3518/D3518M-94). West Conshohocken, PA, United States: ASTM International; 2007.

[33.] Brunbauer J, Pinter G. Effects of mean stress and fibre volume content on the fatigue-induced damage mechanisms in CFRP. International Journal of Fatigue 2015;75:28-38.

[34.] Brunbauer J, Pinter G. On the strain measurement and stiffness calculation of carbon fibre reinforced composites under quasi-static tensile and tension-tension fatigue loads. Polymer Testing 2014;40:256-64.

[35.] ASTM International. Standard Practice for Statistical Analysis of Linear or Linearized Stress -Life (S-N) and Strain-Life (e-N) Fatigue Data (ASTM E739-91). West Conshohocken, PA, United States: ASTM International; 1991.

[36.] Schurmann H. Konstruieren mit Faser-Kunststoff-Verbunden: Mit 39 Tabellen. 2nd ed. Berlin, Heidelberg, New York, NY: Springer; 2007.

[37.] Fuchs PF, Pinter G, Tonjec M. Determination of the orthotropic material properties of individual layers of printed circuit boards. Microelectronics Reliability 2012;52(11):2723-30.

[38.] Brunbauer J, Pinter G. Fatigue life prediction of carbon fibre reinforced laminates by using cycle-dependent classical laminate theory. Composites Part B: Engineering 2015;70:167-74.

[39.] Reifsnider KL, Talug A. Analysis of fatigue damage in composite laminates. International Journal of Fatigue 1980;2(1):3-11.

[40.] Talreja R. Damage Characterization. In: Reifsnider KL, editor. Fatigue of composite materials. Amsterdam, New York: Elsevier; 1991, p. 79103.

[41.] Kirchberger H., Fischmeister S., Dannbauer H., Puchner K., Development of a multifunctional truck cross member made of continuous fiber reinforced plastics, SIMULIA Community Conference, Berlin, Germany, May 2015.

[42.] Guideline VDI 2014 Part 3, Development of FRP components (fibre-reinforced plastics) - Analysis, Dusseldorf, Germany September 2006.

CONTACT INFORMATION

Magna Engineering Center Steyr GmbH & Co KG Steyrer Str. 32

A-4300 St. Valentin

Austria

christian.gaier@magna.com

NOMENCLATURE

Definitions according to [42]:

[R.sup.t.sub.[parallel]], [R.sup.c.sub.[parallel]]- tensile and compressive strength of UD lamina parallel to fiber direction

[R.sup.t.sub.[perpendicular to]], [R.sup.c.sub.[perpendicular to]]- tensile and compressive strength of UD lamina transverse to fiber direction

[R.sub.[parallel][perpendicular to]] - in-plane shear strength of UD lamina

[R.sup.A.sub.[perpendicular to][perpendicular to]]- fracture resistance of an action-plane action parallel to the fiber direction against its fracture due to [[tau].sub.[perpendicular to][perpendicular to]] stressing acting on it

[p.sup.t.sub.[parallel]]. [p.sup.c.sub.[parallel]] - inclination of ([[tau].sub.21], [[sigma].sub.2])-fracture curve at [[sigma].sub.2] = 0 - t for the range [[sigma].sub.2] > 0 (tension) - c for the range [[sigma].sub.2] < 0 (compression)

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Positions and opinions advanced in this paper are those of the author(s) and not necessarily those of SAE International. The author is solely responsible for the content of the paper.

Christian Gaier and Stefan Fischmeister

Magna Engineering Center Steyr GmbH&CoKG

Julia Maier and Gerald Pinter

University Leoben

doi:10.4271/2017-01-0327

Table 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] 107.0 Elastic Poisson's ratio [-] 0.34 Elongation at rupture [%] 1.35 Ultimate tensile strength [MPa] 1550 Ultimate compressive strength [MPa] 549 Table 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 5.5 direction [GPa] Ultimate tensile strength [MPa] 33 Ultimate compressive strength [MPa] 89 Table 3. Quasi-static shear input parameters for FE analysis and fatigue-life prediction evaluated in quasi-static tension tests with [+ or -]45[degrees] specimens. In-plane shear modulus [G.sub.12] [GPa] 3.3 In-plane shear strength [MPa] 74 Table 4. Input S-N curves for the fatigue-life prediction: Slope k, scatter width [T.sub.s] and nominal stress amplitude after 5*[10.sup.6] cycles [[sigma].sub.a,5*10.sup.6] for R = 0.1 and R = -1 evaluated from experimental fatigue tests. Specimen R k [T.sub.s] [[sigma].sub.a,5*10.sup.6] [-] [-] [-] [MPa] UD0[degrees] 0.1 24.9 1/1.14 435.2 UD90[degrees] 0.1 11.1 1/1.15 4.8 [+ or -]45[degrees] 0.1 17.0 1/1.13 29.5 UD0[degrees] -1 13.4 1/1.41 248.4 UD90[degrees] -1 8.0 1/1.58 4.2 [+ or -]45[degrees] -1 13.5 1/1.45 43.4 Table 5. Verification S-N curves for the fatigue-life prediction: Slope k, scatter width [T.sub.s] and nominal stress amplitude after 5*[10.sup.6] cycles [[sigma].sub.a,5*10.sup.6] for R = 0.1 and R = -1 evaluated from experimental fatigue tests. Specimen R k [T.sub.s] [[sigma].sub.a,5*10.sup.6] [-] [-] [-] [MPa] UD45 (0) 0.1 9.2 1/1.12 10.6 Multi-layer 0.1 13.0 1/1.21 109.8 Table 6. Simulation results for UD 0[degrees], UD 90[degrees] and [+ or -]45[degrees]. Specimen R k [[sigma].sub.a,5*10.sup.6] [-] [-] [MPa] UD 0[degrees] 0.1 13.4 414.8 UD90[degrees] 0.1 8.0 5.0 [+ or -]45[degrees] middle 0.1 13.5 29.1 [+ or -]45[degrees] clamping 0.1 13.5 25.8 UD 0[degrees] -1 13.4 235.8 UD90[degrees] -1 8.0 4.1 [+ or -]45[degrees] -1 13.5 43.4 Table 7. Simulation results for UD 45[degrees] and multi-layer composite [0[degrees]/+45[degrees]/-45[degrees]/90[degrees]/symm.]. Specimen R k [[sigma].sub.a,5*10.sup.6] [-] [-] [MPa] UD 45[degrees] middle 04 8.0 10.2 UD 45[degrees] clamping 04 8.0 7.8 Multi4ayer 90[degrees] ply 04 8.0 38.7 Multi4ayer -45[degrees] ply 04 8.0 61.5 Multi4ayer+45[degrees]ply 04 8.0 73.8

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Author: | Gaier, Christian; Fischmeister, Stefan; Maier, Julia; Pinter, Gerald |
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Publication: | SAE International Journal of Engines |

Date: | Apr 1, 2017 |

Words: | 6876 |

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