A computational tool for investigations on the lifetime estimation of multi-directional laminates under multiaxial stress states based on layerwise structural analysis

Mobility and Engineering Research (MER) Volume 1/2018 A computational tool for investigations on the lifetime estimation of multi-directional laminates under multiaxial stress states based on layerwise structural analysis Marc Möller, Jochen Blaurock, Gerhard Ziegmann, Alfons Esderts A computational tool for investigations on the lifetime estimation of multi-directional laminates under multiaxial stress states based on layerwise structural analysis Marc Möllera,∗, Jochen Blaurocka, Gerhard Ziegmannb, Alfons Esdertsc aInstitute for Automotive Engineering, TH Köln, Betzdorfer Straße 2, 50679 Cologne, Germany bInstitute of Polymer Materials and Plastics Engineering, TU Clausthal, Agricolastraße 6, 38678 Clausthal-Zellerfeld, Germany cInstitute for Plant Engineering and Fatigue Analysis, TU Clausthal, Leibnizstraße 32, 38678 Clausthal-Zellerfeld, Germany Abstract In the present paper a calculation tool for the lifetime prediction of composite materials with focus on local multiaxial stress states and different local stress ratios within each lamina is developed. The approach is based on repetitiv, progressive in-plane stress calculations using classical laminate theory with subsequent analysis of the material stressing effort and use of appropriate material degradation models. Therefore experimentally data of S-N curves are used to generate anistropic constant life diagrams for a closer examination of critical fracture planes under any given combination of local stress ratios. The model is verified against various balanced angle plies and multi-directional laminates with arbitrary stacking sequences and varying stress ratios throughout the analysis. Different sections of the model, such as residual strength and residual stiffness, are examined and verified over a wide range of load cycles. The obtained results agree very well with the analyzed experimental data. Keywords: Composite, Multiaxial fatigue, Lifetime prediction, Multi-directional Laminate, Plywise modeling 1. Introduction The growing awareness of climate change and its recognition in many different economic sectors and, associated therewith, the increasing need of strong lightweight materials makes the use of composites become more and more important over a wide range of structural parts. Thus there is a strong demand for efficient design and the continuous improvement of material utilization in the field of fibre-reinforced plastics (FRP). During operation, most structural components are subjected to multiaxial and cycling loads, which make a closer look to the fatigue material behavior of composites even more necessary. To estimate the lifetime of various fibre-reinforced plastic laminates, with arbitrary stacking sequences under complex multiaxial stress states, a mathematical tool based on plywise-modelling is developed. The phenomenological comparability bet- ∗Corresponding author Email addresses: [email protected] (Marc Möller), [email protected] (Jochen Blaurock), [email protected] (Gerhard Ziegmann), [email protected] (Alfons Esderts) ween progressive material failure due to quasi-static loads and failure due to fatigue loads, is the reason why the model is based on extending the classical laminate theory (CLT) to fatigue loads. Extensive research in the field of composite fatigue has been carried out, but, as described in section 2, there is no overall approach or tool for investigations on multiaxial fatigue loads and only few laminates have been previously examined in mechanistic models under mostly only tension-tension loads. To provide an instrument for further investigations on modeling fatigue material behavior of several layups under varying load conditions, different isolated state-of-the-art models are considered and put together in a computational tool written in python. The developed model consists of multiple individualelements, such as formulation of S-N curves, the way of interpolating between stress ratios via constant life diagrams, choice of (quasi-static) failure criteria, formulations for embedded lamina, a concept of shrinking failure envelope due to different combinations of transverse and shear stress ratios and of course stiffness as well as strength degradation models. Considering the level of complexity and the amount of individual concerns within the model, it Nomenclature Ek(ˆ E1)Elastic modulus in k (1)- direction ˆσmGlobal mean stress of laminate E⊥(ˆ E2)Elastic modulus in ⊥ (2)- direction ˆσaGlobal stress amplitude of laminate G⊥k (ˆ G12)In plane shear modulus σkm, σ⊥m, τ⊥kmLamina mean stresses ν⊥k (ˆν21)Major Poisson’s ratio σka, σ⊥a, τ⊥kaLamina stress amplitudes νk⊥ (ˆν12)Minor Poisson’s ratio RσkStress ratio of σkstresses νf⊥k Major Poisson’s ratio of fibre Rσ⊥Stress ratio of σ⊥- stresses EfkParallel elastic modulus of fibre Rτ⊥k Stress ratio of τ⊥kstresses mσfMagnification factor RStress ratio of global stresses ϕFibre volume fraction Qij (Qij)Lamina(te) stiffness matrix σk(ˆσ1) Normal stress in k (1)- direction Aij Shell stiffness matrix σ⊥(ˆσ2) Normal stress in ⊥ (2)- direction Bij Shell-plate interaction matrix τ⊥k (ˆτ21) In-plane shear stress Dij Plate stiffness matrix Xt(ˆ Xt)Parallel tension strength feStress exposure factor Xc(ˆ Xc)Parallel compression strength fwWeakening factor for influence of σk Yt(ˆ Yt)Transverse tension strength p+ ⊥k,p− ⊥k Incline of failure envelope at σ⊥=0: Y0 t(In-situ) Transverse tension strength „+ “ for σ⊥>0and „- “ for σ⊥<0 of embedded lamina p+ ⊥⊥,p− ⊥⊥ Incline of failure envelope at σn=0: Yc(ˆ Yc)Transverse compression strength „+ “ for σ⊥>0and „- “ for σ⊥<0 S⊥k (ˆ S21)In plane Shear strength RA ⊥⊥ Fracture resistance due to τ⊥⊥ S0 ⊥k (In-situ) In plane Shear strength nx,ny,nyx Internal line forces of embedded lamina mx,my,myx Internal line moments Table 1: Nomenclatur for most important symbols in lamina coordinate system related to the principal material axis of the unidirectional lamina (k,⊥,⊥) and in brackets for symbols in laminate coordinate system related to the principal material axis of the laminate (1,2,3) becomes apparent that a lot of coherent experimental data is needed. Because of the lack of own data at that time, the model is validated by comparing the results with experimental data from the well known Composite Material Fatigue Database of the Michigan State University - Department of Energy in cooperation with Sandia National Laboratories (SNL/DOE/MSU - Database) [1] for first investigations. 2. A brief review of progressive and iterative fatigue models for composites The basic idea in terms of a fatigue-extended classical laminate theory is nothing new and has already been propagated before in differing ways. In 2000, Shokrieh and Lessard [2] proposed a progressive model based on the fatigue failure criterion by Hashin [3] and degradation rules for gradual and sudden change in material properties. They verified the model with experimental data of symmetric graphite/epoxy angle-ply laminates ([904/04]s,[04/904]sand [+454/−454]s) under tensiontension loads with constant stress ratio R=0.1 [4]. After the onset of a certain failure mode the specific stiffnessand strength entries were set to zero, while gradual degradation is modeled before any failure occurs. Shokrieh and Lessard also discussed the shortcoming of the Miner’s rule for calculations with more than one block load for the [+454/−454]s-laminate. For the constant block loading the model achieved good results for the fatigue life. Noll, Magin and Himmel [5] examined, in particular,theeffectofanonlinearshearstress-strain(τ⊥k, γ⊥k) - relation within a progressive composite fatigue model and the use of the critical element concept. They used the failure criterion by Puck[6] and a stiffness degradation model, which reduces some entries in the stiffness matrix to a certain value after the onset of a specific failure. Their model was verified calculating the fatigue life of a vinylester/urethane/carbon quasi-isotropic composite laminate ([45/0/−45/90]s) under variable amplitude loading with constant stress ratio R=0.1. The comparison with experimental data showed that the use of linear stress-strain behavior and ignoring stiffness degradation led to overestimation of fatigue life for the examined laminate. Kennedy, Bradaigh and Leen [7] developed an advanced model with the use of Puck’s failure criteria and the s-shaped damage model by Mao and Mahadevan [8] to model stiffness degradation in fibre direction for a unit cell model within a finite element subroutine. Results were examined for a quasi-isotropic (QI) 3 e-glass/epoxylaminate([0/90/45/135]s) under tensiontensionloadswithconstantstressratioR=0.1. The model by Kennedy et al. was seen to predict the fatigue life of the QI-Laminate under R=0.1 very accurately. In 2016, Neumeister et al. [9] presented a model for multi-directional laminates, which is based on the failure mode concept-based strength criteria by Cuntze [10] and calculation of total damage with a linear damage accumulation model. The stiffness degradation follows a nonlinear behavior, which is related to the calculated damage, based on [11]. Neumeister et al. presented first results of the simulation for a [±45/90]s-laminate, but the model itself was not verified to any experimental data within the publication. In 2017, Mejlej, Osorio and Vietor [12] proposed a progressive model for multi-directional laminates under varying stress ratios. The model is build upon the total energy based model by Shokrieh [13], the use of Hashin Failure Criteria and the gradual stiffness degradation by Shokrieh and Lessard [2]. Mejlej et al. applied a relation of fatigue life and total input energy, which is based on fitting parameter to a set of S-N curves, regardless to the stress ratio and fibre orientation. They used the energy failure criterion from Sandhu [14] to calculate fatigue life for various fibre angles and stress ratios. The model was verified against two sets of unidirectional ([0]12) laminate made of carbon/epoxy under both positive and negative stress ratio. The calculation of fatigue life conforms very well to the experimental data under varying stress ratios for unidirectional material. Finally the methodology and tool developed by Vassilopoulos et al. [15] for GFRP lamiantes under complex stress states is briefly mentioned, although it does not really belong to the iterative models. The model uses cycle counting methods for spectrum loads, different fatigue criteria for life prediction, such as e.g. the Fawaz-Ellyin [16] or Hashin-Rotem Criterion [17], and the linear Miner rule for damage summation. The model gets along without the use of stiffness or strength degradation, since it calculates allowable numbers of cycles for each block directly using the mentioned fatigue failure criteria. Differing designs of S-N curves and constant-lifediagrams were examined within the model. Verification was carried out for variable amplitude loading on e-glass/polyester laminates ([0/(±45)2/0]s) for alternating and pulsating stress ratios, focusing mainly on the comparison of used fatigue failure criteria. In short, there are quite a few models for different aspects of fatigue life of composites available. The above mentioned models are based on progressive modeling and vary widely in e.g. used failure criteria or in the way of describing material degradation of strengthand stiffness. For future research in multiple segments of composite fatigue, a general model is presented below and a calculation tool called „Lifetime Estimation ofComposites“ (LEoC - Fig. 1) is developed using the programming language „Python“. Fig. 1: Startup-screen of „LEoC“ for investigations on fatigue of composite 3. Tool and Flow chart for fatigue failure analysis The simulations are performed on the lamina level using a representative volume element (RVE) of the laminate, which represents the corresponding mechanical propertiesof a structural part or component as visualized inFig. 2. Load conditions in terms of internal line forces and moments are applied to the RVE. The general program y x  x y ζ m m m n n n xy y x yx m xy xy Fig. 2: Simulation of 2-D (plane stress) representative volume element (RVE) of exemplary structural component on the lamina level (mesoscale) schedule is demonstrated in Fig. 3, without mentioning which specific models are applied at particular points in the code. The reason for this is that, in principle, diverse models can possibly be applied at every main subject. The divisions along the dashed lines in Fig. 3, which are 4 start n =1 load case static material properties quasistatic material experiments laminate properties multiaxial stress state per layer cyclic material experiments S-N curves Constant Life Diagram S-N extrapolation fracture plain design with extrapolated S-N static failure criteria strength degradation models stiffness degradation FF T,FF C stiffness degradation IFF A stiffness degradation IFF B stiffness degradation IFF C YYY Y Y N NN N overall failure? Y N cyclic properties end n =? N stress ratios per layer failure criteria parameter n = n+1 (update) UD lamina properties Sec.5 Sec.4 Sec.6 Sec.7 Sec.8 in-situ strength of embedded lamina Fig. 3: flow chart for the lifetime estimation of fibre reinforced composites within LEoC 5 labeled from „Sec. 4“ until „Sec. 8“, represent the following sections and the examined areas of focus within this paper: •In section 4, the necessary data from quasi-static material tests for initial conditions of laminate properties is discussed. •Furthermore, input needed from cycling material testsandthe design of constantlifediagrams,forthe extrapolation of unknown S-N curves at arbitrary stress ratios, is explained in section 5. •Using the extrapolated S-N curves, the following strength degradation according to the current multiaxial stress state is addressed in section 6. •In section 7, the failure analysis is considered and the choice of used failure criteria justified. •Models for stiffness degradation after the onset of specific failure modes are discussed and applied to the model in section 8. The failure condition with its different types of failure on the ply-level, as explained in section 7, is checked for every layer in the current load cycle and as far as failure did not occur in every existing ply, the next load cycle is applied to the current laminate condition. The iterative process is carried on until the laminate failed in every ply or the number of intended load cycles is reached. 4. Input from quasi-static material tests One of the decisive initial conditions in these calculations are the static stiffness and strength properties at the ply-level as demonstrated in Fig.3 - field section number 1. Necessary input data of material „D155“ (Fibre: Glass fabric -527 g/m2, Matrix: Orthophthalic Polyester „CoRezyn 63-AX-051“) from [1] is exemplarily shown in table 2. The initial state of the laminate in every cycle n is derived using the CLT. The stiffness matrix of each lamina „k“ in the lamina (local) coordinate system is obtained by Qij,k(n)= Ek(n) 1−∆(n) ν⊥k(n)Ek(n) 1−∆(n)0 E⊥(n) 1−∆(n)0 (sym.)G⊥k(n)  (1) with ∆(n)=ν⊥k(n)νk⊥(n)and experimental data from quasi-static material tests, as shown in table 2. The reduced stiffness matrix of each lamina in the global coordinate system is calculated by Q±ξ ij,k(n)=Tij Qij,k(n)Tji (2) EkE⊥G⊥k ν⊥k [GPa] [GPa] [GPa] [−] x33.60 8.21 4.48 0.29 s 1.80 0.72 1.11 0.01 ϕ0.44 0.44 0.40 0.44 XtXcYtYcS⊥k [MPa] [MPa] [MPa] [MPa] [MPa] x1012.0 653.0 27.00 122.75 73.2 s 29.0 23.3 1.63 9.91 5.8 ϕ0.44 0.41 0.44 0.44 0.44 Table 2: Exemplary static values (x: arithmetic mean, s: standard deviation and ϕ: mean fibre volume fraction) used for simulation input from experimental data of material D155 [1] with use of the transformation matrix Tij = cos2(ξ)sin2(ξ) −sin(2ξ) sin2(ξ)cos2(ξ)sin(2ξ) 0.5sin(2ξ) −0.5sin(2ξ)cos(2ξ). The overall global distortions of the laminate are given by: ˆk(n)=ˆ0 i(n)+z·ˆκ0(n)(3) Herein the distortions are calculated using the strains and curvatures of the combined shell and plate element, which are obtained from ˆ0 i(n) ˆκ0 i(n)=A∗ ij(n)B∗ ij(n) (B∗ ij(n))TD∗ ij(n)·ˆni,bl ˆmi,bl (4) where ˆni,bl ={nx,ny,nxy }and ˆmi,bl ={mx,my,mxy } are the internal line forces and moments of the current block load, as shown in Fig. 2. The extensional- (A∗ ij), bending- (D∗ ij) and coupling- (B∗ ij) compliance matrices are calculated by inverting the ABD matrix with the use of the respective stiffness matrices: 1 χ n Õ k=1 Qij,kzχ k−zχ k−1=        Aij,for χ=1 Bij,for χ=2 Dij,for χ=3 (5) The global distortions and the transformed stiffness matrix are then used to calculate the global stress vector for each lamina. The local in-plane stresses are calculated subsequently with the use of the transformation matrix. ˆσi,k(n)=Qij,k(n) · ˆi,k(n)(6) σi,k(n)=Tij ˆσi,k(n)(7) where σi,k(n)is the local stress vector, with the entries σk,k(n),σ⊥,k(n)and τ⊥k,k(n)for each ply number k. 6 5. Input from fatigue material tests Stress calculations (equation 7) are carried out for two different sets of load conditions ˆni,bl and ˆmi,bl in every block load to gain the multiaxial stress ratios Rσk,Rσ⊥,Rτ⊥k for all three present stress components σk, σ⊥, τ⊥k . The present work will focus mainly on simu0.0 0.5 1.0 1.5 2.0 n 1.0 0.5 0.0 0.5 1.0 1/ max tension compression R=-1.0 R= 0.0 R= R= 0.5 R= 2.0 Fig. 4: Cyclic stresses for one alternating stress ratio (R=-1) and in each case two pulsating tension (R=0, R=0.5) and pulsating compression (R=−∞, R=2) stress ratios lations with stress ratios in pulsating tension (0≤R<1) and pulsating compression range (1<R≤ ∞), as shown in Fig. 4. In the following the four sectors of stress ratio ranges will be referred to as: •T-T: Tension-tension pulsating range (0≤R<1) •T-C: Tension dominated alternating range (−1≤R<0) •C-T: Compression dominated alternating range (−∞ ≤ R<−1) •C-C: Compression-compression pulsating range (1<R≤ ∞) Required relations between stress and endured cycles in terms of S-N curves are obtained by fitting experimental data to the logarithmic linear relation by Basquin: N=C·σ−k a(8) where N is the number of cycles, C is the Y intercept, k is the negative slope and σais the stress amplitude of the current block load for each of the three stress components from equation 7. S-N curves at stress ratio R=0.1 (representative for R=0) and R=10 (representative for R=∞) are at least required to interpolate data at arbitrary stress ratios in pulsating tension-tension and compression-compression range. Fig. 5 exemplarily shows the fitted S-N curves for experimental data of material „D155“ from [1]. 102103104105106107 cycles to failure N (log) / [ ] 100 101 102 stress amplitude Sa (log) / [ MPa ] R = 0.1 kL = -13.0, CL = 2.14e+34 R = 0.1 kL = -27.1, CL = 1.26e+28 R = 0.1 kL = -10.7, CL = 1.42e+19 R = 10 kL = -12.7, CL = 7.55e+33 R = 10 kL = -20.5, CL = 3.10e+37 R = 10 kL = -14.8, CL = 5.56e+27 Fig. 5: Least required S-N curves for investigations in pulsating tension and compression range with experimental data for unidirectional Material D155 [1] One way to receive unknown S-N curves at other stress ratios is by using constant life diagrams (CLD), also often referred to as Haigh-, Goodmanor Smith-Diagrams. A historical view on the development of CLD can be found in [18]. For multiaxial stress states of the lamina, three independent constant life diagrams for parallel, transverse and shear properties are necessary. A comprehensive investigation on the influence of different formulations of CLD for a limited number of multidirectionalglass/polyesterlaminateshasbeencarriedout by [19]. The linear model, which is based on using only one experimental S-N curve, at e.g. stress ratio R=−1, underestimates the strength of the materials in almost all cases for any stress ratio and leads to extremely conservative results. Still it is often used, e.g. in wind turbine certification [20] for the fatigue of FRP, or simply because of the lack of data. The nonlinear fatigue life diagram by Kawai [21], the parametric constant-life model by Harris [22], the multislope model by Boerstra [23] and the piecewise linear (PWL) model by Philippidis and 7 Fig. 6: Exemplary construction of transverse piecewise linear (left) and piecewise nonlinear (right) constant life diagram in LEoC for experimental data from [1] Vassilopoulos [24] showed to give very accurate results for most of the examined fatigue data. Vassilopoulos et. al especially pointed out the sensitivity of Kawai’s model to the choice of input data and the sensitivity of the models by Boerstra and Harris to the fitting of model parameters. Furthermore the model by Kawai tends to often lead to overestimation of the material and therefore non-conservative results. Of all of these models, the PNL seems to be the most stable, since it depicts the S-N behavior by linear interpolation between known S-N curves without any preliminary assumptions. Vassilopoulos et al. also improved their formulation in terms of a piecewise nonlinear (PNL) constant life diagram formulation [25] based on the use of two (PNL-2R) or three (PNL-3R) S-N curves. The accuracy of the nonlinear formulation was very considerable for the same examined data as in [19]. Nonetheless, in the present work the piecewise linear (PWL) model by Philippidis and Vassilopoulos [24] will be preferably used for first investigations on fatigue life (see Fig. 6). Of course, the PNL proves to be only very accurate for a reasonable amount of known S-N curves (at least three S-N curves, each one preferably at the border of every sector (T-T/T-C, T-C/C-T and C-T/C-C). Thus, at least three S-N curves at stress ratios R=0.1, R=- 1 and R=10 are most commonly used for a separation into four sectors . Thus, for the design of three piecewise linear CLD a designated number of S-N curves and the static compression and tension strengths in parallel, transverse and shear direction (Xt,Xc,Yt,Ycand S⊥k) are needed. The calculation of stress amplitude σ∗ aat R∗, if R∗is in between R=1 and the first known stress ratio moving counter clockwise (R1,ccw), is defined as follows: σ∗ a=Xt Xt σa,1,ccw +r∗−(1+R1,ccw) (1−R1,ccw) (9) with σa,1being the stress amplitude at R1,ccw and r∗= (1+R∗)/(1−R∗). If R∗is in between R=1 and the first known stress ratio moving clockwise (R1,cw): σ∗ a=Xc Xc σa,1,cw +r∗−(1+R1,cw) (1−R1,cw) (10) where σa,1is the stress amplitude at R1,cw. If R∗is located in between any of two used stress ratios Riand Ri+1: σ∗ a=σa,i(ri−ri+1) (ri−r∗)σa,i σa,i+1 +(r∗−ri+1) (11) 8 where σa,iand σa,i+1are the stress amplitudes at the two known stress ratios and ri(i+1)=(1+Ri(i+1))/(1−Ri(i+1)). Fig. 7 shows the piecewise linear constant life formulation for a multi-directional laminate with stacking sequence [90/0/±45/0]s. Three S-N curves at stress ratios R=0.1, R=-1 and R=10 are used for linear interpolation, and another three S-N curves at stress ratios R=0.8, R=0.5 and R=2 are plotted for comparison purposes. 600 400 200 0 200 400 600 800 mean stress Sm / [ MPa ] 0 100 200 300 400 stress amplitude Sa / [ MPa ] T-T T-CC-T C-C R = 0 R = 2 R = 0.5 R = 0.8 R = ± R = 1 N=103 N=104 N=105 N=106 Used exp. data Unused exp. data Fig. 7: PWL constant life diagram for laminate „DD16“ with stacking sequence [90/0/±45/0]sfrom [1] 6. Strength degradation First of all, there are possibly two ways to do justice to the degradation of strength in laminates: linking the decrease in strength to a physical meaning like micro mechanical damage, e.g. (inter-)fibre failure, or on the otherhand connecting ittothenumber of cyclesendured. A large number of residual strength models have already been proposed. Most of the state-of-the-art models have been examined by Philippidis and Passipoularidis [26]. A common way to model the residual strength is the use of S-N curves to set a loss in strength and the actual number of endured cycles as well as the sustainable number of cycles into a certain relation. To begin with, the loss in strength in each step is defined as Sr,i=Sr,i−1−∆Sr,i(12) where Sr,iand Sr,i−1represent the five residual strengths {Xt,Xc,Yt,Yc,S⊥k }at the actual and the last step respectively. ∆Sr,iis the drop of residual strength in the i-th cycle and is calculated from ∆Sr,i=("Sst −σmax,i1−ni−1 Niαjβj# −"Sst −σmax,i1−ni Niαjβj#) (13) where Sst is the specific static strength, σmax,iis the maximum stress of the ith cycle, niand ni−1are the number of cycles at the actual and the last step, Niis the maximum number of cycles and αjand βjare material parameters given by curve fitting of residual strength data. By varying values of αjand βjdifferent material behavior can be achieved. With αjand βjbeing both zero, there won’t be any strength degradation of the addressed strength. For αj=1and βj=1equation 12 and 13 can be rearranged to the mostly used linear degradation rule by Broutman and Sahu [27]. Almost all present calculations of residual strength under fatigue loads use the linear model and it is well known that it gives conservative results on the safe side. For arbitrary values of αjwith βj=1the model represents the nonlinear one-parameter model by Schaff and Davidson [28]. The strength degradation is then simulated with either a steep loss of strength at the beginning (“Rapid initial loss in strength“) or at the end (“Sudden death“) of the laminate lifetime. For all other values of αjand βj the equation refers to the „normalized residual strength model (NRSM)“ by Stojkovic et al. [29]. The advantage of the NRSM model is that the typical initial loss of strength followed by slow degradation as well as the sudden decrease in strength at the end can be modeled very well. For the iterative calculation of residual strength within the subroutine, the strength degradation of every ply needs to be defined. For the case of a three piecewise CLD, there are only theoretical 40 parameters eligible in LEoC to overcome all five strength parameters within every sector of the constant life diagram. This of course makes little sense for a practical application. Thus, linking a few comparable degradation behaviors might be an advantage in reducing the amount of experimental data needed. Since the present work focuses only on pulsating stresses, the amount of data is considerably reduced anyway. Particularly because of the lack of experimental data of unidirectional lamina for material used from [1], the linear degradation of all five strength parameters will preferably be used for calculations of fatigue life. Nonetheless, results for all presented residual strength models will be discussed in section 10. Fig. 8 demonstrates the 9 100101102103104105106 0 1 e 4 2 e 4 3 e 4 4 e 4 E 1( n ) a = 124 MPa a = 10.3 MPa a) 1 2 34 Sim. [0/ ± 45/0] s Sim. [90/ ± 45/90] s Exp. data Exp. data 0 15 30 fe 100101102103104105106 0 1 fe b) fe , IFFA in 45 ply fe , IFFA in 90 ply fe , IFFA in 45 ply fe , FFT in 0 ply 100101102103104105106 0 1 2 r c) r in 90 ply r in 45 ply Fig. 15: Comparison of calculated and experimental residual stiffness and for Material #3 from [1] observed in Fig. 15 (b). Afterwards, the stiffness stays nearly constant over a wide range of cycles until a sudden drop in {2}. This is obviously coupled with the increase of fibre failure, as shown in the central figure. Of course, the behavior at later cycles is strongly dependent on the initial values, which indicates that the differences between calculation and experimental data are mainly caused by the overestimated static stiffness. Stiffness degradation of the transverse tested Material #3 is calculated considerably better. The static stiffness as well as stiffness at higher cycles is captured very well. Thecurveis characterizedbytwomain points, numbered as {3} and {4}, too. Due to a steady increase of fe,IFF A in the ±45◦-orientated plies at lower cycles, stiffness is decreased after the onset of failure mode A in {3}. Also, in the ±90◦-orientated plies, there is a rapid rise in fibre failure in {3} due to stress redistribution. Afterwards, there is a continuous decrease in laminate stiffness until global failure in {4}. 11. Conclusions and future work After a brief review of literature, a model for fatigue life analysis is introduced, which is characterized by its iterative and progressive procedure for the calculation of the current laminate conditions at every load cycle. Different sectors and models applied within the methodology are described in detail. Afterwards, various multi-directional fibre reinforced plastic laminates are calculated and presented. Fatigue life for several balanced angle ply laminates under pulsating tension and compression loads is calculated and compared to experimental data. A quantity of multi-directional laminates with arbitrary stacking sequences under different stress ratios are examined and verified, too. Furthermore, residual strength and stiffness of different laminates are considered in more detail. Altogether, the predictions of the model showed a good correlation with most of the experimental data. Any discrepancies or unusual material behaviors are traced back to certain input parameters or known issues, which will be addressed in future work. To sum up, the model tends to be capable of simulating fatigue life of multi-directional laminates under complex multiaxial loads with different stress ratios accurately. Still the findings leave questions for the future, which will be addressed in forthcoming works: •As it is crucial to know the laminates capacity of carrying further loads after a certain time period, a more in-depth analysis of residual strengths models and their application within the model will be carried out next. Detailed strength degradation after certain combined loads, particularly in multiaxially loaded multi-directional laminates, will be addressed. •As the investigations in the present paper have shown, a deeper look on the failure envelope for multiaxial fatigue might be an advantage for further simulations of fatigue life. 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Schürmann, Konstruieren mit Faser-Kunststoff-Verbunden, Springer-Verlag, Berlin ; Heidelberg, 2007. 18 Mobility and Engineering Research (MER) Volume 1/2018 Impressum This document is published within the series ’Mobility and Engineering Research (MER)’. All publications can be downloaded from https://cos.bibl.th-koeln.de/home. Despite thorough revision the information provided in this document is supplied without liability. The document does not necessarily present the opinion of the editors. Date of publication: August, 2018 Editorship of Series Prof. Dr. Jochen Blaurock Prof. Dr. Michael Frantzen Prof. Dr. Patrick Tichelmann Contact editor’s office Prof. Dr. Jochen Blaurock TH Köln Faculty of Automotive Systems and Production Institute for Automotive Engineering Betzdorfer Straße 2 50679 Köln phone: +49 221-8275-2890 email: [email protected]