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Complex stress state effect on fatigue life of GRP laminates. Part II, Theoretical formulation
International Journal of Fatigue 24 (2002) 825–830 www.elsevier.com/locate/ijfatigue
Complex stress state effect on fatigue life of GRP laminates. Part II, Theoretical formulation Theodore P. Philippidis *, Anastasios P. Vassilopoulos Department of Mechanical Engineering and Aeronautics, University of Patras, PO Box 1401, 26500 Patras, Greece Received 23 July 2001; received in revised form 13 December 2001; accepted 20 December 2001
Abstract The synergistic effect of in-plane stress tensor components on fatigue strength is not traditionally considered in the design of thin-wall box-beam structures, e.g. composite rotor blades in general. Fatigue life calculations account only for the normal stresses due to bending and centrifugal forces, neglecting the contribution of shear and transverse normal stresses. The theoretical formulation of a life prediction methodology accounting for all in-plane stress tensor components, through the use of a multiaxial fatigue strength criterion, is presented here. Comparison of theoretical predictions with experimental results from constant amplitude, uniaxial, offaxis tests demonstrates the drastic effect of shear and transverse normal stresses, besides that of axial normal stress, in reducing operational life of a GRP structural laminate. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Composites; Fatigue strength; Multiaxial stress; Fatigue design
1. Introduction In the first part of this work [1], results from a detailed experimental program were presented, which, along with experimental evidence from earlier studies [2–4], highlighted the drastic effect of complex stress states on fatigue strength of glass fiber-reinforced plastic (GFRP) laminates. These investigations were limited to constant amplitude proportional loading. The aim of this research, the results of which are presented in [1] and in this paper, was to investigate the effect of shear and transverse normal stress on the fatigue strength of GRP (glass/polyester) laminates which are mainly stressed axially, in one of their principal material directions. Typically, such stress states arise in box-beam structural members loaded by transverse loads, e.g. aerodynamic forces, producing mainly spanwise distributions of flexural moments, shear forces and occasionally moderate torsional moments. The presence of shear and transverse normal stress relative to the main axial normal stress in the skin of the shell structure is kept small.
* Corresponding author. Tel. and fax: +30-61-997235. E-mail address: [email protected] (T.P. Philippidis).
Predictive formulations for fatigue life, accounting for stress multiaxiality, first appeared in 1970 for composite materials. Most of the proposed criteria were generalizations of static failure criteria to take into account number of cycles, frequency and other fatigue parameters. Hashin and Rotem [5] proposed a fatigue strength criterion for fiber-reinforced materials, based on the different failure modes exhibited. For unidirectional materials two failure modes exist, the fiber and the matrix failure mode. When multidirectional laminates are considered [6], another failure mode, the interlaminar, is encountered. For the application of this criterion, three experimental S–N curves are needed, and it can be implemented only for materials for which failure modes can be clearly discriminated. Owen and Griffiths [2] presented an experimental program consisting of static and fatigue tests to correlate the experimentally obtained fatigue data with predictions from multiaxial static failure criteria. Fujii and Lin [3], in order to validate their experimental, biaxial tension/torsion fatigue data, have interpreted the Tsai– Wu strength criterion [7] for fatigue. However, the adaptation proved to be insufficient to correctly predict fatigue behavior of the material system investigated. Fawaz and Ellyin [8] proposed a multiaxial fatigue failure criterion based on only one experimental S–N
0142-1123/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 2 ) 0 0 0 0 4 - X
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T.P. Philippidis, A.P. Vassilopoulos / International Journal of Fatigue 24 (2002) 825–830
curve, and some static strengths. Multiaxiality is entered through static failure criteria and S–N curves under complex stress states can be predicted. This constitutes a flexible fatigue failure condition, capable of eventually yielding satisfactory predictions. Wide acceptance, however, is constrained by its high sensitivity in the choice of the single experimental S–N formulation required for its application. Jen and Lee [9,10] predicted the fatigue behaviour of off-axis loaded unidirectional composites by introducing an appropriately modified version of the Tsai–Hill criterion. They extended the predictive capability of their theoretical formulation for multidirectional laminates, made of the same prepreg, by means of classical lamination theory and ply-discount considerations. Based on the latter assumptions for stiffness degradation of a failed ply, their theoretical predictions for APC-2 laminates were shown to be corroborated satisfactorily by experimental evidence. An extension of the quadratic version of the failure tensor polynomial, as interpreted by Tsai and Hahn [11], for the prediction of fatigue strength under complex stress states was introduced by the authors [12]. This socalled FTPF (failure tensor polynomial in fatigue) criterion was shown to predict satisfactorily the remaining life or fatigue strength for the entire range of composite materials investigated under off-axis or multiaxial loading. In this paper, the results from the experimental program presented in [1], consisting of uniaxial fatigue tests on coupons cut at several off-axis orientations from a multidirectional GRP laminate, [0/(±45)2/0]T, are used for further validation of the FTPF criterion. Based on this, a theoretical formulation is developed for structural dimensioning and fatigue life prediction of box-beam shell structures. The effect of shear and transverse normal stress on fatigue strength of the laminate loaded mainly by axial normal stress is investigated in depth. The approach differs from previously mentioned studies in the material model used. A direct characterization procedure is adopted for the [0/(±45)2/0]T laminate which is considered as a “homogeneous” anisotropic continuum, thus avoiding uncertainties in the modeling of stiffness degradation of failed layers or interlaminar effects and interactions. This is particularly useful for the GRP laminate investigated, consisting of different glass fabrics with stitched fibers at various orientations. In such cases, simplistic theoretical considerations for load distribution and ply stiffness degradation are no longer applicable.
was called the failure tensor polynomial in fatigue (FTPF) and it was shown to predict successfully fatigue strength under reversed (R=⫺1) complex plane stress states. The failure tensor polynomial for orthotropic media expressed in material symmetry axes, under static plane stress, is given by: F11s21 ⫹ F22s22 ⫹ 2F12s1s2 ⫹ F1s1 ⫹ F2s2
(1)
⫹ F66s26⫺1ⱕ0 where F11 ⫽
1 1 1 1 1 ,F ⫽ ,F ⫽ ,F ⫽ ⫺ , XX⬘ 22 YY⬘ 66 S2 1 X X⬘
(2)
1 1 F2 ⫽ ⫺ . Y Y⬘ X and X⬘ are the tension and compression strengths, respectively, along direction 1 of the material principal system, Y and Y⬘ are the corresponding values for the transverse direction, while S is the shear strength. The specific choice of the off-diagonal term, F12, was shown to lead to completely different failure theories [13]. The form of F12 used in this study is that given by Tsai and Hahn [11]: 1 F12 ⫽ ⫺ 冑F11F22. 2
(3)
FTPF assumes the same functional form as Eq. (1); however, the components of failure tensors are functions of the number of cycles, N, the stress ratio, R, and the frequency, n, of the loading: Fij ⫽ Fij(N,R,n), Fi ⫽ Fi(N,R,n).
(4)
Experimental evidence gained so far for any type of continuous fiber-reinforced polymers strongly suggests the form of functional dependence of failure tensor components shown in Eq. (4). This implies an increased complexity of experimental strength characterization compared to static loading, since it is necessary to discriminate not only between tension or compression and loading rate but also between the same type of loading, e.g. tension, at different R-values or loading frequency, n. Therefore, it was proposed to characterize in-plane fatigue strength of an orthotropic material by means of three experimentally derived S–N curves [12]: 1
X(N,R,n) ⫽ XoN⫺kX
(5)
1
Y(N,R,n) ⫽ YoN⫺kY 1
S(N,R,n) ⫽ SoN⫺kS 2. The fatigue strength criterion
The failure tensor components are defined by:
A modification of the failure tensor polynomial [7] to account for fatigue loading was introduced in [12]. It
F11 ⫽
1 1 ,F ⫽ , X2(N,R,n) 22 Y2(N,R,n)
(6)
T.P. Philippidis, A.P. Vassilopoulos / International Journal of Fatigue 24 (2002) 825–830
F66 ⫽
1 1 ,F ⫽⫺ , F ⫽ F2 S2(N,R,n) 12 2X(N,R,n)Y(N,R,n) 1
⫽ F6 ⫽ 0. Finally, the fatigue strength or life under complex plane stress states and specific R and n cycling parameters is predicted using the equation: s22 s1s2 s26 s21 ⫹ 2 ⫺ ⫹ 2 ⫺1 ⫽ 0. X (N) Y (N) X(N)Y(N) S (N) 2
(7)
It must be noted that the three S–N curves X(N), Y(N) and S(N) in the above equations are derived for the same loading conditions, R, n, as those of the actual stress state {s1, s2, s6}. The experimental characterization of X(N), Y(N) is performed through uniaxial fatigue tests along the respective principal material direction. Experimental determination of fatigue shear strength, S(N), is more complicated as expensive and sophisticated testing equipment and specimens are required. Thus, simple shear characterization methods were devised previously [14–16] and were successfully used to predict stiffness and strength of composite laminates. Shear strength, S(N), in [12] was considered equal to half the value of the fatigue strength of a flat coupon cut off-axis at 45° and loaded uniaxially. This choice yielded satisfactory results for reversed loading, R=⫺1, but its performance proved to be less effective for other loading types such as R=10 and R=0.1. Nevertheless, as it is shown in the sequel, adopting for S(N) the value of 1/2.2 of the fatigue strength of a flat coupon cut off-axis at 45° and loaded uniaxially fits adequately most of the experimental data. The effect of shear stress, s6, and transverse normal stress, s2, on fatigue strength or life is quantified by means of Eq. (7). If, for example, off-axis uniaxial tests are performed, the developed plane stress state depends on the load orientation with respect to the material principal directions. Denoting by si (i=1, 2, 6) the in-plane stress tensor components in the symmetry coordinate system of the multidirectional laminate, and by sx the applied normal stress at a positive off-axis angle q, the following transformation relations are valid: s1 ⫽ sx cos2q; s2 ⫽ sx sin2q; s6 ⫽ sx sin q cos q
(8)
The plane-stress state ratios s2/s1 and s6/s1 as a function of q take values that are equal to tan2q and tan q, respectively. Then, depending on the off-axis direction, the state of stress is either dominated by the axial normal stress, s1, for lower angles q or by the other two stress components, s2, s6, for angles greater than 60°. Substituting relations (8) into Eq. (7) yields the following equation for the off-axis fatigue strength at orientation q: sx(N) ⫽
冋
cos4q sin4q ⫹ 2 ⫹ cos2q sin2q 2 X (N) Y (N)
(9)
冉
冊册
1 1 ⫺ 2 S (N) X(N)Y(N)
⫺
827
1 2
With respect to material properties that need to be determined experimentally in order to use the FTPF criterion (Eqs. (7) and (9)), it is clear that X(N) and Y(N) are the S–N curves determined from on-axis and 90° offaxis coupon tests. These are the fatigue strengths in the two principal, mutually orthogonal, material directions of the “homogeneous” plate [0/(±45)2/0]T. S(N), the shear fatigue strength for the specific stress ratio, R, and the loading frequency should ideally be determined by appropriate tests, e.g. torsion of thin cylindrical tubes. Alternatively, and along the lines followed for the determination of shear response of unidirectional composites [14–16], it is proposed to derive S(N) by dividing the experimental fatigue strength of 45° off-axis coupon by the factor 2.2. This empirical value is shown to fit in an acceptable manner the experimental data from all different tests at various stress ratio, R, values.
3. Materials and testing A comprehensive experimental investigation was carried out, consisting of fatigue tests on straight-edge coupons cut from a multidirectional laminate made of Eglass/polyester. The stacking sequence consists of four layers, two unidirectional laminae of 100% aligned fibers, with a weight of 700 g/m2, and two stitched ±45 of 450 g/m2, 225 g/m2 in each off-axis angle. Rectangular plates were fabricated by a hand lay-up technique and cured at room temperature. Coupons were cut with a diamond wheel at 0°, on-axis, and 15°, 30°, 45°, 60°, 75° and 90° off-axis directions. Test data from on-axis, 45° and 90° off-axis coupons were used as baseline data, while data obtained from tests at other directions were used for the validation of the FTPF criterion. A total of 257 coupons were tested under uniaxial cyclic stress for the determination of the fatigue behaviour at various off-axis directions and loading conditions. Fatigue tests of sinusoidal, constant amplitude waveform were carried out on an MTS machine of 250 kN capacity under load control. In total, 17 S–N curves were determined experimentally, under four different stress ratios, R=10 (C–C), R=⫺1 (T–C), R=0.1 and R=0.5 (T–T). The frequency was kept constant at 10 Hz for all the tests, which were continued until coupon ultimate failure or 106 cycles, whichever occurred first. Especially for the on-axis coupons, tested under reversed loading, R=⫺1, was continued up to 5×106 cycles. An antibuckling jig was used, where appropriate, in this experimental program. More details on the materials, test methods and results can be found in Part I of this work [1].
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Fig. 1. Comparison of theoretical (FTPF) predictions and experimental data from C–C loading, R=10.
Fig. 3. Comparison of theoretical (FTPF) predictions and experimental data from T–T loading, R=0.1.
4. Validation of FTPF The efficient and reliable prediction of fatigue life of any structural component under complex stress states is of paramount importance in design. Such a task can be carried out by means of the FTPF criterion, which for plane stress conditions is expressed by Eq. (7). For the formulation of the criterion in the principal material directions of a laminate possessing similar strength symmetries as the one investigated herein, the S–N curves along the two orthogonal symmetry directions as well as the respective shear fatigue strength must be known. Comparison of FTPF predictions with experimental data from various material systems as well as with theoretical predictions from other strength criteria can be found in [12], where the criterion was shown to predict satisfactorily fatigue strength under uniaxial or multiaxial cyclic loads. For the experimental off-axis data of this program under R=10, ⫺1 and 0.1, respectively, calculations following the aforementioned methodology were performed and the results are presented in Figs. 1– 3. Shear strength S–N formulations are derived for all
Fig. 2. Comparison of theoretical (FTPF) predictions and experimental data from T–C loading, R=⫺1.
cases by dividing the experimental data of 45° off-axis coupons by the factor 2.2. Conservative predictions of the criterion are produced in this way for all but one offaxis directions and loading cases, that of 15°, tested under R=0.1. This is valid especially for low cycle fatigue whereas for N⬎1E+07 the situation is reversed, i.e. the predictions are conservative for this loading case, R=0.1, as well. The results of Figs. 1–3 were derived through Eq. (9). The mathematical expressions for X(N), Y(N) and S(N) correspond to the median survival probability approximately. If a higher reliability level is required, the procedure can be repeated by using values for X(N), Y(N) and S(N), corresponding to that survival probability. It can be concluded from Figs. 1–3 that the predictions of the FTPF criterion for different stress ratios and offaxis orientations such as 30°, 60° or 75° are good and always on the safe side. This means that the material model selected for direct characterization of the laminate investigated, together with the strength criterion itself, forms a successful combination for predicting fatigue strength under complex plane stress states. The application of the method for the case of 15° off-axis coupons under R=0.1 gave non-conservative predictions and this is assigned to the material model selected, i.e. “homogeneous orthotropic” medium. For unidirectional glass/epoxy laminates tested off-axis, it was shown in [12] that predictions of fatigue strength by the FTPF criterion were corroborated satisfactorily by the experimental data for the entire range of off-axis directions. Therefore, it is logical to conclude that the quadratic version of the failure tensor polynomial is adequate for engineering calculations, where the need for safe and reliable predictions is of paramount importance. If higher accuracy is needed, from the material characterization point of view, accurate experimental characterization of the shear fatigue strength, S(N), of the laminated medium and higher order tensor formulation [17] of the criterion
T.P. Philippidis, A.P. Vassilopoulos / International Journal of Fatigue 24 (2002) 825–830
are recommended steps to improve the predictive capability.
冦冧冦 冧 s1a
冦
17.0
4
s2a ⫽ 1.02 MPa at R ⫽ 0.254
5. Effect of complex stress state on fatigue life prediction Based on the results of the experimental study presented in the first part of this work [1] and the FTPF criterion discussed in the previous sections, it is proposed to quantify the so-called “complex stress-state effect” in predicting fatigue life of a thin-wall boxbeam structure. To this end, consider as an example the case of a 20m wind turbine rotor blade made of GRP. Under a realistic load case (power production at rated wind speed), the plane stress field developed in leading-edge inboard regions is typical of that presented in Fig. 4. The part of the Finite Element Method (FEM) model shown corresponds to the cylindrical root and the transition region bridging to the aerodynamic section of the blade. Also shown is part of the inner structure, while grayscale shades correspond to different values of shear flow. Magnified is a typical element and the corresponding inplane stress resultants, Ni (i=1, 2, 6). It is indeed observed that the transverse normal and shear components are only small fractions of the axial stress resultant, albeit not negligible in fatigue life calculations as shown in the sequel. Assuming that the stacking sequence of the multilayer element is similar to that considered in this study, i.e. [0/(±45)2/0]nT, of a thickness 0.02 m approximately, the following average (over the element thickness) stress amplitudes are derived for the specific load case:
Fig. 4.
s6a
1.26
829
(10)
⫺0.61
For ease of calculations, sinusoidal cycling is considered, proportional for each stress component but with different R-ratio values as shown in the above. Application of the criterion by means of Eq. (7) necessitates the definition of strength allowable, X(N), Y(N) and S(N) for each specific R value. This is performed through linear interpolation in the constant life diagrams of [1] and the results for 95%-reliability characteristic values (in MPa) are: X(N,R ⫽ 4) ⫽ 94.130N⫺0.0426
(11)
Y(N,R ⫽ 0.254) ⫽ 22.866N
⫺0.0687
S(N,R ⫽ ⫺0.61) ⫽ 44.212N⫺0.0776 Substituting from Eqs. (10) and (11) into relation (7) and solving the non-linear equation for N yields a number of 5.43E+16 cycles to failure. Repeating the calculations, setting s2a=s6a=0, thus neglecting the effect of transverse normal and shear stress on fatigue life, yields a value for N equal to 2.80E+17 cycles. That means an increase in expected life by 80% approximately or a life period five times greater. Therefore, neglecting transverse normal and shear stresses in fatigue life calculations can lead to erroneous results, overestimating the bearing capability of the multilayer composite. Of course, the numbers given in the above are only indicative. Because of the non-linearity of all the equations involved, one should expect
Detail from GRP rotor blade root. Typical state of in-plane stress resultants.
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T.P. Philippidis, A.P. Vassilopoulos / International Journal of Fatigue 24 (2002) 825–830
large discrepancies, depending on the stress level [highcycle (HCF) or low-cycle (LCF) fatigue]. Usually, the effect of neglecting shear and normal stress components on predicted fatigue life is much higher for HCF regimes.
6. Conclusions Based on the experimental results presented in the first part of this work [1], a life prediction methodology accounting for complex stress states was validated by means of an appropriate fatigue strength criterion. It is usual to neglect transverse normal and shear stresses in fatigue calculations of laminated multicell beam structures such as rotor blades in general, where these stress components are very small fractions of the axial normal stress in the main fiber direction of the laminated composite. Nevertheless, fatigue strengths in the respective directions are also very different and thus, their applied counterparts become important in estimating fatigue life. It was proved by means of a simplified numerical example that neglecting some of the stress components developed in plane-stress states is misleading and overestimates substantially fatigue life of the aforementioned composite structural components.
Acknowledgements Part of this study was supported by the Greek General Secretariat of Research and Technology under contract EPET II#573, by the Center for Renewable Energy Sources (CRES) and by the EC Non-Nuclear Energy Programme under contract number JOR3-CT98-0251. GEOBIOLOGIKI SA prepared the composite plates. The authors gratefully acknowledge their assistance.
References [1] Philippidis TP, Vassilopoulos AP. Complex stress state effect on fatigue life of GRP laminates. Part I, Experimental. Int. J. Fatigue 2002;24(8):813–23. [2] Owen MJ, Griffiths JR. Evaluation of biaxial stress failure surfaces for a glass fabric reinforced polyester resin under static and fatigue loading. J. Mater. Sci. 1978;13:1521–37. [3] Fujii T, Lin F. Fatigue behaviour of a plain-woven glass fabric laminate under tension/torsion biaxial loading. J. Compos. Mater. 1995;29(5):573–90. [4] Philippidis TP, Vassilopoulos AP. Fatigue of composite laminates under off-axis loading. Int. J. Fatigue 1999;21:253–62. [5] Hashin Z, Rotem A. A fatigue criterion for fiber reinforced materials. J. Compos. Mater. 1973;7:448–64. [6] Rotem A. Fatigue failure of multidirectional laminate. AIAA J. 1979;17(3):271–7. [7] Tsai SW, Wu EM. A general theory of strength for anisotropic materials. J. Compos. Mater. 1971;5:58–80. [8] Fawaz Z, Ellyin F. Fatigue failure model for fibre-reinforced materials under general loading conditions. J. Compos. Mater. 1994;28(15):1432–51. [9] Jen M-HR, Lee C-H. Strength and life in thermoplastic composite laminates under static and fatigue loads. Part I: Experimental. Int. J. Fatigue 1998;20(9):605–15. [10] Jen M-HR, Lee C-H. Strength and life in thermoplastic composite laminates under static and fatigue loads. Part II: Formulation. Int. J. Fatigue 1998;20(9):617–29. [11] Tsai SW, Hahn HT. Introduction to composite materials. Technomic, 1980. [12] Philippidis TP, Vassilopoulos AP. Fatigue strength prediction under multiaxial stress. J. Compos. Mater. 1999;33(17):1578–99. [13] Theocaris PS, Philippidis TP. On the validity of the tensor polynomial failure theory with stress interaction terms omitted. Compos. Sci. Technol. 1991;40:181–91. [14] Chamis CC, Sinclair JH. Ten-deg off-axis tests for shear properties in fiber composites. Exp. Mech. 1997;17(9):339–46. [15] Yeow YT, Brinson HF. A comparison of simple shear characterization methods for composite laminates. Composites 1978;9(1):49–55. [16] Lee S, Munro M. Evaluation of in-plane shear test methods for advanced composite materials by the decision analysis technique. Composites 1986;17(1):13–22. [17] Wu EM, Scheublein JK. Laminate strength—a direct characterization procedure in composite materials: testing and design. ASTM STP 546. American Society for Testing and Materials, 1974, pp. 188–206.
Complex stress state effect on fatigue life of GRP laminates. Part II, Theoretical formulation Theodore P. Philippidis *, Anastasios P. Vassilopoulos Department of Mechanical Engineering and Aeronautics, University of Patras, PO Box 1401, 26500 Patras, Greece Received 23 July 2001; received in revised form 13 December 2001; accepted 20 December 2001
Abstract The synergistic effect of in-plane stress tensor components on fatigue strength is not traditionally considered in the design of thin-wall box-beam structures, e.g. composite rotor blades in general. Fatigue life calculations account only for the normal stresses due to bending and centrifugal forces, neglecting the contribution of shear and transverse normal stresses. The theoretical formulation of a life prediction methodology accounting for all in-plane stress tensor components, through the use of a multiaxial fatigue strength criterion, is presented here. Comparison of theoretical predictions with experimental results from constant amplitude, uniaxial, offaxis tests demonstrates the drastic effect of shear and transverse normal stresses, besides that of axial normal stress, in reducing operational life of a GRP structural laminate. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Composites; Fatigue strength; Multiaxial stress; Fatigue design
1. Introduction In the first part of this work [1], results from a detailed experimental program were presented, which, along with experimental evidence from earlier studies [2–4], highlighted the drastic effect of complex stress states on fatigue strength of glass fiber-reinforced plastic (GFRP) laminates. These investigations were limited to constant amplitude proportional loading. The aim of this research, the results of which are presented in [1] and in this paper, was to investigate the effect of shear and transverse normal stress on the fatigue strength of GRP (glass/polyester) laminates which are mainly stressed axially, in one of their principal material directions. Typically, such stress states arise in box-beam structural members loaded by transverse loads, e.g. aerodynamic forces, producing mainly spanwise distributions of flexural moments, shear forces and occasionally moderate torsional moments. The presence of shear and transverse normal stress relative to the main axial normal stress in the skin of the shell structure is kept small.
* Corresponding author. Tel. and fax: +30-61-997235. E-mail address: [email protected] (T.P. Philippidis).
Predictive formulations for fatigue life, accounting for stress multiaxiality, first appeared in 1970 for composite materials. Most of the proposed criteria were generalizations of static failure criteria to take into account number of cycles, frequency and other fatigue parameters. Hashin and Rotem [5] proposed a fatigue strength criterion for fiber-reinforced materials, based on the different failure modes exhibited. For unidirectional materials two failure modes exist, the fiber and the matrix failure mode. When multidirectional laminates are considered [6], another failure mode, the interlaminar, is encountered. For the application of this criterion, three experimental S–N curves are needed, and it can be implemented only for materials for which failure modes can be clearly discriminated. Owen and Griffiths [2] presented an experimental program consisting of static and fatigue tests to correlate the experimentally obtained fatigue data with predictions from multiaxial static failure criteria. Fujii and Lin [3], in order to validate their experimental, biaxial tension/torsion fatigue data, have interpreted the Tsai– Wu strength criterion [7] for fatigue. However, the adaptation proved to be insufficient to correctly predict fatigue behavior of the material system investigated. Fawaz and Ellyin [8] proposed a multiaxial fatigue failure criterion based on only one experimental S–N
0142-1123/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 2 ) 0 0 0 0 4 - X
826
T.P. Philippidis, A.P. Vassilopoulos / International Journal of Fatigue 24 (2002) 825–830
curve, and some static strengths. Multiaxiality is entered through static failure criteria and S–N curves under complex stress states can be predicted. This constitutes a flexible fatigue failure condition, capable of eventually yielding satisfactory predictions. Wide acceptance, however, is constrained by its high sensitivity in the choice of the single experimental S–N formulation required for its application. Jen and Lee [9,10] predicted the fatigue behaviour of off-axis loaded unidirectional composites by introducing an appropriately modified version of the Tsai–Hill criterion. They extended the predictive capability of their theoretical formulation for multidirectional laminates, made of the same prepreg, by means of classical lamination theory and ply-discount considerations. Based on the latter assumptions for stiffness degradation of a failed ply, their theoretical predictions for APC-2 laminates were shown to be corroborated satisfactorily by experimental evidence. An extension of the quadratic version of the failure tensor polynomial, as interpreted by Tsai and Hahn [11], for the prediction of fatigue strength under complex stress states was introduced by the authors [12]. This socalled FTPF (failure tensor polynomial in fatigue) criterion was shown to predict satisfactorily the remaining life or fatigue strength for the entire range of composite materials investigated under off-axis or multiaxial loading. In this paper, the results from the experimental program presented in [1], consisting of uniaxial fatigue tests on coupons cut at several off-axis orientations from a multidirectional GRP laminate, [0/(±45)2/0]T, are used for further validation of the FTPF criterion. Based on this, a theoretical formulation is developed for structural dimensioning and fatigue life prediction of box-beam shell structures. The effect of shear and transverse normal stress on fatigue strength of the laminate loaded mainly by axial normal stress is investigated in depth. The approach differs from previously mentioned studies in the material model used. A direct characterization procedure is adopted for the [0/(±45)2/0]T laminate which is considered as a “homogeneous” anisotropic continuum, thus avoiding uncertainties in the modeling of stiffness degradation of failed layers or interlaminar effects and interactions. This is particularly useful for the GRP laminate investigated, consisting of different glass fabrics with stitched fibers at various orientations. In such cases, simplistic theoretical considerations for load distribution and ply stiffness degradation are no longer applicable.
was called the failure tensor polynomial in fatigue (FTPF) and it was shown to predict successfully fatigue strength under reversed (R=⫺1) complex plane stress states. The failure tensor polynomial for orthotropic media expressed in material symmetry axes, under static plane stress, is given by: F11s21 ⫹ F22s22 ⫹ 2F12s1s2 ⫹ F1s1 ⫹ F2s2
(1)
⫹ F66s26⫺1ⱕ0 where F11 ⫽
1 1 1 1 1 ,F ⫽ ,F ⫽ ,F ⫽ ⫺ , XX⬘ 22 YY⬘ 66 S2 1 X X⬘
(2)
1 1 F2 ⫽ ⫺ . Y Y⬘ X and X⬘ are the tension and compression strengths, respectively, along direction 1 of the material principal system, Y and Y⬘ are the corresponding values for the transverse direction, while S is the shear strength. The specific choice of the off-diagonal term, F12, was shown to lead to completely different failure theories [13]. The form of F12 used in this study is that given by Tsai and Hahn [11]: 1 F12 ⫽ ⫺ 冑F11F22. 2
(3)
FTPF assumes the same functional form as Eq. (1); however, the components of failure tensors are functions of the number of cycles, N, the stress ratio, R, and the frequency, n, of the loading: Fij ⫽ Fij(N,R,n), Fi ⫽ Fi(N,R,n).
(4)
Experimental evidence gained so far for any type of continuous fiber-reinforced polymers strongly suggests the form of functional dependence of failure tensor components shown in Eq. (4). This implies an increased complexity of experimental strength characterization compared to static loading, since it is necessary to discriminate not only between tension or compression and loading rate but also between the same type of loading, e.g. tension, at different R-values or loading frequency, n. Therefore, it was proposed to characterize in-plane fatigue strength of an orthotropic material by means of three experimentally derived S–N curves [12]: 1
X(N,R,n) ⫽ XoN⫺kX
(5)
1
Y(N,R,n) ⫽ YoN⫺kY 1
S(N,R,n) ⫽ SoN⫺kS 2. The fatigue strength criterion
The failure tensor components are defined by:
A modification of the failure tensor polynomial [7] to account for fatigue loading was introduced in [12]. It
F11 ⫽
1 1 ,F ⫽ , X2(N,R,n) 22 Y2(N,R,n)
(6)
T.P. Philippidis, A.P. Vassilopoulos / International Journal of Fatigue 24 (2002) 825–830
F66 ⫽
1 1 ,F ⫽⫺ , F ⫽ F2 S2(N,R,n) 12 2X(N,R,n)Y(N,R,n) 1
⫽ F6 ⫽ 0. Finally, the fatigue strength or life under complex plane stress states and specific R and n cycling parameters is predicted using the equation: s22 s1s2 s26 s21 ⫹ 2 ⫺ ⫹ 2 ⫺1 ⫽ 0. X (N) Y (N) X(N)Y(N) S (N) 2
(7)
It must be noted that the three S–N curves X(N), Y(N) and S(N) in the above equations are derived for the same loading conditions, R, n, as those of the actual stress state {s1, s2, s6}. The experimental characterization of X(N), Y(N) is performed through uniaxial fatigue tests along the respective principal material direction. Experimental determination of fatigue shear strength, S(N), is more complicated as expensive and sophisticated testing equipment and specimens are required. Thus, simple shear characterization methods were devised previously [14–16] and were successfully used to predict stiffness and strength of composite laminates. Shear strength, S(N), in [12] was considered equal to half the value of the fatigue strength of a flat coupon cut off-axis at 45° and loaded uniaxially. This choice yielded satisfactory results for reversed loading, R=⫺1, but its performance proved to be less effective for other loading types such as R=10 and R=0.1. Nevertheless, as it is shown in the sequel, adopting for S(N) the value of 1/2.2 of the fatigue strength of a flat coupon cut off-axis at 45° and loaded uniaxially fits adequately most of the experimental data. The effect of shear stress, s6, and transverse normal stress, s2, on fatigue strength or life is quantified by means of Eq. (7). If, for example, off-axis uniaxial tests are performed, the developed plane stress state depends on the load orientation with respect to the material principal directions. Denoting by si (i=1, 2, 6) the in-plane stress tensor components in the symmetry coordinate system of the multidirectional laminate, and by sx the applied normal stress at a positive off-axis angle q, the following transformation relations are valid: s1 ⫽ sx cos2q; s2 ⫽ sx sin2q; s6 ⫽ sx sin q cos q
(8)
The plane-stress state ratios s2/s1 and s6/s1 as a function of q take values that are equal to tan2q and tan q, respectively. Then, depending on the off-axis direction, the state of stress is either dominated by the axial normal stress, s1, for lower angles q or by the other two stress components, s2, s6, for angles greater than 60°. Substituting relations (8) into Eq. (7) yields the following equation for the off-axis fatigue strength at orientation q: sx(N) ⫽
冋
cos4q sin4q ⫹ 2 ⫹ cos2q sin2q 2 X (N) Y (N)
(9)
冉
冊册
1 1 ⫺ 2 S (N) X(N)Y(N)
⫺
827
1 2
With respect to material properties that need to be determined experimentally in order to use the FTPF criterion (Eqs. (7) and (9)), it is clear that X(N) and Y(N) are the S–N curves determined from on-axis and 90° offaxis coupon tests. These are the fatigue strengths in the two principal, mutually orthogonal, material directions of the “homogeneous” plate [0/(±45)2/0]T. S(N), the shear fatigue strength for the specific stress ratio, R, and the loading frequency should ideally be determined by appropriate tests, e.g. torsion of thin cylindrical tubes. Alternatively, and along the lines followed for the determination of shear response of unidirectional composites [14–16], it is proposed to derive S(N) by dividing the experimental fatigue strength of 45° off-axis coupon by the factor 2.2. This empirical value is shown to fit in an acceptable manner the experimental data from all different tests at various stress ratio, R, values.
3. Materials and testing A comprehensive experimental investigation was carried out, consisting of fatigue tests on straight-edge coupons cut from a multidirectional laminate made of Eglass/polyester. The stacking sequence consists of four layers, two unidirectional laminae of 100% aligned fibers, with a weight of 700 g/m2, and two stitched ±45 of 450 g/m2, 225 g/m2 in each off-axis angle. Rectangular plates were fabricated by a hand lay-up technique and cured at room temperature. Coupons were cut with a diamond wheel at 0°, on-axis, and 15°, 30°, 45°, 60°, 75° and 90° off-axis directions. Test data from on-axis, 45° and 90° off-axis coupons were used as baseline data, while data obtained from tests at other directions were used for the validation of the FTPF criterion. A total of 257 coupons were tested under uniaxial cyclic stress for the determination of the fatigue behaviour at various off-axis directions and loading conditions. Fatigue tests of sinusoidal, constant amplitude waveform were carried out on an MTS machine of 250 kN capacity under load control. In total, 17 S–N curves were determined experimentally, under four different stress ratios, R=10 (C–C), R=⫺1 (T–C), R=0.1 and R=0.5 (T–T). The frequency was kept constant at 10 Hz for all the tests, which were continued until coupon ultimate failure or 106 cycles, whichever occurred first. Especially for the on-axis coupons, tested under reversed loading, R=⫺1, was continued up to 5×106 cycles. An antibuckling jig was used, where appropriate, in this experimental program. More details on the materials, test methods and results can be found in Part I of this work [1].
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Fig. 1. Comparison of theoretical (FTPF) predictions and experimental data from C–C loading, R=10.
Fig. 3. Comparison of theoretical (FTPF) predictions and experimental data from T–T loading, R=0.1.
4. Validation of FTPF The efficient and reliable prediction of fatigue life of any structural component under complex stress states is of paramount importance in design. Such a task can be carried out by means of the FTPF criterion, which for plane stress conditions is expressed by Eq. (7). For the formulation of the criterion in the principal material directions of a laminate possessing similar strength symmetries as the one investigated herein, the S–N curves along the two orthogonal symmetry directions as well as the respective shear fatigue strength must be known. Comparison of FTPF predictions with experimental data from various material systems as well as with theoretical predictions from other strength criteria can be found in [12], where the criterion was shown to predict satisfactorily fatigue strength under uniaxial or multiaxial cyclic loads. For the experimental off-axis data of this program under R=10, ⫺1 and 0.1, respectively, calculations following the aforementioned methodology were performed and the results are presented in Figs. 1– 3. Shear strength S–N formulations are derived for all
Fig. 2. Comparison of theoretical (FTPF) predictions and experimental data from T–C loading, R=⫺1.
cases by dividing the experimental data of 45° off-axis coupons by the factor 2.2. Conservative predictions of the criterion are produced in this way for all but one offaxis directions and loading cases, that of 15°, tested under R=0.1. This is valid especially for low cycle fatigue whereas for N⬎1E+07 the situation is reversed, i.e. the predictions are conservative for this loading case, R=0.1, as well. The results of Figs. 1–3 were derived through Eq. (9). The mathematical expressions for X(N), Y(N) and S(N) correspond to the median survival probability approximately. If a higher reliability level is required, the procedure can be repeated by using values for X(N), Y(N) and S(N), corresponding to that survival probability. It can be concluded from Figs. 1–3 that the predictions of the FTPF criterion for different stress ratios and offaxis orientations such as 30°, 60° or 75° are good and always on the safe side. This means that the material model selected for direct characterization of the laminate investigated, together with the strength criterion itself, forms a successful combination for predicting fatigue strength under complex plane stress states. The application of the method for the case of 15° off-axis coupons under R=0.1 gave non-conservative predictions and this is assigned to the material model selected, i.e. “homogeneous orthotropic” medium. For unidirectional glass/epoxy laminates tested off-axis, it was shown in [12] that predictions of fatigue strength by the FTPF criterion were corroborated satisfactorily by the experimental data for the entire range of off-axis directions. Therefore, it is logical to conclude that the quadratic version of the failure tensor polynomial is adequate for engineering calculations, where the need for safe and reliable predictions is of paramount importance. If higher accuracy is needed, from the material characterization point of view, accurate experimental characterization of the shear fatigue strength, S(N), of the laminated medium and higher order tensor formulation [17] of the criterion
T.P. Philippidis, A.P. Vassilopoulos / International Journal of Fatigue 24 (2002) 825–830
are recommended steps to improve the predictive capability.
冦冧冦 冧 s1a
冦
17.0
4
s2a ⫽ 1.02 MPa at R ⫽ 0.254
5. Effect of complex stress state on fatigue life prediction Based on the results of the experimental study presented in the first part of this work [1] and the FTPF criterion discussed in the previous sections, it is proposed to quantify the so-called “complex stress-state effect” in predicting fatigue life of a thin-wall boxbeam structure. To this end, consider as an example the case of a 20m wind turbine rotor blade made of GRP. Under a realistic load case (power production at rated wind speed), the plane stress field developed in leading-edge inboard regions is typical of that presented in Fig. 4. The part of the Finite Element Method (FEM) model shown corresponds to the cylindrical root and the transition region bridging to the aerodynamic section of the blade. Also shown is part of the inner structure, while grayscale shades correspond to different values of shear flow. Magnified is a typical element and the corresponding inplane stress resultants, Ni (i=1, 2, 6). It is indeed observed that the transverse normal and shear components are only small fractions of the axial stress resultant, albeit not negligible in fatigue life calculations as shown in the sequel. Assuming that the stacking sequence of the multilayer element is similar to that considered in this study, i.e. [0/(±45)2/0]nT, of a thickness 0.02 m approximately, the following average (over the element thickness) stress amplitudes are derived for the specific load case:
Fig. 4.
s6a
1.26
829
(10)
⫺0.61
For ease of calculations, sinusoidal cycling is considered, proportional for each stress component but with different R-ratio values as shown in the above. Application of the criterion by means of Eq. (7) necessitates the definition of strength allowable, X(N), Y(N) and S(N) for each specific R value. This is performed through linear interpolation in the constant life diagrams of [1] and the results for 95%-reliability characteristic values (in MPa) are: X(N,R ⫽ 4) ⫽ 94.130N⫺0.0426
(11)
Y(N,R ⫽ 0.254) ⫽ 22.866N
⫺0.0687
S(N,R ⫽ ⫺0.61) ⫽ 44.212N⫺0.0776 Substituting from Eqs. (10) and (11) into relation (7) and solving the non-linear equation for N yields a number of 5.43E+16 cycles to failure. Repeating the calculations, setting s2a=s6a=0, thus neglecting the effect of transverse normal and shear stress on fatigue life, yields a value for N equal to 2.80E+17 cycles. That means an increase in expected life by 80% approximately or a life period five times greater. Therefore, neglecting transverse normal and shear stresses in fatigue life calculations can lead to erroneous results, overestimating the bearing capability of the multilayer composite. Of course, the numbers given in the above are only indicative. Because of the non-linearity of all the equations involved, one should expect
Detail from GRP rotor blade root. Typical state of in-plane stress resultants.
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large discrepancies, depending on the stress level [highcycle (HCF) or low-cycle (LCF) fatigue]. Usually, the effect of neglecting shear and normal stress components on predicted fatigue life is much higher for HCF regimes.
6. Conclusions Based on the experimental results presented in the first part of this work [1], a life prediction methodology accounting for complex stress states was validated by means of an appropriate fatigue strength criterion. It is usual to neglect transverse normal and shear stresses in fatigue calculations of laminated multicell beam structures such as rotor blades in general, where these stress components are very small fractions of the axial normal stress in the main fiber direction of the laminated composite. Nevertheless, fatigue strengths in the respective directions are also very different and thus, their applied counterparts become important in estimating fatigue life. It was proved by means of a simplified numerical example that neglecting some of the stress components developed in plane-stress states is misleading and overestimates substantially fatigue life of the aforementioned composite structural components.
Acknowledgements Part of this study was supported by the Greek General Secretariat of Research and Technology under contract EPET II#573, by the Center for Renewable Energy Sources (CRES) and by the EC Non-Nuclear Energy Programme under contract number JOR3-CT98-0251. GEOBIOLOGIKI SA prepared the composite plates. The authors gratefully acknowledge their assistance.
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