Benchmarking of lamina failure tests from WWFE-I and WWFE-II with a three parameter micromechanics based matrix failure theory

Transcription

1 International Conference on Future Technologies for Wind Energy October 07-09, 03, Laraie, Wyoing, USA Bencharking of laina failure tests fro WWFE-I and WWFE-II with a three paraeter icroechanics based atrix failure theory Kedar A. Malusare*, Ray S. Fertig III** *Graduate Student, Mechanical Engineering, University of Wyoing, kalusar@uwyo.edu **Assistant Professor, Mechanical Engineering, University of Wyoing, rfertig@uwyo.edu Abstract The use of fiber reinforced coposites in the wind energy industry is growing rapidly. Owing to their increasing iportance, it is critical to develop tools to accurately predict their coplex behavior under different loading conditions. However, their heterogeneous icrostructure and inherent anisotropy ake failure prediction very difficult. Two well-known coposite failure bencharks, World-Wide Failure Exercise I (WWFE-I) and World-Wide Failure Exercise II (WWFE-II) copared leading failure theories fro around the world with one another and with actual test results. This paper presents a three-paraeter icroechanics-based coposite failure theory which uses constituent-level stresses to predict atrix-doinated coposite failure. A representative volue eleent (RVE) of the icrostructure is used to extract constituent stresses for use with the failure theory. The erit of the failure theory lies in its siplistic calibration which requires just three paraeters that can be obtained fro three standard coposite failure tests (transverse tension, transverse copression and in-plane shear). This theory is bencharked against laina failure test data fro WWFE-I and WWFE-II and copared with the failure theories that perfored coparatively well in WWFE-I and WWFE-II. Our results show that for ost of the test cases the predictions of the theory were in close agreeent with the test data. Key Words: Coposite aterials, failure odeling, bencharking, WWFE-I, WWFE-II Introduction Modern coposite aterials constitute a significant portion of engineered aterials arket ranging fro everyday products to sophisticated niche applications []. Unlike conventional hoogeneous aterials like etals, the properties of coposite aterials can be tailored to eet the structural deands of the end product, which results in significant weight and cost savings. One of the sectors where fibrous coposites are a lucrative choice of aterials is the wind energy sector. It is iportant to increase renewable energy production, particularly wind energy generation, to achieve the goal of reducing our dependency on fossil fuels. This can be realized by installing and expanding any on-shore and off-shore wind fars built with large and extralarge wind turbines. The durability and longevity of a wind turbine can be enhanced if the wind blade aterials have high stiffness and strength, and fatigue and environental resistance. Fibrous coposites have been utilized extensively in wind turbines because they offer all the above advantages. Wind turbine blades are exposed to a variety of coplex external loads originating fro wind, gravity and other natural eleents. In order to ensure durability of the turbine blade, reliable failure theories are needed that can predict failure of the blade aterials. The approach in this paper is to use ultiscale odeling to predict constituent-level failure. The focus of this work will be on bencharking an existing atrix failure criterion, coparing it with soe of the leading failure theories, and suggesting areas for enhanceent.

2 International Conference on Future Technologies for Wind Energy October 07-09, 03, Laraie, Wyoing, USA Failure Modeling The Maxiu Distortional Energy, or von Mises, criterion is the ost widely used criterion for predicting yielding in isotropic etals []. Coposite aterials, unlike etals, are anisotropic aking failure prediction a coplicated echanis. For coposite aterials, failure criteria can be broadly categorized into three groups: liit criteria, interactive criteria, and separate ode criteria [3]. The failure theories falling under liit criteria (e.g. axiu stress) predict failure load by coparing laina level stresses (, and ) with corresponding strengths separately. They do not consider interaction aong the different stresses and consequently give the ost conservative results. The interactive failure theories (e.g Tsai-Wu [4] and Tsai Hill) predict failure load using a quadratic or higher order polynoial equation which involves all the stress coponents. The onset of failure is assued when this equation is satisfied. Separate ode criteria are failure theories that have separate atrix failure criterion and fiber failure criterion. Failure load is predicted by equations which depend on either one or ore stress coponents. Hashin [5], Christensen [6] and Puck [7] failure criterion are soe of the exaples of separate ode failure criteria. Failure theories ay also be classified as eso-echanical or icroechanical depending upon the kind of stresses used to predict failure. Meso-echanical failure theories use laina level stresses to predict failure of a laina, whereas icro-echanical failure theories eploy constituent level stresses to predict failure of a constituent of a laina. To copare the different failure theories and assess the aturity of different coposite failure criteria, Soden, Kaddour and Hinton organized the World Wide Failure Exercises [8] [9]. These coposite failure bencharks contain detailed assessents of different theories and their approaches for predicting the failure response of polyer coposite lainates under coplex states of stress. The different failure theories were bencharked against carefully selected test cases after which they were assessed qualitatively. Puck [7], Zinoview [8], Tsai [9] and Bogetti [0] were ranked highest [] in the first World Wide Failure Exercise (WWFE-I) and Carrere [], Pinho [3], Cuntze [4] and Puck [5] were ranked highest [6] in the second World Wide Failure Exercise (WWFE-II) for their predictions and descriptions of failure echanics in coposite aterials. One ajor drawback that is evident in all these theories is the use of a substantially large nuber of input paraeters (fro 50-75) and thus difficult in calibrating the. The icroechanical atrix failure theory used in this paper requires three paraeters, which all have physical eaning, and utilizes volue-average constituent level stresses to predict failure of a constituent (atrix or fiber here) and thereby of the coposite. The erit of the failure theory lies in its siplistic calibration which is required to obtain the three paraeters required to predict failure load under any coposite state of stress. The atrix failure theory is outlined below [7] B t I t 0 B si s BsI s I h () where B,, t Bs Bs represent the coefficients of the stress invariants; I t, Is, Is, Ih represent the invariants of atrix stress tensor, represents the shear strength of the atrix, 0 and represents pressure strengthening due to copressive loading (~ 0.35) [8].

3 International Conference on Future Technologies for Wind Energy October 07-09, 03, Laraie, Wyoing, USA 3 The values of B i are deterined fro three coposite static failure tests: transverse tension, transverse copression, and in-plane shear, all of which involve failure of the atrix constituent. The invariants are coputed fro the volue average atrix stresses as follows I t I I s s () (3) (4) 4 (5) I h 33 where I t corresponds to the axiu tensile stress noral to the fiber, I s is related to the inplane shear, I s is related to the transverse shear, and I h represents the pressure on the axiu transverse shear. The fiber failure criterion is outlined below S f f or f S f (6) where fiber. f S is the longitudinal tensile strength of the fiber and f S copressive strength of the The constituent level stresses were extracted fro a Representative Volue Eleent (RVE) of an idealized icrostructure of a hexagonally packed fiber reinforced coposite, shown in Fig.. The RVE has periodic boundary conditions enforced on all its sides and was subjected to six types of loads,,,,,, which generated stresses in the fiber and the atrix regions. After extracting stresses fro the fiber and the atrix regions, volue average constituent stresses were coputed. There exists a apping between the coposite and constituent stresses, which can be used to copute constituent level stresses for any type of coposite load state. This apping can be coputed as shown below [9] Figure : The RVE with hexagonal fiber packing L f i c L f i i and L i (7) c L

4 International Conference on Future Technologies for Wind Energy October 07-09, 03, Laraie, Wyoing, USA 4 where, the subscript L denotes the load case, i denotes the six coponents of the stress vector, f denotes the fiber, denotes the atrix and c denotes the coposite. For exaple the apping functions for atrix constituent under a pure coposite load state are i c i (8) Thus for any coposite load state, the stress i in a constituent a is given by a i 6 L L a i c j (9) Results The focus of this work is to use the Fertig failure theory to predict failure of coposites and benchark it against laina failure test data [0][] fro well know coposite failure bencharks World Wide Failure Exercise-I (WWFE-I) and World Wide Failure Exercise-II (WWFE-II). The failure exercises contain in total twenty-six carefully selected test cases which include strength envelopes and stress-strain curves for a range of unidirectional and ultidirectional lainates. This failure theory is bencharked against seven strength envelopes for unidirectional lainae. These are ideal for evaluating failure criteria, whereas lainate level tests are appropriate for evaluating the cobination of failure criteria with progressive daage ethodology. The details of the test cases are included in Table. The results obtained were copared with leading theories fro WWFE-I and WWFE-II discussed above. Table Test Laina Material Loading case layup 0 E-glass/LY556 epoxy vs. 0 T300/BSL94C carbon/epoxy vs. 3 0 E-glass/MY750 epoxy vs. 4 0 T300/PR39 vs. ( ) E-glass/MY750 epoxy vs. 3 ( 3) 6 0 S-glass/epoxy vs. 3 ( 3) 7 0 Carbon/epoxy vs. 3 ( 3) The bencharking of the theory and its coparison with the leading failure theories is shown below. Test case : GRP laina under cobined transverse noral and shear loading The failure envelopes predicted by the Fertig failure theory for case are shown in Fig.. Figure (a) shows the failure envelope when the UD values provided by the WWFE-I authors were used as odel inputs. Figure (b) shows the failure envelope obtained when different UD values were

5 International Conference on Future Technologies for Wind Energy October 07-09, 03, Laraie, Wyoing, USA 5 used as inputs to the odel which yielded slightly better results. In both the cases (a) and (b), the theory fits the shape of the test data very well especially in the (, ) quadrant. In case (a), Fertig failure theory is conservative, especially in the (, ) quadrant. By choosing a different transverse copressive strength than the one given by the originators of the exercise, the theory predicts a failure envelope which is a little less conservative. Test case : GRP laina under cobined longitudinal and shear loading The failure envelopes predicted by the Fertig failure theory for test case are shown in Fig. 3. It can be seen that the theory captures the general shape of the experients very well except in the high-shear region. It is known that aterial inhoogeneity in a coposite gives rise to stress and strain fluctuations in the constituents. We have already shown that the volue average atrix stresses do not capture these stress/strain fluctuations in the constituents of the coposite aterial and thus all the strain energy of a constituent is not accounted for []. The bulk of this issing energy (about 30%) is due to fluctuations in the atrix constituent when the coposite is a under shear state of stress. Because our failure theory uses volue average constituent level stresses to predict failure of the constituents of the coposite aterial, the atrix failure in this high shear region is not captured well since the issing strain energy in the atrix constituent is ignored. The atrix failure theory needs to be augented with this issing energy to iprove the predictions for this test case.

6 International Conference on Future Technologies for Wind Energy October 07-09, 03, Laraie, Wyoing, USA 6 Test case 3: CFRP laina under cobined noral and longitudinal loading The failure predictions of our failure theory and a odified failure theory are shown in Fig. 4. Figure 4(a) shows the failure envelope obtained using the original Fertig failure theory. It can be seen that like ost of leading theories, the Fertig failure theory is very conservative in the, quadrant. Figure 4(b) shows the failure predictions of a odified Fertig failure theory which captures the shape of the experients better than any of the leading failure theories in WWFE-I. This odified approach first requires the calculation of the atrix stress concentration factor in an ideal icrostructure at the point of critical atrix failure under transverse tension. Figure 5 shows the fluctuations in the axiu principal atrix stress when the RVE with hexagonal fiber packing was subjected to transverse failure load.

7 International Conference on Future Technologies for Wind Energy October 07-09, 03, Laraie, Wyoing, USA 7 The atrix stress concentration factor ( ) can be coputed using the axiu and noinal (or volue average) stresses as follows (0) ax no where is the atrix stress concentration factor ax and are axiu principal and noinal atrix stresses respectively no The odified atrix failure criteria is as follows ax principal St VM or S VM () where ax principal is the axiu principal atrix stress VMl is the Von Mises stress in the atrix S is the transverse tensile strength of the atrix t S is the Von Mises strength of the atrix VM and is the atrix stress concentration factor The fiber failure criterion reains unchanged. In test case 3, it was observed that after the point of critical failure, the atrix failure was due to large principal stresses in the fiber direction. After this point even though the atrix was failing, the fiber could hold the coposite together but it now it was carrying a larger load. To calculate the resultant fiber stresses after atrix failure, the atrix properties were degraded as follows E 5% E () new new original 0.0% (3) original where new E and new and E original are the new and original Elastic oduli of the atrix respectively and original are the new and original poisons ratio of the atrix respectively. The new resultant fiber stresses were coputed fro the RVE with hexagonal fiber packing using the procedure discussed in the second section (Failure Modeling). The original fiber failure criterion and new fiber stresses were then used to copute failure load which represents catastrophic coposite failure. The odified approach was used to predict failure load in the (, ) and (, ) quadrants while the original Fertig failure theory was used in the reaining two quadrants. Test case 4: Cobined hydrostatic and shear loading The failure envelopes predicted by the Fertig failure theory for test case 4 are shown in Fig. 6. Figure 6(a) shows the failure envelope when the UD values provided by the authors were used as odel inputs and and Fig. 6(b) shows the failure envelope when different UD values were

8 International Conference on Future Technologies for Wind Energy October 07-09, 03, Laraie, Wyoing, USA 8 used as inputs to the odel and35. In both the cases (a) and (b), the theory fits the shape of the test data very well. When the shear strength of the 0 tubes is used as one of the odel inputs, the theory captures the test data of the 0 very well. When the UD values provided by the originators of the exercise are used, the pressure strengthening ter has to be reduced to 0. fro 0.35 to capture the failure of 90 tubes. Test case 5-7: The failure envelopes predicted by the Fertig failure theory for test cases 5-7, which are oitted for space, are very siilar to the other theories that have been copared. The exception is when the priary ode of failure is fiber kinking, where only two theories have been shown to be accurate, but require extensive calibration. Conclusions The failure envelopes fro the Fertig failure theory were copared with the laina failure test data fro WWFE-I and WWFE-II. It was concluded for ost of the test cases, the predictions of the theory were in close agreeent with the test data. The predictions of the theory were also siilar to the leading failure theories fro the WWFE exercises. However, the leading theories fro WWFE-I and WWFE-II require substantially ore input paraeters, which akes calibration very difficult. The erit of Fertig failure theory lies in its siplistic calibration, which requires just three paraeters that can be obtained fro three standard coposite failure tests (transverse tension, transverse copression and in-plane shear). The odified Fertig failure theory, which is a two paraeter theory, perfored better than any of the leading failure theories in predicting the strength of the coposite under cobined transverse and longitudinal loads (Test case 3). Under cobined longitudinal and shear loading (Test case ), the Fertig failure theory did not predict the failure loads when the coposite was under high-shear and lowlongitudinal stresses. It has already been shown that the volue average atrix stresses do not

9 International Conference on Future Technologies for Wind Energy October 07-09, 03, Laraie, Wyoing, USA 9 capture the stress/strain fluctuations in the constituents of the coposite aterial and thus all the strain energy of a constituent is not accounted for especially under shear loading. Moreover, for coonly used fiber volue fractions, alost all of the strain energy is due to the fluctuations in the atrix constituent. Consequently, only the atrix failure needs to be augented with this issing energy to iprove on the predictions for the particular test case. [] Jones, Robert M, vol.. London: Taylor & Francis, 975. [] Gibson, Ronald F, Principles of coposite aterial echanics, vol. Vol. 8. CRC press, 0. [3] I. Daniel, Failure of coposite aterials, Strain, vol. 43, no., pp. 4, 007. [4] S. W. Tsai and E. M. Wu, A general theory of strength for anisotropic aterials, Journal of coposite aterials, vol. 5, no., pp , 97. [5] Z. Hashin and A. Rote, A fatigue failure criterion for fiber reinforced aterials, Journal of coposite aterials, vol. 7, no. 4, pp , 973. [6] R. Christensen, Stress based yield/failure criteria for fiber coposites, International journal of solids and structures, vol. 34, no. 5, pp , 997. [7] A. Puck and H. Schürann, Failure analysis of FRP lainates by eans of physically based phenoenological odels, Coposites Science and Technology, vol. 58, no. 7, pp , 998. [8] P. A. Zinoviev, S. V. Grigoriev, O. V. Lebedeva, and L. P. Tairova, The strength of ultilayered coposites under a plane-stress state, Coposites science and technology, vol. 58, no. 7, pp. 09 3, 998. [9] K.-S. Liu and S. W. Tsai, A progressive quadratic failure criterion for a lainate, Coposites Science and Technology, vol. 58, no. 7, pp , 998. [0] T. A. Bogetti, C. P. Hoppel, V. M. Harik, J. F. Newill, and B. P. Burns, Predicting the nonlinear response and progressive failure of coposite lainates, Coposites science and technology, vol. 64, no. 3, pp , 004. [] M. J. Hinton, A. S. Kaddour, and P. D. Soden, A coparison of the predictive capabilities of current failure theories for coposite lainates, judged against experiental evidence, Copos. Sci. Technol., vol. 6, no. 3, pp , 00. [] N. Carrere, F. Laurin, and J. Maire, Microechanical-based hybrid esoscopic 3D approach for non-linear progressive failure analysis of coposite structures, Journal of Coposite Materials, vol. 46, no. 9 0, pp , 0. [3] S. Pinho, R. Darvizeh, P. Robinson, C. Schuecker, and P. Caanho, Material and structural response of polyer-atrix fibre-reinforced coposites, Journal of Coposite Materials, vol. 46, no. 9 0, pp , 0. [4] R. Cuntze and A. Freund, The predictive capability of failure ode concept-based strength criteria for ultidirectional lainates, Coposites Science and Technology, vol. 64, no. 3, pp , 004. [5] H. M. Deuschle and A. Puck, Application of the Puck failure theory for fibrereinforced coposites under three-diensional stress: Coparison with experiental results, Journal of Coposite Materials, vol. 47, no. 6 7, pp , 03. [6] A. Kaddour and M. Hinton, Maturity of 3D failure criteria for fibre-reinforced coposites: Coparison between theories and experients: Part B of WWFE-II, Journal of Coposite Materials, vol. 47, no. 6 7, pp , 03.

10 International Conference on Future Technologies for Wind Energy October 07-09, 03, Laraie, Wyoing, USA 0 [7] Ray S. Fertig, III, Bridging the gap between physics and large-scale structural analysis: a novel ethod for fatigue life prediction of coposites.. [Accessed: 04- Sep-0]. [8] C. P. Hoppel, T. A. Bogetti, and J. W. Gillespie, Literature Review-Effects of hydrostatic pressure on the echanical behavior of coposite aterials, Journal of Theroplastic Coposite Materials, vol. 8, no. 4, pp , 995. [9] Ray S. Fertig, III, A Coputationally Efficient Method For Multiscale Modeling Of Coposite Materials: Extending Multicontinuu Theory To Coplex 3d Coposites. [0] P. D. Soden, M. J. Hinton, and A. S. Kaddour, Biaxial test results for strength and deforation of a range of E-glass and carbon fibre reinforced coposite lainates: failure exercise benchark data, Coposites Science and Technology, vol. 6, no. 3, pp , Sep. 00. [] M. J. Hinton and A. S. Kaddour, Triaxial test results for fibre-reinforced coposites: The Second World-Wide Failure Exercise benchark data, Journal of Coposite Materials, Sep. 0. [] Kedar A. Malusare and Ray S. Fertig III, On the Use of Volue Average Constituent Stresses for Predicting Failure in Coposites, presented at the 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynaics, and Materials Conference, Boston, Massachusetts, 03.