18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS A MICROMECHANICAL METHODOLOGY FOR FATIGUE LIFE PREDICTION OF POLYMERIC MATRIX COMPOSITES Y. Huang 1 , K. Jin 1 , L. Xu 1 , G. Mustafa 1 , Y. Han 2 , S. Ha 1* 1 Department of Mechanical Engineering, Hanyang University, Ansan, Korea 2 Korea Electric Power Research Institute, 103-16 Munji-dong, Yusong-gu Daejon, 305-380, Korea *Corresponding author (sungha@hanyang.ac.kr) Keywords : polymeric matrix composite (PMC) , micromechanics , fatigue , life prediction characterize its micro structure such that the micro 1 General Introduction stresses can be calculated from ply stresses with Polymeric matrix composites (PMCs) possess reasonable accuracy. The micro structure of a UD superior specific properties to metals, and therefore features longitudinally aligned and transversely are widely used in many applications. However, randomly distributed fibers embedded in polymeric fatigue behavior of composites has been a great matrix. Assuming the actual random fiber concern for years since conventional approaches for arrangement on the cross-section of a UD can be fatigue life prediction of metals are not suitable for replaced by an equivalent regular fiber arrangement, that of composites due to the existence of anisotropy a unit cell consisting of both fiber and matrix can be and the distinction of constituent properties. Despite extracted from the regular fiber array as the basic the fact that so many efforts have been invested into constructing element. Fig. 1 shows three frequently the research on fatigue life prediction of composites cited regular fiber arrays: the square (SQR), [1-10], so far there does not exist a well-established hexagonal (HEX), and diamond (DIA) arrays, as and widely-accepted methodology which can well as their corresponding unit cells. provide satisfactory life prediction for composite In order to correlate ply stresses and micro stresses structures. Micromechanics is a powerful tool in each constituent, a concept called Stress compared with traditional macro-level methods Amplification Factor (SAF) was introduced, so that since it provides insight to the micro stress the micro stresses can be calculated with the formula distribution in each constituent, and consequently shown below [11]: better understanding of fatigue failure mechanism at σ M σ A σ T (1) the constituent level can be developed, which results σ in more reasonable explanation of fatigue behavior where σ is the micro stress at a certain micro point of PMCs as well as more accurate life prediction of within either fiber or matrix, σ being the macro composite structures. In this paper, a (ply-level) stress, Δ T being the temperature micromechanics-based methodology for fatigue life increment, M σ and A σ being the SAF for macro prediction of PMCs was proposed. Theoretical stress and temperature increment, respectively. The prediction of fatigue life of glass-fiber reinforced dimension and value of SAF depend on the location laminates which are intended for wind turbine blade of the micro point [11]. If the micro point resides in application was compared with fatigue test results, fiber or matrix, σ and σ in Eq. (1) are 6×1 matrices and good agreement was obtained. containing six micro and macro stress components, 2 Theory and Approach respectively, while M σ and A σ are in the form of a 6×6 matrix and 6×1 matrix, respectively. For the 2.1 Computation of Micro Stresses fiber-matrix interface, σ becomes a 3×1 matrix The first step towards fatigue analysis at containing three interfacial tractions, i.e. the microscopic level is to obtain micro stresses in each longitudinal traction t x , the tangential traction t t , and constituent, i.e. fiber, matrix, and fiber-matrix the normal traction t n , as indicated by Fig. 2. interface, of a composite laminate under external Accordingly, M σ and A σ become 3×6 and 3×1, loadings. For a continuous fiber reinforced lamina respectively. By applying appropriate boundary (UD), a micromechanical model is required to conditions to the finite element model of a unit cell,
the SAF for any specific micro point can be a material constant. Attention should be paid that the determined numerically. aforementioned plane does not only refer to the After the introduction of regular fiber arrays, it is plane which passes through the given interfacial necessary to validate the equivalence between the point and tangential to the cylindrical outer surface idealized fiber arrangement and the real cases. One of the fiber. Rather, the equivalent stress should be example of verification was provided in Ref. [12]. It calculated for all planes passing through the given was concluded that the SQR and HEX arrays were interfacial point, and the one on which the comparatively superior to the DIA array. equivalent stress attains the maximum is defined as the “critical plane”. The function sign( σ n , τ ) yields 2.2 Constituent Fatigue Models the sign of the one between σ n and τ which has the A UD consists of three constituents: fiber, matrix, greater absolute value. and fiber-matrix interface. Due to the distinct Since the constituent micro stresses vary with time, mechanical properties possessed by each constituent, the constituent equivalent stresses are also time- a UD is microscopically inhomogeneous. Therefore, varying. Thus, the mean value and amplitude of it is rational to propose a fatigue model for each constituent equivalent stresses are readily calculated. constituent, rather than treating the UD as a whole Based on those values, the constituent effective and employing one fatigue governing law. stresses are obtained considering mean stress effect. Despite the microscopic inhomogeneity of the UD, A modified Goodman formulation has been selected each constituent can be assumed homogeneous. So it for the Constant Life Diagram (CLD), and the would be desired to introduce an equivalent micro effective stress at a micro point within a constituent stress for each constituent such that the overall effect is defined as of multi-axial micro stresses is considered. Since amp T fibers undertake most longitudinal loads, the micro eq (5) eff longitudinal stress at a point within fiber σ x,f is used T C T C mean eq as the equivalent stress at that point: 2 2 (2) where T and C are static tensile and compressive eq,f x,f amp strengths of the constituent, respectively. and eq For the matrix, it is regarded as isotropic with mean symbolize the amplitude and mean values of dissimilar tensile and compressive strengths, and the eq equivalent stress is derived from the matrix failure the constituent equivalent stress. With Eq. (5), criterion presented in Ref. [13]: experimentally acquired fatigue test data can be fitted with Basquin’s equation to obtain the S-N 2 2 2 1 I 1 I 4 curve: 1,m 1,m VM,m (3) eq,m 2 log A log N B (6) eff f where β is the ratio between the matrix static Where N f is the number of cycles to failure, A , B compressive strength C m and static tensile strength being constants to be defined by test data. For a T m . I 1,m and σ VM,m are respectively the first stress given constituent effective stress, the number of invariant and the von Mises stress of micro stresses cycles to failure is calculated. The damage caused by at a point within the matrix. that effective stress throughout its duration is The failure occurring at the interface is usually in the obtained following Miner’s rule, so the linear form of debonding due to normal and/or shear cumulative fatigue damage variable D for each tractions. The equivalent stress for the interface is constituent was calculated as follows: defined following a critical plane model: n j D (7) 2 sign , 2 k (4) N eq,i n n j f , j where σ n and τ are the normal and shear stresses on a where n j is the number of cycles of j -th loading, and plane passing through an interfacial point, while k is N f, j is the number of cycles to failure under j -th
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