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15th ASCE Engineering Mechanics Conference June 2-5, 2002, Columbia University, New York, NY EM 2002 INFLUENCE OF GEOMETRICAL FEATURES OF UNIDIRECTIONAL FIBROUS LAMINA ON ITS TRANSVERSE RESPONSES X. Allan Zhong1 and Joe Padovan2 Abstract Dimensional analysis is used in this study to establish general relations between geometric features of unidirectional fibrous lamina and the corresponding transverse responses, including transverse modulus and stress distribution in the matrix. Prediction of transverse response of unidirectional fibrous lamina was made long ago, e.g. the Halpin-Tsai equations. However, specific geometrical features of such composites are typically lumped into the volume fraction of fibers. Using micromechanics via explicit representation of fibers, it is demonstrated explicitly in this paper that the transverse response is dependent on two geometrical parameters, one is the usual fiber volume fraction, and the other is the ratio of fiber diameter to fiber spacing. A new formula for prediction of transverse response of the composite is established based on linear elastic finite element analysis. The new formula is validated via lab test data. The formula is extended to nonlinear analysis by integrating nonlinear matrix behavior in a separable way under certain restrictions. It is also demonstrated that the stress distribution in the matrix and the location of peak stress changes with the aforementioned ratio when fiber diameter and thickness are fixed. The effect of random fiber spacing is also investigated. Keywords: composite geometry, unidirectional lamina, transverse response, dimensional analysis, linear elasticity, and hyperelasticity. 1. BACKGROUND Theories for the mechanical behavior of unidirectional fibrous lamina are of significance in fiber reinforced engineering structures, since they form the foundation of theories for multidirectional laminates in classical lamination theory. The micromechanical models for linear elastic composites have been well established and are summarized extensively in many textbooks, e.g. [1 − 3]. Recent efforts to include material and geometric nonlinearity in estimating effective properties of composites can be found in, e.g. [4, 5]. In this paper we are interested in the geometric effect on the transverse response of unidirectional fibrous lamina, because of their relevance in the design of engineering components or structures, such as the plies in tires, conveyor belts, hoses, etc. In these industrial applications, since the fiber, e.g. steel wire or wire bundle, is much stronger and tougher 1 The Goodyear Tire & rubber company, Akron, OH 44316-001, azhong@goodyear.com 2 Department of Mechanical Engineering, The University of Akron, Akron, OH 44325 than the (polymeric) matrix materials, the fatigue failure occurs in the matrix material only. In this setting, the transverse responses of the lamina of interest include transverse modulus or stiffness, matrix fatigue crack initiation and fatigue propagation of matrix cracks. It has been established that the rule of mixtures can predict the longitudinal modulus of unidirectional fibrous lamina pretty well in [6], and more recently in [5]. The prediction of transverse modulus however is more problematic. In the design of unidirectional fibrous lamina, such as plies in tires, fiber diameter, fiber spacing (usually called EPI, i.e. fiber ends per inch, in tire industry) and lamina thickness are important geometrical parameters. These geometrical parameters influence the constraint of the fiber imposed on the matrix material, and consequently affect the apparent failure properties of the composites. Based on these considerations, it is worthwhile to relate the transverse modulus of unidirectional fibrous lamina to these geometrical parameters explicitly for easy application and interpretation. However, because of the homogenization nature of typical approaches in composite mechanics, effective properties of fiber reinforced composites are related to constituent properties and fiber volume fraction, so theories or equations for the estimation of transverse modulus are not suitable for this type of application. Another response of interest is the peak stress in the matrix between fibers because it is directly related to fatigue crack initiation. The typical approaches of composite mechanics could not predict this quantity due to the inherent averaging nature in these approaches. The same can be said regarding transverse matrix cracking. So, instead of using composite mechanics approaches, micromechanical finite element models are formulated to study the geometric effects on the transverse response of unidirectional composites – transverse modulus (transverse stiffness), and peak stress in the matrix. The matrix cracking related subject will be addressed in a future paper. A 2D-unit cell model is used in the following analysis. As to be shown later, it is found via finite element analysis that the transverse modulus is dependent on two geometric parameters, one is the usual fiber volume fraction, and the other is the ratio of fiber diameter (D) to fiber spacing (d), i.e. D/d, in addition to constituent properties. A new and simple formula for the transverse modulus is obtained under small deformation assumptions. This formula is then extended to large deformation under certain restrictions. The peak stress in matrix is found to be varying with the ratio nonlinearly. It is noted that these findings can not be predicted from composite mechanics approaches, they can not be predicted even from the more sophisticated rebar representation [7] of fibers. 2. TRANSVERSE STIFFNESS AND TRANSVERSE MODULUS Transverse modulus or stiffness represents one aspect of the structural function of the unidirectional fibrous lamina. The simplest calculation of the transverse modulus is obtained by considering the fiber and matrix to act as springs in series [1,2], E f Em E2 = (1) E f (1 − v f ) + E m v f where E f , E m are moduli of fiber and matrix, and v f is the volume fraction of fiber. This is one of the commonly used formulas for estimation of transverse modulus. But it does not account for fiber geometry and spacing. The Halpin-Tsai Equation [6] is frequently used to account for these 2 geometrical features of the lamina. The equation for the transverse modulus is [6], E m [ E f (1 + ξv f ) + ξE m (1 − v f )] E2 = (2) E f (1 − v f ) + ξE m (1 + v f / ξ ) where ξ is a factor depending on fiber geometry and spacing. Similar equations for other moduli are also available. Application of equation (2) is contingent upon the determination of ξ with respect to different fiber spacing and fiber shape. For fibers of nominal circular cross section and E f >> E m , which is the typical case in tire reinforcements, ξ = 2. So equation (2) is reduced to [8], E m (1 + 2v f ) E2 = . (3) (1 − v f ) There are other estimations of transverse modulus of unidirectional fibrous lamina, e.g. [5], which are similar to equation (2) or its refinements. The method of cells [9, 10] is a rather general approach, except that the method requires regular fiber packing. However it is shown in [11], that predictions of transverse response, based on any regular fiber packing pattern considered, do not correlate well with experimental results. Based on these considerations, we will compare our prediction of transverse response to that of the Halpin-Tsai equation, i.e. equation (2) or (3) only. For a lamina with dimension W > > T, under transverse loading (Figure 1), the plane strain assumption of deformation in the lamina is a good approximation of the problem. In a plane strain formulation, only the cross section perpendicular to fiber direction needs to be considered as shown in Figure 2, where D is the effective fiber diameter, d is the spacing between fibers, and T is the lamina thickness. The plane strain problem can be further simplified if there are many fibers in the loading direction (i.e. L >> D + d), which is the case under consideration. F F W T L Figure 1 Schematics of a unidirectional fibrous lamina 3 T D d D+d Figure 2: A unit cell So with the assumptions that W >>T and L >> D + d, a plane strain problem for a unit in the loading direction is appropriate for prediction of the lamina transverse response. The unit used here is the portion bonded by dashed lines in Figure 2. Considering symmetry conditions, only a quarter of the unit is analyzed in a FEA calculation. The fiber is assumed to be linear elastic material and it has Young’s modulus E f and Poisson’s ratio γ f . The matrix can be either a linear elastic material (Young’s modulus E m and Poisson’s ratio γ m = 0.49 (almost incompressible, as is the case for elastomers)) or a hyperelastic material of Neo-Hookean type . The quantities of interest here include transverse modulus E2 of the lamina, and the stiffness S of the lamina. In linear elasticity, the relation between the two quantities is approximated as follow, F σWT composite WT S= = = E2 (4) δ εL L where F is the force applied on the lamina, δ is the corresponding elongation, and W, L and T are the dimensions of the lamina. For the plane strain problem under consideration, we have composite S E2 = (5) T /( D + d ) Here S is the stiffness of the unit in Figure 2. Apparently the transverse modulus of the unidirectional composite is a function of material properties as well as its geometrical features, in general one can write composite E2 = f ( E f , γ f , E m , γ m , D, d , T ) (6) for linear elastic matrix material, and 4 composite E2 (λ ) = f ( E f , γ f , C10 , D, d , T , λ ) (7) for Neo-Hookean matrix material with coefficient C10. The secant transverse modulus or normalized transverse stiffness is dependent on the deformation in the composite. In equation (7), the quantity λ is the stretch ratio in the lamina. Dimensional analysis reduces the number of variables the transverse modulus depends upon. The relations (6) and (7) can be rewritten into the following forms, composite E2 Ef D D = f( , , ,γ f ,γ m ) (8) Em Em d T for linear elastic matrix, and composite E2 (λ ) Ef D D = f( , , , λ,γ f ) (9) C10 C10 d T for hyperelastic matrix material. For easy conversion of FEA results into a concise formula, equation (5), which relates stiffness to transverse modulus, is integrated into equation (8) or (9). From linear elastic analysis of this unit cell model, it is found that for given fiber properties, the relations as given in equation (1) and (3) do not match FEA results, e.g. see Figure 3(a). Normalized transverse modulus of Normalized effective transverse modulus as a unidirectional lamina function of p 12 Normalized transverse Normalized transverse 12 10 10 y = 0.3366x + 1.3339 8 R2 = 0.9992 modulus 8 modulus FEA 6 6 Halpin-Tsai 4 4 2 2 0 0 0 1 2 3 4 0 5 10 15 20 25 30 p_2 p (a) (b) Figure 3: a) Comparison between predictions from the Halpin-Tsai equation and those from FEA, where (1 + 2v ) , f p2 = (1 − v f ) b) comparison of (10) with FEA results, p= 1+ 2⋅v f D 1− v f d 5 Based on dimensional analysis and the Halpin-Tsai equation, it turns out that the following equation can describe the transverse modulus very well composite E2 S 1+ 2⋅vf D = S' = = 0.3366 ⋅ ( ) + 1.339 (10) Em E m (T /( D + d )) 1− vf d The equation is applied to predict the stiffness change of composites studied in [13]. The predictions compare with test results very well, see Figure 4. A similar relation for nonlinear Neo-Hookean matrix material has been obtained. Lamina transverse stiffness: model prediction vs. test results (Gent and Park, 1986) 80 transverse stiffness per unit length 60 test 40 prediction 20 0 0 5 10 15 20 fiber spacing (mm) Figure 4: Effect of fiber spacing on transverse stiffness and illustration of test geometry in [13] 3. INFLUENCE OF GEOMETRICAL FEATURES ON PEAK STRESS IN MATRIX As shown in Figure 5, as d/D decreases, the location of peak stress moves from very close to the fiber/matrix interface to the center between the fibers. As a matter of fact when d/D ≤ 0.4, the peak normal stress is always located at the center between adjacent fibers. It is also found that peak stress in the fiber is amplified for both load and displacement control, see Figure 6. Normal stress along loading direction, d/D=0.1 Normal stress along loading direction, d/D=1 1.05 1.07 1.06 1.04 Normalized stress Normalzied stress 1.05 1.03 1.04 1.02 1.03 1.02 1.01 1.01 1 1 0.99 0.99 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 normalzied distance normalized distance Figure 5: location of peak stress in the matrix between fibers 6 Variation of peak stress between adjacent fibers 35 Normalzied peak stress 30 25 20 15 10 5 0 0 0.2 0.4 0.6 0.8 1 1.2 d/D load control displacement control Figure 6. Normalized peak stress between fibers 4. SENSITIVITY OF THE NEW FORMULA Em Sensitivity of the new formula with respect to variation in and in γ f and γ m , 3D effect, Ef effect of out of plane boundary conditions and Effect of non-uniform fiber spacing, are also studied. Details of this study will be addressed in the full-length paper. 5. SUMMARY Via micromechanical finite element analysis we have demonstrated that 1) The transverse modulus of unidirectional fibrous lamina is dependent on constituent properties, fiber volume fraction and relative fiber spacing (d/D). A simple scaling law is established and the range of applicability is identified. 2) The location and magnitude of the peak stress in the matrix of a unidirectional fibrous lamina under transverse load is shown to be dependent on only one geometric parameter, the relative fiber spacing (d/D) irrespective if fibers are uniformly spaced or not. This is because the peak stress is primarily determined by fiber interaction. The peak stress approaches to a constant with respect to d/D for given load, when d/D is large. Acknowledgements The authors benefited from many suggestions made by Prof. Alan Gent. X.A.Z. also appreciates 7 discussions with S. Kim and Z. Bo. The authors are grateful to The Goodyear Tire & Rubber Company for permission to publish this paper. 6. REFERENCES [1] Jones, R.M (1975), Mechanics of Composite Materials, Washington, Scripta Book Co. [2] Daniel I. M. and Ishai O. (1994), Engineering Mechanics of Composite Materials, Oxford University Press. [3] Nemat-Naser, S. and Hori, M. (1999), Micromechanics: overall properties of heterogeneous solids, Elsevier Science Publisher, 2nd edition. [4] Cataneda, P. P. and Suquet, P. (2000), Nonlinear composites and microstructure evolution, pp, 253 –274, Mechanics for a New Millennium, Proceedings of the 20th International congress of theoretical and applied mechanics, Chicago, Illinois, USA, August 27 – September 2, 2000. [5] Huang, Z-M (2001), Simulation of the mechanical properties of fibrous composites by the bridging micromechanics model, Composites: Part A, 32, pp. 143 – 172. [6] Halpin, J. C. and Tsia, S.W. (1969), Effects of Environmental factors on composite materials, AFML-TR 67-423. [7] Mang, H. A. & Meschke, G. (1991), Finite element analyses of reinforceed and prestressed concret structures”, Engineering Structures, 13, pp. 211-226. See also: Helnwein, P., Liu, C. H., Meschke, G. & Mang, H. A. (1993), A new 3-D finite element model for cord-reinforced rubber composites – application to analysis of automobile tires. Finite elements in analysis and design, 14, pp. 1 – 16. [8] Walter, J. D. (1981), Cord Reinforced Rubber, Chapter 3 of “Mechanics of Pneumatic Tires” edited by S. K. Clark, US department of transportation, National highway traffic safety administration, Washington D.C. 20590. [9] Aboudi, J. (1989), Micromechanical analysis of the composites by the method of cells. Applied Mechanics Review, 42(7), pp. 193 – 221. [10] Aboudi, J. (1996), Micromechanical analysis of the composites by the method of cells - update. Applied Mechanics Review, 49(10), pp. S83 – 91. [11] Brockenbrough, J.R., Suresh, S. and Wienecke, H. A. (1991), Deformation of metal matrix composites with continuous fibers: geometric effect of fiber distribution and shape, Acta Metal Mater, 39(5), pp. 735 – 752. [12] Zhong, X. A. and Knauss, W. G. (2000), Effects of particle interaction and size variation on damage evolution in filled elastomers, Mechanics of composite materials and structures, 7(1), pp. 35-53. [13] Gent, A. N. and Park, B. (1986), Compression of rubber layers bonded between two parallel rigid cylinders or between two rigid spheres, Rubber Chemistry and Technology, 59, pp. 77-88. 8

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