Micro Mechanical Model of 3D Woven Composites

Chu-Wei Zhou

ZHOU Chu-wei

( College of A erospace Engineering, Nanjing University of A eronautics and A stronautics,

Nanjing 210016, China)

Abstract: A combined beam model representing the periodicity of the microstructure and micro deformation of 3D woven composites is developed for predicting mechanical properties. The model considers the effects of off axial tension/compression and bending/shearing couplings as well as the mutual reactions of fiber yarns. The method determining microstructure by using woven parameters is described for a typical 3D woven composite material. An analytical cell, construct«! by a minimum periodic section of yarn and interlayer matrix, is adopted. Micro stresses in the cell under in plane tensile loading are otr tained by using the proposed beam model and macro modulus is then obtained by the averaging method. Material tests and a 2D micro FEM analysis are made to evaluate this model. Analyses reveal that micro stress caused by tensile/ bending coupling effect is not negligible in the stress analysis. Keywords: 3D woven composites; micro mechanics; bending/ shear coupling; off axial effect; combined beam model

2005, 18(1): 40- 46.

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3D woven composites have the advantages over laminate composites in higher damage toler ance and impact resistance due to reinforcement in the thickness direction. They have been finding irr creasing applications in high performance structures such as aerospace, automobile, and sports equipment.

The mechanical properties of 3D woven composites depend upon properties of constituents, fiber volume fraction, microstructure and waviness ratio of fiber yarns. Several models, classified into two categories, analytical estimation or numerical prediction, have been advanced to predict mechanical properties of 3D woven composites'1 13]. Most

analytical models are based on classical laminate theory'2 8]. For example, Whitney and Chou'4] simplified the crimpled fiber yarns in 3D woven composite as inclined straight laminates along the fiber yarn thus the curvature effect was ignored. Yang'5'7 and Yan'6,8] took curvature of fiber yarn into account in their model by viewing fiber yarns as curved laminate, and the average elastic properties were then obtained by integrating stiffness constants along crimple yarn. Yi and Ding'9] defined four sub" unit cells, named w arp, filling, stuffer and binder for textile composites, and the elastic properties of several types of 3D woven composites were obtained by combining these four sub"unit

Received date: 2004 02-22; Revision received date: 2004 10 10 Foundation item: N atural S cience Fundation of Jiangsu Province( KB 2001052)

cells in patterns representing the microstructures. It is found that, however, the existing theoretical models, including those mentioned above, do not take the bending stress of yarns into account, except in Chou's work of studying plane woven com.. [10, 11] posites .

With respect to the numerical category, FEM is most commonly used. For example, a binary model was suggested by Cox et al[12], in which fiber yarns were modeled by bar elements and effective matrix by brick elements. Though 3D brick elements are widely used in analysis of plain woven and 3D braided composites, its applications to 3D woven composites have been rarely seen thus far.

In this paper, a shear/bending coupling combined beam model representing periodic restrain condition of 3D weave composites is suggested. T he element contains two phases, one is a fiber yarn and the other is resin, or transverse fiber yarn, or the combination of them. In the model bending and shear stresses introduced by tension/ bending coupling of fiber yarns are considered. Due to the capability of describing detailed micro stress, this model has potential to be further developed for strength and damage analysis of 3D woven composite.

1 Combined Beam Model with Periodic Restrictions

1.1 Bending/shear coupling of combined beam elements

The microstructure of a typical 3D composite is shown in Fig. 1. In the interior region, the adjacent warps (or wefts) are parallel to each other and packed periodically. For convenience, in the weft

( b) C ross section of w arp Fig. 1 The microstructure of a 3D woven composite

cross section plane shown in Fig. 1(a), warp is referred as phase one while weft and resin as phase two. In warp cross section plane depicted in Fig. 1 (b) the phase one becomes weft while warp and resin compose phase two. In micro view, both structure and deformation under in-plane loads are periodic. T heiefore, repeated unit cell method is used in mechanical analysis. The practical analysis cell is adopted as a half part of the smallest periodic section of the material due to symmetry. The dark color region of A BCD in Fig. 1( a) is the analysis cell in warp direction, while the region of A B C D is the analysis cell in weft direction. The cells contain a half wavelength of the fiber yarn and two layers of a half thickness of phase two. The fiber yarns are in w aves thus introduces bending under in plane tensile or compressive loading. Deformations of analysis cell include both off axial tension/compression and bending characterized by models of u~ nidirectional composites and beams, respectively. A beam element model combining two phases is proposed herein. The element is composed of a section of fiber yarn and two layers of a half thickness of phase two. For simplicity, the beam element de veloped here is straight. Therefore, the curved cell should be segmented into a series of these straight beam elements. The components of phase two in an element depend on the element's location in the cell. As shown in Fig. 1, the beam' s phase two contains only resin if it is in the central part, while the element' s phase two may include transverse fiber yarn only or the combination of fiber and resin if it is located near the end.

w ill introduce additional shear in phase two due to periodicity of deformation. Elastic parameters of fiber yarn can be obtained by the rule of mixture of unidirectional composites, namely Ey 11 = VfE f11 + V m E m Ey22 = E mEf2^ Vf + nvJ HEf22 V m + Em Gy 12 = G m G f^i V f + nv J /( no f12 V m + G, Gy23 = Gm Gf2^ Vf + HV J /[ HGf23 Vm + Gm Vf

where Ef11, Ef22, Gf12 and Gf23 are longitudinal, transverse and shear moduli of the fiber; Em and Gm are tension modulus and shear modulus of matrix material; Vm and Vf are volume fractions of resin and fiber in the fiber yarn. Scalar H is adopted to improve the low estimation tendency of transverse and shear moduli by the mixture rule. Here H takes the value of 0. 5.

Fig. 2 Coupling of yarir bending/matrix- shearing and internal force in a combined beam element

Let the bending deformation curve of the fiber yarn be y( x ), and the rotation angle of the fiber yarn' s cross section be y (x). As shown in Fig. 2, phase two will shear to an angle of Hy (x )/ hm. Here H = hy + hm, is the thickness of the combined beam element. Thus, the shear stress will be

q(x)= G t Hy' (x )/h m = k t y' (x) (2) If phase two is resin, G t= Gm is the shear modulus of resin, and if phase two is transverse fiber yarn, Gt= Gy23 takes the values of transverse shear stiffness of fiber yarn. In the case of mixture phase two, its stiffness is simplified as volume weighted mixture of the two components G t = Gm Gy23( Vy23 + HVr)/( HGy23 Vr + Gm Vy23)

where Vr and Vy23 are volume fractions of fiber yarn and resin in phase two. The moment equilibrium is given as follow s:

- M(x) + S 0 x+ £ Hk t y' (s) ds = M 0 ( 4)

Here M 0 and S 0 are bending moment and shear force at element end of x = 0. U nder irr plane loading, the normal stress distributions on upper and lower bounds of interior fiber yarn are the same due to the periodicity of stress distribution. Therefore, the shear force in element is invariable along the element axis. Since the bending moment is M(x ) = Icy (x), the moment equilibrium yields

Icy"(x) - Hkty(x) = S0X - M0 (5) where I c= ( Ey 11 h3 + E t H 3 - E t h3 )/ 12 is the bending stiffness of the combined element. The general resolution to Eq. (5) is

y(x ) = Cie + C 2e

(M0- S0x)

where a= (Hkt/1 c) 1/2. The four constants of C1, C2, M 0 and S 0 can be determined by applying four boundary conditions. When the flexural curve is known, the internal forces and micro stresses of each phase of the element can be obtained. For ex_ ample, w hen the fiber yarn has a unit rotation at one end while the other end of element is fixed, y(0) = 0, y' (0) = 1, y(L) = 0 and y' (L ) = 0, the internal forces at element ends are expressed as

( 1 + aL) e- aL - ( 1 - aL) e^

(e - e

- 2 aL )

S( 0)= S(L) =

„, - at aL a( e + e

Hk t ( 7)

Here D = a[4- (2+ aL) e- aL - (2 Flexural curve of the element axis is 1

Hk t ( 9) aL) e^y.

y i(x) =

( 1 + aL ) e ^ ax *-*-e +

( 1 - aL ) e"

1 - ax

aL , e)

( 1 + aL)eaL- ( 1 - aL)eaL

D (10)

When one end of the element produces a unit

vertical linear displacement while the other end

= 0, the internal forces at element ends will be

M A(0) = - M A(L) _

_,/ - iL aL

a( e + e

Hk t ( 11)

_2( - aL + aL i

S a(0) _ S Ä(L)_ u ( e D+ e -;Hkt ( 12)

T he flexural function will be

J2(x) _

/ - aL aLi - aL aLi

(e D e } a2 x + (2 - e e } a (13)

1. 2 Off axial effect of the combined beam element

Moduli and Possion' s ratio of the combined beam element can be calculated by using the mixture rule when off axial effect is neglected. They are

¿11 = hTEy 11 + h mE t

¿22 = Ey 22E t (hy + % m )/(E t hy + 22 h m) v21 = (hy! 12 + hmV)/H, V12 = !ik22/k 11_

When phase two of combined element is resin, Et = Em, as the fiber yarn in transverse, Et = Ey22, as the combination of them, Et = VmEm + Vy22 Ey22. Vm and Vy22 are volume fractions of resin and transverse fiber yarn in phase two.

T he off axis effect, usually considered in the analysis of unidirectional composites is also taken into account here. The ratio of normal stress to shear stress caused by off axial tension or compres" sion is determined by the incline angle of fiber yarn. As shown in Fig. 3, when uniaxial tension

Fig. 3 Off axial tension of a combined beam element

load 0x is applied, the rates between components of stress and strain in local coordinate 1-2 can be ex-

°2 2 n #12 n

°1 _ m2' 01 _ m

82 2 S 21 m + 2 S 22 n

81 _ S 11 m 2 + S 12 n 2

%2 S 66 mn

81 " S 11 m 2 + S 12 n2

where m and n denote cos0 and sin0, respective^ ly. Si/ denotes the principal axial flexibilities S 11 = 1 /¿11, S22 = 1/k22, S 12 = - v 12/k22, S 11 = - v 21/k 11, S 66 = 1/k 12

Considering the off axial effect the axial modulus of combined beam element is yielded as 1 + ov 12

E C11 =

1 - v 12 v 21

Using Eqs. ( 7) to (9), (11) to ( 12) and (17) one can construct a beam element to consider the effects of bending/ shear coupling and off axis. 1.3 Micro stress analysis of combined beam element

If end displacements of a combined element are given as (u 0, v 0, (0) and ( u L, v L, (L) , the tension and compression normal stresses in phase one along element axis, °t and 0C, are

t(x)_ Ey 11 (ul - u 0) + h

± 2j-[yV*) (0+

y 1 ( L - X ) (l + y 2(x) v0 + y 2(L - x ) vL7

Normal stress vertical to element axis and shear stress in two phases are the same. They are denoted by 0 and #, and computed by

(v21 + a) ¿22 (uL- u0)

1 - V 12 V 21

nk 11 u l - u 0 m L

+ k«[y' 1(x) ( +

y 1 ( L - x ) (l + y 2(x) v 0 + y 2 ( L - x ) v l7

2 Micro structure Analysis

Determination of microsturcture parameters, such as curvature, incline angle and cross-section shape of fiber yarn, is the basis for mechanical property analysis of 3D woven composites. In

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yai'n becomes flattened. It is assumed as an ellipse, racetrack or double convex lens in published artr cles. These assumptions have been proved being reasonable by experiment observations. To warp and w eft of the 3D woven composite studied in this paper, their' cross section area as well as weave tension and forming pressure applied are the same. Therefore, their cross-section shapes are assumed r dentical after forming. As shown in Fig. 1 and Fig. 4, warp and weft are assumed packing to each other without gaps.

Weave parameters include HT, the thickness of material; Nlt, the number of warp or weft layers in thickness; L W and L L, distances between center lines of neighboring wefts along warp direction and of neighboring warps along weft direction. T he net fiber area of warp and weft are denoted by aL and aT. When resin is soaked into fiber yarns, they become A l = al/fy and A t = at/f y. Here fy is the fiber volume fraction of yarns. Geometric parameters characterizing microstructure of woven composites can be deduced from these weave parameters. In Fig. 4, these geometric parameters are labeled as hL and w L, the height and width of warp; h t and w t, the height and width of weft;

( a) Half part of least periodic section of warp

and z (x ), the curvature function of cross section boundary of fiber yarn. Undulation of fiber yams and inclined angle of straight section of warp can be computed by z ( x ). As is shown in Fig. 4, curve z(x ) should satisfy the following conditions:

( x ) d x = A :

z(0) = ht/2, z(wt/2) = 0, z (0) ^ 0 z ' ( w t/2) = (2H t/ N lt - h t)/(L w - w t)J

Warp and weft of material considered herein have the same cross section area. Their width and thick-

ness are

T = L L ,h L = h T = H T / 2 N lt

( b) Least periodic section of weft

Fig.4 The smallest periodic sections of warp and weft

It should be mentioned that methods of choosing an analytical cell, determination of micro parameters and simplification of the model depend on weave type of woven materials.

3 Results and Discussions

In order to verify the applicability of the combined beam model, a 3rD woven carbon/epoxy composite material is made and its properties are characterized by tension test. Fiber used is 3K T300. Weft density along warp direction is 3.5 rows per centimeter while warp density along weft direction is 10 rows per centimeter. The specimen is 3. 5mm in thickness and contains 12 layers of both warp and weft. Fiber volume fraction of fiber yarn, fy, is 78%. From Fig. 1, one can see that the surface warp has lower curvature than the inte rior one. The surface warp takes the volume fraction of 8. 33% in total warp yarns volume. Using Eqs. (19) to (21) yields the width and thickness of cross"section of both warp and weft as in width and in height. If the function of bound curve of yarn's cross" section is modeled by a third orders polynomial, it will be

z(x)= 6. 4103x3 - 5. 5384x2 + x + 0. 083

The incline angle of the straight section of warp is 0L= 15. 07°. Elastic constants of fiber and resin are listed in Table 1, where unit of modulus is GPa.

warp rKrertion ischosenas

Table 1 Mechanical properties of resin and fiber

e2 G ,2 G 23 ! 12 y ' 23

T 300 220 13. 8 9. 0 4. 8 0. 2 0. 25

T DE- 85 4.5 4. 5 0. 34 0. 34

a half length of the smallest periodic section of warp, enclosed by dotted lines in Fig. 4( a). The analysis cell is then segmented into a series of straight combined beam elements. Tension loading applied to the cell is A= 0. 01 Lw, corresponding to average tensile strain of 1%. If the interior w arp is predominant in volume fraction, the moduli can be found approximately by averaging micro stresses over the interior. Average tensile moduli evaluated by combined beam model with various numbers of element are listed in Table 2, where symbol E denotes average tensile modulus in warp direction and symbols °max and #max denote average and maximum normal stress in fiber yarn and maximum shear stress in second phase, respectively. From T able 2, one can find that the computed average Table 2 Modulis and maximum micro stresses predicted by analytical model with various number of elements

Fig. 5 Analytical cell model with 11 comb in «1 beam elements

Project location of warp cross section/mm Fig. 6 Distributions of normal stress in warp

To verify the analytical results, a 2D FEM analysis is carried out to the cell in warp direction. The FEM mesh and boundary condition are shown in Fig. 7. 2D four nodes element is adopted. The

e /G Pa o/MPa Omax/M Pa #max/ M Pa

3 44. 12 757. 98 966. 48 432. 17

5 43. 16 765. 52 955. 89 458. 43

7 42.96 766. 92 955. 23 463. 49

9 42. 87 767. 42 955. 11 465. 62

11 42. 87 767. 63 954. 63 465. 84

tensile modulus and maximum bending stress decrease with increase of element numbers. This may be due to the fact that the model with more elements provides smaller bending stiffness. However, the peak values of average normal and shear stresses increase with the increase of element numbers. When the element number reaches 11, both average modulus and micro stresses have converged. The 11 elements segment model is pictured in Fig. 5. Distributions of normal stresses on upper boundary, lower boundary and of average of warp are shown in Fig. 6. The maximum bending stress locates at the end of the cell where bending moment is the largest. The maximum normal stress introduced by bending is 27. 50% lager than the average one. Therefore, effect of tension/bending

coupling is not negligible. on each element yields the mean tensile modulus as

Fig. 7 Mesh and boundary condition of 2D micro FEM model

element painted with dark gray is filling transverse yarn and that with bright gray is of resin. The left and right boundaries keep vertical when tensile load applied. To satisfy the periodic deformation condition, the horizontal displacements at pairs of nodes, located at upper and lower boundary and wit h sam e horizontal coordinate, are forced to be the same. While the differences between the vertical displacements at these pair's of nodes keep identical along the axial direction of the cell. The distribution of principal stress is depicted in Fig. 8. Directions of the principal stresses are close to the tangent direction of the yarn. Averaging stresses

Fig. 8 Principal stress distribution obtained by 2D micro FEM model

41. 82GPa, slightly smaller than 42. 87GPa found by using the proposed combined beam model, and 44. 52GPa by experiment. The normal stress distributions predicted by 1D combined beam model and by 2D plane element model agree to each othr er. T he maximum principal stress predicted by the latter is 10% higher than that by the former.

4 Conclusions

A shear/ bending coupling combined beam model is proposed to evaluate the micro stress and effective elastic properties of 3D weave composites. The proposed model takes into account of tension/ bending effect and off axial effect. The method to determine the microsturcture of a typical 3D woven composite from weave parameters is given. Tensile modulus and micro stresses predicted by the proposed model agree well with those obtained by tensile test and 2D FEM analysis. Numerical results indicate that normal stress in yarn caused by tensile/bending coupling is not negligible. This obser vation is important in further strength analysis.

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Biographies:

ZHOU Chu wei Born in 1964, received B. S. from Hehai University in 1986, M.S. from Dalian University of Technology in 1989 and Ph. D. from Ts-inghua University in 1999. He is now an associate professor in Nanjing University of Aeronautics & Astronautics. His research interests include micro mechanics, composite mechanics, and computational mechanics. Tel: 025-84892104 27, E mail: zcw@ nuaa. edu. cn

Micro Mechanical Model of 3D Woven Composites Academic research paper on "Materials engineering"

Abstract of research paper on Materials engineering, author of scientific article — Chu-wei ZHOU

Similar topics of scientific paper in Materials engineering , author of scholarly article — Chu-wei ZHOU

Academic research paper on topic "Micro Mechanical Model of 3D Woven Composites"