Numerical analyses of 3D orthogonal woven composite under three-point bending from multi-scale microstructure approach

Computational Materials Science 79 (2013) 468–477

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Numerical analyses of 3D orthogonal woven composite under three-point bending from multi-scale microstructure approach Xiwen Jia a, Zihui Xia b, Bohong Gu a,⇑ a b

College of Textiles, Key Laboratory of High-Performance Fibers & Products, Ministry of Education, Donghua University, Shanghai 201620, China Department of Mechanical Engineering, University of Alberta, Edmonton, AB T6G 2R3, Canada

a r t i c l e

i n f o

Article history: Received 28 October 2012 Received in revised form 18 June 2013 Accepted 25 June 2013 Available online 24 July 2013 Keywords: 3D orthogonal woven composites (3DOWCs) Repeating unit cells models (RUCs) Three-point bending User-defined material subroutine (UMAT)

a b s t r a c t This paper reports the deformation and damage mechanisms of 3D orthogonal woven composite (3DOWC) under three-point bending based on finite element analysis (FEA) at micro/meso/macro-scale level. One type of micro-scale repeating unit cell model (micro-RUC) has been established with the same fiber volume fraction of the 3DOWC. Two types of meso-scale repeating unit cell models, surface meso-RUC and inner meso-RUC, were established from fiber bundles structures. Combined with the smeared crack failure modes, the mechanical properties of fiber bundles which including the longitudinal and transverse ultimate strengths have been obtained based on properties of the fibers and the resins. Then the ultimate strengths of the surface layer and inner layer of the 3DOWC have been calculated based on the damaged surface meso-RUC and the inner meso-RUC with developed smeared cracks. Finally the 3DOWC beam was modeled with two types of elements, i.e., the surface layer elements and the inner layer elements, at the macro-scale level. The global and local responses of the beam under three-point bending, and the damage initiation and propagation, have been predicted at the full scale beam level. The numerical results were compared with the experimental results and good agreement was found. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Three-dimensional (3D) orthogonal woven composites (3DOWCs) have potential applications to the fields of aircrafts, ships, civil infrastructures and armor protection because of high damage tolerance [1]. There are warp yarns (0°), weft yarns (90°) and Z-yarns interlaced in 3D orthogonal woven fabrics. The Z-yarns run through the thickness direction and bind the 3-D orthogonal woven fabric into a stable structure. Compared with 2-D laminated composites, the 3DOWCs has higher resistance to delamination owing to the existence of Z-yarns [2–4]. The mechanical behaviors of 3DOWCs under quasi-static loadings have been reported in a lot of references. It was found that under tensile [5–12], compressive [13–16] and shear [9,17] loadings, the 3DOWCs were easily damaged with the cracks formed within the resin-rich regions among the fiber tows. The 3DOWCs also have higher in-plane strengths, as well as damage initiation thresholds than that of 2D woven laminated composites. For 3point bending condition, the bending properties can be improved with the existence of Z-yarns [18–20], such as bending modulus, failure load and deflection. The numerical models include the representative volume element (RVE) [21,22], repeated unit cell (RUC) ⇑ Corresponding author. Tel.: +86 21 67792661; fax: +86 21 67792627. E-mail address: [email protected] (B. Gu). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.06.050

[23–32], micromechanics strengths model [33] and other models as reviewed in [34]. These models have made different simplifications on the fabric architecture in the 3DOWCs and may not be able to provide accurate details of local stress distributions at fiber or fiber yarn scales. Furthermore, the engineering constants and ultimate strengths cannot be precisely predicted. More important, the ‘plane–remains–plane’ boundary conditions were applied to the above-mentioned analytical or numerical analyses. In fact, the ‘plane–remains–plane’ boundary conditions are over-constrained condition under shear loadings as indicated in previous work [35,36]. Due to periodic arrays of the RUCs in 3DOWC at both micro- and meso-scales, the periodic boundary condition (PBC) need be applied correctly to RUCs [36], rather than ‘plane–remains–plane’ based on small deformation theory as stated above [21–33]. Xia et al [37,38] and Li [39] developed the periodic boundary condition (PBC) into an explicit form and incorporated into finite element analysis. Then this two-scale analysis method were applied to unidirectional laminates [40–45], cross-ply laminates and angle-ply laminates [33,37,45], filament wound composite tube [46] and 3D interlock woven composite [47], consisting of detailed crack initiation and propagation using post-damage constitutive model on the concept of smear crack [41,44–46]. As an extension of previous research [32], this paper presents a 3-scale finite element strategy including averaged strengths

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2. Experimental 2.1. 3DOWC The 3D orthogonal woven composite was fabricated with the APROLTM INF80501, an unsaturated thermosetting resin, using vacuum assisted resin transfer molding (VARTM) method at room temperature. Fig. 1(a)–(c) display the 3D orthogonal woven composite. The non-crimp weft tows and warp tows are perpendicular to the Z-yarns. The specification of the 3DOWF pre-form (in Fig. 1(a)) is listed in Table 1. And the fiber volume fraction of 43.7% was determined by a burning method. Two kinds of specimens were cut with 200 mm in length along the warp (weft) direction and with 20 mm in width along weft (warp) direction respectively. The thickness for both the samples is 9.8 mm. 2.2. Three-point bending tests Three-point bending tests for the specimens were operated on MTS-810 machine with span of 150 mm between two supporting rollers (with radius of 10 mm), with the load applied by one roller in the middle of specimens. For the sample in length along warp yarns direction, load against deflection (at middle section) curve is shown in Fig. 2. The specimens behaved nearby linearly up to finial fracture, with ultimate load of 4.65 kN and the maximum deflection of 10.5 mm. During the test, the cracks were initiated at bottom surface area. Upon further loading, the weft and warp

Table 1 Specification of 3D orthogonal woven fabric. Yarns

Linear density (tex)

Layers

Weaving density (end/10 cm)

Weft Warp Z-yarns

800 800 56

17 16 N/A

50 47 N/A

5 4.5

Sample 1 Sample 2

4 3.5

Load (kN)

method ranges from fiber/matrix, matrix impregnated fiber bundle to full-scale beam structure of 3DOWCs. From the yarn architecture in the 3DOWC and the principle of equivalent fiber volume fraction in multi-scale level, the micro-RUC and sub-meso-RUCs (inner meso-RUC and surface meso-RUC) were established. The full-scale beam was constructed with interior element (with same size of inner-meso-RUC) and surface element in thickness direction. In the micro-RUC, the fiber and matrix were assumed to be isotropic elastic materials with maximum principal stress failure, respectively. Through FEM analysis on the micro-RUC model, the transversely isotropic elastic properties of fiber bundle and its longitudinal and transverse ultimate strengths were obtained. Then the properties and ultimate strengths of the surface layer and inner layer of the 3DOWC have been obtained through the FEM analyses on the surface meso-RUC and the inner meso-RUC scale. Finally, at the macro-scale, the 3DOWC beam was homogenized by abovementioned two types of elements. The global/local responses and damage initiation and propagation of the beams under 3-point bending cases were calculated. In addition, the different post-damage constitutive relations based on smeared crack concept were introduced to describe the elements behaviors. The constitutive models with damage initiation and evolution were realized by user-defined material subroutines (UMATs) and incorporated into the commercial finite element software package ABAQUS/Standard. The predicted results by the multi-scale FEM analyses were compared to the experimental results and good agreements were found.

3 2.5 2 1.5 1 0.5 0 0

2

4

6

(a)

10

12

14

16

Fig. 2. Load against deflection curves of 3DOWC in length along warp yarns direction.

layers were damaged gradually and propagated in length and thickness direction of sample. With further evolution of damaged zone in 3DOWC, the top surface was also damaged with small area. The specimen was failed ultimately in the central area with matrix cracking and yarns pull out at rear surface and compressive damage at top surface area as shown in Fig. 3. From the final damage morphology, no obvious delamination with layer-to-layer debonding patterns was observed. And also, the 3-point bending behavior of 3DOWC in length along weft yarns direction was tested as shown in Figs. 4 and 5. During the 3-point bending loading, the samples deformed elastically before the initiation of fracture at deflection of 10.1 mm and load of 4.4 kN in Fig. 4. Upon further loading, the samples still could undertake the ultimate load as before, but delamination occurred nearby the bottom layers in Fig. 5. The samples were damaged in form of delamination between the layers rather than cracked patterns in Fig. 3. Due to effect of Z-bundles at 3DOWC in length along different directions, the essential distinctions of mechanical response and damaged morphology for above two types of samples were shown in Figs. 3 and 5. 3. Finite element analyses 3.1. Multi-scale geometric models The 3D orthogonal woven composite consists of fiber bundles (polymer resin impregnated) and pure polymer matrix around fiber bundles in Fig. 1(c). For fiber bundles (warp bundles, weft

x1

x2

8

Deflection (mm)

(b) Fig. 1. 3D orthogonal woven composite (3DOWC).

(c)

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Fig. 3. Damaged morphology of 3DOWC in length along warp yarns direction.

5 Sample 1 Sample 2

Load (kN)

4

3

2

1

0

0

2

4

6

8

10

12

14

16

18

Deflection (mm) Fig. 4. Load against deflection curves of 3DOWC in length along weft yarns direction.

Fig. 5. Damaged morphology of 3DOWC in length along weft yarns direction.

bundles and Z-yarns), each one include hundreds of fibers distributed in the matrix by a randomly pattern. Based on the fiber bundles structure and fiber volume fraction, a multi-scale FE models were established as shown in Fig. 6. With the periodic array structure of 3D orthogonal woven fabric (3DOWF) as displayed in Fig. 1(a), a full-thickness meso-scale repeating unit cell model (meso-RUC) was established firstly in Fig. 6(b). To distinguish the essential structural distinction between the surface layer and the inner layers of meso-RUC, two types of sub meso-RUCs were further defined in Fig. 7. They are called as the inner meso-RUC and surface meso-RUC, respectively. Fig. 7(a) and (b) display the fiber bundle and the matrix respectively. The inner meso-RUC includes 1 full layer of warp bundles and 2 half layers of weft bundles. The surface meso-RUC consists of one inner meso-RUC plus a surface layer at the top. The full

thickness meso-RUC as shown in Fig. 6(b) is constructed by 14 layers of the inner meso-RUC and 2 layers of the surface meso-RUC. Based on the specification of the 3DOWF listed in Table 1, the meso-structural parameters of the 3DOWC were calculated and listed in Table 2, together with Table 3. From weaving density (50 ends/10 cm in weft yarn direction and 47 ends/10 cm in warp yarn direction) and thickness (9.64 mm) in Table 1, the sizes of meso-RUC was calculated with 4.24 mm in weft yarn direction, 4.00 mm in warp direction and 9.64 mm in thickness direction. The fiber volume fraction is about 59.5%. To determine the mechanical properties of the fiber bundles, a micro-scale repeating unit cell model (micro-RUC) was constructed shown in Fig. 6(a). In the micro-RUC model, the fibers were assumed to be distributed in a hexagonal array with fiber volume fraction of 73.5% by the ratio of tested fiber volume fraction of 43.7% to the fiber bundles volume fraction of 59.5%. Table 3 shows its exact sizes. The full-scale macro-scale beam model was established in Fig. 6(c). The beam was meshed with two types of elements homogenized by sub-meso-RUCs in Fig. 7, i.e. the surface elements and the inner elements. The above multi-scale FE models were meshed with 8-node brick elements (C3D8) with 8672 elements for the micro-RUC, 3600 elements for the inner meso-RUC, 6120 elements for the surface meso-RUC and 16,000 elements for macro-scale beam, respectively. The thick sizes of elements for inner layers and surface layers in macro-scale beam in Fig. 6(c) keep same with that of the surface and inner meso-RUCs in Fig. 7(a) and (b), respectively. For accuracy to be insured during the FEA, finer elements sizes were applied and they showed that the current meshed strategy can provide the following results accurately. On the multi-scale FE analysis for 3DOWC in Fig. 6, the mechanical parameters and failure strengths were obtained and transferred from fiber/matrix to micro-RUC, micro-RUC to meso-RUC, and meso-RUC to macro-scale beam, respectively. For the micro/ meso-RUCs in Figs. 6(a) and 7, the periodic boundary conditions were introduced in FE analysis by master node to slave nodes technology. The procedure to apply it with RUCs and how to determine their engineering constants have been discussed in our previous work [32]. Firstly, using the mechanical properties of fiber/matrix, the averaged stiffness and ultimate strengths of micro-RUC can be obtained. Then for meso-scale analysis in Fig. 7, the elements in the warp yarns, weft yarns and Z-yarns were homogenized from micro-RUC and the matrix impregnated fiber bundles (warp yarns, weft yarns and Z-yarns) in length keep parallel with fiber direction

Fig. 6. Multi-scale FE models of 3DOWC.

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Fig. 7. Sub-RUCs of 3DOWC at meso-scale level.

Table 2 Meso-structural parameters of 3DOWC.

Weft bundle Warp bundle Z-bundle Matrix

Table 4 Mechanical properties of fiber and resin.

Length/mm

Width/mm

Thickness/mm

Volume fraction/%

Materials

E/GPa

Poisson’s ratio/m

Failure criteria

4.24 4.00 – 4.24

1.4 1.4 0.4 4.0

0.26 0.26 0.1 9.64

30.93 27.46 1.11 40.5

Glass fiber AROPOLTM INF80501

72.5 3.6

0.22 0.35

2.175 GPa 72 MPa

Table 3 Sizes of micro/meso-RUC.

Micro-RUC Meso-RUC Inner RUC Surface RUC

Length/mm

Width/mm

Thickness/mm

0.1 4.24 4.24 4.24

3.46 4.00 4.00 4.00

2.00 9.64 0.56 0.90

in micro-RUC as displayed in Fig. 6(a). Their averaged responses can be calculated such as tensile modulus, Passion’s ratio and ultimate strengths in principal directions. For macro-scale FE beam analysis in Fig. 6(c), the surface elements and the inner elements were incorporated with averaged mechanical properties of inner/ surface-RUC in Fig. 7, respectively. On the 3-point bending case in Fig. 3, a local Cartesian coordinate system was established in FE beam with 1-direction in length (warp yarn direction), 2-direction in width (weft yarn direction) and 3-direction in thickness. And for another 3-point bending case in Fig. 5, the local coordinate was established with 1-direction in length (weft yarn direction), 2direction in width (warp yarn direction) and 3-direction in thickness in FE beam. In the FEM analyses, a vertical displacement was uniformly applied in the middle with span of 150 mm.

3.2. Multi-scale damage and post-damage constitutive models The mechanical parameters and ultimate strength values were transferred from micro-scale (the micro-RUC contains only fiber and matrix and their properties given in Table 4) to macro-scale

level as shown in Fig. 6. For the matrix/fiber, a maximum principal stress failure criterion was adopted. The meso-RUCs contain fiber bundles and pure matrix. For the matrix, the above failure criterion was still applied. The fiber bundles were transversally isotropic elastic material. For the fiber bundles, a crack will initiate if the normal stress in either longitudinal or transverse directions reaches the ultimate longitudinal or transverse ultimate strength value. The latter ultimate strength values of the fiber bundles were obtained from the FEM analysis on the micro-RUC. Similarly, based on FEM analyses on the inner meso-RUC and the surface mesoRUC, their ultimate strength values in the warp-, weft- and Z-directions could be obtained. And those ultimate strengths would serve as the failure criteria for the corresponding surface layer elements and inner layer elements in the macro-scale FEM analysis. In the above-mentioned multi-scale FEM analyses, if the stress of an element reached a value defined by failure criterion, a crack was assumed to be initiated in the direction perpendicular to the principal/normal direction. Instead of using element death technique commonly provided in commercial FEM codes, a post-damage constitutive model based on smeared crack concept was used in the current analyses. As shown in Fig. 8(a), if a crack initiates in the plane perpendicular to the local principal direction 1, the element loses load bearing capacity in the cracked plane, i.e. the stress components, r11, r12 and r13 must be zeros, but the element can still take other stress components. Therefore, for the fiber or pure matrix element in either microRUC or meso-RUCs, the post-damage relation is expressed in the local principal coordinate system as

fDrgcr ¼ ½DfDegcr  v½Bfrgcr or written in its full form with

ð1Þ

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X. Jia et al. / Computational Materials Science 79 (2013) 468–477

initiated, The stiffness parameters of bZ 1 and bZ 3 degrade by rate of 10-3 in each following iteration, meaning the stress increments, bZ 1 De11 , bZ 3 De12 and bZ3De3 were suppressed by a small value with increase of strain-controlled load. For the stress components, r11 ; r12 ; r13 , they degrade by ratio of 0.2. Naturally the total stress degrades quickly in vertical and tangent directions between surfaces of a crack by constitutive equation (1). As an extension, this post-damage mechanism was introduced to meso/macro-scale models as stated in following chapter. According to the tensile experimental brittle behavior of E-glass fiber and its composite, the value of b and v were determined and be adjusted to and incorporated in FE analysis. For the fiber bundles (at meso-scale level) in Fig. 8(b), three different types of cracks can be initiated independently. Assuming fiber bundles fractured along fiber yarns direction, which means crack 1 was created, its post-damage constitutive model for transversely isotropic fiber bundles can be represented by

(a)

8 9 2 Dr11 >cr bC 1 0 > > > > > 6 0 > > > C2 Dr22 > > > 6 > > > > 6 < 6 0 C3 Dr33 = ¼6 6 0 > 0 > Dr12 > > 6 > > > > 6 > > > > 4 0 0 D r > > 13 > > : ; Dr23 0 0 2 1 0 60 0 6 6 60 0  v6 60 0 6 6 40 0

(b)

0

0

C3

0

C2

0

0

bC 4

0 0

0 0

0 0

0

0 0

0

0 0

0

0 1 0 0 0 1

0 0 0 0

9 38 De11 >cr > > > > > De22 > > > 0 0 7 > > 7> > > > > 7< 0 0 7 De33 = 7 7 0 0 7> > De12 > > > > > 7> > > bC 4 0 5> De13 > > > > > : ; De23 0 C6 9cr 38 0 > r11 > > > > >r > > 07 22 > > > 7> > > > > 7< 0 7 r33 = 7 07 > r12 > > 7> > > > 7> > > 0 5> r > > 13 > > > : ; 0 r23 0

0

0

ð2Þ

where

C1 ¼

(c)

0

0

0

Z2

0

Z1

0

0 0

bZ 3 0

0

0

C3 ¼

0

0 0

0

0 0

0

0 1 0 0 0 1

0 0 0

9 38 De11 >cr > > > > > De22 > > > 0 07 > > 7> > > > 7> < 0 0 7 De33 = 7 0 07 > De12 > > 7> > > > 7> > > > D e bZ 3 0 5> > > 13 > > : ; 0 Z3 De23 9cr 38 0 > r11 > > > > > r22 > > > 07 7> > > > > > 7> < 0 7 r33 = 7 > 07 > 7> > r12 > > > 7> > > > 0 5> r > > 13 > > : ; 0 r23 0

0 0

0

E 1m223 2ð1þm23 ÞE2 m212 1

; C2 ¼

E2 ð1E2 m212 Þ 1

E

1m223 2ð1þm23 ÞE2 m212 1

E

Fig. 8. Local cracks for multi-scale FE analysis respective.

8 9 2 Dr11 >cr bZ 1 0 > > > > > 6 0 Z > Dr22 > > > 1 > > 6 > > > 6 < Dr > = 6 0 Z2 33 ¼6 6 0 > Dr12 > > 0 > 6 > > > > 6 > > > Dr13 > > 4 0 0 > > > : ; 0 0 Dr23 2 1 0 60 0 6 6 60 0  v6 60 0 6 6 40 0

E

E1 ð1m223 Þ

0

where

1m mE Z1 ¼ E; Z 2 ¼ ; ð1 þ mÞð1  2mÞ ð1 þ mÞð1  2mÞ

E Z3 ¼ 2ð1 þ mÞ

The superscript cr indicates the vectors in the local (crack) coordinate system. E is the tensile modulus of the matrix, b (e.g. b = 103  Dt) is a small value to describe the stiffness degradation in these three particular stress directions, and the constant v (e.g. v = 0.2) allows the three stress components to decrease to a near zero value within a few iterations in finite element analysis Once a crack

E2 ðm23 þE2 m212 Þ 1

E

1m223 2ð1þm23 ÞE2 m212

; C 4 ¼ G12 ; C 6 ¼

E2 2ð1þm23 Þ

1

where E1 is the modulus in fiber longitudinal direction while E2 is in transversely direction. For the macro-scale beam (as displayed in Fig. 8(c)) with orthogonal mechanical properties, if a crack was initiated with its surface perpendicular to 1-direction, its post-damage constitutive relation can be defined with

8 9 2 Dr11 >cr > > > > > 6 > > > Dr22 > ½L > > 6 > > > > 6 < = 6 Dr33 ¼6 60 0 0 > > Dr12 > > 6 > > > > 6 > > > 40 0 0 > Dr13 > > > > : ; 0 0 0 Dr23 2 1 0 60 0 6 6 60 0  v6 60 0 6 6 40 0

0 0

0 0

0

0 ½M

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0

9cr 38 0 > De11 > > > > > 7 > > De22 > 0 7> > > > > > 7> < 0 7 De33 = 7 7> De > > 12 > > 7> > 7> > > 5> > De13 > > > > : ; De23 9 38 0 > r11 >cr > > > > > > 07 r22 > > > 7> > > > > 7< 0 7 r33 = 7 7 0 7> > r12 > > > > > 7> > > 1 5> r13 > > > > > : ; 0 r23

In the above 2 16 ½L ¼ 4 H

3 bE1 ð1  E3 =E2 m223 Þ 0 0 7 2 E3 ðm23 þ E2 =E1 m12 m13 Þ 5 0 E2 ð1  E3 =E1 m13 Þ 2 E3 ð1  E2 =E1 m12 Þ 0 E3 ðm23 þ E2 =E1 m12 m13 Þ

ð3Þ

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X. Jia et al. / Computational Materials Science 79 (2013) 468–477

bG12 6 ½M ¼ 4 0

3

0 bG13

0

0

0 7 0 5 G23

1800 1600

with

E1 E2 E3  m223 E1 E23  m212 E22 E3  2m12 m13 m23 E2 E23  m213 E2 E23 H¼ E1 E2 E3

1200 1000 800 600 400

Note that the above Eqs. (1)-(3) were expressed in the local coordinate system. If the local coordinate system was different from the global coordinate system in FEM analyses, the Eqs. (1)(3) must be transformed to the global coordinate system as gl

gl

0

0

S11

1400

Stress (MPa)

2

gl

fDrg ¼ ½D fDeg  v½B frg

200 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Strain (%)

ð4Þ

(a)

with 70

½D0  ¼ ½TT ½D½T

2

l1

m21

n21

l1 m1

m1 n1

n1 l1

3

Stress (MPa)

2

S33

60

½B0  ¼ ½TT ½B½T

7 6 2 7 6 l2 m22 n22 l2 m2 m2 n2 n2 l2 7 6 7 6 2 2 2 7 6 m3 n3 l3 m3 m3 n3 n3 l3 ½T ¼ 6 l3 7 6 2l l 2m m 2n n l m þ l m m n þ m n n l þ n l 7 1 2 1 2 1 2 2 1 1 2 2 1 1 2 2 17 6 12 7 6 4 2l2 l3 2m2 m3 2n2 n3 l2 m3 þ l3 m2 m2 n3 þ m3 n2 n2 l3 þ n3 l2 5 2l3 l1 2m3 m1 2n3 n1 l3 m1 þ l1 m3 m3 n1 þ m1 n3 n3 l1 þ n1 l3 where li mi ni are directional cosines of the local coordinate axes in the global coordinate system. Finally, above multi-scale damage and post-damage constitutive models were realized by user-defined material subroutines (UMATs) and incorporated into commercial finite element software package ABAQUS/Standard. The mechanical responses and damage morphology of multi-scale FE models were obtained, respectively. 4. Results and discussions 4.1. Global and local responses of micro-RUC The elastic engineering constants of micro-RUC were predicted from properties of matrix and E-glass fiber in Table 4. As listed in Table 5, the fiber bundles behave transversely isotropic with much higher tensile modulus along fiber direction (E11 = 54.28 GPa) than transversely (E22 = E33 = 19.13 GPa). After the damage criteria were applied for fiber/matrix in micro-RUC, the longitudinal and transverse loading averaged responses were obtained in Fig. 9. Due to initiation and propagation of cracks in micro-RUC and reinforced effect of fibers, the micro-RUC had much higher ultimate strength in fiber longitudinal direction (r11 = 1599.45 MPa) than that transversely (r33 = 57.91 MPa). Over the above peak value, the microRUC lost its ability to endure load and degraded gradually.

50 40 30 20 10 0

0

0.2

0.4

0.6

0.8

1

1.2

Strain (%)

(b) Fig. 9. Mechanical responses of micro-RUC under normal strain loadings. (a) Under fiber directional loading. (b) Under transverse directional loading.

To verify the accuracy of failure stress in FEA compared to failure criterion for fiber/matrix (in Table 4), the local responses of fiber/matrix in micro-RUC under fiber longitudinal loading were given in Fig. 10. The node 11,741 in fiber and node 4686 in matrix were selected separately (in Fig. 10(a)). The maximum stresses in fiber and matrix were S11 = 2.17499 GPa and S11 = 71.99 MPa, respectively (in Figs. 10(b) and (c)). They both kept enough accuracy for pre-defined failure criterion in Table 4. When cracks initiated, a vibrating shakes occurred for local responses in process of stress degradation in short time, without effect of failure criterions to be carried out normally. 4.2. Global responses of sub-meso-RUCs Fig. 11 shows the global responses of sub-meso-RUCs under axial loadings based on properties of fiber bundles and matrix (in

Table 5 Mechanical properties of micro/meso-RUCs.

Micro-RUC Inner-RUC Surf-RUC

E11/GPa

E22/GPa

E33/GPa

G23/GPa

G13/GPa

G12/GPa

m12

m13

m23

54.28 23.44 17.40

19.13 24.20 21.62

19.13 12.06 9.20

7.11 3.44 2.74

6.97 3.45 2.75

6.97 3.95 3.43

0.2482 0.3256 0.3414

0.2482 0.3265 0.3301

0.3488 0.1330 0.1264

Notes: (1) Micro-RUC: 1-fiber direction, 2,3-transversely cross-section directions. (2) Inner/Surf-RUC: 1-warp yarns direction, 2-weft yarns direction, 3-Z-thickness direction.

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X. Jia et al. / Computational Materials Science 79 (2013) 468–477 Table 6 Failure strength amplitudes of micro/meso-RUCs in normal directions (MPa).

Micro-RUC Inner-RUC Surf-RUC

(a)

1-direction

2-direction

3-direction

1599.45 485.29 300.99

57.91 517.84 482.97

57.91 46.83 N/A

Notes: (1) Micro-RUC: 1-fiber direction, 2,3-transversely cross-section directions. (2) Inner/Surf-RUC: 1-warp yarns direction, 2-weft yarns direction, 3-Z-thickness direction.

ε 11 = 3%

2500 S11

2000

Stress (MPa)

6 In length along weft In length along warp

1500 5

1000

Load (kN)

4

500

3

2

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Strain (%)

1

(b) Node 11741 in fiber 0

80

Stress (MPa)

70

2

4

6

8

10

12

14

16

18

Deflection (mm)

S11

Fig. 12. Load against deflection curves of macro-scale beams for 3DOWC.

60 50 40 30 20 10 0 0

0.5

1

1.5

2

2.5

3

Strain (%)

(c) Node 4686 in matrix Fig. 10. Local mechanical responses in micro-RUC along fiber directional loading.

Inner RUC S22 Inner RUC S11 Surface RUC S22 Surface RUC S11 Inner RUC S33

500

400

300 200

100

0

Tables 4–6). Due to essential distinction in structures between the inner layers and surface layers, the inner-RUC has higher modulus than that of surface-RUC in principal directions (E11in = 23.44 GPa, E11surf = 17.40 GPa and E22in = 24.20 GPa, E22surf = 21.62 GPa, in Table 5) and higher ultimate failure strengths in weft direction than that in warp direction (S11in = 485.29 MPa, S11surf = 300.99 MPa and S22in = 517.84 MPa, S22surf = 482.97 MPa, in Table 6). The numerical results of sub-RUCs feature that the 3DWOC has a little higher tensile modulus in weft direction than that in warp direction due to one more layer of E-glass fiber bundles being included in weft yarns system as shown in Fig. 1(a). It is also observed much higher in-plane stiffness and strength than that of the out-plane ones (in Tables 5 and 6). 4.3. Global responses of macro-scale beam

600

Stress (MPa)

0

0

0.5

1

1.5

2

2.5

3

3.5

Strain (%) Fig. 11. Mechanical responses of sub-RUCs under normal strain loadings.

For macro-scale beam model for 3DOWC in Figs. 6(c) and 8(c), the elastic constants and ultimate strengths of the top and bottom layer elements were from the analysis on the surface meso-RUC and the inner meso-RUC, respectively (in Tables 5 and 6). According to the loading cases (in Figs. 3 and 5), the displacement load was applied at the top middle fields with span of 150 mm between the two supporting rollers. Fig. 12 shows the mechanical behaviors of macro-scale beams both in length along warp yarns direction and weft yarns direction, respectively. Compared to the tested results for sample 1 with ultimate load of 4.53 kN and ultimate deflection of 10.2 mm and for sample 2 with ultimate load of 4.65 kN and ultimate deflection of 11.1 mm as shown in Fig. 2, the predicted load against deflection curve agrees it well with linearly trend for most range with ultimate load of 4.35 kN and ultimate deflection of 11.8 mm. At same time, another 3-point bending analysis was done for samples in

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Fig. 13. Damage evolution for 3DOWC in length along warp yarns direction.

Fig. 14. Local nodes distributions for 3DOWC in length along warp yarns direction (U3 = 10 mm).

4.4. Local damage behaviors of macro-scale beam

560 N8697 N601

480 400

Stress (MPa)

length along weft yarns direction as shown in Fig. 4. Due to the damaged pattern with delamination between layers in Fig. 5, the predicted ultimate load of 5.43 kN is a little higher than tested ultimate load of 4.24 kN with ultimate failure deflection of 12.4 mm, which is a little higher for samples in length along weft yarns direction due to delamination (in Fig. 5). The predicted results imply that the 3-point bending properties for 3DOWC can be further improved with stronger fiber bundles as Z-bindles.

320 240 160

Fig. 13 displays the damage initiation and propagation of macro-scale beam in length along warp direction clearly. Initially, obverse stress concentrations are noticed in middle fields with the top surface fields under compressing loadings and tensile loadings for rear surface fields. Upon further loading, U3 = 8 mm, the middle fields on the rear surface started cracked when maximum stress is satisfied to the ultimate failure strength nearby rear surface layer in warp yarns direction (in Fig. 13(a)). The maximum stress concentrated layer zone in the middle fields occurs nearby rear surface layer zone. The obvious degradation of stress distribution in the middle rear surface fields is found (in Fig. 13(b)). And the damage zone propagated both along thickness and longitudinal directions. The damaged mechanism of macro-scale beam in the middle is with compressive loading form for top surface fields and tensile loading form for rear surface fields, together with damage initiation at rear surface layer initially and evolution to inner layers in the middle fields. The local responses for above macro-scale beam under 3-point bending case are shown in Figs. 14 and 15. Two different locations were chosen by node 8679 in surface element and node 601in interior element in Fig. 14. Their local responses in longitudinal direc-

80 0

0

2

4

6

8

10

12

14

16

Deflection (mm) Fig. 15. Local mechanical responses of inner and surface elements in warp yarn direction (S11).

tion (S11) are shown in Fig. 15. The maximum stress values agree well with the defined ultimate strengths in Table 6. For another macro-scale beam in length along weft yarns direction (in Fig. 5), a similar damage mechanism was revealed. In comparison with tested final damage morphology as shown in Figs. 3 and 5, the predicted damaged macro-scale beam was selected with deflection of U3 = 14 mm in Fig. 16(a) and (a). The damaged morphology in Fig. 16(a) has similar morphology with experimental result in Fig. 3. The damaged zone is larger fields around the loading area. This implies that much more cracks are initiated nearby the fractured fields from the middle area to

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Fig. 16. Final damaged morphology of 3DOWC.

two ends of 3DOWC specimen. But such damaged patterns are not observed for the beam in length along weft yarns direction under the 3-point bending loading as shown in Fig. 5, due to damaged patterns of delaminations in the middle fields. Compared to the predicted the damaged morphology as shown in Fig. 16(b), the sample in length along weft yarns direction should have behaved much higher failure strength with stronger Z-fiber bundles in resistance to delamination as shown in Fig. 5. As shown in Fig. 16(a) and (b), a larger zone nearby the middle rear surface layer is damaged in the beam in length along warp yarns direction than one in length along weft yarns direction at same 3-point bending loading deflection, due to one more layer of E-glass fiber bundles in weft yarns system than that in warp yarns system (in Fig. 1(a)). 4.5. Failure status simulation For 3-point bending analysis of the 3DOWC, a macro-scale beam was used as shown in Fig. 6(c). In this analysis, the inner layer and the surface layer of elements were homogenized from inner meso-RUC in Fig. 7(a) and surface meso-RUC in Fig. 7(b), respectively. The mechanical parameters along principal directions were transferred as follows. For the fiber/resin in the micro-RUC, a maximum principal stress failure criterion was used. In the meso-RUC, the meso-RUCs contain homogenized fiber bundles from the micro-RUC and the pure resin. For the resin, the maximum principal stress failure criterion was still applied. For the fiber bundles, a crack will initiate if the normal stress in either longitudinal or transverse directions reaches the ultimate strengths along longitudinal or transverse direction. In the macro-RUC analysis, the surface meso-RUC and inner meso-RUC were homogenized. And only the mechanical properties were left in surface element and inner element in macro-beam in Fig. 8. Therefore the failure details for fiber and matrix cannot be featured separately in 3-point bending analysis as shown in Figs. 13 and 14. The damage of element in Fig. 16

occurred based on expanded smear crack and post-damage constitutive model as described in Section 3.2. The element damage in Fig. 16 represents the meso-scale unit cell damaged partially or wholly, rather than fiber or resin separately. 5. Conclusions The deformation and damage mechanism of the 3D orthogonal woven composites (3DOWCs) under three-point bending load have been analyzed based on experimental tests and finite element analyses at multi-scale levels. The micro-representative unit cell (RUC) model, representing the fiber bundles with matrix and sub-RUCs (inner meso-RUC and surface meso-RUC), on behalf of the repeated array of inner layers and surface layers for the 3DOWC have been established. The ultimate failure strengths were obtained from properties of fiber tows and matrix. The three-point bending deformations and damage evolutions of the 3DOWCs were calculated using macro-scale cell. The predicted load-deflection curves agree well with the tested results. Furthermore, the detailed damage initiation and propagation including final damage morphologies of the 3DOWC were revealed in the FEA results. The current micro/meso/macro-scale FEA model with damage evolution model can be extended to the deformation and damage analyses of other textile composites under uni-axial or even multi-axial loading. Acknowledgements The authors acknowledge the financial supports from the National Science Foundation of China (Grant No. 11272087). The financial supports from Foundation for the Author of National Excellent Doctoral Dissertation of PR China (FANEDD, No. 201056), Shanghai Rising-Star Program (11QH1400100) and the Fundamental Research Funds for the Central Universities of China are also gratefully acknowledged.

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