1. INTRODUCTION

Composite materials reinforced with woven, braided and knitted fabrics are becoming increasingly popular for various structural applications in the automotive, aerospace, and other industrial sectors. Typical processing techniques include variations of lay-up manufacturing and contact lamination technologies, resin transfer moulding, vacuum/pressure bag moulding and autoclaving of fabric based thermosetting prepregs, and compression/preform moulding of thermoplastic and thermosetting composites. The purpose of this paper is to discuss the developments in the modelling and characterisation of fabric reinforced composite materials and structural components and to demonstrate for particular applications (spatially fabric-reinforced thin-walled structures) how it is possible to combine the basic advantages of fabric composites with those provided by implementation of three-dimensional reinforcement orientation in order to develop efficient designs of lightweight thin-walled composite components. The shape and geometry of the structural components under consideration reasonably implies an application of the approaches developed for thin-walled plates and shells. At the same time, analysis of the three-dimensional stress state of composite material at the level of elementary layers is essential in order to provide a structural integrity of the part at the stages of design and manufacture. The current paper demonstrates an application of corresponding approaches to the analysis of thin-walled composite shells of revolution with spatial orientation of fabric reinforcement. The particular geometry and material structure of the shell reinforced with carbon fabric patterns have been used as an example of the application of the general approach.

2. FABRIC REINFORCEMENT

Textile preforming plays an important role in composite technology providing glass, aramid, carbon, and hybrid fabrics that are widely used as reinforcing materials. The main advantages of woven composites are their cost efficiency and high processability, particularly in lay-up manufacturing of large-scale structures. At the same time, bending of fibres in the process of fabric weaving results in substantial reduction of material strength and stiffness. As can be seen in Figure 1, where a typical

woven structure is shown the warp (lengthwise) and fill (crosswise) yarns forming the fabric make the angle α≥ 0 with the plane of the fabric layer.

Figure 1. Elementary cell of a fabric structure

A demonstration of how the angleα affects the stiffness of the composite materials reinforced with the fabric layers has been presented in [1]. The tensile loading of the structural cell shown in Figure1 in the warp direction has been considered. An apparent modulus of elasticity was expressed in terms of

the moduli of the warp yarns, E , and fill yarns, E_f , as follows

w

E = (EA + EA )/ A (1)

a ff wwa

Here A = h(2t₁ + t ) is the apparent cross-sectional area and

a 2

A_f = h(2t + t )/2 A= h(4t + t )/4

12 w 12

are the areas of the fill and warp yarns in the cross section (see Figure 1). Substitution into equation

(1) yielded

EE (4t + t )

fw 12

E =+ (2)

^a 2 4(2t₁ + t₂)

Since the fibres of the fill yarns are orthogonal to the loading direction (axis 1 in Figure1), the modulus of the fill yarn material could be taken as E = E₂ , where E₂ is the transverse modulus of a

f unidirectional composite reinforced with the same fibres. Compliance of the warp yarn can be decomposed into two parts corresponding to t₁ and t₂ in Figure1, i.e.

2t +t 2tt

12 12

=+ (3)

E_w E₁ E_a

Here, E₁ is the longitudinal modulus of a unidirectional composite, while E_α can be determined with

the aid of the following equation, derived in [1] for the unidirectional anisotropic layer

1 cos⁴ α sin ⁴ α 12ν ₂₁  ₂₂

= + +⎜− sin α cos α (4)

EE E ⎝GE ⎠

α 1 2 12 1

Replacement of 1/ E in equation (3) with its expression from equation (4) and subsequent

a

substitution of the resulting fraction E/(2t +t ) into equation (2) yields

w 12

EE (4t + t )

2 112

E=+ (5)

a

24 ^E14 ^E1 22

4{2t + t [cos α+ sin α+ ( − 2ν )sin α cos α]}

12 21^E2 ^G12

o

For a composite reinforced with the glass fabric with the parameters: α= 12 , t = 2t and elastic

21

constants of a unidirectional material: longitudinal modulus, E₁ = 60GPa , transverse modulus, E = 13 GPa , shear modulus, G = 3.4 GPa , Poisson’s ratio ν= 0.3, the modulus of

2 12 21

oo

elasticity E= 235 GPa. For comparison, a cross-ply [/ laminate made of the same material

_a .090 ] has E = 36 5. GPa [1]. Thus, the modulus of a woven structure is less by 37% than the modulus of the same material but reinforced with straight fibres. The stiffness and strength of fabric composites depend not only on the yarns and matrix properties, but on material structural parameters as well, i.e. on fabric count and weave. The fabric count

specifies the number of warp and fill yarns per inch, while the weave determines how the warp and the fill yarns are interlaced. Typical weave patterns are shown in Figure2 [1].

(a)(b)(c)

Figure 2. Plain (a), twill (b), and triaxial (c) woven fabrics

Various types of material models have been developed for the characterisation of fabric composites with these kinds of reinforcement [2-4]. Being formed from one and the same type of yarns plain, twill, and satin weaves provide approximately the same strength and stiffness of the fabric in the warp and the fill directions. Typical stress-strain diagrams for a fibreglass fabric composite of such a type are presented in Figure 3 [1]. As can be seen, material demonstrates relatively low stiffness and strength under tension at the angle of 45⁰ with respect to the warp or fill directions. To improve these properties, multi-axial woven fabrics (one of this kind is shown in Figure 2(c)), can be used. Fabric materials, whose properties are closer to those of unidirectional composites are made by weaving a great number of larger yarns in a longitudinal direction and fewer and smaller yarns in an orthogonal direction. Such a weave is called unidirectional. This provides materials with high stiffness and strength in one direction, which is specific for unidirectional composites, and high processability, typical for fabric composites.

σ , MPa

Figure 3. Stress-strain curves for fibreglass fabric

ε,% composite loaded in tension at different angles with

respect to the warp direction

012345

Being fabricated as planar structures, fabrics can be shaped on shallow surfaces using the material’s high stretching ability under tension at 45⁰ to the yarns’ directions. Manufacturing of fabric-reinforced composite components with complex three-dimensional geometries needs the implementation of special kinds of technology including mapping between the two-dimensional fabric sheet and the three-dimensional surface, methods for predicting ‘flattened’ patterns used for fitting a sheet to a specific portion of the surface, techniques for inserting one or more ‘darts’ wherever necessary, methods for predicting and preventing defects such as wrinkling, folding, and tearing, and so on [5-6]. There are various approaches to the modelling the draping of woven fabrics onto complex geometries, namely the geometrical fitting, mechanical simulation of the drawing, and so on. The so-called fishnet analysis represents the first group, namely the geometrical fitting. In this case the effect of material properties and interaction with tooling are not taken into account. The fabric is considered as a continuum sheet, and the analysis of the draping is effectively an extension of the deep drawing of metal forming processes for the second group of technologies [7]. A meso-structural modelling of the viscoelastic behaviour of resin combined with unidirectional non-linear elastic behaviour of warp and fill fibres during the shaping process belongs to the third group of methods that could be used as a part of the process design for fabric composite structures [8]. Much more possibilities for such shaping are provided by implementation of knitted fabrics whose strain to failure exceeds 100%. Moreover, a knitting process provides an opportunity to shape the fibrous preform in accordance with the shape of the future composite part.

σ , MPa

250
0o
200
45o
150
100	90o
50
0		ε ,%	Figure 4. Typical stress-strain curves for fibreglass knitted composites loaded in tension at different angles with respect to the material structure primary orientation

Relatively high curvature of the yarns in knitted fabrics and possible fibre breakage in the process of knitting result in materials whose strength and stiffness are less than those of woven fabric composites, but whose processability is higher, and the cost is lower. Typical stress-strain diagrams for composites reinforced by knitted fabrics are presented in Figure 4 [1]. Material properties close to those of woven composites are provided by braided structures which, being usually tubular in form are fabricated by mutual intertwining, or twisting of yarns about each other. Typical braided structures are shown in Figure 5 [1].

01234

(a)(b)

Figure 5. Diamond (a) and regular (b) braided fabric structures

Biaxial braided fabrics in Figure 5 can incorporate longitudinal yarns forming a triaxial braid whose structure is similar to that shown in Figure 2 (d). Braided preforms are characterised with very high processability providing near net-shape manufacturing of tubes, and profiles with various cross-sectional shapes. Fabric reinforced multilayer composites suits very well the manufacture of thin-walled structural components. Under the loading, however, delamination phenomena could occur at the interface between the individual layers due to the low interlaminar shear strength. One of the ways to improve this situation is the implementation of stitched textile composites. Additionally, this technology creates an opportunity to produce textile assembly of complex textile preforms and their further plastics processing. Examples of the fabric composite rotor production based on this technique could be found in [9].

In general, introduction of the out-of-plane reinforcement leads to the development of spatially reinforced composite structures. A natural way the spatial (3D) reinforcement is provided by implementation of three-dimensionally woven or braided fabrics. The similar results could be achieved using the aforementioned processes of sewing and stitching, needle punching, or introduction of short fibres (or whiskers). Another class of spatially reinforced composite materials is represented by bulk composites multi-dimensionally reinforced with fine rectilinear yarns composed from fibres bound with a matrix [10].

Figure 6. Spatial orientation of the fabric reinforcement

Special type of a spatial composite structure (see Figure 6) formed by the fabric composite in which the plies reinforced at angle φ (warp direction) with respect to the x-axis make angles αand β with

the x₁ -axis and the x₂ -axis, respectively could be used for manufacture of thin-walled composite components [11]. This provides a possibility to design and manufacture thin-walled composite components with three-dimensional orientation of the fabric reinforcement and avoid the technological problems related to wrinkling, folding, tearing, etc.

3. MODELLING AND CHARACTERIZATION OF FABRIC COMPOSITES

Although a number of microstructural models of the type shown in Figure 1 and leading to equations similar to equation (5) have been developed to predict stiffness and even strength characteristics of fabric composites (e.g., [12]), for practical design and analysis, these characteristics are usually determined by experimental methods. Elastic constants entering constitutive equations written in the

principal material coordinates, e.g. moduli of elasticity E , E , shear modulus G , and Poisson’s

12 12

ratios νν, are found testing strips cut out of fabric composite plates at different angles with

,

12 21

respect to the orthotropy axes. The 0^o and 90^o specimens are used to determine moduli of elasticity EE and Poisson’s ratios νν (or parameters of non-linear stress-strain diagrams), while the

,,

1 2 1221

in-plane shear stiffness can be obtained with the aid of off-axis tensile test discussed in [1]. As shown

∗

in [1], there exists the angle of the reinforcement orientation φ specified by the following equation

1+ν₂₁ 1

−

E 2G

2 ∗ 1 12

sin φ= (6)

1+ν 1+ν 1

21 12

+−

E EG

1 212

or

1+νyx ₋1+ν₂₁

EE

2 ∗x 1

sin φ= (7) 1+νyx 1+ν 1+ν

21 12

2 −−

E EE

x 12

such that off-axis tension under this angle is not accompanied with shear-extension coupling. For this angle shear modulus is found to be equal to

^Ex

G = (8) ¹² 21 +ν_yx

()

where the modulus E and Poisson’s ratio ν are determined from the off-axis test performed on the

x yx

∗

specimen reinforced under angle φ .

For fabric composites whose stiffness in the warp and the fill directions is the same (E = E ) ,

12

equation (7) yields φ= 45^o . In general case the following equation could be used in order to find the shear modulus

sin ² φ cos² φ

G = (9)

12 44

1 cos φ sin φ 2ν _{21 2}

− −+ cos φ

EE EE

x 1 21

This equation gives the value of the shear modulus corresponding ideal off-axis tensile test when the shear deformation of the specimen ends is not restrained. Applicability of this equation for the real test conditions is discussed in [1] in detail. As shown, reasonably fair results could be obtained for the specimens with l > 3a , where l is the length and a width of the specimen, and the angle φ= 10^o . Thus, the uniaxial tensile test of the anisotropic flayer (off-axis test) can be used to obtain the in-plane shear modulus, G , while the other elastic constants entering constitutive equations written in the

12

principal material coordinates could be determined from conventional tests performed on the standard specimens loaded in the direction of principal material axes. Constitutive equations for the spatially reinforced composite material considered in previous section (see Figure 6.) are:

{σ} = [B]{e} −{S}t (10)

where t is the temperature change. Stiffness, B_ijlk , and thermal, S_ij , coefficients are calculated in

terms of the mechanical and physical properties of the elementary layers as follows

^Bijlk ^{= B}i ^′′j′k′l′^lij′^l ji′^lkl′^llk′ ^{; S}ij ^{= B}i ^′′j′k′l′^lij′^l ji′^αk ^′′l′

where, according to Figure 6 directional cosines are:

l = cos λ cosψ ; l =−sin λ cosψ ; l =−sinψ ;

1112 13

l₂₁ = sin λ cos β− sin β sinψ cos λ ;

l = cos λ cos β+ sin β sin λ sinψ ; l =−sin β cosψ ;

22 23

l₃₁ = sin λ sin β+ cos λ cos β sinψ ;

l = sin β cos λ− cos β sin λ sinψ ; l = cos β cosψ ;

32 33

ψ= tan ⁻¹(tanα cos β ) ; λ =φ + tan⁻¹(sinψ tan β ) ; (11)

The tensor components of the elastic constants B_i ′_{′j′k′l′} and the coefficients of linear thermal expansion

α′ for the elementary fabric layer referred to the system of coordinates x′, x′ , x′ are expressed in

k′l′ 123

terms of moduli of elasticity E , E , E shear moduli G , G , G , Poisson’s ratios ν , ν , ν ,

12 3 121323 121323

and coefficients of linear expansion α , α , α [13-14]. Additional tests are required, in order to

123

determine transverse characteristics of the fabric reinforced composite and complete the material mechanical characterisation in this case.

4. SPATIAL STRUCTURES FORMED BY FABRIC COMPOSITES

As discussed in Section 3, a considerable improvement of the composite laminate properties in transverse direction could be achieved if the material is additionally reinforced with fibres or yarns in the direction orthogonal (or making some angles) to the layer plane. Various technological developments have been implemented in this area including stitched textile assemblies, three-dimensional weaving or braiding, needle punching, spatial multi-dimensional composite reinforcement, etc. [2-4, 10-11]. Similar results could be achieved implementing the special type of a spatial composite structure (see Figure 6) assembled from the spatially oriented fabric layers and using the helical lay-up method [11]. Lightweight thin-walled components (usually with the rotational shapes and geometries) could be designed and fabricated on the basis of this technology. General approach to the modelling and analysis of the thin-walled composite shells reinforced in this way has been developed in [13]. This approach takes into account anisotropic characteristics of the material (due to the three-dimensional layer orientation), additional structural parameters (angles of the layer orientation), and a three-dimensional stress state of an elementary layer.

z

Figure 7. Spatially reinforced shell of revolution

Based on this theory, the analysis of axisymmetrical shell of revolution (see Figure 7) is discussed in this paper in order to demonstrate basic features of the mechanical behaviour, typical for the composite structures with the specified above type of reinforcement. The constitutive equations for the shell presented in Figure 7 have the form

N =hB ^ε −{}₀ , m =1,..., ; n =1,...,5

{}[ ] mn ⎨^ ⎬^ T 3

ψ

⎩

h³

{}M = ₁₂ []B_mn {} {} κ− T , mn, =1,..., 3 (12)

1

⎧ε⎫

{}=_[I _{pq ]}⎨_ψ⎬−{}I_t , p =,...,2 ; q =1,...,5

Q 1

⎩

T TT

where {} { , N , N }, {}{ ,M ,M }Q =QQ }

N =NM =M , and {} { ₁