Understanding and predicting stiffness in advanced fibre-reinforced polymer (FRP) composites for structural applications

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Woodhead Publishing Limited 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 11

R. M. Guedes, university of Porto, Portugal and J. XavieR, CiTaB/uTad, Portugal

DOI: 10.1533/9780857098641.3.298

Abstract: This chapter describes the elastic qualities of advanced

fibre-reinforced composites, in terms of characterization, measurement and prediction from the basic constituents, i.e. the fibre and matrix. The elastic analysis comprises applying micromechanics approaches to predict the lamina elastic properties from the basic constituents, and using classical lamination theory to predict the elastic properties of composite materials composed of several laminae stacked at different orientations. Examples are given to illustrate the theoretical analysis and give a full apprehension of its prediction capability. The last section provides an overview on identification methods for elastic proprieties based on full-field measurements. It is shown that these methodologies are very convenient for elastic characterization of anisotropic and heterogeneous materials.

Key words: elastic modulus, orthotropic, micromechanics, classical

lamination theory, digital image correlation, optical methods.

11.1 Introduction

Polymer matrix-based composites are essentially composed of fibres embedded in polymeric matrices. Polymeric matrices are divided into thermoplastic and thermoset resins. The fibres play an important role by their reinforcing character, enhancing the mechanical performance of these composites to high levels. These arrangements are more than simple combinations of fibres and polymeric matrices, since the synergistic effects are important in their global mechanical performance. In the case of continuous reinforcement, fibres are disposed mainly on the surface ply, either unidirectional or woven fabric, resulting in superior mechanical properties in the ply surface directions. Consequently, in the thickness direction the mechanical properties of composites are low. The complexity of these systems poses some challenges on the models used to predict stiffness evolution, when general loading conditions are applied, including the environmental effects.

The reinforcements can have different geometries – particulate, flake or fibres. The fibres can be continuous, discontinuous and oriented, or randomly

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distributed (see Fig. 11.1). The polymer matrices have much lower stiffness compared to the reinforcements. The matrix holds the reinforcements, protecting them against environmental and external aggression. Simultaneously the matrix allows an effective stress transfer to the reinforcements. This relies on good adhesion between the matrix and the reinforcement. The mechanical performance of fibre-reinforced polymers depends on different factors such as fibre length, fibre orientation and fibre shape. The circular shape is more usual since it is easier to produce, but other shapes are possible.

This chapter is devoted to the analysis of the elastic properties and their characterization for laminated advanced composites. it starts with a general overview of composite stiffness and then moves to lamina analysis, focused on unidirectional reinforced composites. The analysis of laminated composites is addressed through the classical lamination theory (CLT). The last section describes full-field techniques coupled with inverse identification methods that can be employed to measure the elastic constants.

11.2 General aspects of composite stiffness

Composite materials consist of two or more constituents mixed at a nano-, micro- or macroscopic level. These constituents are not soluble and form distinct phases. The reinforcing phase is embedded in the other phase, designated the matrix. Usually the reinforcing material is in the form of continuous or short fibres or particles. Actually, composites provide a more efficient way for using materials in structural applications. For example, they allow mass reduction without decreasing the stiffness and strength of components, by

(a) (b) (c) (d)

11.1 (a) Particulate, (b) oriented discontinuous fibres, (c) randomly

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replacing conventional metal alloys with composite materials. The reduction in weight and number of parts in an assembly largely compensates for the higher composite material costs. Moreover, composite systems often present improved fatigue, impact resistance, thermal conductivity and corrosion resistance.

The lightest structural component for a specified deformation, i.e. strain or deflection, under a specified load is the one with the highest specific modulus. This is calculated as the ratio of the Young’s modulus to the density of the material. For comparison purposes in Fig. 11.2 is plotted the specific modulus for several materials, including typical reinforcing fibres. Carbon-reinforced composites possess higher specific modulus when compared against aluminium and steel [1].

The stiffness of polymer-based composite systems depends on numerous factors such as the stiffness of constituents, the volume fraction of each component, and the size, shape and orientation of reinforcements. as a whole there are three distinct types of polymer composites: continuous fibre-reinforced polymer composites, short fibre-fibre-reinforced polymer composites, and polymer nanocomposites. Theoretical models based on micromechanical models are well developed and provide an adequate representation of composite stiffness. These micromechanical models are formulated based on assumptions of continuum mechanics. However, for nanocomposite materials, with fillers of size approximately 1 nm compared to the typical carbon fibre diameter of 50 mm, the rules and requirements for continuum

Specific modulus (GPa.m 3/kg) 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 Carbon fibre Unidirectional carbon lepoxy Aramid fibre Cross-ply carbon/epoxy Quasi-isotropic carbon/epoxyGlass fibre Steel Aluminium Unidirectional glass/epoxy Cross-ply glass/epoxy Quasi-isotropic glass/epoxy

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modelling are not satisfied. Several studies have attempted to address the applicability of continuum micromechanics to nano-reinforced polymer composites, by taking into account the effects associated with the significant size difference between nano fillers and typical carbon fibre. Nonetheless the classical approach can capture the main effects of nanoparticles on the elastic properties of nanocomposites. Particle clustering is the phenomenon mainly responsible for deviations of experimental data from theoretical predictions.

since the reinforcements possess higher modulus and strength than the polymer matrix, internal mismatches are induced, leading to local stress concentrations. Depending on the loading direction, fibre-aligned cracks may start to form. In the literature the terms matrix microcracks, microcracks, intralaminar cracks, ply cracks and transverse cracks often describe the same phenomenon [2]. These are observed during tensile loading, fatigue loading and thermocycling. Microcracks appear predominantly in plies off-axis to loading directions. This phenomenon leads to degradation in all effective moduli, Poisson’s ratios, thermal expansion coefficients [3] and moisture diffusion coefficients [3, 4]. Furthermore, the nucleation of microcracks may induce delamination [2].

The application of damage mechanics enables one to model matrix degradation in fibre-reinforced polymers due to cracking [5]. This damage is quantified in terms of crack density, i.e. the reciprocal of crack spacing. The model used by Roberts et al. [6], based on the previous work of Zang and Gudmundson [7, 8], applies to microcracks with crack surfaces parallel to the fibre direction and perpendicular to the lamina plane.

The viscoelastic and viscoplastic nature of polymeric materials that constitutes the composite matrix makes their mechanical behaviour time- or rate-dependent. Polymers are also temperature- and moisture-dependent displaying, in certain cases, large stiffness variations. Physical ageing is also an important issue that is often ignored for simplification purposes. A recent overview concerning these important matters is given elsewhere [9]. 11.3 Understanding lamina stiffness

The typical building block of a composite structure is the lamina, with a typical thickness of 0.125 mm. The lamina stress–strain relationships are described for orthotropic, transverse isotropic and isotropic materials. When a lamina is reinforced with unidirectional fibres it can be assumed to be a transversely isotropic material. in this chapter, theoretical determination of lamina elastic properties, assumed to be a transversely isotropic material, using micromechanics approaches is presented and illustrated with experimental data.

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11.3.1 The representative volume element (RVE)

Knowledge of the exact configuration of fibres and matrix in a composite structure is impractical for structural analysis. Observation of a typical micrograph of a cross-section of a unidirectional glass or carbon/epoxy lamina makes it clear that the fibre distribution is not uniform but is fairly random [10]. This provokes irregular gaps and contiguity, i.e. fibres touching in some locations. Similar cross-sections can be computationally randomly generated as shown in Fig. 11.3. detailed discussion on this subject can be found elsewhere [11–13].

The effective properties of a composite material correspond to properties averaged over a repeating representative volume element (Rve). This element should be large enough to represent the microstructure yet sufficiently smaller than the macroscopic structural dimensions. In fibre-reinforced composites, the RVE length scale is several times the fibre diameter. If the Rve dimension is small compared with the characteristic dimensions of the structure, the material can be assumed as homogeneous. Figure 11.4 depicts a schematic example of a RVE with other examples of elements that cannot be considered Rves.

11.3.2 Generalized Hooke’s law

The most general linear elastic relationship between the stresses and strains is given as

11.3 Example of randomly distributed fibres computationally

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Woodhead Publishing Limited 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 s_ij = C_ijkle_kl [11.1]

where skl represent the stresses and ekl the strains, and Cijkl is a

fourth-order tensor with 81 constants called the elastic moduli. This linear elastic stress–strain constitutive law is called the Generalized Hooke’s law. inverting (11.1) the strains are given in terms of stresses in the form

eij = C–1ijklskl [11.2]

where the compliance S_ijkl = C–1

ijkl is defi ned as the inverse of the stiffness.

When the stresses and strains are symmetric the number of independent constants is reduced to 36. Hooke’s law can be written in a contracted notation:

si = Cijej (i, j = 1, 2, …, 6) [11.3]

A three-dimensional representation of the stresses in contract notation is presented in Fig. 11.5.

If there exists a strain density function such as

W C Woodhead Publishing Limited W C Woodhead Publishing Limited W C_{ij i j} W = C W C Woodhead Publishing Limited W C Woodhead Publishing Limited = Woodhead Publishing Limited W C Woodhead Publishing Limited W C W 1C W 1C W ₂C W ₂C W C W C W ₂C W C W C e ee ei ji j [11.4] RVE Not an RVE

11.4 Example of a representative volume element (adapted from [14]).

X3 X1 X2 s3 s4 s5 s2 s4 s6 s6 s5 s1

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Woodhead Publishing Limited 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 and it verifi es si ei W = ∂ ∂ [11.5]

then cij is symmetric and the number of independent elastic constants is

reduced to 21. a material with 21 independent constants is called anisotropic. The stiffness matrix is written as

[ ] = 11 12 13 14 15 16 22 23 24 25 26 Woodhead Publishing Limited 26 Woodhead Publishing Limited 33 34 C [C ] [ ] C11 C C11 C CC1313 CC CC1515 CC C₂₂ C C₂₂ C CC₂₄₂₄ CC C C33 C C33 C ij [ _ij] [ ] C C Woodhead Publishing Limited C C Woodhead Publishing Limited C C C C C C C Woodhead Publishing Limited C Woodhead Publishing Limited C C Woodhead Publishing Limited C C Woodhead Publishing Limited Woodhead Publishing Limited C Woodhead Publishing Limited 35 C35 C C C36 Woodhead Publishing Limited 36 Woodhead Publishing Limited 44 C44 C C C45 Woodhead Publishing Limited 46 Woodhead Publishing Limited 55 C₅₅ C C C Woodhead Publishing Limited 56 Woodhead Publishing Limited Woodhead Publishing Limited 66 Woodhead Publishing Limited . Sym È Î Í È Í È Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÎÍÎ Í Í Í ˘ Woodhead Publishing Limited ˘ Woodhead Publishing Limited ˚ Woodhead Publishing Limited ˚ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˘ ˙ ˘ Woodhead Publishing Limited ˘ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˘ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˚ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˚ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited_[11.6]

The inverted matrix gives the symmetric compliance,

[ ] = [ij ] = [ ]ij [ ] _]_]–1–1 11 12 Woodhead Publishing Limited 13 Woodhead Publishing Limited Woodhead Publishing Limited 14 Woodhead Publishing Limited 15 16 22 Woodhead Publishing Limited 22 Woodhead Publishing Limited Woodhead Publishing Limited 23 Woodhead Publishing Limited 24 Woodhead Publishing Limited 24 Woodhead Publishing Limited 2 S C [ ]S C [ ] = [S = [C [ ]ij [ ]S C [ ]ij [ ] S₁₁ S S₁₁ S S S Woodhead Publishing Limited S S Woodhead Publishing Limited Woodhead Publishing Limited 13 Woodhead Publishing Limited S S Woodhead Publishing Limited 13 Woodhead Publishing Limited S₁₅ S S₁₅ S S S Woodhead Publishing Limited S S Woodhead Publishing Limited 22 S22 S Woodhead Publishing Limited 22 Woodhead Publishing Limited S S Woodhead Publishing Limited 22 Woodhead Publishing Limited S S Woodhead Publishing Limited S S Woodhead Publishing Limited 24 S24 S Woodhead Publishing Limited 24 Woodhead Publishing Limited S S Woodhead Publishing Limited 24 Woodhead Publishing Limited ij 5 2 55 2 5 6 Woodhead Publishing Limited 33 Woodhead Publishing Limited 34 35 36 44 45 46 55 56 66 Woodhead Publishing Limited . Woodhead Publishing Limited S 5 S2 5 2 S S Woodhead Publishing Limited S S Woodhead Publishing Limited Woodhead Publishing Limited 33 Woodhead Publishing Limited S S Woodhead Publishing Limited 33 Woodhead Publishing Limited S35 S S35 S S44 S S44 S S S55 S S55 S S Sy Woodhead Publishing Limited Sy Woodhead Publishing Limited Woodhead Publishing Limited m Woodhead Publishing Limited È Î Í È Í È Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍÍ Í Í Í Í Í Í Í ÍÎÍÎ ÍÍÍ Í ÍÍÍ ˘ ˚ ˙ ˘ ˙ ˘ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙˚˙˚ ˙ ˙ ˙ [11.7]

11.3.3 Material symmetry: orthotropic, transversely isotropic and isotropic materials

Consider a material that is symmetric about two planes: the x1–x2 and x2–x3 planes as shown in Fig. 11.6. It must be expected that the elastic constants in both coordinate systems (unprimed and primed) are identical, i.e. C_ij = C¢ij.

The defi nition of the coordinate systems leads to s1 = s¢1 s4 = –s4¢ s2 = s2¢ s5 = –s5¢ s3 = s3¢ s6 = –s6¢ e1 = e1¢ e4 = –e¢4 [11.8] e2 = e¢2 e5 = –e¢5 e3 = e3¢ e6 = –e¢6

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Expressing s1 = s¢1 in terms of strains and elastic constants using the Generalized Hooke’s law,

C₁₁e1 + C12e2 + C13e3 + C14e4 + C15e5 + C16e6

= C¢11e1 + C¢12e2¢ + C13¢ e¢3 + C¢14e4¢ + C15¢ e5¢ + C16¢ e6¢ [11.9] Now using the previous relationships, the following is obtained:

C₁₄ = C₁₅ = C₁₆ = 0 [11.10]

using similar procedures, it is possible to conclude that

C24 = C25 = C26 = C34 = C35 = C36 = C45 = C46 = C56 = 0 [11.11] The obtained stiffness matrix corresponds to an orthotropic material with nine independent constants. The constitutive equation can be written in matrix form as s s s t t 1 2 3 23 13 12 11 Woodhead Publishing Limited 12 Woodhead Publishing Limited = t Ï Ì Ô Ï Ô Ï Ô Ô Ô Ô Ô Ô Ô Ô Ô Ì Ô Ì ÔÔÔ Ó Ô Ì Ô Ì Ô Ô Ô Ô Ô Ô Ô Ô ÔÓÔÓ Ô Ô Ô ¸ ˝ Ô ¸ Ô ¸ Ô Ô Ô Ô Ô Ô Ô Ô Ô ˝ Ô ˝ ÔÔÔ ˛ Ô ˝ Ô ˝ Ô Ô Ô Ô Ô Ô Ô Ô Ô˛Ô˛ Ô Ô Ô C C Woodhead Publishing Limited C C Woodhead Publishing Limited 11 C₁₁ C Woodhead Publishing Limited C Woodhead Publishing Limited Woodhead Publishing Limited 13 Woodhead Publishing Limited Woodhead Publishing Limited 1 Woodhead Publishing Limited 13 Woodhead Publishing Limited 1 Woodhead Publishing Limited Woodhead Publishing Limited 22 Woodhead Publishing Limited 23 Woodhead Publishing Limited 23 Woodhead Publishing Limited 33 44 55 56 0 0 0 0 0 0 Woodhead Publishing Limited . Woodhead Publishing Limited 0 0 0 0 0 0 Woodhead Publishing Limited C C Woodhead Publishing Limited Woodhead Publishing Limited 22 Woodhead Publishing Limited C C Woodhead Publishing Limited 22 Woodhead Publishing Limited C Woodhead Publishing Limited C Woodhead Publishing Limited C C C Woodhead Publishing Limited Sy Woodhead Publishing Limited Woodhead Publishing Limited m Woodhead Publishing Limited È Woodhead Publishing Limited Î Woodhead Publishing Limited Í È Í È Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Í Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Î Woodhead Publishing Limited Í Woodhead Publishing Limited Î Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited ˘˘˘ ˚ ˙ ˘˘˘ ˙ ˘˘˘ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙˚˙˚ ˙ ˙ ˙ Ï Ì Ô Ï Ô Ï Ô Ô Ô Ô Ô Ô Ô Ô Ô Ì Ô Ì ÔÔÔ Ó Ô Ì Ô Ì 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Woodhead Publishing Limited 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 where t23 = s4, t13 = s5, t12 = s6 and g23 = e4, g13 = e5, g12 = e6.

The stiffness and compliance matrices are described in a very simple manner by the elastic constants. However, the elastic constants are not measured directly. Hence it is important to write the stiffness and compliance matrices as functions of the measured constants called engineering constants. The compliance matrix for an orthotropic material in terms of engineering constants is given as S = 1 _–_– _–_– ₀ – 1 –– 00 – – – – 1 21 2 31 2 12 1 2 32 – 32 0 – 0 3 31 1 E1 E E1 E E E1 E2 E1 E2 E E n21 n n21 n n 1 n n 1 _–n ₀ n12 _–n ₀ n12 n n 0 0 0 0 Woodhead Publishing Limited 0 0 Woodhead Publishing Limited Woodhead Publishing Limited nnn23 2 3 23 Woodhead Publishing Limited 13 Woodhead Publishing Limited 12 1 ₀ 0 1 0 Woodhead Publishing Limited 0 Woodhead Publishing Limited 0 Woodhead Publishing Limited 1 Woodhead Publishing Limited 0 Woodhead Publishing Limited 0 Woodhead Publishing Limited 0 1 E₂ E₃ E₂ E₃ G Woodhead Publishing Limited G Woodhead Publishing Limited G 0 0 Woodhead Publishing Limited 0 0 Woodhead Publishing Limited 0 0 Woodhead Publishing Limited 0 Woodhead Publishing Limited 0 0 0 0 0 Woodhead Publishing Limited 0 0 Woodhead Publishing Limited È Î Í È Í È Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍÍ Í Í Í Í Í Í Í Í ÍÍÍ Í ÍÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÎÍÎ Í Í Í ˘ ˚ ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˘ ˙ ˘ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ ˙ ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙˚˙˚ ˙ ˙ ˙ [11.13]

Since the matrix is symmetric S_ij = S_ji then it must be true that

nij n nij n n n i ji j Ei E Ei E j = ( ( ( ( i,i, ji, ji,i, ji, = 1, 2, 3) j Woodhead Publishing Limited = 1, 2, 3) Woodhead Publishing Limited [11.14] The correspondent stiffness matrix is given as

Woodhead Publishing Limited C = 1– 0 23 Woodhead Publishing Limited 23 Woodhead Publishing Limited Woodhead Publishing Limited 32 Woodhead Publishing Limited Woodhead Publishing Limited 1 Woodhead Publishing Limited 1 31 21 32 1 12 Woodhead Publishing Limited 12 Woodhead Publishing Limited n n Woodhead Publishing Limited n n Woodhead Publishing Limited 23 n n23 Woodhead Publishing Limited 23 Woodhead Publishing Limited n n Woodhead Publishing Limited 23 Woodhead Publishing Limited n n Woodhead Publishing Limited n n Woodhead Publishing Limited 21 n21 n Woodhead Publishing Limited 21 Woodhead Publishing Limited n n Woodhead Publishing Limited 21 Woodhead Publishing Limited n n n nn n2121 n D D Woodhead Publishing Limited D D Woodhead Publishing Limited Woodhead Publishing Limited 1 Woodhead Publishing Limited D D Woodhead Publishing Limited 1 Woodhead Publishing Limited D E E Woodhead Publishing Limited E E Woodhead Publishing Limited E E Woodhead Publishing Limited E E Woodhead Publishing Limited Woodhead Publishing Limited 1 Woodhead Publishing Limited E E Woodhead Publishing Limited 1 Woodhead Publishing Limited 21 E 21 23 E E 23 31E E n n 31E E n n E Woodhead Publishing Limited n n Woodhead Publishing Limited E E Woodhead Publishing Limited n n Woodhead Publishing Limited 21 n21 n E n2121 n n E n E 23n23n E nn E 23n23 31n31E nn E nn +++++++nn n232323 3123n23n23nnnn3131313131E nnnnnnn3131+++++++ _E E + E + E 23 E E + 2323 E + E n n23 E E nn +nn E + E n n n E n E + nn E nn + E 23n23n E nn E + 23n2323n23 E nn + E 23n23 31n31E nn E + nn31313131E nn + E n3131E n ₀ ₀ Woodhead Publishing Limited + + Woodhead Publishing Limited + + Woodhead Publishing Limited Woodhead Publishing Limited + Woodhead Publishing Limited + + Woodhead Publishing Limited + Woodhead Publishing Limited Woodhead Publishing Limited + Woodhead Publishing Limited Woodhead Publishing Limited n n Woodhead Publishing Limited Woodhead Publishing Limited + + Woodhead Publishing Limited n n Woodhead Publishing Limited + + Woodhead Publishing Limited + n n + + n +n + n +n + + n Woodhead Publishing Limited n n Woodhead Publishing Limited Woodhead Publishing Limited + Woodhead Publishing Limited n n Woodhead Publishing Limited + Woodhead Publishing Limited Woodhead Publishing Limited n Woodhead Publishing Limited Woodhead Publishing Limited 13 Woodhead Publishing Limited Woodhead Publishing Limited n n Woodhead Publishing Limited 13 Woodhead Publishing Limited n n Woodhead Publishing Limited Woodhead Publishing Limited + + Woodhead Publishing Limited n n Woodhead Publishing Limited + + Woodhead Publishing Limited 13 Woodhead Publishing Limited + + Woodhead Publishing Limited n n Woodhead Publishing Limited + + Woodhead Publishing Limited Woodhead Publishing Limited 32 Woodhead Publishing Limited Woodhead Publishing Limited + + Woodhead Publishing Limited 32 Woodhead Publishing Limited + + Woodhead Publishing Limited 2 + n n + + 31 + + n n + + + + 13 + + + 2 nn3232 nn + n +n + 32+ + n +n + + 12 31 2 Woodhead Publishing Limited 13 Woodhead Publishing Limited Woodhead Publishing Limited n n Woodhead Publishing Limited 13 Woodhead Publishing Limited n n Woodhead Publishing Limited 3 1 Woodhead Publishing Limited 3 1 Woodhead Publishing Limited Woodhead Publishing Limited n Woodhead Publishing Limited 3 1 Woodhead Publishing Limited n Woodhead Publishing Limited 1– + 1– + + + ₀ D D Woodhead Publishing Limited D D Woodhead Publishing Limited 2 D E2 D E D Woodhead Publishing Limited E E Woodhead Publishing Limited E E + _E _E + + + Woodhead Publishing Limited + + Woodhead Publishing Limited E E Woodhead Publishing Limited + + Woodhead Publishing Limited n n E n n E + n n + + _E _E + + n n + + + 2 E2 31 E E n nn n3131 E E n nn n31 E + n n + + 31 + + n n + + _E _E + + n n + + 31 + + n n + + _E 1313_E + + 13 + + _E _E + + 13 + + + + 1– + + _E _E + + 1– + + + _E ₀ ₀ Woodhead Publishing Limited 2 Woodhead Publishing Limited Woodhead Publishing Limited 222 3 13 3 21 12 3 23 13 0 0 0 0 Woodhead Publishing Limited 0 0 Woodhead Publishing Limited 23 0 ₂₃ 0 0 D D Woodhead Publishing Limited D D Woodhead Publishing Limited 3 D EEE333 D EEE D 23 E 23 13 E E 13 21E E 21E E G 0 G 0 0 0 G n23 n n23 n E n n E E 23 E E n2323 n E E 23 13n13 21nnn21E ––nnnn E n E n E 13 E E 13n13 E n E 13 21E E n2121E n E nn ++nn 21E n E n n E E + E E n n E E nn E 11 nn 00 0 00 0 0 0 0 0 0 0 0 0 00000 GG1313 000000 0 0 0 0 0 G12 È Woodhead Publishing Limited Î Woodhead Publishing Limited Í È Í È Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Í Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Î Woodhead Publishing Limited Í Woodhead Publishing Limited Î Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited ˘ ˚ ˙ ˘ ˙ ˘ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙˚˙˚ ˙ ˙ ˙ [11.15] where D = 1 – (n31n13 + n12n12 + n23n32 + n23n31n12 + n21n13n32)

(10)

Figure 11.7 gives an example of an orthotropic fibre-reinforced material where the cross-section of the fibres is oval.

When the cross-section of fibre-reinforced composite is a plane of isotropy, it is called a transversely isotropic material, as described in Fig. 11.8. Since the elastic properties are isotropic in the plane transverse to the fibres, the plane x₂–x₃ (Fig. 11.8), additional simplifications on elastic constants are obtained as follows:

C22 = C33, C12 = C13, C55 = C66, C44 = (C22 – C23)/2 [11.16] The stiffness matrix corresponds to a transversely isotropic material with five independent constants. The constitutive equation can be written in matrix form as

X3

X2 X1

11.7 Example of an orthotropic composite (adapted from [14]).

X2 X3

X1 Transversely _{isotropic plane}

(11)

Woodhead Publishing Limited 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 s s s t t t 1 2 3 23 13 12 11 12 = Ï Ì Ô Ï Ô Ï Ô Ô Ô Ô Ô Ô Ô Ô Ô Ì Ô Ì ÔÔÔ Ó Ô Ì Ô Ì Ô Ô Ô Ô Ô Ô Ô Ô ÔÓÔÓ Ô Ô Ô ¸ ˝ Ô ¸ Ô ¸ Ô Ô Ô Ô Ô Ô Ô Ô Ô ˝ Ô ˝ ÔÔÔ ˛ Ô ˝ Ô ˝ Ô Ô Ô Ô Ô Ô Ô Ô Ô˛Ô˛ Ô Ô Ô C11 C C11 C C131131 22 23 22 22 23 66 66 0 0 0 2 0 0 0 0 0 0 0 0 0 C₂₂ C C₂₂ C C C22 C C22–C C –C C Symm.. CC È Î Í È Í È Í Í Í Í Í Í Í Í ÍÍÍ Í Í Í Í Í Í Í Í ÍÍÍ Í ÍÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Î Í Î Í Í Í ˘ ˚ Woodhead Publishing Limited ˚ Woodhead Publishing Limited ˙ ˘ ˙ ˘ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˚ ˙ ˚ ˙ ˙ ˙ Ï Ì Ô Ï Ô Ï Ô Ô Ô Ô Ô Ô Ô Ô Ô Ì Ô Ì ÔÔÔ Ó Woodhead Publishing Limited Ó Woodhead Publishing Limited Ô Ì Ô Ì Ô Ô Ô Ô Ô Ô Ô Ô ÔÓÔÓ Woodhead Publishing Limited Ó Woodhead Publishing Limited Ô Woodhead Publishing Limited Ó Woodhead Publishing Limited Ô Ô Ô ¸ ˝ Woodhead Publishing Limited ˝ Woodhead Publishing Limited Ô ¸ Ô ¸ e e e g g Woodhead Publishing Limited g Woodhead Publishing Limited Woodhead Publishing Limited g Woodhead Publishing Limited 1 2 3 23 Woodhead Publishing Limited 23 Woodhead Publishing Limited Woodhead Publishing Limited 13 Woodhead Publishing Limited Woodhead Publishing Limited 12 Woodhead Publishing Limited ÔÔÔ Ô Ô Ô Ô Ô Ô Ô Ô ÔÔÔ Ô ÔÔÔ Ô ˝ Ô ˝ Woodhead Publishing Limited ˝ Woodhead Publishing Limited Ô Woodhead Publishing Limited ˝ Woodhead Publishing Limited ÔÔÔ Woodhead Publishing Limited˛ Woodhead Publishing Limited Woodhead Publishing Limited Ô Woodhead Publishing Limited Woodhead Publishing Limited ˝ Woodhead Publishing Limited Ô Woodhead Publishing Limited ˝ Woodhead Publishing Limited Woodhead Publishing Limited Ô Woodhead Publishing Limited Woodhead Publishing Limited Ô Woodhead Publishing Limited Ô Woodhead Publishing Limited Ô Woodhead Publishing Limited Woodhead Publishing Limited Ô Woodhead Publishing Limited Woodhead Publishing Limited Ô Woodhead Publishing Limited Ô Woodhead Publishing Limited Ô Woodhead Publishing Limited Woodhead Publishing LimitedÔ Woodhead Publishing Limited Woodhead Publishing Limited˛ Woodhead Publishing LimitedÔ Woodhead Publishing Limited˛ Woodhead Publishing Limited Woodhead Publishing Limited Ô Woodhead Publishing LimitedÔ Woodhead Publishing Limited Ô Woodhead Publishing Limited [11.17]

The compliance matrix for a transversely isotropic material in terms of engineering Constants is given as

S = 1 –– –– 0 1 _–_– ₀₀ 1 Woodhead Publishing Limited 0 Woodhead Publishing Limited 2(1+ Woodhead Publishing Limited 1+ Woodhead Publishing Limited Woodhead Publishing Limited ) Woodhead Publishing Limited 1 12 2 12 1 2 23 – 23 0 – 0 2 2 Woodhead Publishing Limited 2 Woodhead Publishing Limited Woodhead Publishing Limited 23 Woodhead Publishing Limited E1 E E1 E E E₂ E E₂ E E n12 n n12 n n –n 0 – 0 Woodhead Publishing Limited n Woodhead Publishing Limited 0 0 Woodhead Publishing Limited 0 0 Woodhead Publishing Limited Woodhead Publishing Limited 0 0 Woodhead Publishing Limited Woodhead Publishing Limited 0 Woodhead Publishing Limited Woodhead Publishing Limited Woodhead Publishing Limited E Woodhead Publishing Limited E Woodhead Publishing Limited E Woodhead Publishing Limited G Sym G Woodhead Publishing Limited 2 Woodhead Publishing Limited 12 12 0 0 Woodhead Publishing Limited 0 0 Woodhead Publishing Limited 1 ₀ . 1 È Î Í È Í È Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Î Í Î Í Í Í Woodhead Publishing Limited ˘ Woodhead Publishing Limited ˚ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˘ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˘ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ Woodhead Publishing Limited ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙˙˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙˙˙ ˙ ˙˙˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˚ ˙ ˚ ˙ ˙ ˙ [11.18]

The correspondent stiffness matrix is given as

Woodhead Publishing Limited Woodhead Publishing Limited C = Woodhead Publishing Limited = Woodhead Publishing Limited 1– – + 23 Woodhead Publishing Limited 23 Woodhead Publishing Limited 1 Woodhead Publishing Limited 1 Woodhead Publishing Limited Woodhead Publishing Limited 1 Woodhead Publishing Limited 1 Woodhead Publishing Limited 1 Woodhead Publishing Limited 2 12 1 2 1 1 Woodhead Publishing Limited 1 1 Woodhead Publishing Limited 2 2 2 1 2 – 1 2 + – 3+ – 3+ – + n n n Woodhead Publishing Limited n n Woodhead Publishing Limited Woodhead Publishing Limited 12 Woodhead Publishing Limited n n Woodhead Publishing Limited 12 Woodhead Publishing Limited n n –n n + –n n + 1 n1 n 1––––––––nnnn12 22 22 22 2 1 21 21 21 2nnnn ++++++++ D D Woodhead Publishing Limited D D Woodhead Publishing Limited 1 D1 D Woodhead Publishing Limited 1 Woodhead Publishing Limited D D Woodhead Publishing Limited 1 Woodhead Publishing Limited Woodhead Publishing Limited 1 Woodhead Publishing Limited D D Woodhead Publishing Limited 1 Woodhead Publishing Limited D D E E Woodhead Publishing Limited E E Woodhead Publishing Limited Woodhead Publishing Limited E E Woodhead Publishing Limited Woodhead Publishing Limited 1 Woodhead Publishing Limited E E Woodhead Publishing Limited 1 Woodhead Publishing Limited Woodhead Publishing Limited 12 Woodhead Publishing Limited E E Woodhead Publishing Limited 12 Woodhead Publishing Limited n n E n E n Woodhead Publishing Limited n n Woodhead Publishing Limited E E Woodhead Publishing Limited n n Woodhead Publishing Limited Woodhead Publishing Limited 12 Woodhead Publishing Limited n n Woodhead Publishing Limited 12 Woodhead Publishing Limited E E Woodhead Publishing Limited 12 Woodhead Publishing Limited n n Woodhead Publishing Limited 12 Woodhead Publishing Limited E E E Woodhead Publishing Limited E E Woodhead Publishing Limited – + E – E + Woodhead Publishing Limited – + Woodhead Publishing Limited E E Woodhead Publishing Limited – + Woodhead Publishing Limited 1 1 E1 1 E Woodhead Publishing Limited 1 1 Woodhead Publishing Limited E E Woodhead Publishing Limited 1 1 Woodhead Publishing Limited – + 1– 1 + E1––n1 E n ++ E ––nn E nn ++ E11––––––nn11 E nn ++++++ E11––––––––nn112 22 2E nn ++++++++ E ––––––––nnnnn 2 22 2E EEEnnnnn ++++++++ 0 0 0 2 E 2E E n E n E n_n2E n_n E ––nn E nn ++ E ––_–_–n_n2E n_n ++₊₊ E ––n E n ++nnn n 12 2 2 1 n1 1 n12 2 2 2 23 12 12 0 0 1 1+ 0 0 2 2 0 0 0 0 2 0 E E n E E1 n1 E E₁1 nn₁1 E E1 nn12 2E E₁₁– ₁₁_{2 2}E E₁₁– ₁₁2E E 2E E G G D D È Î Woodhead Publishing Limited Î Woodhead Publishing Limited Í È Í È ÍÍÍ Í Í Í Í Í Í Í Í ÍÍÍ Í ÍÍÍ Í Í Í Í Í Í Í Í Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Í Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Î Í Î Woodhead Publishing Limited Î Woodhead Publishing Limited Í Woodhead Publishing Limited Î Woodhead Publishing Limited Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited Í Woodhead Publishing Limited ˘ ˚ ˙ ˘ ˙ ˘ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˚ ˙ ˚ ˙ ˙ ˙ [11.19]