Transversely isotropic composite

Maimi P, Mayugo J A and Camanho P P (2008) A three-dimensional damage model for transversely isotropic composite laminates , / Comp Mater, 42(25), 2717-2745. 

Approximate formulae for four E, E, v i, G12) of the five elastic properties of a transversely isotropic composite can be developed using simple approaches based on the strength of materials concepts. These concepts do not necessarily satisfy in full all the elasticity requirements. The RVE considered consists of a uniform arrangement of straight, continnons fibres. 

The thickness of the PCM surrounding the cells in the middle and deep zones was assumed to be 2.0 pm. Middle and deep zone cells are also surrounded by the PC which was modeled as a biphasic, homogeneous but linear transversely isotropic composite with thickness of 1.0 pm [27,28]. 

The elasticity approaches depend to a great extent on the specific geometry of the composite material as well as on the characteristics of the fibers and the matrix. The fibers can be hollow or solid, but are usually circular in cross section, although rectangular-cross-section fibers are not uncommon. In addition, fibeie rejjsuallyjsotropic, but can have more complex material behavior, e.g., graphite fibers are transversely isotropic. 

A variation on the exact soiutions is the so-caiied seif-consistent modei that is explained in simpiest engineering terms by Whitney and Riiey [3-12]. Their modei has a singie hollow fiber embedded in a concentric cylinder of matrix material as in Figure 3-26. That is, only one inclusion is considered. The volume fraction of the inclusion in the composite cylinder is the same as that of the entire body of fibers in the composite material. Such an assumption is not entirely valid because the matrix material might tend to coat the fibers imperfectiy and hence ieave voids. Note that there is no association of this model with any particular array of fibers. Also recognize the similarity between this model and the concentric-cylinder model of Hashin and Rosen [3-8]. Other more complex self-consistent models include those by Hill [3-13] and Hermans [3-14] which are discussed by Chamis and Sendeckyj [3-5]. Whitney extended his model to transversely isotropic fibers [3-15] and to twisted fibers [3-16]. 

One of the most important properties which control the damage tolerance under impact loading and the CAI is the failure strain of the matrix resin (see Fig. 8.8). The matrix failure strain influences the critical transverse strain level at which transverse cracks initiate in shear mode under impact loading, and the resistance to further delamination in predominantly opening mode under subsequent compressive loading (Hirschbuehler, 1987 Evans and Masters, 1987 Masters, 1987a, b Recker et al., 1990). The CAI of near quasi-isotropic composite laminates which are reinforced with AS-4 carbon fibers of volume fractions in the range of 65-69% has... 

Upper and lower bounds on the elastic constants of transversely isotropic unidirectional composites involve only the elastic constants of the two phases and the fiber volume fraction, Vf. The following symbols and conventions are used in expressions for mechanical properties Superscript plus and minus signs denote upper and lower bounds, and subscripts / and m indicate fiber and matrix properties, as previously. Upper and lower bounds on the composite axial tensile modulus, Ea, are given by the following expressions ... 

Fig. 11.1 Examples of transversely isotropic whisker-reinforced ceramic composites.

Insertion of Eqn. (19) into Eqn. (4) along with Eqn. (20) through Eqn. (23) yields a noninteractive reliability model for a three-dimensional state of stress in a transversely isotropic whisker-reinforced ceramic composite. The noninteractive representation of reliability for orthotropic ceramic composites would follow a similar development. The analytical details for this can be found in Duffy and Manderscheid.20... 

In a transverse direction, the thermal conductivity of a unidirectionally aligned fiber composite (i.e. transversely isotropic) can be approximated by the action-in-series model. This would give... 

The quantification of the thermal residual stresses will be shortly highlighted. Then, the occurrence of the first transverse crack will be analysed using the level of stress reached as well as fracture mechanics principles. For both these techniques, the composite beams are considered from a macroscopic point of view, where the layers are assumed to behave in a transversal isotropic way. 

Equation 20.1 assumes that anisotropic fillers such as fibers or platelets are all aligned in the direction of tensile deformation. If the orientation distribution differs from this transversely isotropic alignment, the composite modulus would be obtained in the three principal axis directions by orientational averaging. This equation (along with other equations which are not listed here), when combined with orientational averaging, can thus provide an estimate of the reinforcement expected from any orientational distribution of discrete fibers or platelets. 

The efficiency of reinforcement is related to the fiber direction in the composite and to the direction of the applied stress. The maximum strength and modulus are realized in a composite along the direction of the fiber. However, if the load is applied at 90° to the filament direction, tensile failure occurs at very low stresses, and this transverse strength is not much different than the matrix strength. To counteract this situation, one uses cross-pKed laminates having alternate layers of unidirectional libers rotated at 90°, as shown in Figure 3.47c. (A more isotropic composite results if 45° plies are also inserted.) The stress-strain behavior for several types of fiber reinforcement is compared in Figure 3.48. 

A multilayer laminate is composed of several laminas (or layers, or plies). Classical laminate theory (CLT) describes the linear elastic response of a thin laminated composite subjected to in-plane loads and bending moments see, e.g., Eckold (1994) and Herakovich (1998). Individual layers are assumed to be homogeneous, orthotropic, or transversely isotropic and in a state of plane stress. The constitutive relation for a thin multilayer laminated composite is given as ... 

For two-dimensional randomly oriented fibers in a composite, approximating theory of elasticity equations with experimental results yielded this equation for the planar isotropic composite stiffness and shear modulus in terms of the longitudinal and transverse moduli of an identical but aligned composite system with fibers of the same aspect ratio ... 

Even quasi-isotropic composition carbon FRPs, where the matrix performance has a more dominant role, can show a two—to four—fold improvement in fatigue resistance over steel and aluminium. It should be noted however that when stressed in the direction transverse to the fibres or in compression the fatigue life is substantially reduced. 

As a general rule, the thermal conductivity of a composite material is a complex function of the thermal conductivity of the matrix kj and that of the reinforcement (k). In the particular case of an orthotropic composite material, the thermal conductivity of each component (i.e., matrix, reinforcement) is a tensor quantity [k..] with only three components fcjp k and fcjj along major axes, that is, one in the axial direction (fcjj) and two in the transversal directions k and k ). In the particular case of transversely isotropic materials such as fiber reinforced composites, the axial thermal conductivity of the material, expressed in... 

Consider a subunit that comprised of fully aligned short fibers. The subunit is said to be transversely isotropic, if it is isotropic in all planes perpendicular to the fiber direction, and the fiber axis is the axis of asymmetry. For a transversely isotropic unidirectional stiffness tensor Cijki, orientation averaging can be done to give the composite stiffness, Cijki), as follows (Advani and Tucker 1987) ... 

Lin et al. (2004) adopted the same approach but they extended the five-constant equation to a nine-constant equation, which relaxes the transversely isotropic assumption of the properties for the composite of aligned inclusions, and can be use for more general orthotropic properties. Eduljee et al. (1994) presented an orientation averaging approach capable of distinguishing between dispersed and aggregated microstructures in short-fiber composites. 

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. 

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. 

The engineering properties of interest are the elastic constants in the principal material coordinates. If we restrict ourselves to transversely isotropic materials, the elastic properties needed are Ei, Ei, v, and G23, i.e. the axial modulus, the transverse modulus, the major Poisson s ratio, the in-plane shear modulus and the transverse shear modulus, respectively. All the elastic properties can be obtained from these five elastic constants. Since experimental evaluation of these parameters is costly and time-consuming, it becomes important to have analytical models to compute these parameters based on the elastic constants of the individual constituents of the composite. The goal of micromechanics here is to find the elastic constants of the composite as functions of the elastic constants of its constituents, as... 

The strength of materials approach provides fonr of the five elastic properties of transversely isotropic nnidirectional composites. Two properties Ey, V12) are well predicted by this simple approach, i.e. nsing the law of mixtures. The other two ( , G 2) require more accurate micromechanics models. The main reason for this is that E i and V12 are independent of fibre packing while d E and G i2 depend strongly on fibre arrangement. 

Kriz and Stinchcomb [32] published experimental data for unidirectional graphite/epoxy composites. These results illustrate the case when the fibres are transversely isotropic. The elastic properties of the matrix are = 5.28 GPa and T = 0.354, and for the fibres E = 232 GPa, E = 15 GPa, 0(2 = 24 GPa, v 2 = 0.279 and v 3 = 0.49. In Figs 11.21-11.25 are plotted the predictions against the experimental data for , , G12, G23 and V23, i.e. the longitudinal or axial modulus, the transverse modulus, the in-plane shear modulus, the transverse shear modulus and the transverse Poisson s ratio, respectively. 

For the transverse shear modulus, the approach designated self-consistent was based on the formula obtained by the self-consistent method for the plane-strain bulk modulus (11.61), on the transverse modulus calculated using the Chamis approach (11.49b) and the in-plane Poisson s ratio given by the rule of mixtures. Except when used to predict the axial modulus and the major Poisson s ratio, the rule of mixtures underestimates the remaining composite elastic properties. The Bridging Model proved to be a very effective theory to account for all five elastic properties for unidirectional composites that are transversely isotropic. 

This chapter began by describing briehy the elasticity of anisotropic materials, providing the fundamental relationships and the allowed simplihcations by the existence of material planes of symmetry. The current unidirectional composites are usually classihed as transversely isotropic materials. In this case, only hve independent elastic constants are necessary to fully characterize unidirectional composites. The micromechanics provides the analytical and numerical approaches to predict the elastic constants based on the elastic properties of the composite constituents. Several classical closed formulas are revisited and compared with experimental data. Finally, stiffness and compliance transformations are given in the context of unidirectional composites. Experimental data are used to assess theoretical predictions and illustrate the off-axis in-plane elastic properties. 

Kriz R. D. and Stinchcomb W. W. (1979) Elastic moduli of transversely isotropic graphite fibres and their composites - Equations used to calculate the complete set of elastic transversely isotropic properties for unidirectional fibre-reinforced materials having transversely isotropic fibres are experimentally verified by using improved ultrasonic techniques. Experimental Mechanics, 19(2), 41 9. 

In a transversally isotropic material, there is a plane in which all properties are isotropic. Perpendicular to this plane, the properties differ. One example for such a material is a hexagonal crystal which is transversally isotropic with respect to its mechanical properties.Other technically important materials may also be transversally isotropic, for example directionally solidified metals in which the grains have a preferential orientation (see also section 2.5), or composites (chapter 9) with fibres oriented in one direction, but aligned arbitrarily (or hexagonally) in the perpendicular plane. 

Composites with uniformly distributed unidirectional fibers aligned with a principal axis additionally possess a plane of isotropy in the transverse direction, where the material behavior is invariant to rotations. For such transversely isotropic properties, the number of constants is reduced to five. When the concerned axis of rotation is oriented in the es-direction,... 

Remark 5.7. For composites in accordance with Remark 5.2 consisting of aligned transversely isotropic fiber and matrix materials, the overall material behavior is expected to be at most orthotropic and, in the case of identical directional fiber fractions i/i and 1/2, to be transversely isotropic. 

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