Micromechanics elasticity approach

The mechanics of materials approach to the micromechanics of material stiffnesses is discussed in Section 3.2. There, simple approximations to the engineering constants E., E2, arid orthotropic material are introduced. In Section 3.3, the elasticity approach to the micromechanics of material stiffnesses is addressed. Bounding techniques, exact solutions, the concept of contiguity, and the Halpin-Tsai approximate equations are all examined. Next, the various approaches to prediction of stiffness are compared in Section 3.4 with experimental data for both particulate composite materials and fiber-reinforced composite materials. Parallel to the study of the micromechanics of material stiffnesses is the micromechanics of material strengths which is introduced in Section 3.5. There, mechanics of materials predictions of tensile and compressive strengths are described. 

A strong background in elasticity is required for solution of problems in micromechanics of composite materials. Many of the available papers are quite abstract and of little direct applicability to practical analysis at this stage of development of elasticity approaches to micromechanics. Even the more sophisticated bounding approaches are a bit obscure. 

The micromechanical behavior of a lamina was treated in Chapter 3. Both a mechanics of materials and an elasticity approach were used to predict the fundamental lamina stiffnesses that were compared to measured stiffnesses. Mechanics of materials approaches were used to predict some of the fundamental strengths of a lamina. 

In this section, a micromechanics-based approach for randomly oriented discrete elastic isotropic spheroid particles randomly dispersed in a continuous elastic isotropic medium is presented. The present micromechanical model uses a self-consistent scheme based on the double-inclusion model to account for both the inter-particle and particle-matrix interactions. 

The objective of aii micromechanics approaches is to determine the eiastic moduli or stiffnesses or compiiances of a composite materiai in terms of the eiastic moduii of the constituent materiais. For example, the elastic moduii of a fiber-reinforced composite materiai must be determined in terms of the properties of the fibers and the matrix and in terms of the reiative voiumes of fibers and matrix ... 

Also thermocapillarity can be utilized to control the flow of the droplet (or bubble) phases Baroud et al. demonstrated control over formation of droplets and guiding them with a laser beam [52]. The conceptually simplest approach to valving at small scales is to construct micromechanical valves. These approach, however, needs not be simple in terms of fabrication of the devices. Mechanical microvalves have two most popular varieties, both utilizing the deformation of elastic membranes in soft (PDMS) systems [53] proposed by Quake et al, and for rigid chips [54] by Grover et al. 

We consider an elastic solid weakened by a set of microcracks. The elastic free energy, used as thermodynamic potential, can be estimated by using micromechanics approaches (Krajcinovic 1989, Pensee and Kondo 2001). In this work, we assume an isotropic distribution of microcracks. We limit the present study to the case of fully open microcracks. However we account for an energy coupling between damage evolution and plastic flow. Therefore, the thermodynamic potential for dry material is obtained ... 

In a multiscale analysis, the localization relations, that is, stress and strain concentration tenors, bridge the microscopic and macroscopic mechanical fields. When the medium behaves elastically, these relations are exact [56]. The main difficulty arises when nonlinearity is introduced in the mechanical behavior of the subphases, such as inelastic deformation or damage [56,62], which is the case for SMPFs. In general, three approaches within the micromechanics framework are available to establish the localized relations in the presence of such nonlinear processes. 

This view of traditional composite micromechanics, underlies the widely accepted rule-of-mixtures approach to modeling fiber reinforced composite materials. It states that the modulus of the composite is a linear combination of the moduli of the materials from which it is composed, and weights each modulus with the volume fraction of that component. Its basis lies in continuity of parallel strain between the fibers and matrix provided a linearly elastic response of the composite occurs for small strains. 

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. 

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. 

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. 

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. 

There have been a number of attempts to produce more appropriate predictions of the Young s modulus of particulate reinforced composites, without having such widely-separated bounds on the predictions. They have also been reviewed recently by Young et a/. A number of years ago, Halpin and Tsai developed an approach based upon the self-consistent micromechanics method of Hill that enabled prediction the elastic behaviour of a composite for a variety of both fibre and particulate geome-... 

Considering PET at ambient temperature which shows a low contrast in terms of elastic behavior between amorphous and crystalline phases, several models have proven to provide similar results (Figure 1.20). Here, micromechanics models are only slightly sensitive to the crystallite shape ratio parameters. It explains why models with no crystallite shape ratio, such as the U-inclusion model [183], still apply. Therefore, the obtained results in the case of glassy amorphous phases ai e contrasted. In terms of absolute values, the Yomig s modulus is decently approached by the models but in terms of the shape of the cm ve, the convexity of the experimental data is poorly represented. 

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