Mechanics of materials 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. 

The apparent Young s modulus, E2, of the composite material in the direction transverse to the fibers is considered next. In the mechanics of materials approach, the same transverse stress, 02, is assumed to be applied to both the fiber and the matrix as in Figure 3-9. That is, equilibrium of adjacent elements in the composite material (fibers and matrix) must occur (certainly plausible). However, we cannot make any plausible approximation or assumption about the strains in the fiber and in the matrix in the 2-direction. 

The in-plane shear modulus of a lamina, G12. is determined in the mechanics of materials approach by presuming that the shearing stresses on the fiber and on the matrix are the same (clearly, the shear deformations cannot be the samel). The loading Is shown in the representative volume element of Figure 3-15. By virtue of the basic presumption,... 

The foregoing are but examples of the types of mechanics of materials approaches that can be used. Other assumptions of physical behavior lead to different expressions for the four elastic moduli for a unidirectionally reinforced lamina. For example, Ekvall [3-2] obtained a modification of the rule-of-mixtures expression for and of the expression for E2 in which the triaxial stress state in the matrix due to fiber restraint is accounted for ... 

Use a mechanics of materials approach to determine the apparent Young s modulus for a composite material with an inclusion of arbitrary shape in a cubic element of equal unit-length sides as In the representative volume element (RVE) of Figure 3-17. Fill in the details to show that the modulus is... 

The mechanics of materials approach to the estimation of stiffness of a composite material has been shown to be an upper bound on the actual stiffness. Paul [3-4] compared the upper and lower bound stiffness predictions with experimental data [3-24 and 3-25] for an alloy of tungsten carbide in cobalt. Tungsten carbide (WC) has a Young s modulus of 102 X 10 psi (703 GPa) and a Poisson s ratio of. 22. Cobalt (Co) has a Young s modulus of 30x 10 psi (207 GPa) and a Poisson s ratio of. 3. 

The constituent material properties are substituted in Equations (3.61) and (3.57) to obtain the upper bound on E of the composite material and in Equation (3.47) to obtain the lower bound on E. In addition, the mechanics of materials approach studied in Problems 3.2.1 through... 

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. 

The mechanics of materials approach is the simplest and least useful. It assumes that each phase is subjected either to the same strain (the Voigt model Equation 9.4) or the same stress (the Reuss model Equation 9.5). This yields the relationships ... 

The mechanics of materials approach is taken in developing micromechanical relationships between constituent and composite material properties. The basic assumptions inherent in this approach are the following ... 

The micromechanics analysis which follows will use the mechanics of materials approach to obtain equations relating these composite elastic properties to the engineering elastic moduli of the fibers and matrix (Fp, Gf, J,f, and E, G, The key featnre of the mechanics of materials approach is the simplifying assnmption that planes perpendicular to the axis of the fibers remain plain during elastic deformation. 

FIGURE 8.5 Mechanics of materials approach in viewing deformation of representative heterogeneous volume elements, (a) Deformation assumption and (b) force balance. 

The mechanics of materials (or strength of materials or resistance of materials) approach embodies the usual concept of vastly simplifying assumptions regarding the hypothesized behavior of the mechanical system. The elasticity approach actually is at least three approaches (1) bounding principles, (2) exact solutions, and (3) approximate solutions. 

FIG. 35. Schematic diagram of the proposed mechanism of material transfer from a Cu-covered STM tip to the Au substrate induced by an appropriate tip approach toward the substrate. (From Ref. 525.)... 

K. Allesu, A. Ghosh, M.K. Muju Preliminary qualitative approach of a proposed mechanism of material removal in electrical machining of glass. European Journal Mech. Eng. 36 (1992), p. 202. 

High fibre volume is essential for good aircraft stmcture performance. It is also important that distribution of both fibre and resin is uniform throughout the component. To illustrate, the simple mle of mixtures (ROM) approach for calculation of longitudinal modulus falls into the mechanics of materials category the modulus of elasticity in the longitudinal direction ( l), which is the direction parallel fibres, is given as ... 

Hutchinson, J.W. and Evans, A.G. (2000) Mechanics of materials topdown approaches to fracture. Acta Materialia, 48, 125-135. 

The classical (Galilei, Newton) approach of the mechanics of material bodies is based on several fundamental principles of time, space, mass, event, d5mamics. They are best explained in the treatise Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) by the following assertions (classical postulates, CP) ... 

Benveniste Y. (1987) A new approach to the application of Mori-Tanaka s theory in composite materials. Mechanics of Materials, 6(2), 147-157. 

C. Li, T.W. Chou, Modeling of elastic buckling of carbon nanotubes by molecular structural mechanics approach. Mechanics of Materials, 36, 1047-1055, (2004). 

It is clear that in continuous sliding tests. Equation (21) is still valid but toff is zero. Such an approach can thus result in a protocol that provides information on the evolution of the surface with testing time, and the identification of the resulting mechanisms of material loss and surface degradation (Pourbaix, 1974). 

There are other examples of material approaches that have contributed to our interpretation of elastomer-to-metal bonding mechanisms, but all can be rationalized under one or more of the processes in the following schematic (Figure 6). Here, we have tried... 

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