Mechanics of materials

A simplistic mechanics of materials analysis of the shear modulus Gn gives... 

At this point, we recall from elementary mechanics of materials the transformation equations for expressing stresses in an x-y coordinate system in temis of stresses in a 1-2 coordinate system. 

Glenn Murphy, Advanced Mechanics of Materials, McGraw-Hill, New York, 1946, p. 83. 

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. 

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... 

For particulate-reinforced composite materials, Paul derived upper and lower bounds on the composite modulus [3-4]. His approximate mechanics of materials solution agrees fairly well with experimental data for tungsten carbide particles in cobalt. 

Classical lamination theory consists of a coiiection of mechanics-of-materials type of stress and deformation hypotheses that are described in this section. By use of this theory, we can consistentiy proceed directiy from the basic building block, the lamina, to the end result, a structural laminate. The whole process is one of finding effective and reasonably accurate simplifying assumptions that enable us to reduce our attention from a complicated three-dimensional elasticity problem to a SQlvable two-dimensinnal merbanics of deformable bodies problem. 

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. 

We have observed that the kinematics and the kinetics of the plate (and beam) problem are not consistent. However, such inconsistencies are an inherent part of mechanics of materials which must contain some inconsistencies othenvise, mechanics of materials would be elasticity ... 

It has been shown that the thermodynamic foundations of plasticity may be considered within the framework of the continuum mechanics of materials with memory. A nonlinear material with memory is defined by a system of constitutive equations in which some state functions such as the stress tension or the internal energy, the heat flux, etc., are determined as functionals of a function which represents the time history of the local configuration of a material particle. 

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.)... 

To optimize direct labor costs versus investment costs by the mechanization of material-handling operations... 

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