Elements of elasticity theory

Appendix E Elements of elasticity theory E.5.3 Solid with cubic symmetry... 

Roscoe60 and Laws and McLaughlin30 have considered the problem of linearly viscoelastic elements, where the extremum theorems of elasticity theory do not apply. Roscoe considers linear viscoelasticity and uses the complex modulus comparing the material with an elastic one of the same phase geometry. He shows that the real parts of the overall moduli of the viscoelastic composite are not less than the corresponding overall moduli of the elastic composite when its phases have moduli equal to the real parts of the moduli of the corresponding phases in the viscoelastic composite. Similarly for the imaginary parts. 

Viscoelastic constitutive equations are used to model material properties. Viscoelastic theory combines the elements of elasticity and Newtonian fluids. The theory of viscoelasticity was developed to describe the behavior of materials which show intermediate behavior between solids and fluids. 

The characteristic features of a cord—mbber composite have produced the netting theory (67—70), the cord—iaextensible theory (71—80), the classical lamination theory, and the three-dimensional theory (67,81—83). From stmctural considerations, the fundamental element of cord—mbber composite is unidirectionaHy reinforced cord—mbber lamina as shown in Figure 5. From the principles of micromechanics and orthotropic elasticity laws, engineering constants of tire T cord composites in terms of constitutive material properties have been expressed (72—79,84). The most commonly used Halpin-Tsai equations (75,76) for cord—mbber single-ply lamina L, are expressed in equation 5 ... 

There are currently no ISO standard methods for biaxial extension and such measurements are rarely made in industrial laboratories. However, biaxial stressing is of value in the consideration of the theory of elasticity and is preferred by many for producing data for input to finite element programmes, as well as being involved in certain practical applications of rubber. The British standard for finite element analysis on rubber19 outlines the two approaches, equibiaxial stretching of a flat sheet and inflation of a flat sheet. The principles of these are illustrated in Figure 8.14. 

After an introductory chapter we review in Chap. 2 the classical definition of stress, strain and modulus and summarize the commonly used solutions of the equations of elasticity. In Chap. 3 we show how these classical solutions are applied to various test methods and comment on the problems imposed by specimen size, shape and alignment and also by the methods by which loads are applied. In Chap. 4 we discuss non-homogeneous materials and die theories relating to them, pressing die analogies with composites and the value of the concept of the representative volume element (RVE). Chapter 5 is devoted to a discussion of the RVE for crystalline and non-crystalline polymers and scale effects in testing. In Chap. 6 we discuss the methods so far available for calculating the elastic properties of polymers and the relevance of scale effects in this context. 

The elastic shear constant, ci - C 2, as obtained in the Bond Orbital Approximation of Eq. (8-14), as obtained by Sokcl (1976) and including the bonding-antibonding matrix elements from perturbation theory, and experimental values all are in units of 10 erg/enr ... 

We describe next the main elements of the Greenwood and Williamson theory [10,11], which is frequently used to model elastic contact between rough surfaces. In the simplest version of this theory, the load is assumed to be light enough so that deformation of the asperities is elastic and asperity tips can be... 

The mechanical response of viscoelastic materials to mechanical excitation has traditionally been modeled in terms of elastic and viscous components such as springs and dashpots (1-3). The corresponding theory is analogous to the electric circuit theory, which is extensively described in engineering textbooks. In many respects the use of mechanical models plays a didactic role in interpreting the viscoelasticity of materials in the simplest cases. However, it must be emphasized that the representation of the viscoelastic behavior in terms of springs and dashpots does not imply that these elements reflect the molecular mechanisms causing the actual relaxation... 

In the following results are presented for the application of the Boundary Finite Element Method both for the case of the laminate free-edge effect and for the case of a single transverse matrix crack in the framework of linear elasticity theory. 

The surface forces act on the surface elements conceived inside an elastic body or on its boundary. Let us imagine a flat infinitesimally small site ds at some point in an elastic body, and let us draw a unit normal n towards it, taking some direction of that normal as positive. If there is an elastic stress in the body, the parts of the body positioned on the different sides of the site nds will act on one another with a certain force. Let us denote by d the force acting on the site nds. Within the framework of elastic oscillation theory, this force is linearly proportional to the site nds ... 

The interest in multicomponent materials, in the past, has led to many attempts to relate their mechanical behaviour to that of the constituent phases (Hull, 1981). Several theoretical developments have concentrated on the study of the elastic moduli of two-component systems (Arridge, 1975 Peterlin, 1973). Specifically, the application of composite theories to relationships between elastic modulus and microstructure applies for semicrystalline polymers exhibiting distinct crystalline and amorphous phases (Andrews, 1974). Furthermore, as discussed in Chapter 4, the elastic modulus has been shown to be correlated to microhardness for lamellar PE. In addition, H has been shown to be a property that describes a semicrystalline polymer as a composite material consisting of stiff (crystals) and soft, compliant elements. Application of this concept to lamellar PE involves, however, certain difficulties. This material has a microstructure that requires specific methods of analysis involving the calculation of the volume fraction of crystallized material, crystal shape and dimensions, etc. (Balta Calleja et al, 1981). 

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