Elastic constants 216 Subject

A plastic component was subjected to a series of step changes in stress as follows. An initial constant stress of 10 MN/m was applied for 1000 seconds at which time the stress level was increased to a constant level of 20 MN/m. After a further 1000 seconds the stress level was decreased to 5 MN/m which was maintained for 1000 seconds before the stress was increased to 25 MN/m for 1000 seconds after which the stress was completely removed. If the material may be represented by a Maxwell model in which the elastic constant = 1 GN/m and the viscous constant rj = 4000 GNs/m, calculate the strain 4500 seconds after the first stress was applied. 

The measured relationships between piezoelectric polarization and strain for x-cut quartz and z-cut lithium niobate are found to be well fit by a quadratic relation as shown in Fig. 4.4. In both materials a significant nonlinear piezoelectric effect is indicated. The effect in lithium niobate is particularly notable because the measurements are limited to much smaller strains than those to which quartz can be subjected. The quadratic polynomial fits are used to determine the second- and third-order piezoelectric constants and are summarized in Table 4.1. Elastic constants determined in these investigations were shown in Chap. 2. 

According to the model, a perturbation at one site is transmitted to all the other sites, but the key point is that the propagation occurs via all the other molecules as a collective process as if all the molecules were connected by a network of springs. It can be seen that the model stresses the concept, already discussed above, that chemical processes at high pressure cannot be simply considered mono- or bimolecular processes. The response function X representing the collective excitations of molecules in the lattice may be viewed as an effective mechanical susceptibility of a reaction cavity subjected to the mechanical perturbation produced by a chemical reaction. It can be related to measurable properties such as elastic constants, phonon frequencies, and Debye-Waller factors and therefore can in principle be obtained from the knowledge of the crystal structure of the system of interest. A perturbation of chemical nature introduced at one site in the crystal (product molecules of a reactive process, ionized or excited host molecules, etc.) acts on all the surrounding molecules with a distribution of forces in the reaction cavity that can be described as a chemical pressure. 

The cited relationship for all classes of regular configuration are just the simplest. The subject is discussed comprehensively by Nye (1962), and practically by Yushkin (1971). Some authors recommend Su, the hardness constant, and some Sik, the elasticity constant. 

L. B. Freund Crack Propagation in an Elastic Solid Subjected to General Loading—I. Constant Rate of Extension, Journal of Mechanics and Physics of Solids, 20, 129-140 (1972). 

Used to determine the elastic constants of crystals (that may be subject to different T, P, etc.)... 

These considerations are amply supported by experiment. By making x-ray measurements on materials subjected to known stresses, we can determine the stress constant K experimentally. The values of K so obtained can differ substantially from the values calculated from the mechanically measured elastic constants. Moreover, for the same material the measured values of K usually vary with the indices hkl) of the reflecting planes. 

Many composites are elastically anisotropic and, thus, more than the two elastic constants are needed to describe their elastic behavior. This is a very large topic and, for this text, just some of the basic ideas that apply to unidirectional fiber composites will be discussed. Consider a two-phase material, with the two geometries shown in Fig. 3.15, being subjected to a uniaxial tensile stress. For the structure in Fig. 3.15(a), the two phases are subjected to equal strain whereas. 

The elastic or mechanical properties of a perfect crystal can be related to changes in the structural energy when the crystal is subjected to a strain field. It is possible to determine such properties by direct calculations using either quantum, or classical methods. The elastic constants of a solid at ambient pressure can be determined from ... 

The correspondence principle states that for problems of a statically determinate nature involving bodies of viscoelastic materials subjected to boundary forces and moments, which are applied initially and then held constant, the distribution of stresses in the body can be obtained from corresponding linear elastic solutions for the same body subjected to the same sets of boundary forces and moments. This is because the equations of equilibrium and compatibility that are satisfied by the linear elastic solution subject to the same force and moment boundary conditions of the viscoelastic body will also be satisfied by the linear viscoelastic body. Then the displacement field and the strains derivable from the stresses in the linear elastic body would correspond to the velocity field and strain rates in the linear viscoelastic body derivable from the same stresses. The actual displacements and strains in the linear viscoelastic body at any given time after the application of the forces and moments can then be obtained through the use of the shift properties of the relaxation moduli of the viscoelastic body. Below we furnish a simple example. 

The NisAl-based solid solutions are the subject of Chapter 10. Experimental methods and also the computer simulating technique are used for these technologically important intermetallic compounds. An increase in elastic constants as a result of the replacement of aluminum atoms by the atoms of 3d and 4d transition elements is described and discussed. We demonstrate the electron density distributions that evidence delocalization of electrons in alloyed intermetallic compounds. 

Elastic constants Material, when subjected to stress in one of three modes produces three elastic constants. 

As noted, the field of molecular simulation is relatively new, and a detailed review of it is beyond the scope of this text and we introduce here a few of the more relevant references. One of the first applications of molecular mechanics to polymers was by Theodorou and Suter (93,94), who modeled atactic polypropylene as an amorphous cell subjected to a range of stress conditions (hydrostatic pressure, pure strain, and uniaxial strain). Such modeling generally gives reasonable estimates of the elastic constants of a material [within 15% (79)], providing the density of the glass is correctly modeled. 

It was observed empirically by Hooke that, for many materials under low strain, stress is proportional to strain. Young s modulus may then be defined as the ratio of stress to strain for a material under uniaxial tension or compression, but it should be noted that not all materials (and this includes polymers) obey Hooke s law rigorously. This is particularly so at high values of strain but this section only considers the linear portion of the stress-strain curve. Clearly, reality is more complicated than described previously because the application of stress in one direction on a body results in a strain, not only in that direction, but in the two orthogonal directions also. Thus, a sample subjected to uniaxial tension increases in length, but it also becomes narrower and thinner. This quickly leads the student into tensors and is beyond the scope of this chapter. The subject is discussed elsewhere [21-23]. There are four elastic constants usually used to describe a macroscopically isotropic material. These are Young s modulus, E, shear modulus, G, bulk modulus, K, and Poisson s ratio, v. They are defined in Figure 9.2 and they are related by Equations 9.1-9.3. 

There have been no similar attempts to determine comprehensive sets of elastic constants for oriented fibres or monofilaments. Kawabata [21] devised an apparatus that used a linear differential transformer to measure diametral changes of 0.05 iim in single fibres of diameter 5 fxm subjected to transverse compression. Equation (7.2) above was then used to calculate the transverse modulus El = l/ ii. Results were obtained for poly(p-phenylene terephthalamide) (Kevlar) and high-modulus polyethylene (Tekmilon) fibres. Values of E were in the range 2.31 -2.59 GPa for Kevlar and a value of 1 -2 GPa was found for Tekmilon. 

Bone is mineralized tissue that constitutes part of the vertebral skeleton. Its function is to transmit and bear the loads to which the body is constantly subjected, protect the inner organs, and produce blood cells. From a mechanic point of view, the osseous tissue is an anisotropic viscoelastic material with properties that depend on direction and velocity of the applied load, as well as on the mineral content. Indeed, although the minerals confer rigidity and hardness to this tissue, the collagen imparts some elasticity, which ultimately results in its limited tensile strength and resilience. As a consequence of its composition, bone is an essentially brittle material (Fig. 17.1). Several pathologies can affect bone, including fractures, arthritis, infections, osteoporosis, and tumors, and may require adjuvant biomaterial devices. 

A perfectly elastic solid subjected to large shear deformations also exhibits normal stress differences. At equilibrium these are constant and strictly speaking outside the scope of viscoelasticity. The simplest extension beyond infinitesimal deformations leads to the relation ... 

The sensitivity of neutrons to light atoms in the presence of heavier ones has made this technique of special interest in the study of actinide defect structures. This subject is somewhat tangential to the scientific area covered in this review, so we will just mention the large body of work on UO2 and Th02 (Clausen et al. 1984, 1989) and on defect UO2.13. Most of the measurements have been done at high temperature, up to 2700K in the case of UO2, and give information on the elastic constants... 

This method is the subject of a classic textbook by Born and Huang [59], and has beenfoUowedextensively by Japanese researchers [60] (for a review see reference [61]) to calculate full sets of elastic constants for polymer crystals. 

Table 2 Elastic constants C = (Cm - Cn) 2 and C in the S" row in the Periodic Table. Upright numbers refer to experiments. Numbers in italics are mosdy from ab initio electron structure calculations. Parentheses denote that a subjective evaluation of available information is made in the present paper. For Cu in bcc Hf, Os and Pt, and in fee Os, not enough seems to be known to motivate an estimate. See the text for references and discussion on entries in the table.

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