Elastic body

Figure 3.6 Definition of variables to define the shear deformation of an elastic body.

The equilibrium problem for an elastic body occupying the domain fig can be formulated as follows. We want to find a function W = u,v,w)... 

Banichuk N.V. (1980) Shape optimization of elastic bodies. Nauka, Moscow (in Russian). 

Khludnev A.M. (1983) A contact problem of a linear elastic body and a rigid punch (variational approach). Appls. Maths. Mechs. 47 (6), 999-1005 (in Russian). 

The new approach to crack theory used in the book is intriguing in that it fails to lead to physical contradictions. Given a classical approach to the description of cracks in elastic bodies, the boundary conditions on crack faces are known to be considered as equations. In a number of specific cases there is no difflculty in finding solutions of such problems leading to physical contradictions. It is precisely these crack faces for such solutions that penetrate each other. Boundary conditions analysed in the book are given in the form of inequalities, and they are properly nonpenetration conditions of crack faces. The above implies that similar problems may be considered from the contact mechanics standpoint. 

Figure 3.2. Application of stress to a highly elastic body. Rate of chain uncoiling with time...

The J value is defined as the elastic potential difference between the linear and nonlinear elastic bodies with the same geometric variables [52,53]. The elastic potential energy for a nonlinear elastic body is expressed by ... 

Contact mechanics deals with the deformation of solids in contact. Consider two elastic bodies, shown schematically in Fig. 3, of radii of curvature R[ and Rt, Young s moduli E and E2, and Poisson s ratios and V2. Define... 

R. A. Beth, Statics of Elastic Bodies, In Handbook of Physics, (Edited by E. U. Condon and H. Odishaw). McGraw-Hill, New York (1958). 

S. G. Lekhnitskil, Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, San Francisco, 1963. 

The resulting finite difference equations constitute a set of nonho-mogeneous linear algebraic equations. Because there are three dependent variables, the number of equations in the set is three times the number of material points. Obviously, if a large number of points is required to accurately represent the continuous elastic body, a computer is essential. 

S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, Government Publishing House for Technical-Theoretical Works, Moscow and Leningrad, 1950. Also P. Fern (Translator), Holden-Day, San Francisco, 1963. 

Bernoulli and Euler dominated the mechanics of flexible and elastic bodies for many years. They also investigated the flow of fluids. In particular, they wanted to know about the relationship between the speed at which blood flows and its pressure. Bernoulli experimented by puncturing the wall of a pipe with a small, open-ended straw, and noted that as the fluid passed through the tube the height to which the fluid rose up the straw was related to fluid s pressure. Soon physicians all over Europe were measuring patients blood pressure by sticking pointed-ended glass tubes directly into their arteries. (It was not until 1896 that an Italian doctor discovered a less painful method that is still in widespread... 

With respect to the harmonic content of the strain signal, a correction method was therefore developed for torque harmonics, based on observations made when testing an ideal elastic body, for instance, the calibration spring. T(n(x>/ (a) data are corrected according to... 

If the process is conducted reversibly, dQ = TdS where S is the entropy of the elastic body. Substitution of this expression for dQ in Eq. (5) will require dW to represent the element of reversible work. In order to comply with this requirement, the coefficients P and / in Eq. (6) must be assigned their equilibrium values. In particular, / will henceforth represent the equilibrium tension for a given state of the system, which may be specified variously by aS, F, and L, by T, F, and L, or by T, P, and L. Then... 

Fig. 84.—The retractive force / of a hypothetical elastic body plotted against the absolute temperature at constant pressure.

Considerably better agreement with the observed stress-strain relationships has been obtained through the use of empirical equations first proposed by Mooney and subsequently generalized by Rivlin. The latter showed, solely on the basis of required symmetry conditions and independently of any hypothesis as to the nature of the elastic body, that the stored energy associated with a deformation described by ax ay, az at constant volume (i.e., with axayaz l) must be a function of two quantities (q +q +q ) and (l/a +l/ay+l/ag). The simplest acceptable function of these two quantities can be written... 

This is Mooney s equation for the stored elastic energy per unit volume. The constant Ci corresponds to the kTvel V of the statistical theory i.e., the first term in Eq. (49) is of the same form as the theoretical elastic free energy per unit volume AF =—TAiS/F where AaS is given by Eq. (41) with axayaz l. The second term in Eq. (49) contains the parameter whose significance from the point of view of the structure of the elastic body remains unknown at present. For simple extension, ax = a, ay — az—X/a, and the retractive force r per unit initial cross section, given by dW/da, is... 

The mechanical properties of a material play an important role in powder flow and compaction by influencing particle-particle interaction and cohesion, that is to say, by influencing the true area of contact between particles. For example, Hertz [26] demonstrated that both the size and shape of the zone of contact followed simply from the elastic properties of a material. Clearly then, the true area of contact is affected by elastic properties. From the laws of elasticity, one can predict the area of contact between two elastic bodies. More recent work has demonstrated, however, that additional factors must be taken... 

For our purposes, work is done when a displacement occurs under the influence of a force the amount of work is taken as the product of a force by the displacement. Because force and displacement can be given suitable operational significance, the term work also will share this characteristic. The measurement of the displacement involves experimental determinations of a distance, which can be carried out, in principle, with a measuring rod. The concept of force is somewhat more complicated. It undoubtedly originated from the muscular sensation of resistance to external objects. A quantitative measurement is obtained readily with an elastic body, such as a spring, whose deformation can be used as a measure of the force. However, this definition of force is limited to static systems. For systems that are being accelerated, additional refinements must be considered. Because these considerations would take us far... 

For quality cured thermoset resins, approximately one percent of the mass is soluble when subjected to long-term leaching with tetrahydrofuran. Equilibrium is approached in two weeks resin swell is not visually noticeable. The monomeric, chemical structures are such that the hydrocarbon resins exhibit more pronounced viscoelastic properties whereas, the epoxy resins are similar to elastic bodies when subjected to tensile testing at room temperature. Therein, LRF 216 is less sensitive to flaws and is more nonlinear in tensile or compressive stress-strain analysis. 

This is the fundamental for phenomenologic analyses of elastic bodies. 

Refs 1)Beil 2,849(327) [6931 2)Org-SynthjCollVol 2(1943). 12 Acoustics. A branch of science which treats of the phenomena and laws of sound waves (including their production, transmission and effects), and other vibrations in elastic bodies(See also Ultrasonics)... 

On the first line, SX is regarded as a function of X. On the second line, the first term is the work exerted from the outside on the surface, da and rij being the surface element and the outward unit normal vector, while the second term is the change within the elastic body due to mechanical disequilibrium. We divide the stress into two parts,... 

Note that e was defined as e = K/fi + 1/3 in Eq. (4.17) for the isotropic case c = 1. As a simplifying result, these conditions are of the same form as those for the usual isotropic elastic bodies [36] except for the difference in the definition of e. 

According to the theory of elasticity, the pure shear and torsion are accompanied only by a change in shape of an elastic body but its volume remains unchanged. The equation of state of the Hookean body for pure shear can be represented in the following form 7,10)... 

Thus, this consideration shows that the thermoelasticity of the majority of the new models is considerably more complex than that of the phantom networks. However, the new models contain temperature-dependent parameters which are difficult to relate to molecular characteristics of a real rubber-elastic body. It is necessary to note that recent analysis by Gottlieb and Gaylord 63> has demonstrated that only the Gaylord tube model and the Flory constrained junction fluctuation model agree well with the experimental data on the uniaxial stress-strain response. On the other hand, their analysis has shown that all of the existing molecular theories cannot satisfactorily describe swelling behaviour with a physically reasonable set of parameters. The thermoelastic behaviour of the new models has not yet been analysed. 

---