Modified Hooke s law

The force field approach assumes that it is possible to describe the geometry of a molecule by a set of mechanical equivalents. Changes in bond length are, for example, represented by- a modified Hook s law V(r) = 0.5 x 1 x (r — r0)2 for a spring or bending of a bond angle is modelled by a bending potential VB = kB x (0 — 0O)2, where kB, ks, r0, and 0O are parameters that are dependent on the incorporated atom... 

The first four terms of the function are commonly found in molecular mechanics strain energy functions, and they are modified Hooke s law functions. The last term has been added to insure the proper stereochemistry about asymmetric atoms. A model is refined by minimizing the highly nonlinear strain energy function with respect to the atomic coordinates. An adaptive pattern search routine is used for the strain energy minimization because it does not require analytical derivatives. The time necessary to obtain good molecular models depends on the number of atoms in the molecule, the flexibility of the structure, and the quality of the starting model. 

Modified Hooke s law corrected with cubic (as in the MM2-based force fields [8]) and further extensions to quartic terms (as in MM3 [9], CFF [10], and MMFF [11] force fields see Eq. (3) [9]) or other expansions [12] have been developed to mimic the Morse potential and are used to speed up convergence in very distorted starting geometries, while keeping a proper description of the potential energy. 

It is knoi n that if an electrically conducting magnetoelastic solid is subjected to electromagnetic forces while immersed in a magnetic field, the electromagnetic field is still governed by Maxwell s equations with a modified Ohm s law, while the elastic deformation field is determined by a modified Hooke s law, see e.g. [13]. 

In brief, the coupling between the elastic strain and hthium concentration, Cu, in the particle, the local stress, Oy, can be elucidated in terms of the modified Hooke s law for chemically active solid based on the formulation by Garcia et al. [41] ... 

Chapter 4 outlines operations of symmetry on ideal solids that show how the number of independent components of the modulus tensor diminishes as the number of symmetry elements in the solid increases. This analysis leads to the formulation of the generalized Hooke s law utilizing both elastic modulus and elastic compliances for amorphous solid materials. These relationships, conveniently modified, are further used in viscoelasticity. In this chapter the generalized law of Newton for ideal liquids is also stated. 

Figure 2.9 shows an overlay of the stress-strain curve of natural rubber and an impact-modified nylon 6,6. At the top of the nylon curve is the yield point or elastic limit of the nylon. At this point, the strain is no longer recoverable and the sample exhibits plastic deformation. Of course, below this point, the material obeys Hook s law (F = -kx), where k is the spring constant, x is the distance traveled when force is applied, and F is that applied force. 

Unfortunately, Hooke s Law does not accurately enough reflect the stress-strain behavior of plastics parts and is a poor guide to good successful design. Assuming that plastics obey Hookean based deformation relationships is a practical guarantee of failure of the part. What will be developed in this chapter is a similar type of basic relationship that describes the behavior of plastics when subjected to load that can be used to modify the deformation equations and predict the performance of a plastics part. UnUke the materials that have been used which exhibit essentially elastic behavior, plastics require that even the simplest analysis take into account the effects of... 

Both Eqs. (11.1) and (11.2) account for the effect of transverse strain on plastic strain intensity factor characterized by the modified Poisson s ratio, V. In Eq. (11.1), this is accounted for by the ratio Sy/Sa, whereas in Eq. (11.2) the ratio Eg/E serves the same purpose as will be shown later. The modified Poisson s ratio in each case is intended to account for the different transverse contraction in the elastic-plastic condition as compared to the assumed elastic condition. Therefore this effect is primarily associated with the differences in variation in volume without any consideration given to the nonlinear stress-strain relationship in plasticity. Instead the approaches are based on an equation analogous to Hooke s law as obtained by Nadai. Gonyea uses expression (rule) due to Neuber to estimate the strain concentration effects through a correction factor, K, for various notches (characterized by the elastic stress concentration factor, Kj). Moulin and Roche obtain the same factor for a biaxial situation involving thermal shock problem and present a design curve for K, for alloy steels as a function of equivalent strain range. Similar results were obtained by Houtman for thermal shock in plates and cylinders and for cylinders fixed to a wall, which were discussed by Nickell. The problem of Poisson s effect in plasticity has been discussed in detail by Severud. Hubei... 

The first bracketed term is merely Hooke s law. Any modification of ( 8) must be so chosen that the Hookean form is maintained in the limit of vanishing strain. Apart from this restriction however there are an infinity of choices that can be made to modify ( 8) to account for compressibility. From the second bracketed teimi we see however that terms which arise from this modification... 

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