Strain energy Subject

When two material layers adhere to one another and one layer differentially expands or contracts relative to the other, the layers bend in order to minimize the strain energy. Subject to various constraints, such as locally uniform layer thicknesses, and that the stiffest layer be linear elastic, the local bending can be described by the arc of a circle of radius, R. If the constraints are valid across an entire sample, the entire sample will indeed bow in the form of an arc of a circle. This is the operating principle of the bending beam apparatus and is illustrated in Figure 1. 

As mentioned in the Introduction, trimethyltriphosphirane 3 and tetramethyl-tetraphosphetane 4 have almost the same strain energies. We subjected the parent molecules, triphosphirane 14 and tetraphosphetane 15, to the bond model analysis... 

It has been asserted that the choice of thermochemical reference states, including those for strain energy considerations, is inherently subjective and so this question cannot be objectively answered see, J. F. Liebman and D. Van Vechten, in Molecular Structure and Energetics Physical Measurements (Eds. J. F. Liebman and A. Greenberg), VCH, New York, 1987. 

In our experience, the introduction of "extra potentials" is a particularly useful technique when molecular conformations other than the minimum energy one must be explored. In this method, potentials are added which make it prohibitively expensive (in energy terms) for the molecule not to assume the desired structural feature. The total energy—strain energy plus "extra potential" energy—is minimized, giving the minimum energy conformation of the molecule subject to the constraint imposed by the "extra potentials". 

Cyclodecene is the smallest cycloalkene, which can accommodate a trans double bond without significant deformation of bond angles and/or dihedral angles. The strain energies and structures of smaller fraws-cycloalkenes have been the subjects of considerable research over the years. 

Any molecular parameter which, in a trial structure, has a value at variance with the characteristic electronic standard, adds to the strain energy. It is considered a steric effect and subject to optimization. At convergence the actual molecular structure is recovered, providing that all empirical constants had been specified correctly. 

In the previous lesson we have shown how to generate the structures of the four conformational isomers of [Co(en)3]3+. We will now subject each of these structures to energy minimization using MOMEC and use the strain energies we obtain to calculate the isomer distribution. 

Evaluation of the strain energy in SPC was a subject of numerous calculations, which provided substantially different results. Unfortunately, very limited experimental data on the heats of formation of these compounds preclude the evaluation of the validity of the calculations. The results of nonempirical (4-31G//STO-3G) and molecular mechanics (MM2) calculations were in reasonable agreement with experimental data, while the semiempiri-cal methods resulted in substantial deviations. Even better results were obtained by Aped and Allinger using the MM3 method Available experimental data on the heats of formation of SPC and matching calculated values are collected in Table 1. 

The idea behind bond localization in annelated benzenes is exemplified by compound 66, tricyclopropabenzene. Of the two standard Kekule structures of 66, one places the double bonds within the three-member ring (endo) and the other places the double bonds outside (exo) of the small ring. Since cyclopropene is much more strained than cyclopropane, the avoidance of strain energy suggests that the double bonds might be localized into the exo positions. Compounds 66 and 67 are the prime test subjects for this idea of strain-induced bond localization (SIBL). ... 

A perfectly elastic solid subjected to a non-destructive shear force will deform almost instantaneously an amount proportional to its shear modulus and then deform no further, strain energy being stored in the bonds of the material. A fluid, on the other hand, continues to deform under the action of a shear stress, the energy imparted to the system being dissipated as flow. 

The unstable fracture of epoxies has been shown by Mai and Atkins to be accompanied by a negative change of the strain energy release rate, G, tvith crack velocity, a. This is in contrast to the positive dG/da which they find characterizes stable fracture. Whether a negative dG/da is the cause or the consequence of unstable fracture is, however, subject to debate... 

Equation 3 can be solved for minimum ring strain energy (fiRSmin) subject to the geometrical constraint associated with Eq. 4. The flexibility (i.e. ease of deformation) of a ring can thus be qualitatively assessed by examining the relationship between Rsmin and a. 

Another interaction is responsible for the recovery of the material after it is subjected to stress. " Rubber bridging the neighboring particles of filler is an example. Some particles are connected through several rubber chains which makes their association more permanent and assures filler-filler contact. These filler-filler contacts are responsible for the recovery since, unlike chain-filler contacts, they store the strain energy which is then used in the recovery process. Chain-filler contacts can easily debond or rearrange in different location and this process does not result in recovery of the initial shape. 

If a crystal is subjected to small strain elastic deformation it is convenient to imagine the energetics of the strained solid in terms of the linear theory of elasticity. As we noted in chap. 2, the stored strain energy may be captured via the elastic strain energy density which in this context is a strictly local quantity of the form... 

The other extreme of behavior involves the "phantom chain" approximation. Here, it is assumed that the individual chains and crosslink points may pass through one another as if they had no material existence that is, they may act like phantom chains. In this approximation, the mean position of crosslink points in the deformed network is consistent with the affine transformation, but fluctuations of the crosslink points are allowed about their mean positions and these fluctuations are not affected by the state of strain in the network. Under these conditions, the distribution function characterizing the position of crosslink points in the deformed network cannot be simply related to the corresponding distribution function in the undeformed network via an affine transformation. In this approximation, the crosslink points are able to readjust, moving through one another, to attain the state of lowest free energy subject to the deformed dimensions of the network. 

The most commonly used direct method for determination of surface stress is the bending beam method in which a lever bends when subjected to a change in surface stress in one of its faces in order to minimize its stored strain energy. The relationship between the deflection of a cantilever and the different stresses in its smfaces was first determined by Stoney [25] and in surface science it has been used to measure the stress changes associated with the reconstructions of semiconductor surfaces [26 - 27]. 

---