Strain energy distribution

Saitta and Klein studied the mechanochemistry of isolated hydrocarbon polymer with and without a knot by quantum-chemical calculation [282, 283]. They investigated two model chains (C35H72 and C28H58) and employed a trefoil knot for each strand. Figure 42 sketches the strain energy distribution in the knotted chains when they are stretched. It is quite different compared to the corresponding distribution in unknotted chains. The bonds at the midpoint of the knot are almost luistrained... 

Fig. 42 Strain energy distribution in a knotted hydrocarbon strand with (a) AT = 35 and (b) AT =28 carbon atoms. (Adopted with permission bom Saitta et al. [282]. Copyright 1999 Nature Publishing Group)...

Stress and strain-energy distributions within diffusion-controlled insertion-electrode particles subjected to periodic potential excitations. J. Ekctrochem. Soc., 156 (11), A927-A937. 

Although in principle the substrate could also distort, the vast majority of the strain of a thin layer on a thick substrate is in the fihn. For materials with roughly equal modulus, the strain is distributed according to the ratio of the film and substrate thicknesses. Thus, a 1 micron film on a 100 micron thick substrate will have 100 times more strain in the fihn than in the substrate. Because the strain energy depends upon the strain squared, the strain energy distribution goes as the ratio of the squares of the thicknesses. Thus the vast majority of the strain energy is in the film. 

Fig. 11.40 Distribution of strain energy is two knotted polymer chains containing 35 (left) and 28 (right) carbon atoms. The strain energy is localised and most of the bonds immediately outside the entrance point to the knot. (Figure redrawn from Saitta A M, P D Sooper, E Wasserman and M L Klein 1999. Influence of a knot on the strenght of a polymer strand. Nature 399 46-48.)...

The total area under the curve A—D, shown as shaded in Figure 1, is the strain energy stored in a body. This energy is not uniformly distributed throughout the material, and it is this inequaUty that gives rise to particle failure. Stress is concentrated around the tips of existing cracks or flaws, and crack propagation is initiated therefrom (Fig. 2) (1). 

The strain-energy-release rate was expressed in terms of stresses around a crack tip by Inwin. He considered a crack under a plane stress loading of a , a symmetric stress relative to the crack, and x°° a skew-symmetric stress relative to the crack in Figure 6-12. The stresses have a superscript" because they are applied an infinite distance from the crack. The stress distribution very near the crack can be shown by use of classical elasticity theory to be, for example. 

Owing to the chain orientation distribution the fracture mechanism of oriented polymer fibres is different from that of isotropic fibres. The presence of this distribution leads to a non-uniform distribution of the strain energy between the domains. The strain energy is defined by... 

Fig. 12 The meridianal orientation function p(0) for a Gaussian distribution, the distribution function p(0)sin0,the shear strain energy function tan20sin0p(0) and the function tan20...

In conclusion, the initiation of fracture in a polymer fibre preferably occurs in the domains in the tail of the orientation distribution. The reasons are (1) in these domains the local shear stress will exceed the critical shear stress first, (2) the release of the strain energy is most effectively brought about by fracture of these domains and (3) the Griffith length in these domains adopts its lowest value. 

The strain energy of the molecule, estimated by Boyd 7 9> to be 31 kcal/mol, is obviously not confined to part of the molecule but distributed over the whole molecular skeleton 10>. Attribution of this... 

An important, implicit assumption is that the potential energy function holds only if the nuclear displacement from a position of equilibrium is not too large. Hence, highly strained molecules are not the best targets. Whereas the MM calculations give the steric energy distribution within a molecule, which no other computational method can provide, one must be aware that this distribution is correct for the model of the molecule, not necessarily for the molecule itself (110-... 

All these methods have found applications in theoretical considerations of numerous problems more or less directly related to solvent extraction. The MM calculated structures and strain energies of cobalt(III) amino acid complexes have been related to the experimental distribution of isomers, their thermodynamic stability, and some kinetic data connected with transition state energies [15]. The influence of steric strain upon chelate stability, the preference of metal ions for ligands forming five- and six-membered chelate rings, the conformational isomerism of macrocyclic ligands, and the size-match selectivity were analyzed [16] as well as the relation between ligand structures, coordination stereochemistry, and the thermodynamic properties of TM complexes [17]. 

He and Hutchinson (1989) considered a crack approaching an interface as a continuous distribution of dislocations along a semi-infinite half space. The effect of mismatch in elastic properties on the ratio of the strain energy release rates, Gi/Gj, is related to two non-dimensional parameters, the elastic parameters of Dundurs, a and /f (Dundurs, 1968) ... 

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