Near field stress analysis of a spot weld between elastic plates

The mechanics of a circular, spot weld between two identical elastic plates is studied. A method of solution similar to that used by the authors in a paper concerned with the influence of an elastic layer on the tangential compliance of bodies in contact is used. In the present analysis the relation between tangential load and deflection is lost due to the nature of the problem; hence, physical quantities are determined in terms of the total tangential load applied in the weld region.     

N-Functions and integration by substitution

By use of a new result concerning the validity of the chain rule for the derivative of a composite function, the author succeeds in extending the classical change of variable formula for Lebesgue and Denjoy/Perron integrals to the case in which the change is effected by continuousN-functions of unbounded variation. A novelty of the resulting formula is that the integration is only carried out over the set where theN-function is differentiable, and that set may fail to have full measure, as an example shows.     

Copolyesteramides—IV. Anionic copolymers of ε-caprolactam with ε-caprolactone. Molecular and chain structure

The chain structures of copolyesteramides formed by anionic ring-opening copolymerisation of ε-caprolactam with ε-caprolactone have been investigated. The copolymers are shown to possess random multiblock structures with chains composed of O(CH₂)₅CO[NH(CH₂)₅CO]_n groups (having relatively small values of n) intercalated with sequences of O(CH₂)₅CO groups. The distributions of structural units in the chains are significantly affected by reaction time and by the method of introducing lactone into the copolymerising system. A mechanism is proposed to explain the formation of the multiblock structures. The nature of an additional absorbance in the NH-stretching region of the i.r. spectra of the copolyesteramides has been investigated with the aid of model compounds and attributed to hydrogen-bonding between amide and ester groups in the amorphous regions of the copolymers. This bonding may be the cause of the anomalous mechanical properties found at high ester contents.     

A theorem of ordered duality

We associate, to any ordered configuration of n points or ordered arrangement of n lines in the plane, a periodic sequence of permutations of [1, n] in a way which reflects the order and convexity properties of the configuration or arrangement, and prove that a sequence of permutations of [1, n] is associated to some configuration of points if and only if it is associated to some arrangement of lines. We show that this theorem generalizes the standard duality principle for the projective plane, and we use it to derive duals of several well-known theorems about arrangements, including a version of Helly's theorem for convex polygons.     

Lie algebras of unbounded derivations

In this paper some new results on analytic domination of operators and on integrability of Lie algebras of operators are proved and then our methods are applied to the study of Lie algebras of unbounded derivations in $\text{C}{}^1$ algebras.