Deformational covariance of graphs

Graphs on n vertices are classified into equivalence classes under the linear group L(n). All graphs representing the same operator on a vector space V _n belong to the same class. Graphs of different operators may or may not belong to the same class as ascertained by rules that were given by the author. Graphs in the same class are structurally covariant. If, in addition, two graphs can be continuously deformed into each other in the sense of varying line strengths while remaining structurally covariant throughout, then the two graphs are termed deformationally covariant along such paths. Applications in the quantum theory of chemistry and to the dynamic stability theory of coupled reaction systems which occur in various fields are indicated.     

Generalized Deformations and Hochschild Cohomology

We present a generalization of Gerstenhaber's theory of deformations. We no longer assume that the deformation parameter t acts in its usual free and symmetric way on the elements of the original algebra A, but in the following manner: t · a = (a)t and a · t = (a)t, where and are endomorphisms of A. We develop the cohomological framework adapted to these deformations.     

The Elastic Interpretation of Electrodynamics

In his famous treatise, Maxwell presumed explicitly, without developing it, an elastic interpretation of his electrodynamics; the relevant quotation is given in the introduction. It is in that spirit that we have developed the theory of the elastic interpretation here presented. We show that the Maxwell equations and the electrodynamic forces can formally be obtained from the theory of elasticity. An electromagnetic field can be considered as a deformation of a certain elastic medium . This deformation is created by volumetric densities of deforming couples and of irrotational deforming forces that act on . A distribution of electric charges is proportional to the divergence of the deforming couples. The electrodynamic forces are due to the interactions between such deformations. Even though the consistency of the elastic interpretation of electrodynamics is demonstrated by the fact that it gives back Maxwell's equations, and the electrodynamic forces, we show that the elastic interpretation is compatible with Einstein's special relativity. Then we demonstrate that an adequate density of irrotational deforming forces can produce the same effects on light propagation as those of a Schwarzschild metric background. This fact suggests a possible extension to take into account many of the effects attributed to the geometrical structure of the general relativity. Thus, we have given an elastic interpretation to the action at distance, and contributed to eliminate the inconsistency mentioned by Einstein, implying in his view, an incompleteness of the Maxwell theory. In future works, we hope to give an elastic interpretation for additional aspects of the particle waves and of general relativity and for the electroweak and strong fields.     

Induced formal deformations and the Cohen-Macaulay property

The main result states: if is a module finite extension of excellent local normal domains which is unramified in codimension two and if represents a deformation of the completion of , then there is a corresponding -algebra deformation such that the ring homomorphism represents a deformation of . The main application is to the ascent of the arithmetic Cohen-Macaulay property for an étale map of smooth projective varieties over an algebraically closed field.     

Mechanics of polymeric membranes subjected to chemical exposure

The mechanics of polymeric hyperelastic membranes that are subjected to uniform transverse pressure loading are discussed. The paper also focuses on the membrane behaviour when there is loss of hyperelasticity resulting from the removal of plasticizer from the polymeric material as a result of chemical exposure. Constitutive models presented describe the influence of both hyperelasticity and rate-sensitivity on the mechanical behaviour of the polymeric membrane in its natural and chemically exposed states. The constitutive models developed through experimental investigations are implemented in computational techniques to develop solutions to the membrane deformation problems.     

用于三维裂纹扩展分析的有限变形不可逆内聚元

为了能准确和高效的跟踪动态裂纹扩展,我们发展了三维有限变形的内聚元和一系列不可逆内聚力关系.该内聚元通过采用不可逆内聚力关系来控制裂纹两侧物质的逐渐分离和形成自由表面,这一点可类比于传统有限元对块体材料的离散化.为了展示该方法的预测能力及便于灵活使用的特点,我们模拟了Zehnder和Rosakis$^{[1]}$所做的重物落下动态断裂实验,值得注意的是该方法可以近似模拟出裂尖的轨迹.      

On Compressible Materials Capable of Sustaining Axisymmetric Shear Deformations. Part 4: Helical Shear of Anisotropic Hyperelastic Materials

Conditions on the form of the strain energy function in order that homogeneous, compressible and isotropic hyperelastic materials may sustain controllable static, axisymmetric anti-plane shear, azimuthal shear, and helical shear deformations of a hollow, circular cylinder have been explored in several recent papers. Here we study conditions on the strain energy function for homogeneous and compressible, anisotropic hyperelastic materials necessary and sufficient to sustain controllable, axisymmetric helical shear deformations of the tube. Similar results for separate axisymmetric anti-plane shear deformations and rotational shear deformations are then obtained from the principal theorem for helical shear deformations. The three theorems are illustrated for general compressible transversely isotropic materials for which the isotropy axis coincides with the cylinder axis. Previously known necessary and sufficient conditions on the strain energy for compressible and isotropic hyperelastic materials in order that the three classes of axisymmetric shear deformations may be possible follow by specialization of the anisotropic case. It is shown that the required monotonicity condition for the isotropic case is much simpler and less restrictive. Restrictions necessary and sufficient for anti-plane and rotational shear deformations to be possible in compressible hyperelastic materials having a helical axis of transverse isotropy that winds at a constant angle around the tube axis are derived. Results for the previous case and for a circular axis of transverse isotropy are included as degenerate helices. All of the conditions derived here have essentially algebraic structure and are easy to apply. The general rules are applied in several examples for specific strain energy functions of compressible and homogeneous transversely isotropic materials having straight, circular, and helical axes of material symmetry.     

A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part I: Formulation

In this Part I, of a two-part paper, we present a detailed continuum-mechanical development of a thermo-mechanically coupled elasto-viscoplasticity theory to model the strain rate and temperature dependent large-deformation response of amorphous polymeric materials. Such a theory, when further specialized (Part II) should be useful for modeling and simulation of the thermo-mechanical response of components and structures made from such materials, as well as for modeling a variety of polymer processing operations.     

Two-dimensional sediment transport models for sand-bed rivers

The problem of determining the specific mass flow of sediment entrained by a liquid flow passing above the sand bottom is studied. The boundary-value problem for a two-phase mixture of the liquid and solid particles in the active bottom layer is solved, and a general formula for determining the specific mass flow of sediment is derived. Constraints imposed on the rheological model of a moving mixture, which allow the phenomenological parameter (concentration of particles in the active layer of the mixture) to be eliminated from the model, are found. Within the framework of the proposed rheological model, the equation of riverbed deformations in the case of a sand bottom is obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 131–139, May–June, 2009.