Variants of the local strains theory

By means of the proposed new variants of the local strains theory it is possible to solve a broad class of complex loading problems for initially isotropic materials. The basic equations for a series of variants of incompressible and compressible materials are derived. The author investigates the possibility of using the relations obtained to solve complex loading problems for materials with irreversible plastic strains or with different physical reversible or irreversible creep relations depending on the loading path.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 233–240, March–April, 1970.     

Loading conditions in the local strains theory

Use of the local strains theory [7, 8] makes it possible to extend the loading criterion to complex loading paths if the hardening conditions are related to the local strains on individual small areas. Then the components of the total plastic strain increment are determined by integration over only those local areas on which the given loading condition is satisfied [9]. It is shown that the components of the strain tensor depend significantly on the loading history and the loading condition employed. The changes in the loading zones on the unit sphere are investigated for particular cases of complex loading as a function of the various loading conditions and the loading parameter.Mekhanika Polimerov, Vol. 3, no. 1, pp. 29–34, 1967     

Investigation of loading surfaces in the local strains theory

Variants of the application of isotropic, kinematic, and combined strain hardening in the local strains theory are analyzed. The subsequent yield surfaces are determined for biaxial tension and combined torsion and tension and the results are compared with experiment. The theory predicts convex yield surfaces and the dependence of the angle (corner point) of the yield surface at the loading point on the loading path. The expediency of employing the concept of a plastic strain probability factor is demonstrated.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 796–800, September–October, 1971.     

A variant of the theory of local strains

A variant of the theory of local strains giving the stress tensor as a function of a given strain tensor is formulated. The stresses in an orthogonal coordinate system are established by means of a functional that averages the local stresses expressed by the local stress function. This function is determined by the given strain program. It is shown that in certain practical problems these relations are more convenient than those previously proposed.Mekhanika Polimerov, Vol. 3, No. 5, pp. 800–802, 1967     

Basis of the local strains theory (review 2)

The evolution of the local strains theory since 1965 is reviewed.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 5, No. 1, pp. 22–29, January–February, 1969.     

Physical relations for an orthotropic material in the local strains theory

Compressible and incompressible orthotropic materials are investigated. A method of constructing variants of the local strains theory describing the deformation properties of an orthotropic material is developed on the basis of the concept of an elastic potential.Institute of Polymer Mechanics, AS LatSSR, Riga. Translated from Mekhanika Polimerov, Vol. 4. No. 6, pp. 970–975, November–December, 1968.     

Relation between local strains theory and deformation theory in the problem of stability of inelastic shells

In accordance with the theory of local plastic strains and the deformation theory of plasticity the coefficients of the compliance matrices are determined for the problem of stability of inelastic shells in the uniaxial precritical stress state. A comparison of the compliance matrices shows that for integration over a complete sphere the results obtained using the local strains theory and deformation theory are very close.Mekhanika Polimerov, Vol. 1, No. 6, pp. 54–60, 1965     

Basis of the local strains theory (review. 1)

A study is made of the relationship between the stress and strain tensors, temperature, time and the material characteristics in the most general case for a complex loading path. The failure and loss of stability of actual structures corresponds to a complex loading path, when the change in the stress tensor components is not proportional to a single parameter. The same applies to the microphenomena of failure, when the destruction of the material is preceded by a complex redistribution of internal stresses and strains. The phenomenological basis of the theory is developed. This makes it possible to solve a number of practical problems more correctly than has so far proved possible. These problems include problems of the buckling of beam sections and shells. The local strains theory is a variant of the theory of plasticity that allows the solution of the complex loading problem and makes it possible to unify the theory of plasticity and the theory of deformation of theonomic materials. The review is based on more than twenty publications dealing with questions relating to the local strains theory.Mekhanika Polimerov, Vol. 1, No. 4, pp. 12–27, 1965     

Elastic potential in local strains theories

Expressions for the elastic potential V in local strains theories are examined. The relation between the elastic potentials in the moving and fixed coordinate system is established. The expressions for the function V of an incompressible material at high degrees of nonlinearity are investigated.Mekhanika Polimerov, Vol. 4, No. 3, pp. 406–412, 1968     

Analysis of possible combinations of local stresses in local strain theory

In determining the strain tensor e_ij instead of all the components of the local stress deviator, it is possible to use only the shear stress _xz acting on a small local area. This fact makes it easy to establish the incremental loading conditions in solving complex loading problems for a plastic material. It is shown that in the local strain theory, distinct from the deformation theory, at degrees of nonlinearity n>3 the effect of the third, as well as the second, invariant of the stress deviator is taken into account.Mekhanika Polimerov, Vol. 3, No. 4, pp. 636–644, 1967     

Yield surfaces in local strain theory

The initial yield surfaces for biaxial tension and combined tension and torsion are determined on the basis of the local strain theory. The limit surfaces of the resultant stress on a local plane in tension (torsion) are obtained. A plastic strain probability factor is introduced and its values are calculated for various loading paths.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 594–598, July–August, 1971.     

Finite axiomatizability of local set theory

The need for modifying axiomatic set theories was caused, in particular, by the development of category theory. The ZF and NBG axiomatic theories turned out to be unsuitable for defining the notion of a model of category theory. The point is that there are constructions such as the category of categories in naïve category theory, while constructions like the set of sets are strongly restricted in the ZF and NBG axiomatic theories. Thus, it was required, on the one hand, to restrict constructions similar to the category of categories and, on the other hand, adapt axiomatic set theory in order to give a definition of a category which survives restricted construction similar to the category of categories. This task was accomplished by promptly inventing the axiom of universality (AU) asserting that each set is an element of a universal set closed under all NBG constructions. Unfortunately, in the theories ZF + AU and NBG + AU, there are toomany universal sets (as many as the number of all ordinals), whereas to solve the problem stated above, a countable collection of universal sets would suffice. For this reason, in 2005, the first-named author introduced local-minimal set theory, which preserves the axiom AU of universality and has an at most countable collection of universal sets. This was achieved at the expense of rejecting the global replacement axiom and using the local replacement axiom for each universal class instead. Local-minimal set theory has 14 axioms and one axiom scheme (of comprehension). It is shown that this axiom scheme can be replaced by finitely many axioms that are special cases of the comprehension scheme. The proof follows Bernays’ scheme with significant modifications required by the presence of the restricted predicativity condition on the formula in the comprehension axiom scheme.     

Model theory via set theory

We develop a technique for applying models of set theory to obtain results in the model theory of infinitary languages. Some results on Δ-logics are also discussed.     

Some results of the local theory of smooth functions

We present results of the investigation of the local behavior of smooth functions in neighborhoods of their regular and critical points and prove theorems on the mean values of the functions considered similar to the Lagrange finite-increments theorem. We also study the symmetry of the derivative of an analytic function in the neighborhood of its multiple zero, prove new statements of the Weierstrass preparation theorem related to the critical point of a smooth function with finite smoothness, determine a nongradient vector field of a function in the neighborhood of its critical point, and consider one critical case of stability of an equilibrium position of a nonlinear system. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 2, pp. 231–267, February, 2007.