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.     

Model of viscoplasticity in the local strains theory

A model of viscoplasticity is proposed within the framework of the local strains theory. The proposed model, which applies to a small local area, takes into account the interaction of rheonomic and plastic strains, the Bauschinger effect, and recovery of the material after unloading.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, pp. 1126–1128, November–December, 1969.     

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     

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.     

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.     

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     

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     

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     

On integral conditions in the mapping theory

Interconnections between various integral conditions that play an important role in the theory of space mappings and in the theory of degenerate Beltrami equations in the plane are established.     

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.     

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     

Embedding problems with local conditions

LetY→X be a connectedG-Galois cover of affine varieties in characteristicp, and supposeG=Γ/P for somep-groupP. We show that there is a connected Γ-Galois coverZ→X dominatingY→X, and thatZ→X can be chosen to have prescribed behavior over a given closed subset ofX. There are several versions of this result, depending on whether ramification is permitted, and whether adelic behavior is prescribed. The results are deduced from a general assertion about embedding problems, which is proven for profinite groups. Supported in part by NSF Grant DMS94-00836.     

Initial conditions in quasi-classical field theory

We investigate the problem of divergences and renormalizations in the Hamiltonian formalism of quasiclassical field theory. This approach is known to involve divergences in the leading term of the expansion. Proposals have been made to eliminate the divergences by using nonequivalent representations of the canonical commutation relations at different moments of time. In this paper, we consider the Schrödinger equation with ultraviolet and infrared cutoffs. In order to remove the cutoffs, conditions are imposed on the initial state of the regularized theory in addition to the conditions imposed on the counterterms in the Hamiltonian. In the leading order of the quasi-classical expansion, we give the explicit form of these conditions, which is invariant under the evolution. This allows us to show that this approximation does not require the introduction of nonunitary evolution transformations.     

Common properties of the operator products in local spectral theory

Let X, Y be Banach spaces, A, D : X → Y and B, C : Y → X be the bounded linear operators satisfying operator equation set AC D = DBD,DBA = AC A.In this paper, we show that AC and BD share some basic operator properties such as the injectivity and the invertibility. Moreover, we show that AC and BD share many common local spectral properties including SVEP, Bishop property(β) and Dunford property(C).