Development of the local strain function and the slip function in statistical theories of plasticity

The development of the concept of a local strain function is examined. The principal forms and methods of approximation of the local strain function are described and compared. A generalization of the local strains theory in differential form is proposed. Yosimura's criticism of the slip theory is shown to be invalid. Certain aspects of the local strains and slip theories are compared.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 422–430, May–June, 1969.     

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.     

Effect of a local load on an orthotropic glass-reinforced plastic shell

The deformation of a layered orthotropic cylindrical shell under a local normal load is investigated on the basis of equations that do not depend on the hypothesis of straight normals. The solutions of the analogous classical problems were analyzed in [3]; a solution based on equations that take transverse shear strains approximately into account was proposed in [4]. The high degree of variability of the state of stress created by local loads indicates that it is quite important to take transverse shear strains rigorously into account in problems of this class. An attempt is made to estimate the error introduced by the hypothesis of straight normals and to calculate the load leading to debonding of the shell.S. Ordzhonikidze Moscow Aviation Institute. Translated from Mekhanika Polimerov, No. 1, pp. 95–101, January–February, 1970.     

Modeling of transverse shears of piecewise homogeneous composite bars using an iterative process with account of tangential loads

A nonclassical model for the stress-strain state of a piecewise homogeneous composite bar is proposed. The model is based on an iterative process and takes into account the deplanation of cross sections of the bar caused by transverse shears. Based on the shear strains of some particular approximation, higher approximation models are constructed. The model accounts for both the normal and tangential loads.Presented at the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000).Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 487–500, July–August, 2000.     

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     

Stress relaxation in glassy polymers

Stress relaxation in amorphous linear polymers has been investigated below the glass transition point at various strain rates, temperatures, and strains. A model of a nonlinear viscoelastic body is proposed as a means of describing the effect. It is shown that stress relaxation is well described by the model considered and is correlated with the deformation characteristics of the polymers.A. F. Ioffe Physico-Technical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 6, pp. 987–994, November–December, 1971.     

Yield surfaces of composite media

Yield surfaces with different probabilities of the appearance of plastic strains are obtained for a composite material reinforced in one and two directions. These surfaces (local yield surface density) are obtained using the local strains theory and on the assumption of the inextensibility of the composition material in the directions of reinforcement. The theoretically obtained limit surfaces are shown to be in satisfactory agreement with the experimental data.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 35–40, January–February, 1976.     

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.     

Reasons for the lowering of the lifetime of polymers under cyclic loading

A model is proposed which explains the lowering of the lifetime of polymers on transition from static to cyclic loading by the interaction of local stresses from an external load with residual stresses which arose after preceding loading cycles. The results predicted on the basis of the model have been checked experimentally on polymethyl methacrylate, and a good agreement has been shown.M. I. Kalinin Leningrad Polytechnic Institute. Translated from Mekhanika Polimerov, No. 2, pp. 279–283, March–April, 1976.     

Variant of the variable-modulus theory of elasticity

A variant of the variable-modulus theory — a generalization of the ideas of the classical theory of elasticity in which the observed difference in the moduli of elasticity in uniaxial tension and compression and homogeneous shear is taken into account — is considered. Quasilinear expressions are proposed for the stresses in terms of the strains and the strains in terms of the stresses.Tula Polytechnic Institute. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 363–365, March–April, 1969.     

Stress potential in the nonlinear theory of viscoelasticity

By introducing a functional derivative and the line integral of a functional the author supplies an energy proof of the nonlinear theory of viscoelasticity. The stress potential for small strains, analogous to the potential proposed by Coleman for large strains [7], is considered for comparison.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 4, pp. 633–642, July–August, 1970.     

Problems of the theory of deformation of polymeric materials under complex loading

Research on problems of the theory of deformation of polymeric materials under complex loading based on a synthesis of creep theories and the local strains theory is reviewed.Presented at the 2nd All-Union Conference on Polymer Mechanics, Riga, November 10–12, 1971.M. V. Lomonosov Moscow State University. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 434–443, May–June, 1972.     

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.     

Elastic constants of monotropic plastic foams. 1. Deformation parallel to foam rise direction. A mathematical model

A mathematical model of the deformative properties and structure of lightweight, monotropic (or isotropic in the limiting case) plastic foams with a pronounced strut-like structure has been elaborated in the linear theory of deformation. A selection of five independent elastic constants is described. For the integral characterization of the deformative properties of plastic foams as micrononhomogeneous composite materials, the elastic constants are introduced as the effective constants. In order to describe the plastic foam structure, a local model consisting of two parts is proposed, i.e., a model of a continuous medium for the calculation of stresses and a local structure model. Considering deformation parallel to the foam rise direction when the semiaxes hypothesis is assumed, the Young modulus and Poisson's ratio are determined.Institute of Polymer Mechanics. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 6, pp. 719–733, November–December, 1997.     

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     

System of equations describing the flow of viscous liquid and gas in a wide range of reynolds numbers

A mathematical model is proposed for describing stationary and various self-exciting nonstationary (including turbulent) flows of viscous liquid and gas. The model does not assume that the stressed state at the points of the medium is described by a symmetric stress tensor. The model thus allows for internal moments of elementary viscous liquid and gas volumes relative to their centers of mass and for unsymmetric strains.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 17–26.     

Statistical derivation of the equations of viscoelasticity of rubbery polymers for homogeneous finite deformations

The statistical derivation of the equations of linear viscoelasticity [1, 2] is extended to the slow finite homogeneous deformations of rubbery polymers. It is shown with reference to a Gaussian subchain network model that the time correlation functions of the momentum fluxes (relaxation moduli) in the deformed and undeformed states are the same. The relations previously proposed on the basis of purely phenomenological considerations [15] are obtained for the viscous stresses in uniaxial tension. Comparison with experiment has shown that the proposed relations approximately describe the existing data on the stretching of elastomers in a certain region of finite strains.Institute of Chemical Physics, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 980–986, November–December, 1969.     

Relation between the vectors E and S for a plastic material in succesive biaxial tension

The local strains theory is used to determine the values of the components of the vector E{E₁, E₂} in five-dimensional space for a material subjected to successive biaxial tension at degrees of nonlinearity n=3 and n=5. The relation between the vectors E and S and the tangents to the loading and strain trajectories is considered together with the relation between the moduli E of the strains for simple and complex loading.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, pp. 975–979, November–December, 1969.     

Torsional mode of buckling in nonlinear creep with allowance for the difference in loading and unloading laws

The torsional mode of creep buckling of a thin-walled element of material with nonlinear creep is investigated. The local strains theory is used for the physical relations.Mekhanika Polimerov, Vol. 3, No. 1, pp. 142–145, 1967     

Relations between the strain and stress tensors in successive biaxial tension

The manner of applying the load is one of the main factors governing the formation of the principal strains and their limiting values. Starting from the theory of local strains, relations between the strain and stress tensors are formulated for both simple and complex loading. It is shown that the principal strains associated with a given stress state vary depending upon the loading path.Mekhanika Polimerov, Vol. 1, No. 3, pp. 43–51, 1965