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     

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.     

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     

A possible approach to the construction of a relation between the stress and strain tensors and temperature in rheonomic media

So-called averaging functionals have been determined and used to extend and generalize the local strains theory in relation to the solution of a problem of the mechanics of rheonomic media-the construction of the strain tensor as a function of the stress tensor and temperature in time.Mekhanika Polimerov, Vol. 3, No. 1, pp. 104–110, 1967     

Stress-strain relationships with nonlinear deformation of orthotropic glass-reinforced plastics

A theory of nonlinear elasticity of anisotropic bodies of the flow theory type is used to investigate nonlinear strains in orthotropic glass-reinforced plastics. Its applicability is experimentally confirmed (for two types of GRP) and the strain-hardening function employed in the theory is plotted. Simplified relationships are derived for direct loading and the relations obtained are compared with the strain theory of nonlinearelastic anisotropic bodies.Mekhanika Polimerov, Vol. 1, No. 4, pp. 28–34, 1965     

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     

Analytic investigation of the stress-strain relation for an arbitrary loading trajectory

The determination of the strain trajectory from a known arbitrary loading trajectory is described within the framework of the theory of slip. Specific calculations have been made on an electronic computer.Institute of Physics and Mathematics, Academy of Sciences of the Kirgiz SSR, Frunze. Translated from Mekhanika Polimerov, No. 6, pp. 981–986, November–December, 1971.     

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.     

Effect of initial imperfections on the equilibrium states of elastoplastic sheets

The local strains theory provides the basis for an analysis of the equilibrium states of initially imperfect compressed sheets of elastoplastic strain-hardening material. In the presence of initial imperfections of the sheet geometry there is no clearly expressed limit at which the deflection increases sharply. Such a limit, corresponding to the critical stress in the linear theory of plates and shells, emerges as the initial imperfection parameter tends to zero. The difference between the total transverse deflections of the initially imperfect and ideal sheets reaches a maximum at the stress level corresponding to this limit. Beyond this limit the initial imperfection effect is important at leastic and weakly developed plastic strains but diminishes considerably as the plastic strain level develops.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 897–905, September–October, 1968.     

Effect of combined loading on the components of the compliance tensor in shell stability problems

The local strains theory is used to establish relations for determining the components of the compliance tensor in shell buckling with allowance for combined loading of the material at the buckling point and as a function of the precritical loading path in stress space.Mekhanika Polimerov, Vol. 2, No. 3, pp. 387–391, 1966     

Complex loading of a polymer material with nonlinear creep

The nonlinear tensor stress, strain, and time relations for a memory-type medium under complex loading are examined using degenerate kernels. The basic expressions for simple loading and the material parameters were determined in [5]. The local strains theory is used to find expressions for the strain components in the presence of stepwise complex variation of the stress components, and these expressions are shown to be in satisfactory agreement with the experimental data for high-density polyethylene.Mekhanika Polimerov, Vol. 3, No. 3, pp. 421–426, 1967     

Simple loading of a polymer material with nonlinear creep

Nonlinear tensor relations between strain, stress, and time are examined for a memory-type medium using degenerate kernels. The material parameters are determined from creep tests in a simple state of stress. Expressions for the strain associated with a complex state of stress and simple loading, found on the basis of the local strains theory, are in satisfactory agreement with the experimental data obtained for specimens of high-density polyethylene.Mekhanika Polimerov, Vol. 3, No. 2, pp. 236–242, 1967     

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.     

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     

Deformation properties of polymers

The basic relations of the local strains theory are used to construct expressions for the deformation properties of polymers. The local strains function is assumed to be piecewise-linear, and nonlinear relations ₁₁(e₁₁) qualitatively describing the deformation of polymers are obtained.Mekhanika Polimerov, Vol. 1, No. 6, pp. 49–53, 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     

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.     

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.     

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.     

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.