Exact solution of problems of flow theory with isotropic-kinematic hardening. Part 1. Setting the loading trajectory in the space of stresses

For an arbitrary isotropic and linear kinematic hardening and loading paths given in the form of arbitrary multisection polygonal lines in the five-dimensional deviator space of stresses, we studied analytically an initially isotropic elastoplastic von Mises material and the associated flow rule. The solutions obtained are valid for arbitrary relationships governing the variation of the spherical component of the stress tensor. Explicit solutions are obtained for several important cases of material behavior. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 6, pp. 81–92, November–December, 1999.     

Asymptotic solution for mode III crack growth in J 2-elastoplasticity with mixed isotropic-kinematic strain hardening

Mode III fracture propagation is analyzed in a J ₂-flow theory elastoplastic material characterized by a mixed isotropic/kinematic law of hardening. The asymptotic stress, back stress and velocity fields are determined under small-strain, steady-state, fracture propagation conditions. The increase in the hardening anisotropy is shown to be connected with a decrease in the thickness of the elastic sector in the crack wake and with an increase of the strength of the singularity. A second order analytical solution for the crack fields is finally proposed for the limiting case of pure kinematic hardening. It is shown that the singular terms of this solution correspond to fully plastic fields (without any elastic unloading sector), which formally are identical to the leading order terms of a crack steadily propagating in an elastic medium with shear modulus equal to the plastic tangent modulus in shear.     

Prediction of the fracture of metals in processes of cold plastic deformation. Part 2. Effect of anisotropic hardening and experimental verification of the model of plastic deformation and fracture

On the basis of the experimental investigation of the Bauschinger effect, in steels with developed prestrains, we generalize the model of plastic deformation to the case of anisotropic hardening. Within the framework of the model, we show that for high strains, the Bauschinger effect is caused by dislocations. We present the results of the experimental verification of the developed semiphenomenological model of the joint process of plastic deformation and fracture of metals under the conditions of cold deformation. It is shown that this model gives adequate predictions (in good agreement with the experimental data) of the probability of fracture of the metal caused by exhaustion of the plasticity margin in the processes of plastic deformation realized under the conditions of both simple and complex loading. Ufa State Aviation Technical University, Ufa, Russia. Translated from Problemy, Prochnosti, No. 2, pp. 74–84, March–April, 1999.     

Simulation of the fading memory of the form of trajectory in the theory of simple materials with elastoplastic behavior. Part 2. Infinitely small strains

For the processes of deformation close to proportional and slightly different from the stress- and strain-free configurations, in which plastic strains are formed immediately after the application of loading and monotonically increase in the process of deformation, we develop a mathematical theory of rigorous construction and specialization of determining relations for hardening elastoplastic materials with fading memory of the first order form of trajectory. The strains are regarded as infinitely small. The type of symmetry of the material is arbitrary. We use the determining relations of the linear theory of elastoplasticity for finite strains established earlier by the author. The condition of smallness of the measures of strains for the entire history is accepted. Special attention is given to the case of isotropic materials. The conditions of reduction of the constructed relations to one of the existing versions of the endochronic theory of plasticity are established.Translated from Problemy Prochnosti, No. 6, pp. 87–98, November–December, 2004     

On the moment theory of elasticity. Plane deformation. Part 2

For the problem of bending of a plate under the action of a concentrated force represented as pressure distributed over a small surface, we obtain a solution by methods of the moment theory of elasticity, the principal statements of which were given in Part 1. Moscow State Institute of Food Industry, Moscow, Russia. Translated from Problemy Prochnosti, No. 4, pp. 106–115, July–August, 1998.     

On the concept of relative and plastic spins and its implications to large deformation theories. Part II: Anisotropic hardening plasticity

Summary By resorting to both microscopic and macroscopic considerations, including the concept of single slip, dislocation stress, and a scale invariance argument we show that the notion and formalism of the relative spin introduced in Part I reduces to that of plastic spin previously recognized in the literature. The central feature of this reduction is the possibility of obtaining physically based constitutive equations for the plastic spin along with appropriate evolution equations for the dislocation or back stress. When these constitutive models are incorporated in the analysis of existing data on tension-torsion tests, we find satisfactory agreement between theory and experiment. In particular, a theoretical interpretation of the torsionally induced axial strain, as observed for example by Swift, Bailey et al., Hart and Chang, and others, is provided. Moreover, the recent experiments of Montheillet et al. on torsionally induced axial stresses are discussed in the light of the presently proposed models of large inelastic deformation inelasticity accounting for anisotropy and texture effects.

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With 12 Figures     

Exact solution to the problems of massive freezing in a solution

The problem of massive freezing in a hypoeutectic solution is analyzed under various boundary conditions at the free surface.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 24, No. 5, pp. 921–928, May, 1973.     

Problem of modeling hardening in the theory of flow

An analysis of relationships used to define isotropic and kinematic hardening in a theory of flow previously proposed by the author is presented. Familiar hardening laws are obtained using appropriate simplifications of the relationships examined.Institute of Strength Problems, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Problemy Prochnosti, No. 12, pp. 44–47, December, 1989.     

Prediction of the fracture of metals in the processes of cold plastic deformation. Part 1. Approximate model of plastic deformation and fracture of metals

We propose an approximate semiphenomenological model of the joint process of cold plastic deformation and fracture of metals. Within the framework of this model, for 10kp, 20kp, 20G2R, and 38KhGNM steels, we show that moving dislocations overcome barriers through a force process. The formation of nascent microcracks is also realized through a force process, i.e., local stresses in the “head” of an arrested dislocation pileup attain the levels of theoretical strength. We also suggest a general algorithm of the application of the proposed model to the prediction of fracture of metals in technological processes of plastic metal working. Ufa State Aviation Technical University, Ufa, Russia. Translated from Problemy Prochnosti, No. 1, pp. 76–85, January–February, 1999.     

On the moment theory of elasticity. Plane deformation. Part 3

By the methods of the moment theory of elasticity, we solve the problem of tension of a plate with restraint. We suggest equations determining the positions of the principal planes and the level of principal stresses and present expressions for the stress and strain tensors in the plane problem of the moment theory of elasticity. Moscow State Extra-Mural Courses of the Institute of the Food Industry, Moscow, Russia. Translated from Problemy Prochnosti, No. 5, pp. 82–88, September–October, 1998.     

On the moment theory of elasticity. Plane deformation. Part I

We propose a model of the moment theory of elasticity in which equations of the classical (momentless) theory of elasticity are supplemented with equilibrium equations (momental equations) and boundary conditions for couple stresses and shear strains are determined by tangential and couple stresses. As a result, the basic relationships between strains and displacements and, hence, the continuity equations for strains preserve the same form as in the momentless theory. The proposed moment theory of elasticity completely includes all basic postulates of the classical theory and supplements it in the case where the law of parity of tangential stresses is not valid. We obtain solutions of some problems which cannot be solved within the framework of the classical theory of elasticity. Moscow State Postal-Tuition Institute of the Food Industry, Moscow, Russia. Translated from Problemy Prochnosti, No. 3, pp. 103–113, May–June, 1998.     

Reserves of strength in plastic deformation. Report 2

The results are presented of experimental and theoretical investigation of transformation of the frontal portion of the loading surface for four series of repeated loadings after a preliminary one complex in trajectory, the first section of which was axial tension and the second orthogonal loading in the plane of main stresses. In three series, repeated radial loadings were made after three types of unloading, complete unloading along a straight line, complete unloading along the trajectory of preliminary loading, and partial unloading along the second section of this trajectory to different values of the parameter of the form of stressed state. In the fourth series, return movement along the section to the point of the break in the loading trajectory and subsequent repeated loadings along straight lines were used. It was shown that the form of the loading surface is substantially determined by the form of the stressed state at the initial moment of repeated loading, the direction of this loading, and the history of unloading. An analysis of the form of the loading surfaces constructed in different series is given and their difference is explained from viewpoints of the theory of plasticity of anisotropically hardening media based on the shear mechanism of deformation.Translated from Problemy Prochnosti, No. 11, pp. 9–13, November, 1991.     

Modelling of the plastic deformation and primary creep of metals coupled with DC in terms of the synthetic theory of irrecoverable deformation

The paper deals with modelling of the plastic and creep deformation of metals coupled with current. The passage of DC manifests itself in the increase in creep deformation and leads to primary creep time shortening. With plastic deformation, a short electric impulse results in the step-wise decrease of stress (stress-drop) on the stress–strain diagram. To catch these phenomena, we utilize the synthetic theory of recoverable deformation. The constitutive equation of this theory is supplemented by a term taking into account the intensity of DC. Further, we introduce DC intensity into the function governing transient creep. As a result, we predict the parameters of transient creep and calculate the stress-drop as a function of current intensity. The model results show good agreement with experimental data.

     

Formulation of the plastic source method for plane inelastic problems Part 1: Green's functions for plane inelastic deformation

Summary Green's function representation of the residual stress caused by any plane inelastic strain is given for the infinite region with a simple defect such as a crack or an elliptic hole. The plane inelastic strain developed within an infinitesimal region is represented by a double couple whose Green's functions (or complex potentials) are derived using the analytic function theory. The inelastic strain over a finite region is, then, represented by a continuous distribution of such double couples and its complex potentials by area integrals. Closed form expression for the stress field arising from the uniform inelastic strain distribution is given for the general polygonal region withn sides.With 5 FiguresThis paper is dedicated to the memory of Aris Phillips, founding Co-Editor of Acta Mechanica, and was presented at the Aris Phillips Memorial-Symposium, Gainesville, Fla., 1987.     

Formulation of the plastic source method for plane inelastic problems Part 2: Numerical implementation for elastoplastic problems

Summary In Part 1 of this paper, Green's function representation for the residual stress, arising from any plane inelastic strain distribution under no applied load, has been given. In Part 2 (i.e., the current paper) we focus our attention to the plastic strain arising in plane elastoplastic problems subject to applied loads. A numerical procedure, based on the Green's function approach, to determine the unknown plastic strain distribution under the applied load is established. The numerical result for an elastoplastic steady crack growth problem demonstrates, overwhelmingly, the advantage of the proposed method over the finite element method approach for the same problem.With 5 FiguresThis paper is dedicated to the memory of Aris Phillips, founding Co-Editor of Acta Mechanica, and was presented at the Aris Phillips Memorial-Symposium, Gainesville, Fla., 1987.