Strain localization analysis using a large deformation anisotropic elastic–plastic model coupled with damage

Sheet metal forming processes generally involve large deformations together with complex loading sequences. In order to improve numerical simulation predictions of sheet part forming, physically-based constitutive models are often required. The main objective of this paper is to analyze the strain localization phenomenon during the plastic deformation of sheet metals in the context of such advanced constitutive models. Most often, an accurate prediction of localization requires damage to be considered in the finite element simulation. For this purpose, an advanced, anisotropic elastic–plastic model, formulated within the large strain framework and taking strain-path changes into account, has been coupled with an isotropic damage model. This coupling is carried out within the framework of continuum damage mechanics. In order to detect the strain localization during sheet metal forming, Rice’s localization criterion has been considered, thus predicting the limit strains at the occurrence of shear bands as well as their orientation. The coupled elastic–plastic-damage model has been implemented in Abaqus/implicit. The application of the model to the prediction of Forming Limit Diagrams (FLDs) provided results that are consistent with the literature and emphasized the impact of the hardening model on the strain-path dependency of the FLD. The fully three-dimensional formulation adopted in the numerical development allowed for some new results – e.g. the out-of-plane orientation of the normal to the localization band, as well as more realistic values for its in-plane orientation.     

Heterogeneity of Plastic Flow of Zirconium Alloys with a Parabolic Law of Strain Hardening

The specific features of plastic–strain macrolocalization at the stage of the parabolic law of strain hardening in samples from industrial zirconium–based alloys are considered. It is shown that in predeformed blanks, zones with a different character of plastic–strain localization are formed. It is also shown that the strain–localization macropattern can be used as a characteristic of the susceptibility of a material to further plastic form–changing, for example, upon tube rolling. The sign of fracture of alloys upon plastic deformation is revealed. The scale effect in the formation of localizedplastic–flow zones is shown and studied.     

Propagation of shear perturbations in dilatant liquids

Processes of shear-perturbation propagation in dilatant liquids are investigated. The evolution over time of the carrier of the Cauchy-problem solution for the quasilinear parabolic equation describing such processes is analyzed. Nonlinear effects of finite velocity and spatial localization of the shear perturbations are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 60–65, November–December, 1978.     

Numerical simulation of plastic-flow localization for simple shear

An algorithm was developed to numerically simulate plastic-flow localization for simple shear of a thermally plastic and viscoplastic material. The algorithm is based on solving the partial differential equations describing continuum flow. The closing equation is the constitutive relation known in the literature as the power law linking the plastic-strain rate to the flow stress, temperature, and accumulated plastic strain. Calculated relations for the time evolution of the shear-band width and the temperature and plastic strains localized in it agree satisfactorily with experimental relations. Good agreement with experimental results is also obtained for the sample temperature distribution at the developed stage of the localization process.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 173–180, January–February, 2005     

Nonlinear interactions of langmuir waves in a weakly inhomogeneous plasma

Kinetic equations for the scattering of the waves of the one-dimensional spectrum by plasma particles are obtained for a weakly inhomogeneous plasma. The equation for the evolution of the spectrum of the short waves [k² > (m_e/m_i) D_e ^–2] trapped in the inhomogeneities of the plasma density differs significantly from the kinetic equation for the waves in a homogeneous plasma. The problem of localization on the spectrum of the Langmuir waves in regions near the minima of the plasma density is also considered. A solution of the kinetic equation for the waves, which describes this process, is obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 6–13, November–December, 1972.In conclusion, the author thanks A. S. Kingsep for suggesting the problem and for directing the work.     

The nonlinear evolution of two-dimensional and three-dimensional waves in mixing layers

The authors consider problems connected with stability [1–3] and the nonlinear development of perturbations in a plane mixing layer [4–7]. Attention is principally given to the problem of the nonlinear interaction of two-dimensional and three-dimensional perturbations [6, 7], and also to developing the corresponding method of numerical analysis based on the application to problems in the theory of hydrodynamic stability of the Bubnov—Galerkin method [8–14].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhldkosti i Gaza, No. 1, pp. 10–18, January–February, 1985.     

Determination of mechanical properties of fiber–matrix interface from pushout test

For ceramic matrix composites, the pushout test is the most widely used test for finding the two mechanical properties of the fiber–matrix interface – (1) the coefficient of friction and (2) the residual radial stress. Experimental measurements from the pushout test do not directly give the values of these two mechanical properties of the fiber–matrix interface, but need to be regressed to theoretical models. Currently, approximate theoretical models based on shear–lag analysis are used for regression. In this paper, the adequacy of the shear–lag analysis model in accurately finding the mechanical properties of the fiber–matrix interface is discussed. An elasticity solution of the pushout test based on boundary element method is developed. Regressing one set of available experimental data from a pushout test to both shear–lag analysis and boundary element method models gives values differing by 15% for the coefficient of friction but similar values for the residual radial stress. Parametric studies were also conducted to show the difference between the shear–lag analysis and boundary element method results for factors such as fiber to matrix elastic moduli ratios, coefficient of friction and fiber volume fractions.     

Population inversion and radiation density in a Q-switched CO2 laser

The mechanism of lasing in a steady-state CO₂ laser has been investigated in [1–3]. In this paper we present a numerical analysis of the processes which occur in a CO₂ laser when the resonator is rapidly Q-switched. It is shown that the transition of the laser into the state with a new Q has an oscillatory form.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 18–23, November–December, 1972.We wish to thank R. I. Soloukhin for his interest and help.     

On the measurement of some parameters of planetary atmospheres by means of the radiation generated by the descent of landers

The results are given of investigations into the possibility of determining the parameters of planetary atmospheres (the element composition and density) by measuring the spectrum of the radiation from the high-temperature gas formed by a lander when it enters the atmosphere of a planet at hypersonic velocity in the altitude range H = 70–150 km. The example of Venus demonstrates the possibility of determining these parameters under both night and day conditions of observation. The results are recommended for investigating such parameters at the time of aerodynamic deceleration in the descent of the landers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–45, January–February, 1981.     

Numerical simulation of shock wave intersections

A large number of papers, generalized and classified in [1, 2], have been devoted to unsteady gas flows arising in shock wave interaction. Experimental results [3–5] and theoretical analysis [6–9] indicate that the most interesting and least studied types of interaction arise in cases when there are several shock waves. At the same time, nonlinear effects, which depend largely on the nature of the shock wave intersections, become appreciable. Regions of existence of different types, of plane shock wave intersections have been analyzed in [10–13]. It has been shown that in a number of cases the simultaneous existence of different types of intersections is possible. The aim of the present paper is to study unsteady shock wave intersections in the framework of a numerical solution of the axisymmetric boundary-value problem that arises in the diffraction of a plane shock wave on a cone in a supersonic gas flow. Flow regimes that augment the experimental data of [3–5] and the theoretical analysis of [9] are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 134–140, September–October, 1986.     

Estimating pressure in nonsteady reflection of a shock wave

A comparative analysis of the results of physical [1–4] and mathematical experiments [5–8] is used to elucidate the mechanism of additional pressure lift at a Mach wavefront. The possibility and range of application of simplified flow models for the estimation of the pressure at certain characteristic points of the reflection surface (for example, a prism, a cylinder, or a sphere) was investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 100–106, March–April, 1977.     

Three-dimensional instability of an elliptic Kirchhoff vortex

The instability of a Kirchhoff vortex [1–3] with respect to three-dimensional perturbations is considered in the linear approximation. The method of successive approximations is applied in the form described in [4–6]. The eccentricity of the core is used as a small parameter. The analysis is restricted to the calculation of the first two approximations. It is shown that exponentially increasing perturbations of the same type as previously predicted and observed in rotating flows in vessels of elliptic cross section [4–9] appear even in the first approximation. As distinct from the case of plane perturbations [1-3], where there is a critical value of the core eccentricity separating the stable and unstable flow regimes, instability is predicted for arbitrarily small eccentricity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 40–45, May–June, 1988.     

Effect of Localization of Pulsed Energy Addition on the Wave Drag of an Airfoil in a Transonic Flow

The possibility of controlling the aerodynamic characteristics of airfoils with the help of local pulsed-periodic energy addition into the flow near the airfoil contour at transonic flight regimes is considered. By means of the numerical solution of two-dimensional unsteady equations of gas dynamics, changes in the flow structure and wave drag of a symmetric airfoil due to changes in localization and shape of energy-addition zones are examined. It is shown that the considered method of controlling airfoil characteristics in transonic flow regimes is rather promising. For a zero angle of attack, the greatest decrease in wave drag is obtained with energy addition at the trailing edge of the airfoil.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 60–67, September–October, 2005.     

Nonisothermal instability of high-velocity elastoplastic flows

An important feature of the high-velocity deformation of solids is the localization of deformation, one of the causes of which may be the nonisothermal instability of plastic flow [1–6]. In connection with the intensive development of high-velocity technology in the treatment of materials, the investigation of the criteria for nonisothermal stability of processes of plastic deformation is of fundamental interest, since in certain cases they determine the optimum technological regimes [5]. The critical values of deformation velocities, above which the effects of thermal instability becomes decisive in the process of deformation of solids, are estimated by semiempirical methods in [1]. The non-boundary-value problem of the criteria for nonisothermal instability is analyzed in [2] for the point of view of flow stability in the so-called coupled formulation. The latter means that the heat-conduction equation is added to the basic equations determining the dynamics of an elastoplastic medium. The problem is solved in [6] in an analogous formulation, but for flow averaged over the spatial coordinate. The solution of the boundary-value problem for one-dimensional flow in this formulation is given in the present paper.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 133–138, May–June, 1986.     

Electroconvective stability of a weakly conducting liquid

In inhomogeneous electric fields, at sufficiently high field strengths, a weakly conducting liquid becomes unstable and is set in motion [1–4]. The cause of the loss of stability and the motion is the Coulomb force acting on the space charge formed by virtue of the inhomogeneity of the electrical conductivity of the liquid [4–13]. This inhomogeneity may be due to external heating [4–6], a local raising of the temperature by Joule heating [2, 7, 8], and nonlinearity of Ohm's law [9–13]. In the present paper, in the absence of a temperature gradient produced by an external source, a condition is found whose fulfillment ensures that the influence of Joule heating on the stability can be ignored. Under the assumption that this condition is satisfied, a criterion for stability of a weakly conducting liquid between spherical electrodes is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–142, July–August, 1979.     

Shock-initiated implosion of an elliptical cavity and detonation in a liquid layer

Many papers [1–9] have been devoted to the dynamical analysis of bubble implosion in a liquid layer. Experiments have shown that an initially circular cavity is displaced or transformed into an elliptical cavity during the implosion process due to instability, whereupon its further contraction produces cumulative jets. This problem is important in the study of surface wear in cavitation flow [7] and in the analysis of the impact sensitivity of liquid explosives [1–6]. The onset of accumulation is conveniently investigated by starting with an elliptical cavity or by displacing a circular cavity relative to the impact axis, thereby creating an asymmetrical pressure field about the center of the cavity. In the present article certain theoretical notions are advanced with regard to the onset of the cumulative jet in an elliptical or displaced cavity and its influence on the ignition of liquid explosives due to the formation of minute droplets [4] in the adiabatically heated gas inside the cavity. Experimental data on the jet formation time and the frequency of nitroglycerin detonations qualitatively support the theoretical predictions.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 78–85, September–October, 1971.     

Nonisothermal Couette flow of a non-Newtonian fluid under the action of a pressure gradient

Nonisothermal Couette flow has been studied in a number of papers [1–11] for various laws of the temperature dependence of viscosity. In [1] the viscosity of the medium was assumed constant; in [2–5] a hyperbolic law of variation of viscosity with temperature was used; in [6–8] the Reynolds relation was assumed; in [9] the investigation was performed for an arbitrary temperature dependence of viscosity. Flows of media with an exponential temperature dependence of viscosity are characterized by large temperature gradients in the flow. This permits the treatment of the temperature variation in the flow of the fluid as a hydrodynamic thermal explosion [8, 10, 11]. The conditions of the formulation of the problem of the articles mentioned were limited by the possibility of obtaining an analytic solution. In the present article we consider nonisothermal Couette flows of a non-Newtonian fluid under the action of a pressure gradient along the plates. The equations for this case do not have an analytic solution. Methods developed in [12–14] for the qualitative study of differential equations in three-dimensional phase spaces were used in the analysis. The calculations were performed by computer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 26–30, May–June, 1981.     

Linear and nonlinear stability of combined plane-parallel flow of a film of liquid and gas

The problem of the linear stability of a layer of liquid entrained by a gas has been investigated for some special cases in [1–7]. In [8], the linear problem was solved numerically and the solution compared with some analytic solutions for special cases of the flow. In the present paper, the results of linear analysis are presented more comprehensively; the problem of finite-amplitude stability of the film is posed and solved numerically; the results of the linear and nonlinear analysis are compared with data of an experiment performed by the authors and by other experimentalists.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–42, January–February, 1979.We are grateful to V. Ya. Shkadov for supervising the work, to all the participants of G. I. Petrov's seminar for helpful discussion, and also to E. L. Kokon for assistance in evaluating the experimental data.     

Plastic Flow Localization in Commercial Zirconium Alloys

The behavior of plastic flow curves and patterns of plastic strain localization were studied for tension of samples of Zr — 1% Nb (É110 alloy) and Zr — 1% Nb — 1.3% Sn — 0.4% Fe (É635 alloy) were studied. The relationship of the localization kinetics with the strain hardening law in plastic flow and transition to fracture is established. The dislocation microstructure of the alloys in strain localization and prefracture zones is examined.     

Dynamic problem for a two-layer medium with a vibrational source at the boundary interface

The present study is concerned with an analysis of gravitational and acoustic waves which are excited by a vibrational source deeply placed in a liquid covered by ice. An analysis of the rigidity characteristics of ice modeled by an elastic layer or by a Kirchhoff plate is done by factorization of the solution to the integral equation equivalent to an initially combined boundary value problem. The uncombined boundary condition is used to solve problems for unrestricted ice fields in [1–3], whereas combined conditions with vibrational sources positioned at the boundary of the medium are used in [4].Translated from Zhurnal Prikladnoi Mekhaniki, No. 3, pp. 125–129, May–June, 1986.