Viscoplastic flow between two approaching or parallel circular plates

The problem of the flow of a viscoplastic medium between two parallel circular plates in translatory coaxial relative motion is solved. The Bingham model [1] of a viscoplastic medium is assumed. The problem is solved in the inertialess thin layer approximation [2] for arbitrary values of the viscosity coefficient and yield stress.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 9–17, January–February, 1996.     

Plane viscoplastic flow in narrow channels with deformable walls

The equations of motion of a continuum in a thin layer are derived for a given functional dependence of the stress tensor on the strain rate tensor. The general problem of viscoplastic flow is considered in the thin-layer approximation for boundary surface material points travelling in the lateral direction in a predetermined fashion.The projections of the continuum point velocity, pressure, flow rate through a cross-section of the channel, and the power of external forces are expressed as functions of the boundary deformation law. The problem of determining the channel boundary deformation law is formulated for a given boundary pressure distribution. The expressions for the continuum flow rate and pressure and the power of external forces written as functionals of the channel width allow formulation of the problems of controlling viscoplastic flows in thin layers and optimizing the processes.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 23–31, March–April, 1996.     

Three-dimensional problem for entry of a disk into a compressible liquid

The unsteady problem for the oblique entry of a disk into water is solved. The water is assumed a perfect compressible liquid and the flow is assumed adiabatic. The flow and state parameters are determined during the numerical integration of the system of nonlinear equations which describe the given flow by means of a three-dimensional finite-difference scheme [1]. The variation in time of the drag coefficient as a function of the Mach number and the angles of entry and attack, the pressure distribution and the shape of the free surface formed behind the disk are investigated. The oblique entry of a disk into water and its subsequent motion have mainly been studied for velocities at which the compressibility of the water is negligible [2–4]. The influence of compressibility on the duration of the rise time and the impact load was investigated experimentally in the range of Mach numbers 0 < M0 <–0.3 [5]. Semiempirical dependences are obtained for the maximum of the drag coefficient and its rise time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 17–20, January–February, 1988.     

Bounds for the effective properties of heterogeneous plates

This paper presents new bounds for heterogeneous plates which are similar to the well-known Hashin–Shtrikman bounds, but take into account plate boundary conditions. The Hashin–Shtrikman variational principle is used with a self-adjoint Green-operator with traction-free boundary conditions proposed by the authors. This variational formulation enables to derive lower and upper bounds for the effective in-plane and out-of-plane elastic properties of the plate. Two applications of the general theory are considered: first, in-plane invariant polarization fields are used to recover the “first-order” bounds proposed by Kolpakov [Kolpakov, A.G., 1999. Variational principles for stiffnesses of a non-homogeneous plate. J. Meth. Phys. Solids 47, 2075–2092] for general heterogeneous plates; next, “second-order bounds” for n-phase plates whose constituents are statistically homogeneous in the in-plane directions are obtained. The results related to a two-phase material made of elastic isotropic materials are shown. The “second-order” bounds for the plate elastic properties are compared with the plate properties of homogeneous plates made of materials having an elasticity tensor computed from “second-order” Hashin–Shtrikman bounds in an infinite domain.     

Pressure-shear waves in 6061-T6 aluminum and alpha-titanium

Thepressure-shear plate impact technique is used to study material behavior at high rates of deformation. In this technique, plastic waves of combined pressure and shear stresses are produced by impact of parallel plates skewed relative to their direction of approach. Commercially pure alpha-titanium and 6061-T6 aluminum are tested under a variety of pressure and shear tractions by using different combinations of impact velocities and angles of inclination. A laser interferometer system is used to monitor simultaneously the normal and transverse components of motion of a point at the rear surface of the target plate. The experimental results are compared with numerical solutions based on an elastic/viscoplastic model of the material. Both isotropic and kinematic strain hardening models are used in the computations. The results indicate that unlike the normal velocity profiles, the transverse velocity profiles are sensitive to the dynamic plastic response and, thus, can be used to study material behavior at high strain rates. For the materials tested the results suggest that the flow stress required for plastic straining increases markedly with increasing strain rate at strain rates above 10⁴s^?1. Hydrostatic pressure of the order that exists in the tests (up to 2 GPa) does not affect the plastic flow in 6061-T6 aluminum and appears to have at most a minor effect on the deformation of the titanium.     

Nonstationary shear flow of a viscoplastic medium

We consider problems involving nonstationary shear flow of a viscoplastic medium between two parallel plates and also in a cylindrical tube under the action of a time-varying shear stress applied to the walls of the passage.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 133–137, July–August, 1972.     

Local stability loss and failure of cracked plates during the plastic deformation of materials

The phenomena of local stability loss of thin cracked plates under tension, which develops for certain ratios between crack length and plate thickness, and its effect on the structure and strength characteristics of the plates, is not well understood. Guz' and Dyshel' [1] provide the most complete review of research performed. Virtually all experimental investigations, however, have been conducted on plates, the material of which was in the elastic stage of deformation right up to the start of the failure process, i.e., the influsence exerted by the material's plastic properties on the critical and failing loads has not been investigated. This report cites the results of tests of cracked plates found in the elasto-plastic and plastic stages of deformation.S. P. Timoshenko Mechanics Institute, Ukraine Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 1, pp. 51–55, January, 1994.     

Evolution of plastic zones in dynamically loaded plates using different elastic–viscoplastic laws

In the framework of viscoplastic theory many different laws were developed, accounting for material behaviors like creep, relaxation or evolution of overstresses. Though each model is able to predict in uni-axial material tests the values of stresses depending on plastic strains and plastic strain rates the question is if solutions of simulations are still realistic if the viscoplastic law is applied on structural deformations. In the present study strain rate sensitive metal plates are subjected to shock waves. The purpose is to compare simulation results obtained with different elastic–viscoplastic laws to experiments in order to determine the most appropriate material model. By subjecting circular metal plates experimentally to shock wave loadings realistic deformation histories are measured. The measurements are compared to simulation results obtained with different viscoplastic laws. The aim is to find out the accuracy of each model concerning the predictions of displacements, shape formings, spread of plastic zones and evolutions of inner bending moments.     

Stability of an elastoplastic sphere under external pressure

The stability of ideally plastic, elastoplastic, and reinforced elastoviscoplastic bodies subjected to large subcritical strains was investigated in [1–4], The problems solved in these papers were related to the stability of systems in which homogeneous stress and strain fields arise in the initial state. The stability of an elastic thick-walled spherical shell subjected to external pressure leading to large subcritical strains was investigated in [5]. The stability of an axisymmetric sphere of elastoplastic material subjected to large plastic strains is examined below.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 155–159, September–October, 1977.     

Oscillation of flat layered shells with local elastic supports

The study of oscillation in thin-walled construction elements on elastic supports is of great practical interest. Various aspects of the problems of mechanics arising in this regard have been considered by many authors, especially in recent years. The authors of [4, 8, 13, 14, 17–19, 22] have presented voluminous graphical and tabular material for solid beams with elastic supports and for rectangular plates supported on rigid point supports along the edges and in the inner area; moreover, the authors of [22] present results relating to linear supports and circular plates, while in [5, 13, 18, and 22] the results reported have to do with the forms of the fundamental oscillation. In [5] the elastic bond is modelled by means of a Vinkperovskii foundation with a discontinuous bed coefficient. Cylindrical shells are examined in [1, 6], while in [21], for a spherical shell with elastic supports, an analytical solution is constructed. The authors of [23, 24] investigate the effect of an attached mass and a linear support for a circular and a rectangular plate, and a comparison with experimental data is made for a lower frequency. The close connection between the problems in question with those involving oscillation of shells with attached masses is reflected in [3, 7, 11, 16]. Analysis of the results obtained in the works mentioned above and in others shows that, unlike the case of beam systems, numerical results for plates and shells are significantly more difficult to obtain. Therefore, in the overwhelming majority of publications, thin plates and shells are examined, while to describe the process of their deformation classical models are used; here the supports, as a rule, are assumed to be absolutely rigid. The oscillation of anisotropic and, in particular, layered construction elements on elastic supports with further consideration of the bending rigidity of the latter clearly has not been studied sufficiently, which makes further research in this field timely. The present article examines layered, flat, orthotropic shells on a rectilinear layout, for which a solution of the static problem has been obtained previously [9, 10]. The basic assumptions of the computation method, developed for calculating the stress-deformed state (SDS) arising during driven oscillation of these objects far from the resonance points, as well as for determining the fundamental oscillation frequencies (FOF), are presented. Unlike traditional approaches, this method realizes the possibility of calculating, along with the normal reactions of elastic supports, reactive moments and tangential forces; in describing the movement of the system, the relations of the improved theory of shells are used [2].S. P. Timoshenko Institute of Mechanics of the Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 55–60, February, 1994.     

On a finite strain viscoplastic theory based on a new internal dissipation inequality

This work is focused on the theoretical development and numerical implementation of a viscoplastic law. According to the second law of thermodynamics a dissipation inequality described in the rotated material coordinate system is developed. Based on this dissipation inequality and the principle of maximum dissipation a finite strain viscoplastic model described also in the rotated material coordinate system is formulated. The evolution equations are expressed in terms of the material time derivatives of the rotated elastic logarithmic strain, the accumulated plastic strain and the strain-like tensor conjugate to the rotated back stress. The mathematical structure of this theory is concise and similar to that of the infinitesimal viscoplastic theory. These characteristics make the numerical implementation of this theory easy. The stress integration algorithm and the algorithmic tangent moduli for the infinitesimal theory can be applied to the numerical implementation of the present finite strain theory with a little reformulation. The complicated algorithmic formulations for most of other finite plastic laws can be therefore circumvented. In order to check the effectivity of the present finite strain theory a set of numerical examples under strict deformation conditions are presented. These numerical examples prove the excellent performance of the present viscoplastic material law at describing the finite strain elastoplastic and viscoplastic problems.     

材料的弹粘塑性本构关系和应力波的演化

试验表明，大多数工程材料在冲击载荷作用之下的变形一般都同时包含有可恢复的瞬态性弹性变形和不可恢复的粘滞性塑性变形，即其本构关系可以用弹粘塑性模型来描述。本文从内变量理论出发，探讨了时率相关材料的弹粘塑性本构关系的一般特性，建立了增量型的弹粘塑性本构关系的一般理论框架和普适的表达式，并且对两种最常用的本构模型——Bodner-Partom模型和Johnson-Cook模型给出了在一维应变条件下的具体形式。通过计算和讨论一维应变粘塑性靶板中冲击波的衰减机制和应力波的演化规律，特别是考察各种粘塑性本构模型中的材料参数对冲击波的衰减和应力波的演化的影响，得出了一些可以直接应用或具有一定借鉴价值的结果，为研究应力波的其他衰减机制以及在人防工程中智能防护层设计时新材料的选取奠定了基础。     

The effect of transverse pressure on the stability of a plate

The stability problem of a centrally compressed infinite plate is solved with allowance for the transverse normal deformation caused by uniform load for various boundary conditions at the edges. The linearized nonlinear equations of elastic deformation of thin plates taking into account transverse shear and transverse normal deformation are used. The obtained critical loads are compared with existing solutions.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 170–178, March–April, 2005.     

Well production indicators in a deformable reservoir interacting with rocks

Steady-state flow towards a well through a thin porous deformable two-layer reservoir with allowance for deformation of the surrounding rocks is investigated. The permeability of the reservoir is considered to be a function of the displacements of its top and bottom. The effect of deformation on the well production indicators is studied. The results obtained agree qualitatively with the data of full-scale experiments. Earlier, in [1–5], in considering the self-consistent processes of flow through porous media and their deformation attention was concentrated on the analysis of the stress-strain state of the rocks and reservoir and on unsteady problems within the framework of the nonlocally elastic flow regime.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 86–93, January–February, 1995.     

Pulse shaping techniques for testing brittle materials with a split hopkinson pressure bar

We present pulse shaping techniques to obtain compressive stress-strain data for brittle materials with the split Hopkinson pressure bar apparatus. The conventional split Hopkinson pressure bar apparatus is modified by shaping the incident pulse such that the samples are in dynamic stress equilibrium and have nearly constant strain rate over most of the test duration. A thin disk of annealed or hard C11000 copper is placed on the impact surface of the incident bar in order to shape the incident pulse. After impact by the striker bar, the copper disk deforms plastically and spreads the pulse in the incident bar. We present an analytical model and data that show a wide variety of incident strain pulses can be produced by varying the geometry of the copper disks and the length and striking velocity of the striker bar. Model predictions are in good agreement with measurements. In addition, we present data for a machineable glass ceramic material, Macor, that shows pulse shaping is required to obtain dynamic stress equilibrium and a nearly constant strain rate over most of the test duration.     

Hydraulic impact of “exponential” and nonlinearly viscoplastic media in pipes made of a viscoplastic material

Differential equations are derived and the hydraulic impact process for “exponential” and nonlinearly viscoplastic media in pipes made of a viscoelastic material is analyzed. Hydraulic impact problems for actual media in pipes has been repeatedly treated in the literature [1–4]. The hydraulic impact of a viscous and linearly viscoplastic media in pipes made of an elastic and viscoelastic material was studied in this work. It is well known [5] that many media in the region of low and moderate shear rates reveal a nonlinearity of the flow curve (oil, drilling fluids, polymer solutions and melts, loaded fuels, fuel mixtures, blood, etc.). It should be noted that flexible pipes made of natural materials (pipe boreholes made of polymer materials, membranes of blood vessels, etc.) are described by complicated rheological equations of state for viscoelastic media. Thus a calculation of the influence of nonlinearity of these media and of the viscoelastic properties of the pipe material on the hydraulic impact process is of theoretical and practical interest in many engineering problems.     

On shakedown theory for elastic-plastic materials and extensions

The idea that an elastic-plastic structure under given loading history may shake down to some purely elastic state (and hence to a safe state) after a finite amount of initial plastic deformation, can apply to many sophisticated material models with possible allowable changes of additional material characteristics, as has been done in the literature. Despite some claims to the contrary, it is shown; however, that the shakedown theorems in a Melan-Koiter path-independent sense have been extended successfully only for certain elastic-plastic hardening materials of practical significance. Shakedown of kinematic hardening material is determined by the ultimate and initial yield stresses, not the generally plastic deformation history-dependent hardening curve between. The initial yield stress is no longer the convenient one (corresponding to the plastic deformation at the level of 0.2%) as in usual elastic-plastic analysis but to be related to the shakedown safety requirement of the structure and should be as small as the fatigue limit for arbitrary high-cycle loading. Though the ultimate yield strength is well defined in the standard monotonic loading experiment, it also should be reduced to the so-called “high-cycle ratchetting” stress for the path-independent shakedown analysis. A reduced simple form of the shakedown kinematic theorem without time integrals is conjectured for general practical uses. Application of the theorem is illustrated by examples for a hollow cylinder, sphere, and a clamped disk, under variable (including quasiperiodic dynamic) pressure.     

Bending of Composite Plates under Static and Dynamic Loading

The stationary bending of a two-dimensional elastic system of joined rectangular plates with different mechanical properties and boundary conditions is studied. A technique for solving the corresponding problem is proposed. It is based on approximate approaches in combination with a generalization of the force method. It is established that the static strain state of the system is essentially dependent on the difference between the elastic moduli of the constituent plates. It is also shown that with certain boundary conditions, the dependence of the first resonant frequency on the relative position of plates along the contact line is nonmonotonic__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 77–84, January 2005.     

Accounting for the elastic properties of a non-Newtonian material under its viscosimetric flow

This paper considers the deformation and viscoplastic flow of a non-Newtonian material enclosed between coaxial rigid cylindrical surfaces, each of which performs a rotation followed by a stop and a rotation in the opposite direction. The problem is solved using the model of large elastoviscoplastic deformations, in contrast to the classical solutions obtained using the model of a rigid viscoplastic body. The parameters of the viscosimetric process are calculated in both the region of viscoplastic flow developed and the region of elastic deformation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 143–151, March–April, 2008.