A generalization of Prandtl’s and Spencer’s solutions on axisymmetric viscous flow

New analytical solutions for axisymmetric deformation of a viscous hollow circular cylinder on a rigid fibre are given. One of the solutions generalizes the famous Prandtl’s solution for compression of a rigid perfectly plastic layer between two rough, parallel plates and the other is a modification of Spencer’s solution for compression of an axisymmetric rigid perfectly plastic layer on a rigid fibre. All equations are satisfied exactly whereas some boundary conditions are approximated in a standard manner. Special attention is devoted to frictional interface conditions since these conditions result in additional limitations of the applicability of the solution when compared to that based on a rigid perfectly plastic models. In particular, difficulties with the convergence of numerical solutions under certain conditions can be explained with the use of results obtained. Therefore, the solutions can serve as benchmark problems for verifying numerical codes. The solutions are also adopted to predict the brittle fracture of fibres by means of an approach used in previous studies and confirmed by experiment.     

Finite element computation of torsional plastic waves in a thin-walled tube

Summary  A finite element technique is presented for the analysis of one-dimensional torsional plastic waves in a thin-walled tube. Three different nonlinear consitutive relations deduced from elementary mechanical models are used to describe the shear stress–strain characteristics of the tube material at high rates of strain. The resulting incremental equations of torsional motion for the tube are solved by applying a direct numerical integration technique in conjunction with the constitutive relations. The finite element solutions for torsional plastic waves in a long copper tube subjected to an imposed angular velocity at one end are given, and a comparison with available experimental results to assess the accuracy of the constitutive relations considered is conducted. It is demonstrated that the strain-rate dependent solutions show a better agreement with the experimental results than the strain-rate independent solutions. The limitations of the constitutive equations are discussed, and some modifications are suggested. Received 9 February 1999; accepted for publication 28 March 2000     

Crack tip singular fields in ductile crystals with taylor power-law hardening. I: Anti-plane shear

Asymptotic singular solutions of the HRR type are presented for anti-plane shear cracks in ductile crystals. These are assumed to undergo Taylor hardening with a power-law relation between stress and strain at sufficiently large strain. Results are given for several crack orientations in fcc and bcc crystals. The neartip region divides into angular sectors which are the maps of successive flat segments and vertices on the yield locus. Analysis is simplified by use of new general integrals of crack tip singular fields of the HRR type. It is conjectured that the single crystal HRR fields are dominant only over part of the plastic region immediately adjacent to the crack tip, even at small scale yielding, and that their domain of validity vanishes as the perfectly plastic limit is approached. This follows from the fact that while in the perfectly plastic limit the HRR stress states approach the correct discontinuous distributions of the complete elasticideally plastic solutions for crystals (Rice and Nikolic, J. Mech. Phys. Solids33, 595 (1985)), the HRR displacement fields in that limit remain continuous. Instead, the complete elastic-ideally plastic solutions have discontinuous displacements along planar plastic regions emanating from the tip in otherwise elastically stressed material. The approach of the HRR stress fields to their discontinuous limiting distributions is illustrated in graphical plots of results. A case examined here of a fcc crystal with a crack along a slip plane is shown to lead to a discontinuous near-tip stress state even in the hardening regime.Through another limiting process, the asymptotic solution for the near-tip field for an isotropic material is also derived from the present single crystal framework.     

Series solutions of unsteady MHD flows above a rotating disk

The series solutions of unsteady flows of a viscous incompressible electrically conducting fluid caused by an impulsively rotating infinite disk are given by means of an analytic technique, namely the homotopy analysis method. Using a set of new similarity transformations, we transfer the Navier–Stokes equations into a pair of nonlinear partial differential equations. The convergent series solutions are obtained, which are uniformly valid for all dimensionless time 0 ≤ τ < ∞ in the whole spatial region 0 ≤ η < ∞. To the best of our knowledge, such kind of series solutions have never been reported. The effect of magnetic number on the velocity is investigated.     

Further improvement on fundamental solutions of plane problems for orthotropic materials

On the basis of the ecisting fundamental solutions of displacements, further improvement is made, and then the general fundamental solutions of both plane elastic and plane plastic problems for orthotropic materials are obtained. Two parameters based on material constants α₁ = α₁ are used to derive the relevant expressions in a real variable form. Additionally an analytical method of solving the singular integral for the internal stresses is introduced, and the corresponding results are given. If α₁ = α₁ = 1, all the expressions obtained for orthotropy can be reduced to the corresponding ones for isotropy. Because all these expressions and results can be directly used for both isotropic problems and orthotropic problems, it is convenient to use them in engineering with the boundary element method (BEM).     

Rheological properties of materials from the point of view of physical kinetics

Zhurkov and Kauzmann rheological bodies, reflecting the physical regularities of plastic deformation of materials, are studied. Solutions of differential equations of flow for composite rheological bodies are obtained. Examples of using structural models of materials that consist of new rheological bodies and describe the inelastic behavior of alloys for various types of temperature-force loading are given. The obtained solutions of the differential equations are used for analysis of the creep of a structurally unstable alloy. Chaplygin Siberian Aviation Research Institute, Novosibirsk 630051. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 1, pp. 119–128, January–February, 1998.     

Experimental investigation of circular cantilever beams subjected to impulsive loading

This paper reports the experimental results of quadrantal circular cantilever beam carrying a mass at its tip which is subjected to radial impact. It was observed that after impact a travelling plastic segment moved from the tip towards the root in a very short time, and when specimen reached the largest deformation, the beam whipped out evidently in the reverse direction. The curvature change of the curved beam reaches a peak value near the middle region of the beam and the maximum is at the root. The experimental results are compared with the solutions for the rigid-perfectly plastic model given in [12–13].The authors would like to thank Professor Yang Gui-tong for guidance and help.     

The use of the “trident” model in the analysis of plastic zones near crack tips and corner points

The paper deals with calculation of a plastic zone near a crack tip in a homogeneous elastoplastic solid and near a corner point of the boundary of this solid. The calculations are conducted for a solid subject to plane strain and within the framework of models with plastic strips. It is shown that in comparison with the widely used model with two straight slip-lines, the process of plastic deformation is described by the “trident” model more accurately. The results of calculations of the plastic zone by the “trident” model that correspond to different stages of the development of plastic deformation are given for a crack of normal separation in a quasibrittle material. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 95–100, March, 2000.     

THE EXACT SOLUTIONS OF ELASTIC－PLASTIC CRACK LINE FIELD FOR MODE ⅡPLANE STRESS CRACK

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An analysis of the plane-strain compression of a three-layer strip

Summary  The problem considered here is that of the plane-strain compression of a long symmetric strip of a three-layered material between rigid, parallel, rough plates. Two combinations of layers are examined: (a) a viscoplastic material placed between two layers of a rigid/perfectly plastic material, and (b) a rigid/perfectly plastic material placed between two layers of a viscoplastic material. Closed-form solutions are presented for each combination, and qualitative differences between these solutions and solutions obtained for homogeneous materials are discussed. A possible effect of asymptotic behaviour of the solution in the vicinity of maximum-friction surfaces on the general structure of the solution is mentioned. Received 24 July 2000; accepted for publication 6 February 2001     

The near tip fields and temperature distribution around a crack in a body of hardening material containing small damage

In this paper, the following conclusions are reached: The influence of damage on the stress and strain feilds can be neglected in an asymptotic sense for the solutions of damage field in a plastic solid containing small damage. The formulation of the problem is simplified with an uncoupled approach. Based on experimental results of plastic damage, most of the damage in the material are considered as small damage with the critacal damage variable ω _c ≪1. Using this approach, closed form expressions of the near tip damage fields for mode III, mode I and the temperature distribution induced by plastic dissipation in a hardening material containing damage are deduced. We point out that the temperature distribution in the process zone is strongly dependent on the damage of materials even for the small damage case. The results of the predicted value of the temperature rise near the tip region ignoring the damage effect is appreciably higher than the observed data. The main reason of this discrepancy is the presence of damage dissipation and the fact that its influence on the calculation of plastic dissipation have not been appropriately taken account of. The calculation is improved by taking into account the damage effect on the temperature rise, then theT _max value is in better accord with the experimental value. The project supported by the National Natural Science Foundation of China.     

Exact solution of rotating FGM shaft problem in the elastoplastic state of stress

Plane strain analytical solutions to estimate purely elastic, partially plastic and fully plastic deformation behavior of rotating functionally graded (FGM) hollow shafts are presented. The modulus of elasticity of the shaft material is assumed to vary nonlinearly in the radial direction. Tresca’s yield criterion and its associated flow rule are used to formulate three different plastic regions for an ideal plastic material. By considering different material compositions as well as a wide range of bore radii, it is demonstrated in this article that both the elastic and the elastoplastic responses of a rotating FGM hollow shaft are affected significantly by the material nonhomogeneity.     

Minimum theorems for displacement and plastic strain rate histories in structural elastoplasticity

Summary The finite element method approach is used to obtain formulations of analysis problems relative to elastic-plastic structures when subjected to prescribed programmes of loads, and under the restrictive hypotheses:a) the yielding surfaces are piecewise linearized, andb) the plastic flow-laws are supposed to be of holonomic type within a single “finite” time interval. For mulations are given as linear complementarity problems and quadratic programming problems: one pair of formulations in terms of velocity and plastic multiplier rate histories, and another pair in terms of plastic multiplier rate histories only. The solutions are shown to be characterized by two minimum principles for displacement and plastic strain rate histories. After some general remarks about computational procedures, the paper is concluded with some suggestions for future developments.

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Sommario Si usa il metodo degli elementi finiti per formulare problemi di analisi relativi a strutture elasto-plastiche soggette a prescritti programmi di carico, sotto le ipotesi restrittive:a) le superfici di plasticizzazione sono linearizzate a tratti, eb) la legge del flusso plastico è olonoma all'interno del singolo intervallo di tempo “finito”. Si danno formulazioni come problemi di complementarità lineare e come problemi di programmazione quadratica: due formulazioni sono in termini di storia delle velocità e dei coefficienti di attivazione plastica, altre due sono in termini di storia dei coefficienti di attivazione plastica soltanto. Si dimostra che le soluzioni sono caratterizzate da due principi di minimo per la storia delle velocità di deformazione. Dopo alcune osservazioni generali sui procedimenti di calcolo, il lavoro si conclude con dei suggerimenti per futuri sviluppi.

     

Some properties of three-dimensional Biltrami flows

The investigation of Beltrami flows is important for the research on the mechanism of turbulent structure. In this paper the general solutions of the Beltrami flows are given, which depend explicitly on the solutions of three independent Helmholtz equations with scalar unknowns. Velocity fields of Beltrami flows can then be obtained explicitly after the application of some curl operations on the solutions of Helmholtz equations. On the basis of the exact solutions of Euler equations given above, we obtain one kind of exact solutions of non-steady Navier-Stokes equations which are also the Beltrami flows. Some interesting examples of Beltrami flows other than “ABC flows”, “Kolmogolov flows”, “Rayleigh-Bernard flows”, “Q-flows” are given. The detailed analytic results of these examples will be published in the near future.     

Physically and Geometrically Nonlinear Deformation of Spherical Shells with an Elliptic Hole

The elastoplastic state of thin spherical shells with an elliptic hole is analyzed considering that deflections are finite. The shells are made of an isotropic homogeneous material and subjected to internal pressure of given intensity. Problems are formulated and a numerical method for their solution with regard for physical and geometrical nonlinearities is proposed. The distribution of stresses (strains or displacements) along the hole boundary and in the zone of their concentration is studied. The results obtained are compared with the solutions of problems where only physical nonlinearity (plastic deformations) or geometrical nonlinearity (finite deflections) is taken into account and with the numerical solution of the linearly elastic problem. The stress—strain state in the neighborhood of an elliptic hole in a shell is analyzed with allowance for nonlinear factors __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 95–104, June 2005.     

Experimental study of a metal-matrix composite

A metal-matrix specimen was investigated using moiré interferometry with 2400 ℓ/mm (60,960 ℓ/in.). The specimen was a 6-ply [0/±45] _s boron-aluminum tensile coupon with a central slot. The unequal deformations in fibers and matrix were evident. Special attention was given to a plastic slip zone in the matrix. Normal and shear strains were calculated in the slip zone; the shear-strain concentration factor rose dramatically with the onset of plastic slip and continued to rise gradually with load to 95 percent of the failure load. The precipitous change from maximum tensile strain in one fiber to zero tensile strain in the neighboring fiber was accommodated by the very high shear strain in the matrix in the slip zone. Experimental considerations are discussed: shear strains are independent of rigid-body rotations; simplified strain calculations are justified; averaging across the narrow slip region and the influence of finite thickness of the specimen grating contribute to underestimation of peak strains. Paper was presented at the 1986 SEM Spring Conference on Experimental Mechanics held in New Orleans, LA on June 8–13.     

Thermo–elasto–plastic stresses in functionally graded materials under consideration of the fabrication process

Summary  The present study analyzes elasto–plastic thermal stresses in some particle-reinforced functionally graded material plates (FGP) by taking into consideration residual stresses of the fabrication process. For the FGP, the region near the cooling metal surface consists of distributed ceramic particles in a metal matrix, while the region near the heating ceramic surface contains distributed metal particles in a ceramic matrix. We use the thermo–elasto–plastic constitutive equation of a particle-reinforced composite, taking into consideration temperature changes and damage as well as the reinforcing effect of particles. Elasto–plastic thermal stresses are discussed here with the goal of reducing the thermal stresses. Two kinds of particle-reinforced FGP are considered: the first kind (FGP1) represents distributed ceramic particles in the metal matrix, and the second one (FGP2) represents distributed metal particles in the ceramic matrix. We modify the thermo–elasto–plastic constitutive equation of a particle-reinforced composite for the FGP2 by taking into consideration temperature changes and damage as well as the reinforcing effect of particles. Using the temperature-dependent material properties, three cases of temperature conditions are studied. The first one is the cooling from the fabrication temperature to the room temperature, the second one is the heating from the room temperature, and the last one is the heating after cooling from the fabrication temperature. The particle volume fraction is assumed to vary according to a power function in the thickness direction of the FGPs. Using the finite element method, the effects of the distribution parameter of the composition on the macroscopic stress, the stress in the matrix and the stress in the particle in the FGPs are discussed. Also, the effects of the particle volume fraction and the fabrication temperature on the maximum tensile matrix stress are discussed. Received 22 November 2000; accepted for publication 24 April 2001     

A generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials

In this paper, a generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials is derived. The evolution equation for the active yield surface with reference to the memory yield surface is obtained by considering the continuous expansion of the active yield surface during the unloading/reloading process. The incremental constitutive relation based on the associated flow rule is then derived for a general yield function for pressure insensitive and sensitive materials. Detailed incremental constitutive relations for materials based on the Mises yield function, the Hill quadratic anisotropic yield function and the Drucker–Prager yield function are derived as the special cases. The closed-form solutions for one-dimensional stress–plastic strain curves are also derived and plotted for materials under cyclic loading conditions based on the three yield functions. In addition, the closed-form solutions for one-dimensional stress–plastic strain curves for materials based on the isotropic Cazacu–Barlat yield function under cyclic loading conditions are summarized and presented. For materials based on the Mises and the Hill anisotropic yield functions, the stress–plastic strain curves show closed hysteresis loops under uniaxial cyclic loading conditions and the Masing hypothesis is applicable. For materials based on the Drucker–Prager and Cazacu–Barlat yield functions, the stress–plastic strain curves do not close and show the ratcheting effect under uniaxial cyclic loading conditions. The ratcheting effect is due to different strain ranges for a given stress range for the unloading and reloading processes. With these closed-form solutions, the important effects of the yield surface geometry on the cyclic plastic behavior due to the pressure-sensitive yielding or the unsymmetric behavior in tension and compression can be shown unambiguously. The closed form solutions for the Drucker–Prager and Cazacu–Barlat yield functions with the associated flow rule also suggest that a more general anisotropic hardening theory needs to be developed to address the ratcheting effects for a given stress range.     

Inelastic deformation of flexible cylindrical shells with a curvilinear hole

The elastoplastic state of thin cylindrical shells weakened by a curvilinear (circular) hole is analyzed considering finite deflections. The shells are made of an isotropic homogeneous material. The load is internal pressure of given intensity. The distributions of stresses (strains, displacements) along the hole boundary and in the zone of their concentration are studied. The results obtained are compared with solutions that account for physical (plastic strains) or geometrical (finite deflections) nonlinearity alone and with a numerical linear elastic solution. The stress-strain state around a circular hole is analyzed for different geometries in the case where both nonlinearities are taken into account __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 12, pp. 115–123, December, 2006.     

Fields near a rapidly propagating crack-tip in an elastic perfectly-plastic material

Dynamic effects are investigated for the steady-state fields of stress and deformation in the immediate vicinity of a rapidly propagating crack-tip in an elastic perfectly-plastic material. Both the cases of antiplane strain and in-plane strain have been considered. The governing equations in the plastic regions are hyperbolic in nature. Simple wave solutions together with uniform fields provide explicit asymptotic expressions for the stresses and the strains in the near-tip regions. The dynamic solutions describe a region of plastic loading which completely surrounds the propagating crack-tip.