SLIP-LINE FIELD THEORY OF PLANE PLASTIC STRAIN DEALING WITH MOHR''''S CRITERION EXPRESSED BY QUADRATIC LIMITING CURVES

In this paper, the slip-line field theory of plane plastic strain dealing with Mohr's criterion expressed by quadratic limiting curves is preliminarily established. It takes the classical slip-line field theory as its special case, and it can be applied to the analysis of plane-strain problems in metal processing, rock and soil mechanics and tectonomechanics. As preliminary application, the slip-line field and limiting loads of flat punch indenting problem are determined by numerical solution, and the Slip-line field of bedded medium gravity-sliding problem is determined and discussed.     

Stability of the crossflow pattern around a slender and influence of disturbance

Topological structure and stability of a slender cross flow is discussed by the stability theory of dynamic system. The inner boundary of flow field was limiting streamline and it was proved that the topological structure connected saddles by limiting streamline is stable. It is proved that the development of slender vortices leads to the change of topological structure about cross flow. And it is the change from stable and symmetrical vortices flow pattern to unstable and symmetrical vortices flow pattern, and then to stable and asymmetrical vortices flow pattern due to little disturbance which leads to the development of asymmetrical slender vortices. The influence of disturbance to flowfield structure was discussed by unfolding theory too.     

AN UPPER BOUND SOLUTION FOR THE FORCE OF COMBINED BACKWARD-FORWARD EXTRUSION

A rigid-triangle velocity field for combined backward-forwardextrusion based on the experiments and the slip-line field isproposed in this paper.The flow separation point in the ri-gid-triangle velocity field is defined in accordance with theslip-line theory.A formula of minimum upper bound solutionfor the punch pressure of the combined extrusion is derived.The values from this formula are compared with those from theslip-line solution and with experimental results.The formulaof upper bound solution can be used in practice.     

Diffraction of elastic waves in the plane multiply-connected region and dynamic stress concentration

This paper deals with the problem of diffraction of elastic waves in the plane multiply-connected regions by the theory of complex functions. The complete function series which approach the solution of the problem and general expressions for boundary conditions are given.’ Then the problem is reduced to the solution to infinite series of algebraic equations and the solution can be directly obtained by using electronic computer. In particular, for the case of weak interaction, an asymptotic method is presented here, by which the problem ofp waves diffracted by a circular cavities is discussed in detail. Based on the solution of the diffracted wave field the general formulas for calculating dynamic stress concentration factor for a cavity of arbitrary shape in multiply-connected region are given.     

DIFFRACTION OF ELASTIC WAVES IN THE PLANE MULTIPLY-CONNECTED REGION AND DYNAMIC STRESS CONCENTRATION

This paper deals with the problem of diffraction of elastic waves in the plane multiply-connected regions by the theory of complex functions.The complete function series whichapproach the solution of the problem and general expressions for boundary conditions aregiven.Then the problem is reduced to the solution to infinite series of algebraic equationsand the solution can be directly obtained by using electronic computer.In particular,for thecase of weak interaction,an asymptotic method is presented here,by which the problem of pwaves diffracted by a circular cavities is discussed in detail.Based on the solution of thediffracted wave field the general formulas for calculating stress concentrationfactor for a cavity of arbitrary shape in multiply-connected region are given.     

AN INVESTIGATION ON ASYMPTOTIC NEAR-TIP FIELDS OF DYNAMIC CRACK IN AN ELASTIC-PERFECTLY PLASTIC COMPRESSIBLE MATERIAL

The stress and deformation fields near the tip of a mode-I dynamic crack steadilypropagating in an elastic-perfectly plastic compressible material are considered under plane strain condi-tions. Within the framework of infinitesimal displacement gradient theory, the material is character-ized by the Von Mises yield criterion and the associated J_2 flow theory of plasticity. Through rigorousmathematical analysis, this paper eliminates the possibilities of elastic unloading and continuousasymptotic fields with singular deformation, and then constructs a fully continuous and boundedasymptotic stress and strain field. It is found that in this solution there exists a parameter (?)_0 whichcannot be determined by asymptotic analysis but may characterize the effect of the far field. Lastly thevariations of continuous stresses, velocities and strains around the crack tip are given numerically fordifferent values of (?)_0.     

ELASTO-PLASTIC CONSTITUTIVE MODEL OF SOIL-STRUCTURE INTERFACE IN CONSIDERATION OF STRAIN SOFTENING AND DILATION

The behavior of soil-structure interface plays a major role in the definition of soil-structure interaction. In this paper a bi-potential surface elasto-plastic model for soil-structure interface is proposed in order to describe the interface deformation behavior,including strain softening and normal dilatancy. The model is formulated in the framework of generalized potential theory,in which the soil-structure interface problem is regard as a two-dimensional mathematical problem in stress field,and plastic state equations are used to replace the traditional field surface. The relation curves of shear stress and tangential strain are fitted by a piecewise function composed by hyperbolic functions and hyperbolic secant functions,while the relation curves of normal strain and tangential strain are fitted by another piecewise function composed by quadratic functions and hyperbolic secant functions. The approach proposed has the advantage of deriving an elastoplastic constitutive matrix without postulating the plastic potential functions and yield surface. Moreover,the mathematical principle is clear,and the entire model parameters can be identified by experimental tests. Finally,the predictions of the model have been compared with experimental results obtained from simple shear tests under normal stresses,and results show the model is reasonable and practical.     

Exact solution for the rotational flow of a generalized second grade fluid in a circular cylinder

This paper deals with the rotational flow of a generalized second grade fluid, within a circular cylinder, due to a torsional shear stress. The fractional calculus approach in the constitutive relationship model of a second grade fluid is introduced. The velocity field and the resulting shear stress are determined by means of the Laplace and finite Hankel transforms to satisfy all imposed initial and boundary conditions. The solutions corresponding to second grade fluids as well as those for Newtonian fluids are obtained as limiting cases of our general solutions. The influence of the fractional coefficient on the velocity of the fluid is also analyzed by graphical illustrations.     

DAMPING OF VERTICALLY EXCITED SURFACE WAVE IN WEAKLY VISCOUS FLUID

In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions is developed in weakly viscous fluids to investigate the motion of single free surface standing wave by linearizing the Navier-Stokes equation. The fluid field is divided into an outer potential flow region and an inner boundary layer region. The solutions of both two regions are obtained and a linear amplitude equation incorporating damping term and external excitation is derived. The condition to appear stable surface wave is obtained and the critical curve is determined. In addition, an analytical expression of damping coefficient is determined. Finally, the dispersion relation, which has been derived from the inviscid fluid approximation, is modified by adding linear damping. It is found that the modified results are reasonably closer to experimental results than former theory. Result shows that when forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when forcing frequency is high, the surface tension of the fluid is prominent.     

Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation

In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.     

An upper bound solution for the force of combined backward-torward extrusion

A rigid-triangle velocity field for combined backward-forward extrusion based on the experiments and the slip-line field is proposed in this paper. The flow separation point in the rigid-triangle velocity field is defined in accordance with the slip-line theory. A formula of minimum upper bound solution for the punch pressure of the combined extrusion is derived. The values from this formula are compared with those from the slip-line solution and with experimental results. The formula of upper bound solution can be used in practice.     

A note on calculations concerning the plastic compression of thin material between smooth plates under conditions of plane strain

Summary The solution to this problem was originally given by Green and consists of two distinct slip-line field solutions, and these fields he determined in a relatively complex way using the velocity equations. This note gives a method of approach to the problem which is easy to understand, and gives results demonstrating almost all the features mentioned in his paper. It is very simple to apply, and though it falls short in one important respect, it is offered as an interesting method of attacking the problem.     

Slip-line field theory of transversely isotropic body

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Plane strain compression between rough inclined plates

Summary The normal pressure distribution on rough plates, inclined at various angles, and the mean vertical pressure to effect a sideways expulsion of metal in terms of the plane-strain compressive yield stress of the material have been calculated using slip-line field solutions and conventional technological theory; the results are plotted for immediate use. A comparison of the results afforded by both these approaches is also made.     

金属滑动接触的滑移线场理论之发展现状

本文综述介绍了金属滑动接触的滑移线场理论的基本原理及其发展现状。该理论以一个硬质楞在较软金属表面的滑动作为金属滑动摩擦的基本模式,用滑移线场分析方法考察了滑动过程中的应力和应变,导出了摩擦系数的计算公式,揭示了在粘着理论中被隔离开来的粘着与犁削之间的内在联系。试验结果与理论计算结果有较好的一致性。该理论还提出了非疲劳磨损与疲劳磨损的不同变形和断裂机制,揭示了Archard磨损系数与微凸体发生疲劳断裂的临界受荷周期数之间的定量关系。文章还对该理论的进步意义及其发展趋势作了简要的评述。     

The applicability of slip-line field theory to contained elastic-plastic flow around a notch

Recent calculations by J.R. Griffiths and D.R.J. Owen (1971) on the growth of the elastic-plastic stresses for the plane strain bending of a V-notched bar reveal an interesting phenomenon : the stress maximum lies some way before the elastic-plastic interface, inside the plastic zone. Later calculations have confirmed this effect, for both work-hardening and perfectly-plastic von Mises and Tresca materials. At low applied loads the calculated stresses conflict with plastic slip-line field theory. This result is important, because it means that notch stresses before general yield cannot readily be deduced by etching up plastically-yielded zones. This paper explains the conflict analytically.     

A modified upper bound approach to limit analysis for plane strain deeply cracked specimens

The classical upper bound approach of limit analysis is based on assumption of rigid blocks of deformation that move between lines of tangential displacement discontinuity. This assumption leads to considerable simplification but often at cost of higher estimate of the actual load. Moreover, in many cases, it does not give a correct shape of the plastic field. In order to overcome these limitations a modified upper bound approach is proposed in this article. The proposed approach is basically an energetic approach but unlike the classical upper bound approach it is capable of including presence of statically governed stress field. As an application, of proposed approach, theoretical plane strain solutions are presented for deeply cracked fracture mechanics specimens (single edge cracked specimen in pure bending – SE (PB), single edge cracked specimen in three-point bending – SE (B), and compact tension – C (T) specimens). Plane strain plasticity problem in rigid elastic–plastic mono-material (homogeneous) was solved to evaluate useful parameters like limit load, plastic eta function (η_p) and plastic rotation factor (r_p) and in bi-material (mismatch welds) to evaluate mismatch limit load, for deeply cracked specimens. New kinematically admissible velocity fields are proposed for SE (B) and C (T) specimens. Proposed theoretical solutions were confirmed by classical slip-line field solutions, wherever available, and by detailed elastic–plastic finite element analysis with Von-Mises yield criterion. Good agreement was found between proposed solutions and results obtained from the classical slip-line field theory and finite element analysis.     

Strain-hardening extrusion—I. Construction of slip-line fields from experimental flow patterns

A possible method of solving problems in strain-hardening flows is by perturbation of known perfectly-plastic solutions. It is shown that vertex singularities, which are possessed by most such solutions, are not admissible in steady hardening flows. The structure of regular local solutions, both perfectly-plastic and strain-hardening, is investigated, and it is shown how vertex singularities can be replaced by regular local corner solutions.A scheme for constructing strain-hardening slip-line fields based on experimental flow patterns is described, which uses the maximum shear strain-rate directions calculated from the digitized and smoothed flow pattern to perturb local Hencky-Prandtl nets, which are then patched together in conformity with the topology of the solution to form a complete slip-line field. This method has been implemented in a computer program to construct slip-line fields from flow patterns for extrusion through wedge-shaped dies, and some fields computed by the program are presented.     

The calculation of Cauchy principal values in integral equations for boundary value problems of the plane and three-dimensional theory of elasticity

The integrals in certain singular integral equations of the theory of elasticity are defined in the sense of the Cauchy principal value. The existence of the Cauchy principal value has been proved for the plane problem by numerous authors (see Muskhelishvili [1], [2]) and for the three-dimensional problem by Kupradze and co-workers [3]. The knowledge of the limiting values of the integrands at the test point is essential for the numerical treatment. In this paper it is shown that the limiting values of the integrands are essentially determined by the curvature of the surface of the elastic body and by the gradient of the solution of the integral equation. A special regard is payed to test points at which the curvature and the gradient are discontinuous.     

Plane-strain crack-tip stress field for anisotropic perfectly-plastic materials

Plane-strain crack-tip stress solutions for anisotropic perfectly-plastic materials are presented. These solutions are obtained using the plane-strain slip-line theory developed by Rice (1973). The plastic anisosotropy is described by the Hill quadratic yield condition. The crack-tip stress solutions under symmetric (Mode I) and anti-symmetric (Mode II) conditions agree well with the low-hardening solutions for the corresponding power-law hardening materials. The crack-tip stress solutions under mixed Mode I and II conditions are also presented. All the solutions indicate that the general features of the slip-line field near a crack tip in orthotropic plastic materials with the elliptical yield contours in the Mohr plane are the same as those associated with isotropic plastic materials. However, the angular variations of the crack-tip stress fields for the materials with large plastic orthotropy differ substantially from those for isotropic plastic materials. Modifications due to polygonal yield contours are outlined and implications of solutions to the fracture analysis of ductile composite materials containing macroscopic flaws are discussed.