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CONTENTS
Volume 58, Number 4, May25 2016
 

Abstract
A mechanical engineer involved in analysis and design of structures and processes usually tends to focus on numerical methods such as FEM. However, there are at least three reasons that justify the development of qualitative and analytical methods for engineering applications. Firstly, programming errors easily occurs and accurate benchmark results are essential for the verification of existing numerical codes and the development of new numerical codes. Secondly, there are numerous boundary value problems that cannot be solved by means of standard numerical techniques and the development of new numerical techniques that might be capable of solving these problems requires an intensive analytical treatment before a reliable numerical technique can be proposed. Thirdly, even when obtainable in principle, accurate numerical solutions covering useful ranges of all relevant parameters would often not be worth computing for economical reasons by comparison with less expensive semi-analytic solutions that account for most important features of this or that structure or process. This is of special importance for design problems. Furthermore, neither the material, nor the geometry (at least, in process design) nor the friction can ever be closely specified. Therefore, mathematically accurate numerical solutions are not really demanded for some practical purposes. The present special issue is devoted to qualitative and analytical methods in elasticity and plasticity. The papers included in this issue deal with both exact semi-analytic solutions and asymptotic analyses that can in general be used in conjunction with numerical techniques.

Key Words
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Address
Institute for Problems in Mechanics

Abstract
Contact interaction of a beam (flexible element) with an elastic half-plane is considered, when a soft inhomogeneous (functionally graded) interlayer is present between them. The beam is bent under the action of a distributed load applied to the surface and a reaction of the elastic interlayer and the half-space. Solution of the contact problem is obtained for different values of thickness and parameters of inhomogeneity of the layer. The interlayer is assumed to be significantly softer than the underlying halfplane; case of 100 times difference in Young\'s moduli is considered as an example. The influence of the interlayer thickness and gradient of elastic properties on the distribution of the contact stresses under the beam is studied.

Key Words
bending of a beam; analytic solution; dual integral equation; functionally graded layer; soft layer; elastic half-plane

Address
Sergey M. Aizikovich, Boris I. Mitrin and Nikolai M. Seleznev: Research and Education Center \"Materials\", Don State Technical University, Rostov-on-Don 344000, Russia
Yun-Che Wang: Department of Civil Engineering, National Cheng Kung University, Tainan 70101,Taiwan
Sergey S. Volkov: Research Institute for Mechanics, Lobachevsky State University of Nizhni Novgorod, Nizhni Novgorod 603950, Russia

Abstract
The analysis of thermally induced stresses in engineering structures is a very important and necessary task with respect to design and modeling of pressurized containers, heat exchangers, aircrafts segments, etc. to prevent them from failure and improve working conditions. So, the purpose of this study is to investigate elasto-plastic thermal stresses and deformations in a thin annular plate embedded into rigid container. To this end, analytical research devoted to mathematically and physically rigorous stress/strain analysis is performed. In order to evaluate the effect of logarithmic thermal gradients, commonly applied to structures which incorporate thin plate geometries, different thermal parameters such as temperature mismatch and varying constraint temperature were introduced into the model of elastic perfectly-plastic annular plate obeying the von Mises yield criterion with its associated flow rule. The results obtained may be used in sensitive to temperature differences aircraft structures where the thermal effects on equipment must be kept in mind.

Key Words
logarithmic thermal gradient; plane stress; perfect plasticity; analytical solution

Address
Nelli N. Aleksandrova: Centre of Exact Sciences and Engineering, Madeira University, 9020-105 Funchal, Madeira, Portugal

Abstract
This paper deals with the pure bending of incompressible elastic perfectly plastic two-layer sheets under plane strain conditions at large strains. Each layer is classified by its yield stress, shear modulus of elasticity and its initial percentage thickness in relation to the whole sheet. The solution found is semianalytic. In particular, a numerical technique is only necessary to solve transcendental equations. The general solution is cumbersome because different analytic expressions for the radial and circumferential stresses should be adopted in different regions of the whole sheet. In particular, there are several alternative ways a plastic region (or plastic regions) can propagate. However, for any given set of material and process parameters the solution to the problem consists of a sequence of rather simple analytic expressions connected by transcendental equations. The general solution is illustrated by a simple example.

Key Words
plane strain bending; bimetallic sheet; elastic/perfectly plastic material; large strains; analytic solution

Address
Sergei E. Alexandrov: Laboratory for Strength and Fracture of Materials and Structures, Institute for Problems in Mechanics, 101-1 Prospect Vernadskogo, 119526, Moscow, Russia
Nguyen D. Kien, Dinh V. Manh: Institute of Mechanics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam
Fedor V. Grechnikov: Metal Forming Department, Samara State Aerospace University, 34 Moskovskoye shosse, 443086, Samara, Russia

Abstract
This paper presents an elastic/plastic model that neglects strain hardening during loading, but accounts for the Bauschinger effect. These mathematical features of the model represent reasonably well the actual behavior of several materials such as high strength steels. Previous attempts to describe the behavior of this kind of materials have been restricted to a class of boundary value problems in which the state of stress in the plastic region is completely controlled by the yield stress in tension or torsion. In particular, the yield stress is supposed to be constant during loading and the forward plastic strain reduces the yield stress to be used to describe reversed yielding. The new model generalizes this approach on plane stress problems assuming that the material obeys the von Mises yield criterion during loading. Then, the model is adopted to describe reversed yielding in thin hollow discs subject to external pressure.

Key Words
bauschinger effect; Mises yield criterion; thin disc; new material model

Address
Sergei E. Alexandrov: Laboratory for Strength and Fracture of Materials and Structures, Institute for Problems in Mechanics, 101-1 Prospect Vernadskogo, 119526 Moscow, Russia
Alexander R. Pirumov: Technical Mechanics Department, Moscow Technological University, 78 Prospect Vernadskogo,
119454, Moscow, Russia
Yeau-Ren Jeng: Department of Mechanical Engineering and Advanced Institute of Manufacturing with High-tech Innovations, National Chung Cheng University, 62102 Chia-Yi, Taiwan

Abstract
On the basis of the energy approach it is reported a development of the yield function and the constitutive equations for the orthotropic material with consideration of the crystal lattice constants and parameters of the crystallographic texture for the general stress state. For practical use in sheet metal forming analysis it is considered different loading scenarios: plane stress and plane strain states. Using the proposed yield function, the influence of single ideal components on the shape of yield surface was analyzed. The six texture components investigated here were cube, Goss, copper, brass, S and rotated cube, as these components are typically observed in rolled sheets from FCC alloys.

Key Words
anisotropy; plasticity; yield function; texture; crystallographic orientation; plane stress; plane strain; yield surface; rolled sheet

Address
Yaroslav A. Erisov, Fedor V. Grechnikov and Sergei V. Surudin: Metal Forming Department, Samara State Aerospace University, 34 Moskovskoye shosse, Samara, 443086, Russia

Abstract
The Drucker-Prager yield criterion is combined with an equilibrium equation to provide the elastic-plastic stress distribution within rotating annular hyperbolic discs and the residual stress distribution when the angular speed becomes zero. It is verified that unloading is purely elastic for the range of parameters used in the present study. A numerical technique is only necessary to solve an ordinary differential equation. The primary objective of this paper is to examine the effect of the parameter that controls the deviation of the Drucker-Prager yield criterion from the von Mises yield criterion and the geometric parameter that controls the profile of hyperbolic discs on the stress distribution at loading and the residual stress distribution.

Key Words
rotating annular disc; variable thickness; plastic yielding; Drucker-Prager yield criterion

Address
Woncheol Jeong: Department of Materials Science and Engineering, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea
Kwansoo Chung: Department of Materials Science and Engineering, Research Institute of Advanced Materials, Engineering Research Institute, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea

Abstract
The main objective of the present paper is to investigate, by means of a boundary value problem permitting a semi-analytic solution, qualitative behaviour of solutions for two pressure-dependent yield criteria used for plastically incompressible polymers. The study mainly focuses on the regime of friction (sticking and sliding). It is shown that the existence of the solution satisfying the regime of sticking depends on other boundary conditions. In particular, there is such a class of boundary conditions depending on the yield criterion adopted that the regime of sliding is required for the existence of the solution independently of the friction law.

Key Words
strength-differential effect; friction; sticking; sliding; polymers

Address
Elena A. Lyamina: Laboratory for Strength and Fracture of Materials and Structures, Institute for Problems in Mechanics, 101-1 Prospect Vernadskogo, 119526 Moscow, Russia
Prashant P. Date: Metal Forming Laboratory, Department of Mechanical Engineering, IIT Bombay, Powai, Mumbai 400076, India

Abstract
The main objective of the present paper is to study the effect of the yield criterion on the magnitude of the strain rate and plastic work rate intensity factors in axisymmetric flow of isotropic incompressible rigid perfectly plastic material by means of a problem permitting a closed-form solution. The boundary value problem consisting of the axisymmetric deformation of a plastic tube is solved. The outer surface of the tube contracts. The radius of the inner surface does not change. The material of the tube obeys quite a general yield criterion and its associated flow rule. The maximum friction law is assumed at the inner surface of the tube. Therefore, the velocity field is singular near this surface. In particular, the strain rate and plastic work rate intensity factors are derived from the solution. It is shown that the strain rate intensity factor does not depend on the yield criterion but the plastic work rate intensity factor does.

Key Words
strain rate intensity factor; plastic work rate intensity factor; generalised yield criterion; axisymmetric flow

Address
Elena A. Lyamina: Laboratory for Strength and Fracture of Materials and Structures, Institute for Problems in Mechanics, 101-1 Prospect Vernadskogo, 119526, Moscow, Russia
Thanh Nguyen: Institute of Mechanics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam

Abstract
Plastic behaviors, based on the von Mises yield criterion, of circular discs containing a purely elastic, circular inclusion under uniform temperature loading are studied with the finite element analysis. Temperature-dependent mechanical properties are considered for the matrix material only. In addition to analyzing the plane stress and plane strain disc, a 3D thin disc and cylinder are also analyzed to compare the plane problems. We determined the elastic irreversible temperature and global plastic collapse temperature by the finite element calculations for the plane and 3D problem. In addition to the global plastic collapse, for the elastically hard case, the plane stress problem and 3D thin disc may exhibit a local plastic collapse, i.e. significant pile up along the thickness direction, near the inclusion-matrix interface. The pileup cannot be correctly modeled by the plane stress analysis. Furthermore, due to numerical difficulties originated from large deformation, only the lower bound of global plastic collapse temperature of the plane stress problem can be identified. Without considerations of temperature-dependent mechanical properties, the von Mises stress in the matrix would be largely overestimated.

Key Words
plasticity; finite element analysis; temperature-dependent material properties; composite circular disc; elastic inclusion

Address
Somayeh Bagherinejad Zarandi, Yun-Che Wang: Department of Civil Engineering, National Cheng Kung University, 1 University Road, Tainan, 70101, Taiwan
Olga V. Novozhilova: Bauman Moscow State Technical University, 2nd Baumanskay street 5, 105005, Moscow, Russia


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