Stress analysis of the strain rate hardening materials in axially symmetric field

This paper presents an analysis of a circular hollow cylinder composed of strain rate hardening plastic material subjected to a sudden internal pressure loading. Materials satisfying the constitutive relation, and Levy-Mises equation are considered in the analysis. Dynamic and static analyses of axially symmetric material loaded under plane strain condition are described. The paper discusses the effect of strain rate hardening exponent, m, on the dynamic and static plastic response in the axially symmetric medium. A method is presented for the determination of the strain rate hardening exponent by measuring the hoop stress on the outer surface of a thin cylindrical specimen using the static solution.     

Couette flows of rigid/plastic solids: analytical examples of the interaction of constitutive and frictional laws

The exact analytical solution under plane strain conditions is found for a hollow circular cylinder subjected to rotational friction on the inner surface and an uniform pressure on the outer surface. Two constitutive laws, a rigid/plastic hardening model with a saturation stress (Voce-Palm material) and a rigid/viscoplastic model (Bingham material), are considered as well as three interfacial laws, sticking, Coulomb sliding friction and Tresca shearing friction. The study focuses on general behavior of the solutions for various pairs of constitutive and interfacial models, such as existence and uniqueness and qualitative differences in solution. For the Voce-Palm material, it is shown that solutions do not exist for certain boundary conditions if only sticking or Coulomb friction are permitted. On the other hand, multiple solutions exist for certain boundary conditions if only sticking and Tresca friction are permitted. Existence is achieved under all boundary conditions if sticking and both frictional laws permitted simultaneously. Uniqueness and unambiguity are achieved if two interface selection principles are invoked: (1) sticking must occur if it is possible and (2) Coulomb sliding must occur if it is possible and sticking is not. Geometrical interpretation of results in the form of friction maps is provided to illustrate possible regimes of interfacial behavior for different boundary conditions.     

Deformation analysis of plastic strain rate hardening material using stress function under plane strain condition

The two-dimensional plane strain equation of plastic flow in accordance with the Levy-Mises constitutive relation is expressed in terms of stress functions of complex variables. Expressions for the stress, strain rate and velocity are derived for plastic flow in a non linear viscous medium assuming the stress function in the form of both the summation and product of conjugate stress functions. The plastic states are derived, also using a mixed mode solution expressed in terms of non-seperable, independent conjugate complex variables.The analysis of the block indentation associated with a nonlinear viscous (strain rate hardening) material under plane strain condition using the product form solution is performed. The effect of the variation in the strain-rate hardening exponent on the contact stress is investigated. The predicted behavior of the vertical component of the contact stress suggests the possibility of the development of a specially instrumented plane strain block indentation tests, for the rapid determination of the strain-rate sensitivity of the real material. The vertical contact stress and strain-rate obtained from the product of the complex conjugate stress function are compared with those obtained from the summation form of the complex conjugate stream function.     

Elastic-plastic stress distribution in a rotating hyperbolic disk with rigid inclusion

The elastic-plastic stress distribution of a rotating hyperbolic annular disk mounted on a circular rigid shaft is investigated in this work. The exact solution presented is based on the usual assumptions of plane stress, Tresca's yield condition, its associated flow rule and linear strain hardening material behaviour. According to the present analysis the plastic core consists of three parts with different forms of the yield condition. The theory presented is illustrated by numerical results. The results show that the plastic flow and maximum stresses are influenced by the thickness parameter.     

Axisymmetric deformation of circular elastic-plastic tubes under axial tension and internal pressure

The axisymmetric bifurcation and post-bifurcation behaviour of circular tubes subjected to the combined action of axial tension and internal pressure are investigated numerically using the finite element method. It is assumed that the tubes are made of elastic-plastic strain hardening material with a smooth yield surface and that they deform without shear stress at both ends under the proportional stress path based on the stress values averaged over the cross-section.The bifurcation point and associated mode shape are obtained for each stress path. The initial to medium nonuniform deformation was studied for several specific stress paths and the growth of necking of the tube wall and swelling of the middle plane due to bifurcation are clarified.     

Constrained zones at singular points of inclusion contours

Stress concentration fields at singular points of the boundary of rigid inclusions in an elastic matrix under conditions of generalized plane stress were investigated by the method of caustics. Inclusion shapes were simulated by closed curves exhibiting cuspidal points, generated from the unit circle by means of a suitable transformation. It was proved that caustics developed at such points are generalized epicycloids, whose size is directly proportional to the respective stress intensity factor, while their orientation corresponds to the directions of the principal stress axes. A straightforward method for the experimental determination of these parameters was developed, important for the design of actual composites.     

直角平面内圆形刚性动夹杂对边界出平面点源载荷的动态响应

采用复变函数法和多极坐标移动技术研究二维直角平面区域内可动圆形刚性夹杂在边界出平面点源载荷作用下的动态响应问题.首先构造出直角平面区域内不含有夹杂时满足直角边界应力条件的格林(Green)函数解;其次求解波动方程边值问题,建立直角平面区域内含有夹杂时满足直边界应力自由条件的散射波解,利用叠加原理写出问题的总波场.借助于夹杂边界处的位移条件和夹杂运动的动力学条件,确定夹杂运动的位移幅度和散射波解中的未知系数.给出的算例结果表明本文方法的有效实用性.     

Elastic-plastic analysis of a rolling disk by finite elements

An elastic-plastic finite element method is employed to determine the stress distribution in a circular disk rolling on a rigid track and subjected to hub loads. This model can be assumed to represent a railroad wheel. Of particular interest are the stresses that exceed the initial yield limit near the point of contact. The material constitutive laws considered herein represent elastic-perfectly plastic and strain hardening materials that obey Mises' yield condition. It is shown that unloading of the previously yielded elements due to rotation can be easily handled by the direct stiffness tangent modulus approach. Total and plastic strains, and the plastic work in the disk model are given for the entire rolling history of the disk. It is also demonstrated that the disk will shake down to a purely elastic steady state for load values beyond the initial yield if the material of the disk exhibits even a small amount of strain hardening.     

The stability of a rigid rotor in ruptured finite journal bearings

The plane motion stability of a rigid rotor supported in fluid film journal bearings having a finite length and subjected to a steady external load and an unbalance load is theoretically analysed. Oil whirl is investigated for a ruptured film under separation boundary conditions. The non-linear equations of motion are solved by using numerical methods. Journal centre trajectories are obtained for different conditions of loading and speed. The stability threshold curve is plotted and compared with the curves of previous investigators obtained under other assumptions and boundary conditions.     

Bifurcation and post bifurcation behaviour of internally pressurized elastic-plastic circular tubes under plane strain conditions

The nonaxisymmetric bifurcation and post bifurcation behaviours of circular tubes subjected to internal pressure are investigated numerically in terms of the finite element method. It is assumed that tubes are made of elastic-plastic strain hardening material with smooth yield surface and that they deform under plane strain condition. Hill's uniqueness theory along with the Prandtl-Reuss equation and the separation of the variables are employed to obtain the bifurcation point and corresponding mode. The results indicate that the mode with longest wave length is critical and always follows the maximum pressure point for all cases investigated here.

For thin to thick tubes, the post bifurcation behaviours are numerically investigated. The development of unloading region, stress and strain distributions, localization of the deformation and thinning of the tube are clarified.     

三维压电介质中不同电边界条件的裂纹分析

在不可导通、可导通和半可导通等三种电边界条件下，系统研究并给出三维无限横观各向同性压电介质中平行于各向同性面的、任意形状的平片裂纹在任意载荷作用下的不连续位移和不连续电势边界积分方程方法。无论平片裂纹的形状如何，结果都表明：对不可导通裂纹，应力强度因子只与机械载荷有关，电位移强度因子只与电载荷有关；而可导通裂纹的应力强度因子和电位移强度因子只与机械载荷有关，电位移强度因子由机械载荷通过压电效应产生。半可导通裂纹所对应的边界积分方程组为非线性方程组，文中给出一种迭代解法。在均布的机械载荷和电载荷作用下。裂纹腔内的电位移为一常数，只与外加载荷有关，而与裂纹形状无关。     

Application of complex variables and pseudo-stress function to power-law materials and stress analysis of single rigid inclusion in power-law materials subjected to simple tension and pure shear

The two-dimensional compatibility equation for time-dependent materials described by the power law is expressed in terms of the second derivative of the stress function with respect to complex conjugate variables. The equation is solved by introducing the pseudo-stress function which satisfies the biharmonic equation resulting from the compatibility equation. The relationship between the second derivative of the stress function and the pseudo-stress function is established. The mixed derivative of the stress function associated with the dilatational stress is expressed by an integral of the complicated pseudo-stress function. The velocity and strain-rate components are expressed in terms of the pseudo-stress function. Therefore, responses of power-law creep materials subjected to a boundary condition can be obtained. Using the pseudo-stress function, the stress distribution in power-law materials containing a single rigid inclusion subjected to a uniaxial tensile and a simple shear stress is found. The results obtained by using the pseudo-stress function are compared with those obtained by another investigator. The results show that there are discrepancies between the two results in the neighborhood of the inclusion surface, and the deviation diminishes in the region away from the inclusion surface. Significantly, the discrepancies of the stress components associated with the dilatational stress, such as σ_r and σ_θ are predominant while σ_rθ, σ_r - σ_θ and gs agree reaso the region away from the inclusion surface, approximately 1.5R (R is the inclusion radius). The reasons creating the discrepancies between the two results are found, rationalized and discussed. The discrepancies are caused by the singularities involved in the integrand associated with the dilatational stress expression which is integrable deleting the singularities, and the effects of the singularities are predòminant for low strain-rate hardening exponents, m. The effects of the singularities on the stress components may not have been considered by the previous investigator and the reason is discussed. Through the analyses, it is found that: (1) the pseudo-stress function is a valid function for two-dimensional problems and it can be used for an analysis of power-law materials; and (2) the effects of the strain-rate exponent on the stress components and the stress concentration factor are found.     

Rigid viscoplastic finite element analysis of the gas-pressure constrained bulging of superplastic circular sheets into cone disk shape dies

This paper describes the development of a rigid viscoplastic finite element formulation for analysing the gas-pressure constrained bulging processes of superplastic circular sheets in cone disk shape dies. In this formulation, the effects of strain hardening and strain-rate sensitivity of materials are included, and the boundary friction condition is introduced into the formulation in the form of friction functional. The finiteelement model based on the membrane theory is developed, and then applied to simulate superplastic constrained bulging processes. The solutions by the rigid viscoplastic finite element method are compared with existing experimental data. The influences of the geometrical parameters of the dies, the friction factor in the friction functional, the strain hardening and the strain-rate sensitivity on the inhomogeneity of thickness distribution are studied in detail.     

Influence of slip velocity in Herschel-Bulkley fluid flow between parallel plates - A mathematical study

This theoretical study investigates three types of basic flows of viscous incompressible Herschel-Bulkley fluid such as (i) plane Couette flow, (ii) Poiseuille flow and (iii) generalized Couette flow with slip velocity at the boundary. The analytic solutions to the nonlinear boundary value problems have been obtained. The effects of various physical parameters on the velocity, flow rate, wall shear stress and frictional resistance to flow are analyzed through appropriate graphs. It is observed that in plane Poiseuille flow and generalized Couette flow, the velocity and flow rate of the fluid increase considerably with the increase of the slip parameter, power law index, pressure gradient. The fluid velocity is significantly higher in plane Poiseuille flow than in plane Couette flow. The wall shear stress and frictional resistance to flow decrease considerably with the increase of the power law index and increase significantly with the increase of the yield stress of the fluid. The wall shear stress and frictional resistance to flow are considerably higher in plane Poiseuille flow than in generalized Couette flow.

     

Two-dimensional elastic analysis of doubly periodic circular holes in infinite plane

Two-dimensional elastic analysis of doubly periodic circular holes in an infinite plane is given in this paper. Two cases of loading, remote tension and remote shear, are considered. A rectangular cell is cut from the infinite plane. In both cases, the boundary value problem can be reduced to a complex mixed one. It is found that the eigenfunction expansion variational method is efficient to solve the problem. Based on the deformation response under certain loading, the notched medium could be modeled by an orthotropic medium without holes. Elastic properties for the equivalent orthotropic medium are investigated, and the stress concentration along the hole contour is studied. Finally, numerical examples and results are given.     

Radial stressing of thin sheets with plastic anisotropy

The effect of transverse plastic orthotropy in uniform radial stressing at a circular hole, embedded in an infinite sheet, is studied. Material behaviour is modelled by a particular version of the anisotropic flow rule proposed by Hill [Math. Proc. Camb. Phil. Soc. 85, 171 (1979)], along with a linear-hardening characteristic. Analytical solutions are derived within the usual assumptions of plane stress and small strain plasticity. Some further results are presented for arbitrary hardening. The main finding is that parameter R has an appreciable influence of the plastic stress profile. The related stress concentration problem, of a circular hole embedded in an infinite sheet under remote uniform tension, is analyzed for rigid-plastic materials with arbitrary hardening. Comparison with experimental results shows a reasonable agreement for the stress field.     

双材平面中环形界面环向裂纹问题的超奇异积分方程方法

从研究环形界面双相材料平面任点处沿径向、环向作用单位力时的弹性力学基本解出发,利用Betti定律、几何关系和虎克定律得到双材料平面环向裂纹问题的位移场和应力场表达式,经代入裂纹岸应力边界条件,导出极坐标下以裂纹岸位移间断为基本未知量的超奇异积分方程组;通过适当的积分变换,用有限部积分原理处理方程组中所包含的两类奇异积分—Cauchy奇异积分和超奇异积分,解决极坐标下环形界面双材料平面环向裂纹问题用超奇异积分方程法的理论描述与数值算法。在嵌入物半径足够大时,计算结果与已发表文献对直线界面情况下平行于界面裂纹问题的计算结果一致。     

平面刚性夹杂问题的边界积分方程方法

研究了含刚性夹杂的平面弹性力学问题。由Ｓｏｍｉｇｌｉａｎａ公式出发，将问题归结为一组混合型边界积分方程的求解。其中外侧边界仍为常规的边界积分方程，而夹杂线上则是典型的Ｃａｕｃｈｙ型奇异积分方程，结合使用边界元及奇异积分方程算法［１］．本文对若干具有工程意义的例子作了数值计算，得到了其应力强度因子；有关结果还与文［２］作了比较，从而证明本文方法是完全可靠的，它可进一步用于解决更一般的问题。     

二维直角平面内偏心圆形衬砌对稳态入射平面SH波的散射

利用复变函数法、多极坐标变换及傅里叶级数展开技术求解二维直角平面内偏心圆形衬砌对稳态入射平面SH(shearing horizontal)波的散射问题.首先构造出介质内不存在偏心圆形衬砌时的入射波场和反射波场;其次建立介质内存在偏心圆形衬砌时由衬砌外边界产生的能够自动满足直角边应力自由条件的散射波解和衬砌外边界向衬砌介质内的折射波解以及衬砌内边界的散射波解,从而利用叠加原理可写出衬砌介质内外的总波场.利用衬砌外边界处应力位移的连续条件和内边界处应力自由条件以及傅里叶级数展开方法列出求解波解中未知系数的无穷代数方程组,在满足计算精度的前提下通过有限项截断,得到相应有限代数方程组的解,最后通过算例具体讨论衬砌内边界处的动应力集中系数和水平直边界位移幅度比及其相位随无量纲波数、入射波入射角、衬砌位置及其偏心度的不同而变化的情况,结果表明文中算法的有效实用性.