Plane stress state problems for notched bodies whose plastic properties depend on the form of the stress state

We consider one possible approach to the problem of describing the dependence of material plastic strain characteristics on the stress hydrostatic component arising in many porous, fractured, and other inhomogeneous materials. The plastic strain of the media under study is investigated under the plasticity assumption in the corresponding generalized form with the use of the form parameter of the stress state. The plasticity constitutive relations are stated on the basis of the plastic flow law associated with the accepted plasticity condition. For the conditions of plane stress state in the framework of the material rigid-plastic model, a system of partial differential equations is obtained and conditions for its hyperbolicity are determined. The relations for determining the stress fields and velocity fields in plastic domains are obtained, and their properties are investigated. The problem of tension of a strip with symmetric angular notches is solved, where the stress fields are determined and the continuous displacement rate field is constructed. The problem of uniform symmetric tension of a plane with a circular hole is considered. The stress fields in a strip with symmetric circular notches are examined. A comparison with solutions for plastically incompressible media whose properties are invariant with respect to the form of the stress state is performed.     

Analysis of circular hole expansion with generalized yield criteria

Expansion of a circular hole, embedded in an infinite elastoplastic sheet, is studied within the framework of large strain plane-stress plasticity. Material response is modeled by deformation type theories with two families of generalized isotropic yield criteria. Two distinct problems are examined in detail: hole enlargement under internal pressure and hole expansion under remote tension. Strain hardening and elastic compressibility are fully accounted for.Numerical illustrations reveal constitutive sensitivity of stress and deformation profiles. For the internally pressurized hole the specific power needed to create a new volume unit reaches an asymptotic level practically independent of yield criteria. The specific cavitation power is used to derive a simple relation for the ballistic limit in quasi-static plate perforation, showing good agreement with experimental results. Under remote tension the hole expands spontaneously when external stress approaches a limit which is found to be highly sensitive to the yield criteria.     

形状记忆合金椭圆孔口问题的有限单元法

基于Lagoudas形状记忆合金(SMA)三维本构模型,假设材料为各向同性,推导了SMA平面应力状态的增量型本构方程,继而编写了ABAQUS用户自定义材料(UMAT)子程序,研究了在双向拉伸情况下,外载荷、温度、椭圆孔口长短轴之比对超弹性SMA椭圆孔口板中应力诱发马氏体相变区的影响。数值结果表明:应力诱发马氏体相变首先发生在椭圆孔口长轴端点部位,在外加载荷作用下逐渐扩展到板内,并由内向外形成马氏体相区、相变混合区和奥氏体相区;SMA板内应力诱发马氏体完全相变区面积与施加外载荷成正相关,与温度成负相关;随着椭圆孔口长短轴之比增大,SMA板内应力诱发马氏体完全相变区面积呈现出先减小后增大的趋势;拉应力差值相同时,相较于拉应力沿椭圆孔口长轴方向较大的情况,当拉应力沿椭圆孔口短轴方向较大时,SMA板内完全相变区面积较大,椭圆孔口周边应力集中现象更明显。     

拉压不同弹性模量薄板上圆孔的应力分析

采用弹性理论研究了拉压不同弹性模量薄板上圆孔的孔边应力集中问题．采用广义虎克定律推导出了拉压不同弹性模量薄板上圆孔边的应力平衡方程,并联合利用应力函数及边界条件得到了拉压不同弹性模量薄板上圆孔边的应力表达式．算例分析表明,当薄板材料的拉压弹性模量相差较大时,采用经典弹性理论研究薄板上圆孔的孔边应力是不合适的,当经典弹性理论与拉压不同弹性模量弹性理论的计算结果间的差别超过工程允许误差5%时,应该采用拉压不同弹性模量弹性理论进行计算．     

A finite element investigation of the effect of crack tip constraint on hole growth under mode i and mixed mode loading

In this work, the effect of constraint on hole growth near a notch tip in a ductile material under mode I and mixed mode loading (involving modes I and II) is investigated. To this end, a 2-D plane strain, modified boundary layer formulation is employed in which the mixed mode elastic K–T field is prescribed as remote boundary conditions. A finite element procedure that accounts for finite deformations and rotations is used along with an appropriate version of J₂ flow theory of plasticity with small elastic strains. Several analyses are carried out corresponding to different values of T-stress and remote elastic mode-mixity. The interaction between the notch and hole is studied by examining the distribution of hydrostatic stress and equivalent plastic strain in the ligament between the notch tip and the hole, as well as the growth of the hole. The implications of the above results on ductile fracture initiation due to micro-void coalescence are discussed.     

A mechanical model on elastic cavitation in solid under hydrostatic tension

In this paper, a mechanical model is proposed to study cavitation in solids. The material is elastic and obeys Hook's law. Logarithm strain and Cauchy stress in nonlinear elasticity is used. A sphere with a central hole is loaded under hydrostatic tension on surface. In spherical symmetry, closed form solution is got. As radius of the hole tends to zero, a pitchfork bifurcation appears on load-displacement curve. The pitchfork bifurcation indicates that there is a cavitation at center of solid sphere as load reaches a critical value.     

一维正方准晶椭圆孔口平面弹性问题的解析解

利用复变方法,引入广义保角映射,研究了一维正方准晶中具有椭圆孔口的平面弹性问题,给出了各应力分量的复变表示,并在特殊情况下转化为Griffith裂纹,得到该裂纹尖端处的应力强度因子的解析解.当准晶体的对称性增加时,正方准晶椭圆孔口平面弹性问题退化为一维四方准晶中具有椭圆孔口的平面弹性问题,同样在特殊情况下转化为Griffith裂纹,得到裂纹尖端处的应力强度因子的解析解.     

Contact Transform for the Biharmonic Equation Applicable to Plane Stress Elastoplastic Elliptical Hole Problems

An analytical technique is developed that reduces the unknown elastic-plastic boundary of a linear elastic-perfectly plastic material containing an elliptical hole under tensile plane stress loading conditions into an equivalent mathematical problem with known boundaries. This mathematical transformation may facilitate this problem’s solution by either analytical or numerical means. Yield is assumed to occur in this analysis under the Tresca yield criterion. An example elastic-plastic problem illustrating this method is drawn from existing literature in the form of a perturbation solution for an elliptical hole derived by a series expansion about a circular boundary.     

Finite element analysis for steady-state hydride-induced fracture in metals by composite model

Delayed hydride cracking, which is observed in hydride-forming metals, due to the precipitation of hydrides near the crack tip, is investigated under conditions of constant temperature and crack velocity, plane strain and small-scale hydride-precipitation. The coupling of the operating physical processes of hydrogen-diffusion, hydride precipitation and material deformation is taken into account. The material is assumed to be an elastic composite made of hydrides and solid solution, with properties depending locally on the volume fraction of the hydrides. In the present analysis, the composite elastic properties have been derived by a generalized self consistent model for particulate composites. With respect to hydride-precipitation, two cases have been considered: (i) precipitation in a homogeneous medium with elastic properties, equal to the effective properties of the composite and (ii) precipitation in an inhomogeneous medium, where the expanding hydride has different elastic properties than those of the surrounding solid solution. The differences between the near-tip field distributions, produced by the two precipitation models, are relatively small. The effect of the hydrogen concentration far from the crack tip, on the near-tip field is also studied. It is shown that for small crack growth velocities, near the threshold stress intensity factor, the remote hydrogen concentration weakly affects the normalized stress distribution in the hydride-precipitation zone, which is controlled by the thermodynamically required hydrostatic stress, under hydrogen chemical equilibrium. However, for values of the applied stress intensity factor and the crack tip velocity, away from the threshold stress intensity factor and crack arrest, the effect of remote hydrogen concentration on the normalized near-tip stress field is strong. Reduction of the remote hydrogen concentration generally leads to reduction of the hydride-precipitation zone and increase of the near-tip stresses. Also reduction of the remote hydrogen concentration leads to distributions closer to those under hydrogen chemical equilibrium.     

复杂加载下混凝土的弹塑性本构模型

混凝土材料在不同应力路径下或复杂加载条件下会表现出差异性显著的应力应变关系,在小幅循环加载条件下,其应力应变关系会表现出类似于弹性变形的滞回曲线.在不同应力水平下,混凝土的应力应变关系以及破坏特性都具有静水压力相关特点,即随着静水压力增大,各向异性强度特性弱化.此外,混凝土受压及受拉破坏机理不同,因而对应于混凝土硬化损伤亦有不同,即可分为受压硬化损伤,受拉硬化损伤及两者的混合硬化损伤类型.基于Hsieh模型,对该模型进行了三点改进.（1）针对小幅循环加载下混凝土无塑性变形的试验规律,而模型中在应力水平较低的循环加载条件下始终存在塑性变形的预测问题,采用在边界面模型框架下,设置了应力空间的弹性域,初始屈服面与后续临界状态屈服面几何相似的假定.（2）基于广义非线性强度准则将原模型采用变换应力方法将其推广为三维弹塑性本构模型,采用变换后模型可合理的考虑不同应力路径对于子午面以及偏平面上静水压力效应形成的影响,并避免了边界面应力点奇异问题.（3）分别对拉压两种加载损伤模式建议了相应的硬化参数表达式,可分别用于描述上述加载中产生的应变软化及强度退化行为.基于多种加载路径模拟表明:所建立的三维弹塑性本构模型可合理地用于描述混凝土的一般应力应变关系特性.      

On neck propagation in amorphous glassy polymers under plane strain tension

Unlike metals, necking in polymers under tension does not lead to further localization of deformation, but to propagation of the neck along the specimen. Finite element analysis is used to numerically study necking and neck propagation in amorphous glassy polymers under plane strain tension during large strain plastic flow. The constitutive model used in the analyses features strain-rate, pressure, and temperature dependent yield, softening immediately after yield and subsequent orientational hardening with further plastic deformation. The latter is associated with distortion of the underlying molecular network structure of the material, and is modelled here by adopting a recently proposed network theory developed for rubber elasticity. Previous studies of necking instabilities have almost invariably employed idealized prismatic specimens; here, we explicitly account for the unavoidable grip sections of test specimens. The effects of initial imperfections, strain softening, orientation hardening, strain-rate as well as of specimen geometry and boundary conditions are discussed. The physical mechanisms for necking and neck propagation, in terms of our constitutive model, are discussed on the basis of a detailed parameter study.     

Stress concentration factor expression for tension strip with eccentric elliptical hole

Two explicit expressions of the stress concentration factor for a tension finite-width strip with a central elliptical hole and an eccentric elliptical hole, respectively, are formulated by using a semi-analytical and semi-empirical method. Accuracy of the results obtained from these expressions is better, and application scope is wider, than the results of Durelli’s photo-elastic experiment and Isida’s formula. When eccentricity of the elliptical hole is within a certain range, the error is less than 8%. Based on the relation between the stress concentration factor and the stress intensity factor, a stress intensity factor expression for tension strips with a center or an eccentric crack is derived with the obtained stress concentration factor expressions. Compared with the existing formulae and the finite element analysis, this stress intensity factor expression also has sufficient accuracy.     

A plane stress anisotropic plastic flow theory for orthotropic sheet metals

A phenomenological theory is presented for describing the anisotropic plastic flow of orthotropic polycrystalline aluminum sheet metals under plane stress. The theory uses a stress exponent, a rate-dependent effective flow strength function, and five anisotropic material functions to specify a flow potential, an associated flow rule of plastic strain rates, a flow rule of plastic spin, and an evolution law of isotropic hardening of a sheet metal. Each of the five anisotropic material functions may be represented by a truncated Fourier series based on the orthotropic symmetry of the sheet metal and their Fourier coefficients can be determined using experimental data obtained from uniaxial tension and equal biaxial tension tests. Depending on the number of uniaxial tension tests conducted, three models with various degrees of planar anisotropy are constructed based on the proposed plasticity theory for power-law strain hardening sheet metals. These models are applied successfully to describe the anisotropic plastic flow behavior of 10 commercial aluminum alloy sheet metals reported in the literature.     

高炉炉壳开孔位置对应力集中的影响

考虑到冶金行业中高炉炉壳开孔的实际情况，本文应用有限元分析的方法研究受远场均匀拉伸载荷，二维有限区域内菱形分布的圆孔中间不同开孔位置下的应力分布，得到最大应力集中系数随孔位置变化的三维变化曲面。此外，孔沿座标轴及沿原孔边缘位置变化对应力集中系数的影响被详细研究，从而为合理设计炉壳开孔提供了理论依据。     

A new yield function and a hydrostatic stress-controlled void nucleation model for porous solids with pressure-sensitive matrices

A macroscopic yield function for porous solids with pressure-sensitive matrices modeled by Coulomb's yield function was obtained by generalizing Gurson's yield function with consideration of the hydrostatic yield stress of a spherical thick-walled shell and by fitting the finite element results of the yield stresses of a voided cube. The macroscopic yield function is valid for the negative hydrostatic stress as well as for the positive hydrostatic stress. From the yield function, a plastic potential function for the porous solids was derived either for plastic normality flow or for plastic non-normality flow of the pressure-sensitive matrices. In addition, void nucleation was modeled by a normal distribution function with the macroscopic hydrostatic stress regarded as a controlling stress. This set of constitutive relations was implemented into a finite element code abaqus as a user material subroutine to analyze the cavitation and the deformation behavior of a rubber-modified epoxy around a crack tip under the Mode I plane strain conditions. By comparing the cavitation zone and the plastic zone obtained in the analysis with those observed in an experiment, the mean stress and the standard deviation for the void nucleation model could be determined. The cavitation and the deformation behavior of the rubber-modified epoxy were also analyzed around notches under four-point bending. The size and shape of the cavitation zone and the plastic zone were shown to be in good agreement with those observed in an experiment.     

Determining Stress–Strain in Rubber with In-Plane Biaxial Tensile Tester

The nominal stress–strain relationships of industrial rubber materials under multiaxial deformation are essential for precisely determining the constitutive laws of those materials. This paper proposes a new method for precisely estimating the nominal stress–stretch relationships of carbon-black-filled styrene butadiene rubbers (SBRs) under uniaxial tension, pure shear, and equibiaxial tension by using an in-plane biaxial tensile tester. The proposed method employs sheet-shaped rubber samples with notches for the pure-shear and equibiaxial tension tests to mitigate the influence of non-uniform deformation around the clamps. Finite element analysis and biaxial tensile tests were performed to verify the effectiveness of the proposed method. Performance evaluations based on both numerical calculations and experiments revealed that the proposed method enabled the precise calculation of the nominal stress–stretch relationship for uniform deformation from a tensile load and deformation of the reference area defined at the center of the samples.     

On the behavior of viscoelastic free jets with elliptical cross section

In [1,2] we develop a comprehensive theory of one space dimensional closure models (closed systems of 1-D equations for the unknown, or modal, variables) for free viscoelastic jets. These closure models are derived via asymptotics from the full 3-D boundary value problem under the conditions of a Von Kármán-like flow geometry, a Maxwell-Jeffreys constitutive model, elliptical free surface cross section, and a slender jet scaling. The focus of the present paper is to determine the consequences and predictions of the lowest order system of equations in this asymptotic analysis. For the special cases of elliptical inviscid and Newtonian free jets, subject to the effects of surface tension and gravity, our model predicts oscillation of the major axis of the free surface elliptical cross section between perpendicular directions with distance down the jet, and draw-down of the cross section, in agreement with observed behavior. In the absence of surface tension the transformation from a cross section with major axis in one direction to a cross section with major axis in the perpendicular direction occurs only once, in agreement with the observation of Taylor [4]. In viscoelastic regimes, our model predicts swell of the elliptical extrudate and distortion of the elliptical extrudate cross section from the dimensions of the die aperture.