Irreversible deformation and subsequent unloading of a spherical elastoviscoplastic layer

Analytical solutions of a number of one-dimensional quasi-static problems that describe the processes of elastic deformation of the material of a hollow sphere and of generation and development of the plastic flow in this material with increasing pressure on the external boundary are presented. The process of unloading during slow removal of the loading pressure is studied. Stress fields, fields of elastic and plastic strains in the material of the spherical layer, the law of motion of the elastoplastic boundary, and residual stresses are determined. It is demonstrated that (in contrast to the ideal plasticity case) the allowance for the viscous properties of the material during its plastic flow eliminates the possibility of plastic flow emergence during unloading.     

Finite deformation analysis of crack tip fields in plastically compressible hardening–softening–hardening solids

Crack tip fields are calculated under plane strain small scale yielding conditions. The material is characterized by a finite strain elastic–viscoplastic constitutive relation with various hardening–softening–hardening hardness functions. Both plastically compressible and plastically incompressible solids are considered. Displacements corresponding to the isotropic linear elastic mode I crack field are prescribed on a remote boundary. The initial crack is taken to be a semi-circular notch and symmetry about the crack plane is imposed. Plastic compressibility is found to give an increased crack opening displacement for a given value of the applied loading. The plastic zone size and shape are found to depend on the plastic compressibility, but not much on whether material softening occurs near the crack tip.On the other hand, the near crack tip stress and deformation fields depend sensitively on whether or not material softening occurs. The combination of plastic compressibility and softening(or softening–hardening) has a particularly strong effect on the near crack tip stress and deformation fields.     

Crack-tip fields in steady crack-growth with linear strain-hardening

Singular stress and strain fields are found at the tip of a crack growing steadily and quasi-statically into an elastic-plastic strain-hardening material. The material is characterized byJ₂ flow theory together with a bilinear effective stress-strain curve. The cases of anti-plane shear, plane stress and plane strain are each considered. Numerical results are given for the order of the singularity, details of the stress and strain-rate fields, and the near-tip regions of plastic loading and elastic unloading.     

功能梯度材料的黏弹性断裂问题

功能梯度材料(FGM)是一种不同于传统复合材料的新型工程复合材料 [1], 国内外关于FGM的断裂力学方面的研究发展非常迅速. 关于FGM静态裂纹问题,学者们研究了不同类型裂纹尖端场的应力强度因子 [2-5], 探讨了有限长裂纹在不用载荷作用下的传播等问题. 而关于动态裂纹问题,也已经取得很大成就 [6-9]. FGM一个很重要的应用是高温结构材料,在强大的热环境中,很多材料都呈现出黏弹性. 因此,研究FGM的黏弹性断裂力学非常具有实际价值.对此,众多研究 [10-14]提出不同的分析模型,并在不同受载条件,通过理论计算,分析了黏弹性裂纹尖端场的力学 行为.本文考查了功能梯度材料板条中界面裂纹垂直于梯度方向时的黏弹性断裂问题,首先利用有限元法求解线弹性功能梯度材料板条的裂纹尖端场,然后根据黏弹性的对应性原理,求解出黏弹性功能梯度材料板条裂纹问题的应力场强度因子.      

Two-dimensional planar flow of a viscoelastic plastic medium

The present study is concerned with finite element simulation of the planar entry flow of a viscoelastic plastic medium exhibiting yield stress. The numerical scheme is based on the Galerkin formulation. Flow experiments are carried out on a carbon black filled rubber compound. Steady-state pressure drops are measured on two sets of contraction or expansion dies having different lengths and a constant contraction or expansion ratio of 4:1 with entrance angles of 90, 45 and 15 degrees. The predicted and measured pressure drops are compared. The predicted results indicate that expansion flow has always a higher pressure drop than contraction flow. This prediction is in agreement with experimental data only at low flow rates, but not at high flow rates. The latter disagreement is possibly an indication that the assumption of fully-developed flow in the upstream and downstream regions is not realistic at high flow rates, even for the large length-to-thickness ratio channels employed. The evolution of the velocity, shear stress, and normal stress fields in the contraction or expansion flow and the location of pseudo-yield surfaces are also calculated.     

A three-dimensional numerical study of mode I crack tip fields in pressure sensitive plastic solids

In this paper, a finite element study of 3D crack tip fields in pressure sensitive plastic solids (such as polymers or metallic glasses) under mode I, small scale yielding conditions is performed. The material is assumed to obey a small strain, Extended Drucker–Prager yield condition. The roles of pressure sensitive yielding, plastic dilatancy and yield locus shape on the 3D plastic zone development and near-crack front fields are systematically studied. It is found that while pressure sensitivity leads to a significant drop in the hydrostatic stress all along the 3D crack front, it enhances the plastic strain and crack opening displacements. The implications of these contrasting trends on ductile fracture processes are discussed in the light of some recent micro-mechanical simulations.     

黏弹性接触界面端附近的奇异应力场

研究蠕变加载条件下线黏弹性材料接触界面端附近的奇异应力场问题.考虑接触界面的摩擦,假设界面端的滑移方向不改变,相对滑移量微小,且其与位移同量级,由此线性化局部边界条件,根据对应原理得到Laplace变换域中的界面端应力场,导出时域中奇异应力场的卷积积分表达式.对卷积积分核函数进行数值反演,考虑接触材料的两类组合,一是持久模量具有量级上的差异,另一是持久模量接近相同.算例结果证实核函数可以用准弹性法求得的解析式较准确地近似.在此基础上,利用积分中值定理,并引入各应力分量的修正系数,得到黏弹性奇异应力场的简化式.结合核函数的数值反演结果分析修正系数表达式的取值范围,得到如下结论,若两相接触材料的持久模量相差很大,可以采用准弹性解的解析式较准确地描述界面端的奇异应力场;一般情况下,应力场不存在统一的奇异值和应力强度系数,当采用类似于准弹性解的表达式近似给出黏弹性应力场时,可以估计此近似描述的误差限.文中最后采用有限元分析黏弹性板端部嵌入部位的应力场,算例包括了黏弹性板与弹性金属支承、黏弹性板与黏弹性垫层所形成的滑移接触界面端,利用黏弹性有限元的数值结果验证理论分析所得结论的有效性.      

A two-dimensional dislocation dynamics model of the plastic deformation of polycrystalline metals

Two-dimensional dislocation dynamics (2D-DD) simulations under fully periodic boundary conditions are employed to study the relation between microstructure and strength of a material. The material is modeled as an elastic continuum that contains a defect microstructure consisting of a preexisting dislocation population, dislocation sources, and grain boundaries. The mechanical response of such a material is tested by uniaxially loading it up to a certain stress and allowing it to relax until the strain rate falls below a threshold. The total plastic strain obtained for a certain stress level yields the quasi-static stress-strain curve of the material. Besides assuming Frank-Read-like dislocation sources, we also investigate the influence of a pre-existing dislocation density on the flow stress of the model material. Our results show that - despite its inherent simplifications - the 2D-DD model yields material behavior that is consistent with the classical theories of Taylor and Hall-Petch. Consequently, if set up in a proper way, these models are suited to study plastic deformation of polycrystalline materials.     

A plasticity model for pressure-dependent anisotropic cellular solids

The initial and subsequent yield surfaces for an anisotropic and pressure-dependent 2D stochastic cellular material, which represents solid foams, are investigated under biaxial loading using finite element analysis. Scalar measures of stress and strain, namely characteristic stress and characteristic strain, are used to describe the constitutive response of cellular material along various stress paths. The coupling between loading path and strain hardening is then investigated in characteristic stress–strain domain. The nature of the flow rule that best describes the plastic flow of cellular solid is also investigated. An incremental plasticity framework is proposed to describe the pressure-dependent plastic flow of 2D stochastic cellular solids. The proposed plasticity framework adopts the anisotropic and pressure-dependent yield function recently introduced by Alkhader and Vural [Alkhader M., Vural M., 2009a. An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations. J. Mech. Phys. Solids 57(5), 871–890]. It has been shown that the proposed yield function can be simply calibrated using elastic constants and flow stresses under uniaixal loading. Comparison of stress fields predicted by continuum plasticity model to the ones obtained from FE analysis shows good agreement for the range of loading paths and strains investigated.     

A finite element analysis of laminar unsteady flow of viscoelastic fluids through channels with non-uniform cross-sections

The effects of non-Newtonian behaviour of a fluid and unsteadiness on flow in a channel with non-uniform cross-section have been investigated. The rheological behaviour of the fluid is assumed to be described by the constitutive equation of a viscoelastic fluid obeying the Oldroyd-B model. The finite element method is used to analyse the flow. The novel features of the present method are the adoption of the velocity correction technique for the momentum equations and of the two-step explicit scheme for the extra stress equations. This approach makes the computational scheme simple in algorithmic structure, which therefore implies that the present technique is capable of handling large-scale problems. The scheme is completed by the introduction of balancing tensor diffusivity (wherever necessary) in the momentum equations. It is important to mention that the proper boundary condition for pressure (at the outlet) has been developed to solve the pressure Poisson equation, and then the results for velocity, pressure and extra stress fields have been computed for different values of the Weissenberg number, viscosity due to elasticity, etc. Finally, it is pertinent to point out that the present numerical scheme, along with the proper boundary condition for pressure developed here, demonstrates its versatility and suitability for analysing the unsteady flow of viscoelastic fluid through a channel with non-uniform cross-section.     

金属材料在复杂应力状态下的塑性流动特性及本构模型

工程应用中,金属材料和结构往往处于复杂应力状态。材料的塑性行为会受到应力状态的影响,要精确描述材料在复杂应力状态下的塑性流动行为,必须在本构模型中考虑应力状态效应的影响。然而,由于在动态加载下材料的应变率效应和应力状态效应相互耦合、难以分离,给应力状态效应的研究和模型的建立造成很大困难。通过对Ti-6Al-4V钛合金材料开展不同加载条件下的力学性能测试,提出了一个包含应力三轴度和罗德角参数影响的新型本构模型,并通过VUMAT用户子程序嵌入ABAQUS/Explicit软件。分别采用新提出的塑性模型和Johnson-Cook模型对压剪复合试样的动态实验进行了数值模拟。结果表明,新模型不仅在对材料本构曲线的拟合方面具有较强的优势,而且由该模型所得到的透射脉冲和载荷-位移曲线均更加准确。因此,该模型能够更精确地描述和预测金属材料在复杂应力状态下的塑性流变行为。     

Effect of yield surface vertex on crack-tip fields in mode III

An asymptotic crack-tip analysis of stress and strain fields is carried out for an antiplane shear crack (Mode III) based on a corner theory of plasticity. Because of the nonproportional loading history experienced by a material element near the crack tip in stable crack growth, classical flow theory may predict an overly stiff response of the elastic plastic solid, as is the case in plastic buckling problems. The corner theory used here accounts for this anomalous behavior. The results are compared with those of a similar analysis based on the J₂ flow theory of plasticity.     

Modeling the effects of material non-linearity using moving window micromechanics

In the analysis of materials with random heterogeneous microstructure the assumption is often made that material behavior can be represented by homogenized or effective properties. While this assumption yields accurate results for the bulk behavior of composite materials, it ignores the effects of the random microstructure. The spatial variations in these microstructures can focus, initiate and propagate localized non-linear behavior, subsequent damage and failure. In previous work a computational method, moving window micromechanics (MW), was used to capture microstructural detail and characterize the variability of the local and global elastic response. Digital images of material microstructure described the microstructure and a local micromechanical analysis was used to generate spatially varying material property fields. The strengths of this approach are that the material property fields can be consistently developed from digital images of real microstructures, they are easy to import into finite element models (FE) using regular grids, and their statistical characterizations can provide the basis for simulations further characterizing stochastic response. In this work, the moving window micromechanics technique was used to generate material property fields characterizing the non-linear behavior of random materials under plastic yielding; specifically yield stress and hardening slope, post yield. The complete set of material property fields were input into FE models of uniaxial loading. Global stress strain curves from the FE–MW model were compared to a more traditional micromechanics model, the generalized method of cells. Local plastic strain and local stress fields were produced which correlate well to the microstructure. The FE–MW method qualitatively captures the inelastic behavior, based on a non-linear flow rule, of the sample continuous fiber composites in transverse uniaxial loading.     

金属切屑塑性流动的稳定性

对金属正交切削过程中切屑形成机制和材料塑性流动行为进行实验研究和理论分析. 通过对4 种常用金属材料正交切削过程的实验研究和切屑形貌的微观观察,确定了连续切屑转变成锯齿切屑的临界速度. 结果表明该临界速度与材料性能相关. 在实验观察基础上,提出描述材料正交切削过程的二维分析模型. 该模型假设切屑形成区为包括主剪切区和次剪切区的一个平行四边形. 载荷有主剪切区中的剪应力和次剪切区中的正压力;通过量纲分析得到描述材料正交切削过程的无量纲主控参数和无量纲形式的基本控制方程;应用线性稳定性分析方法建立平面应变状态下评价材料塑性流动稳定性的普遍准则;求得切屑形成区内材料塑性变形的速度和应力近似解. 讨论切屑形成、形貌转变以及相关的塑性失稳机制. 分析结果表明, 表征材料惯性与阻尼之比的无量纲参数— 雷诺数可以作为主控参数描述金属切削过程以及切屑材料塑性流动的稳定性.      

Experimental Energy Balance During the First Cycles of Cyclically Loaded Specimens Under the Conventional Yield Stress

This paper, as an extension of Maquin and Pierron (Mech Mater 41(8):928–942, 2009), presents an experimental procedure developed to macroscopically estimate the energy balance during the very first cycles of a uniaxially loaded metallic specimen at low stress levels. This energy balance is performed by simultaneously measuring the plastic input energy using a load cell and a strain gauge, and the dissipative energy using the temperature field provided by an infrared camera. Some experimental limitations led to restrain the present procedure to positive stress ratios, and to complement this energy balance by a second measurement while the material plastic work per cycle is negligible compared to the dissipative energy. Some results obtained on a cold rolled low carbon steel specimen are presented. First, a sensitivity study is undertaken to precisely determine the detection threshold on both thermal and plastic energies. Then, after having verified the homogeneity of the dissipative source fields, energy balances have been performed at different stress levels. It was thus confirmed that the slow variations of the dissipative sources occurring during the first cycles are due to micro-plastic adaptation, and that the dissipative sources remaining after some hundreds of cycles are due to viscoelastic (internal friction) phenomena. This procedure provides a better understanding of dissipation based approaches to fatigue found in the literature and an advanced tool to study viscoelastic phenomena in uniaxial loading.     

Phenomenological modeling of the response of a dense colloidal suspension under dynamic squeezing flow

The split Hopkinson pressure bar experimental technique is used to evaluate the squeezing flow response of a concentrated, discontinuously thickening colloidal suspension of spherical silica particles loaded at high stresses/strain rates. These results provide insight into the transitional behavior of these materials, as well as the post-transitional response under compressive loading. A method of analyzing the strain and strain rate dependent behavior is presented to identify modes of material response (viscous, elastic, etc.). Experimental results are presented as stress–strain–strain rate plots and a surface fitting approach is used to develop a phenomenological model describing the overall response. From this model, it is possible to identify regions of elastic and viscous behavior using a gradient analysis approach. It was found that, after an initial period of viscous deformation, the suspension behaves like a viscoelastic material – this regime corresponds well with transition in which large clusters of particles percolate. This is followed by a third, viscous regime in which the material undergoes viscous deformation. At the highest stresses, a plateau region of plastic deformation has been identified. This approach and the conditions under which it may be applied are described in detail in the paper.     

The transmission of turbulent boundary layer unsteady pressure and shear stress through a viscoelastic layer

The transmission of unsteady pressure and shear stress, generated by a turbulent boundary layer in water, through a viscoelastic layer backed by a rigid plate is investigated. Analytical models are used to estimate the unsteady pressure and shear stress from 10 to 1000 Hz for a flat plate boundary layer with zero pressure gradient. Additionally, models for the transfer of the unsteady pressures and shear stress through the viscoelastic layer are developed. The models are used to predict the unsteady pressure fluctuations, or flow noise, which would be seen by a finite size sensor embedded under the elastomer layer. The unsteady pressure levels are found to be 20 dB greater than the unsteady shear stress levels across all frequency ranges computed, in agreement with recent measurements. The unsteady pressure transfer functions have a peak at the shear wavenumber and are larger than the shear stress transfer magnitudes from 10 to 50 Hz. The unsteady shear stress transfer functions have a peak at the acoustic wavenumber and are larger than the pressure transfer magnitudes from 50 to 1000 Hz. Over the frequency range examined, the unsteady pressures were found to be the dominant contributor to the sensor flow noise due to the considerably larger magnitude of the unsteady pressures on the top of the viscoelastic layer.     

The yield surface of viscoelastic and plastic fluids in a vane viscometer

Vane viscometers are often used to investigate the low shear rate properties of plastic fluids. The shear stress is determined by assuming that the material is held in the space between the vane blades so that it behaves like a rigid cylinder. Experimental evidence supports this assumption and the aim of the present study is to model numerically the yield process in a vane rheometer using viscoelastic and plastic fluids. The finite element method has been used to model the behavior of Herschel-Bulkley (Bingham), Casson and viscoelastic (Maxwell type) fluids. The penalty function approach for the pressure approximation and a rotating reference frame are used together with fine meshes containing more than 1300 elements. The results show that for Herschel-Bulkley (Bingham), and Casson fluids a rotating rigid cylinder of fluid is trapped inside the periphery of the vane, the shear stress is uniformly distributed over the surface of the cylinder. Finally a modified second order fluid is used to simulate the viscoelastic behaviour, anticipated to be an intermediate between the elastic deformation and the plastic flow, to provide a more realistic simulation of the yield process about a vane. In this case, contrast with the concentration of the elastic strain rate at the blade tips, a nearly uniform distribution of the plastic shear rate is still found. This implies that the plastic shear always distributes uniformly during the entire yielding process. Evidently the assumption of uniform shear on a rotating cylinder of material occluded in the blades of a vane is a valid and useful model for many types of fluid possessing a yield stress.     

Computation of viscoelastic cable coating flows

A viscoelastic analysis is presented for model tube tooling, draw-down and combined geometry flows encountered in the cable coating industries. The work investigates the development of stress fields and studies the effect of varying entry flow stress boundary conditions. The analysis takes into account tube tooling and draw-down flow sections individually, and in combination. The flow behaviour of cable-coating grade low density polyethylene is studied assuming a viscoelastic, isothermal flow, and employing a Taylor–Petrov–Galerkin finite element scheme with an exponential Phan-Thien–Tanner constitutive model. © 1998 John Wiley & Sons, Ltd.     

Strain-hardening extrusion—II. Analysis of slip-line fields based on experimental flow patterns

A method of constructing slip-line fields based on experimental extrusion flow patterns was described in a previous paper. Such fields provide a basis for constructing representative solutions of strain-hardening flow problems. In this paper, after proving some theoretical results on the structure of such solutions, a method of analyzing slip-line fields is described and some typical results presented. The analysis comprises computation of the velocity solution and construction from it of a set of streamlines for comparison with the original flow pattern; computation of the total strain by integration along the streamlines, its graduation and adjustment to meet the boundary conditions; estimation of the stress field (shear stress k and mean pressure p) and check for consistency; and computation of the forces on the boundary of the plastic region and their resultant.