弹性-应变软化粘塑性材料反平面剪切动态扩展裂纹尖端的渐近解*

本文首先给出了一种用于描述材料软化,并存在有粘塑性的材料模型.用这种模型对反平面剪切型动态扩展状态下,裂纹尖端的弹粘塑性场进行了渐近分析,给出了弹性－应变软化粘塑性材料反平面剪切动态扩展裂纹尖端的渐近解方程.分析结果表明,在裂纹尖端应变具有(ln(R/r))1/(n＋1)的奇异性,应力具有(ln(R/r))-n/(n＋1)的奇异性.从而本文揭示了应变软化粘塑性材料反平面剪切动态扩展裂纹尖端的渐近行为.     

简单剪切有限塑性内时理论分析

在有限塑性内时理论中引入Jaumann率、广义Jaumann率、扶率及Wu率,并以此分析了简单剪切大变形问题．结果验证了简单剪切变形中,采用次弹性或内时刚塑性材料的Jaumann率客观模型,随单调递增的剪切变形剪切应力和法向应力都会出现振荡现象．这说明振荡现象的出现不取决于弹塑性模型,而与选取不同的客观率有很大的关系．同时指出在简单剪切大变形时,法向应力并不为零．     

Constitutive Modelling of Resins in the Compliance Domain

A rheological HWKK/H model for resins is developed taking into consideration the up-to-date analyses of experimental results. Constitutive compliance equations of linear are formulated for this model in the shear/bulk form, which describes, among other things, the first-rank reversible isothermal creep. The shear (distorsional) deformations are simulated with three independent stress history functions of fractional and normal exponential types. The volume deformations are simulated as perfectly elastic. The model is described by two elastic and six viscoelastic constants, namely three long-term creep coefficients and three retardation times.The constitutive compliance equations of viscoealsticity for resins are also formulated in the coupled form. Formulae for converting the constants of shear/bulk (uncoupled) viscoelasticity into the constants of coupled viscoelasticity are given too.An algorithm for identifying the material constants, based on the creep of uniaxially tensioned bar samples, is formulated in a way that gives unique results. The material constants are fiund for Epidian 53 epoxy and Polimal 109 polyester resins. The creep processes, simulated based on the experimental data, are presented graphically for both the resins examined.     

中医滚法推拿对血液流动影响的数值研究

研究了中医滚法推拿的血液动力学机制．用狭窄轴向运动来模拟滚法推拿,通过轴对称非线性模型和含网格重分算法的任意欧拉-拉格朗日有限元方法研究狭窄轴向运动的轴对称刚性管中的粘性流动．流量和管壁切应力通过数值求解Navier-Stokes方程得到．数值结果表明,狭窄运动的频率,也就是滚法推拿的频率对流量和管壁切应力有很大的扰动作用．滚法推拿中另一个可变参数——刻划狭窄严重程度的狭窄度,对流量和管壁切应力同样表现出显著影响．这些数值结果可以为推拿的临床应用提供一些值得参考的数据．     

Falling film flow of a viscoelastic fluid along a wall

In this paper, we study the heat transfer in the fully developed flow of a viscoelastic fluid, a slag layer, down a vertical wall. A new constitutive relation for the stress tensor of this fluid is proposed, where the viscosity depends on the volume fraction, temperature, and shear rate. For the heat flux vector, we assume the Fourier's law of conduction with a constant thermal conductivity. The model is also capable of exhibiting normal stress effects. The governing equations are non‐dimensionalized and numerically solved to study the effects of various dimensionless parameters on the velocity, temperature, and volume fraction. The effect of the exponent in the Reynolds viscosity model is also discussed. The different cases of shear‐thinning and shear‐thickening, cooling and heating, are compared and discussed. The results indicate that the viscous dissipation and radiation (at the free surface) cause the temperature to be higher inside the flow domain. Copyright © 2013 John Wiley & Sons, Ltd.     

Quadruple Trigonometrical Series Equations and Their Application to an Inclusion Problem in the Theory of Elasticity

Closed form solution of quadruple series equations involving cosine kernels has been obtained by reducing the series equations into triple Abel's type integral equations which in turn are reduced to a single integral equation. Making use of finite Hilbert transforms the solution of the single integral equation is obtained in closed form. This solution is used to solve an electrostatic problem. The results of this paper have also been used in a two-dimensional elastostatic problem under anti-plane shear and the effect of rigid line inclusions with thickness on the Griffith cracks has been examined. The expressions for shear stress and stress intensity factor at the tip of the crack are obtained. Finally, some numerical results for the stress intensity factor and shear stress distribution are obtained.     

Micro-mechanical analyses of the effect of stress triaxiality and the Lode parameter on ductile damage and failure

The paper deals with the effect of different stress states on plastic deformations, damage and fracture of ductile materials. To be able to model these effects a continuum damage model has been introduced taking into account the dependence of stress-state on the constitutive equations. The model is based on the introduction of damaged and fictitious undamaged configurations. All parameters appearing in the constitutive equations are stress-state-dependent which can be characterized by the stress intensity, the stress triaxiality and the Lode parameter. Only experiments are not adequate enough to determine all constitutive parameters. Thus, additional series of three-dimensional micro-mechanical simulations of representative volume elements have been performed to get more insight in the complex damage mechanisms. These simulations cover a wide range of stress triaxialities and Lode parameters in tension, shear and compression domains. After all, the results from the micro-mechanical simulations are used to suggest the damage equations and to identify corresponding parameters. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-layer flows of generalized immiscible second grade fluids in a rectangular channel

Unsteady one-dimensional flows of two incompressible and immiscible generalized second grade fluids in a rectangular channel are studied. A constant pressure gradient acts in the flow direction, while the channel walls have oscillating translational motions in their planes. The generalization considered in this paper consists into a mathematical model based on constitutive equations of second grade fluid with Caputo time-fractional derivative in which the history of the shear stress influences the velocity gradient. The velocity and shear stress fields in the Laplace transform domain are obtained. Numerical solutions for the real velocity and shear stress have been found by employing the Stehfest numerical algorithm for the inverse Laplace transform. The influence of the fractional parameters on the velocity and shear stress has been studied by numerical simulations and graphical illustrations. It is found that the memory effects are significant only for small values of the time t.     

Constitutive equations of non-isothermal polymer melt

Constitutive equations of non-isothermal polymer melt are presented by the analysis of entropic free energy contribution of the macromolecular chains, which are treated as elastic dumbbell models. With describing non-isothermal dumbbell spring, as the function of temperature, the non-linear elastic coefficient expression causes the appearance of temperature gradient in stress constitutive equations. Following the constitutive equation of Hookean dumbbell model, non-isothermal stress constitutive equations of FENE and FENE-P models are derived. In deriving process of constitutive equations, the second moment approximation is used to closure FENE model. Using the non-isothermal constitutive equations, numerical simulations of polymer flow through shear cavity and planar contraction cavity are presented. And the distributions of correlative stress functions and the effects of different temperatures on stress functions are discussed. The present results are shown to explore the non-isothermal constitutive equations of elastic dumbbell models, and to search more accurately describing way of non-isothermal polymer melt.     

考虑非局部剪切效应的碳纳米管弯曲特性研究

基于Hamilton(哈密顿)变分原理和非局部连续介质弹性理论,建立了新型非局部Timoshenko(铁木辛柯)梁模型（ANT）,推导了碳纳米管（CNT）的ANT弯曲平衡方程以及两端简支梁、悬臂梁和简支 固定梁的边界条件表达式,分析了剪切变形效应和非局部微观尺度效应对碳纳米管弯曲特性的影响.数值计算结果显示,碳纳米管的弯曲刚度随着小尺度效应的增强而升高.其次,这种小尺度效应对自由端受集中力的悬臂梁碳纳米管有明显作用,其刚度变化规律和其它约束条件的碳纳米管一样,这一点是ANT模型区别于普通非局部纳米梁模型的主要特点.经分子动力学模拟验证,ANT模型是合理分析碳纳米管力学特性的有效方法.     

Nonlinear flexural vibrations of composite shallow open shells

This paper addresses geometrically nonlinear flexural vibrations of open doubly curved shallow shells composed of three thick isotropic layers. The layers are perfectly bonded, and thickness and linear elastic properties of the outer layers are symmetrically arranged with respect to the middle surface. The outer layers and the central layer may exhibit extremely different elastic moduli with a common Poisson's ratio ν. The considered shell structures of polygonal planform are hard hinged supported with the edges fully restraint against displacements in any direction. The kinematic field equations are formulated by layerwise application of a first order shear deformation theory. A modification of Berger's theory is employed to model the nonlinear characteristics of the structural response. The continuity of the transverse shear stress across the interfaces is specified according to Hooke's law, and subsequently the equations of motion of this higher order problem can be derived in analogy to a homogeneous single-layer shear deformable shallow shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

主动脉弓及分支血管内非稳态血流分析

运用流体力学中的三维非定常Navier-Stokes方程作为血液流动的控制方程,并采用计算流体力学方法对人体主动脉弓及分支血管内非Newton(牛顿)血液黏度模型下血流进行瞬态数值模拟.分析了一个心动周期内不同时刻血流动力学特征参数的分布对动脉粥样硬化斑块形成的影响,并与Newton血液黏度模型下的血管壁面压力和壁面切应力特征参数进行对比.结果表明:与Newton血液模型相比,非Newton血液模型下血流分布更符合真实血流特性;在心动收缩期,分支血管外侧壁附近存在面积较大的低速涡流区,该区域内血管壁面压力与壁面切应力具有较大的变化量,血液中的血小板、脂质和纤维蛋白等易沉积,血管内壁易疲劳损伤并发生血管重构,促使动脉粥样硬化斑块形成;而在心动舒张期,分支血管内血流速度分布均匀,血管壁面压力与壁面切应力变化量较小,血管壁受到较小的应力作用,对动脉粥样硬化斑块形成的作用较小.     

A geometrically nonlinear (3,2) reduced degree-of-freedom unified zigzag laminated beam element for large deformation analysis

A geometrically nonlinear (3,2) unified zigzag beam element is developed with a reduced number of degree-of-freedom for the large deformation analysis. The main merit of the beam element model is the Kirchhoff and Cauchy shear stress solution for large deformation and large strain analysis is more accurate. The geometrically nonlinearity is considered in the calculation of the zigzag coefficients. Thus, the results of shear Cauchy stress are matching well with solid element analysis in case of the beam with aspect ratio greater than 20 under large deformation. The zigzag coefficients are derived explicitly. The Green strain and the second Piola Kirchhoff stress are used. The second Piola Kirchhoff shear stress is continuous at the interface between adjacent layers priori. The bottom surface second Piola Kirchhoff shear stress condition is used to determine the zigzag coefficient and the top surface second Piola Kirchhoff shear stress condition is used to reduce one degree-of-freedom. The nonlinear finite element equations are derived. In the numerical tests, several benchmark problems with large deformation are solved to verify the accuracy. It is observed that the proposed beam has accurate solution for beam with aspect ratio greater than 20. The second Piola Kirchhoff and Cauchy shear stress accuracy is also good. A convergence study is also presented.     

Identification of the Material Parameters of a Micropolar Model for Granular Materials

Granular frictional materials show a complex stress‐strain behaviour depending on the stress state and the load history. Furthermore, biaxial experiments exhibit the occurrence of shear band phenomena as the result of the localization of plastic strains. It is well known that the onset of shear bands is associated with microrotations of the granular microstructure, which has a significant influence on the macroscopic behaviour. Consequently, the macroscopic material must result in a micropolar model, which incorporates rotational degrees of freedom. After the formulation of the constitutive equations and the numerical implementation, it is necessary to determine all required material parameters. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling thermoplastic material behaviour of dual-phase steels on a microscopic length scale - numerical treatment

On a microscopic length scale dual-phase steels exhibit a polycrystalline microstructure consisting of ferrite and martensite. In this work it is assumed that the martensitic phase behaves purely thermoelastic while for the ferritic phase a thermoplastic material model was developed based on the assumption that the driving mechanism for persistent deformation is the movement of dislocations on preferred planes in preferred directions. The necessary shear stress to move dislocations at a certain temperature and deformation rate is assumed to possess contributions from the atomic lattice, alloying atoms and the dislocation structure. To consider the influence of the dislocation structure, dislocation densities are introduced as state variables for which temperature and deformation rate dependent evolution equations are formulated. Since for general loading histories the model equations cannot be integrated analytically, a time discretized form of the model equations with an appropriate solution algorithm is presented. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

反平面塑性V形切口尖端应力和位移渐近解

提出了一种确定幂硬化材料反平面V形切口尖端应力和位移渐近解的主导项和高阶项的有效方法。首先通过在弹塑性理论基本方程中引入V形切口尖端应力场和位移场的渐近级数展开,建立以应力和位移为特征函数的非线性和线性常微分方程组。然后采用插值矩阵法求解常微分方程组,可得到多阶应力特征指数和其相对应的特征函数。该方法具有通用性强、精度高等优点,可处理任意开口角度和应变硬化指数的V形切口。典型算例验证了该方法的准确性和有效性。     

The steady axisymmetric flow of ideally plastic materials in a conical channel

The problem of the asymmetric flow of an ideally plastic medium is formulated within the framework of the von Mises model and the total plasticity condition, using the invariant condition of compatibility for the deviator component of the stress tensor. Flow in a converging conical channel, on the boundary of which the shear stresses are specified, is considered. First-order differential equations are obtained, describing the shear-stress distribution in the moving medium, one of which corresponds to the von Mises model, and the other to the total plasticity condition. It is established from an analysis of the solution in the neighbourhood of singular points, that the minus sign in front of the radical in these equations corresponds to positive shear stresses and vice versa. The problem of the shear stresses reaching a maximum value on the specified boundary surface of the channel is investigated. The aperture angle of the channel, beginning from which this value is reached, is determined. It is established that the value of the angle, following from the total plasticity condition, somewhat exceeds its value obtained within the framework of the von Mises model.     

Pulsatile flow of Herschel–Bulkley fluid through catheterized arteries – A mathematical model

The pulsatile flow of blood through catheterized artery has been studied in this paper by modeling blood as Herschel–Bulkley fluid and the catheter and artery as rigid coaxial circular cylinders. The Herschel–Bulkley fluid has two parameters, the yield stress θ and the power index n. Perturbation method is used to solve the resulting quasi-steady nonlinear coupled implicit system of differential equations. The effects of catheterization and non-Newtonian nature of blood on yield plane locations, velocity, flow rate, wall shear stress and longitudinal impedance of the artery are discussed. The existence of two yield plane locations is investigated and their dependence on yield stress θ, amplitude A, and time t are analyzed. The width of the plug core region increases with increasing value of yield stress at any time. The velocity and flow rate decrease, whereas wall shear stress and longitudinal impedance increase for increasing value of yield stress with other parameters held fixed. On the other hand, the velocity, flow rate and wall shear stress decrease but resistance to flow increases as the catheter radius ratio (ratio of catheter radius to vessel radius) increases with other parameters fixed. The results for power law fluid, Newtonian fluid and Bingham fluid are obtained as special cases from this model.     

The effective shear modulus for an n-layered composite sphere

This work presents the derivation of the effective shear modulus for a heterogeneous material composed of multi-layered composite spheres embedded in a linear elastic matrix. It is based on the composite spheres model known from the literature. In contrast to previous publications the effective shear modulus is obtained by equating the results of two models: In the first model, a heterogeneous sphere is embedded in an equivalent homogeneous material, whereas in the second model, the heterogeneous sphere is replaced by an equivalent homogeneous sphere. In the context of both, a shear stress approach and a shear deformation approach, this results into an overdetermined system of equations which is solved with the least squares method. In a numerical study our results are compared to effective moduli and bounds from the literature. Furthermore, a convincing agreement with experimental data for glass microspheres embedded in a polyester matrix is demonstrated. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of blood flow through a modelled artery with an aneurysm

The intention of the present work is to carry out a systematic analysis of flow features in a tube, modelled as artery, having a local aneurysm in presence of haematocrit. The arterial model is treated to be axi-symmetric and rigid. The blood, flowing through the modelled artery, is treated to be Newtonian and non-homogeneous. For a thorough quantitative analysis of the flow characteristics such as wall pressure, flow velocity, wall shear stress, the unsteady incompressible Navier-Stokes equations in cylindrical polar co-ordinates under the laminar flow conditions are solved by using the finite-difference method. Finally, the numerical illustrations presented in this paper provide an effective measure to estimate the combined influence of haematocrit and aneurysm on flow characteristics. It is found that the magnitude of wall shear stress and also the length of separation increase with increasing values of the haematocrit parameter. The length of flow separation increases but the peak value of wall shear stress decreases with the increasing length of aneurysm. The peak value of wall shear stress as well as the length of separation increases with the increasing height of the aneurysm.