Recoverable creep deformation and transient local stress concentration due to heterogeneous grain-boundary diffusion and sliding in polycrystalline solids

Numerical simulations are used to investigate the influence of heterogeneity in grain-boundary diffusivity and sliding resistance on the creep response of a polycrystal. We model a polycrystal as a two-dimensional assembly of elastic grains, separated by sharp grain boundaries. The crystal deforms plastically by stress driven mass transport along the grain boundaries, together with grain-boundary sliding. Heterogeneity is idealized by assigning each grain boundary one of two possible values of diffusivity and sliding viscosity. We compute steady state and transient creep rates as functions of the diffusivity mismatch and relative fractions of grain boundaries with fast and slow diffusion. In addition, our results show that under transient conditions, flux divergences develop at the intersection between grain boundaries with fast and slow diffusivity, which generate high local stress concentrations. The stress concentrations develop at a rate determined by the fast diffusion coefficient, and subsequently relax at a rate determined by the slow diffusion coefficient. The influence of the mismatch in diffusion coefficient, loading conditions, and material properties on the magnitude of this stress concentration is investigated in detail using a simple model problem with a planar grain boundary. The strain energy associated with these stress concentrations also makes a small fraction of the plastic strain due to diffusion and sliding recoverable on unloading. We discuss the implications of these results for conventional polycrystalline solids at high temperatures and for nanostructured materials where grain-boundary diffusion becomes one of the primary inelastic deformation mechanisms even at room temperature.     

The instability of the diffusion-controlled grain-boundary void in stressed solid

As atoms migrate along a void surface and grain-boundary, driven by various thermodynamic forces, the grain-boundary void changes its shape and volume. When the void changes its configuration, the free energy of the system also changes. In this article, the free energy is calculated for an evolving grain-boundary void filled with gas in a stressed solid. Then the instability conditions and the equilibrium shape of the void are determined as a function of the grain-boundary and surface energies, the void volume, the externally applied stresses, as well as the internal pressure built up by the gas filled in the void. The project supported by the National Natural Foundation of China (10272075 and 19972053)     

Diffusionally assisted grain-boundary sliding and viscoelasticity of polycrystals

Motivated by recent attenuation experiments on finely grained samples, we reanalyse the Raj-Ashby model of grain-boundary sliding. Two linearly elastic layers having finite thickness and identical elastic constants are separated by an interface (grain boundary) whose location is a given periodic function of position. Dissipation is confined to that interfacial region. It is caused by two mechanisms: a slip (boundary sliding) viscosity, and grain-boundary diffusion, with corresponding Maxwell relaxation times t_v and t_d. Owing to the assumption of a given, time-independent interface, the resulting boundary-value problem (b.v.p.) is linear and time-separable. The response to time-periodic forcing depends on angular frequency ω, on the ratio M=t_v/t_d of Maxwell times, and on the characteristic interface slope. The b.v.p. is solved using a perturbation method valid for small slopes. To relate features of the mechanical loss spectrum previously studied in isolation, we first discuss the solution as a function of M. Motivated by experiments, we then emphasize the case M?1 in which the relaxation times are widely separated. The loss spectrum then always has two major features: a frequency band 1?ωt_d?M^-1 within which the loss varies relatively weakly with ω; and a loss maximum at ωt_d∼M^-1 due to the slip viscosity. If corners on the interface are sufficiently rounded, those two universal features are separated by a third feature: between them, there is a strong minimum whose location is (entirely) independent of slip viscosity. The existence of that minimum has not previously been reported. These features are likely to occur even in solutions for finite interface slopes, because they are a consequence of the separation of timescales. The precise form of the spectrum in the weakly varying band must, however, be slope-dependent because it is controlled by stress singularities occurring at corners, and the strength of those singularities depends on the angle subtended by the corner.     

DYNAMICAL FORMATION OF CAVITY FOR COMPOSED THERMAL HYPERELASTIC SPHERES IN NONUNIFORM TEMPERATURE FIELDS

Dynamical formation and growth of cavity in a sphere composed of two incompressible thermal-hyperelastic Gent-Thomas materials were discussed under the case of a non-uniform temperature field and the surface dead loading. The mathematical model was first presented based on the dynamical theory of finite deformations. An exact differential relation between the void radius and surface load was obtained by using the variable transformation method. By numerical computation, critical loads and cavitation growth curves were obtained for different temperatures. The influence of the temperature and material parameters of the composed sphere on the void formation and growth was considered and compared with those for static analysis. The results show that the cavity occurs suddenly with a finite radius and its evolvement with time displays a non-linear periodic vibration and that the critical load decreases with the increase of temperature and also the dynamical critical load is lower than the static critical load under the same conditions.     

幂强化材料和超弹性材料组合球体中孔穴的动态生成

在简单加载条件下,研究幂强化材料和超弹性材料组合球体中的动态孔穴生成和增长问题,首先在有限变形动力学的框架下建立了相应的非线性数学模型,得到了应力的表达式,利用变量变换的方法求得了外加载荷和孔穴半径之间的一个精确的微分关系式,证明了当突加载荷超过其临界值时,球体内部有孔穴的突然生成,并随时间呈现非线性的周期振动．通过数值计算,分析了材料参数和球体的半径比对孔穴生成和增长的影响,并与相应的静态结果进行了比较．结果发现,惯性力的影响降低孔穴生成的临界载荷,而且材料的塑性对孔穴生成和增长有明显的影响．     

Simplified model for mass diffusivity estimation based on dynamic pendant drop volume analysis

We present a simplified correlation for calculating the dissolved gas moles in a pendant drop during the diffusion time, for several drop shapes. After this correlation is determined, the Yang and Gu (Ind Eng Chem Res 44:4474–4483, 2005) dynamic pendant drop volume analysis (DPDVA) method for calculation of mass diffusivity from the pendant drop volume variation against time can be used. We solved the differential equation in cylindrical coordinates for the mass transfer model of the gas diffusion into the liquid inside the pendant drop, using a different characteristic length (L_C), instead of the outer radius of the syringe needle (r_n) used in Yang and Gu (Ind Eng Chem Res 44:4474–4483, 2005) for defining the dimensionless variables. L_C is the relationship between the pendant drop volume and its mass transfer surface area at the initial conditions. The generalized correlation saves time, simplifies the method application and the deviations in the diffusion coefficient calculation respect to the complete Yang and Gu model are below 6%.     

Simulations of the spreading of a vesicle on a substrate surface mediated by receptor-ligand binding

A continuum model was introduced for the adhesion of vesicles to substrate surfaces. In the model, the vesicle membrane was assumed to be a closed shell with hyperelasticity. The vesicle cavity is filled with a liquid of fixed volume. The receptors on the membrane are mobile and initially uniformly distributed while the ligands on the substrate surface are fixed and also uniformly distributed. The formation of localized regions of tight binding between receptors and ligands, results in vesicle adhesion to the substrate surface. An adhesive model was introduced to describe the adhesive interaction between the receptors and the ligands. The growth of the adhesion area occurs via recruiting receptors from the non-adhered region through diffusion. Finite-element methods were used to solve the governing equations for the deformation of the vesicle and the receptor diffusion on the membrane surface. Effects of the membrane stiffness, the cohesive parameters and the receptor density on the adhesion kinetics of the vesicle were studied. In addition, the instability of the advancing front of the adhesion was also analyzed.     

Low-frequency axisymmetric waves in blood vessels of constant cross-section: an asymptotic approach

The asymptotic methods of shell theory are used to study the propagation of axisymmetric waves in blood vessels of constant cross-section. The initial equations are simplified using the assumption that the shell radius is small compared with the wave length. We show that the terms corresponding to the shell inertia cannot be omitted if it is required to describe not only the pressure wave but also the longitudinal wave. We study the influence of external fixation on the pressure wave. In this case, we compare the following two models: in the first model, the ambient medium is modelled by elastic and damping elements uniformly distributed over the shell exterior surface and by additional masses; in the second model, the ambient medium is represented by an infinite elastic space with a cylindrical cavity where the vessel is placed. On the interface between the elastic space and the vessel, we pose the full contact conditions. We show that, from the qualitative standpoint, both models lead to the same result: the pressure wave in the first approximation is a wave in the shell whose walls cannot move in the longitudinal direction. We asymptotically integrate the original equations and hence obtain a one-dimensional equation for the bulk blood flow.     

Diffusion interaction of growth steps in vacuum deposition

The interaction of two straight parallel growth steps under saturation conditions was studied using a modified Langmuir adsorption model. It is shown that as the growth steps approach each other, the rate of their motion decreases, which explains the effect of hindered coalescence of condensate islands during vacuum deposition with layer-by-layer filling of atomic layers. The cause of this phenomenon is found to be the overlap of surface diffusion layers which are a source of material for the growth steps. Features of surface diffusion and nucleation under conditions close to the saturation conditions are determined. It is shown that the nucleation of islands of a new atomic layer at supersaturation and the nucleation of islands of vacancies in the underlying layer at unsaturation are asymmetric. The results can be used to develop a technology to produce thin films by vacuum deposition.     

Complex variable solution for boundary value problem with X-shaped cavity in plane elasticity and its application

A new type of displacement pile, the X-section cast-in-place concrete(XCC)pile, has recently been developed in China. Extensive field tests and laboratory experiments are undertaken to evaluate its performance and quantify the non-uniform deformation effect(NUDE) of the X-shaped cross section during installation. This paper develops a simplified theoretical model that attempts to capture the NUDE. Based on the theory of complex variable plane elasticity, closed-form solutions of the stress and displacement for the X-shaped cavity boundary value problem are given. Subsequently, the analytical solution is used to evaluate the NUDE, the concrete filling index(CFI), and the perimeter reduction coefficient of the XCC pile cross section. The computed results are compared with field test results, showing reasonable agreement. The present simplified theoretical model reveals the deformation mechanism of the X-shaped cavity and facilitates application of the newly developed XCC pile technique in geotechnical engineering.     

Creep flow, diffusion, and electromigration in small scale interconnects

This paper proposes a three-dimensional electromigration model for void evolution in small scale interconnects. Concurrent kinetics of creep flow and surface diffusion as well as the effect of the surrounding material is considered to provide better understanding of the evolution process. The multiple kinetics and energetics are incorporated into a diffusive interface model. A semi-implicit Fourier spectral method and the preconditioned biconjugate-gradient method are proposed for the computations to achieve high efficiency and numerical stability. We systematically studied kinetic processes in diffusion dominated to creep dominated regime. Which process dominates, as revealed by the analysis, is determined by a combination of viscosity, mobility, interconnect thickness, and void radius. Previous studies on electromigration suggest that a circular void subjected to an electron wind force and surface diffusion is always stable against any small shape perturbation. Our simulations show that a shape that is stable in surface diffusion can become unstable in a creep dominated process, which leads to a quite different void morphology. A spherical void can evolve into a bowl shape and further break into smaller voids. It is also shown that the interconnect geometry has an important effect.     

Magma flow through elastic-walled dikes

A convection–diffusion model for the averaged flow of a viscous, incompressible magma through an elastic medium is considered. The magma flows through a dike from a magma reservoir to the Earth’s surface; only changes in dike width and velocity over large vertical length scales relative to the characteristic dike width are considered. The model emerges when nonlinear inertia terms in the momentum equation are neglected in a viscous, low-speed approximation of a magma flow model coupled to the elastic response of the rock.Stationary- and traveling-wave solutions are presented in which a Dirichlet condition is used at the magma chamber; and either a (i) free-boundary condition, (ii) Dirichlet condition, or (iii) choked-flow condition is used at the moving free or fixed-top boundary. A choked-flow boundary condition, generally used in the coupled elastic wave and magma flow model, is also used in the convection–diffusion model. The validity of this choked-flow condition is illustrated by comparing stationary flow solutions of the convection–diffusion and coupled elastic wave and magma flow model for parameter values estimated for the Tolbachik volcano region in Kamchatka, Russia. These free- and fixed-boundary solutions are subsequently explored in a conservative, local discontinuous Galerkin finite-element discretization. This method is advantageous for the accurate implementation of the choked flow and free-boundary conditions. It uses a mixed Eulerian–Lagrangian finite element with special infinite curvature basis function near the free boundary and ensures positivity of the mean aperture subject to a time-step restriction. We illustrate the model further by simulating magma flow through host rock of variable density, and magma flow that is quasi-periodic due to the growth and collapse of a lava dome.     

Cavitation in finite elasticity with surface energy effects

Cavity formation in incompressible as well as compressible isotropic hyperelastic materials under spherically symmetric loading is examined by accounting for the effect of surface energy. Equilibrium solutions describing cavity formation in an initially intact sphere are obtained explicitly for incompressible as well as slightly compressible neo-Hookean solids. The cavitating response is shown to depend on the asymptotic value of surface energy at unbounded cavity surface stretch. The energetically favorable equilibrium is identified for an incompressible neo-Hookean sphere in the case of prescribed dead-load traction, and for a slightly compressible neo-Hookean sphere in the case of prescribed surface displacement as well as prescribed dead-load traction. In the presence of surface energy effects, it becomes possible that the energetically favorable equilibrium jumps from an intact state to a cavitated state with a finite cavity radius, as the prescribed loading parameter passes a critical level. Such discontinuous cavitation characteristics are found to be highly dependent on the relative magnitude of the surface energy to the bulk strain energy.     

正向爆轰驱动高焓激波风洞的数值模拟

对充满氢氧可燃气体、带扩容腔的正向爆轰驱动的激波风洞进行了数值模拟。计算采用了欧拉方程,频散可控耗散差分格式（DCD）和改进的二阶段化学反应模型。在扩容腔附近采用二维轴对称计算模型,而在驱动段和被驱动段的直管道部分则采用一维计算模型。本文分析了爆轰波在管道中的传播、反射和绕射过程。计算结果表明扩容腔的尺寸对爆轰波的传播、反射、汇聚等起着决定性的作用;带扩容腔的正向爆轰驱动的激波风洞能够得到平稳的持续时间较长的气流,提高了实验的精确度和可重复性。     

悬索桥结构分析中的索-鞍座单元

为在悬索桥结构的有限元分析中真实、简洁、高效地模拟索鞍,本文建立了一类新的单元。新单元包括索段的一端固定在与其接触部分为单一半径圆弧的索鞍上,另一端分别位于索鞍两侧的两节点“左索-鞍座单元”和“右索-鞍座单元”,以及索段两端点分别位于索鞍两侧,中间一点固定于鞍座上的三节点“索-鞍座单元”,后者的鞍槽可为两不同半径圆弧的组合。根据要求的成桥状态几何参数确定结构的无应力状态时,可利用前二者进行悬索桥的单跨分析。新单元通过自动调整索与鞍座的脱离点而处于平衡状态,从而简化了计算。单元算法的推导基于有限元分析的基本原理和弹性悬链线的精确解,并利用了处于平衡状态时索与鞍座之间的内力关系。新单元可象常规单元一样直接用于成桥状态或施工过程中悬索桥结构的有限元分析。设计的算例验证了新单元的正确性,并举例说明了新单元在悬索桥结构分析中的应用。     

混凝土靶体计及压实效应的球腔动态膨胀模型和侵彻计算

建立了一混凝土靶体的球腔动态膨胀的侵彻模型。靶体的静水压力一体积应变关系考虑了不同的密实混凝土模型。也研究了混凝土作为多孔材料在高压下的压实效应。混凝土有拉伸强度极限值，其剪切强度与压力相关为Mohr—Coulomb定律。球腔动态膨胀模型考虑球对称空腔在无限介质里从零半径开始以常速度膨胀，产生一塑性区和一弹性区。当膨胀速度足够小时，由于介质的低拉伸强度，塑性区和弹性区之间可存在一径向裂纹区。可由解析方法获得空腔表面压力和弹塑性区界面传播速度随空腔表面扩张速度变化的曲线，再由球腔动态膨胀模型所得的空腔表面压力分布用于计算混凝土靶体的刚性杆弹侵彻深度。本文模型的计算结果与经验公式，不可压缩和线性可压缩混凝土模型的进行了比较。     

Analysis of cavitation problem of heated elastic composite ball

The cavitation problem of a composite ball under a uniform temperature is investigated, and the ball is composed of two elastic solid materials. The nonlinear mathematical model of the problem is established with the finite logarithmic strain measure for a large geometric deformation and by the Hooke law for elastic materials. The analytic solutions in a parametric form are derived for the thermal dilatation of the composite ball with a large elastic deformation. Solution curves are given to describe the variations of the critical temperature in the cavitation with the geometric and material parameters. The bifurcation curve is also given to reveal the cavity growth after void nucleation. The numerical results for a computational example indicate that the radius of the cavity will rapidly grow above the critical temperature, and the loop stress will become infinite when void nucleation. This means that the materials near the cavity will produce a plastic deformation leading to local failure and fracture if the material of the internal ball is elastoplastic. In addition, the cavitation of the composite ball appears at a low temperature if the elastic property of the material of the internal ball is nearly uncompressible.     

Modeling Chemotactic Waves in Saturated Porous Media using Adaptive Mesh Refinement

Bacterial transport is heavily influenced by chemical gradients and interfaces that exist in the subsurface. The main aim of this article is to describe a method of simulating the propagation of a traveling bacterial wave in a contaminated region and the resulting degradation of the contaminant. The presence of the chemotactic term and the relatively small bacterial diffusion means that the wave contains a very sharp wavefront. We, therefore, use an upwind conservative numerical scheme to obtain accurate and numerically stable solutions. The accuracy of the method is verified by comparisons with an exact one-dimensional solution of a simplified problem to give the same wavespeed. The method is then used to simulate the propagation of a realistic chemotactic wave in one dimension. We then use adaptive mesh refinement (AMR) to compute the propagation of chemotactic waves in two dimensions using the simplified model calibrated to give the same wavespeed as the full model.     

Ein Stoffgruppenmodell für die Berechnung der Diffusion in polynÄren Gasgemischen

A component group model for a simplified calculation of diffusion in multicomponent gaseous mixtures is developed. It is assumed, that the components of the mixture may be classified into groups in such way, that the binary diffusion coefficients between components of a group and also the binary diffusion coefficients between components of different groups may be approximated by averages. The application of the component group model is demonstrated for the mixture of dissociated air and for a 14-component H-C-N-O mixture. For many technically important gaseous mixtures a model with two or three groups of components yields a sufficient accurate approximation.