Size-dependent yield strength of thin films

Biaxial strain and pure shear of a thin film are analysed using a strain gradient plasticity theory presented by Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52, 1379–1406]. Constitutive equations are formulated based on the assumption that the free energy only depends on the elastic strain and that the dissipation is influenced by the plastic strain gradients. The three material length scale parameters controlling the gradient effects in a general case are here represented by a single one. Boundary conditions for plastic strains are formulated in terms of a surface energy that represents dislocation buildup at an elastic/plastic interface. This implies constrained plastic flow at the interface and it enables the simulation of interfaces with different constitutive properties. The surface energy is also controlled by a single length scale parameter, which together with the material length scale defines a particular material.Numerical results reveal that a boundary layer is developed in the film for both biaxial and shear loading, giving rise to size effects. The size effects are strongly connected to the buildup of surface energy at the interface. If the interface length scale is small, the size effect vanishes. For a stiffer interface, corresponding to a non-vanishing surface energy at the interface, the yield strength is found to scale with the inverse of film thickness.Numerical predictions by the theory are compared to different experimental data and to dislocation dynamics simulations. Estimates of material length scale parameters are presented.     

Interfacial gradient plasticity governs scale-dependent yield strength and strain hardening rates in micro/nano structured metals

The effect of the material microstructural interfaces increases as the surface-to-volume ratio increases. It is shown in this work that interfacial effects have a profound impact on the scale-dependent yield strength and strain hardening of micro/nano-systems even under uniform stressing. This is achieved by adopting a higher-order gradient-dependent plasticity theory [Abu Al-Rub, R.K., Voyiadjis, G.Z., Bammann, D.J., 2007. A thermodynamic based higher-order gradient theory for size dependent plasticity. Int. J. Solids Struct. 44, 2888–2923] that enforces microscopic boundary conditions at interfaces and free surfaces. Those nonstandard boundary conditions relate a microtraction stress to the interfacial energy at the interface. In addition to the nonlocal yield condition for the material’s bulk, a microscopic yield condition for the interface is presented, which determines the stress at which the interface begins to deform plastically and harden. Hence, two material length scales are incorporated: one for the bulk and the other for the interface. Different expressions for the interfacial energy are investigated. The effect of the interfacial yield strength and interfacial hardening are studied by analytically solving a one-dimensional Hall–Petch-type size effect problem. It is found that when assuming compliant interfaces the interface properties control both the material’s global yield strength and rates of strain hardening such that the interfacial strength controls the global yield strength whereas the interfacial hardening controls both the global yield strength and strain hardening rates. On the other hand, when assuming a stiff interface, the bulk length scale controls both the global yield strength and strain hardening rates. Moreover, it is found that in order to correctly predict the increase in the yield strength with decreasing size, the interfacial length scale should scale the magnitude of both the interfacial yield strength and interfacial hardening.     

Modeling the interfacial effect on the yield strength and flow stress of thin metal films on substrates

It is shown in this paper that interfacial effects have a profound impact on the scale-dependent yield strength and strain hardening rates (flow stress) of metallic thin films on elastic substrates. This is achieved by developing a higher-order strain gradient plasticity theory based on the principle of virtual power and the laws of thermodynamics. This theory enforces microscopic boundary conditions at interfaces which relate a microtraction stress to the interfacial energy at the interface. It is shown that the film bulk length scale controls the size effect if a rigid interface is assumed whereas the interfacial length scale dominates if a compliant interface is assumed.     

Compliant interfaces: A mechanism for relaxation of dislocation pile-ups in a sheared single crystal

Discrete dislocation plasticity models and strain-gradient plasticity theories are used to investigate the role of interfaces in the elastic–plastic response of a sheared single crystal. The upper and lower faces of a single crystal are bonded to rigid adherends via interfaces of finite thickness. The sandwich system is subjected to simple shear, and the effect of thickness of crystal layer and of interfaces upon the overall response are explored. When the interface has a modulus less than that of the bulk material, both the predicted plastic size effect and the Bauschinger effect are considerably reduced. This is due to the relaxation of the dislocation stress field by the relatively compliant surface layer. On the other hand, when the interface has a modulus equal to that of the bulk material a strong size effect in hardening as well as a significant reverse plasticity are observed in small specimens. These effects are attributed to the energy stored in the elastic fields of the geometrically necessary dislocations (GNDs).     

测量界面软铬层力学性质的纳米压入法

为了测量双层铬的界面软铬层力学性质,提出了化学腐蚀基体法,通过溶解掉基体制备没有基体支撑的自由铬层,将在横截面内线状显示的界面转化为界面表面(铬层与基体相连接的面),避免了横截面不能显示界面表面的缺点。对界面表面进行纳米压入实验和借助于表征薄膜力学性质的表面压入能量法,测得了描述界面软铬层力学性质的弹性模量和压入弹、塑性功等参数。     

非平整、多孔介质海底上波浪传播的复合方程

为了反映近岸区域实际存在的多孔介质海底效应，并且考虑到波浪在刚性海底上传播模型的 最新研究进展，运用Green第二恒等式建立了波浪在非平整、多孔介质海底上传播的复合方 程. 假设水深和多孔介质海底层厚度均由两种分量组成：慢变分量，其水平变化的长度尺度大于 表面波的波长；快变分量，其水平变化的长度尺度与表面波的波长等阶，但其振幅小于表面 波的振幅. 另外，多孔介质层下部边界的快变分量比水深的快变分量小1个量级. 针对水体层和多孔介质层，选择Green第二恒等式方法给出了波浪传播和渗透的复合方程， 它在交接面上满足压力和垂直渗透速度的连续性条件,可充分考虑波数变化的一般连续性， 并包含了某些著名的扩展型缓坡方程.     

Direct numerical simulation of a freely decaying turbulent interfacial flow

Whereas Large Eddy Simulation (LES) of single-phase flows is already widely used in the CFD world, even for industrial applications, LES of two-phase interfacial flows, i.e. two-phase flows where an interface separates liquid and gas phases, still remains a challenging task. The main issue is the development of subgrid scale models well suited for two-phase interfacial flows. The aim of this work is to generate a detailed data base from direct numerical simulation (DNS) of two-phase interfacial flows in order to clearly understand interactions between small turbulent scales and the interface separating the two phases. This work is a first contribution in the study of the interface/turbulence interaction in the configuration where the interface is widely deformed and where both phases are resolved by DNS. To do this, the interaction between an initially plane interface and a freely decaying homogeneous isotropic turbulence (HIT) is studied. The densities and viscosities are the same for both phases in order to focus on the effect of the surface tension coefficient. Comparisons with existing theories built on wall-bounded or free-surface turbulence are carried out. To understand energy transfers between the interfacial energy and the turbulent one, PDFs of the droplet sizes distribution are calculated. An energy budget is carried out and turbulent statistics are performed including the distance to the interface as a parameter. A spectral analysis is achieved to highlight the energy transfer between turbulent scales of different sizes. The originality of this work is the study of the interface/turbulence interactions in the case of a widely deformed interface evolving in a turbulent flow.     

Spontaneous Inertial Imbibition in Porous Media Using a Fractal Representation of Pore Wall Rugosity

Considering the separable phenomena of imbibition in complex fine porous media as a function of timescale, it is noted that there are two discrete imbibition rate regimes when expressed in the Lucas–Washburn (L–W) equation. Commonly, to account for this deviation from the single equivalent hydraulic capillary, experimentalists propose an effective contact angle change. In this work, we consider rather the general term of the Wilhelmy wetting force regarding the wetting line length, and apply a proposed increase in the liquid–solid contact line and wetting force provided by the introduction of surface meso/nanoscale structure to the pore wall roughness. An experimental surface pore wall feature size regarding the rugosity area is determined by means of capillary condensation during nitrogen gas sorption in a ground calcium carbonate tablet compact. On this nano size scale, a fractal structure of pore wall is proposed to characterize for the internal rugosity of the porous medium. Comparative models based on the Lucas–Washburn and Bosanquet inertial absorption equations, respectively, for the short timescale imbibition are constructed by applying the extended wetting line length and wetting force to the equivalent hydraulic capillary observed at the long timescale imbibition. The results comparing the models adopting the fractal structure with experimental imbibition rate suggest that the L–W equation at the short timescale cannot match experiment, but that the inertial plug flow in the Bosanquet equation matches the experimental results very well. If the fractal structure can be supported in nature, then this stresses the role of the inertial term in the initial stage of imbibition. Relaxation to a smooth-walled capillary then takes place over the longer timescale as the surface rugosity wetting is overwhelmed by the pore condensation and film flow of the liquid ahead of the bulk wetting front, and thus to a smooth walled capillary undergoing permeation viscosity-controlled flow.     

A mathematical basis for strain-gradient plasticity theory—Part I: Scalar plastic multiplier

Strain-gradient plasticity theories are reviewed in which some measure of the plastic strain rate is treated as an independent kinematic variable. Dislocation arguments are invoked in order to provide a physical basis for the hardening at interfaces. A phenomenological, flow theory version of gradient plasticity is constructed in which stress measures, work-conjugate to plastic strain and its gradient, satisfy a yield condition. Plastic work is also done at internal interfaces and a yield surface is postulated for the work-conjugate stress quantities at the interface. Thereby, the theory has the potential to account for grain size effects in polycrystals. Both the bulk and interfacial stresses are taken to be dissipative in nature and due attention is paid to ensure that positive plastic work is done. It is shown that the mathematical structure of the elasto-plastic strain-gradient theory has similarities to conventional rigid-plasticity theory. Uniqueness and extremum principles are constructed for the solution of boundary value problems.     

Reprint of “Size dependence and orientation dependence of elastic properties of ZnO nanofilms” [In. J. Solids Struct. 45 (2008) 1730–1753]

The elastic properties of ZnO nanofilms with different film thickness, surface orientations and loading directions are investigated by using molecular mechanics (MM) method. The size dependence of elastic properties is relevant to both the film surface crystallographic orientation and loading direction. Both atomic structure analysis and energy calculation are employed to identify the mechanisms of the size-dependent elastic properties, under different loading directions and surface orientations. Upon small axial deformation, the relationship between intralayer and interlayer bond length variation and film elastic stiffness is established; it is found that the atomic layers with larger bond length variation have higher elastic stiffness. The strain energies of atomic layers of ZnO nanofilm and bulk are decoupled, from which the stiffness of film surface, intralayers, and interlayers are derived and compared with their bulk counterparts. The surface stiffness is found to be much lower than that of the interior layers and bulk counterpart, and with the decrease of film thickness, the residual tension-stiffened interior atomic layers are the main contributions of the increased elastic modulus of ZnO nanofilms.     

Size dependence and orientation dependence of elastic properties of ZnO nanofilms

The elastic properties of ZnO nanofilms with different film thickness, surface orientations and loading directions are investigated by using molecular mechanics (MM) method. The size dependence of elastic properties is relevant to both the film surface crystallographic orientation and loading direction. Both atomic structure analysis and energy calculation are employed to identify the mechanisms of the size-dependent elastic properties, under different loading directions and surface orientations. Upon small axial deformation, the relationship between intralayer and interlayer bond length variation and film elastic stiffness is established; it is found that the atomic layers with larger bond length variation have higher elastic stiffness. The strain energies of atomic layers of ZnO nanofilm and bulk are decoupled, from which the stiffness of film surface, intralayers, and interlayers are derived and compared with their bulk counterparts. The surface stiffness is found to be much lower than that of the interior layers and bulk counterpart, and with the decrease of film thickness, the residual tension-stiffened interior atomic layers are the main contributions of the increased elastic modulus of ZnO nanofilms.     

Formulation of strain gradient plasticity with interface energy in a consistent thermodynamic framework

In this work, the strain gradient formulation is used within the context of the thermodynamic principle, internal state variables, and thermodynamic and dissipation potentials. These in turn provide balance of momentum, boundary conditions, yield condition and flow rule, and free energy and dissipative energies. This new formulation contributes to the following important related issues: (i) the effects of interface energy that are incorporated into the formulation to address various boundary conditions, strengthening and formation of the boundary layers, (ii) nonlocal temperature effects that are crucial, for instance, for simulating the behavior of high speed machining for metallic materials using the strain gradient plasticity models, (iii) a new form of the nonlocal flow rule, (iv) physical bases of the length scale parameter and its identification using nano-indentation experiments and (v) a wide range of applications of the theory. Applications to thin films on thick substrates for various loading conditions and torsion of thin wires are investigated here along with the appropriate length scale effect on the behavior of these structures. Numerical issues of the theory are discussed and results are obtained using Matlab and Mathematica for the nonlinear ordinary differential equations (NODE) which constitute the boundary value problem.This study reveals that the micro-stress term has an important effect on the development of the boundary layers and hardening of the material at both hard and soft interface boundary conditions in thin films. The interface boundary conditions are described by the interfacial length scale and interfacial strength parameters. These parameters are important to control the size effect and hardening of the material. For more complex geometries the generalized form of the boundary value problem using the nonlocal finite element formulation is required to address the problems involved.     

A unified treatment of strain gradient plasticity

A theoretical framework is presented that has potential to cover a large range of strain gradient plasticity effects in isotropic materials. Both incremental plasticity and viscoplasticity models are presented. Many of the alternative models that have been presented in the literature are included as special cases. Based on the expression for plastic dissipation, it is in accordance with Gurtin (J. Mech. Phys. Solids 48 (2000) 989; Int. J. Plast. 19 (2003) 47) argued that the plastic flow direction is governed by a microstress q_ij and not the deviatoric Cauchy stress σ_ij′ that has been assumed by many others. The structure of the governing equations is of second order in the displacements and the plastic strains which makes it comparatively easy to implement in a finite element programme. In addition, a framework for the formulation of consistent boundary conditions is presented. It is shown that there is a close connection between surface energy of an interface and boundary conditions in terms of plastic strains and moment stresses. This should make it possible to study boundary layer effects at the interface between grains or phases. Consistent boundary conditions for an expanding elastic-plastic boundary are as well formulated. As examples, biaxial tension of a thin film on a thick substrate, torsion of a thin wire and a spherical void under remote hydrostatic tension are investigated.     

A Continuum Theory of Multispecies Thin Solid Film Growth by Chemical Vapor Deposition

A continuum theory for the chemical vapor deposition of thin solid films is proposed, in which a flowing, chemically reacting, gaseous mixture is coupled to the bulk of a growing thin film via the equations that govern the morphological evolution of the interface separating them. The vapor-film interface is viewed as a surface of zero thickness capable of sustaining mass and endowed with thermodynamic variables that account for its distinct structure. We consider situations in which species diffusion and heat conduction occur in all three phases (vapor, bulk and surface), with the former mechanism augmented by the convective transport of particles in the gas. Special attention is given to the chemical reactions that occur both in the vapor and on the film surface. Ours is a conceptual framework based on conservation laws for chemical species, momentum and energy, together with a separate balance of configurational forces. These balances are supplemented by an appropriate version of the second law which is used to develop suitable constitutive relations for each of the phases. In particular, we investigate the case of an elastic film, deposited on a rigid substrate and in contact with a reacting, multispecies, ideal vapor, whose surface behaves like an anisotropic, chemically reactive, multicomponent, ideal lattice gas. In addition to recovering the standard equations that describe the behavior of the gas and film phases, we derive the coupled PDE's that govern the interfacial morphological, chemical, and thermal evolution. In particular, the constitutively augmented interfacial configurational force balance provides a “kinetic relation” linking the thermodynamic “driving force” at the film surface to the growth rate. The special cases of (i) negligible interfacial species densities, and (ii) local (mechanical) equilibrium of both multi- and single-species films are investigated.     

Length scale of interface heterogeneities selects propagation mechanism of frictional slip fronts

We present three-dimensional finite-element simulations showing the propagation of slip fronts at striped heterogeneous interfaces. The heterogeneous area consists of alternating stripes of weaker and stronger frictional properties, which is equivalent to a lower and higher fracture energy, respectively. By comparing the slip front propagation at interfaces that differ solely by the length scale of the heterogeneous pattern, we illustrate that two different propagation regimes exist. Interfaces with wide stripes present slip fronts with propagation speeds that transition from sub-Rayleigh to inter-sonic. Thinner stripes are, however, characterized by the propagation of sub-Rayleigh slip fronts, which are preceded by slip pulses of negligible slip in the weaker stripes. From a macroscopic point of view, an interface with a smaller heterogeneous pattern appears to be stronger than the equivalent coarser interface even though both have the same average properties. The numerical results as well as a theoretical approach based on fracture-mechanics considerations suggest that the origin of these two distinct propagation mechanisms lies in the interaction between the length scales of the cohesive zone and the heterogeneous configuration. We further show by estimating the relevant length scales that the occurring propagation mechanism is influenced by the friction weakening rate of the interface as well as the shear modulus of the bulk material.     

Balance equations and structural models for phase interfaces

The general balance equations are developed for an interface represented by a dividing surface and for a moving common line represented as an intersection of dividing surfaces. The surface excess variables associated with a dividing surface are expressed both in terms of those variables describing the three-dimensional interfacial region of finite thickness and in terms of those variables describing bulk phases that extend up to the dividing surface.A structural model for the interface is suggested in which a suspension of solid bodies representing surfactant molecules is distributed about a singular surface separating two adjacent bulk solvent phases. The suspension is required to have the same average behavior as the interfacial region. This is interpreted as meaning that the general jump balance for a continuum dividing surface represented by an interfacial suspension is a local area average. Specific results are derived for two structural models, each in the same simple shear field. One consists of a dilute suspension of neutrally buoyant spheres floating with their centers restricted to the dividing surface. The other is a dilute suspension of chains of neutrally buoyant spheres with the sphere at one end of the chain floating in the dividing surface.     

基于能量等效原理的应变局部化分析:Ⅰ.一维解析解

基于热力学第一定律和非局部塑性理论,提出了一种求解应变局部化问题的非局部方法.对材料的每一点定义了局部和非局部两种状态空间,局部状态空间的内变量通过非局部权函数映射到非局部空间,成为非局部内变量.在应变软化过程中,局部状态空间中的塑性变形服从正交流动法则,材料的软化律在非局部状态空间中被引入.通过两个状态空间的塑性应变能耗散率的等效,得到了应变软化过程中明确定义的局部化区域以及其中的塑性应变分布.应用本方法导出了一维应变局部化问题的解析解.解析解表明,应变局部化区域的尺寸只与材料内尺度有关;对于高斯型非局部权函数,局部化区域的尺寸大约是材料内尺度的6倍.一维算例表明,局部化区域的塑性应变分布以及载荷-位移曲线仅与材料参数和结构几何尺寸有关,变形局部化区域的尺寸随着材料内尺度的减小而减小,同时塑性应变也随着材料内尺度的减小变得更加集中.当内尺度趋近于零时,应用本文方法得到的解与采用传统的局部塑性理论得到的解相同.     

Direct calculations of material parameters for gradient plasticity

Length scale parameters introduced in gradient theories of plasticity are calculated in closed form with a continuum dislocation based theory. The similarity of the governing equations in both models for the evolution of plastic deformation of a constrained thin film makes it possible to identify parameters of the gradient plasticity theory with the dislocation based model. A one-to-one identification is not possible given that gradient plasticity does not account for individual dislocations. However, by comparing the mean plastic deformation across the film thickness we find that the length scale parameter, l, introduced in the gradient plasticity theory depends on the geometry as well as material constants.     

Thermodynamics and continuum fracture mechanics for nonlocal-elastic plastic materials

Nonlocal elasticity is used as an improved elasticity model which engenders no crack-tip stress singularities and thus makes applicable the classical stress-based failure criteria. Considering nonlocal-elastic plastic materials exposed to softening by particle decohesion in a process surface and to subsequent surface separation by fracture, fracture mechanics is addressed within the framework of irreversible internal-variable thermodynamics in the hypothesis of small strains and arbitrary (but sufficiently regular) fracture surface (crack surface plus process surface). The state equations and the energy dissipation densities are derived for the bulk material and for the process surface, for both of which thermodynamically consistent evolutive equations are also proposed. The energy consumption for the formation of the unit crack area is evaluated as the sum of two contributions, one as free energy released by the process surface microstructure, the other as mechanical work done by the surrounding bulk material. Basing on the second thermodynamics law, a crack local stability criterion is provided in terms of crack front characteristics, i.e. fracture force vector, fracture resistance vector and fracture (symmetric) stiffness matrix, all defined at the points of the crack front through the response fields and the response sensitivities to virtual crack front advancements. Stability is guaranteed if, at every point of the crack front and for any virtual advancement of the latter, the fracture resistance is greater than the fracture force, or in case of equality, the fracture stiffness matrix is positive definite. The limit case of perfectly brittle fracture is considered, also in the case of local elasticity.