Material voids in elastic solids with anisotropic surface energies

This work discusses the role of highly anisotropic interfacial energy for problems involving a material void in a linearly elastic solid. Using the calculus of variations it is shown that important qualitative features of the equilibrium shape of the void may be deduced from smoothness and convexity properties of the interfacial energy.     

Diffusion induced void nucleation in SnPb solder joints

Towards a simple and robust model for void-based fatigue prediction, we investigate the interaction of voids with its surrounding by using a multi-field method. We couple the concentration fields of tin c₁ and lead c₂ with an additional field c₃, where the latter is assigned with a void field. The interaction potential manifests three stable states. Two are obtained by experimental results of tin-lead (SnPb) and the void stable state is postulated by construction. The logarithmic form of the thermodynamically consistent configurational entropy is approximated within this study by a fourth order polynom. It has been shown that the interfacial energy coefficient is independent of void's size, but rather depends numerically on the mesh size, which is used in the model presented here. Both governing equations follows a Cahn-Hilliard-type equation to mimic the microstructural changes. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Closure laws for a two-fluid two-pressure modelLois de fermeture pour un modèle à deux pressions d'écoulement diphasique

Closure laws for interfacial pressure and interfacial velocity are proposed within the frame work of two-pressure two-phase flow models. These enable us to ensure positivity of void fractions, mass fractions and internal energies when investigating field by field waves in the Riemann problem. To cite this article: F. Coquel et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 927–932.     

Numerical modelling of bubbly flows with and without heat and mass transfer

In this paper, modelling gas–liquid bubbly flows is achieved by the introduction of a population balance equation combined with the three-dimensional two-fluid model. For gas–liquid bubbly flows without heat and mass transfer, an average bubble number density transport equation has been incorporated in the commercial code CFX5.7 to better describe the temporal and spatial evolution of the geometrical structure of the gas bubbles. The coalescence and breakage effects of the gas bubbles are modelled according to the coalescence by the random collisions driven by turbulence and wake entrainment while for bubble breakage by the impact of turbulent eddies. Local radial distributions of the void fraction, interfacial area concentration, bubble Sauter mean diameter, and gas and liquid velocities, are compared against experimental data in a vertical pipe flow. Satisfactory agreements for the local distributions are achieved between the predictions and measurements. For gas–liquid bubbly flows with heat and mass transfer, boiling flows at subcooled conditions are considered. Based on the formulation of the MUSIG (multiple-size-group) boiling model and a model considering the forces acting on departing bubbles at the heated surface implemented in the computer code CFX4.4, comparison of model predictions against local measurements is made for the void fraction, bubble Sauter mean diameter, interfacial area concentration, and gas and liquid velocities covering a range of different mass and heat fluxes and inlet subcooling temperatures. Good agreement is achieved with the local radial void fraction, bubble Sauter mean diameter, interfacial area concentration and liquid velocity profiles against measurements. However, significant weakness of the model is evidenced in the prediction of the vapour velocity. Work is in progress through the consideration of additional momentum equations or developing an algebraic slip model to account for the effects of bubble separation.     

Effects of electric and mechanical loads on the morphological evolution of a void in piezoelectric films

This paper reports the result of an investigation into the effect of electric and mechanical loads on the morphological evolution of a void in piezoelectric materials based on a model for the morphological evolution of a void, the thermodynamics potential and energy principle. Thus, the path and the bifurcation condition of the morphological evolution of the void in piezoelectric materials are described, which gives some insight into the reliability of piezoelectric films under electric and mechanical loads.     

超弹性矩形板单向拉伸时微孔的增长*

本文研究了含中心微孔的超弹性矩形板在单向拉伸时的有限变形和受力分析.为了考察微孔的存在对矩形板变形和应力的影响,将问题化成一个超弹性环形板的变形和受力分析,并用最小势能原理得到变分近似解.进行了数值计算,分析了微孔的增长情况.     

Void dynamics in lead-free Sn-Ag-Cu solder joints

As a result of thermal and mechanical loading of Sn-Ag-Cu solder joints pores, flaws and voids may nucleate and grow. Such void nucleation is studied here by means of a phase-field approach which accounts for the decomposition of the solder into phases described by its concentration c. In this investigation, the void growth results from boundary inward flux only, effects due to grain size and interstitials are not involved, cf. [1]. The free energy functional is approximated by a second order Taylor expansion and thus composed of a bulk free energy density and a Ginzburg-type gradient term. The bulk free energy density ϕ follows a classical Ginzburg-Landau double-well potential. The gradient-energy coefficient κ depends on ϕ and is calculated similar to [2]. Experimental data have been adapted for the modeling of the temporal evolution of the concentration of voids located on a square domain in 2D. The simulation is based on a B-spline finite element analysis. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical scheme for regularized anisotropic curve shortening flow

Realistic interfacial energy densities are often non-convex, which results in backward parabolic behavior of the corresponding anisotropic curve shortening flow, thereby inducing phenomena such as the formation of corners and facets. Adding a term that is quadratic in the curvature to the interfacial energy yields a regularized evolution equation for the interface, which is fourth-order parabolic. Using a semi-implicit time discretization, we present a variational formulation of this equation, which allows the use of linear finite elements. The resulting linear system is shown to be uniquely solvable. We also present numerical examples.     

АГ-Convergence Result for Thin Curved Films Bonded to a Fixed Substrate with a Noninterpenetration Constraint

The behavior of a thin curved hyperelastic film bonded to a fixed substrate is described by an energy composed of a nonlinearly hyperelastic energy term and a debonding interfacial energy term. The author computes the F-limit of this energy under a noninterpenetration constraint that prohibits penetration of the film into the substrate without excluding contact between them.     

A Г-Convergence Result for Thin Curved Films Bonded to a Fixed Substrate with a Noninterpenetration Constraint

Abstract The behavior of a thin curved hyperelastic film bonded to a fixed substrate is described by an energy composed of a nonlinearly hyperelastic energy term and a debonding interfacial energy term. The author computes the Γ-limit of this energy under a noninterpenetration constraint that prohibits penetration of the film into the substrate without excluding contact between them.     

InP-to-InGaAs interfacial strain grown by using tertiarybutylarsine and tertiarybutylphosphine

Lattice-matched InGaAs/lnP heterostructures have been grown by using metalorganic vapor phase epitaxy (MOVPE) with tertiarybutylarsine (TBAs), tertiarybutylphosphine (TBP) as the group V sources. The results of X-ray diffraction on InGaAs/lnP single herterostructure show that there is a compressive-strained interfacial layer at the InP-to-InGaAs interface. X-ray diffraction of InGaAs/ InP superlattices is successfully simulated by using the same interfacial layer. TBAs purging of InP surface has a significant influence on the interfacial strain. A novel gas switching sequence, which switches group III to the run line earlier than TBAs, is proposed to reduce this interfacial strain. As a result, the average compressive strain of superlattices decreases, and a blue shift of photoluminescence ( PL) peak energy and narrowing in PL width are obtained.     

Hysteresis and Stick-Slip Motion of Phase Boundaries in Dynamic Models of Phase Transitions

Summary. We investigate hysteretic behavior in two dynamic models for solid-solid phase transitions. An elastic bar with a nonconvex double-well elastic energy density is subjected to time-dependent displacement boundary conditions. Both models include inertia and a viscous stress term that provides energy dissipation. The first model involves a strain-gradient term that models interfacial energy. In the second model this term is omitted. Numerical simulations combined with analytical results predict hysteretic behavior in the overall end-load versus end-displacement diagram for both models. The hysteresis is largely due to metastability and nucleation; it persists even for very slow loading when viscous dissipation is quite small. In the model with interfacial energy, phase interfaces move smoothly. When this term is omitted, hysteresis is much more pronounced. In addition, phase boundaries move in an irregular, stick-slip fashion. The corresponding load-elongation curve exhibits serrations, in qualitative agreement with certain experimental observations in shape-memory alloys. Received August 4, 1998; revised December 11, 1998     

A justification of the theory of martensitic thin films in the absence of an interfacial energy

In the framework of Nonlinear Elasticity, the asymptotic behavior of the free energy of a martensitic material is studied as the height of the sample tends to zero. The effective thin film energy is identified in terms of parametrized probability measures, which allows for a justification of the kinematic compatibility conditions proposed in Bhattacharya and James' theory of thin films of martensitic materials, in the absence of higher order interfacial energy contributions to the three-dimensional bulk energy.     

Doping states of x PPP/yPA diblock copolymers

A tight-binding model describing the xPPP/yPA diblock copolymer has been adopted.The ground states, doping states, configurations, electronic distributions and the types of excitations have been studied. It is found that the ratios and interfacial couplings of the components have effect on the properties of excitations in xPPP/yPA diblock copolymers. The interface between two homopolymers acts as an energy barrier which increases with the weakening of the interfacial coupling.     

Doping states of x PPP/ y PA diblock copolymers

A tight-binding model describing the xPPP/yPA diblock copolymer has been adopted. The ground states, doping states, configurations, electronic distributions and the types of excitations have been studied. It is found that the ratios and interfacial couplings of the components have effect on the properties of excitations in xPPP/yPA diblock copolymers. The interface between two homopolymers acts as an energy barrier which increases with the weakening of the interfacial coupling.     

A two-gradient approach for phase transitions in thin films

Motivated by solid-solid phase transitions in elastic thin films, we perform a Γ-convergence analysis for a singularly perturbed energy related to second order phase transitions in a domain of vanishing thickness. Under a two-wells assumption, we derive a sharp interface model with an interfacial energy depending on the asymptotic ratio between the characteristic length scale of the phase transition and the thickness of the film. In each case, the interfacial energy is determined by an explicit optimal profile problem. This asymptotic problem entails a nontrivial dependance on the thickness direction when the phase transition is created at the same rate as the thin film, while it shows a separation of scales if the thin film is created at a faster rate than the phase transition. The last regime, when the phase transition is created at a faster rate than the thin film, is more involved. Depending on growth conditions of the potential and the compatibility of the two phases, we either obtain a sharp interface model with scale separation, or a trivial situation driven by rigidity effects.     

Doping states of xPPP/yPA diblock copolymers

A tight-binding model describing the xPPP/yPA diblock copolymer has been adopted. The ground states, doping states, configurations, electronic distributions and the types of excitations have been studied. It is found that the ratios and interfacial couplings of the components have effect on the properties of excitations in xPPP/yPA diblock copolymers. The interface between two homopolymers acts as an energy barrier which increases with the weakening of the interfacial coupling.     

An inhomogeneous,anisotropic, elastically modified Gibbs­Thomson law as singular limit of a diffuse interface model

We consider the sharp interface limit of a diffuse phase-field model with prescribed total mass taking into account a spatially inhomogeneous, anisotropic interfacial energy and an elastic energy. The main aim is to establish a weak formulation of an inhomogeneous, anisotropic, elastically modified Gibbs-Thomson law in the sharp interface limit. To this end we pass to the limit in the weak formulation of the Euler-Lagrange equation of the diffuse phase-field energy. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dewetting Dynamics of Anisotropic Particles: A Level Set Numerical Approach

Applications of Mathematics - We extend thresholding methods for numerical realization of mean curvature flow on obstacles to the anisotropic setting where interfacial energy depends on the orien-...     

一种预测界面裂纹形成的新方法

本文提出了一种预测界面裂纹形成的新方法,称为“β=0法”.这是一种变换方法,通过调整有关材料弹性常数,把振荡应力场变换为非振荡应力场.界面韧性曲线将随调整后新的材料性质而改变.本文方法预测的临界荷载与界面断裂力学预测的临界荷载严格一致.最后通过几个实例说明了保持能量释放率不变的情况下,怎样实施β=0法.