Microstructural changes and material parameter development

At previous GAMM-meetings the first two authors reported on the spinodal decomposition and coarsening in lead-containing as well as lead-free solders. They derived an extended diffusion equation for the corresponding computer modelling on the basis of a multi-component theory of mixtures. Consequently, they performed numerical studies that are capable of predicting quantitatively the development of SnPb and AgCu microstructures. Knowing this development is quite important from a technological point-of-view in order to quantify the current joining capability of the solder material (“Resteigenschaften”). On the basis of these results this paper now attempts to predict the corresponding change of the materials properties, in particular of the elastic constants. This is done by means of a homogenization technique developed by the last two authors. In this paper we will briefly summarize the results related to the extended diffusion equation, present numerical simulations of the coarsening process, outline the theory underlying the homogenization and, finally, show how the material properties behave over time. Wherever possible, alternative analytical solutions for the homogenized constants are also presented (e.g., by laminate theory). (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Void nucleation by vacancy condensation

A computational simulation of Kirkendall voiding in metallic materials is presented. After a brief explanation of the phenomena and its consequences on the reliability of microelectronic components we introduce a constitutive model for void nucleation and growth which is based on vacancy diffusion and rate–dependent inelastic deformation. This model can be used to predict the temporal development of voids in solder during thermal cycling and/or impact loading. A numerical study illustrates the potential of the model for the failure analysis. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determining Material Parameter of Solder Alloys by Nanoindentation

Environmental awareness motivates the replacement of traditional plumbiferous solder joints in microelectronic devices by new material alloys. In our group mechanical properties of some lead-free alloys are determined by nanoindentation experiments. We use a Microsystems-Nanoindenter Machine with Berkovich tip and apply standard techniques to extract the material properties from the measured load-displacement curves. To assess the quality of our experimental results we model and analyze the setup by finite element computations. By comparing the real (input) material data with the results determined from the load-displacement curves we analyze the obtained data in dependence of strength and stiffness of the materials under consideration. For low-strength material we point out deviations. By inverse analysis we adapt numerically elastic modulus and yield stress to experimentally measured load-displacement curves. To obtain information on the material's work hardening we suggest the use of a blunt indenter tip, e.g., a spherical indenter. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of model parameters of lead‐containing and lead‐free solders

In this paper mechanical parameters of tin‐lead and lead‐free microelectronic solder materials will be determined. Specifically, the emphasis is on SnPb37 and SnAg3.8Cu0.7Bi2. The investigation is based on an experimental technique known as the small punch test (SPT) using different SPT displacement rates in combination with an appropriate numerical non‐linear Finite Element (FE) analysis. The elastic‐plastic creep characteristics of solder materials are modeled and investigated.     

PCB焊点热循环失效分析和改进设计

针对PCB（印制电路板）焊点在高低温热循环下的失效,建立了传统结构的三维模型和在芯片边角下加锡块的改进设计模型,通过实验测得FR-4的弹性模量和热膨胀系数,用ANSYS计算了PCB在高低温循环下的应力应变,并用修正的Coffin-Manson经验方程计算了焊点的热循环寿命。结果表明,通过在芯片边角下加锡块,PCB焊点的最大等效塑性应变显著降低,其热疲劳寿命得到明显提高.     

Modelling of Microstructure Evolution in Two‐Phase Materials using Generalized Driving Forces

In two‐phase materials, such as nickel base alloys or zirconia, the macroscopic material response is strongly influenced by the morphology of the microstructure. In nickel alloys microstructural rearrangements due to diffusion occur at elevated temperatures. A continuum mechanical model is presented that takes elastic inhomogeneity and eigenstrains together with an interface energy into account. The driving force for the diffusion process is identified and used to simulate morphology evolution and equilibrium shapes. The numerical simulation is done with a 3D Boundary Element Method applicable to anisotropic materials.     

A higher gradient theory for solid mixtures

During use the microstructure of solders will change over time. To enable us to determine the lifetime and reliability of these materials, a theory to predict such changes is needed. In this paper two kinds of changes are considered: The forming of intermetallic compounds due to chemical reactions of two components and spinodal decomposition of a binary alloy followed by coarsening. A theoretical approach is presented and evaluated numerically. For this purpose, an extended diffusion equation is needed and derived within the framework of undisputed concepts of the entropy principle. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermomigration in Sn-Pb solder bumps: Modelling and simulation

It is well known a homogeneous binary alloy subjected to a temperature gradient will become inhomogeneous. This decomposition phenomenon is called thermomigration or thermotransport or Soret effect, which is a cross-effect in irreversible processes between heat conduction and atomic diffusion. In the present contribution we focus on a diffuse interface model for separation processes affected by temperature gradients coupled with a heat diffusion equation in order to describe thermomigration effects caused by joule-heating. To be specific, we focus on phase separation in solder bumps consisting of tin and lead. Therefore we discuss the modelling of a Gibbs' configurational free energy density as well as the modelling of the chemical mobility, mobility of thermotransport and Dufour coefficient as sufficiently smooth functions in particle concentration and absolute temperature. The resulting set of partial differential equations involves spatial derivatives of fourth order. Consequently, the variational formulation of the problem mandates approximation functions which are at least C¹-continuous. In order to fulfill this requirement a B-Spline based finite element scheme is provided. One of the main advantages of B-Splines is the possibility to represent complex geometries exactly. However, it has been shown that especially curved B-Spline geometries can not be treated in a straight forward manner in finite element analysis. For this reason we demonstrate the implementation of boundary conditions to avoid the arising numerical perturbations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase Separation in the Advective Cahn–Hilliard Equation

The Cahn–Hilliard equation is a classic model of phase separation in binary mixtures that exhibits spontaneous coarsening of the phases. We study the Cahn–Hilliard equation with an imposed advection term in order to model the stirring and eventual mixing of the phases. The main result is that if the imposed advection is sufficiently mixing, then no phase separation occurs, and the solution instead converges exponentially to a homogeneous mixed state. The mixing effectiveness of the imposed drift is quantified in terms of the dissipation time of the associated advection–hyperdiffusion equation, and we produce examples of velocity fields with a small dissipation time. We also study the relationship between this quantity and the dissipation time of the standard advection–diffusion equation.

     

A Variational Homogenization Approach to Electromechanical Hystereses Caused by Domain Wall Movement

In recent years, increasing interest in so-called smart materials such as ferroelectric polymers and ceramics has been shown. Those materials are used in various actuators, sensors, and also in medical devices. In this paper, we outline a micro-macro approach to the modeling of macroscopic hystereses which directly takes into account the microstructural evolution of electrically poled domains. To this end, an incremental variational formulation for a gradient-type phase field model is developed and exploited for the simulation of electromechanically coupled problems. The formulation determines the hysteretic response of the material in terms of an energy-enthalpy and a dissipation function which both depend on the microscopic remanent polarization treated as an order parameter. The gradient-type balance law for the phase field can be considered as a generalization of Biot's equation for standard dissipative materials and may be related to the classical Ginzburg-Landau equation. Furthermore, the variational formulation serves as natural starting point for a compact and symmetric finite element implementation of the coupled micromechanical problem covering the displacement, the electric potential, and the microscopic polarization vector. For this three-field scenario we develop a variational-based homogenization method which determines the overall macroscopic hysteretic properties of a polycrystalline aggregate. The proposed computational method can be used as a numerical laboratory for the improvement of microstructural properties. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Regularized Sharp Interface Model for Phase Transformation Accounting for Prescribed Sharp Interface Kinetics

The macroscopic mechanical behavior of multi-phasic materials depends on the formation and evolution of their microstructure by means of phase transformation. In case of martensitic transformations, the resulting phase boundaries are sharp interfaces. We carry out a geometrically motivated discussion of the regularization of such sharp interfaces by use of an order parameter/phase-field and exploit the results for a regularized sharp interface model for two-phase elastic materials with evolving phase boundaries. To account for the dissipative effects during phase transition, we model the material as a generalized standard medium with energy storage and a dissipation function that determines the evolution of the regularized interface. Making use of the level-set equation, we are thereby able to directly translate prescribed sharp interface kinetic relations to the constitutive model in the regularized setting. We develop a suitable incremental variational three-field framework for the dissipative phase transformation problem. Finally, the modeling capability and the associated numerical solution techniques are demonstrated by means of a representative numerical example. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Special solutions for an equation arising in sand ripple dynamics

We study a nonlinear fourth order evolution equation arising in the context of sand ripple dynamics. We analyse the set of stationary solutions and travelling waves in order to recover the observed phenomenology such as different wavelengths ripples, travelling waves, coarsening and time scales. Moreover, we construct an approximate solution which describes the early stages of the dynamics and which suggests the existence of coarsening and of time scales with different dynamical behaviour.     

Cahn-Hilliard inpainting and the Willmore functional

The Cahn-Hilliard equation has its origin in material sciences and serves as a model for phase separation and phase coarsening in binary alloys. A new approach in the class of fourth order inpainting algorithms is inpainting of binary images using the Cahn-Hilliard equation. Like solutions of the Cahn-Hilliard equation converging to two main values during the phase separation process, the grayvalues inside the missing part of the image are oriented towards the binary states black and white. We present stability/instability results for solutions of the Cahn-Hilliard equation and their connection to the Willmore functional. In particular we will consider the Willmore functional as a quantity to find the optimal scale of the inpainting result. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heat Conduction in a Phase Field Model for Martensitic Transformation

We study the martensitic transformation with a phase field model, where we consider the Bain transformation path in a small strain setting. For the order parameter, interpolating between an austenitic parent phase and martensitic phases, we use a Ginzburg-Landau evolution equation, assuming a constant mobility. In [1], a temperature dependent separation potential is introduced. We use this potential to extend the model in [2], by considering a transient temperature field, where the temperature is introduced as an additional degree of freedom. This leads to a coupling of both the evolution equation of the order parameter and the mechanical field equations (in terms of thermal expansion) with the heat equation. The model is implemented in FEAP as a 4-node element with bi-linear shape functions. Numerical examples are given to illustrate the influence of the temperature on the evolution of the martensitic phase. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupled chemomechanics and phase field modeling of failure in electrode materials of Li-ion batteries

We propose a canonical finite strain theory for diffusion-mechanics coupling for the intercalation induced stress generation in Li-ion electrode particles. The intrinsic coupling arises from both mechanical pressure gradient-induced diffusion of Li-ion particles and diffusion induced swelling/shrinkage leading to mechanical stresses. In addition, we extend the finite strain theory for diffusion-mechanics coupling to chemomechanical fracture of electrode particles by introducing a nonlocal crack phase field which replaces a sharp crack topology with a smooth diffuse interpolation between the intact and broken states of the material. We employ a semi-implicit Galerkin-type finite element method for the solution of resulting set of differential equations. In addition to the mechanical, chemical and crack phase field, we introduce the pressure as an independent field variable in order to reduce the smoothness requirements on the interpolation functions. We illustrate characteristic features of the proposed model by means of representative initial-boundary value problems. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Composite Waves for a Cell Population System Modelling Tumor Growth and Invasion

In the recent biomechanical theory of cancer growth, solid tumors are considered as liquid-like materials comprising elastic components. In this fluid mechanical view, the expansion ability of a solid tumor into a host tissue is mainly driven by either the cell diffusion constant or the cell division rate, with the latter depending on the local cell density (contact inhibition) or/and on the mechanical stress in the tumor. For the two by two degenerate parabolic/elliptic reaction-diffusion system that results from this modeling, the authors prove that there are always traveling waves above a minimal speed, and analyse their shapes. They appear to be complex with composite shapes and discontinuities. Several small parameters allow for analytical solutions, and in particular, the incompressible cells limit is very singular and related to the Hele-Shaw equation. These singular traveling waves are recovered numerically.     

One-dimensional stochastic heat equation with discontinuous conductance

We introduce a new stochastic partial differential equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by a space–time white noise. Such equation could be used in mathematical modeling of diffusion phenomena in medium consisting of two kinds of materials and undergoing stochastic perturbations. We prove the existence of the solution and we present explicit expressions of its covariance and variance functions. Some regularity properties of the solution sample paths are also analyzed.     

On M-Wright transforms and time-fractional diffusion equations

In this paper, we consider the Mittag-Leffler operator as an analytical solution of time-fractional diffusion equation in the Caputo sense. This solution is presented by an integral representation in terms of the M-Wright functions and the exponential operators. Further, we study the Mittag-Leffler operators associated with the Legendre and Bessel diffusion equations. Finally, we extend the obtained integral representation for the time-fractional diffusion equation of distributed order.