W.H. Müller

A study of the coarsening in tin/lead solders

Abstract This paper presents a model, which is capable to simulate the coarsening process observed during thermo-mechanical treatment of binary tin–lead solders. Fourier transforms and spectral theory are used for the numerical treatment of the thermo-elastic as well as of the diffusion problem encountered during phase separation in these alloys. More specifically, the analysis is based exclusively on continuum theory and, first, relies on the numerical computation of the local stresses and strains in a representative volume element. Second, this information is used in an extended diffusion equation to predict the local concentrations of the constituents of the solder. Besides the classical driving forces for phase separation, as introduced by Fick and Cahn–Hilliard, this equation contains an additional term which links the mechanical to the thermodynamical problem. It connects internal and external stresses, strains, temperature, as well as concentrations and allows for a comprehensive study of the coarsening and aging process.

Modeling diffusional coarsening in eutectic tin/lead solders: a quantitative approach

Abstract This paper presents a quantitative simulation of the phase separation and coarsening phenomenon in eutectic tin/lead (SnPb) solders. The computer modeling is based on continuum theory and phase field models that were evaluated using the most recently available data for the free energy of the tin/lead system, diffusional and mobility coefficients, elastic constants as well as surface tensions of both phases. The model presented allows us to study the influence as well as the interaction between classical diffusion of the Fickean type, surface energies according to Cahn and Hilliard, as well as stresses and strains on phase separation and coarsening. An attempt is made to compare the temporal development of a eutectic SnPb microstructure at different temperature levels and subjected to different stress levels as predicted by the model to actual experiments.

Strengthening mechanisms in Al2O3/SiC nanocomposites

Abstract A strengthening mechanism merely arising from internal (residual) microstresses due to thermal expansion mismatch is proposed for explaining the high experimental strength data measured in Al2O3/SiC nanocomposites. Upon cooling, transgranular SiC particles undergo lower shrinkage as compared to the surrounding matrix and provide a hydrostatic “expansion” effect in the core of each Al2O3 grain. Such a grain expansion tightens the internal Al2O3 grain boundaries, thus shielding both weakly bonded and unbonded (cracked) grain boundaries. It is shown that the shielding effect by intragranular SiC particles is more pronounced than the grain-boundary opening effect eventually associated with thermal expansion anisotropy of the Al2O3 grains, even in the “worst” Al2O3-grain cluster configuration. Therefore, an improvement of the material strength can be found. However, a large stress intensification at the grain boundary is found when intergranular SiC particles are present, which can produce a noticeable wedge-like opening effect and trigger grain-boundary fracture. The present model enables us to explain the experimental strength data reported for Al2O3/SiC nanocomposites and confirms that the high strength of these materials can be explained without invoking any toughening contribution by the SiC dispersion.

Computer modeling of the coarsening process in tin–lead solders

Abstract The temporal progress of phase separation and coarsening observed in eutectic tin/lead (SnPb) solders was investigated using computer modeling. The modeling framework is based on an extended diffusion equation of the phase-field model type that includes classical Fickean diffusion, effects of surface tensions according to the Cahn–Hilliard formalism, as well as diffusive morphology changes due to local thermo-mechanical stresses and strains. The numerical solution of the extended diffusion equation for a eutectic tin/lead microstructure is performed by means of discrete Fourier transforms (DFTs). In particular, a one-dimensional computer modeling procedure will be presented in detail. The phase separation and the coarsening process are discussed, and it is investigated how these depend on different initial conditions and on different temperature levels. The results are in fair agreement with experimental observation. Moreover, the weight and volume fractions of the α- and β-phase, as well as an average “α-particle size” are determined numerically and their temporal development is studied.

Linear and elastic–plastic fracture mechanics revisited by use of Fourier transforms – theory and application

Abstract This paper investigates whether and how discrete Fourier transforms (DFT) can be used to compute the local stress/strain distribution around holes in externally loaded two-dimensional representative volume elements (RVEs). To this end, the properties of DFT are first reviewed and then applied to the solution of linear elastic and time-dependent elastic plastic material response. The equivalent inclusion method is used to derive a functional equation which allows for the numerical computation of stresses and strains within an RVE with heterogeneities of arbitrary shape and stiffness. This functional equation is then specialized to the case of circular and elliptical holes of different minor axes which eventually degenerate into Griffith cracks. The numerically predicted stresses and strains are compared to the corresponding analytical solutions for a single circular as well as an elliptical hole in an infinitely large plate under tension as well as to finite element calculations (for time-independent elastic/plastic material response).

Hypermedia teaching of mechanics— MechANIma

In this paper we describe the mechANIma project, a joint venture between the department of computer science and the laboratory for mechanics at the University of Paderborn, Germany.

Mixed mode fracture in epicycloid specimens II. Point force loading

The method of complex potentials is used to obtain an analytical solution for the stresses in epicycloidal specimens due to point force loading. The solution is used to obtain an analytical expression for the stress intensity factors of cusp-like cracks in such specimens which can be considered as a generalization of the well-established concept of Griffith cracks. It is shown that by suitable positioning of the point forces negative mode I stress intensity factors will result. This illustrates the potential of epicycloid specimens for the determination of fracture properties under compressive loading where frictional contact of the crack surfaces is a priori avoided.

Mixed mode fracture in epicycloid specimens. I—Thermal inclusions

Abstract The technique of complex potentials is used to derive an analytical solution for the stresses that develop in epicycloidal specimens subjected to hot spot loading. The solution gives stress intensity factors for cusp-like cracks in such specimens which can be considered as a generalization of the traditional Griffith crack. It is shown that by suitable positioning of the hot spot the complete range of mode mixety, as well as negative mode I stress intensity factors, can be obtained. This illustrates the potential of epicycloid specimens for testing of bimaterial fracture properties as well as failure under compressive loading without frictional contact of crack surfaces. The paper concludes with a first experimental measurement of K1 in order to validate this new concept.

Mixed mode fracture in epicycloid specimens iii. Dislocations

Complex stress potentials are derived to obtain an analytical solution for the stresses in epicycloidal specimens which contain a dislocation. The solution is used to obtain an analytical expression for the stress intensity factors of cusp-like cracks in such specimens which can be considered as a generalization of the well established concept of Griffith cracks. It is shown that by suitable positioning of the dislocation, both positive and negative mode I stress intensity factors will result. This illustrates the potential of epicycloid specimens for determination of fracture properties under compressive loading where frictional contact of the crack surfaces is a priori avoided.