Christina Bjerkén

A numerical method for calculating stress intensity factors for interface cracks in bimaterials

Abstract This paper presents a method for obtaining the complex stress intensity factor (or alternatively the corresponding energy release rate and mode mixity) for an interface crack in a bimaterial using a minimum number of computations. A crack closure integral method for homogeneous materials developed by Rybicki and Kanninen has been modified to include mismatch in material properties. This was achieved directly from the nodal forces at the crack tip and the displacements near the tip as obtained from a finite element analysis using only four-node constant strain elements. Numerical calculations for tensile and mixed mode loading showed good agreement with results from corresponding analytical solutions. The main advantages of this method are that it is straightforward and easy to use and that the number of calculations needed to obtain the stress intensity factors can be held to a minimum.

A tool to model short crack fatigue growth using a discrete dislocation formulation

A method is presented that combines the modelling of cracks by distributed dislocation dipoles with developing plasticity represented by discrete dislocations moving along slip bands. Crack growth is due to the emission of dislocations from the crack tip along preferred slip planes. Eventual annihilation of dislocations occurs by reunion with the corresponding displacement steps of the crack surface. Crack surface overlap is not allowed. The equilibrium state for each load increment is solved iteratively, allowing various crack geometries. The method is applied to the problem of a short edge crack growing in mode I due to fatigue loading. It is shown that the development of a local plastic zone and the propagation of the crack can be monitored in detail.

In situ hydrogen loading on zirconium powder

Commercial-grade Zr powder loaded with hydrogen in situ and phase transformations between various Zr and ZrHx phases have been monitored in real time.

Phase ordering kinetics of second-phase formation near an edge dislocation

The time-dependent Ginzburg-Landau (TDGL) equation for a single component non-conservative structural order parameter is used to study the spatio-temporal evolution of a second phase in the vicinity of an edge dislocation in an elastic crystalline solid. A symmetric Landau potential of sixth-order is employed. Dislocation field and elasticity modify the second-order and fourth-order coefficients of the Landau polynomial, respectively, where the former makes the coefficient singular at the origin. The TDGL equation is solved numerically using a finite volume method, where a wide range of parameter sets is explored. Computations are made for temperatures both above and below the transition temperature of a defect-free crystal . In both cases, the effects of the elastic properties of the solid and the strength of interaction between the order parameter and the displacement field are examined. If the system is quenched below , a steady state is first reached on the compressive side of the dislocation. On the tensile side, the growth is held back. The effect of thermal noise term in the TDGL equation is studied. We find that if the dislocation is introduced above , thermal noise supports the nucleation of the second phase, and a steady state will be attained earlier than if the thermal noise was absent. For a dislocation-free solid, we have compared our numerical computations for a mean-field (spatially averaged) order parameter versus time with the late time growth of the ensemble-averaged order parameter, calculated analytically, and find that both results follow upper asymptotes of sigmoid curves.

Phase structural ordering kinetics of second-phase formation in the vicinity of a crack

The formation of a second phase in presence of a crack in a crystalline material is modelled and studied for different prevailing conditions in order to predict and a posteriori prevent failure, e.g. by delayed hydride cracking. To this end, the phase field formulation of Ginzburg–Landau is selected to describe the phase transformation, and simulations using the finite volume method are performed for a wide range of material properties. A sixth order Landau potential for a single structural order parameter is adopted because it allows the modeling of both first and second order transitions and its corresponding phase diagram can be outlined analytically. The elastic stress field induced by the crack is found to cause a space-dependent shift in the transition temperature, which promotes a second-phase precipitation in vicinity of the crack tip. The spatio-temporal evolution during nucleation and growth is successfully monitored for different combinations of material properties and applied loads. Results for the second-phase shape and size evolution are presented and discussed for a number of selected characteristic cases. The numerical results at steady state are compared to mean-field equilibrium solutions and a good agreement is achieved. For materials applicable to the model, the results can be used to predict the evolution of an eventual second-phase formation through a dimensionless phase transformation in the crack-tip vicinity for given conditions.

Oriented ordering near line defects in crystals

Abstract General properties of directed ordering near line defects, in particular an edge dislocation, in elastic crystals undergoing phase transition are studied using the two-component time-dependent Ginzburg–Landau equation in two dimensions or 2D-XY model. The associated Landau potential comprises a sixth-order term, cubic anisotropy terms and the field of the dislocation. In thermodynamic equilibrium, the phase diagram for the model is delineated. Upon quenching the system below its transition point, the temporal evolution of the order parameter components in the vicinity of the defect is numerically evaluated. The development of vortices, emanated from the model, is explored and their interaction with the dislocation is examined. The dislocation produced a vortex free circular region whose diameter grew almost linearly with time. The time-dependence of vortex density for various settings of the Landau potential coefficients are evaluated. The vortex density (in 2D) decreased inversely with time, albeit faster in the absence of dislocation. By computing the two-point correlation function, we established that the dynamic scaling law is satisfied for the considered model if the distance is scaled by or by its half-width for a dislocation free crystal. Finally, phase transitions in improper ferroelectrics in the context of the model are discussed.

Fracture mechanisms of a thin elastic plastic laminate

16th European Conference of Fracture (ECF16), Failure Analysis of Nano and Engineering Materials and Structures Alexandroupolis, Greece, July 3-7, 2006