In the present paper, we investigate the influence of corrosion driving forces and interfacial toughness for a coated material subjected to mechanical loading. If the protective coating is cracked, the substrate material may become exposed to a corrosive media. For a stress corrosion sensitive substrate material, this may lead to detrimental crack growth. A crack is assumed to grow by anodic dissolution, inherently leading to a blunt crack tip. The evolution of the crack surface is modelled as a moving boundary problem using an adaptive finite element method. The rate of dissolution along the crack surface in the substrate is assumed to be proportional to the chemical potential, which is function of the local surface energy density and elastic strain energy density. The surface energy tends to flatten the surface, whereas the strain energy due to stress concentration promotes material dissolution. The influence of the interface energy density parameter for the solid-fluid combination, interface corrosion resistance and stiffness ratios between coating and substrate is investigated. Three characteristic crack shapes are obtained; deepening and narrowing single cracks, branched cracks and sharp interface cracks. The crack shapes obtained by our simulations are similar to real sub-coating cracks reported in the literature.
Directional crack growth criteria in compressed elastic–plastic materials are considered. The conditions at the crack tip are evaluated for a straight stationary crack. Remote load is a combined hydrostatic stress and pure shear, applied via a boundary layer assuming small scale yielding. Strains and deformations are assumed to be small. Different candidates for crack path criteria are examined. Maximum non-negative hoop stress to judge the risk of mode I and maximum shear stress for mode II extension of the crack are examined in some detail. Crack surfaces in contact are assumed to develop Coulumb friction from the very beginning. Hence, a condition of slip occurs throughout the crack faces. The plane in which the crack extends is calculated using a finite element method. Slip-line solutions are derived for comparison with the numerically computed asymptotic field. An excellent agreement between numerical and analytical solutions is found. The agreement is good in the region from the crack tip to around halfway to the elastic–plastic boundary. The relation between friction stress and yield stress is varied. The crack is found to extend in a direction straight ahead in shear mode for sufficiently high compressive pressure. At a limit pressure a kink is formed at a finite angle to the crack plane. For lower pressures the crack extends via a kink forming an angle to the parent crack plane that increases with decreasing pressure.
Abstract. A directional crack growth prediction in a compressed homogenous elastic isotropic material under plane strain conditions is considered. The conditions at the parent crack tip are evaluated for a straight stationary crack. Remote load is a combined biaxial compressive normal stress and pure shear. Crack surfaces are assumed to be frictionless and to remain closed during the kink formation wherefore the mode I stress intensity factor KI is vanishing. Hence the mode II stress intensity factor KII remains as the single stress intensity variable for the kinked crack. An expression for the local mode II stress intensity factor k2 at the tip of a straight kink has been calculated numerically with an integral equation using the solution scheme proposed by Lo (1978) and refined by He and Hutchinson (1989). The confidence of the solution is strengthened by verifications with a boundary element method and by particular analytical solutions. The expression has been found as a function of the mode II stress intensity factor KII of the parent crack, the direction and length of the kink, and the difference between the remote compressive normal stresses perpendicular to, and parallel with, the plane of the parent crack. Based on the expression, initial crack growth directions have been suggested. At a sufficiently high non-isotropic compressive normal stress, so that the crack remains closed, the crack is predicted to extend along a curved path that maximizes the mode II stress intensity factor k2. Only at an isotropic remote compressive normal stress the crack will continue straight ahead without change of the direction. Further, an analysis of the shape of the crack path has revealed that the propagation path is, according the model, required to be described by a function y = cxγ , where the exponent γ is equal to 3/2. In that case, when γ = 3/2, predicts the analytical model a propagation path that is selfsimilar (i.e. the curvature c is independent of any length of a crack extension), and which can be described by a function of only the mode II stress intensity factor KII at the parent crack tip and the difference between the remote compressive normal stress perpendicular to, and parallel with, the parent crack plane. Comparisons with curved shear cracks in brittle materials reported in literature provide limited support for the model discussed.
Stress intensity factors are calculated for long plane cracks with one tip interacting with a region of graded material characteristics. The material outside the region is considered to be homogeneous. The analysis is based on assumed small differences in stiffness in the entire body. The linear extent of the body is assumed to be large compared with that of the graded region. The crack tip, including the graded region, is assumed embedded in a square-root singular stress field. The stress intensity factor is given by a singular integral. Solutions are presented for rectangular regions with elastic gradient parallel to the crack plane. The limiting case of infinite strip is solved analytically, leading to a very simple expression. Further, a fundamental case is considered, allowing the solution for arbitrary variation of the material properties to be represented by Fourier's series expansion. The solution is compared with numerical results for finite changes of modulus of elasticity and is shown to have a surprisingly large range of validity. If an error of 5% is tolerated, modulus of elasticity may drop by around 40% or increase with around 60%.
Premature failures in metals can arise from the local reduction of the fracture toughness when brittle phases precipitate. Precipitation can be enhanced at the grain and phase boundaries and be promoted by stress concentration causing a shift of the terminal solid solubility. This paper provides the description of a model to predict stress-induced precipitation along phase interfaces in one-phase and two-phase metals. A phase-field approach is employed to describe the microstructural evolution. The combination between the system expansion caused by phase transformation, the stress field and the energy of the phase boundary is included in the model as the driving force for precipitate growth. In this study, the stress induced by an opening interface crack is modelled through the use of linear elastic fracture mechanics and the phase boundary energy by a single parameter in the Landau potential. The results of the simulations for a hydrogenated (α+β) titanium alloy display the formation of a precipitate, which overall decelerates in time. Outside the phase boundary, the precipitate mainly grows by following the isostress contours. In the phase boundary, the hydride grows faster and is elongated. Between the phase boundary and its surrounding, the matrix/hydride interface is smoothened. The present approach allows capturing crack-induced precipitation at phase interfaces with numerical efficiency by solving one equation only. The present model can be applied to other multi-phase metals and precipitates through the use of their physical properties and can also contribute to the efficiency of multi-scale crack propagation schemes.
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.
The stress driven growth of an expanding precipitate at a crack tip is studied. The material is assumed to be linearly elastic, and the expansion is considered to be isotropic or transversely isotropic. The extent of the precipitate is expected to be small as compared with the crack length and distance to boundaries. The problem has only a single length scale given by the squared ratio of the stress intensity factor and a critical hydrostatic stress that initiates the growth of the precipitate. Therefore, the growth occurs under self-similar conditions. The equations on non-dimensional form show that the free parameters are expansion strain, degree of anisotropy and Poisson's ratio. It is found that the precipitate, once initiated, grows without remote load for expansion strains above a critical value. The anisotropy of the expansion strongly affects the shape of the precipitate but does not have a large effect on the crack tip shielding.
To honour the memory of Knut Bertram Broberg, and to promote continuing work on the many ideas which he shared so generously with students and colleagues throughout his lifetime, a symposium was arranged at University College of Dublin in 2007. The idea was to create an informal atmosphere, and to keep the num- ber of participants fairly low. This has now become a bi-annual symposium with the highlight being one or two lectures by outstanding world-class scientists. The second Broberg Memorial Symposium was arranged by the Lund Institute of Technology in Sweden (LTH) in May 2009. The venue of the conference was a renais- sance castle, Trolleholm, that was put to our disposal by the courtesy of the Rector of LTH. Trolleholm was once the home of Sophia Brahe whose careful obser- vations of planetary orbits later was used by Johannes Kepler to develop his laws of planetary motion. Kepler was at the time working as an assistant to Sophia’s brother Tycho Brahe.