Acta Metall Research Articles

Evaluation of critical fracture energy parameter Gfr and assessment of its transferrability

Prediction of maximum load bearing capacity and crack growth for ductile materials using existing models like J-R curve approach has the problem of transferability and the use of micro-mechanical model (e.g. Gurson Tvergaard, and Needleman [Tvergaard V, Needleman A. Analysis of cup cone fracture in a round tensile bar. Acta Metall 1984;32:157–169]) are limited by the requirements of the huge computation time and large numbers of critical metallurgical parameters as input to analysis. Marie and Chapuliot [Marie S, Chapuliot S. Ductile tearing simulation based on local energy criterion. Fatigue Fract Engng Mater Struct 1998;21:215–227] of CEA, France, proposed a simple but convenient ductile crack growth model using critical fracture energy ( G fr) for crack growth and J i for initiation, both of which are material parameters. They also proposed several schemes, namely, graphical and slope of modified plastic J-integral vs crack growth, J M-pl − Δa methods for the evaluation of the value of G fr from specimens as well as from components. In all these methods the role of non-crack displacement in the crack growth process was not considered. The necessary modifications due to non-crack displacement in the above methods to evaluate the values of G fr was studied and published [Acharyya S, Dhar S, Chattopadhyay J. (2003). The effect of non-crack component on Critical fracture energy on ductile material. Int J Pressure Vessels Piping 2004;81:345–353] by the authors earlier. In this paper, the modified methods and formulation have been applied to evaluate the values of G fr from experimental and FE simulated results for compact tensile (CT), three point bend (TPB) specimens and also from components like pipes and elbows. Then statistical estimation is done from these G fr values to assess whether G fr can be accepted as constant value material parameter. Finally, the mean value of G fr obtained from statistical computation is used as material constant along with crack initiation toughness parameter ( J i) SZW to consider crack growth for FE simulation of load vs load-line–displacement (LLD) and load vs crack growth curves for different specimens and components. Finite element simulated results are compared with the experimental results and good matching between the two for several components are found and maximum error in prediction of maximum load is found to be within 12%.

Strain nonuniformity in GaAs heteroepitaxial films on Si(001) studied by x-ray diffraction

High-resolution x-ray diffraction measurements are used to fully characterize the strain state of relaxed highly mismatched GaAs films, grown on vicinal Si (001) substrates by molecular beam epitaxy. The nonuniformity of the misfit dislocation network at the GaAs∕Si (001) interface is studied by analyzing the profiles of x-ray diffraction peaks and the reciprocal space maps for different reflections. The detailed analysis of the peak positions shows a dependence of the relaxation on the crystallographic direction, with the relaxation being larger in the direction perpendicular to the α-dislocation lines. Based on analytical expressions for the full width at half maximum in the longitudinal and transverse sections, an advanced version of the Williamson-Hall plot [Acta Metall. 1, 22 (1953)] is proposed that takes into account the geometry of dislocation distribution and the scattering geometry. We show that this type of analysis can reveal both the type and density of misfit dislocations. The measured peak widths are attributed to random uncorrelated 60°-type misfit dislocations with density much smaller than the total dislocation density required for lattice-mismatch relaxation. The major part of the GaAs∕Si lattice mismatch is accommodated by periodic arrays of edge-type perfect dislocations that do not cause nonuniform strain in the film. The applied theoretical and experimental analysis is easily applicable on other zinc blende highly lattice-mismatched systems.

Sintering of particles of different sizes

A computer simulation study of the sintering process of cylindrical particles is presented following an earlier work by Pan et al. [Pan J, Le H, Kucherenko S, Yeomans JA. A model for the sintering of spherical particles of different sizes by solid state diffusion. Acta Metall 1998;46:4671–90]. The current paper focuses on the effect that the size of the particles has on the sintering kinetics. The computer simulation revealed a sophisticated behaviour for this apparently well-understood problem. Firstly, it is shown that a smaller particle can grow temporarily at the expense of its larger neighbours, in contradiction to conventional wisdom. There are two conditions for this to happen: (a) the particles must be nano-sized and (b) the smaller particle must be placed between two larger neighbours. Secondly, a pair of nanoparticles of different size can separate from each other in the later stage of their co-sintering process. Thirdly, the extent to which particles approach each other is strongly influenced by degree of constraint that the particles are subjected to by the neighbouring particles. This sophisticated behaviour revealed by the computer simulation questions the relevance of the oversimplistic sintering models to a real powder compact.

Multiscale model for microstructure evolution in multiphase materials: Application to the growth of isolated inclusions in presence of elasticity

We present a multiscale model based on the classical lattice time-dependent density-functional theory to study microstructure evolution in multiphase systems. As a first test of the method, we study the static and dynamic properties of isolated inclusions. Three cases are explored: elastically homogeneous systems, elastically inhomogeneous systems with soft inclusions, and elastically inhomogeneous systems with hard inclusions. The equilibrium properties of inclusions are shown to be consistent with previous results: both homogeneous and hard inclusions adopt a circular shape independent of their size, whereas soft inclusions are circular below a critical radius and elliptic above. In all cases, the Gibbs-Thomson relation is obeyed, except for a change in the prefactor at the critical radius in soft inclusions. Under growth conditions, homogeneous inclusions exhibit a Mullins-Sekerka shape instability [W. Mullins and R. Sekerka, J. Appl. Phys. 34, 323 (1963)], whereas in inhomogeneous systems, the growth of perturbations follows the Leo-Sekerka model [P. Leo and R. Sekerka, Acta Metall. 37, 3139 (1989)]. For soft inclusions, the mode instability regime is gradually replaced by a tip-growing mechanism, which leads to stable, strongly out-of-equilibrium shapes even at very low supersaturation. This mechanism is shown to significantly affect the growth dynamics of soft inclusions, whereas dynamical corrections to the growth rates are negligible in homogeneous and hard inclusions. Finally, due to its microscopic formulation, the model is shown to automatically take into account phenomena caused by the presence of the underlying discrete lattice: anisotropy of the interfacial energy, anisotropy of the kinetics, and preferential excitation of shape perturbations commensurate with the rotational symmetry of the lattice.

High temperature deformation of spinel–zirconia composites: Effect of zirconia content

Creep experiments have been performed on a spinel matrix and two composites, containing 20 and 30 wt.% of zirconia particles, respectively, in stress and temperature ranges 8–200 MPa and 1350–1410 °C. The creep rates may be described as the result of two sequential processes that occur at low and high stress, respectively. In addition, a threshold stress was observed that strongly complicated the determination of the stress and grain size exponents and of the activation energy at low stress. The plastic flow of composites may be deduced from that of the spinel matrix by an inclusion model that considers the zirconia grains as soft inclusions. This description reinforces the role of spinel–spinel boundaries in the deformation of the composites, in agreement with measurements of grain boundary sliding capacity for the two kinds of boundaries present in the composites. A model developed by Artz et al. [E. Artz, M.F. Ashby, R.A. Verrall, Interface controlled diffusional creep, Acta Metall. 31 (1983) 1977–1989] may account for the creep rates of the matrix at low stress with a good accuracy. This model supposes that the boundary dislocation density is the factor that limits the creep rates in this low stress range. The grain boundary sliding is probably accommodated by grain boundary diffusion in the whole stress range.

Note on correction factor for estimating the diameter of embedded cylindrical fibres from metallographic sections

This article recalculates the correction factor for estimating the diameter of aligned cylindrical fibres from random metallographic sections as put forward in the paper of Lewis and Withers [Acta Metall. Mater. 43 (1995) 3685]. Their assertion may contain typographic and arithmetic errors (leading to an error of a factor of 2). The diameter must be estimated from longitudinal metallographic sections where the fibre diameters are partially embedded and therefore cannot be measured directly. In view of the high level of citations of the original paper, it is important to address this problem accurately and completely to ensure the successful application of their suggested method by others. The purpose of this short note is to correct their results.

Numerical analysis of a three-dimensional radially symmetric grain attached to a free crystal surface

We study a three-dimensional bicrystal containing an axially symmetric shrinking grain which is initially a spherical segment attached along a circular groove root to the flat exterior surface of a second grain. Following Mullins [Mullins W. J Appl Phys 1956;27:900; Mullins W. J Appl Phys 1956;28:333; Mullins W. Acta Metall 1958;6:414], a time-dependent problem is formulated for the coupled motion of the grain boundary, the groove root, and the external surface. Numerical solutions calculated using an implicit finite difference scheme indicate that the grain shrinks and disappears in finite time, no non-trivial limiting motion is seen, and there is no pinning of the grain boundary that might be associated with so-called “jerky” motion. Surprisingly enough, the surface groove seems only to minimally affect the grain boundary motion, the groove depth varies non-monotonically in time, and after annihilation of the grain the exterior surface has a profile which depends on the system’s history and contains certain features which can be interpreted as so-called “ghost lines.”

Mechanical behavior of δ-phase Pu–Ga alloys. Part I: Constitutive model

We developed a constitutive model, based on the mechanical threshold strength (MTS) model reported by Follansbee and Kocks [P.S. Follansbee, U.F. Kocks, Acta Metall. 36 (1988) 81], to predict the stress/strain behavior of face-centered cubic (fcc) δ-stabilized plutonium–gallium (Pu–Ga) alloys. Input to the model is derived from our previous work on other fcc metals and published test data for Pu–Ga alloys. The model accounts for the effects of temperature, strain rate, grain size, and alloy content on the constitutive behavior. In Part I, we describe the development of the model and present all the pertinent equations and parameters. In Part II, we validate the model against existing literature data and demonstrate how the model is used to quantify the effects of test condition, grain size, alloy content, and impurity levels.

Acoustic Emission as a Probe of the Kinetics of the Martensitic Transformation in a Shape Memory Alloy

The kinetics of the martensitic transformation in a CuAlMn shape memory alloy (SMA) has been studied using the acoustic emission (AE) technique. It is demonstrated that the volume fraction of martensite as a function of time and temperature can be derived from the measured AE power. The fraction data obtained can be described by the Koistinen and Marburger (Acta Metall. 7 (1959) 59) equation with high accuracy, which indicates that the nucleation of martensite takes place heterogeneously and that the average volume of martensite crystals is constant over the extent of the transformation. The martensite-start temperature determined from the measured AE data is in good agreement with the value found by differential scanning calorimetry (DSC). Furthermore, the results of AE experiments on the SMA are compared with optical Confocal Laser Scanning Microscopy (CLSM) observations of the surface of the SMA. The observations show that both small and large martensite plates are formed both at the beginning and at the end of the transformation, which is in agreement with the assumption of a constant average volume of martensite crystals used in the Koistinen and Marburger model.

Experimental and numerical study on splitting failure of brittle solids containing single pore under uniaxial compression

Under uniaxial compression, brittle solids are frequently observed to fail by splitting parallel to the direction of loading. Although this failure has long been recognized, considerable confusion regarding how such splitting mechanisms develops still remains. Axial propagation of a crack-like flaw inclined to the loading direction of compression is one of the accepted mechanisms. In this paper, the axial propagation of a pore-like flaw as one of the mechanisms of splitting failure is approached. A series of physical and numerical tests of uniaxial compression on samples containing a single hole with varied diameters and sample widths are carried out to investigate the splitting failure, the failure modes and strength characterization due to crack growth from such flaw. Samples made from the Hong Kong granite are used in the physical tests to study the inference of heterogeneity of solid on crack growth. A Material Failure Process Analysis code (MFPA 2D) is used as a numerical simulation tool to compare and explain the mode of failure from experiment. Sammis and Ashby [Sammis, C.G., Ashby M.F., 1986. The failure of brittle porous solids under compressive stress states. Acta Metall. 34, 511–526] theoretical crack model is employed to study the inference of sample size and diameter of hole on peak strength.

Crossover from diffusion-limited to kinetics-limited growth of ice crystals

Experimental study of the free growth of an ice crystal into a supercooled pure water film is presented. We have performed the measurements of main parameters of growing dendrite such as the tip velocity υ t , the main distance z ¯ SB between the tip and the position of the first sidebranch, and the fractal dimension d f of the whole dendrite as functions of initial supercooling of water. The experimental results, i.e. υ t , z ¯ SB , and d f , have been compared with analytical studies [J.S. Langer, H. Müller-Krumbhaar, Acta Metall. 26 (1978) 1681; E. Brener, D. Temkin, Phys. Rev. E 51 (1995) 351], and numerical simulations [J. Nitmann, H.E. Stanley, J. Phys. A 20 (1987) L981]. It is found that at high supercooling there is an essential disagreement between present results and predictions of diffusional models of dendritic growth. We show that the disagreement between experiment and theory is due to crossover from diffusion-limited to kinetics-limited growth of ice crystal. The anisotropic growth kinetics of ice crystals is discussed.

Effects of microstructural inhomogeneity on dynamic grain growth during large-strain grain boundary diffusion-assisted plastic deformation

Mesoscale simulations of grain-boundary (GB) diffusion creep in which GB migration-induced static grain growth is suppressed were carried out based on the variational principle of dissipated power. Assuming that the boundaries exhibit no sliding resistance in response to shear stress, the variation of the normal-stress distribution and the diffusive fluxes along the grain boundaries during Coble creep were analysed. The effects of microstructural inhomogeneity, including topological and physical inhomogeneity (i.e. distributions in the grain sizes and GB diffusivities) were investigated. We find that because of the lack of GB migration as an accommodation process to relax the stress concentrations in the microstructure, a topologically inhomogeneous microstructure becomes unphysical at high strains (of typically between 50–100%). Consistent with earlier simulations by Pan and Cocks (1993 Comput. Mater. Sci. 1 95), we find that even in the absence of static grain growth an inhomogeneous microstructure exhibits dynamic grain growth induced by grain-switching induced grain-disappearance events. Our simulations also reveal that in a topologically inhomogeneous microstructure, the diffusive fluxes along any given GB can be in the same direction at both delimiting triple points; i.e. qualitatively different from a homogeneous (i.e. regular hexagonal) microstructure in which, according to Spingarn and Nix (1978 Acta Metall. 26 1389), these fluxes oppose each other.

Pinch-off maps for the design of morphologically stable multilayer thin films with immiscible phases

Design maps are developed for the time to pinch-off via grain boundary grooving in multilayer thin films consisting of two immiscible phases. Variables include the (isotropic) interfacial/grain boundary energies, interfacial diffusivities, initial grain aspect ratio, and imposed in-plane strain rate. The maps are based on a new 2D analysis of grooving in multilayer thin films that extends earlier work by Thouless [Thouless MD. Acta Metall Mater 1993;41:1057] for single phase films. A modified Mullins [Mullins WW, J Appl Phys 1957;28:333] parameter B ∼ multi and time scale τ multi are found to control grooving kinetics, with initial groove depth scaling as t 1/4, t 3/4, or t, depending on the magnitude of in-plane strain rate and geometry. The maps predict that nano-scale films may or may not be more unstable to pinch-off at elevated temperature compared to micro-scale counterparts, depending on morphology, interfacial/grain boundary energies, and in-plane strain rate.

Effect of tetragonal distortion introduced by anti-site defect configuration on additional hardening in off-stoichiometric Al-rich Ni 3Al alloys

A theoretical approach to the hardening mechanism of L1 2–Ni 3Al compound with Al-rich off-stoichiometric composition is attempted based on the classical elastic interaction model between an edge dislocation and a unit cell with an equi-atomic composition embedded in Ni 3Al. A tetragonal distortion is expected to arise from a local anisotropic atomic configuration in the unit cell which is recognized as a face-centered tetragonal-L1 0 or distorted B2 NiAl structure around an anti-site Al atom. According to the study by Cochardt et al. (Acta Metall 1955;3:533), it is found that the maximum point force F m on an edge dislocation from the local L1 0 unit cell increases with decreasing the c i / a i ratio of the L1 0 structure toward the stable B2 structure, which is expected from the authors’ recent electronic structure calculations on the stability of L1 0–NiAl relative to B2-NiAl. This increment of F m by decreasing the c i / a i ratio accounts for a larger hardening in Al-rich Ni 3Al than that in Ni-rich Ni 3Al at 77 K.

Heterogeneity Influence on Electric Field Induced Piezoelectric Microfracture

Spatial variations in piezoelectric material properties can influence localized residual stresses under electro-mechanical loading, which has been shown to contribute to microfractures (Jiang, Q., Subbarao, E.C. and Cross, L.E. 1994. ''Grain Size Dependence of Electric Fatigue Behavior of Hot Pressed PLZT Ferroelectric Ceramics,'' Acta Metall. Mater., 42(11):3687–3694; Lynch, C.S. 1998. ''Fracture of Ferroelectric and Relaxor Electro-ceramics: Influences of Electric Field,'' Acta Mater., 46(2):599–608; Wang, Z., Jiang, Q., White, G.S. and Richardson, A.K. 1998. ''Processing Flaws in PZT Transducer Rings,'' Smart Mater. Struct., 7:867–873). The effect of residual stress on piezoelectric microfracture has been modeled by introducing a crack at the edge of a piezoelectric elliptic inclusion with dissimilar piezoelectric matrix material properties. Piezoelectric weight functions were used to assess changes in intensity factors and energy release rates when an inclusion is present. The shape of the elliptic inclusion is shown to have an effect on local driving forces. Additionally, comparison of impermeable and permeable crack face boundary conditions illustrate the importance of applying the more 'physical' permeable conditions to achieve positive flaw-localized driving forces under electrical loading.

Measurement of the size effect in the yield strength of nickel foils

There is a growing consensus that materials become stronger in small volumes and in the presence of large strain gradients. It has not been clear whether this is due to increased resistance to the motion of dislocations, fewer dislocations, or increased difficulty of multiplying dislocations in these situations. A classic experiment by Stölken and Evans (J.S. Stölken and A.G. Evans, Acta metall. 46 5109 (1998)) showed that thin nickel foils under bending display increased strengthening at large plastic strain values and, correspondingly, large plastic strain gradients. We have adapted their technique to small strains, and report preliminary data for the stress–strain curves of thin nickel foils through the elastic–plastic transition. These data show unambiguously that the yield strength is greater in the thinner foils. The strengthening is additive to the Hall–Petch effect, and is consistent with a size effect at the onset of plastic deformation.

Distribution of solute atoms round a moving dislocation

Cottrell and Jaswon [A.H. Cottrell, M.A. Jaswon, Proc. R. Soc. Lond. A 199 (1949) 104–114] gave the detailed theory of the distribution of solute atoms which interact weakly with a slowly gliding dislocation. Without giving complete solutions, we show how the analysis can be modified to treat a climbing dislocation. As already indicated by Takeuchi and Argon [S. Takeuchi, A.S. Argon, Acta Metall. 24 (1976) 883–889], the results are similar for glide and for climb. Attention is also drawn to the case in which the interaction between the dislocation and a solute atom is large in comparison with thermal energies, and the solute atmosphere approximates a line of excess concentration rather than a concentration dipole. The dilatation-free stress field of this line of dilatation interacts with solute atoms which produce uniaxial strains.

Viscous grain-boundary sliding and grain rotation accommodated by grain-boundary diffusion

When the sliding of a viscous grain boundary is accommodated by grain-boundary diffusion, we evaluate the sliding rate and the stress distribution on the boundary, by employing the energy-balance method developed by Mori et al. [Mori T, Nakasone Y, Taya M, Wakashima K. Phil Mag Lett 1997;75:359 and Mori T, Koda M, Monzen R, Mura T. Acta Metall 1983;31:275]. The phenomena analyzed are the sliding of non-planar grain boundaries, the sliding of boundaries containing particles and the rotation of polygonal grains in two dimensions. For given grain-boundary configurations or grain shapes, we obtain respective analytical expressions for the sliding and rotation rates in a steady state, as a function of the viscosity for the boundary sliding. The obtained sliding and rotation rates are different from the existing ones. In particular, the sliding rates of non-planar grain boundaries have different expressions from the classical predictions by Raj and Ashby [Raj R, Ashby MF. Metall Trans 1971;2:1113]. The sliding/rotation rate and the normal stress on the grain boundary decrease with an increase in the boundary viscosity. As the boundary viscosity becomes sufficiently large, the deformation rate is dominated by viscous grain-boundary sliding, and the contribution of grain-boundary diffusion becomes negligible.

Space charge characterisation by EDS microanalysis in spinel MgAl 2O 4

It was already showed that spinel presents a grain boundary sliding deformation accommodated by diffusion during creep at high temperature [Béclin, F., Duclos, R., Crampon, J. and Valin, F., Microstructural superplastic deformation in MgO·Al 2O 3 spinel. Acta Metall. Mater. 1995, 43, 2753–2760; Addad, A., Etude de la plasticité haute température de matériaux céramiques spinelle-zircone. Ph.D. thesis, Laboratoire de Structure et Propriétés de l’Etat solide, Villeneuve d’Ascq, 1995]. A space charge layer at grain boundary in ionic ceramics can explain the observed interface reaction controlling diffusion at low stress (less than 60 MPa). The nonstoichiometric area due to ionic defects should create an electrostatic potential between the grain surface and inside grain. The purpose of this work is to study the grain boundary region stoichiometry to confirm this theory. Fine-grained spinel (average grain size under micron), which presents an interface reaction at low stresses, is studied by TEM nano-scale microanalysis. The cation ratio Al/Mg variation is well described. It varies from 2.34 at grain boundary to 2.10 at 100 nm inside the grain. The ratio Al/O shows no variation across the grain boundary and suggests that the stoichiometry defect observed is due to an excess of magnesium vacancies located at the grain boundary.

Grain Boundary Area, Grain Boundary Curvature and Particle Pinning in an Al-1mass%Mn Alloy Containing Al<sub>6</sub>Mn Precipitates

The usual expression to calculate the limiting grain radius due to particle pinning is RL=ar/f, where r is the particle radius, f is the volume fraction of the particles, RL is the limiting grain radius and a is a constant equal to 1/6 according to Rios[Acta Metall. Vol. 35 (1987) p. 2805]. This form is not convenient for comparison with experimental measurements. In this work two alternative expressions are proposed: 1)SVL=3SVP where SV is the grain boundary area per unit of volume and SVP is the interface area per unit of volume and 2)H=SVP where H is the total grain boundary curvature. These expressions are compared with experimental measurements carried out in a high purity Al-1mass%Mn alloy containing Al6Mn precipitates annealed for 3600 s at temperatures ranging from 500 to 620 oC. Good agreement is obtained between theory and experiment. The relative merits and shortcomings of each expression are discussed in detail from a fundamental and experimental point of view. It is concluded that the first expression is probably the best choice for practical purposes.