Stress intensity factors at tips of branched cracks under various loadings

Several papers have been published on branched cracks by using various analytical methods, but most of them are concerned with special crack geometries or special loading conditions, and often give unreliable values for cracks with short branches or with small branching angles. The purpose of this paper is to give reliable formulae and new results of the stress intensity factors of various branched cracks in a wide plate. The analysis is based on the body force method combined with a perturbation procedure, and the stress intensity factors at the tips of all the branches and the main crack are given by power series formulae. Numerical results for typical branched cracks are discussed.     

Strong interactions of morphologically complex cracks

Previous works on crack morphology have focused on such cracks as a kinked crack, a branched crack, and an inclined array of identical branched cracks. In this paper, the strong interactions between two cracks in two-dimensional solids under remote tension are investigated. Three morphological types are considered: kinks, branches and zigzags. The method of analysis follows the singular integral equation approach in which the deviations from the main cracks are modeled by distributions of dislocations. Investigations are made on the dependence of the stress intensity factors on the asymmetry of the crack configuration, the crack separation, and the shape of the cracks. The results show that (i) strong interactions can have significant effects on the mode mixity of the stress intensity factors, (ii) a small asymmetry of the crack configuration can cause significant changes to the stress intensity factors, and (iii) zigzag cracks with rectangular steps reduce the stress intensity factors more efficiently than those with triangular or trapezoidal steps.     

Plane strain stress intensity factors for branched cracks

A method of calculating stress intensity factors for branched and bent cracks embedded in an infinite body has been developed. The branches are always assumed to be sharp cracks and are modelled by dislocation distributions. The original crack may be either sharp or of elliptical cross-section with finite root radius. Hence, the method which has a precision ±2%, is also applicable to the study of crack branches emanating from elliptical holes and, approximately, also from notches. The following detailed calculations have been made assuming mode I loading: branched sharp crack with branches of equal and different length, bent sharp crack, and one and two crack branches emanating from the crack with a finite root radius. Bending of a sharp crack under mixed mode loading has also been studied. The criteria of maximum tensile stress and maximum energy release rate used in the study of direction of crack propagation are discussed.     

T-stress based fracture model for cracks in isotropic materials

A modified version of the T-stress based fracture model, developed by Cotterell and Rice, is proposed. In this version an experimentally determined T_crit value is included in the model. The model is then applied to a branched crack in an isotropic material, and the direction of growth of the crack is predicted qualitatively. The branched crack problem is solved using the method of dislocations and a singular integral equation is obtained. The singular integral equation is solved using three different numerical techniques and their respective merits are discussed. The stress intensity factors and the T-stress in front of the branched crack tip are evaluated numerically. It is shown that the T-stress and the stress intensity factors are insensitive to the order of the singularity assumed at the reentrant wedge corner of the branched crack. For an uniaxial load and short kink length it is demonstrated that the kink will turn from its initial direction and realign with the main crack. However if the loading is biaxial then the direction of kink growth depends strongly on the applied transverse stress σ_xx. For a longer initial kink length the direction of kink growth depends on both the kink angle and loading.     

Hierarchical weeping willow nano-tree growth and effect of branching on dye-sensitized solar cell efficiency

In this paper we have demonstrated the simple, low cost, low temperature, hydrothermal growth of weeping willow ZnO nano-trees with very long branches to realize high efficiency dye-sensitized solar cells (DSSCs). We also discuss the effects of branching on solar cell efficiency. By introducing branched growth on the backbone ZnO nanowires (NWs), the short circuit current density and the overall light conversion efficiency of the branched ZnO NW DSSCs increased to almost four times that for vertically grown ZnO NWs. The efficiency increase is attributed to the increase in surface area for higher dye loading and light harvesting and also to reduced charge recombination through direct conduction along the crystalline ZnO branches. As the length of the branches increased, the branches became flaccid and the increase in solar cell efficiency slowed down because the effective surface area increase was hindered by branch bundling during the drying process and subsequent decrease in the dye loading.     

Fast growth of branched nickel monosilicide nanowires by laser-assisted chemical vapor deposition

Branched nickel monosilicide (NiSi) nanowires (NWs), for the first time, have been synthesized on Ni foams by laser-assisted chemical vapor deposition using disilane precursor molecules. Studies indicate that 600?°C is the threshold temperature for the growth of a large number of branched NiSi NWs with 100-500 nm long branches extending from the main stems. Below the threshold temperature, unbranched NiSi NWs were obtained. The density of the branched NiSi NWs is relatively higher in comparison to that of the unbranched ones. The growth rate of the branched NiSi NWs at 700?°C is estimated up to 10 μm min(-1). High-resolution transmission electron microscopy and energy-dispersive x-ray spectroscopy of the branched NiSi NWs suggest that the formation of these branched nanostructures is ascribed to the Ni-dominant diffusion process. These NiSi NWs with branched nanostructures could bring them new opportunities in nanodevices.     

Controlled synthesis of ZnO branched nanorod arrays by hierarchical solution growth and application in dye-sensitized solar cells

We demonstrate the controlled synthesis of ZnO branched nanorod arrays on fluorine-doped SnO₂-coated glass substrates by the hierarchical solution growth method. In the secondary growth, the concentration of Zn(NO₃)₂/hexamethylenetetramine plays an important role in controlling the morphology of the branched nanorod arrays, besides that of diaminopropane used as a structure-directing agent to induce the growth of branches. The population density and morphology of the branched nanorod arrays depend on those of the nanorod arrays obtained from the primary growth, which can be modulated though the concentration of Zn(NO₃)₂/hexamethylenetetramine in the primary growth solution. The dye-sensitized ZnO branched nanorod arrays exhibit much stronger optical absorption as compared with its corresponding primary nanorod arrays, suggesting that the addition of the branches improves light harvesting. The dye-sensitized solar cell based on the optimized ZnO branched nanorod array reaches a conversion efficiency of 1.66% under the light radiation of 1000 W/m². The branched nanorod arrays can also be applied in other application fields of ZnO.     

MIXED-MODE FATIGUE THRESHOLDS

Abstract— Near threshold fatigue crack growth under mixed-mode loading and elastic plane-strain conditions has been studied in 316 stainless steel in laboratory air at room temperature. Particular emphasis was placed on the influence of the mode II component. Crack growth from the starter crack, although initially coplanar, branches to be perpendicular to the maximum normal stress. However the threshold for the branched crack growth is controlled not only by mode I displacement, but also by the mode II component. Upper and lower bound curves are obtained for the threshold condition and discussed in terms of crack tip reversed plastic deformation, crack surface rubbing and oxide-induced closure. A theoretical method for predicting the lower bound curve is proposed and compared with the maximum normal stress and strain energy density criteria. The new theory shows the best agreement with experimental results, giving a safe prediction for design purposes.     

Secondary afterfailure fractures due to transversely reflected waves

The method of caustics was employed to study dynamic crack propagation and after failure fracture in PMMA plates containing an almost longitudinal internal crack, forming a small angle with the loading axis. In order to initiate fracture a high strength longitudinal stress pulse was applied producing two symmetric transverse crack branches running under mode-I deformation with very high velocities. After complete fracture of the plate, the re-entrant corners between the initial slant crack and the two transverse crack branches become centers of emanation of new cracks under the influence of the transversely reflected stress pulses created during the rupture of the longitudinal boundaries. The secondary crack branches are strongly curved, propagate under mixed-mode conditions and arrest. During a third step of fracture the arrested secondary branches accelerate again forming new mixed-mode cracks, which are curved, zig-zaged and presenting sometimes local branchings.     

Nanoforest of hydrothermally grown hierarchical ZnO nanowires for a high efficiency dye-sensitized solar cell

In this paper, in order to increase the power conversion efficiency we demonstrated the selective growth of "nanoforest" composed of high density, long branched "treelike" multigeneration hierarchical ZnO nanowire photoanodes. The overall light-conversion efficiency of the branched ZnO nanowire DSSCs was almost 5 times higher than the efficiency of DSSCs constructed by upstanding ZnO nanowires. The efficiency increase is due to greatly enhanced surface area for higher dye loading and light harvesting, and also due to reduced charge recombination by providing direct conduction pathways along the crystalline ZnO "nanotree" multi generation branches. We performed a parametric study to determine optimum hierarchical ZnO nanowire photoanodes through the combination of both length-wise growth and branched growth processes. The novel selective hierarchical growth approach represents a low cost, all solution processed hydrothermal method that yields complex hierarchical ZnO nanowire photoanodes by utilizing a simple engineering of seed particles and capping polymer.     

HPLC and MALDI-TOF MS analysis of highly branched polystyrene: resolution enhancement by branching

Temperature gradient interaction chromatography (TGIC) and matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF MS) were applied for the characterization of highly branched polystyrenes (PS) prepared by linking living polystyryl anions using 4-chlorodimethylsilylstyrene. Reversed-phase (RP)-TGIC showed an unexpectedly high resolution according to the number of branches despite significant overlap of the molecular weight as confirmed by MALDI-TOF MS. The enhancement of the resolution is ascribed to the contribution of the nonpolar groups in the branched PS: the dimethylsilyl groups in the branching unit as well as the sec-butyl initiator groups. As the number of branches increases, the number of nonpolar groups increases, which in turn increases the RP-TGIC retention synergistically with increasing molecular weight. In contrast, a poorer resolution was found in normal-phase-TGIC, in which the nonpolar groups reduce the retention. The resolution in RP-TGIC appears superior to that of liquid chromatography at the chromatographic critical condition (LCCC) of PS. It is seemingly due to the synergistic contribution of the incremental PS molecular weight to the functionality in the branched PS in RP-TGIC while only the functionality contributes to the separation in LCCC. This type of resolution enhancement could be utilized efficiently for the analysis of highly branched polymers such as dendrimers or hyperbranched polymers.     

Seed-mediated synthesis of branched gold nanoparticles with the assistance of citrate and their surface-enhanced Raman scattering properties

We report an easy synthesis of highly branched gold particles through a seed-mediated growth approach in the presence of citrate. The addition of citrate in the growth solution is found to be crucial for the formation of these branched gold particles. Their size can be varied from 47 to 185?nm. The length of the thumb-like branch is estimated to be between about 5 and 20?nm, and changes slightly as the particle size increases. Owing to these obtuse and short branches, their surface plasmon resonance displays a marked red-shift with respect to the normal spherical particles. These branched gold particles exhibit stronger SERS activity than the non-branched ones, which is most likely related to these unique branching features.     

Synthesis and structural characterization of single-crystalline branched nanowire heterostructures

We report the synthesis of three-dimensional single-crystalline branched nanowire heterostructures, where the backbones and branches are assembled with ZnS and CdS, respectively. Growth of branch and backbones with control over the compositions was enabled via sequential seeding of gold nanocluster catalysts. Elemental mapping data confirmed that branched nanowire heterostructures were synthesized with the intended chemical modulation, CdS branches on ZnS backbones. Transmission electron microscopy studies showed that the growth of heterostructure branches occurs epitaxially from the backbone while maintaining single-crystalline structure. This unique class of heterostructures holds great potential in assembling electronics and photonics in three dimensions.     

Superposition method for calculating singular stress fields at kinks, branches and tips in multiple crack arrays

A method is developed for calculating stresses and displacements around arrays of kinked and branched cracks having straight segments in a linearly elastic solid loaded in plane stress or plain strain. The key idea is to decompose the cracks into straight material cuts we call `cracklets', and to model the overall opening displacements of the cracks using a weighted superposition of special basis functions, describing cracklet opening displacement profiles. These basis functions are specifically tailored to induce the proper singular stresses and local deformation in wedges at crack kinks and branches, an aspect that has been neglected in the literature. The basis functions are expressed in terms of dislocation density distributions that are treatable analytically in the Cauchy singular integrals, yielding classical functions for their induced stress fields; that is, no numerical integration is involved. After superposition, nonphysical singularities cancel out leaving net tractions along the crack faces that are very smooth, yet retaining the appropriate singular stresses in the material at crack tips, kinks and branches. The weighting coefficients are calculated from a least squares fit of the net tractions to those prescribed from the applied loading, allowing accuracy assessment in terms of the root-mean-square error. Convergence is very rapid in the number of basis terms used. The method yields the full stress and displacement fields expressed as weighted sums of the basis fields. Stress intensity factors for the crack tips and generalized stress intensity factors for the wedges at kinks and branches are easily retrieved from the weighting coefficients. As examples we treat cracks with one and two kinks and a star-shaped crack with equal arms. The method can be extended to problems of finite domain such as polygon-shaped plates with prescribed tractions around the boundary.     

Branched TiO₂ nanorods for photoelectrochemical hydrogen production

We report a hierarchically branched TiO(2) nanorod structure that serves as a model architecture for efficient photoelectrochemical devices as it simultaneously offers a large contact area with the electrolyte, excellent light-trapping characteristics, and a highly conductive pathway for charge carrier collection. Under Xenon lamp illumination (UV spectrum matched to AM 1.5G, 88 mW/cm(2) total power density), the branched TiO(2) nanorod array produces a photocurrent density of 0.83 mA/cm(2) at 0.8 V versus reversible hydrogen electrode (RHE). The incident photon-to-current conversion efficiency reaches 67% at 380 nm with an applied bias of 0.6 V versus RHE, nearly two times higher than the bare nanorods without branches. The branches improve efficiency by means of (i) improved charge separation and transport within the branches due to their small diameters, and (ii) a 4-fold increase in surface area which facilitates the hole transfer at the TiO(2)/electrolyte interface.     

Moving Griffith crack in an orthotropic material

In this paper, an integral transform technique is employed to solve the plane elastodynamic problem of a crack of fixed length propagating at a constant speed in a uniformly stressed medium. It is assumed that the crack is located in a plane of elastic symmetry of the material. The stresses and strains ahead of the crack tip are determined explicitly and the conditions governing the initial growth of the crack are investigated using two current theories of fracture (Maximum normal stress and Minimum strain-energy density). Based on these theories, it appears that, depending upon the particular orthotropy of the material, the crack may extend in a straight line for all velocities or may immediately branch out at low velocities (compared to the shear wave velocity of the material) or may start propagating along its initial position for small velocities and then, as the velocity increases, may curve and branch out.     

A hybrid finite element formulation of the linear biphasic equations for hydrated soft tissue

A hybrid finite element method has been developed for application to the linear biphasic model of soft tissues. The biphasic model assumes that hydrated soft tissue is a mixture of two incompressible, immiscible phases, one solid and one fluid, and employs mixture theory to derive governing equations for its mechanical behaviour. These equations are time dependent, involving both fluid and solid velocities and solid displacement, and will be solved by spatial finite element and temporal finite difference approximation. The first step in the derivation of this hybrid method is application of a finite difference rule to the solid phase, thus obtaining equations with only velocities at discrete times as primary variables. A weighted residual statement of the temporally discretized governing equations, employing C° continuous interpolations of the solid and fluid phase velocities and discontinuous interpolations of the pore pressure and elastic stress, is then derived. The stress and pressure functions are chosen so that the total momentum equation of the mixture is satisfied; they are jointly referred to as an equilibrated stress and pressure field. The corresponding weighting functions are chosen to satisfy a relationship analogous to this equilibrium relation. The resulting matrix equations are symmetric. As an illustration of the hybrid biphasic formulation, six-noded triangular elements with complete linear, several incomplete quadratic, and complete quadratic stress and pressure fields in element local co-ordinates are developed for two dimensional analysis and tested against analytical solutions and a mixed-penalty finite element formulation of the same equations. The hybrid method is found to be robust and produce excellent results; preferred elements are identified on the basis of these results.     

Singular behavior of the stress field at the wedge-shaped corners of branching cracks

Summary The method of homogeneous solutions using the self-similarity nature of certain field variables is employed to investigate the behavior of the stress field in the vicinity of wedge-shaped corners of branching cracks, a problem which poses difficulty when attempted by other methods. Closed form solutions are obtained by using the theory of complex variables. Two examples are studied; 1) when an existing crack branches off with constant velocity under an arbitrary angle and 2) when an existing crack bifurcates with different velocities at arbitrary angles. The method presented here can also be extended to study the stress field behavior at the wedge-shaped corners created by any number of branching cracks.     

The 3-dimensional morphology of dendrite during equiaxed solidification of an Al-5 wt.% Cu alloy

In a sample quenched during equiaxed solidification of an Al-5 wt.% Cu alloy, the multi-scales 3-dimensional morphology of equiaxed dendrite was observed. The slim primary stem and secondary branches constitute the frame of dendrite, and rows of dense tertiary branches further divide the 3-dimensional space. In the divided space, the quartic branches grow further. The dendritic branches,which are perpendicular to each other, can change their growth directions and coalesce into a whole. In the tertiary branches and quartic branches, the formation of double branch structures is induced by competitive growth. The branch that wins in the competitive growth will produce a cabbage-like structure by wrapping the failed branches. In addition, the side branch can also wrap the original parent branch to produce cabbage-like structures. Depending on the historical growth direction, the dendritic arms can form vein-like and spicate structures, and the shapes of single dendritic arm may be the cylinder, plate and trapezoid platform. According to the compositions and etching morphology, the single dendritic arm in the final solidification structures should coalesce from a fine porous structure. The porous structures at different length-scales are principally induced by the preferential growth. Based on 3-dimensional morphology of equiaxed dendrite, a new research object for the investigation of microsegregation was suggested.     

Dissolution driven crack branching in polycarbonate

Stress corrosion, in the form of chemically assisted crack growth, in polycarbonate is examined with focus on crack branching characteristics. Cracks with finite width are observed; this is to be expected for dissolution driven cracking. The cracks branched repeatedly and crack widths before and after branching are measured. Both symmetric and asymmetric branching is found. The dissolution rate is assumed to be a linear function of the strain along the crack surface. In the literature, it is proposed that the crack width is proportional to the square of the mode I stress intensity factor. Energy considerations lead to that the sum of branch widths must equal the width of the unbranched crack. The results from this study correspond fairly well with this assumption. The branching angle is found to be 32°± 12°, which is in line with results for sharp cracks reported in the literature. The mean growth direction of the branches is found to deviate slightly from the expected straight. No significant correlation between angles and crack widths is found. The scatter in results is mainly addressed to the inherent perturbation sensitivity of stress corrosion cracking. Also numerically simulations of crack branching is performed. These results show promising agreement with the experiments.