A solution to the thermo-elastic interface crack branching in dissimilar anisotropic bi-material media

An interface crack or delamination may often branch out of the interface in a laminated composite due to thermal stresses developing around the delamination/crack tip when the media is exposed to heat flow induced by environmental events such as a sudden short-duration fire. In this paper, the thermo-elastic problem of interface crack branching in dissimilar anisotropic bi-media is studied by using the theory of Stroh’s dislocation formalism, extended to thermo-elasticity in matrix notation. Based on the complex variable method and the analytical continuation principle, the thermo-elastic interface crack/delamination problem is examined and a general solution in compact form is derived for dissimilar anisotropic bi-media. A set of Green’s functions is proposed for the dislocations (conventional dislocation and thermal dislocation/heat vortex) in anisotropic bi-media. These functions may be more suitable than those which have appeared in the literature on addressing thermo-elastic interface crack branching in dissimilar anisotropic bi-materials. Using the contour integral method, a closed form solution to the interaction between the dislocations and the interface crack is obtained. Within the scope of linear fracture mechanics, the thermo-elastic problem of interface crack branching is then solved by modelling the branched portion as a continuous distribution of dislocations. The influence of thermal loading and thermal properties on the branching behavior is examined, and criteria for predicting interface crack branching are suggested, based on the extensive numerical results from the study of various cases.     

Analysis of deformation-induced crack tip toughening in ductile single crystals by nano-indentation

This paper reports on the experimental examination of the deformation characteristics near a crack tip in a cyclically work-hardened copper single crystal using a 2D surface scans with nano-indentation. The experimental methodology enables the characterization of the primary deformation field near a crack tip via the modulation of the imposed secondary deformation field by a nano-indentation. In a heavily deformed 4-point bend specimen, the measurements showed an existence of an asymptotic field around the crack tip at a distance of R ⩾ 2.5J/σ₀. The measurements also showed the qualitative details of toughness evolution within the high-gradient deformation field around the crack tip. The nature of dislocation distribution (i.e. statistically distributed vs. distributions necessitated by geometry) around the crack tip is quantified. The measurements indicate the dominance of the geometrically necessary dislocation within the finite deformation zone ahead of the tip up to a distance of R ≈ 3J/σ₀. Thereafter, it is confined in radial rays coinciding with the sector boundaries around the crack tip. These measurements elucidate the origin of the inhomogeneous hardening and the size dependent macroscopic response close to crack tip.     

Surface responses induced by point load or uniform traction moving steadily on an anisotropic half-plane

Surface responses induced by point load or uniform traction moving steadily with subsonic speed on an anisotropic half-plane boundary are investigated. It is found that the effects of the material constant on surface displacements are through matrices L^?1(v) and S(v)L^?1(v), while those on surface stress components are through matrices Ω(v) and Γ(v). Explicit expressions for the elements of these four matrices are expressed in terms of elastic stiffness for general anisotropic materials. The special cases of monoclinic materials with symmetry plane at x₁ = 0, x₂ = 0 and x₃ = 0, and the case for orthotropic materials are all deduced. Results for isotropic material may be recovered from present results. For monoclinic materials with a plane of symmetry at x₃ = 0, two of the elements of matrix Ω(v) are found to be independent of subsonic speed.     

Influence of temperature on a carbon–fibre epoxy composite subjected to static and fatigue loading under mode-I delamination

In this paper, interlaminar crack initiation and propagation under mode-I with static and fatigue loading of a composite material are experimentally assessed for different test temperatures. The material under study is made of a 3501-6 epoxy matrix reinforced with AS4 unidirectional carbon fibres, with a symmetric laminate configuration [0°]_16/S. In the experimental programme, DCB specimens were tested under static and fatigue loading. Based on the results obtained from static tests, fatigue tests were programmed to analyse the mode-I fatigue behaviour, so the necessary number of cycles was calculated for initiation and propagation of the crack at the different temperatures. G–N curves were determined under fatigue loading, N being the number of cycles at which delamination begins for a given energy release rate. G_ICmax–a, a–N and da/dN–a curves were also determined for different G_cr rates (90%, 85%, 75%, etc.) and different test temperatures: 90 °C, 50 °C, 20 °C, 0 °C, ?30 °C and ?60 °C.     

Transient motion of an anisotropic elastic bimaterial due to a line source

The transient motion of an anisotropic elastic bimaterial due to a line force or a line dislocation is studied. The bimaterial is assumed to be at rest and stress-free for t < 0. The line source is applied at t = 0 and maintained for t > 0. A formulation which is an extension to Stroh’s formalism for anisotropic elastostatics is employed. The general solution is expressed in terms of the eigenvalues and eigenvectors of a related eigenvalue problem. The method is used to obtain the analytic solutions without the need of performing integral transforms. Numerical examples of the GaAs bimaterial due to a line force or dislocation are presented for illustration.     

Effects of multi-dopants (Zn and Ho) in stabilizing spinel structure for cathode materials in lithium rechargeable batteries—Novel chelated sol–gel synthesis

Multi-doped spinels, namely LiMn₂O₄ and LiZn_xHo_yMn_2−x−yO₄ (x = 0.10–0.18; y = 0.02–0.10), for use as cathode materials for lithium-ion rechargeable batteries were synthesized via sol–gel method, using lauric acid as the chelating agent, to obtain micron-sized particles. The physical properties of the synthesized samples were investigated using differential thermal analysis, Fourier-transform infrared spectroscopy, X-ray diffraction (XRD), scanning electron microscopy (SEM), transmission electron microscopy, energy-dispersive X-ray analysis, and electrochemical methods. XRD showed that LiMn₂O₄ and LiZn_xHo_yMn_2−x−yO₄ have high degrees of crystallinity and good phase purities. The SEM images of LiMn₂O₄ showed an ice-cube morphology with particles of size 1 μm. Charge–discharge studies showed that undoped LiMn₂O₄ delivered the discharge capacity of 124 mA h/g with coulombic efficiency of 95% during the first cycle, whereas doped spinels delivered discharge capacities of 125, 120, and 127 mA h/g in the first cycle with coulombic efficiencies of 96%, 91%, and 91%, respectively.     

Dynamic fracture of concrete – compact tension specimen

The behavior of concrete structures is strongly influenced by the loading rate. Compared to quasi-static loading concrete loaded by impact loading acts in a different way. First, there is a strain-rate influence on strength, stiffness, and ductility, and, second, there are inertia forces activated. Both influences are clearly demonstrated in experiments. Moreover, for concrete structures, which exhibit damage and fracture phenomena, the failure mode and cracking pattern depend on loading rate. In general, there is a tendency that with the increase of loading rate the failure mode changes from mode-I to mixed mode. Furthermore, theoretical and experimental investigations indicate that after the crack reaches critical speed of propagation there is crack branching. The present paper focuses on 3D finite-element study of the crack propagation of the concrete compact tension specimen. The rate sensitive microplane model is used as a constitutive law for concrete. The strain-rate influence is captured by the activation energy theory. Inertia forces are implicitly accounted for through dynamic finite element analysis. The results of the study show that the fracture of the specimen strongly depends on the loading rate. For relatively low loading rates there is a single crack due to the mode-I fracture. However, with the increase of loading rate crack branching is observed. Up to certain threshold (critical) loading rate the maximal crack velocity increases with increase of loading rate, however, for higher loading rates maximal velocity of the crack propagation becomes independent of the loading rate. The critical crack velocity at the onset of crack branching is found to be approximately 500 m/s.     

Turbulent boundary layers over sparsely-spaced rod-roughened walls

Direct numerical simulations (DNSs) of spatially developing turbulent boundary layers (TBLs) over sparsely-spaced two-dimensional (2D) rod-roughened walls were performed. The rod elements were periodically arranged along the streamwise direction with pitches of p_x/k = 8, 16, 32, 64 and 128, where p_x is the streamwise spacing of the rods, and k is the roughness height. The Reynolds number based on the momentum thickness was varied from Re_θ = 300–1400, and the height of the roughness element was k = 1.5θ_in, where θ_in is the momentum thickness at the inlet. The characteristics of the TBLs, such as the friction velocity, mean velocity, and Reynolds stresses over the rod-roughened walls, were examined by varying the spacing of the roughness features (8 ⩽ p_x/k ⩽ 128). The outer-layer similarity between the rough and smooth walls was established for the sparsely-distributed rough walls (p_x/k ⩾ 32) based on the profiles of the Reynolds stresses, whereas those are not for p_x/k = 8 and 16. Inspection of the interaction between outer-layer large-scale motions and near-wall small-scale motions using two-point amplitude modulation (AM) covariance showed that modulation effect of large-scale motions on near-wall small-scale motions was strongly disturbed over the rough wall for p_x/k = 8 and 16. For p_x/k ⩾ 32, the flow that passed through the upstream roughness element transitioned to a smooth wall flow between the consecutive rods. The strong influence of the surface roughness in the outer layer for p_x/k = 8 and 16 was attributed to large-scale erupting motions by the surface roughness, creating both upward shift of the near-wall turbulent energy and active energy production in the outer layer with little influence on the near-wall region.     

Constant moving crack in a magnetoelectroelastic material under anti-plane shear loading

Analytical solutions for an anti-plane Griffith moving crack inside an infinite magnetoelectroelastic medium under the conditions of permeable crack faces are formulated using integral transform method. The far-field anti-plane mechanical shear and in-plane electrical and magnetic loadings are applied to the magnetoelectroelastic material. Expressions for stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived. Field intensity factors for magnetoelectroelastic material are obtained. The stresses, electric displacements and magnetic inductions at the crack tip show inverse square root singularities. The moving speed of the crack have influence on the dynamic electric displacement intensity factor (DEDIF) and the dynamic magnetic induction intensity factor (DMIIF), while the dynamic stress intensity factor (DSIF) does not depend on the velocity of the moving crack. When the crack is moving at very lower or very higher speeds, the crack will propagate along its original plane; while in the range of M_c1 < M < M_c2, the propagation of the crack possibly brings about the branch phenomena in magnetoelectroelastic media.     

Interaction between a screw dislocation and a viscoelastic piezoelectric bimaterial interface

The complex variable method is employed to derive analytical solutions for the interaction between a piezoelectric screw dislocation and a Kelvin-type viscoelastic piezoelectric bimaterial interface. Through analytical continuation, the original boundary value problem can be reduced to an inhomogeneous first-order partial differential equation for a single function of location z = x + iy and time t defined in the lower half-plane, which is free of the screw dislocation. Once the initial, steady-state and far-field conditions are known, the solution to the first order differential equation can be obtained. From the solved function, explicit expressions are then derived for the stresses, strains, electric fields and electric displacements induced by the piezoelectric screw dislocation. Also presented is the image force acting on the screw dislocation due to its interaction with the Kelvin-type viscoelastic interface. The derived solutions are verified by comparing with existing solutions for the simplified cases, and various interesting features are observed, particularly for those associated with the image force.     

Use of equivalent body forces for acoustic emission from a crack in a plate

A method to determine acoustic emission of surface waves from a crack near the free edge of a plate, is presented, in terms of the function f(t), which defines the time dependence of the crack opening process, the crack opening volume per unit thickness of the plate, and the elastic constants of the plate. The determination of the time-varying displacement is based on the use of equivalent body forces, which are shown to be two double forces. The acoustic emission of the crack, or the equivalent radiation from the double forces, has been obtained by a novel use of the elastodynamic reciprocity theorem. It is of interest that the normal surface-wave displacement at a position x₀ of the free edge comes out as depending on df/dt evaluated at x₀ for t > x₀/c_R, where c_R is the velocity of surface waves on the free edge.     

An experimental study of circular sand–water wall jets

Laboratory experiments were carried out to study the effects of sand particles on circular sand–water wall jets. Mean and turbulence characteristics of sand particles in the sand–water wall jets were measured for different sand concentrations c_o ranging from 0.5% to 2.5%. Effects of sand particle size on the centerline sand velocity of the jets were evaluated for sand size ranging from 0.21 mm to 0.54 mm. Interesting results with the range of measurements are presented in this paper. It was found that the centerline sand velocity of the wall jets with larger particle size were 15% higher than the jets with smaller particle size. Concentration profiles in the vertical direction showed a peak value at x/d = 5 (where x is the longitudinal distance from the nozzle and d is the nozzle diameter) and the sand concentration decreased linearly for x/d > 5. Experimental results showed that the turbulence level enhanced from the nozzle to x/d = 10. For sand–water wall jets with a higher concentration (c_o = 1.5–2.5%), the turbulence intensity became smaller than the corresponding single-phase wall jets by 34% due to turbulent modulation. A modified logarithmic formulation was introduced to model the longitudinal turbulent intensity at the centerline and along the axis of the jet.     

Synthesis of layered cathode materials Li[CoxNiyMn1−x−y]O2 from layered double hydroxide precursors

Cathode materials Li[Co_xNi_yMn_1?x?y]O₂ for lithium secondary batteries have been prepared by a new route using layered double hydroxides (LDHs) as a precursor. The resulting layered phase with the α-NaFeO₂ structure crystallizes in the rhombohedral system, with space group R-3m having an interlayer spacing close to 0.47 nm. X-ray photoelectron spectroscopy (XPS) was used to measure the oxidation states of Co, Ni and Mn. The effects of varying the Co/Ni/Mn ratio on both the structure and electrochemical properties of Li[Co_xNi_yMn_1?x?y]O₂ have been investigated by X-ray diffraction and electrochemical tests. The products demonstrated a rather stable cycling behavior, with a reversible capacity of 118 mAh/g for the layered material with Co/Ni/Mn = 1/1/1.     

An improved atomistic simulation based mixed-mode cohesive zone law considering non-planar crack growth

A novel and improved atomistic simulation based cohesive zone law characterizing interfacial debonding is developed which explicitly accounts for the non-planarity of the crack propagation. Group of atoms in the simulation constituting cohesive zones which are used to obtain local stress and crack opening displacement data are determined dynamically during the non-planar crack growth as they cannot be determined apriori. The methodology is used to study the debonding of Σ5 (2 1 0)/[0 0 1] symmetric tilt grain boundary interface in a Cu bicrystal under several mixed mode loading conditions. Simulations show that such bicrystalline specimen exhibits three types of energy dissipative mechanisms – shear coupled GB migration (SCM) away from the crack-tips, change in spacial orientation of GB structural units rendering highly disordered grain boundary near the crack tips and brittle intergranular fracture. Which combination of these three deformation mechanism will be active influencing the degree of non-planarity of the crack propagation at various stages of loading depends on the loading mode-mixity. As the ratio of shear component of the loading parallel to the GB plane and normal to the tilt axis with respect to the normal loading increases (thereby increasing the mode-mixity), overall strain-to-failure also increases and SCM tends to become the dominant deformation mechanism. Through this framework, analytical functional forms and parameters describing cohesive laws for both normal and shear traction as a function of the mode-mixity of the loading and crack opening displacement are predicted.     

Dynamic fracture of concrete compact tension specimen: Experimental and numerical study

Compared to quasi-static loading concrete loaded by higher loading rates acts in a different way. There is an influence of strain-rate and inertia on resistance, failure mode and crack pattern. With increase of loading rate failure mode changes from mode-I to mixed mode. Moreover, theoretical and numerical investigations indicate that after the crack reaches critical velocity there is progressive increase of resistance and crack branching. These phenomena have recently been demonstrated and discussed by O?bolt et al. (2011) on numerical study of compact tension specimen (CTS) loaded by different loading rates. The aim of the present paper is to experimentally verify the results obtained numerically. Therefore, the tests and additional numerical studies on CTS are carried out. The experiments fully confirm the results of numerical prediction discussed in O?bolt et al. (2011). The same as in the numerical study it is shown that for strain rates lower than approximately 50/s the structural response is controlled by the rate dependent constitutive law, however, for higher strain rates crack branching and progressive increase of resistance is observed. This is attributed to structural inertia and not the rate dependent strength of concrete. Maximum crack velocity of approximately 800 m/s is measured before initiation of crack branching. The comparison between numerical and experimental results shows that relatively simple modeling approach based on continuum mechanics, rate dependent microplane model and standard finite elements is capable to realistically predict complex phenomena related to dynamic fracture of concrete.     

Electrostatic stress analysis of an anisotropic piezoelectric half-plane under surface electromechanical loading

The generalized stress components on an anisotropic piezoelectric half-plane boundary under surface electromechanical loading are investigated. It is found that the behaviors of generalized stress components are related to matrices Γ and Ω, which have the same form as those for the purely elastostatic problem. Matrices Γ and Ω contain all the electro-mechanical coupling phenomena of the generalized stress components. All elements of matrices Γ and Ω are expressed explicitly in terms of generalized elastic stiffness for monoclinic piezoelectric materials with the plane of symmetry at x₃ = 0 and for transversely isotropic piezoelectric materials in which the coupled effects between the mechanical (electrical) deformations induced by electrical (mechanical) loadings are studied analytically. A numerical example of the electro-mechanical coupling behavior for PZT-4 is also given.     

Stress singularity due to traction discontinuity on an anisotropic elastic half-plane

In a half-plane problem with x₁ paralleling with the straight boundary and x₂ pointing into the medium, the stress components on the boundary whose acting plane is perpendicular to x₁ direction may be denoted by t₁ = [σ₁₁, σ₁₂, σ₁₃]^T. Stress components σ₁₁ and σ₁₃ are of more interests since σ₁₂ is completely determined by the boundary conditions. For isotropic materials, it is known that under uniform normal loading σ₁₁ is constant in the loaded region and vanishes in the unloaded part. Under uniform shear loading, σ₁₁ will have a logarithmic singularity at the end points of shear loading. In this paper, the behavior of the stress components σ₁₁ and σ₁₃ induced by traction-discontinuity on general anisotropic elastic surfaces is studied. By analyzing the problem of uniform tractions applied on the half-plane boundary over a finite loaded region, exact expressions of the stress components σ₁₁ and σ₁₃ are obtained which reveal that these components consist of in general a constant term and a logarithmic term in the loaded region, while only a logarithmic term exists in unloaded region. Whether the constant term or the logarithmic term will appear or not completely depends on what values of the elements of matrices Ω and Γ will take for a material under consideration. Elements for both matrices are expressed explicitly in terms of elastic stiffness. Results for monoclinic and orthotropic materials are all deduced. The isotropic material is a special case of the present results.     

Periodic group crack problems in an infinite plate

This paper investigates periodic group crack problems in an infinite plate. The periodic group crack is composed of infinite groups with numbering from j = −∞, …, −2, −1, 0, 1, 2, …, to j = ∞, and the groups are placed periodically. The same loading condition and the same geometry are assumed for cracks in all groups. A singular integral equation is used to solve the problems. The singular integral equation is formulated on cracks of the 0th group (or the central group) with the collection of influences from the infinite groups. The influences of many neighboring groups to the central group are evaluated exactly. Meantime, the influences of many remote groups to the central group can be summed up into one term approximately. The stress intensity factors at crack tips can be evaluated from the solution of the singular integral equation. It is found from some sample problems that the obtained results are very accurate. Finally, several numerical examples are presented and interaction among the group cracks is addressed.