Comments on the evaluation of the stress intensity factor for a general re-entrant corner in anisotropic bi-materials

This paper illustrates an efficient contour integral procedure to obtain stress intensity factors in combination of the asymptotic analysis with finite element analysis. Note that this set-up is very general: the material can be anisotropic elastic, and the specimen can be built as a bi-material system, notches of arbitrary opening angle can be analyzed (γ = 0 → crack, γ = 180° → free edge).The purpose of this technical note is to comment on three issues in the notch mechanics: the interpretation of the eigenvalue equation, the definition of stress intensity factors, and the effect of the outer contour location on H-integral evaluations.     

Edge cracks in a transversely isotropic hollow cylinder

The analytical solution for the linear elastic, axisymmetric problem of inner and outer edge cracks in a transversely isotropic infinitely long hollow cylinder is considered. The z = 0 plane on which the crack lies is a plane of symmetry. The loading is uniform crack surface pressure. The mixed boundary value problem is reduced to a singular integral equation where the unknown is the derivative of the crack surface displacement. An asymptotic analysis is done to derive the generalized Cauchy kernel associated with edge cracks. It is shown that the stress intensity factor is a function of three material parameters. The singular integral equation is solved numerically. Stress intensity factors are presented for various values of material and geometric parameters.     

Stress intensity factors for a wide range of long-deep semi-elliptical surface cracks, partly through-wall cracks and fully through-wall cracks in tubular members

To calculate the rate of fatigue crack growth in tubular members, one approach is to make use of the fracture mechanics based Paris law. Stress intensity factors (SIF) of the cracked tubular members are prerequisite for such calculations. In this paper, stress intensity factors for circumferential deep semi-elliptical surface crack (a/t > 0.8), semi-elliptical partly through-wall crack and fully through-wall crack cracks in tubular members subjected to axial tension are presented. The work has produced a comprehensive set of equations for stress intensity factors as a function of a/T, c/πR and R/T for deep surface cracks. For the partly through-wall cracks and fully through-wall cracks, two sets of bounding stress intensity factor equations were produced based on which all stress intensity factors within the range of parameters can be obtained by interpolation.     

Thermal mismatch stress of a cylindrical inclusion in a cubic crystal

Thermal mismatch stress of a [0 0 1] cylindrical inclusion in a cubic crystal is theoretically investigated by using complex variable formulation of anisotropic elasticity. The closed-form expressions for the interface stress in terms of material constants are derived. Both the stresses in the inclusion and in the matrix are determined. Some numerical results are provided to illustrate the general features of the stress distribution. The stress intensity factors associated with radial cracks emanating from the inclusion are also determined. Based on these results brief discussions on the possible damage modes of the composite due to thermal mismatch are given.     

A through interface crack between a transversely isotropic pair of materials (+30°/−60°, −30°/+60°)

This study focuses on a delamination between two layers of a fiber-reinforced composite material oriented in the directions θ/(θ − 90°). Two specific interfaces are examined: the +30°/−60° interface and −30°/+60° interface. The delamination in these cases is treated effectively as a crack between two monoclinic materials. The behavior of the stress and displacement fields near the crack tip is studied. The first term of the asymptotic expansion for the stress and displacement fields are found by means of the Stroh and Lekhnitskii formalisms. A general solution is obtained for an interface crack in the x₂ = 0 plane. The crack is between two monoclinic materials with x₂ = 0 a symmetry plane.In order to calculate the stress intensity factors, a three-dimensional interaction energy or conservative M-integral is extended and implemented in conjunction with the finite element method. For the M-integral, the auxiliary fields used are particular cases of the stress and displacement fields obtained earlier. The displacement extrapolation method is also extended for this case. The crack surface displacements obtained from a finite element analysis are employed. The methods are independent of each other; hence, they may be used for validation of the results determined.Three test cases are analyzed to examine the accuracy of the results obtained by means of the M-integral method. In addition, two problems of a central crack in a symmetric composite under different loadings are solved. Those loadings are tension and in-plane shear. Stress intensity factors and the interface energy release rate are obtained along the crack front for all cases.     

Dominant stress region for crack initiation at interface edge of microdot on a substrate

In order to examine the mechanics of crack initiation at the free interface edge of a microcomponent on a substrate, delamination tests are carried out for two specimen shapes of Cr microdots on a SiO₂ substrate. The microdots of the first specimen are shaped like the frustum of a round cone. The Cr microdots are successfully delaminated from the SiO₂ substrate in a brittle manner and the critical load is measured by atomic force microscopy (AFM) with a lateral loading apparatus. Stress analysis reveals that a singular stress field exists near the interface edge and the strength for the crack initiation is governed by the intensified normal stress field. The critical stress intensity parameter is evaluated as K_σC ≈ 0.24 MPa m^0.39. Similar delamination tests are conducted for microdots shaped like the frustum of an oval cone. The stress distributions at the crack initiation of this specimen shape show a higher normal stress than the first specimen shape in the region near the interface edge of about x < 40 nm, while it is lower in the region of about x > 50 nm (x: distance from the edge). This suggests a limitation of conventional fracture mechanics: namely, the crack initiation in these specimens is not uniquely governed by the intensity of the singular field. It is found that the delamination crack is initiated when the averaged stress σ_ya in the region of 90-130 nm reaches 190-270 MPa, regardless of the specimen shape. This indicates that the dominant stress region of crack initiation is roughly estimated as 90-130 nm and the criterion is given in terms of the averaged stress in the region.     

The effect of aspect ratio scaling on hydrostatic stress in passivated interconnects

In this study, numerical work using ANSYS and analytical work based on Eshelby models were performed to examine the effect of aspect ratio scaling on the hydrostatic stress in passivated metal interconnects. Aluminium and copper interconnects passivated with phosphosilicate glass (PSG) with aspect ratios ranging from 1 × 10^− 4 to 100 were studied. Copper interconnects in damascene structure were also studied. The results from analytical models agreed well with numerical results and relevant experimental results. The results showed a decreasing trend of hydrostatic stress with aspect ratio for narrow interconnects, and increasing trend of hydrostatic stress for wide interconnects, with a maximum hydrostatic stress at an aspect ratio of 1. It was observed that there is a large scaling effect. For example, in the case of aluminium interconnect, stress values vary between 50 MPa and 463 MPa. It was also observed from the hydrostatic stress contours that the regions of highest stress do not correspond to the void locations seen experimentally. This implies that it is insufficient to look only at hydrostatic stress for determination of failure sites. Another factor that should be examined is the stress gradient.     

Stress concentration around a strongly oblate spheroidal cavity in a transversely isotropic medium

Expressions for the Eshelby tensor of a strongly oblate spheroidal region in a transversely isotropic medium are given explicitly. Based on the equivalent inclusion method, three dimensional stress concentration around the spherodal cavity subjected to remote uniform loading is analyzed and the associated stress concentration factor is determined. Analogously to two-dimensional blunted cracks, the so called stress rounding factor is introduced so that the connection between the crack tip stress and the stress intensity factor in linear fracture mechanics is established. Numerical values of the stress rounding factor for several representative cases of transversely isotropic symmetry are given.     

Cracking behavior of evaporated amorphous silicon films

The residual stress in amorphous silicon films deposited by evaporation is investigated with different substrate temperatures. The stress measured from all the films studied in this paper is tensile. The level of stress decreases from 580 MPa to 120 MPa with increasing substrate temperature from 60 °C to 350 °C. When the film becomes thicker, strain increases and cracks are formed for stress relaxation. 10 µm thick amorphous Si films are deposited at 350 °C without cracks. This cracking behavior is theoretically studied and confirmed by experiment.     

Increase of stress intensity near interface edge of elastic-creep Bi-material under a sustained load

The transition from small-scale creep to large-scale creep ahead of a crack tip or an interface edge with strong elastic stress singularity at the loading instant causes stress relaxation and the decrease of stress intensity in general. However, this study shows that the stress near the interface edge of bi-material with no or weak elastic stress singularity increases after the loading instant and brings about the stress concentration during the transition. In addition, the creep strain distribution of this bi-material after the loading instant is different from that occurred in the transition of an interface edge with strong elastic stress singularity or a crack tip (notch root). The criterion for the increase or decrease of stress intensity near the interface edge proved by the finite element method is proposed in this study. The stress intensity near the interface edge increases when the elastic stress singularity is lower than the creep stress singularity (λ^el < λ^cr) and vice versa.     

Magnetron sputter deposition of low-stress, carbon-containing cubic boron nitride films using Ar―N2―CH4 gas mixtures

Cubic boron nitride (c-BN) films produced by PVD and plasma-assisted CVD techniques typically exhibit undesired high compressive stresses. One of the effective and feasible methods to reduce stress and hence improve film adhesion has been a controlled addition of a third element into the film during deposition. In the present study, BN films were grown on to silicon substrates using reactive magnetron sputtering with a hexagonal BN target. An auxiliary flow of methane was mixed into argon and nitrogen as the working gas. The deposition was conducted at various methane flow rates at 400 °C substrate temperature, 0.2 Pa total working pressure, and − 250 V r.f. substrate bias. The microstructure of the deposited films was then examined in dependence of the methane flow rate. With increasing methane flow rate from 0 to approx. 2.0 sccm, the fraction of the cubic BN phase in the deposited films decreased gradually down to approx. 75 vol.%, whereas the film stress was reduced much more rapidly and almost linearly in relation to the methane flow rate. At 2.1 sccm methane, the stress became approx. 3 times reduced. Owing to the significantly decreased film stress, adherent, micrometer thick, cubic-phase dominant films can be allowed to form on silicon substrate. The microstructure of the films will be illustrated through FTIR and XRR.     

Effects of stress on the interfacial reactions of metal thin films on (0 0 1)Si

The influences of stress on the interfacial reactions of Ti and Ni metal thin films on (0 0 1)Si have been investigated. Compressive stress present in the silicon substrate was found to retard significantly the growth of Ti and Ni silicide thin films. On the other hand, the tensile stress present in the silicon substrate was found to enhance the formation of Ti and Ni silicides. For Ti and Ni on stressed (0 0 1)Si substrates after rapid thermal annealing, the thicknesses of TiSi₂ and NiSi films were found to decrease and increase with the compressive and tensile stress level, respectively. The results clearly indicated that the compressive stress hinders the interdiffusion of atoms through the metal/Si interface, so that the formation of metal silicide films was retarded. In contrast, tensile stress facilitates the interdiffusion of atoms. As a result, the growth of Ti and Ni silicide is promoted.     

Analysis of cracked plates using an iterative hybrid technique of boundary element method and distributed dislocation method

An iterative hybrid technique of boundary element method (BEM) and distributed dislocation method (DDM) is introduced for solving two dimensional crack problems. The technique decomposes the problem into (n + 1) subsidiary problems where n is the number of crack branches. The required solution will be the sum of these (n + 1) solutions. The first subsidiary problem is to find the stress distribution induced in the plate in the absence of the crack using BEM. All of the remaining subsidiary problems, are stress disturbance ones that will be solved using DDM. The results will be added and compared with the boundary conditions of the original problem. Iteration will be performed between the plate boundaries and crack faces until all of the boundary conditions are satisfied.     

Investigation of failure criteria for a sharp notch

In this study, a method of evaluating the static strength of a V-shaped notch based on the singular stress field at the notch tip is studied. The singular stress fields is defined by two parameters, and , which correspond to the intensities of symmetric stress field and the skew-symmetric field, respectively. Four kinds of fracture criteria are considered; two of them are based on the tensile strength σ _B and the other two are based on the fracture toughness K _IC . The usefulness of the criteria is investigated through the experimental results carried out on plane specimens of acrylic resin having a sharp notch for various notch configurations such as the opening angle, the inclined angle and the notch depth. It is shown that the criteria using stress intensity factor and the energy release rate not sensitive to the length of the virtual crack .     

Stress concentration factors of a general triaxial ellipsoidal cavity

The equivalent inclusion method is applied to solve the stress concentration problem concerning the disturbing effect of a general triaxial ellipsoidal cavity on an otherwise uniform normal stress state. Several useful solutions in simple form for limiting cases are derived and numerical results for general cases are obtained. These findings show the general features of the stress concentration factors around the base equator of the cavity. It is found that (1) when Poisson's ratio of the material is zero, the stress concentration factor does not vary along the equator of the cavity; (2) When the aspect ratio c/b of the cavity is very small, the stress concentration factor is also constant along the equator; (3) In general, the variation of stress concentration factors around the equator is less than 0.17 regardless of Poisson's ratio of the material. Thus, it is concluded that the stress concentration factor may be treated as constant around the equator of an ellipsoidal cavity with only a slight error.     

Instability of threshold voltage under constant bias stress in pentacene thin film transistors employing polyvinylphenol gate dielectric

The instability of threshold voltage and mobility of pentacene thin film transistors using a poly(4-vinylphenol) gate dielectric have been investigated under constant bias stress. The mobility was very stable in vacuum by exhibiting 2% variation after 6 h stress even under the high gate bias stress of V_GS = − 20 V. Meanwhile, we observe a negative shift of threshold voltage under stress in vacuum. This shift is attributed to charges trapped in deep electronic states in pentacene near the gate interface. We propose a model for the negative shift of the threshold voltage and extract the hole concentration, 4.5 × 10¹¹ cm^− 2, needed to avoid the onset of stress effects, resulting in a design rule of the channel width to length ratio larger than 40.     

Intrinsic stress effect on fracture toughness of plasma enhanced chemical vapor deposited SiNx:H films

The apparent fracture toughness for a series of plasma enhanced chemical vapor deposition SiN_x:H films with intrinsic film stress ranging from 300 MPa tensile to 1 GPa compressive was measured using nanoindentation. The nanoindentation results show the measured fracture toughness for these films can vary from as high as > 8 MPa⋅√m for films in compression to as low as < 0.5 MPa⋅√m for the films in tension. Other film properties such as density, Young's modulus, and hydrogen content were also measured and not observed to correlate as strongly with the measured fracture toughness values. Various theoretical corrections proposed to account for the presence of intrinsic or residual stresses in nanoindent fracture toughness measurements were evaluated and found to severely underestimate the impact of intrinsic stresses at thicknesses ≤ 3 μm. However, regression analysis indicated a simple linear correlation between the apparent fracture toughness and intrinsic film stress. Based on this linear trend, a stress free/intrinsic fracture toughness of 1.8 ± 0.7 MPa⋅√m was determined for the SiN_x:H films.     

Development of a high harmonic gyrotron with an axis-encircling electron beam and a permanent magnet

A gyrotron with an axis-encircling electron beam is useful for high frequency operation, because the high beam efficiency is kept even at high harmonic of electron-cyclotron resonance. We have designed and constructed such a gyrotron with a permanent magnet. The gyrotron has already succeeded in operation at the third harmonic and the fourth harmonic resonances. The operation frequencies are 89.3 and 112.7 GHz, respectively. Operation cavity modes are TE₃₁₁ and TE₄₁₁. The permanent magnet system consists of many magnet elements made of NbFeB and additional coils for controlling the field intensities in cavity and electron gun regions. The magnetic field at the cavity region can be varied from 0.97 to 1.18 T. At the optimum condition of the magnetic field intensity, the output power at the third harmonic operation is 2.5 kW. The operation is pulsed, the pulse length is 1 ms and the repetition frequency is 1 Hz. The beam energy and current are 40 kV and 1.2 or 1.3 A. Starting current, beam efficiency and emission pattern also have been measured. In this paper, the operation results of the gyrotron and comparison with the computer simulation results are described.     

Effects of deposition conditions of the Al film in Al/glass specimens and annealing conditions on internal stresses and hillock formations

In the present study, 25 kinds of specimen with five Al-film thicknesses were prepared to investigate the relation between the internal stress formed during the annealing process and the hillocks. In the preparation of specimens, the governing factors including deposition conditions, annealing temperature, and annealing time, were arranged following the orthogonal table of five-level and six-factorial (L₂₅(5⁶)) design. Stoney's formula is applied to describe the internal stresses before and after annealing (σ₀ and σ_f), respectively. The internal stress arising during the annealing process (σ_an) is evaluated using the model developed by Flinn et al. [1]. Then, the response surface methodology (RSM) is used to express the three stress parameters in terms of influential factors. The incipient σ_an value for hillocks appearing in the specimens was found to be between − 28.7 MPa and − 32 MPa in a compressive form. The annealing temperature, time, and Al-film thickness are the three major factors, affecting internal stress σ_an. An increase in the annealing time reduces the tensile stress or increases the compressive stress, or both. The tensile stress decreases and the compressive stress increases during the annealing process with increasing Al film thickness and annealing temperature. The number of hillocks formed in a unit of area is linearly proportional to both σ_an and (σ_f − σ_an).