Weight function calculations for mixedmode fracture problems with the virtual crack extension technique

An efficient finite element method is presented for calculating the stress intensity factors (K_I and K_II) and the weight functions for mixed-mode cracks with one virtual crack extension. The computational efficiency is enhanced through the use of singular elements and the application of colinear virtual crack extension (VCE) technique to symmetric mesh in cracktip neighborhood. This symmetric mesh in crack-tip vicinity permits the analytical separation of strain energy release rate into G_I for Mode I and G_II for Mode II for the mixed fracture problems with the colinear virtual crack extension.

Rice's displacement derivative representation of weight function vector for symmetric crack has been extended to the mixed fracture mode at nodal location (x_i,y_i) with crack length (a) and inclination angle (β) as h_I(II)(x_i, y_i, a, β) = (H/2K_I(II)(∂U_I(II)(x_i, y_i, a, β/∂a).

This equation permits explicit determination of weight functions for the entire structure of a given asymmetric crack geometry with colinear VCE technique. The explicit weight functions for mixed fracture mode depend strongly on the constraint conditions. The method of obtaining the required stress intensity factors of a given asymmetric crack geometry, from the weight function concept under the selected constraint conditions, which are different from constraint conditions used in the available weight functions for the same crack geometry, is also presented in this paper. This is accomplished by combining the predetermined explicit weight functions with the self-equilibrium forces at their application locations. These self-equilibrium forces include both the applied surface tractions and the reaction forces induced from the constraint conditions.     

Magneto-mechanical properties evolution of Fe–C alloy during precipitation process

Evolution of magneto-mechanical properties of 160 ppm Fe–C alloy due to carbides precipitation during isothermal annealing at 473 K (up to t=50×10³ s) was studied by means of classical Barkhausen noise (HBN) and mechanical Barkhausen noise (MBN) effects. The MBN was measured for the torsion mode of load with a torque motor. Also the B(H) hysteresis loop and the coercive field H_c were evaluated using a low-frequency magnetisation set. Magnetic hysteresis losses ΔW₁ were compared with the integral ΔW₂ of HBN intensity over one period of magnetisation and the integral ΔW₃ of MBN intensity over one period of the mechanical load. The internal stress distribution function and the resulting mean level of internal stress parameter σ_i were evaluated from the MBN ‘first load’ data. It was revealed that a correlated increase of ΔW₁ and ΔW₂ parameters exists. However, the relative increase of ΔW₃ is much lower than the relative increase of ΔW₁. The relationship between H_c and σ_i was found to be parabolic. This dependence explained by Néel’s model of the impact of the residual stress level on H_c. The presence of precipitates of type was confirmed by scanning electron microscopy.     

Stress intensity factors and energy release rates of delaminations in composite laminates

Due to the oscillatory characteristics of stresses near interface crack tips, the stress intensity factor K_i, i = I, II, III, should be modified and the energy release rate G_i, i = 1, 2, 3, of each fracture mode calculated by the virtual crack closure method may not exist. Based upon a near-tip solution for interface cracks between dissimilar anisotropic media, a proper definition for the stress intensity factors and energy release rates for general anisotropic bimaterial interface cracks is provided in this paper, which is applicable for the delaminated composites. Moreover, this definition can be reduced to the classical definition for a crack tip in homogeneous media when the two materials become the same. A simple quadratic relation between K_i and _Gi is derived, which is further reduced explicitly for orthotropic bimaterials. The influence of fiber orientation and the coupling among opening, shearing and tearing mode fracture are studied numerically. The results show that the classical stress intensity factors and energy release rates are still the dominant stress intensity and energy release rate of the mixed mode condition induced by the interface.     

Measurement of continuous damage parameter

The present paper introduces several measuring methods of continuous damage parameter derived by the classical definition D = 1 − (A_e/A_a): (1) The measuring method on the basis of D = 1 − (E'/ E); (2) The measuring method on the basis of D = 1 − (ε₁/ε₂); (3) The measuring method on the basis of D = −Δρ_o/ρ_o, (4) The measuring method on the basis of D = A_d/A_a, and comments on these measuring methods.     

Experimental micromechanical evaluation of strength of granular materials: Effects of particle rolling

Biaxial compression tests have been performed on assemblies of oval cross-sectional rods, in an effort to evaluate the effects of interparticle friction, particle shape, and initial fabric on the overall strength of granular materials. The variation in the spatial arrangement of the particles (fabric) and particle rolling and sliding are monitored by taking photoelastic pictures at various stages during the course of deformation. Based on this, the following conclusions are obtained. (1) Particle rolling appears to be a major microscopic deformation mechanism, especially when interparticle friction is large. (2) There are relatively few contacts at which relative sliding is dominant, and this seems to be true even when the assembly reaches the overall failure state; this observation is in contradiction to the common assumption that particle sliding is the major microscopic deformation mode. (3) During the course of deformation and up to the peak stress, new contacts are continually formed in such a manner that the contact unit normals tend to concentrate more in a direction parallel to the maximum principal compression. This concentration of unit normals seems to be closely related to the formation of new column-like load paths which carry the increasing axial stress under constant lateral force. After the peak stress, such a column-like microstructure disappears and considerable rearrangement of the load paths takes place, leading to a more diffused (homogeneous) microstructure in the critical state. (4) If a fabric tensor F_ij, i, J = 1,2,3, is defined to be proportional to the volume average of the quantity m_im_j, where m_i are the rectangular Cartesian components of a unit vector along a vector that connects the centroids of two typical contacting granules, then it appears that the overall stress with components σ_ij tends to become coaxial with the fabric tensor F_ij, as the overall deformation continues. For two-dimensional granules the result σ_ij = _OF_ij + β_OF_jkF_kj (k summed) obtained by Mebrabadi, Nemat-Nasser and Oda (1980) by microchemical modeling is confirmed experimentally; _O and β_O are material parameters.     

Development of Bi-oxide superconducting tapes and coils

We have developed Bi-2212 and 2223 tapes. For Bi-2212, two double stacked pancake type coils were fabricated using Bi-2212/Ag tapes prepared by a combination of the continuous dip-coating process and melt-solidification. A small coil (13 mm inner bore, 46.5 mm outer diameter) was inserted in a conventional superconducting magnet system. In a bias field of 20.9 T, the generated field of the coil was 0.9 T, at an I_c of 310 A (criterion 10⁻¹³ Ωm) at 1.8 K. Thus, the superconducting magnet system achieved the generation of a field of 21.8 T in the full superconducting state. A large coil (20 mm inner bore, 94 mm outer diameter) generated a field of 2.6 T (I_c = 385 A (10⁻¹³ Ωm)) at 4.2 K and 1.53 T (I_c = 225 A (10⁻¹³Ωm)) at 20 K in self-field. For Bi-2223, tapes were prepared by the powder-in-tube technique using Ag-10% Cu-x%M (x = 0–1.0, M = Ti, Zr, Hf or Au) alloy sheaths. The high J_c values of 5–7 × 10⁴ A cm⁻² at 4.2 K and 14 T were obtained for the tapes doped with x = 0.03–0.1 at.% Ti, 0.1 at.% Zr, 0.1 at.% Hf or 0.3% Au. These tapes have a modified Bi-2223 grain structure at the sheath/core interface and also a dense and more aligned microstructure, resulting in higher J_c values.     

Probabilistic description of fatigue crack growth in polycrystalline solids

A stochastic model describing the crack evolution and scatter associated with the crack propagation process has been built on the basis of the discontinuous Markovian process. The evolution and scatter are identified in terms of constant probability curves whose equation is derived as In P_r(i) = B(e^KI₀ − e^Ki), i ≥ I₀, where i is the number of cycles, B and K are crack-length-dependent variables, P_r(i) is the probabiliity of the crack being at position r along the fracture surface after i cycles elapse and I₀ is the minimum number of cycles required for the crack to advance from one position on the fracture surface to the next. The validity of the model is established by comparing the crack growth curves generated for Al 2024-T3 at a specific loading condition with those experimentally obtained.     

Some experimental studies of fatigue slip bands and persistent slip bands during fatigue process of low-carbon steel

Initiation and propagation behaviours of fatigue slip bands and persistent slip bands in lowcarbon steel have been investigated: fatigue slip bands were expressed quantitatively as a volume fraction which was products of the slip bands area (Σ A_s/A₀), the depth (h_s/h₀) and the number of slipped grains (G_s/G₀).

It was shown that distributions of fatigue slip bands and persistent slip bands showed the maximum at the angle division of 0° ˜ 10° (at and near the normal to the stress axis) and decreased gradually with increasing of the angle. And the normalized volume fraction of fatigue slip bands increased linearly in proportion to the number of cycles independently of the stress amplitude σ_a and the angle θ. The increment in volume fraction of fatigue slip bands meant the increment in fatigue damage and the fatigue life decreased in proportion to the increment in volume fraction of fatigue slip bands until its value reached a certain content. Then, the initiation rates of fatigue slip bands, persistent slip bands and stage I microcracks showed the increasing tendency with increasing of stress amplitude σ_a, and the propagating rates from fatigue slip bands to persistent slip bands and from fatigue slip bands to stage I micro-cracks were also similar increasing tendency with stress amplitude.     

Statistical investigation of surface fatigue cracks in large-sized turbine rotor shaft steel

Distribution properties of an initiation life N_i and a propagation life N_p of surface cracks, statistical characteristics of a crack growth rate dl/dN, and a relationship between a scatter of the distributions and a gradient a of S-log N curves in rotating bending fatigue tests were investigated for Ni-Cr-Mo-V steel, using for a large-sized turbine rotor shaft. The distributions of N_i and N_p were expressed as Weibull distributions, and the scatter of them for smooth specimens and for lower stress amplitude σ_a tests were larger than those for notched ones and for higher σ_a tests, respectively. The statistical properties of crack propagation rate were almost similar in both smooth and notched specimens. The relationship between the a and a coefficient of variation η for the distributions of N_i, N_p and a final fracture life N_f was expressed as η = c(a)^−b, where c and b are constants.     

The theory of non-steady state fracture kinetics analysis—1: General theory of crack propagation

The kinetics theory of thermally activated time dependent crack propagation is extended to describe the crack size distribution in non-steady state. The distribution is represented by a series of n differential equations, each expressing the rate of crack tip concentration change over the system of n consecutive energy barriers. The general solution for the set of homogeneous linear first order differential equations developed in this report is of the form ρ_i = Σ_jc_i,je^{−λj_t + C_i}, where ρ_i is the crack tip concentration in the ith valley. The theory takes into consideration the discrete character of solids in contrast with the usual continuum models of fracture studies. The analysis is readily applicable to Regions I and II of stress corrosion cracking.     

Computer simulation of martensitic textures

We consider a Ginzburg-Landau model free energy F(ε, e₁, e₂) for a (2D) martensitic transition, that provides a unified understanding of varied twin/tweed textures. Here F is a triple well potential in the rectangular strain (ε) order parameter and quadratic e₁², e₂² in the compressional and shear strains, respectively. Random compositional fluctuations η(r) (e.g. in an alloy) are gradient-coupled to ε, ˜ − ∑_rε(r)[(Δ_x² − Δ_y²)η(r)] in a “local-stress” model. We find that the compatibility condition (linking tensor components ε(r) and e₁(r), e₂(r)), together with local variations such as interfaces or η(r) fluctuations, can drive the formation of global elastic textures, through long-range and anisotropic effective ε-ε interactions. We have carried out extensive relaxational computer simulations using the time-dependent Ginzburg-Landau (TDGL) equation that supports our analytic work and shows the spontaneous formation of parallel twins, and chequer-board tweed. The observed microstructure in NiAl and Fe_xPd_{1 − x} alloys can be explained on the basis of our analysis and simulations.     

A general method for determining mixed-mode stress intensity factors from isochromatic fringe patterns

A general method is presented for determining mixed-mode stress intensity factors K_I and K_II from isochromatic fringes near the crack tip. The method accounts for the effects of the far-field, non-singular stress, σ_ox. A non-linear equation is developed which relates the stress field in terms of K_I, K_II, and σ_ox to the co-ordinates, r and θ, defining the location of a point on an isochromatic fringe of order N.

Four different approaches for the solution of the non-linear equation are given. These include: a selected line approach in which data analysis is limited to the line θ = π and the K---N relation can be linearized and simplified, the classical approach in which two data points at (r_m, θ_m) are selected where r_m/θ = 0; a deterministic method where three arbitrarily located data points are used; and an over-deterministic approach where m (>3) arbitrarily located points are selected from the fringe field.

Except for the selected line approach, the method of solution involves an iteractive numerical procedure based on the Newton-Raphson technique. For the over-deterministic approach, the method of least squares was employed to fit the K-N relation to the field data.

All four methods provide solutions to 0.1% providing that the input parameters r, θ, and N describing the isochromatic field are exact. Convergence of the iterative methods is rapid (3–5 iterations) and computer costs are nominal. When experimental errors in the measurements of r and θ are taken into consideration, the over-deterministic approach which utilizes the method of least squares has a significant advantage. The method is global in nature and the use of multiple-point data available from the full-field fringe patterns permits a significant improvement in accuracy of K_I, K_II, and σ_ox determinations.     

On the formation of radiation-induced defects in fluoroaluminate glasses

The spectra of electron paramagnetic resonance (EPR) of fluoroaluminate glass (FAG-36) based on mineral usovite Ba₂CaMgAlF₁₄ were studied. The paramagnetic centers responsible for EPR signals were induced by ion bombardment of the substrates prepared from this glass. The N⁺, O⁺, Ar⁺ and Pb + ions with energy E = 150 keV were used. The integrated dose D was 2 × 10¹⁶ ions/cm². It is shown by means of isochronal anneal experiments and computer simulation of the EPR spectra that they contain four components: broad Gaussian line (GL) with g = 2.016 and σ oscillating in the range 30–40; two anisotropic spectra with g_z = 2.016, g_y = 2.009; g_x = 2.001 (FA₁) and g_z = 2.045; g_y = 2.010; g_x = 1.98 (FA₂) as well as narrow isotropic line of Lorentzian shape with g = 2.0025 and ΔH = 0.6 mT. The comparison of obtained results with literature data for γ-irradiated fluoride glasses and ion-implanted oxide glasses of different compositions permitted to conclude that GL is due to hole defects typical of fluoride glasses and localized on several anions (fluorines and oxygen(s)); anisotropic FA₁- and FA₂-spectra are attributed to molecular 0₂⁻-ions, and narrow isotropic signal is supposedly assigned to big molecular ions (O₂O⁻, 0₄⁻ , CO⁺, CO⁻) located in voids of damaged implantation layer.     

Influence of stress ratio at baseline loading on delayed retardation phenomena of fatigue crack growth in A553 steel and A5083 aluminium alloy

The delayed retardation phenomena of fatigue crack growth following a single application of tensile overload were investigated under the baseline loading with the stress ratio, R = σ_min/σ_max, ranging from −1 to 0.5 for A553 steel and A5083 aluminium alloy. Two different overload cycles were applied; the one is the case that the ratio of peak stress range to baseline stress range, r = Δσ₂/Δσ₁, is equal to two and the other is the case that the ratio of maximum peak stress to maximum baseline stress, σ_2max/σ_1max, is equal to two. The retardation took place stronger in aluminium than in steel. Under the condition of r = 2 the normalized number of cycles, N_D/N_C, (N_D: the number of cycles during retardation, N_C: the number of cycles required for propagation through the overload-affected-zone size) decreased slightly as the R ratio increased from −1 to 0.5, while under the condition of σ_2max/σ_1max = 2 the N_D/N_C-values increased drastically as the R ratio increased from −1 to 0 (or the overload ratio, r, increased from 1.5 to 2) in both the materials. These retardation behaviors were expressed theoretically according to the model proposed by Matsuoka and Tanaka [1, 3] by using four parameters: the overload ratio, r, the exponent in Paris equation, m, the overload-affected-zone size, ω_D, and the distance at the inflection point, ω_B.     

Hardness, internal stress and fracture toughness of epitaxial AlxGa1−xAs films

Vickers microhardness indentations of 10 μm (001) oriented epilayers of Al_xGa_1−xAs on GaAs substrates have been utilized to evaluate the hardness H_v, the internal stress, and the fracture toughness K_Ic of the layers as a function of their composition parameter x. The hardness H_v varies linearly according to: (6.9-2.2x) GPa and K_Ic increases linearly with x according to: K_1c = (0.44+1.30x) MPa m^1/2. The influence of the substrate on these measurements was found to be negligible for the layer thickness (10 μm) and the indentation load (0.25 N) used, disregarding internal stresses.

Internal film stresses were evaluated by the bimorph buckling method, and were found to depend on the composition parameter according to σ = 0.13x GPa. These stresses did not notably affect the H_v measurements, but for K_Ic corrections as large as 25% had to be made.

The radial cracks observed were of the shallow Palmqvist type. In contradiction to previous reports on this type of cracking, it was found to initiate during unloading, not during loading, and a physical explanation for this deviation is given. No deep radial/median cracks were observed. It was found important to use expressions based on the correct crack geometry in the K_Ic evaluation. Also, a simple theory for the influence of internal stresses on the K_Ic results has been developed.     

Optical transitions near the band edge in bulk CuInxGa1−xSe2 from ellipsometric measurements

From the analysis of the variation of optical absorption coefficient with incident photon energy between 0.8 and 2.6 eV, obtained from ellipsometric data, the energy E_G of the fundamental absorption edge and E_G′ of the forbidden direct transition for CuIn_xGa_1−xSe₂ alloys are estimated. The change in E_G and the spin-orbit splitting Δ_SO=E_G′−E_G with the composition x can be represented by parabolic expression of the form E_G(x)=E_G(0)+ax+bx² and Δ_SO(x)=Δ_SO(0)+a′x+b′x², respectively. b and b′ are called “bowing parameters”. Theoretical fit gives a=0.875 eV, b=0.198 eV, a′=0.341 eV and b′=−0.431 eV. The positive sign of b and negative sign of b′ are in agreement with the theoretical prediction of Wei and Zunger [Phys. Rev. B 39 (1989) 6279].     

Stress relaxation in the interface between creep particle and elastic matrix-strengthening mechanism

The effect of a power law creep particle on interface behavior between the particle and elastic matrix is investigated by stress analysis. Using the results obtained through the stress analysis, the forces due to interaction of an applied stress and stress concentration with an edge dislocation are determined. The direct interaction between the edge dislocation and the creeping particle is studied under fully relaxed stress conditions. Through the investigation the following results are derived. Stress relaxation in the interface can be caused by power law creep along or by diffusion, or a combination of both mechanisms. The degree of stress relaxation caused by diffusion can be defined in terms of the relaxation time for both boundary diffusion and volume diffusion. The amount of stress relaxation caused by the power law creep particle is characterized by the quantity ₂ which is a function of Γ₀ = 2(1/√3)^{1 + m} × (σ_∞/2μ)^m(σ₀/σ_∞t^m), where m is strain rate hardening exponent, σ_∞ is applied stress, μ is the shear modulus, σ₀ is the material constant of the power law creep particle, and t is elapsed time. The value ₂ = 1.0 corresponds to the fully relaxed condition and ₂ = −0.6 corresponds to the initial state. The time to reach a fully relaxed condition is very sensitive to the strain rate exponent, with the smaller m values leading to longer times. The stress state of complete relaxation in the elastic matrix is equivalent to the solution of a void in an elastic matrix superposed on the solution of positive surface traction on the void. This result is identical to that obtained by Srolovitz et al. [Acta. Metall.32, 1979 (1984)]. When the stress is completely relaxed in the particle, all stress components (σ_r, σ_θand σ_rθ) are relaxed, while in the matrix relaxations are observed only for σ_rand σ_θ. The critical resolved shear stress and critical stress to climb the dislocation in the neighborhood of the particle exceed the Orowan stress. Also, the particle attracts the dislocation. Therefore the strengthening of a power law creep particle in an elastic matrix is caused by the Orowan mechanism and by attraction of the dislocation.     

The electronic structure of colossal magnetoresistive manganites

The ferromagnetic manganites A_{1 − x}B_xMnO₃ (A a trivalent element and B divalent) have been the subject of intensive study in the past few years. These manganites exhibit colossal magnetoresistance (CMR), i.e. their resistance can drop several orders of magnitude under influence of an external magnetic field. Electronic structure calculations show that these manganites are half-metallic: they are conducting for one spin direction exclusively. The possible relation between half-metallic magnetism and colossal magnetoresistance is discussed.     

Magneto-elastic interchange in ferromagnetic alloys

The internal friction δ, exchange integral A, magnetocrystalline anisotropic constant K_I and saturation magnetization M_s of Fe–Cr–Al and Fe–Cr–Al–Si alloys annealed at 1373 and 1473 K are measured. The energy density and volume fraction of domain walls (DWs) of these alloys are calculated based on the theories of ferromagnetism and the magnetic parameters measured. The physical process of irreversible movement of 90° DWs is suggested. The results indicate the dissipated elastic energy per unit volume due to the irreversible movements of 90° DWs is equal in value to the energy density of DWs, that is γ_w/δ_w=λ_sE/2. It is an effect of magneto-elastic interchange in ferromagnetic alloys.     

AlxGa−xAs/GaAs heterostructure characterization by wet chemical etching

Selective wet chemical etching of the Al_xGa_1−xAs/GaAs system has been applied to heterostructure characterization. Samples of LPE grown AlGaAs/GaAs laser double-heterostructures and separate confinement heterostructures as well as antiresonant reflecting optical waveguides heterostructures were treated with “I2 solution” (I₂:KI:H₂O) and hydrochloric acid. These compounds selectively etch the ternary Al_xGa_1−xAs layers, but with different “threshold composition” x_th values (the x value is that above which the etching rate of a given compound increases sharply). Selectively etched samples have been examined by SEM. The experimental dependence of etching rate on the x value for I2 solution has been derived. From this dependence, the x composition of any ternary layer can be estimated simply. Observations were made of the “microscopic” properties of the heterostructure, such as the smoothness of the interfaces and the uniformity of layers. All imperfections resulting from the growth process, such as interface perturbations or compositional nonuniformity of layers, are clearly seen. An additional advantage of this etching technique is its simplicity. It allows quick examination of grown heterostructure for the selection of wafers for further processing.