Cracked semi-infinite cylinder and finite cylinder problems

This work considers the analysis of a cracked semi-infinite cylinder and a finite cylinder. Material of the cylinder is assumed to be linearly elastic and isotropic. One end of the cylinder is bonded to a fixed support while the other end is subjected to axial tension. Solution of this problem can be obtained by superposition of solutions for an infinite cylinder subjected to uniformly distributed tensile load at infinity (I) and an infinite cylinder having a penny-shaped rigid inclusion at z = 0 and two penny-shaped cracks at z = ±L (II). General expressions for the perturbation problem (II) are obtained by solving Navier equations with Fourier and Hankel transforms. When the radius of the inclusion approaches the radius of the cylinder, the end at z = 0 becomes fixed and when the radius of the cracks approach the radius of the cylinder, the ends at z = ±L become cut and subject to uniform tensile load. Formulation of the problem is reduced to a system of three singular integral equations. By using Gauss–Lobatto and Gauss–Jacobi integration formulas, these three singular integral equations are converted to a system of linear algebraic equations which is solved numerically.     

An enhanced global–local higher-order theory for the free edge effect in laminates

According to the double-superposition hypothesis proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Methods Eng 1997;40:1197–212], an enhanced global–local higher-order theory (EGLHT-mn) is developed to analyze the edge-effect problems in laminates. The in-plane displacement field consists of mth-order (9  m  3) polynomial in global coordinate z along the thickness direction and order 3 power series in local coordinate ζ within each layer whereas the transverse deflection is represented by a nth-order (9  n  3) polynomial of global coordinate z. By imposing the free surface conditions and the geometric and the stress continuity conditions at interfaces, the number of variables of the higher-order theory is independent of the number of layers of the laminates. As an improvement to the global–local theory proposed by Li and Liu (1997), the present higher-order theory takes into account all the effects of shear and normal stresses. A three-node triangular element satisfying the requirement of C¹ weak-continuity conditions between elements is also presented. Comparing to previous published results, it is found that the present higher-order theory is capable of treating free-edge problems in symmetric and unsymmetric laminates under extension, bending and thermal loading. Other characteristic of the present theory is that transverse shear stresses can be accurately computed directly from the constitutive equations without smoothing. However, to determine transverse normal stresses, the local equilibrium equation approach has to been adopted.     

SENSITIVITY ANALYSIS OF PROCESSING PARAMETERS BY DIRECT DIFFERENTIATION METHOD FOR LASER SURFACE HARDENING TREATMENT

The present study deals with an application of the finite element method to the sensitivity analysis of processing parameters for laser surface hardening treatment. Computing the sensitivity of temperature distributions to changes in processing parameters allows one to determine the more effective input parameters for laser surface hardening treatment. The interesting processing parameters are taken as the characteristic beam radius (r_b and the laser scan velocity (v). A state equation governing the heat flow in laser surface treatment is analyzed using a three-dimensional finite element method. The response sensitivities of the temperature T versus r_b and v distributions were obtained using a direct differentiation method. To verify the numerical analysis results, hardened layer dimensions of the numerical analysis are compared with experimental results.     

Molecular dynamics simulations of oxygen ion diffusion and superionic conduction in ytterbia-stabilized zirconia

Superionic conduction of oxygen ions in 10 mol% ytterbia-stabilized zirconia (YbSZ) at different temperatures is studied employing molecular dynamics simulations. Eventhough discrete hopping of one or two oxygen ions starts at about 675 K, onset of superionic conduction occurs at about 1200 K when almost all the oxygen ions participate in the hopping process. The activation energy for oxygen ion diffusion is found to be 53.25 kJ mol⁻¹. At this temperature and above, oxygen ionic conductivity exceeds 0.1 Ω⁻¹ cm⁻¹ thereby confirming superionic conduction in the YbSZ material. For 675 K < T < 1200 K, the material acts as a normal ionic conductor. The ionic conductivity values, obtained through our simulation compare well with experimental results. But activation energy for oxygen ion conduction, found from the Arrhenius plot of our simulation is 52.71 kJ mol⁻¹ which is 35% less than experimental value. Radial distribution functions, g(r) show that there is no sharp structural phase transition and no oxygen ion sub-lattice melting in YbSZ material at superionic transition. However, the reduction in, broadening and shifting of the peaks in g(r) for all ionic pairs, at higher temperatures, indicate a volume expansion of the crystal.     

A theory of suppressed nitrogen solubility in Fe–Z–N ternary alloy systems in the relatively high range of concentration of Z (Z=Si or C)

It was shown in earlier work that, for the FeZ_zN_x in the relatively low range of z, the solubility pattern of N as a function of temperature T and nitrogen partial pressure p(N₂) can be interpreted in terms of decreased number θ of available interstitial sites for occupation by N atoms per Fe atom in the Fe lattice in proportion to increased z under the assumption of negligible interaction between two interstitial constituents, N and Z (Z=Si or C). In the present work, solubility reduction of N in ternary FeZ_zN_x with increasing z in the relatively high range of z were analysed on the basis of statistical thermodynamics. The present analysis for molten FeSi_zN_x in the relatively high range of z showed that a correction to the θ versus z relation for the relatively high range of z was desirable taking into account overlapping of blocked interstitial sites for occupation of N atoms in order to reproduce the observed equilibrium pressure–temperature–composition (PTC) relation for this system while the interaction between N and Si remains negligible. On the other hand, for the simulation of equilibrium PTC relation for austenitic FeC_zN_x alloy over the entire range of z analysed, significant N–C interaction was important in addition to the blockage effect for the available interstitial sites for occupation of N atoms.     

Bending and free vibration analysis of shear deformable laminated composite beams by finite element method

In the present investigation a higher-order shear deformation theory and the conventional first-order theory are used to develop a finite element method to analyse accurately the bending and free vibration behaviour of laminated composite beams, using nine-noded isoparametric elements. The higher-order theory assumes all the displacement components, u, v and w, which contain variation up to a cubic power of z. The effects of various parameters such as fibre orientation, stacking sequence, span-to-thickness ratio and support condition on the non-dimensionalised deflections, stresses and fundamental frequencies are investigated. Cases where only the higher-order theory is likely to yield accurate results are highlighted.     

Stress intensity factors for corner cracks emanating from fastener holes under tension

In this paper, previous work associated with the stress intensity factor for corner cracks at fastener holes in finite thickness plates is briefly reviewed. The stress intensity factors for two symmetric quarter-elliptical corner cracks subjected to remote tension are evaluated by using both the quarter-point displacement and J-integral methods based on three-dimensional finite element analyses. The geometry ratios analyzed cover a wide range, i.e. depth ratio a/t: 0.2–0.95, aspect ratio a/c: 0.2–5, and hole radius ratio r/t: 0.5–3. Analysis of the J-integral path independence and mutual comparison of the stress intensity factor results between the two methods demonstrate that the present results are of good numerical accuracy. Deviation of the present results from some other solutions found in the literature is also revealed, particularly from Newman and Raju's equations. It is shown that the difference among these results obtained by the different methods is generally within a reasonable bound of error, but Newman and Raju's equations systematically underestimate (up to 15%) the stress intensity factor for cracks of depth ratio larger than 0.8.     

Structural correlations in light irradiated Ge36Se64 amorphous films—Radial distribution function study

The effect of laser irradiation on the electronic structure of amorphous Ge₃₆Se₆₄ films has been detected by studying the variation of the bond length (r) and the coordination number C_N. The total distribution function T(r) of the as deposited film is characterized by the first coordination sphere corresponding to the superposition of the correlation Ge–Se and Se–Se situated at 2.53 Å. The average estimated C_N is 2.519. The second peak ascribed to the correlation Se–Se lies at 3.85 Å showing good agreement with other published data. After irradiation, the first peak of T(r) shows a considerable shift towards a small r and a reduction of C_N. On the contrary, the second neighbor data shows a slight increase of r and a great increment of C_N value (5.11 before irradiation against 6.59 after irradiation). Study of the variation of both r and C_N values induced by subsequent annealing of the film is also given. The relative concentrations of the GeGe, GeSe and SeSe bonds, as well as, the number of the GeSe₄ tetrahedral per atom are calculated using the continuous random network (CRN) and the chemically order continuous random network (COCRN) models. These calculations argue the presence of Ge₂(Se_1/2)₆ ethane like unit in addition to Ge(Se₄)_1/2 even with the COCRN model. The formation of dynamical bonds during irradiation of the film under study is suggested. Correlation to volume changes during illumination studied by tight binding molecular dynamics computer simulation has been also considered.     

Electro-optic properties of c-axis oriented LiNbO3 films grown on Si(1 0 0) substrate

We developed a spectroscopic–ellipsometric approach to evaluate the electro-optic coefficient of highly c-axis oriented LiNbO₃ films on an Si(1 0 0) substrate grown by electron cyclotron resonance plasma sputtering. Applying an electric field between the TiN transparent top electrode and Si substrate, the interference fringe appearing in the tan Ψ spectrum was slightly modulated by phase retardation in the wavelength domain. The change in effective wavelength was converted to refractive index change, yielding dispersion in the Pockels coefficient (r₃₃) between 0.3 and 0.8 μm. At 633 nm, we obtained an r₃₃ that was 57% of the bulk LN crystal value.     

Micromechanics based analysis of randomly distributed fiber reinforced composites using simplified unit cell model

The geometry of the simplified unit cell (SUC) model [Aghdam MM, Smith DJ, Pavier MJ. Finite element micromechanical modeling of yield and collapse behavior of metal matrix composites. J Mech Phys Solids 48 (2000) 499–528] is extended to study effects of random fiber arrangement on the mechanical and thermal characterizations of unidirectional composites. The representative volume element (RVE) considered in the model consists of an r × c unit cells in which fibers are surrounded randomly by matrix cells. The presented model is general and can be used to predict the behavior of a fibrous composite subjected to thermal and mechanical, normal and shear, loading. The model also is capable of analyzing various combinations of these loading conditions such as off-axis test of unidirectional coupons. Both random and repeating fiber arrays can be considered in the model. Results for the overall thermal and elastic properties of a SiC/Ti metal matrix composite (MMC) show good agreement with both the finite element and other analytical models with repeating fiber arrays. Results of transverse properties also revealed that hexagonal array assumption for fiber arrangement is more realistic than square array assumption.     

Finite element analysis of an interface crack with large crack-tip contact zones

There is considerable ambiguity regarding the limiting values of the strain energy release rate (SERR) components at the tips of a crack lying along the interface between two dissimilar isotropic media. In this paper this aspect is examined using finite element analysis and Modified Crack Closure Integral (MCCI) for a problem in which the material properties are chosen so as to cause a large size crack-tip contact zone. By careful choice of this problem, interpenetration of the crack faces in the crack-tip contact zones is observed for the first time in the finite element analysis. Earlier solutions primarily on remote mode 1 loading reported that SERR components do not converge as the virtual crack extension Δa → 0 and that these components show an oscillatory nature when Δa is less than the contact zone size r_c. In the present work, multipoint constraints are imposed on crack face normal displacements in the contact zone and meaningful results are generated for both remote tension and shear loading cases. The apparent nonconvergence of the SERR components as Δa → 0 can be explained if these components are considered as functions of Δa, and Δa is considered as the actual crack growth step size.     

A fracture criterion for three-dimensional crack problems

A criterion for predicting the growth of three-dimensional cracks is developed on the basis of the strain energy density concept which has been used successfully for treating two-dimensional crack problems. Fracture is assumed to initiate from the nearest neighbor element located by a set of spherical coordinates (r, θ, φ) attached to the crack border. The new fracture surface is described by a locus of these elements whose locations correspond to the strain energy function, dW/dV, being a minimum. The function dW/dV is found to be singular of the type 1/r and is of quadratic form in the three stress intensity factors k₁, k₂ and k₃ expressed through the strain energy density factor S. It is postulated that unstable crack propagation initiates from a region where S reaches a critical value S_cr = r₀(dW/dV)_cr. The locations of failure lying on the fracture surface is determined by holding (dW/dV)_cr = S_min/r₀ constant. The quantity S_min stands for the value of S minimized with respect to θ and φ and r₀ is a radial distance measured from the crack border.

An example of failure prediction for an embedded elliptical crack subjected to both normal and shear loads is presented. According to the S-criterion, fracture initiation takes place at the ends of the minor axis. An unexpected result is that for a narrow elliptical crack and Poisson's ratio of 1/3 the lowest failure load occurs when the uniaxial tensile load makes an angle of approximately 60° with the crack surface and is in the plane of the major axis. This is in contrast to the expectation that the lowest critical load occurs when the uniaxial tension is perpendicular to the crack surface. In the limit as the elliptical crack becomes increasingly narrower, the result reduces to the two dimensional line crack case of Mode I and III loading. The S-criterion is also applied to the failure prediction of three dimensional cracks under compressive loads.     

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.     

Total energy calculation for the uniform electron gas in the GW approximation

Expressing the self-energy operator, Σ, in the GW approximation as GW (where G is the Green function of a free electron and W is the screened interaction potential), we have calculated the total energy of a uniform electron gas in terms of one-particle state energies and normalization factors describing the one-particle state contributions to the ground state of the system, at different densities r_s. The results of our calculations are in rough coincidence with the RPA (random phase approximation) total energies for r_s = 0.5, 0.7, 1.5, 2.5, 3.0, 4.0 and are in good coincidence for r_s = 1.0, despite differences in basic assumptions. r_s is determined according to 4πr_s³a₀³/3 = 1/n where a₀ is the Bohr radius and n is the number of particles per unit volume. At low densities, r_s > 1, we find there are unphysical solutions of Dyson's equation, with negative normalization factors. They were ignored in our total energy calculations. Such solutions were also obtained by Hedin, et al in 1967 but were not correctly interpreted. Our results have implications for the anticipated application of this methology to real materials.     

Experimental and modeling approaches of MR behaviors under large step strains

Rheological properties of MR fluids under large step strain shear are presented in this paper. The experiments were carried out using a rheometer with parallel-plate geometry. Under the large step strain shear, MR fluids behave as nonlinear viscoelastic properties, where the stress relaxation modulus, G(t, γ), shows a decreasing trend with step strain. The experimental results indicate that G(t, γ) obeys time-strain separability. Thus, a mathematical form based on finite exponential serials is proposed to predict MR behavior. In this model, G(t, γ) is represented as the product of a linear stress relaxation, G(t), and the damping function, h(γ), i.e. G(t, γ)=G(t) h(γ). G(t) is simply represented as a three-parameter exponential serial and h(γ) has a sigmoidal form with two parameters. The parameters are identified by adopting an efficient optimization method proposed by Stango et al. The comparison between the experimental results and the model-predicted values indicates that this mathematical model can accurately predict MR behavior.     

Elasticity, shell theory and finite element results for the buckling of long sandwich cylindrical shells under external pressure

The buckling of a sandwich cylindrical shell under uniform external hydrostatic pressure is studied in three ways. The simplifying assumption of a long shell is made (or, equivalently, ‘ring’ assumption), in which the buckling modes are assumed to be two-dimensional, i.e. no axial component of the displacement field, and no axial dependence of the radial and hoop displacement components. All constituent phases of the sandwich structure, i.e. the facings and the core, are assumed to be orthotropic. First, the structure is considered a three-dimensional (3D) elastic body, the corresponding problem is formulated and the solution is derived by solving a set of two linear homogeneous ordinary differential equations of the second-order in r (the radial coordinate), i.e. an eigenvalue problem for differential equations, with the external pressure, p the parameter/eigenvalue. A complication in the sandwich construction is due to the fact that the displacement field is continuous but has a slope discontinuity at the face-sheet/core interfaces, which necessitates imposing ‘internal’ boundary conditions at the face-sheet/core interfaces, as opposed to the traditional two-end-point boundary value problems. Second, the structure is considered a shell and shell theory results are generated with and without accounting for the transverse shear effect. Two transverse shear correction approaches are employed, one based only on the core, and the other based on an effective shear modulus that includes the face-sheets. Third, finite element results are generated by use of the ABAQUS finite element code. In this part, two types of elements are used: a shear deformable shell element and a solid 3D (brick) element. The results from all these three different approaches are compared.     

Buckling analysis of functionally graded plates subjected to uniaxial loading

Elastic bifurcational buckling of functionally graded plates under in-plane compressive loading is studied. It is supposed that the gradients of material properties throughout the structure are produced by a spatial distribution of the local reinforcement volume fraction v_f = v_f(x, y, z). To analyze the problem, a method based on a combination of micromechanical and structural approaches is employed. This establishes the effective constitutive behavior at every point of a nonhomogeneous composite plate and provides a buckling criterion. The derived criterion enables one to calculate the critical buckling load R_x^cr for a given distribution v(x, y, z).

Furthermore, with the aim to improve the buckling resistance of the functionally graded plate, the functional R_x^cr(v_f) is maximized. This yields an optimal spatial distribution v_f(x, y, z) of the reinforcement phase.

Results are presented for both short- and long-fiber SiC/Al plates in which the fibers are nonuniformly distributed in the x-, y-, or z-directions. The effects of length-to-width ratio of the plate, and of different types of boundary conditions are studied. Buckling load improvements of up to 100%, as compared to the corresponding uniformly reinforced structure, are shown.     

Structure quantitative map in application for AB2X4 system

Structure maps are constructed for AB₂X₄ (A = metal, B = three-valency element and X = O, S, Se, Te) system (>490 compounds). A way to improve the resolution of the maps is proposed. Crystal structure characteristic (symmetry) is represented on a quantitative scale (coordinate axe Z). The 3D structure diagram with approximating integral plane is constructed and a field of high symmetry compounds is specified with ion radii ratio R(A)/R(X) ≈ 0.60–0.85 and R(B)/R(X) ≈ 0.55–0.70.     

Characterization of mechanical properties of thin films using nanoindentation test

Using finite element modeling (FEM), this work investigates using finite element modeling (FEM) the mechanical behavior of film on substrate composites during the penetration of a rigid tip. In order to understand the magnitude of the substrate effect, the difference of strain gradient through the thickness of a given layer, deposited first on a softer substrate and then on an harder substrate can be observed. In this specific case, up to a critical ratio (h/t) = 0.35 (with h the indentation depth and t the film thickness), the mechanical behavior of the layer is quite similar. But, for h/t > 0.35, two different behaviors may be observed: (i) in the first case H_f/H_S  1 (with H_f and H_S, respectively, the film and substrate hardness values), the total strain remains contained within the film thickness up to a ratio h/t close to 1 and (ii) in the second case H_f/H_S  1, the total strain extends deeply into the substrate. These results show that the empirical 10% rule is not valid, even for a hard film on a softer substrate. The main error is caused by a wrong estimation of the contact depth between the indenter tip and the film surface. Indeed, the simulation runs exhibit the formation of pile-up depending on the ratios (h/t) and Y_f/Y_S (with Y_f and Y_S, respectively, the film and substrate yield stress values). As a function of the used model for calculating the contact depth, at least three variation of hardness may be found from load–displacement curves obtained by FEM. In these conditions, it seems ambiguous to try to determine a weighting function to extract meaningful mechanical properties of the thin film. Another way to determine film properties consists in using the loading phase. A relationship between the applied load (P) and the indentation depth (h) is studied during the loading phase. For the case of a soft film on harder substrate (H_f/H_S  1), it is possible to determine the yield stress of the film, from the previous relationship. This approach is applied to experimental amorphous Al₂O₃ films formed by electron beam evaporation on silicon substrate.     

Crack growth resistance characterized by the strain energy density function

Stable ductile fracture of a typical metal alloy is found to be governed by the condition dS/da = const., i.e. the rate change of the strain energy density S with crack length 2a (or a) remained constant. Since fracture and/or yielding are load rate dependent, the incremental theory of plasticity is employed for analyzing crack growth where unloading in the material near the crack can take place. Attention is focused on the energy per unit volume, dW/dV, stored along the prospective path of crack growth. The nearest neighbor continuum element must necessarily be at a finite distance r from the crack front. This leads to the general relation dW/dV = S/R. The critical value (dW/dV)_c representing the area under the uniaxial true stress and strain curve is assumed to correspond with failure of material elements. If yielding and unloading occurred locally, a certain amount of irrecoverable energy will not be available for dissipation during macrocracking. Hence, the threshold energy density must be modified to read as (dW/dV)_c^* < (dW/dV)_c. The quantity (dW/dV)_c may be regarded as the crack growth resistance whose magnitude decreases with increasing distance from the crack tip at which point yielding is most intensified.

The results are displayed graphically and shown that the condition dS/da = const. provides a rational means of collating and interpreting ductile fracture data.