Mixed-mode crack-tip stress fields for orthotropic functionally graded materials

Quasi-static mixed mode stress fields for a crack in orthotropic inhomogeneous medium are developed using asymptotic analysis coupled with Westergaard stress function approach. In the problem formulation, the elastic constants E ₁₁, E ₂₂, G ₁₂, ν ₁₂ are replaced by an effective stiffness ${E=\sqrt {E_{11} E_{22}}}$ , a stiffness ratio ${\delta =\left({{E_{11}}\mathord{\left/ {\vphantom {{E_{11}} {E_{22}}}}\right. \kern-0em} {E_{22}}} \right)}$ , an effective Poisson’s ratio ${\nu =\sqrt {\nu_{12}\nu _{21}} }$ and a shear parameter ${k=\left({E \mathord{\left/ {\vphantom {E {2G_{12}}}}\right. \kern-0em} {2G_{12}}}\right)-\nu }$ . An assumption is made to vary the effective stiffness exponentially along one of the principal axes of orthotropy. The mode-mixity due to the crack orientation with respect to the property gradient is accommodated in the analysis through superposition of opening and shear modes. The expansion of stress fields consisting of the first four terms are derived to explicitly bring out the influence of nonhomogeneity on the structure of the mixed-mode stress field equations. Using the derived mixed-mode stress field equations, the isochromatic fringe contours are developed to understand the variation of stress field around the crack tip as a function of both orthotropic stiffness ratio and non-homogeneous coefficient.     

The Effect of History on Hydrogen Assisted Cracking: 1. Coupling of hydrogenation and crack growth

This paper analyzes the effect of process history on K-dominance condition in hydrogen assisted cracking. The performed study of stress-strain assisted diffusion reveals that hydrogenation of the fracture process zone is not a purely K-driven local autonomous phenomenon. Hydrogenation and crack growth are shown to be coupled processes, one influencing the other. Consequently, particular histories of stress intensity factor K and crack size evolutions affect crack growth rate ν that can occur at the same instantaneous K-value in a given material. Thus the crack growth kinetics curve ν=ν(K) is not unique as an intrinsic material property must be. On leave from: Pidstryhach Institute for Applied Mechanics and Mathematics, Lviv, Ukraine This revised version was published online in August 2006 with corrections to the Cover Date.     

Fracture mechanics of piezoelectric materials

This paper presents an analysis of crack problems in homogeneous piezoelectrics or on the interfaces between two dissimilar piezoelectric materials based on the continuity of normal electric displacement and electric potential across the crack faces. The explicit analytic solutions are obtained for a single crack in an infinite piezoelectric or on the interface of piezoelectric bimaterials. For homogeneous materials it is found that the normal electric displacement D₂, induced by the crack, is constant along the crack faces which depends only on the remote applied stress fields. Within the crack slit, the perturbed electric fields induced by the crack are also constant and not affected by the applied electric displacement fields. For bimaterials, generally speaking, an interface crack exhibits oscillatory behavior and the normal electric displacement D₂ is a complex function along the crack faces. However, for bimaterials, having certain symmetry, in which an interface crack displays no oscillatory behavior, it is observed that the normal electric displacement D₂ is also constant along the crack faces and the electric field E₂ has the singularity ahead of the crack tip and has a jump across the interface. Energy release rates are established for homogeneous materials and bimaterials having certain symmetry. Both the crack front parallel to the poling axis and perpendicular to the poling axis are discussed. It is revealed that the energy release rates are always positive for stable materials and the applied electric displacements have no contribution to the energy release rates. This revised version was published online in July 2006 with corrections to the Cover Date.     

Fatigue crack propagation in short-carbon-fiber reinforced plastics evaluated based on anisotropic fracture mechanics

The influence of fiber orientation on crack propagation was studied with single edge-notched specimens cut from injection-molded plates of fiber-reinforced polyphenylene sulfide (PPS). Fracture mechanics parameters were calculated by FEM based on anisotropic elasticity. For mode I crack propagation in specimens parallel (MD) and perpendicular (TD) to molding direction, difference in crack propagation rate, dc/dN, among specimens becomes small when correlated to a crack-tip-opening radius parameter, H_IΔG_I, where H_I is a compliance parameter. Including crack propagation under mixed loading, all the data tend to merge a single relation when correlated to total energy-release-rate range divided by Young’s modulus, ΔG_total/E_θ.     

Optical and structural properties of flash evaporated HgTe thin films

Thin films of HgTe were thermally flash evaporated onto glass and quartz substrates at room temperature. The structural investigations showed that stoichiometric and amorphous films were produced. The transmittance, T, and reflectance, R, of thin films of HgTe have been measured over the wavelength ranges 300–2500 nm. From analysis of the transmittance and reflectance results, the refractive index, n, and the extinction coefficient, k, has been studied. Analysis of the refractive index yields a high frequency dielectric constant, ɛ_∞, and the energy of the effective oscillator, E_o, the dispersion energy, E_d, the covalent value β and the M₋₁ and M₋₃ moments of the imaginary dielectric function of optical spectrum. Also, the dependence of the real part dielectric constant ɛ₁(hν) on its imaginary part ɛ₂(hν) of HgTe films can be used to determine the molecular relaxation time τ, the distribution parameter α^\ and the macroscopic (electronic) relaxation time τ_o. The graphical representations of surface and volume energy loss functions, dielectric constant, the optical conductivity as well as the relaxation time as a function of photon energy revealed three transitions at 0.63, 2.21 and 2.76 eV.     

Preparation,microstructure and electrorheological property of nano-sized TiO<Subscript>2</Subscript> particle materials doped with metal oxides

New nano-sized TiO₂ electrorheological (ER) materials doped with different metal (M = Na, Zr, Ce, Al, Ca, Zn) oxides have been prepared. Relationships between the composition, microstructure, conductivity, dielectric property and ER effect of these materials have been studied. The results show that doping Na₂O, ZrO₂, Al₂O₃ or CeO₂ can enhance the ER performance of the TiO₂ material, whereas, doping CaO or ZnO would decrease the ER activity of the material. The shear stress (τ_E) of the suspension (25 wt%) of Na-doped TiO₂ in dimethyl silicone oil reaches 1.6 kPa at the electric field strength E = 4.2 kV/mm and shear rate γ = 300 s⁻¹, and its τ_r value of 54.6 (τ_r = τ_E/τ₀, where τ₀ is the shear stress at no electric field) is seven times higher than that of pure TiO₂ suspension. This high τ_r value is very advantageous to the use. The dielectric loss tangent (tanδ) plays a dominant role in influencing the ER performance of a particle material, and the effect of the surface area (pore volume, especially) and grain size should be taken into account.     

Strain Measurement on Composites: Effects due to Strain Gauge Misalignment

Abstract: The present work analyses the errors affecting the strains measured by misaligned strain gauges installed on orthotropic‐composite laminae. Various analytical relationships are derived showing that, besides the fibre and strain gauge orientations, the misalignment error in unidirectional off‐axis orthotropic composite samples depends also on the lamina stiffness properties (E₁, E₂, ν₁₂, G₁₂). If the fibres are aligned with the loading axis, it is found that the higher Poisson's ratio ν₁₂ is the only elastic property influencing the misalignment error. Experimental results are shown confirming the theoretical predictions.     

Static and energetic fracture parameters for rocks and concretes

The object of the paper is to determine the fracture toughness parameters K_1C,G _1C and J_1C for some aggregative materials. Values of the J-integral are calculated from load-displacement curves, following the procedure suggested by Begley and Landes for steel alloys. Some recurring experimental incoherences are explained applying Buckingham's Theorem for physical similitude and scale modeling to Fracture Mechanics. Thus a non-dimensional parameter can be defined (the test brittleness number), which governs the fracture-sensitivity phenomenon. The fracture parameters K_1C and J_1C are connected by a fictitious Young's modulus E^*, which is lower than the real modulus E and represents the stiffness of the damaged material near the crack tip before the extension. When the specimen sizes are so small that the material becomes fracture insensitive, then E^* appears higher than E.     

Determination of the effective mode-I toughness of a sinusoidal interface between two elastic solids

A finite element model of crack propagation along a sinusoidal interface with amplitude A and wavelength λ between identical elastic materials is presented. Interface decohesion is modeled with the Xu and Needleman (J Mech Phys Solid 42(9):1397, 1994) cohesive traction–separation law. Ancillary calculations using linear elastic fracture mechanics theory were used to explain some aspects of stable and unstable crack growth that could not be directly attained from the cohesive model. For small aspect ratios of the sinusoidal interface (A/λ ≤ 0.25), we have used the analytical Cotterell–Rice (Intl J Fract 16:155–169, 1980) approximation leading to a closed-form expression of the effective toughness, K _Ic , given by where is the work of separation, E is Young’s modulus, and ν is Poisson’s ratio. For A/λ > 0.25, both the cohesive zone model and numerical J-integral estimates of crack tip stress intensity factors suggest the following linear relationship: Parametric studies show that the length of the cohesive zone does not significantly influence K _Ic , although it strongly influences the behavior of the crack between the initiation of stable crack growth and the onset of unstable fracture. An erratum to this article can be found at     

On fast fracture in an elastic-(plastic)-viscoplastic solid Part II – The motion of crack

The motion of a crack in an elastic-(plastic )-viscoplastic medium is studied in terms of an energetic analysis. Combined with the stress and velocity fields obtained in Part 1, Kishimoto's energy integral, , is used as a crack driving force to determine its motion. The major results obtained are: (1) dependence of crack speed on a modified near-field parameter, K _I tip, (or equivalently, a modified dynamic energy release, G _I tip), which is different from the usual stress intensity factor K _I of an elastic crack-tip field but is related to it; (2) influence of inelastic effect, such as the viscoplastic exponent n, on the motion of the crack; and (3) stability condition of crack motion. In particular, for the last point, it has been found that, for a given loading and material coefficients, there exist two possible motions of the crack: one is stable crack growth and the other is unstable fracture. The lower and upper bounds of crack motion are also discussed. It is finally shown that the maximum crack velocity is lower than the Rayleigh wave speed, and is dependent on the viscoplastic exponent of the material.     

Numerical modeling of fracture coalescence in a model rock material

The crack pattern, as well as crack initiation, -propagation and -coalescence observed in experiments on gypsum specimens with pre-existing fractures in uniaxial, biaxial, and tensile loading are satisfactorily predicted with the numerical model presented in this paper. This was achieved with a new stress-based crack initiation criterion which is incorporated in FROCK, a Hybridized Indirect Boundary Element method first developed by Chan et al. (1990). The basic formulation of FROCK is described, and the code verified for both open and closed pre-existing fractures either with only friction or with friction and cohesion. The new initiation criterion requires only three material properties: σ_crit, the critical strength of the material in tension; τ_crit, the critical strength of the material in shear; r₀, the size of the plastic zone. The three parameters can be determined with the results from only one test. Predictions using this model are compared with experiments on gypsum specimens with pre-existing fractures loaded in uniaxial and biaxial compression performed by the authors. Specifically, wing crack and shear crack initiation, crack propagation, coalescence stress and -type as well as the crack pattern up to coalescence can be modeled. The model can also duplicate experimental results in compression and tension obtained by other researchers. These results show that stress-based criteria can be effectively used in modeling crack initiation and crack coalescence. This revised version was published online in August 2006 with corrections to the Cover Date.     

A modular integration and multiresolution framework for image interpretation

In this paper, we give a generalized formulation for a vision problem in the framework of modular integration and multiresolution. The developed framework is used to solve the high-level vision problem of scene interpretation. The formulation essentially involves the concept of reductionism and multiresolution, where the given vision taskν is broken down into simpler subtasksν ₁,ν ₂, …,ν _m. Moreover, instead of solving the vision taskν ^Ω =ν at the finest resolution Θ, we solve the synergetically coupled vision subtasks at coarser resolutions (Ω −N) for Θ ≥N>0 and use the results obtained at resolution (Θ −N) to solveν ^{Ω −N+1}, the vision task at resolution (Θ −N+1). Image interpretation is a two-phased analysis problem where some salient features or objects in an image are identified by segmenting the image and the objects in the segmented image are interpreted based on their spatial relationships. We present a solution to the joint segmentation and interpretation problem in the proposed generalized framework. For the interpretation part we exploit the Markov Random Field (MRF) based image interpretation scheme developed by Modestino and Zhang. Experimental results on both indoor and outdoor images are presented to validate the proposed framework.     

The Effect of Crack Spacing Distribution on Stiffness Reduction of Cross-ply Laminates

The effect of transverse crack distribution on the effective mechanical properties of cross-ply laminates is considered. Young’s modulus and Poisson’s ratio dependence on the transverse ply crack density is obtained experimentally for glass fiber/epoxy laminates of lay-ups [0₂/90₂]_s, [0/90₂]_s, and [0/90₄]_s subjected to uniaxial tensile loading. Crack spacing distributions at the edge of the specimen are also measured at a predefined applied strain. Mechanical property reduction is evaluated for two crack spacing distributions: uniform spacing routinely considered in theoretical derivations and the experimental crack spacing distribution; the results are compared with test data.     

Fracture toughness of ceramics and semi-brittle alloys using a miniaturized disk-bend test

∞ is the crack resistance at ”infinite” crack length. It is convincingly shown that this so-called R-curve equation correctly predicts K_∞, which is comparable to the conventionally measured Mode I plain-strain fracture toughness, K_Ic, of the same material. The fundamental constants in the fracture-mechanics-based equations are discussed, emphasizing the aspects pertinent to the small specimens used in the MDBT. Results are presented on 8 materials: ZnS, glass-ceramic, Si₃N₄, Ti₅Si₃, SiC, Ni₃Ge, NiAl and Ti-46.5A1-2.1Cr-3.0Nb-0.2W. All are brittle except for the latter two, which undergo slight plastic deformation before fracturing. The resulting values of K_∞ are in excellent agreement with published values derived from conventional measurements, providing considerable confidence in the method. where Q is a constant and K_∞ is the crack resistance at ”infinite” crack length. It is convincingly shown that this so-called R-curve equation correctly predicts K_∞, which is comparable to the conventionally measured Mode I plain-strain fracture toughness, K_Ic, of the same material. The fundamental constants in the fracture-mechanics-based equations are discussed, emphasizing the aspects pertinent to the small specimens used in the MDBT. Results are presented on 8 materials: ZnS, glass-ceramic, Si₃N₄, Ti₅Si₃, SiC, Ni₃Ge, NiAl and Ti-46.5A1-2.1Cr-3.0Nb-0.2W. All are brittle except for the latter two, which undergo slight plastic deformation before fracturing. The resulting values of K_∞ are in excellent agreement with published values derived from conventional measurements, providing considerable confidence in the method. Received: 13 October 1999 / Reviewed and accepted: 9 November 1999     

Computation of the relaxation and creep functions of elastomers from harmonic shear modulus

The purpose of this paper is to compute the relaxation and creep functions from the data of shear complex modulus, G ^∗(iν). The experimental data are available in the frequency window ν∈[ν_min,ν_max] in terms of the storage G′(ν) and loss G″(ν) moduli. The loss factor h( n) = \fracG"( n)G￠(n)\eta( \nu) = \frac{G'( \nu )}{G'(\nu )} is asymmetrical function. Therefore, a five-parameter fractional derivative model is used to predict the complex shear modulus, G ^∗(iν). The corresponding relaxation spectrum is evaluated numerically because the analytical solution does not exist. Thereby, the fractional model is approximated by a generalized Maxwell model and its rheological parameters (G _k ,τ _k ,N) are determined leading to the discrete relaxation spectrum G(t) valid in time interval corresponding to the frequency window of the input experimental data. Based on the deterministic approach, the creep compliance J(t) is computed on inversing the relaxation function G(t).     

Thermal stress intensity factors of an infinite orthotropic layer with a crack

Stress intensity factors are determined for a crack in an infinite orthotropic layer. The crack is situated parallel to the plane surfaces of the layer. Stresses are solved for two kinds of the boundary conditions with respect to temperature field. In the first problem, the upper surface of the layer is heated to maintain a constant temperature T ₀, while the lower surface is cooled to maintain a constant temperature –T ₀. In the other problem, uniform heat flows perpendicular to the crack. The surfaces of the crack are assumed to be insulated. The boundary conditions are reduced to dual integral equations using the Fourier transform technique. To satisfy the boundary conditions outside the crack, the difference in temperature at the crack surfaces and differences in displacements are expanded in a series of functions that vanish outside the crack. The unknown coefficients in each series are evaluated using the Schmidt method. Stress intensity factors are then calculated numerically for a steel layer that behaves as an isotropic material and for a tyrannohex layer that behaves as an orthotropic material.     

Fatigue crack propagation behavior of friction stir welded 5083-H32 Al alloy

Fatigue crack propagation (FCP) behavior of friction stir welded (FSWed) 5083-H32 Al alloy was examined with the fatigue crack growing either along the dynamically recrystallized zone (DXZ) at variable ΔK or perpendicular to the DXZ at a constant ΔK value of 10, 13, 15, and 17 MPa√m, respectively. The FCP behavior of FSWed 5083-H32 specimen is substantially influenced by the presence of FSW zone, the trend of which is discussed based on residual stress measurement and fractographic observation.     

Estimate of the potential relief (rippling) of a surface associated with adsorption of d-metal atoms on d-substrates

A cohesion approach developed previously by Davydov and Tikhonov to describe adsorption properties [Surf. Sci. 371, 157 (1997)] is used to calculate the ripple parameter Ω=E _dif/E _des (where E _dif and E _des are the activation energy of surface diffusion and the adsorption energy) for atoms of d-metals adsorbed on a W(110) surface. The results of the calculations show good agreement with the experimental data. Pis’ma Zh. Tekh. Fiz. 24, 70–74 (December 12, 1998)     

The fracture energy of brittle crystals

An expression is derived for the fracture energy, γ, in brittle crystals, namely, γ = kd ₀ E, d ₀ being the lattice spacing, E Young’s modulus perpendicular to the fracture plane, and k is a constant. The value of γ obtained through this expression is compared to experimental data for cubic crystals. Despite the fit, we conclude that because the fracture energy is dominated by the elastic constant, comparisons between a computed γ and experimental data cannot be used to distinguish between bonding functions.     

Propagation of a Mode-II crack in a viscoelastic medium with different bulk and shear relaxation

A solution for a crack propagating under shear-loading in an isotropic viscoelastic medium with different relaxation under volume and shear deformations is presented. The medium is infinite and the semi-infinite crack propagates along the x ₁-axis at constant speed V, which may take any value up to the speed of dilatational waves. The requisite Riemann–Hilbert problem for the steady-state case has been solved and the asymptotics of the stress component σ₁₂ directly ahead of the crack and at infinity have been obtained.