Experimental GH singularity field around growing crack-tip

The displacement fieldsu _x ,u _y at growing crack tip of LY12-M specimens with double edge cracks are measured using moire method. The experimental singularity fields are compared with GH theoretical field [12–14]. The size and shape of the experimental GH singularity fields are obtained. The error in both the experimental and theoretical evaluations is controlled within ±10%. The experiments show that there is singularity dominant around a growing crack tip. The shape of this dominant region ranges from butterfly wing to oblate and circular. Inside GH-field, there is a 3-D deformed damage zone where no GH singularity exists. The project suppoted by National Natural Science Foundation of China     

Analysis of a permeable interface crack in elastic dielectric/piezoelectric bimaterials

A permeable interface crack between elastic dielectric material and piezoelectric material is studied based on the extended Stroh’s formalism. Motivated by strong engineering demands to design new composite materials, the authors perform numerical analysis of interface crack tip singularities and the crack tip energy release rates for 35 types of dissimilar bimaterials, respectively, which are constructed by five kinds of elastic dielectric materials: Epoxy, Polymer, Al₂O₃, SiC, and Si₃N₄ and seven kinds of practical piezoelectric ceramics: PZT-4, BaTiO₃, PZT-5H, PZT-6B, PZT-7A, P-7, and PZT-PIC 151, respectively. The elastic dielectric material with much smaller permittivity than commercial piezoelectric ceramics is treated as a special transversely isotropic piezoelectric material with extremely small piezoelectricity. The present investigation shows that the structure of the singular field near the permeable interface crack tip consists of three singularities: and , which is quite different from that in the impermeable interface crack. It can be concluded that different far field loading cases have significant influence on the near-tip fracture behaviors of the permeable interface crack. Based on the present theoretical treatment and numerical analysis, the electric field induced crack growth is well explained, which provides a better understanding of the failure mechanism induced from interface crack growth in elastic dielectric/piezoelectric bimaterials. The project supported by the National Natural Science Foundation of China (10572110), Doctor Foundation of the Chinese Education Ministry and Doctorate Foundation of Xi’an Jiaotong University. The English text was polished by Yunming Chen.     

Independence of the Presence of Cracks or Inclusions of the Net Interaction Force Acting on a Dislocation by a Skewed Line Singularity

Orlov and Indenbom [1] have shown that the net (integrated) interaction force F between two skew dislocations with Burgers vectors separated by a distance h in an infinite anisotropic elastic medium is independent of h. Nix [2] computed numerically the net interaction force F between two skew dislocations that are parallel to the traction-free surface X2=0 of an isotropic elastic half-space. His numerical results showed that F was independent of h; a partial result of what Barnett [3] called Nix"s theorem. The separation-independence portion of Nix"s theorem has been proved to hold for a general anisotropic elastic half-space with a traction-free, rigid, or slippery surface, and for bimaterials [3-5]. In this paper, we show that the net interaction force is independent of the presence of inclusions. We will consider the case in which the line dislocation b is a more general line singularity which can include a coincident line force with strength f per unit length of the line singularity. An inclusion is an infinitely long dissimilar anisotropic elastic cylinder of an arbitrary cross-section whose axis is parallel to the line singularity (f, b). The (skew) line dislocation does not intersect the inclusion. The special cases of an inclusion are a void, crack, or rigid inclusion. There can be more than one inclusion of different cross sections and different materials. The line singularity (f, b) can be outside the inclusions or inside one of the inclusions. The inclusions and the matrix need not have a perfect bonding. One can have a debonding with or without friction. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Remarkable Nature of Cylindrically Anisotropic Elastic Materials Exemplified by an Anti-Plane Deformation

A material is cylindrically anisotropic when its elastic moduli referred to a cylindrical coordinate system are constants. Examples of cylindrically anisotropic materials are tree trunks, carbon fibers [1], certain steel bars, and manufactured composites [2]. Lekhnitskii [3] was the first one to observe that the stress at the axis of a circular rod of cylindrically monoclinic material can be infinite when the rod is subject to a uniform radial pressure (see also [4]). Ting [5] has shown that the stress at the axis of the circular rod can also be infinite under a torsion or a uniform extension. In this paper we first modify the Lekhnitskii formalism for a cylindrical coordinate system. We then consider a wedge of cylindrically monoclinic elastic material under anti-plane deformations. The stress singularity at the wedge apex depends on one material parameter γ. For a given wedge angle α, one can choose a γ so that the stress at the wedge apex is infinite. The wedge angle 2α can be any angle. It need not be larger than π, as is the case when the material is homogeneously isotropic or anisotropic. In the special case of a crack (2α=2π) there can be more than one stress singularity, some of them are stronger than the square root singularity. On the other hand, if γ < there is no stress singularity at the wedge apex for any wedge angle, including the special case of a crack. The classical paradox of Levy [6] and Carothers [7] for an isotropic elastic wedge also appears for a cylindrically anisotropic elastic wedge. There can be more than one critical wedge angle and, again, the critical wedge angle can be any angle. This revised version was published online in August 2006 with corrections to the Cover Date.     

Three-dimensional analysis of defects in a piezoelectric material

In this paper, the Green's function technique is used to develop a solution of an infinite, piezoelectric medium containing either an ellipsoidal cavity or a flat elliptical crack. The coupled elastic and electric fields both inside the cavity and on the boundary of the cavity are obtained, and the stress intensity factor and the electric field intensity factor are also obtained for an elliptical crack. It is found that; (1) the coupled elastic and electric fields inside the cavity keep uniform when the external elastic field and electric field are constant; (2) the behavior of the stress and electric field components in the neighborhood of the crack tip shows the classical type of singularity. The project supported by National Natural Science Foundation of China     

Dynamics of a Non-Autonomous ODE System Occurring in Coagulation Theory

We consider a constant coefficient coagulation equation with Becker–D?ring type interactions and power law input of monomers J ₁(t) = α t ^ω, with α > 0 and . For this infinite dimensional system we prove solutions converge to similarity profiles as t and j converge to infinity in a similarity way, namely with either or constants, where is a function of t only. This work generalizes to the non-autonomous case a recent result of da Costa et al. (2004). Markov Processes Relat. Fields 12, 367–398. and provides a rigorous derivation of formal results obtained by Wattis J. Phys. A: Math. Gen. 37, 7823–7841. The main part of the approach is the analysis of a bidimensional non-autonomous system obtained through an appropriate change of variables; this is achieved by the use of differential inequalities and qualitative theory methods. The results about rate of convergence of solutions of the bidimensional system thus obtained are fed into an integral formula representation for the solutions of the infinite dimensional system which is then estimated by an adaptation of methods used by da Costa et al. (2004). Markov Processes Relat. Fields 12, 367–398.      

Transient response of a finite crack in an orthotropic strip under the in-plane shear impact

The transient response of a central crack in an orthotropic strip under the in-plane shear impact loading is studied by using the dual integral equation method proposed by Copson and Sih. The general formula for the shear stress intensity factor near the crack tip is derived. Numerical results of with in various cases are obtained by solving the second kind Fredholm integral equation and by performing the inverse Laplace transform.     

A novel impact three-point bend test method for determining dynamic fracture-initiation toughness

A novel impact three-point bend test method has been developed for determining the dynamic fracture-initiation toughness,K _Id, over the range of loading rates . The split-Hopkinson pressure-bar technique is used to measure dynamic loads applied to a bend specimen with a fatigue precrack. The stress-intensity-factor histories for the bend specimen are evaluated by means of a dynamic finite-element technique and the standard formula (ASTM E 399-83) based on the measured dynamic loads. The time of crack initiation is determined using a strain gage mounted near a crack tip. The results for 7075-T6 aluminum alloy and Ti−6A1−2Sn−4Zr−6Mo alloy indicate that the reliableK _Id data can only be obtained by evaluation procedures which take the inertial effects into account. It is shown that the novel impact bend test method in conjunction with dynamic finite-element analysis provides an effective means of characterizing the dynamic fracture-toughness parameterK _Id.     

Experimental investigation of GH singularity field of the growing crack-tip for different ductile metals

The displacementsu _x ,u _y inx, y directions at growing crack tip of the specimens with double edge cracks for four different alumium alloys and two coppers are measured by using moire method and optical spatial filtering technique. From experimental displacement fields, the displacement singularity fields are obtained and compared with GH theoretical field. Unknown constantsA and _yo in theoretical solution are determined from experimental data. The theoretical singularity field thus compared is given for plane-stress, mode-I, strain-hardening materials. The error in both the experimental and theoretical evaluations is within ± 10%. The experiments show that there exists dominant singularity region around a growing crack-tip. In the experiments, the strain hardening indexn amounts from 3.158 to 14. The shape of this dominant region ranges from butterfly wing to oblate or circular shape. The size and shape of GH dominant region depend on the material property, the specimen geometry and loading type. Inside GH-field, there is a 3-D deformed damage zone, where no GH singularity exists. Very near to the crack-tip, there is a fracture process zone.The project is supported by the National Science Foundation of China.     

A method for measuring mode I crack tip constraint under static and dynamic loading conditions

A novel experimental technique for measuring crack tipT-stress, and hence in-plane crack tip constraint, in elastic materials has been developed. The method exploits optimal positioning of stacked strain gage rosette near a mode I crack tip such that the influence of dominant singular strains is negated in order to determineT-stress accurately. The method is demonstrated for quasi-static and low-velocity impact loading conditions and two values of crack length to plate width ratios (a/W). By coupling this new method with the Dally-Sanford single strain gage method for measuring the mode I stress intensity factorK _I , the crack tip biaxiality parameter is also measured experimentally. Complementary small strain, static and dynamic finite element simulations are carried out under plane stress conditions. Time histories ofK _I andT-stress are computed by regression analysis of the displacement and stress fields, respectively. The experimental results are in good agreement with those obtained from numerical simulations. Preliminary data for critical values ofK _I and β for dynamic experiments involving epoxy specimens are reported. Dynamic crack initiation toughness shows an increasing trend as β becomes more negative at higher impact velocities.     

Plane stress dynamic fields near a propagating crack-tip in a power-law material

Based on the plastic-dynamic equations, the asymptotic behaviour of the near-tip fields for a plane stress tensile crack propagating in a power-law material has been studied in this paper. It is shown that the stress and strain singularities are, respectively, of the order and , whereA is a constant which is related to the size of plastic region,r is the distance to the crack tip,n is the power-law exponent. Projects sponsored by the National Science Foundation.     

The Asymptotic Finite-dimensional Character of a Spectrally-hyperviscous Model of 3D Turbulent Flow

We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in which hyperviscous terms are added spectrally to the standard incompressible Navier–Stokes equations (NSE). Let P _m be the projection onto the first m eigenspaces of A =−Δ, let μ and α be positive constants with α ≥3/2, and let Q _m =I − P _m , then we add to the NSE operators μ A _φ in a general family such that A _φ≥Q _m A ^α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers past a cutoff λ _m0 where m ₀ ≤ m, so that for large enough m ₀ the inertial-range wavenumbers see only standard NSE viscosity. We first obtain estimates on the Hausdorff and fractal dimensions of the attractor (respectively and ). For a constant K _α on the order of unity we show if μ ≥ ν that and if μ ≤ ν that where ν is the standard viscosity coefficient, l ₀ = λ₁^−1/2 represents characteristic macroscopic length, and is the Kolmogorov length scale, i.e. where is Kolmogorov’s mean rate of dissipation of energy in turbulent flow. All bracketed constants and K _α are dimensionless and scale-invariant. The estimate grows in m due to the term λ _m /λ₁ but at a rate lower than m ^3/5, and the estimate grows in μ as the relative size of ν to μ. The exponent on is significantly less than the Landau–Lifschitz predicted value of 3. If we impose the condition , the estimates become for μ ≥ ν and for μ ≤ ν. This result holds independently of α, with K _α and c _α independent of m. In an SVV example μ ≥ ν, and for μ ≤ ν aspects of SEV theory and observation suggest setting for 1/c within α orders of magnitude of unity, giving the estimate where c _α is within an order of magnitude of unity. These choices give straight-up or nearly straight-up agreement with the Landau–Lifschitz predictions for the number of degrees of freedom in 3D turbulent flow with m so large that (e.g. in the distinguished-class case for m ₀ large enough) we would expect our solutions to be very good if not virtually indistinguishable approximants to standard NSE solutions. We would expect lower choices of λ _m (e.g. with a > 1) to still give good NSE approximation with lower powers on l ₀/l _ε, showing the potential of the model to reduce the number of degrees of freedom needed in practical simulations. For the choice , motivated by the Chapman–Enskog expansion in the case m = 0, the condition becomes , giving agreement with Landau–Lifschitz for smaller values of λ _m then as above but still large enough to suggest good NSE approximation. Our final results establish the existence of a inertial manifold for reasonably wide classes of the above models using the Foias/Sell/Temam theory. The first of these results obtains such an of dimension N > m for the general class of operators A _φ if α > 5/2. The special class of A _φ such that P _m A _φ = 0 and Q _m A _φ ≥ Q _m A ^α has a unique spectral-gap property which we can use whenever α ≥ 3/2 to show that we have an inertial manifold of dimension m if m is large enough. As a corollary, for most of the cases of the operators A _φ in the distinguished-class case that we expect will be typically used in practice we also obtain an , now of dimension m ₀ for m ₀ large enough, though under conditions requiring generally larger m ₀ than the m in the special class. In both cases, for large enough m (respectively m ₀), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics, and in particular trajectories on are controlled by essentially NSE dynamics.      

An Example of Finite-time Singularities in the 3d Euler Equations

Let be the exterior of the closed unit ball. Consider the self-similar Euler system

Analysis and control of a class of uncertain linear periodic discrete-time systems

Feedback control problems for linear periodic systems （LPSs） with interval- type parameter uncertainties are studied in the discrete-time domain. First, the stability analysis and stabilization problems are addressed. Conditions based on the linear matrices inequality （LMI） for the asymptotical stability and state feedback stabilization, respec-tively, are given. Problems of L2-gain analysis and control synthesis are studied. For the L2-gain analysis problem, we obtain an LMI-based condition such that the autonomous uncertain LPS is asymptotically stable and has an L2-gain smaller than a positive scalar γ. For the control synthesis problem, we derive an LMI-based condition to build a state feedback controller ensuring that the closed-loop system is asymptotically stable and has an L2-gain smaller than the positive scalar γ. All the conditions are necessary and sufficient.     

Effect of electric current on migration of point defects near a crack tip

The effect of direct current on migration of point defects dissolved in a crystal near the tip of a crack in tension is estimated. Calculations are carried out with allowance for the plastic strain near the crack tip of a loaded specimen, caused by the motion of dislocations in the active slip planes of the crystal, the Joule heat released, and the effect of gas exchange on the crack edges on the evolution of distribution of interstitial impurity atoms. A numerical analysis is performed for an Fe crystal.     

Control for going from hovering to small speed flight of a model insect

The longitudinal steady-state control for going from hovering to small speed flight of a model insect is studied, using the method of computational fluid dynamics to compute the aerodynamic derivatives and the techniques based on the linear theories of stability and control for determining the non-zero equilibrium points. Morphological and certain kinematical data of droneflies are used for the model insect. A change in the mean stroke angle （δФ） results in a horizontal forward or backward flight; a change in the stroke amplitude （δФ） or a equal change in the down- and upstroke angles of attack （δα1） results in a vertical climb or decent; a proper combination of δФ and δФ controls （or δФ and δα1 controls） can give a flight of any （small） speed in any desired direction.     

Effect of electric current on the evolution of plastic strain near a crack tip

The effect of direct current on the evolution of plastic strain near the tip of a crack in a crystal in tension is studied. The plastic strain near the crack tip in the loaded specimen is the result of motion of dislocations in the active slip planes of the crystal under the action of shear stresses caused by external loading and electric current. The Joule heat, Thomson effect, and electron wind (electroplastic effect) are taken into account in calculations. The plastic strain and stress distributions near the crack tip are determined at different moments of time for a given magnitude ofelectric current. The effect of the plastic zone on the stressintensity factor of the crack is studied. It is found that the plastic strain is affected largely by the Joule heat released upon passage of the electric current. A numerical analysis is performed for an Fe crystal.     

An analysis on three-dimensional effects near the tip of a crack in an elastic plate

An infinite plate containing a finite through crack under tensile loading is analysed by Fourier transform based on the Kane-Mindlin kinematic assumptions for the quasi-three-dimensional deformation of plates in extension. The asymptotic expressions of stress and displacement fields near the crack tip, the variation of the stress intensity factor with the plate-thickness and the three-dimensional deformation zone near the crack tip are investigated. The results of the analysis show that, (a) the crack-tip stress and displacement fields accounting for the plate-thickness effects are different from the plane stress solutions and this is true even for extremely small parameter (=1–vh/6 a). In a very small region near the crack tip, plane strain solutions prevail; (b) the ratio of the stress intensity factor K_I to the corresponding plane stress one K_I, K_I/K _I ^o , approaches 1/(1–v²) as tends to zero; (c) plane stress solutions can give satisfactory results for points a distance from the crack tip greater than about three-fourths of the plate-thickness; (d) the linear elastic result for the zone of three-dimensional effects is approximately valid for an elasto-plastic material with linear strain-hardening when the plastic tangential mudulus E_t is not very small.The Project Supported by National Natural Science Foundation of China.     

On the Rank 1 Convexity of Stored Energy Functions of Physically Linear Stress-Strain Relations

The rank 1 convexity of stored energy functions corresponding to isotropic and physically linear elastic constitutive relations formulated in terms of generalized stress and strain measures [Hill, R.: J. Mech. Phys. Solids 16, 229–242 (1968)] is analyzed. This class of elastic materials contains as special cases the stress-strain relationships based on Seth strain measures [Seth, B.: Generalized strain measure with application to physical problems. In: Reiner, M., Abir, D. (eds.) Second-order Effects in Elasticity, Plasticity, and Fluid Dynamics, pp. 162–172. Pergamon, Oxford, New York (1964)] such as the St.Venant–Kirchhoff law or the Hencky law. The stored energy function of such materials has the form