Re-entrant corner problems for isotropic materials and layered composites

Modern materials like composites in different types require new methods in computational fracture mechanics. Besides of classical fracture mechanics we have to solve the asymptotic solution around crack tips. This can be done on the basis of the Kondratiev s theorem with the Pietrov-Galerkin method to solve the unknown eigenvalues for those problems. Additionally, we have to de.ne instead classical modes I, II and III, mode I* up to mode IV* to interpret the computational results for the unknown eigenvalues. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ⅰ型动态扩展裂纹尖端的弹粘-理想塑性场

由于材料在扩展裂纹尖端的粘性效应的存在,考虑粘性效应并假设粘性系数与塑性等效应变率的幂次成反比,对理想塑性材料中平面应变扩展裂纹尖端场进行了弹粘塑性渐近分析,得到了不含间断的连续解,并讨论了Ⅰ型裂纹数值解的性质随各参数的变化规律．分析表明,应力和应变均具有幂奇异性,通过分析使尖端场的弹、粘、塑性可以合理匹配．对于Ⅰ型裂纹,裂尖场不含弹性卸载区．趋于极限情况时,裂纹尖端处于一种超粘性状态,并积聚了大量的能量,在各个受压应力状态下裂纹扩展．     

Investigations of interlaminar fracture toughness of laminated polymeric composites

The behavior of interlaminar fracture of fiber reinforced laminated polymeric composites has been investigated in modes I, II, and different mixed mode I/II ratios. The experimental investigations were carried out by using conventional beam specimens and the compound version of the CTS (compact tension shear) specimen. In this study, a compound version of the CTS specimen is used for the first time to determine the interlaminar fracture toughness of composites. In order to verify the results obtained by the CTS tests, conventional beam tests were also carried out. In the beam tests, specimens of double cantilever beam (DCB) and end notched flexure (ENF) were used to obtain the critical rates of the energy release for failure modes I and II. The CTS specimen is used to obtain different mixed mode ratios, from pure mode I to pure mode II, by varying the loading conditions. The highest mixed mode ratio obtained in the experiment was G _I /G _II =60. The data obtained from these tests were analyzed by the finite element method. The separated critical rates G _I and G _II of the energy release were calculated by using the modified virtual crack closure integral (MVCCI) method. The experimental investigations were performed on a unidirectional glass/epoxy composite. The results obtained by the beam and CTS tests were compared. It was found that the interlaminar fracture toughness G _IC ^init of mode I at crack initiation and the corresponding value G _II ^Cinit of mode II obtained by the conventional beam and the CTS tests were in rather good agreement. The experimental results of interlaminar fracture of mixed mode were used to obtain the parameters required for the failure criterion. The two different failure criteria were compared. The best correlation with the experimental data was obtained by using the failure criterion proposed by Wu in 1967 containing linear and quadratic terms of the rates of the energy release.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 307–322, May–June, 1998.     

Two-dimensional analysis of a crack in quasicrystal materials

The two-dimensional problem of a crack in three-dimensional quasicrystals subject to far field loadings is studied. The analysis is based on the generalized Lekhnitskii's formalism. The analytical expressions for both the entire fields and the asymptotic fields near the crack tip are determined. The fracture quantities of quasicrystals, i.e., field intensity factors, energy release rates and so on, is a prerequisite. Numerical results for a Griffith crack under phason loading Mode I and II conditions are poltted. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Average projection type weighted Cramér- von Mises statistics for testing some distributions

This paper addresses the problem of testing goodness-of-fit for several important multivariate distributions: (I) Uniform distribution on p-dimensional unit sphere; (II) multivariate standard normal distribution; and (III) multivariate normal distribution with unknown mean vector and covariance matrix. The average projection type weighted Cramér-von Mises test statistic as well as estimated and weighted Cramér-von Mises statistics for testing distributions (I), (II) and (III) are constructed via integrating projection direction on the unit sphere, and the asymptotic distributions and the expansions of those test statistics under the null hypothesis are also obtained. Furthermore, the approach of this paper can be applied to testing goodness-of-fit for elliptically contoured distributions.     

Investigation of the prefracture zone at the tip of a mode I crack reaching a nonsmooth interface of an elastic media

We perform calculation of the initial prefracture zone at the tip of a mode I crack that reaches a nonsmooth interface of two dissimilar elastic media at its corner point by the Wiener–Hopf method. The zone is modeled by a line of normal displacement fracture on a crack continuation. Expressions for the length of the prefracture zone and the potential energy accumulated in it are obtained. Their numerical values are compared with the corresponding values for the prefracture zone in a bonding material on the interface of the media, on the basis of which we draw a conclusion on the possible direction of development of the zone.     

Scattering of a plane wave on a periodic curve with trapping domains

The problem of scattering on a periodic curve is considered. The asymptotic solution of the problem is constructed, and its principal terms are presented. The justification of the asymptotic solution found is provided. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 75–85. Translated by I. V. Kamotskii     

Short-wave asymptotic behavior of space-time creeping waves in elasticity theory

We consider the elastic space-time (ST) wave on an unstressed convex surface in a deep shadow zone. The uniform high-frequency asymptotic expansion of the wave field is constructed as the sum of the caustic expansion for the longitudinal (transverse) wave containing the Airy function and the space-time ray series for the transverse (longitudinal) wave. The contribution of the ray expansion with the transverse eikonal is comparable to the contribution of the longitudinal creeping wave to the wave field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 176–189, 1985.I would like to thank V. M. Babich for suggesting the topic and for discussion of results.     

弹性-幂硬化蠕变性材料Ⅱ型界面裂纹准静态扩展的渐近分析

建立了弹性-幂硬化蠕变性材料Ⅱ型界面裂纹准静态扩展的力学模型,求得了在裂纹表面自由和裂纹面有摩擦接触两种情况下,裂纹尖端应力场分离变量形式的渐近解．求解结果表明:Ⅱ型界面裂纹问题的应力、应变具有相同的奇异性;Ⅱ型界面裂纹尖端场不存在振荡奇异性;材料的幂硬化指数n和弹性模量比对裂纹尖端应力场幂硬化蠕变性材料区有着显著的影响,而弹性区仅受幂硬化指数n的影响,当n很大时,蠕变变形占主导地位,应力场趋于稳定,不随n的变化而变化;泊松比对裂纹尖端应力场的影响不明显．     

Operational asymptotic stochastic dominance

Levy (2016) proposes asymptotic first-degree stochastic dominance as a distribution ranking criterion for all non-satiable decision makers with infinite investment horizons. Given Levy’s setting, this paper defines and offers the equivalent distributional conditions for asymptotic second-degree stochastic dominance, as well as operational asymptotic first- and second-degree stochastic dominance. Interestingly, the operational asymptotic stochastic dominance provides a full rank over assets with lognormal returns and different means. Empirical applications show that our conditions can be readily implemented in practice.     

Driving force acting on a running crack

The asymptotic analysis of the dynamic crack problem for the anti‐plane shear mode is provided. The field near the crack tip is studied in detail for a nonlinear elastic incompressible material whose stored energy behaves asymptotically as a power of the first invariant of the strain tensor at large strains. It is shown that the hardening parameter characterizes fully the singularity degree of the near‐crack‐tip field. Based on the latter knowledge the driving force acting on the crack tip is calculated. Possible scenarios of the crack propagation are discussed.     

Long-time asymptotics of the focusing Kundu–Eckhaus equation with nonzero boundary conditions

The long-time asymptotics of the focusing Kundu–Eckhaus equation with nonzero boundary conditions at infinity is investigated by the nonlinear steepest descent method of Deift and Zhou. Three asymptotic sectors in space–time plane are found: the plane wave sector I, plane wave sector II and an intermediate sector with a modulated one-phase elliptic wave. The asymptotic solutions of the three sectors are proposed by successively deforming the corresponding Riemann–Hilbert problems to solvable model problems. Moreover, a time-dependent g-function mechanism is introduced to remove the exponential growths of the jump matrices in the modulated one-phase elliptic wave sector. Finally, the modulational instability is studied to reveal the criterion for the existence of modulated elliptic waves in the central region.     

Type (II) regions between curves of the Fucik spectrum

The Fucik spectrum for a semilinear problem with asymptotic linearities has been shown to consist, at least locally, of curves emanating from a sequence of points in the plane. Regions between curves emanating from different points (referred to as type (I) regions in this paper) have a different nature than those between curves emanating from the same point (referred to as type (II) regions). Problems for which asymptotic limits fall in regions of type (I) have been solved by several authors, but not those for which the limits fall in a type (II) region. In the present paper we solve problems in which the asymptotic limits fall in type (II) regions. Received March 5, 1996     

A Game Related to the Number of Hides Game

In this paper, we study a discrete search game on an array of N ordered cells, with two players having opposite goals: player I (searcher) and player II (hider). Player II has to hide q objects at consecutive cells and player I can search p consecutive cells. The payoff to player I is the number of objects found by him. In some situations, the players need to adopt sophisticated strategies if they are to act optimally.     

Characterisation of multiple conducting permeable objects in metal detection by polarizability tensors

Realistic applications in metal detection involve multiple inhomogeneous‐conducting permeable objects, and the aim of this paper is to characterise such objects by polarizability tensors. We show that, for the eddy current model, the leading order terms for the perturbation in the magnetic field, due to the presence of N small conducting permeable homogeneous inclusions, comprises of a sum of N terms with each containing a complex symmetric rank 2 polarizability tensor. Each tensor contains information about the shape and material properties of one of the objects and is independent of its position. The asymptotic expansion we obtain extends a previously known result for a single isolated object and applies in situations where the object sizes are small and the objects are sufficiently well separated. We also obtain a second expansion that describes the perturbed magnetic field for inhomogeneous and closely spaced objects, which again characterises the objects by a complex symmetric rank 2 tensor. The tensor's coefficients can be computed by solving a vector valued transmission problem, and we include numerical examples to illustrate the agreement between the asymptotic formula describing the perturbed fields and the numerical prediction. We also include algorithms for the localisation and identification of multiple inhomogeneous objects.     

Mode I,mode II,and mixed-mode I/II interlaminar fracture toughness of GFRP influenced by fiber surface treatment

The interlaminar fracture behavior of unidirectional glass fiber reinforced composites with fiber surface treatment has been investigated in modes I and II and for fixed mode I to mode II ratio of 1.33. The data obtained from these tests have been analyzed by using different analytical approaches. The present investigation is focused on the influence of the glass fiber surface treatment on the interlaminar fracture toughness of unidirectional laminates. Glass fibers with two different fiber surface treatments have been investigated. fiber surface treatment was carried out by using a polyethylene or silane coupling agent in combination with modifying agents. The glass fibers were embedded in the brittle epoxy matrix. Mode I, mode II, and mixed-mode I/II tests were performed in order to determine critical strain energy release rates. Double cantilever beam (DCB), end-notched flexure (ENF), and mixed-mode flexure (MMF) specimens were used. For both types of fiber surface treatment about the same values of mode I initiation fracture toughness G_IC ^init were obtained. It was observed that in mode I interlaminar crack growth in the DCB test for the composite sized by polyethylene, the crack propagation is accompanied by extensive fiber bridging. For both fiber surface treatments interlaminar fracture toughness increases considerably with increasing of crack length. For the fiber surface treatment with the silane coupling agent, the value of mode II initiation fracture toughness G_IIC ^init was about 2.5-times higher in comparison with that of a composite sized by polyethylene. For both types of fiber surface treatments the mixed-mode I/II test has shown a similar behavior to the mode I DCB test.     

Subcritical and critical states of a crack with failure zones

The problem on the stress–strain state near a mode I crack in an infinite plate is solved in the frame of a cohesive zone model. The complex variable method of Muskhelishvili is used to obtain the crack opening displacements caused by the cohesive traction, which models the failure zone at the crack tip, as well as by the external load. The finite stress condition and logarithmic singularity of the derivative of the separation with respect to the coordinate at the tip of a physical crack are taken into account.The cohesive traction distribution is sought in a piecewise linear form, nodal values of which are being numerically chosen to satisfy the traction-separation law. According to this law, the cohesive traction is coupled with the corresponding separation and fracture toughness. The tips of the physical crack and cohesive zone (geometric variables) along with the discrete cohesive traction are used as the problem parameters determining the stress-strain state. If the crack length is included in the set, then the critical crack size can be found for the given loading intensity.The obtained determining system of equations is solved numerically. To find the initial point for a standard numerical algorithm, the asymptotic determining system is derived. In this system, the geometric variables can be easily eliminated, which make it possible to linearize the system.In the numerical examples, the one-parameter traction-separation laws are used. Influence of the shape parameters of the law on the critical crack size and the corresponding cohesive length is studied. The possibility of using asymptotic solutions for determining the critical parameters is analysed. It is established that the critical crack length slightly depends on the shape parameter, while the cohesive length shows a strong dependence on the shape of cohesive laws.     

High order asymptotic approach to the evaluation of near field,radiated by ultrasonic transducer to the layered elastic plates

It is well known that one of the most time consuming problem in non-destructive evaluation (NDE) is the near-field modeling. In our approach the near field radiated by an ultrasonic transducer (UT) to the first Fresnel zone is approximated in the frequency domain by a closed form asymptotic solution based on the 3-D generalized orthogonality conditions of Lamb waves and on the stationary phase method. As shown, these asymptotics permit us to elucidate the physics and to work out a numerical algorithm which is about 10⁴ times faster than any direct way of computation. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Diffraction on a wedge-shaped domain with curved sides

The problem of diffraction of side waves overrunning a wedge-shaped body ω with curved sides is considered. The incidence wave is given by a ray asymptotic expansion. The scattered field is found as the sum of a geometric-optical field and a diffraction field. Bibliography: 2 tittles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 263–273. Translated by N. Ya. Kirpichnikova.