Theoretical analysis of three-dimensional ground displacements induced by shield tunneling

In this study, an approach to estimate three-dimensional ground displacements induced by shield tunneling is proposed by using the superposition method, advocating consideration of the ground loss as well as the working loads during shield tunneling. The ground movements caused by different factors, i.e., applied support pressures, shield shell frictions and cutter head rotation, are investigated in detail, which can provide a better understanding of how to set the shield advancing parameters for ground disturbance controlling. The main advantage of this approach is that it can well reproduce the uplift and settlement characteristics of the ground. It is also of great significance for assessing the safety of closely spaced structures when the ground differential settlement is taken into account. To estimate the ground movements induced by twin tunneling, an additional ground loss ratio is introduced to describe the disturbance caused by the first tunneling. The proposed approach in this paper is tested against the field data at a construction site and provides a satisfactory estimate of the ground movement.     

The penny-shaped crack in three-dimensional electrostriction

Zusammenfassung Zur Untersuchung der Spannungs- und Verschiebungsfelder in einem unendlichen elastischen Dielektrikum verwenden wir die dreidimensionale lineare ungekoppelte Theorie der Elektrostriktion.Von besonderem Interesse ist das Wesen der Spannungs-Singularität am Rande eines flachen kreisförmigen Spaltes innerhalb der begrenzenden Geometrie. Ein zweifacher Sachverhalt wurde festgestellt: 1. diese Spannungs-Singularität kann von höherer Ordnung sein als diejenige, welche mit den klassischen Umgebungen mechanischer und thermischer Art verknüpft ist;2. diese Singularität neigt nicht dazu, den Spalt in bezug auf eine spannungslose Spaltoberfläche auszubreiten. Die obigen Ergebnisse stimmen mit unseren früheren zweidimensionalen Analysen überein.

---

---

This work was supported by the United States Atomic Energy Commission.     

On some relations between different interface crack models

Exact analytical solutions for an interface crack with an artificial contact zone in isotropic, anisotropic and piezoelectric materials have been analyzed and their comparison with the correspondent classical results has been performed. This analysis showed the way of a presentation of the main results for an artificial contact zone model in the manner very similar to the classical model, and due to this phenomenon the essential simplification of the investigation of the contact zone model has been attained. The application of the obtained results to interface cracks in anisotropic and piezoelectric bimaterials can be demonstrated. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hypersingularities in three-dimensional crack configurations in composite laminates

Three-dimensional crack configurations in composite laminates are studied by means of the Scaled Boundary Finite Element Method (SBFEM) particularly regarding stress singularities and their associated deformation modes. The SBFEM is an efficient semi-analytical method that permits solving linear elastic mechanical problems. Only the boundary needs to be discretized while the problem is considered analytically in the direction of the dimensionless radial coordinate pointing from the scaling center to the boundary . An important advantage is that it requires no additional effort for the characterization of existing stress singularities. The situation of two meeting inter-fiber cracks is investigated in detail, considering different materials and fiber / crack orientations. It is shown that in three-dimensional crack configurations in composite laminates so-called hypersingularities can occur, i.e. stress singularities which are even stronger than the classical crack singularity. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic response of planar interface crack between viscoelastic bodies

IntroductionThe dynamic stress intensity factor (SIF) plays an important role in dynamic fracture underboth harmonic and transient loads. It predicts whether or not the fracture toughness of thematerial will be exceeded and catastrophic crack propagation will follow. The dynamic SIFof interfaCe cracks between two dissimilar elastic materials has been studied widely. Kundull]studied the dynamic SIF of interface crack under transient loading with the method based onBetti's reciprocal theore…     

Substructuring FE–XFE approaches applied to three-dimensional crack propagation

Two substructuring methods are investigated in order to allow for the use of the eXtended Finite Element Method (X-FEM) within commercial finite element (FE) codes without need for modifying their kernel. The global FE problem is decomposed into two subdomains, the safe domain and the cracked domain based on the value of the level sets representing the crack. The safe domain is treated by the host FE software while the cracked domain is treated by an independent XFE code. The first substructuring method consists of calculating the Schur matrix of a cracked super-element with the XFE code. The second technique introduces the finite element tearing and interconnecting method (FETI) which ensures the compatibility of the displacements at the interface between the cracked and safe subdomains. The stiffness matrices and nodal forces are provided by the XFE and FE codes for the cracked and safe subdomains, respectively. The solutions obtained with these two techniques are rigorously equivalent to those computed with the stand-alone XFE code. First, the computational efficiency of the two approaches is demonstrated. Second, a validation is proposed towards comparison with reference values of the stress intensity factors in simple 3D cracked geometries. Finally, this contribution presents an application of the FE–XFE–FETI method to the computation of the stress intensity factor induced by a crack inside a hydraulic cylinder under internal pressure.     

An improved boundary element Galerkin method for three-dimensional crack problems

In this paper we analyze the solution of crack problems in three-dimensional linear elasticity by equivalent integral equations of the first kind on the crack surface. Besides existence and uniqueness we give sharp regularity results for the solution of these pseudodifferential equations. Two versions of Eskin's Wiener-Hopf technique are presented: the first one requires the factorization of matrix-valued symbols which is avoided in the second case. Based on these regularity results we show how to improve the boundary element Galerkin method for our integral equations by using special singular trial functions. We apply the approximation property and inverse assumption of these elements together with duality arguments and derive quasi-optimal asymptotic error estimates in a scale of Sobolev spaces.Dedicated to Prof. Dr.-Ing. W. L. Wendland on the occasion of his 50th birthday.A part of this work was done while the first author was a guest at the Georgia Institute of Technology and while the second author was partially supported by the NSF grant DMS-8501797.     

Extension of a piezoceramic bimorph with a crack crossing the interface

We consider the boundary-value problem of electroelasticity for a composite plate weakened by a crack crossing the line joining the media. The initial boundary-value problem, is reduced to a mixed system of singular integral and algebraic equations. We present the calculation results characterizing the variation in the stress intensity factors as a function of the opening angle of a segmented crack for different types of loading.Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 482–488, July–August 1997.     

An interface crack moving between magnetoelectroelastic and functionally graded elastic layers

A constant crack moving along the interface of magnetoelectroelastic and functionally graded elastic layers under anti-plane shear and in-plane electric and magnetic loading is investigated by the integral transform method. Fourier transforms are applied to reduce the mixed boundary value problem of the crack to dual integral equations, which are expressed in terms of Fredholm integral equations of the second kind. The singular stress, electric displacement and magnetic induction near the crack tip are obtained asymptotically and the corresponding field intensity factors are defined. Numerical results show that the stress intensity factors are influenced by the crack moving velocity, the material properties, the functionally graded parameter and the geometric size ratios. The propagation of the moving crack may bring about crack kinking, depending on the crack moving velocity and the material properties across the interface.     

Boundary integral equations for an interface linear crack under harmonic loading

The present study is devoted to application of boundary integral equations to the problem of a linear crack located on the bimaterial interface under time-harmonic loading. Using the Somigliana dynamic identity the system of boundary integral equations for displacements and tractions at the interface is derived. For the numerical solution the collocation method with piecewise constant approximation on each linear continuous boundary elements is used. The distributions of the displacements are computed for different values of the frequency of the incident tension-compression wave. Results are compared with static ones.     

A boundary integral equation method for three-dimensional crack problems in elasticity

This paper presents a solution procedure for three-dimensional crack problems via first kind boundary integral equations on the crack surface. The Dirichlet (Neumann) problem is reduced to a system of integral equations for the jump of the traction (of the field) across the crack surface. The calculus of pseudodifferential operators is used to derive existence and regularity of the solutions of the integral equations. With the concept of the principal symbol and the Wiener-Hopf technique we derive the explicit behavior of the densities of the integral equations near the edge of the crack surface. Based on the detailed regularity results we show how to improve the boundary element Galerkin method for our integral equations. Quasi-optimal asymptotic estimates for the Galerkin error are given.     

Problem of thermoelectromagnetoelasticity for a piezoelectric/piezomagnetic bimaterial with interface crack

We consider a plane strain problem for a piezoelectric/piezomagnetic bimaterial space with a crack in the region of the interface of the materials. At infinity, tensile and shear stresses and heat, electric, and magnetic flows are set. Using representations for all mechanical, thermal, and electromagnetic factors in terms of piecewise analytic functions, we formulate problems of linear conjugation that correspond to a model of an open crack and models taking into account the contact zone in the vicinity of a crack tip. Exact analytic solutions of the indicated problems are constructed. Expressions for stresses, the electric and magnetic inductions, jumps of derivatives of displacements, and electric and magnetic potentials on the interface are written. The coefficients of intensities of the indicated factors are presented. We derive a transcendental equation for the determination of the real length of the contact zone. The dependences of this length and the coefficients of intensity on the set external influences are investigated.     

Singularities at the tip of a crack on the interface of piezoelectric bodies

Singularities of elastic and electric fields are investigated at the tip of a crack on the interface of two anisotropic piezoelectric media under various boundary conditions on the crack surfaces. The singularity exponents form the spectrum of a certain polynomial pencil, and although explicit formulas are not available, this spectrum is described completely though. The mathematical results apply to problems in fracture mechanics. In this way the Griffith formulas are obtained for increments of energy functionals due to the growth of the crack, and the notion of energy release matrix is introduced. Normalization conditions for bases of singular solutions are proposed to adapt them to energy, stress, and deformation fracture criteria. Connections between these bases are determined, and additional properties of the deformation basis related to the notion of electric surface enthalpy are established. Bibliography: 44 titles. Dedicated to Vsevolod Alekseevich Solonnikov Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 241–271.     

The interface crack in anisotropic bodies. stress singularities and invariant integrals

For the problem of the deformation of a composite anisotropic plate with a crack (in a linear formulation, with no assumption of symmetry), all possible power solutions are listed and general relations between the ordinary and singular solutions are revealed. The asymptotic form of the increment of the potential energy of deformation is computed for the cases of the rectilinear propagation of the crack, deviation of a shoot or branching. The form obtained for the final formula is the same as the classical version of the Griffiths formula and involves two invariant integrals. Two methods of determining the modes of radical singularities of the stress-strain state near the crack tip, associated with the use of force and energy criteria, are proposed.     

The interface crack with Coulomb friction between two bonded dissimilar elastic media

We study a model of interfacial crack between two bonded dissimilar linearized elastic media. The Coulomb friction law and non-penetration condition are assumed to hold on the whole crack surface. We define a weak formulation of the problem in the primal form and get the equivalent primal-dual formulation. Then we state the existence theorem of the solution. Further, by means of Goursat-Kolosov-Muskhelishvili stress functions we derive convergent expansions of the solution near the crack tip.     

Regularization of a discrete scheme governing the three-dimensional evolution of a fluid-fluid interface

A regularized discrete scheme is developed that describes the three-dimensional evolution of the interface between fluids with different viscosities and densities in the Leibenzon-Muskat model. The regularization is achieved by smoothing the kernel of the singular integral involved in the differential equation governing the moving interface. The discrete scheme is tested by solving the problem of a drop of one fluid evolving in a translational flow of another.