Timefractional heat conduction in a plane with two. In the system of integral equations of the first kind, the fundamental solution is the weight function of the chebyshev polynomials of first or second kind. These factors are odd and even functions, respectively, with respect to the deepest point of the crack. Relevant work on a pennyshaped crack at the interface of a bimaterial under compressive force and shear is very limited. Exact and closed form fundamental solutions are expressed in terms of elementary functions. Previous question next question transcribed image text from this question. Elastodynamic fields for a growing pennyshaped crack under tension have been obtained by the time domain boundary integral equation method. Assume that this value is for planestrain conditions. Feb 08, 2015 for example, a transient thermal fracture problem corresponding to a semi infinite medium with a surface crack was studied in, a penny shaped crack in a piezoelectric material was studied in, transient thermal cracking associated with nonclassical heat conduction in cylindrical coordinate system was studied in and the thermal shock. Catastrophic fracture occurs when a stress of 700 mpa is applied. Spatial distribution of raowiltonglisson rwg functions. In this work, we consider a twodimensional dynamical problem of an infinite space with finite linear modei crack and employ a recently proposed heat conduction model.
Suppose that is a pennyshaped crack, with radius so that the crack occupies the region where and are polar coordinates, and. Numerical solution for an epicycloid crack fracture. Nonlinear behavior and critical state of a pennyshaped. The analysis is based on a nonhypersingular time domain boundary integral equation method and is fully threedimensional. The representation integral expressing the scattered displacement field has been reduced to an integral equation for the unknown crackopening displacement. For example, a transient thermal fracture problem corresponding to a semiinfinite medium with a surface crack was studied in, a pennyshaped crack in a piezoelectric material was studied in, transient thermal cracking associated with nonclassical heat conduction in cylindrical coordinate system was studied in and the thermal shock. The representation integral expressing the scattered displacement field has been reduced to an integral equation for the unknown crack opening displacement. A complete closed form solution was obtained for a pennyshaped crack in an elastic space, subjected to arbitrary internal tractions. Numerical results for a pennyshaped interface crack in al2o3pmma. Based upon integral transform technique, the crack boundary value problem. An extended model for electrostatic tractions at crack faces in piezoelectrics. The electroelastic response of a pennyshaped crack in a piezoelectric cylinder of finite radius is investigated in this study. The calculation of tstress along the crack front using domain integral. In this paper, we discuss the temperature distribution and thermal stresses in a semiinfinite cylinder whose lower and upper surfaces are free of traction and subjected to a given axisymmetric temperature distribution with the help of lord shulman theory and classical coupled theory of thermoelasticity using integral transform technique.
A crack of radius r 1, is initially in equilibrium with a static tensile stress applied at infinity. The electroelastic response of a penny shaped crack in a piezoelectric cylinder of finite radius is investigated in this study. A cylindrical crack in an infinite elastic medium under impact loading was. The use of these formulae is illustrated by a consideration of the special case in which the. A mathematically similar problem arises in the consideration of a penny shaped crack subjected to an arbitrary normal pressure.
A bar of size 2w1 u 2 w2 u 2h, containing a central pennyshaped crack, is subjected to the heaviside traction load v t v 0 h t acting on the ends, as shown in fig. Continuum mechanics studies the foundations of deformable body mechanics from a mathematical perspective. This study aims to investigate the interactions of multiple parallel cracks in a semiinfinite domain in both deterministic and probabilistic ways by using an automated finite element modeling procedure and the monte carlo simulation. Transient elastodynamic analysis of a pennyshaped crack. The body is strainless at the initial moment, which means there are no initial displacements of the points of the body. Timefractional heat conduction in a plane with two external. For example, a transient thermal fracture problem corresponding to a semi infinite medium with a surface crack was studied in, a penny shaped crack in a piezoelectric material was studied in, transient thermal cracking associated with nonclassical heat conduction in cylindrical coordinate system was studied in and the thermal shock. In this paper, the dynamic response of a piezoelectric layer with a pennyshaped crack is investigated. Dynamic fracture analysis of a pennyshaped crack in a. In both cases, two geometrical models of cracks are examined and discussed.
Suppose that is a pennyshaped crack, with radius so that the crack occupies the region where and are polar coordinates, and now, let be a simply connected domain in the plane defined as whose boundary has the polar equation, where is bounded and piecewise continuous and is a small positive parameter. A penny crack in an infinite solid is subjected to a uniform shear on one of its crack surface as depicted in figure 6. A mathematical formulation is presented for the dynamic stress intensity factor mode. In this paper, the field equation of micropolar fluid with general lubrication theory assumptions is simplified into two systems of coupled ordinary differential equation. The method relies on the construction of virtual diskshaped integral domains at points. Fundamental electroelastic field in an infinite transversely. A boundary integral equation method in the frequency domain. Deterministic and probabilistic investigation on multiple. This book deals with the mechanics and physics of fractures at various scales. Thermal shock fracture mechanics analysis of a semi.
A generation of special triangular boundary element shape. The problem is solved by use of the boundary integral equations in the frequency domain, and the components of the solution are presented by the fourier exponential series. Multiple pennyshaped cracks interaction in a finite body. Boundary integral equations in elastodynamics of interface. In this paper, the dynamic response of a piezoelectric layer with a penny shaped crack is investigated. The investigation of multiple crack interactions in fracture mechanics is important to predict the safety and reliability of structures. Interface cracks in piezoelectric materials iopscience. Potential theory, mixed boundary value problems of. The material of the medium is considered to be homogeneous and isotropic. Herein, we consider the axisymmetric problem of a pennyshaped crack in an elastic material sandwiched between other materials. Neumann problem is prescribed at the remaining part.
A formula is derived for the stress intensity factor at the rim of a pennyshaped crack in an infinite solid in which there is an axisymmetric distributing of body forces acting in a direction normal to the original crack surfaces. The scattering of normally incident elastic waves by an embedded elliptic crack in an infinite isotropic elastic medium has been solved using an analytical numerical method. Surface crack subject to mixed mode loading harvard john a. The crack is assumed to be electrically permeable and is subjected to mechanical loadings applied symmetrically with respect to the crack face. Xiao zm, luo j 2004 on the dynamic interaction between a pennyshaped crack and an expanding spherical inclusion in 3d solid. The analytical solutions of velocity and microrotat ion velocity are obtained. Most of these studies investigate interactions of the elastic waves with a 2d slit or 3d penny shaped fracture in an infinite isotropic elastic solid. Arutyunyan, torsion of a composite sphere with an annular or penny shaped crack, mechanics of solids mtt, 1991, 26, no 4, 8395.
This paper concerns the fracture mechanics problem for elastic. Stress intensity factors for embedded cracks within. Fluidsaturated pennyshaped crack in a poroelastic solid. The present work is concerned with resolving a dynamical problem of an infinite type iii thermoelastic space weakened by a finite linear mode i crack. The governing integrodifferential equation takes the form. This paper examines the problem of a penny shaped crack in a thermoporoelastic body. The problem a10 can be interpreted as an electrostatic problem of a charged disc inside an infinite grounded diaphragm. The boundary of the crack is subjected to a prescribed stress distribution and temperature. Acoustic emission and neartip elastodynamic fields of a. Gross d transient elastodynamic analysis of a penny shaped crack. The timefractional heat conduction equation with the caputo derivative is solved for an infinite plane with two external halfinfinite slits with the prescribed heat flux across their surfaces. An expression for the surface displacement of the crack is also given. Zappalorto, a threedimensional stress field solution for pointed and sharply radiused vnotches in plates of 1 c.
Pennyshaped crack in an infinite domain under uniaxial tension. Uniform uniaxial stress if the crack is located centrally in a finite plate of width 2 b \displaystyle 2b and height 2 h \displaystyle 2h, an approximate relation for the stress intensity factor is 4. A twodimensional problem of a mode i crack in a type iii. Fourier and hankel transforms are used to reduce the problem to the. The choice of the incompressible isotropic powerlaw constitutive relations can be justified by the following reasons.
My main research interests are mathematical modeling in dynamic fracture mechanics, penetration mechanics, aeroelasticity, fluid dynamics supercavitating flow, and scattering of sound and electromagnetic waves. The two xfem boundary value problem classes are as follows. Monitoring fatigue crack growth in plate structures based on remote strain fields and the solution of multiparametric inverse problems. Axisymmetric problems of a pennyshaped crack at the interface of a. Pdf elastic tstress solution for pennyshaped cracks under. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. An investigation on a twodimensional problem of modei. In this paper, the transient response of a pennyshaped crack embedded in a.
The wavenumber domain solutions are calculated for the same wavenumbers for the same stations as in the. The penny shaped crack in an infinite block of material subjected to remote shear. May 22, 2012 for the first time, exact and fundamental solutions are developed for an infinite transversely isotropic piezoelectric medium, weakened by a circular external crack. This paper examines the problem of a pennyshaped crack in a thermoporoelastic body. Asymptotics of solutions to the poisson problem in a perforated domain with corners j. Nevertheless, an elastic analysis of a homogeneous material with a pennyshaped crack under radial shear can be found in kassir and sih, 1975, lee, 20. Based on advanced continuum mechanics of heterogeneous media, it develops a rigorous mathematical framework for single macrocrack problems as well as for the effective properties of microcracked materials. Antipov, analytic solution of a twodimensional problem for a halfplane with a crack that is close to the boundary, studies in elasticity and plasticity, 1990, no.
Micropolar fluid lubrication reynolds equation is deduced. For the case of a penny shaped crack situated in an infinite isotropic medium with the crack faces subjected to arbitrary tractions, the integral equations are solved explicitly. Mixed boundary value problems can be encountered in almost any branch of engineering and are among the most difficult to solve. On the basis of the recently developed general solutions for thermoporoelasticity, appropriate potentials are suggested and the governing equations are solved in view of the similarity to those for pure elasticity. In this case, the correction factor for a round crack is simply given by actually, eq. Other readers will always be interested in your opinion of the books youve read. Transient elastodynamic analysis of a stationary pennyshaped crack in an infinite elastic solid is presented. An investigation on a twodimensional problem of modei crack. Transient response of an elastic solid containing a pennyshaped crack due to mechanical and thermal impact loadings using coupled theory of. The pennyshaped crack in an infinite body of a powerlaw.
Implicitly restarted arnoldi with purification for the shiftinvert transformation k. The dynamic stress intensity factors are computed for different stress pulses and compared with those. Beteq2010 part 1 boundary element method mathematical. In this paper, we discuss the temperature distribution and thermal stresses in a semi infinite cylinder whose lower and upper surfaces are free of traction and subjected to a given axisymmetric temperature distribution with the help of lord shulman theory and classical coupled theory of thermoelasticity using integral transform technique. Dynamic stress intensity factor mode i of a permeable penny. Dirichlet problem is given at one part of the boundary, and a neumanntype boundary condition cf. We refer to as the equivalent mises stress and s is the deviator of the stress tensor t.
Based on these theoretical models, the effects of multiple dry fractures can be studied using the noninteraction approximation proposed by foldy 1945. For the crack front elements, special shape functions with asymptotic. Dynamic problem of generalized thermoelasticity for a semi. However, the boundary of the crack is subjected to a prescribed temperature and stress distributions. A boundary integral equation method in the frequency domain for. The kind of boundary value problems where the dirichlet boundary condition cf. Timedomain bem for 3d transient elastodynamic problems with. The crack then propagates with a given rupture velocity, but stops abruptly when its radius reaches r 2 r 1. Axisymmetric dynamic response of a pennyshaped crack to a. A penny crack of radius a in an infinite body, with applied remote. The timefractional heat conduction equation with the caputo derivative is solved for an infinite plane with two external half infinite slits with the prescribed heat flux across their.
Jan 28, 2016 this paper concerns the fracture mechanics problem for elastic cracked materials under transient dynamic loading. We present a new single expansion framework which describes the interaction between several physical processes, namely viscosity. Multiple pennyshaped cracks interaction in a finite body and their effect on stress intensity factor. As an example, consider an elastic space weakened by a flat crack of general shape, subjected to an arbitrary normal traction. An internal pennyshaped crack in an infinite thermoelastic solid. A diskshaped domain integral method for the computation of stress. Xiao zm, luo j 2004 on the dynamic interaction between a penny shaped crack and an expanding spherical inclusion in 3d solid. Boundary element discretization of a pennyshaped crack in an infinite space. Critical load for a modei crack reinforced by bridging fibres n. Stress intensity factor wikipedia republished wiki 2. It also acts as a base upon which other applied areas such as solid mechanics and fluid mechanics are developed.
Thermal shock fracture mechanics analysis of a semiinfinite. The thermoelastic medium is taken to be homogeneous and isotropic. Timedomain bem for 3d transient elastodynamic problems. The dynamics of hydraulic fracture, described by a system of nonlinear integrodifferential equations, is studied through the development and application of a multiparameter singular perturbation analysis. Siam journal on applied mathematics society for industrial. The problem of a penny shaped interface crack between a functionally graded piezoelectric layer and a homogeneous piezoelectric layer under. Fracture, mathematical problems of encyclopedia of.
Scattering attenuation, dispersion and reflection of sh. Further application of tranters method, in expressing unknown functions as an infinite series of bessel functions, reduces the equations to an infinite set of linear algebraic equations whose solution in the laplace transform domain is inverted numerically to yield the values of the dynamic stressintensity factor, k 1 t. Pdf analysis of multiple axisymmetric annular cracks. Pwave dispersion and attenuation due to scattering by. The response of a single pennyshaped crack to transient elastic waves also was. Propagation of a penny shaped crack under the irwin criterion analysis, numerics and applications of differential and integral equations. Thermal impact response of a thermoelastic solid with a finite crack. Exact and complete fundamental solutions for pennyshaped. Scattering from an elliptic crack by an integral equation. Now, let be a simply connected domain in the plane defined as whose boundary has the polar equation, where is bounded and piecewise continuous and is a small positive parameter. It will be proved also that one of the derived relationships is valid not only for a pennyshaped crack but also for an arbitrary crack, thus making it possible for the first time to consider nonelliptical cracks subjected to shear loading analytically. Axisymmetric problems of a pennyshaped crack at the.
The piezoelectric layer is subjected to an axisymmetrical action of both mechanical and electrical impacts. The stress intensity factor at the tip of a pennyshaped crack of radiusthe stress intensity factor at the tip of a pennyshaped crack of radius a in an infinite domain under uniaxial tension sigma view the full answer. Transient response of a piezoelectric layer with a penny. In this work, we solve a dynamical problem for an infinite thermoelastic solid with an internal penny shaped crack, which is subjected to prescribed temperature and stress distributions. It is assumed that the central substance is composed of an elastic layer held between two semi infinite bodies with different elastic constants, and that the crack is situated in the central plane of the.
Identification of crack parameters and stress intensity factors in finite and semiinfinite plates solving inverse problems of linear elasticity. The important conclusion of this paper was that for antiplane shear problems the stress intensity factors depend upon the lateral part of the nonhomogeneity in the directions parallel to the crack. An analytical solution for the axisymmetric problem of a. Using the properties of the related orthogonal polynomials, approximate solutions of systems of simultaneous singular integral equations are obtained, in which the essential features of the singularity of the unknown functions are preserved. Thermal impact response of a thermoelastic solid with a. The other crack surface is subjected to the same load, but in the opposite direction. The timefractional heat conduction equation follows from the law of conservation of energy and the corresponding timenonlocal extension of the fourier law with the longtail power kernel.
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