Three-Dimensional Analysis of Cracking in a Multilayered Composite

1995 ◽  
Vol 62 (2) ◽  
pp. 273-281 ◽  
Author(s):  
C. H. Kuo ◽  
L. M. Keer
Keyword(s):  
Integral Equations ◽  
Stiffness Matrix ◽  
Crack Opening ◽  
Crack Problem ◽  
Three Dimensional ◽  
Single Layer ◽  
Singular Integral Equations ◽  
Material System ◽  
Multilayered Composite ◽  
Penny Shaped Crack

The three-dimensional problem of a multilayered composite containing an arbitrarily oriented crack is considered in this paper. The crack problem is analyzed by the equivalent body force method, which reduces the problem to a set of singular integral equations. To compute the kernels of the integral equations, the stiffness matrix for the layered medium is formulated in the Hankel transformed domain. The transformed components of the Green’s functions and derivatives are determined by solving the stiffness matrix equations, and the kernels are evaluated by performing the inverse Hankel transform. The crack-opening displacements and the three modes of the stress intensity factor at the crack front are obtained by numerically solving the integral equations. Examples are given for a penny-shaped crack in a bimaterial and a three-material system, and for a semicircular crack in a single layer adhered to an elastic half-space.

Analytic Elements from Singular Integral Equations

2020 ◽  
pp. 227-284
Author(s):  
David R. Steward
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Three Dimensional ◽  
Single Layer ◽  
Singular Integral Equations ◽  
Tangential Component ◽  
Interface Problems ◽  
Far Field ◽  
Line Segments ◽  
Analytic Elements

Solutions to interface problems may be developed using analytic elements with mathematical solutions to the Laplace equation developed by singular integral equations. This formulation leads to solutions with discontinuities occurring across line segments, where the potential or stream function is discontinuous across double layer elements in Section 5.2, and the normal or tangential component of the vector field is discontinuous across single layer elements in Section 5.3. Examples illustrate a broad range of solutions to interface conditions possible with these elements. Series expansions are used to represent the far-field at larger distances from elements in Section 5.4, which leads to higher-order elements with nearly exact solutions and also provides a simpler representation for contiguous strings of adjacent elements. Such strings of elements are used with polygon elements in 5.5 to solve conditions along the interfaces of heterogeneities, and to provide a common series expansion to represent the far-field for a group of neighboring elements. Methods are extended to analytic elements with curvilinear geometry using conformal mappings (Section 5.6) and to three-dimensional fields in Section 5.7.

Three-Dimensional Analysis of Cracking Through the Boundary of a Two-Phase Material

1989 ◽  
Vol 56 (4) ◽  
pp. 850-857 ◽  
Author(s):  
M. T. Hanson ◽  
W. Lin ◽  
L. M. Keer
Keyword(s):  
Stress Intensity ◽  
Phase Boundary ◽  
Stress Intensity Factors ◽  
Crack Opening ◽  
Three Dimensional ◽  
Singular Integral Equations ◽  
The Body ◽  
Intensity Factors ◽  
Two Phase ◽  
Penny Shaped Crack

The penetration through a two-phase boundary by a biplanar (kinked) crack of arbitrary shape is considered in this paper. The two-phase boundary is modeled as the interface between two perfectly-bonded elastic, isotropic, homogeneous half spaces with different elastic constants. The planar crack on either side of the interface may be arbitrarily orientated with respect to the interface boundary. The body-force method is used to derive a set of coupled two-dimensional singular integral equations which are solved numerically. The solution yields the three crack opening displacements as well as the three modes of stress intensity factors along the crack contour. Numerical results are given for a penny-shaped crack symmetrically oriented with respect to the interface. Mode I stress intensity factors are given for the biplanar crack that experiences a kink when passing through the interface.

Stress Concentrations in Three-Dimensional Electrostriction

1967 ◽  
Vol 34 (2) ◽  
pp. 431-436 ◽  
Author(s):  
T. E. Smith
Keyword(s):  
Displacement Vector ◽  
Spherical Inclusion ◽  
Crack Problem ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Uniform Electric Field ◽  
Inclusion Problem ◽  
Stress Concentrations ◽  
Displacement Potential ◽  
Penny Shaped Crack

Using the techniques employed in developing a Papkovich-Neuber representation for the displacement vector in classical elasticity, a particular integral of the kinematical equations of equilibrium for the uncoupled theory of electrostriction is developed. The particular integral is utilized in conjunction with the displacement potential function approach to problems of the theory of elasticity to obtain closed-form solutions of several stress concentration problems for elastic dielectrics. Under a prescribed uniform electric field at infinity, the problems of an infinite elastic dielectric having first a spherical cavity and then a rigid spherical inclusion are solved. The rigid spheroidal inclusion problem and the penny-shaped crack problem are also solved for the case where the prescribed field is parallel to their axes of revolution.

scholarly journals A mixed boundary value problem of a cracked elastic medium under torsion

2021 ◽  
pp. 10-10
Author(s):  
Belkacem Kebli ◽  
Fateh Madani
Keyword(s):  
Boundary Value Problem ◽  
Elastic Medium ◽  
Integral Equations ◽  
Integral Transformation ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Crack Problem ◽  
Fredholm Integral Equations ◽  
Mixed Boundary Value Problem ◽  
Penny Shaped Crack

The present work aims to investigate a penny-shaped crack problem in the interior of a homogeneous elastic material under axisymmetric torsion by a circular rigid inclusion embedded in the elastic medium. With the use of the Hankel integral transformation method, the mixed boundary value problem is reduced to a system of dual integral equations. The latter is converted into a regular system of Fredholm integral equations of the second kind which is then solved by quadrature rule. Numerical results for the displacement, stress and stress intensity factor are presented graphically in some particular cases of the problem.

The Axisymmetric Crack Problem in a Nonhomogeneous Medium

1993 ◽  
Vol 60 (2) ◽  
pp. 406-413 ◽  
Author(s):  
M. Ozturk ◽  
F. Erdogan
Keyword(s):  
Stress Intensity ◽  
Shear Modulus ◽  
Mixed Mode ◽  
Crack Opening ◽  
Crack Problem ◽  
Singular Integral Equations ◽  
Nonhomogeneous Medium ◽  
Intensity Factors ◽  
Simple Simulation ◽  
Graded Properties

In this paper, the axisymmetric crack problem for a nonhomogeneous medium is considered. It is assumed that the shear modulus is a function of z approximated by μ = μ0eαz. This is a simple simulation of materials and interfacial zones with intentionally or naturally graded properties. The problem is a mixed-mode problem and is formualated in terms of a pair of singular integral equations. With fracture mechanics applications in mind, the main results given are the stress intensity factors as a function of the nonhomogeneity parameter a for various loading conditions. Also given are some sample results showing the crack opening displacements.

A Survey of Macro-Microcrack Interaction Problems

2000 ◽  
Vol 53 (5) ◽  
pp. 117-146 ◽  
Author(s):  
Vera Petrova ◽  
Vitauts Tamuzs ◽  
Natalia Romalis
Keyword(s):  
Three Dimensional ◽  
Elastic Solid ◽  
Singular Integral Equations ◽  
Crack Location ◽  
Penny Shaped Crack ◽  
Shaped Crack ◽  
Microcrack Interaction ◽  
The Impact ◽  
The Continuum ◽  
Model Approach

The results obtained on the problem of the interaction between a large crack and an array of microcracks or other microdefects are reviewed. The following problems are considered: interaction of main crack with microcracks in the two-dimensional case at tensile, shear or combined stress state; a closure of macro or microcracks as a result of their interaction, and the influence of this phenomenon on the stress intensity factor; the thermal cracking of an elastic solid caused by the macro-microcracks interaction and cracks closure; the interaction of a crack with an array of small pores or rigid inclusions; three-dimensional problems of the interaction of a penny-shaped crack with small penny-shaped microcracks. Discussed analytical results are based on the asymptotic analysis and the series solution to systems of singular integral equations describing the interaction of the macrocrack and microdefects. The series solutions were obtained with respect to the small parameter representing the ratio of micro- to macrocrack sizes. Throughout the review, the known solutions on the crack interaction are surveyed. The comparison with solutions to other relevant problems such as an interaction of semi-infinite crack with an array of finite cracks is given. The impact of a close crack location, and a comparison with relevant results of the continuum model approach are discussed. This review article includes 332 references.

Singular integrals in axisymmetric problems of elastostatics

2020 ◽  
Vol 11 (01) ◽  
pp. 2050003
Author(s):  
Artem Karaiev ◽  
Elena Strelnikova
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Singular Integrals ◽  
Integral Transformation ◽  
Logarithmic Singularity ◽  
Three Dimensional ◽  
Singular Integral Equations ◽  
Elliptic Integrals ◽  
Surface Integral ◽  
Axisymmetric Problems

Singular integral equations arisen in axisymmetric problems of elastostatics are under consideration in this paper. These equations are received after applying the integral transformation and Gauss–Ostrogradsky’s theorem to the Green tensor for equilibrium equations of the infinite isotropic medium. Initially, three-dimensional problems expressed in Cartesian coordinates are transformed to cylindrical ones and integrated with respect to the circumference coordinate. So, the three-dimensional axisymmetric problems are reduced to systems of one-dimensional singular integral equations requiring the evaluation of linear integrals only. The thorough analysis of both displacement and traction kernels is accomplished, and similarity in behavior of both kernels is established. The kernels are expressed in terms of complete elliptic integrals of first and second kinds. The second kind elliptic integrals are nonsingular, and standard Gaussian quadratures are applied for their numerical evaluation. Analysis of external integrals proved the existence of logarithmic and Cauchy’s singularities. The numerical treatment of these integrals takes into account the presence of this integrable singularity. The numerical examples are provided to testify accuracy and efficiency of the proposed method including integrals with logarithmic singularity, Catalan’s constant, the Gaussian surface integral. The comparison between analytical and numerical data has proved high precision and availability of the proposed method.

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