Solution for a Crack Embedded in Multiply Confocally Elliptical Layers in Antiplane Elasticity

2015 ◽  
Vol 31 (3) ◽  
pp. 261-267 ◽  
Author(s):  
Y.-Z. Chen
Keyword(s):  
General Solution ◽  
Complex Potentials ◽  
Mapping Technique ◽  
Infinite Matrix ◽  
Exact Relation ◽  
Free Condition ◽  
Antiplane Elasticity ◽  
Correct Form ◽  
Remote Loading ◽  
Crack Faces

ABSTRACTThis paper provides a general solution for a crack embedded in multiply confocally elliptical layers in antiplane elasticity. In the problem, the elastic medium is composed of an inclusion, many confocally elliptical layers and the infinite matrix with different elastic properties. In addition, the remote loading is applied at infinity. The complex variable method and the conformal mapping technique are used. On the mapping plane, the complex potentials for the inclusion and many layers are assumed in a particular form with two undetermined coefficients. The continuity conditions for the displacement and traction along the interface between two adjacent layers are formulated and studied. By enforcing those conditions along the interface, the exact relation between two sets of two undetermined coefficients in the complex potentials for j-th layer and j + 1-th layer can be evaluated. From the traction free condition along the crack faces, the correct form of the complex potential for the cracked inclusion is obtained. Finally, many numerical results are provided.

AN INNOVATIVE SOLUTION IN CLOSED FORM AND NUMERICAL ANALYSIS FOR DISSIMILAR ELLIPTICAL INCLUSION IN PLANE ELASTICITY

2014 ◽  
Vol 06 (06) ◽  
pp. 1450080 ◽  
Author(s):  
Y. Z. CHEN
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Plane Elasticity ◽  
Form Solution ◽  
Complex Potentials ◽  
Infinite Matrix ◽  
Elliptical Inclusion ◽  
The Matrix ◽  
Innovative Solution ◽  
Remote Loading

This paper provides a closed form solution for dissimilar elliptical inclusion in plane elasticity. A dissimilar elliptical inclusion is embedded in the infinite matrix with different elastic properties. The infinite matrix is applied by the constant remote loading. Complex variable method is used and two sets of the complex potentials are assumed in the analysis. One set is used for the matrix portion, and other for the inclusion portion. Catching the idea from the eigenstrain problem, we can assume the stresses in the inclusion to be constant. From the continuity conditions for stresses and displacements along the interface, we can get the two sets of the complex potentials in a closed form. In the analysis, an adequate form of the complex potential defined in the elliptical inclusion portion is analyzed in detail.

Certain cases of elastic equilibrium of a composed wedge with an interface crack

2020 ◽  
Vol 25 (12) ◽  
pp. 2199-2209
Author(s):  
Konstantin B Ustinov
Keyword(s):  
Half Plane ◽  
Edge Crack ◽  
Elastic Equilibrium ◽  
Restrictive Condition ◽  
Uniform Loading ◽  
The Matrix ◽  
Remote Loading ◽  
The Common ◽  
Inclined Edge Crack ◽  
Crack Faces

Problems of interface cracks starting from the common corner points of pairs of perfectly glued wedges of different isotropic elastic materials are addressed. It is demonstrated that for a few particular configurations and a restrictive condition imposed on values of elastic constants (corresponding to vanishing of the second Dundurs parameter), the problem of elastic equilibrium may be solved by Khrapkov’s method. These configurations are: (i) the wedges forming a half-plane; (ii) the wedges forming a plane; (iii) one of the wedges being a half-plane. In all cases, the external boundaries are supposed to be free of stresses. By applying Mellin’s transform for all three configurations the problem has been reduced to vector Riemann’s problem, and the matrix coefficient has been factorized for the case of the mentioned restrictive condition. The first configuration, i.e. the problem of an inclined edge crack located along the boundary separating two wedges of different elastic isotropic materials forming a half-plane is considered in more detail. The solution has been obtained for both uniform (corresponding to remote loading) and non-uniform (loading applied at the crack faces) problems. Numerical results are presented and compared with the available results obtained by other authors for particular cases. The obtained solutions appear especially valuable for analysing extreme cases of parameters.

Uniform strain fields inside multiple inclusions in an elastic infinite plane under anti-plane shear

2016 ◽  
Vol 22 (1) ◽  
pp. 114-128 ◽  
Author(s):  
Ming Dai ◽  
CQ Ru ◽  
Cun-Fa Gao
Keyword(s):  
Holomorphic Function ◽  
Internal Strain ◽  
System Of Nonlinear Equations ◽  
Infinite Matrix ◽  
Infinite Plane ◽  
Plane Shear ◽  
Strain Fields ◽  
Admissible Range ◽  
Remote Loading ◽  
Two Inclusions

This paper constructs multiple elastic inclusions with prescribed uniform internal strain fields embedded in an infinite matrix under given uniform remote anti-plane shear. The method used is based on the sufficient and necessary conditions imposed on the boundary values of a holomorphic function, which guarantee the existence of the holomorphic function in a multiply connected region. The unknown shape of each of the multiple inclusions is characterized by a polynomial conformal mapping with a finite number of unknown coefficients. With the aid of Cauchy’s integral formula and Faber series, these unknown coefficients are determined by a system of nonlinear equations. Detailed numerical examples are shown for multiple inclusions with various prescribed uniform internal strain fields, for symmetrical inclusions and for inclusions whose shapes are independent of the remote loading, respectively. It is found that the admissible range of uniform internal strain fields for multiple inclusions is moderately larger than the admissible range of the uniform internal strain field for a single elliptical inclusion under the same remote loading. In particular, specific conditions on the prescribed uniform internal strain fields and elastic constants of the multiple inclusions are derived for the existence of symmetric inclusions and rotationally symmetrical inclusions. Moreover, for any two inclusions among multiple inclusions of shapes independent of the remote loading, it is shown that the ratio between the uniform internal strain fields inside the two inclusions equals a specific ratio determined by the shear moduli of the two inclusions and the matrix.

Numerical Solution for Dissimilar Confocally Elliptic Layers in Antiplane Elasticity

2016 ◽  
Vol 08 (05) ◽  
pp. 1650071 ◽  
Author(s):  
Y. Z. Chen
Keyword(s):  
Boundary Condition ◽  
Stress Distribution ◽  
General Solution ◽  
Numerical Solution ◽  
Complex Variable ◽  
Final Solution ◽  
Shear Moduli ◽  
Numerical Examples ◽  
Antiplane Elasticity ◽  
The Matrix

This paper provides a general solution for confocally elliptic layers in antiplane elasticity. The studied medium is composed of many layers with different shear moduli. The remote stresses are applied at infinity. Complex variable method is used to study the problem. The continuity conditions for the displacement and the resultant force along the interfaces are suggested. By using the complex variable, the matrix transfer technique, and the boundary condition, the final solution is obtainable. Numerical examples are carried out to show the influence of the different shear moduli defined on different layers to the stress distribution.

Stress-Intensity Factors and Complex Path-Independent Integrals

1980 ◽  
Vol 47 (2) ◽  
pp. 342-346 ◽  
Author(s):  
P. S. Theocaris ◽  
N. I. Ioakimidis
Keyword(s):  
Stress Intensity ◽  
Stress Intensity Factors ◽  
Complex Analysis ◽  
Complex Stress ◽  
Complex Potentials ◽  
Intensity Factors ◽  
Antiplane Elasticity ◽  
Elasticity Problems ◽  
Crack Tips

Path-independent integrals about crack tips may be used to estimate stress-intensity factors at crack tips in plane and antiplane elasticity problems. In this paper a new class of such integrals is established by using complex stress functions and the trivial application of the Cauchy theorem of complex analysis. Both the simple Westergaard complex potentials of plane and antiplane elasticity and the more general Muskhelishvili complex potentials will be used for the construction of appropriate path-independent integrals. Two applications of these integrals to the theoretical determination of stress-intensity factors at crack tips are presented. An optical method for the experimental determination of stress-intensity factors at crack tips, based on the use of appropriate complex path-independent integrals, is also proposed.

Analytical Solution for Size-Dependent Elastic Field of a Nanoscale Circular Inhomogeneity

2006 ◽  
Vol 74 (3) ◽  
pp. 568-574 ◽  
Author(s):  
L. Tian ◽  
R. K. N. D. Rajapakse
Keyword(s):  
Analytical Solution ◽  
Elastic Field ◽  
Infinite Matrix ◽  
Surface Residual Stress ◽  
Residual Surface Stress ◽  
Stress Effects ◽  
Size Dependent ◽  
Closed Form Analytical Solution ◽  
Remote Loading ◽  
Interface Elasticity

Two-dimensional elastic field of a nanoscale circular hole/inhomogeneity in an infinite matrix under arbitrary remote loading and a uniform eigenstrain in the inhomogeneity is investigated. The Gurtin–Murdoch surface/interface elasticity model is applied to take into account the surface/interface stress effects. A closed-form analytical solution is obtained by using the complex potential function method of Muskhelishvili. Selected numerical results are presented to investigate the size dependency of the elastic field and the effects of surface elastic moduli and residual surface stress. Stress state is found to depend on the radius of the inhomogeneity/hole, surface elastic constants, surface residual stress, and magnitude of far-field loading.

A General Treatment of the Elastic Field of an Elliptical Inhomogeneity Under Antiplane Shear

1992 ◽  
Vol 59 (2S) ◽  
pp. S131-S135 ◽  
Author(s):  
S. X. Gong ◽  
S. A. Meguid
Keyword(s):  
General Solution ◽  
Laurent Series ◽  
Elastic Field ◽  
Antiplane Shear ◽  
Complex Potentials ◽  
Geometrical Parameters ◽  
Elliptical Inhomogeneity ◽  
Surrounding Matrix ◽  
Antiplane Problem ◽  
Isotropic Elastic Medium

A general solution to the antiplane problem of an elliptical inhomogeneity in an isotropic elastic medium is provided. The proposed analysis is based upon the use of conformal mapping and Laurent series expansion of the corresponding complex potentials. The general expressions of the complex potentials are derived explicitly in both the elliptical inhomogeneity and the surrounding matrix. Several specific solutions are provided in closed form which are verified by comparison with existing ones. The effect of material and geometrical parameters upon the change of the elastic energy, due to the presence of an elliptical inhomogeneity, has also been considered.

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