Inhomogeneous materials with doubly periodic nonuniform cracks under antiplane shear is dealt with. Consequently, antiplane elastic cloaks have been designed successfully based on the transformation method 17,26. Methods and applications roland schinzinger electrical engineering department, university of california, irvine, ca 92717, u. An analytic function maps a point on the complex plane to an image. A unified solution approach for a large variety of antiplane. The proposed conformal mapping method as well as the solving process are presented in sections 3 and 4. Antiplane shear torsion crack notch stress distribution abstract in this work, the exact solution for the stress. Here we look at some examples of analytic functions that illustrate that they are conformal maps. In this work, we consider the exact sifs and errs of an orthotropic dcb specimen under antiplane delaminating in order to provide the. In proof of 6, consider a circular cylinder of radius a loaded on its boundary by the shear stress. Isotoxal starshaped polygonal voids and rigid inclusions in.
An effective method is developed and used to investigate the elastic field and electric field of a rigid line in a confocal elliptic piezoelectric inhomogeneity embedded in an infinite piezoelectric medium. The problem is first reduced to a nonhomogeneous riemannhilbert. Antiplane problem of periodically stacked parallel cracks. Numerical examples show the influences of some microstructure parameters of crack distribution on stress intensity factor.
We investigate the anti plane shear problem of a curvilinear crack lying along the interface of an arbitrarily shaped elastic inhomogeneity embedded in an infinite matrix subjected to uniform stresses at infinity. A unified solution approach for a large variety of antiplane shear and. Chapter 3 conformal mapping technique an overview 3. Evaluation of the degenerate scale for bie in plane. Neutrality of coated holes in the presence of screw. A new approach for electroelastic analysis of piezoelectric. Worked examples conformal mappings and bilinear transformations example 1 suppose we wish to. It is not difficult to ascertain that in this case the halflines parallel to the sides of the angle are transformed into halfparabolas with a common focus at o figure 3. Article information, pdf download for uniform stress state inside a. A general solution to the antiplane problem of an elliptical inhomogeneity in an isotropic elastic medium is provided. The method is based on the complex potential approach for antiplane elasticity combined with conformal mapping. This provides an efficient way to finding the exact elasticity solution for antiplane composite dcb specimens.
Isotoxal starshaped polygonal voids and rigid inclusions. Worked examples conformal mappings and bilinear transfor. Lubarda where a0 n and b n 0 are the corresponding fourier coef. In addition, shahani 56 derived the analytical expressions by the conformal mapping techniques for mode iii stress intensity factor of circular shafts with edge cracks, bonded half planes. Zappalorto international journal of solids and structures 102103 2016 1020 11 fig. Linear elasticity division of engineering brown university. All the work in the literature led to fracture criteria such as stress intensity.
It follows from this that the only nonzero stress components are determined via. We develop a twoparameter conformal mapping function for a doubly connected domain to solve the inverse problem in antiplane and plane elasticity associated with a nonelliptical inhomogeneity. Hypocycloidalinclusionsinnonuniformoutofplane elasticity. Conformal mapping is a field in which pure and applied mathematics are both involved. Analytic continuation and conformal mapping techniques are applied to establish that the state of stress inside a nonelliptical elastic inhomogeneity can remain uniform despite the presence of a nearby irregularly shaped hole when the surrounding matrix is subjected to uniform remote antiplane shear stresses. Antiplane shear mode stress intensity factor for a slightly perturbed circular crack subject to shear load 101 figure 1. For small strains, the strain tensor under antiplane shear can be written as where the plane is the plane of interest and the direction is perpendicular to. A crack emanating from the apex of an infinite wedge in an anisotropic material under antiplane shear is investigated. Linear elasticity division of engineering brown university 9. The crack is assumed to be either electrical impermeable or permeable. Singularities and stress intensities at the corner point of a. International journal of solids and structures 102103 2016 1020. Suppose that the upper and lower edges of the plate are stressed by antiplane shear forces so that in which is the known dirac delta function.
Furthermore, although ground cloaks have been considered in the acoustic context they have not yet been. Different numerical approaches for inverse antiplane problems were employed in 12. Conformal mapping is used to transform the integral\ud equation into a similar equation over a circular region, d. Nullfield approach for the multiinclusion problem under. One of the exceptions however in the context of elastodynamics is the antiplane shear wave case, since the elastic displacement in this twodimensional scalar problem is governed by helmholtzs equation, similar to acoustics. A circular eshelby inclusion interacting with a nonparabolic. The mapping is said to be conformal at any point where. This paper deals with a slightly perturbed circular crack.
Method of automorphic functions for an inverse problem of antiplane. Lee and earmme 8 examined an interfacial edge crack in an anisotropic bimaterial under antiplane singularity. Pdf threephase parabolic inhomogeneities with internal. A wedge crack in an anisotropic material under antiplane shear. A general complex variable approach is presented for anti plane problem for a system of one anisotropic elliptic inhomogeneity embedded in an infinite anisotropic medium matrix. Dynamic analysis of cracks running at a constant velocity. For small strains, the strain tensor under antiplane shear can be written as. Two imperfectly bonded halfplanes with an arbitrary. Electroelastic interaction between a screw dislocation. The explicit expressions of the complex potentials are derived in both the inhomogeneity and the surrounding matrix using conformal mapping and the perturbation techniques.
An isotropic wedge crack subjected to concentrated forces is first solved by using the conformal mapping technique. The general solution to the problem is obtained through the use of conformal mapping technique and laurent series expansion of the associated complex potentials. This paper evaluates the degenerate scale for bie in plane elasticity and antiplane elasticity by using the complex variable and the conformal mapping. Authors personal copy we rst introduce the following conformal mapping function. Instead of achieving conformal mapping by using microstructural units. Theory and examples, international journal of solids and structures on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. In this work, the exact solution for the stress fields ahead of cracks initiated at sharp notch tips under antiplane shear and torsion loadings is derived in close form, leveraging conformal mapping and the complex potential method for antiplane elasticity.
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths more formally, let and be open subsets of. To fully capture the circular geometries, separable expressions of fun. Uniform stress state inside a nonelliptical inhomogeneity near an. Using the complex variable function method and the technique of conformal mapping, the fracture mechanics of two symmetric collinear cracks originating from an elliptical hole in a onedimensional 1d hexagonal piezoelectric quasicrystals qcs are investigated under antiplane shear loading and electric loading. Elementary mathematical theory for stress singularities at.
Usually these solutions show stress singularities at the inclusion corners. Considering now the conformal mapping equation 42 of part i, the. By definition, a conformal mapping of a domain is required to be continuous and conformal only at the interior points of. A unified solution approach for a large variety of. By constructing proper westergaard stress function and using the periodicity of the hyperbolic function, the antiplane problem of the periodic parallel cracks. Theory and examples, international journal of solids and structures on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at. The antiplane shear problem of two symmetric cracks. A line inhomogeneity in an elastic half plane under anti. For i l2 with different choices of c with the mapping 3, eq.
Pdf antiplane shear stresses in orthotropic plates with. Conformal mapping is used to transform the integral equation into a similar equation over a circular. On the other hand, it is well known that mode iii antiplane shear deformation is also important in the mechanics of fracture 8, then, we consider the orders of stress singularity and the stress intensities k for the corner of a polygonal hole and a rigid polygonal inclusion under antiplane shear by the conformal mapping, where the regular. Making use of the complex potentials and conformal mapping techniques, we show that the multiple coatings can be analyzed through a recurrence procedure in the transformed domain, while remaining explicit in detail and transparent overall. By using the complex potential and the conformal mapping. Antiplane shear or antiplane strain is a special state of strain in a body. On the circumferential shear stress around circular and. Using the complex variable function method and the technique of conformal mapping, the fracture mechanics of two symmetric collinear cracks originating from an elliptical hole in a onedimensional 1d hexagonal piezoelectric quasicrystals qcs are investigated under anti plane shear loading and electric loading. The analytical solution is obtained using the conformal mapping and the theorem of analytic continuation. Basic equations in the complex variable method of plane elasticity are compactly addressed.
To the novice, it may seem that this subject should merely be a simple reworking of standard real variable theory that you learned in. The proposed analysis is based upon the use of conformal mapping and laurent series expansion of the corresponding complex potentials. Dec 15, 2016 read a unified solution approach for a large variety of antiplane shear and torsion notch problems. This work focuses on a major large class of multiple inclusions characterized by a simple condition. Through the use of a conformal mapping, the cracked. Analytic continuation and conformal mapping techniques are applied to establish that the state of stress inside a nonelliptical elastic inhomogeneity can remain uniform despite the presence of a nearby irregularly shaped hole when the surrounding matrix is. Chalkboard photos, reading assignments, and exercises solutions pdf 2. Nullfield approach for the multiinclusion problem under antiplane shears in this paper, we derive the null. Laura universidad nacional del sur, 8000 bahia bianca, argentina and institute of applied mechanics conicet elsevier amsterdam oxford new york tokyo 1991. A line inhomogeneity in an elastic half plane under anti plane shear loading george, n, emenogu.
The idea is through conformal transformations satisfying the conditions requested of the problem make this an easier problem to deal,but i dont know which be this transformation. The antiplane problem is turned into the boundary value problem of partial differential equation. We focus first on the simple case of antiplane shear deformation. Examples of conformal maps and of critical points we know that an analytic function fz is conformal preserves angles and orientation at all points where the derivative fz is not zero. The outofplane elastic problem can therefore be solved through a complex potential representation g. The problem of the analytical evaluation of the stress. Based on complex variable and conformal mapping method, the dynamic stress concentration factors and the electric field concentration factors at the boundary of the noncircular cavity are obtained by applying the orthogonal function expansion technique. The solution is obtained using the complex potential technique, with conformal mapping 23, 24, 25, which leads to a full. By choosing an appropriate mapping function, the analyst can transform the inconvenient geometry into a much more convenient one. A method of laurent series and a conformal mapping from an annulus to a doubly connected domain to recover the profile of two inclusions with uniform stresses was applied in. Barber department of mechanical engineering, university of michigan, usa. Due to complicated boundary conditions in the interfacial edge crack problems, a number of investigators used conformal. As working with ellipses is unpleasant, we want to nd a conformal mapping that maps the region outside the ellipse to a region outside a circle.
The problem of a confocally multicoated elliptical inclusion in an unbounded matrix subjected to an antiplane shear is studied. November 20, 2008 conformal same form or shape mapping is an important technique used in complex analysis and has many applications in di erent physical situations. Conformal mapping is used to transform the integral equation into a similar equation over a circular region, d. Pdf analytical solution with validity analysis for an elliptical void. So lets look at an in nite plane with an elliptic hole, subject to antiplane shear. The antiplane problem of the periodic parallel cracks in an infinite linear elastic orthotropic composite plate is studied in this paper. Comparison of the numerical solutions with the existing asymptotic solutions show a good.
A confocally multicoated elliptical inclusion under antiplane. Complex variable and conformal mapping techniques are used to derive an analytical solution in series form. Complex analysis and conformal mapping the term complex analysis refers to the calculus of complexvalued functions fz depending on a single complex variable z. This state of strain is achieved when the displacements in the body are zero in the plane of interest but nonzero in the direction perpendicular to the plane. The matrix is subjected to remote antiplane shear and inplane electric fields. In this work, a unified solution approach is proposed for the analytical evaluation of the stress fields close to notches under antiplane shear and torsion loadings, which allows a large variety of notch problems to be tackled. This approach would appear to eliminate the requirement of metamaterials with inhomogeneous anisotropic shear moduli and density. The system is solved numerically for the unknown coefficients, which will later be used in determining the antiplane shear mode stress intensity factor. We develop a twoparameter conformal mapping function for a doubly connected domain to solve the inverse problem in anti plane and plane elasticity associated with a nonelliptical inhomogeneity. Nonlinear prestress for cloaking from antiplane elastic. Uniform stress state inside a nonelliptical inhomogeneity. An arbitrarily shaped eshelby inclusion interacting with a circular. Chapter 3 conformal mapping technique various techniques have been used to calculate the conductor loss, including wheelers incremental inductance rule 26, closedform formulae based on rigorous numerical techniques and interpolation 27, perturbation methods combined with the. This has often been solved by conformal mapping, integral transforms, perturbation methods, numerical techniques and ergen value techniques among the techniques.
Murdoch surfaceinterface model, an antiplane shear problem of a magnetically and electrically impermeable nano. A theory is presented showing that cloaking of objects from antiplane elastic waves can be achieved by employing nonlinear elastic prestress in a neohookean elastomeric material. A general treatment of the elastic field of an elliptical. An closed form solution for antiplane problem of doubly.
Read a unified solution approach for a large variety of antiplane shear and torsion notch problems. Analytic continuation and conformal mapping techniques are applied to establish that. After validating the proposed model and method, the normal shear stress and the tangential electric field distributions outside the piezoelectric inclusion, and the effective moduli are obtained as shown in section 5. U1standard model where the dimensional parameter in the higgs potential is replaced.
We consider a coated nonelliptical inhomogeneity interacting with a nearby circular eshelby inclusion inside an infinite elastic matrix subjected to antiplane shear deformations and uniform. Method of automorphic functions for an inverse problem of. A method of laurent series and a conformal mapping from an annulus to a doubly connected domain to recover the profile of two inclusions with uniform stresses. The matrix is subjected to uniform remote antiplane shear.
From the formulation of an exterior problem in plane elasticity, the. An effective method is developed and used to investigate the antiplane problem of a rigid line in a confocal elliptic inhomogeneity embedded in an infinite medium. In the case of antiplane shear deformations of a hexagonal piezoelectric material with poling direction. Rigid line in a confocal elliptic inhomogeneity embedded. The system is subject to farfield shear stresses in the matrix at infinity and eigenstrains inside the. The results reveal that when the inhomogeneity reduces to a cavity, the electric field. Conformal mapping article about conformal mapping by the. Considering now the conformal mapping equation 42 of part i, the relationship between. The problem of finding the resulting shear forces can be formulated as a hypersingular integral equation over a considered domain. Antiplane shear waves, galerkin numerical method, colinear cracks, halfspace, strip 1 introduction the study of elastodynamic problems of moving or. Dynamic analysis of cracks running at a constant velocity in a strip. By using conformal mapping technique and elliptic function theory, the stress field and stress intensity factor at the tip of each crack are derived in closed form. In this case, there is only one field quantity to be computed, and the stresses and strains are related by the twodimensional expressions.
The matrix is subjected to the remote antiplane shear and inplane electric field. Debonding of an elastic inhomogeneity of arbitrary shape. In this case, there is only one field quantity to be computed, and the stresses and strains ar. You are unlikely to make a living by solving problems involving circular disks or holes subjected to prescribed displacements or tractions. This book tries to bridge the gulf that many times divides these two disciplines by combining the theoretical and practical approaches to the subject. The stress state in this cylinder is the same as the state of stress inside the circle r a of the in. A unified solution approach for a large variety of antiplane shear and torsion notch problems. A generalized and unified treatment is presented for the antiplane problem of an elastic elliptical inclusion undergoing uniform eigenstrains and subjected to arbitrary loading in the surrounding matrix. Exact solution for the mode iii stress fields ahead of. Conformal mapping and its applications suman ganguli1 1department of physics, university of tennessee, knoxville, tn 37996 dated. A confocally multicoated elliptical inclusion under. A unified treatment of the elastic elliptical inclusion under.