Multiple circular nano-inhomogeneities and/or nano-pores in one of two joined isotropic elastic half-planes

The paper considers the problem of multiple interacting circular nano-inhomogeneities or/and nano-pores located in one of two joined, dissimilar isotropic elastic half-planes. The analysis is based on the solutions of the elastostatic problems for (i) the bulk material of two bonded, dissimilar elastic half-planes and (ii) the bulk material of a circular disc. These solutions are coupled with the Gurtin and Murdoch model of material surfaces [Gurtin ME, Murdoch AI. A continuum theory of elastic material surfaces. Arch Ration Mech Anal 1975;57:291–323; Gurtin ME, Murdoch AI. Surface stress in solids. Int J Solids Struct 1978;14:431–40.]. Each elastostatic problem is solved with the use of complex Somigliana traction identity [Mogilevskaya SG, Linkov AM. Complex fundamental solutions and complex variables boundary element method in elasticity. Comput Mech 1998;22:88–92]. The complex boundary displacements and tractions at each circular boundary are approximated by a truncated complex Fourier series, and the unknown Fourier coefficients are found from a system of linear algebraic equations obtained by using a Taylor series expansion. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-planes and inside the nano-inhomogeneities. Numerical examples demonstrate that (i) the method is effective in solving the problems with multiple nano-inhomogeneities, and (ii) the elastic response of a composite system is profoundly influenced by the sizes of the nano-features.     

Axisymmetric solution for a semi-infinite elastic circular cylinder of one material indenting an elastic half-space of another material

The axisymmetric bonded contact problem of a semi-infinite right circular cylinder of one elastic material indenting a half-space of a different elastic material is reduced to a system of singular integral equations of the second kind. The kernels of the integral equations are found to contain Cauchy and generalized Cauchy-type singularities. The index of the singularity for various material parameters combinations is determined by solving a characteristic determinant, which is obtained by considering the dominant part of the kernels. Using a modification of the method employed in [1], the system of singular integral equations is reduced to a system of simultaneous algebraic equations. The latter may then be solved numerically as in [1].     

A hybrid complex-variable solution for piezoelectric/isotropic elastic interfacial cracks

In many practical applications, piezoelectric ceramics are bonded to non-piezoelectric and insulating isotropic elastic materials such as polymer. Since the conventional form of Stroh’s formulation, on which almost all of existing works on interfacial cracks in piezoelectric media have been based, breaks down or becomes complicated for isotropic elastic materials, many solutions available in the literature cannot be directly applied to interfacial cracks between a piezoelectric material and an isotropic elastic material. The present paper is devoted to a hybrid complex-variable method which combines the Stroh’s method of piezoelectric materials with the well-known Muskhelishvili’s method of isotropic elastic materials. This method is illustrated in detail for an insulating interfacial crack between a piezoelectric half-plane and an isotropic elastic half-plane, although interface cracks between piezoelectric and isotropic elastic conductor can be analyzed in a similar way. The solution obtained generally exhibits oscillatory singularity, in agreement with a previous known result based on the Stroh’s formulation. A simple explicit condition is obtained for the bimaterial constants under which the oscillatory singularity disappears. It is expected that the hybrid complex-variable method could more conveniently handle other possible complications (such as a hole or an inclusion) inside the isotropic elastic material, because it offers explicit solutions of a single complex variable rather than several different complex-variables associated with the Stroh’s formulation.     

A semi-analytical solution for free vibration of variable thickness two-directional-functionally graded plates on elastic foundations

The present paper investigates free vibration of variable thickness two-directional-functionally graded circular plates, resting on elastic foundations. The results are obtained for clamped, free, and simply supported edge conditions. Variations of the material and geometrical parameters are monitored by five distinct exponential functions. Therefore, the resulted non-dimensional solution may be used for a wide range of the practical problems. Mindlin’s plate theory and the differential transformation technique are used to obtain the governing equations of the natural frequencies of the circular plates. Effects of variations of the material properties in the radial and thickness directions, geometric parameters (e.g., the thickness-to-radius ratio in the center of the plate), stiffness parameters of the foundation, and various boundary conditions on the natural frequencies are investigated. Results reveal that by choosing a suitable combination of the material properties, the free vibration behavior of the thick plates may be enhanced without the need to change the geometric parameters.     

A Galerkin boundary integral method for multiple circular elastic inclusions

The problem of an infinite, isotropic elastic plane containing an arbitrary number of circular elastic inclusions is considered. The analysis procedure is based on the use of a complex singular integral equation. The unknown tractions at each circular boundary are approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using the classical Galerkin method and the Gauss–Seidel algorithm is used to solve the system. Several numerical examples are considered to demonstrate the effectiveness of the approach. Copyright © 2001 John Wiley & Sons, Ltd.     

Axisymmetric solution of the end-problem for a semi-infinite elastic circular cylinder and its application to joined dissimilar cylinders under uniform tension

Using transform methods, axisymmetric end-problem for a semi-infinite elastic circular cylinder is reduced to a system of singular integral equations. The kernels of the integral equations are found to contain Cauchy as well as generalized Cauchy-type singularities. The dominant part of the equations is separated and analyzed to determine the index of the singularity for differing boundary conditions at the end. An approximate method is used to obtain a system of simultaneous algebraic equations from the system of singular integral equations. As an application, axisymmetric solution for joined dissimilar elastic semi-infinite cylinders under uniform tension is solved and various physical quantities of interest, such as normal and shear stresses at the interface, are obtained.     

Incomplete mechanical contact between two elastic half-planes made of orthotropic materials

Summary A planar case in elastic theory is considered for incomplete contact between orthotropic half-planes; complex Kolosov-Muskhelishvili potentials are used. A singular integral equation is derived and solved analytically. The results are analyzed.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 26, No. 3, pp. 65–69, May–June, 1990.     

A boundary integral method for multiple circular holes in an elastic half-plane

This paper presents a semi-analytical method for solving the problem of an isotropic elastic half-plane containing a large number of randomly distributed, non-overlapping, circular holes of arbitrary sizes. The boundary of the half-plane is assumed to be traction-free and a uniform far-field stress acts parallel to that boundary. The boundaries of the holes are assumed to be either traction-free or subjected to constant normal pressure. The analysis is based on solution of complex hypersingular integral equation with the unknown displacements at each circular boundary approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using a Taylor series expansion. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-plane. Several examples available in the literature are re-examined and corrected, and new benchmark examples with multiple holes are included to demonstrate the effectiveness of the approach.     

Mass sensitivity of thickness-shear modes in an isotropic elastic circular cylinder

Theoretical prediction of the mass sensitivity of axial and tangential thickness-shear modes in an isotropic elastic circular cylinder due to a thin surface mass layer is presented. Results suggest the possibility of new mass sensors with certain advantages.     

Boundary element analysis of three-dimensional crack problems in two joined transversely isotropic solids

The present paper presents a boundary element analysis of penny-shaped crack problems in two joined transversely isotropic solids. The boundary element analysis is carried out by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The conventional multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of the stress intensity factors are obtained by using crack opening displacements. The numerical scheme results are verified with the closed-form solutions available in the literature for a penny-shaped crack parallel to the plane of the isotropy of a homogeneous and transversely isotropic solid of infinite extent. The new problem of a penny-shaped crack perpendicular to the interface of a transversely isotropic bi-material solid is then examined in detail. The crack surfaces are subject to the three normal tractions and the uniform shear traction. The associated stress intensity factor values are obtained and analyzed. The present results can be used for the prediction of the stability of composite structures and the hydraulic fracturing in deep rock strata and reservoir engineering.     

Analytical solution of stability problem for two composite half-planes compressed along interfacial cracks

The plane stability problem for two composite half-planes compressed along the interface, which contains an arbitrary number of cracks, is considered. An exact analytical solution of the problem is found for elastic and elastic–plastic, isotropic and orthotropic, compressible and incompressible half-planes in the common form for finite and small deformations. This solution was developed using complex potentials within the exact approach based on equations of the three-dimensional linearised theory of deformable bodies’ stability. Critical loads are rigorously proved to be independent of the number and disposition of interfacial cracks.     

Interaction of elastic waves in two bonded dissimilar elastic half-planes having Griffith crack at interface—I

The paper deals with the problem of finding the stress distribution near a Griffith crack located at the interface of two bonded dissimilar elastic half-spaces. The crack is opened by the interaction of a plane harmonic elastic wave, incident normally on the crack. The problem is first reduced to a set of simultaneous dual integral equations which are further transformed to a set of simultaneous singular integral equations. These are solved numerically by reducing them to a set of algebraic equations. The solution is used to calculate the stress intensity factors.

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Résumé Le mémoire relatif au problème de la détermination de la distribution des tensions au voisinage d'une fissure de Griffith localisée à l'interface de 2 demi-espaces élastiques dissemblables et collés. La fissure est ouverte par l'interaction d'une onde élastique plane et harmonique dont l'incidence est normale par rapport à la fissure. Le problème est, en premier lieu, ramené à une série d'équations intégrales doubles simultanées qui sont ensuite transformées en une série d'équations intégrales singulières simultanées. Ces équations sont résolues par voie numérique en les ramenant à des séries d'équations algébriques. La solution est utilisée pour calculer les facteurs d'intensité des contraintes.

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This work was supported by CSIR Grant No. 23 (87)/74-GAU-II.     

Finite difference solution for circular plates on elastic foundations

In the present work the authors have developed a finite difference method of analysis for any circular plate with any kind of loading on semi-infinite elastic foundations. No assumption regarding the contact pressure distribution has been made. The equations have been developed in non-dimensional form and also the results have been obtained in non-dimensional form. These results have been compared with the available experimental results and the agreement between them is found to be much better than that of the previous works. The same method with slight modification can be applied for Winkler type foundations and problems of circular plates with varying thickness.     

An elasticity solution for transversely isotropic,functionally graded circular plates

This article presents a new elasticity solution for transversely isotropic, functionally graded circular plates subject to axisymmetric loads. It is assumed that the material properties vary along the thickness of a circular plate according to an exponential form. By extending the displacement function presented by Plevako to the case of transversely isotropic material, we derived the governing equation of the problem studied. The displacement function was assumed as the sum of the Bessel function and polynomial function to obtain the analytical solution of a transversely isotropic, functionally graded circular plate under different boundary conditions. As a numerical example, the influence of the graded variations of the material properties on the displacements and stresses was studied. The results demonstrate that the graded variations have a significant effect on the mechanical behavior of a circular plate.     

Nearly isochoric finite torsion of a compressible isotropic elastic circular cylinder

Summary Finite torsion of a circular bar of isotropic compressible hyperelastic material is considered. A procedure suggested by Truesdell is used to obtain solutions for a particular strain energy function although the method used is not restricted to this particular form. The procedure is applicable if the volume strain is small. Results for finite twist with the length prevented from changing and with the length allowed to change so that the resultant longitudinal force is zero are presented.

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Fast-isochore endliche Torsion eines kompressiblen, isotropen, elastischen Kreiszylinders

Zusammenfassung Betrachtet wird die endliche Verdrehung eines Stabes mit Kreisquerschnitt aus einem isotropen, kompressiblen, hyperelastischen Werkstoff. Eine von Truesdell vorgeschlagene Vorgangsweise wird zur Bestimmung der Lösungen, für eine spezielle Verzerrungsenergiefunktion, verwendet, obwohl die verwendete Methode nicht auf diese spezielle Form beschränkt ist. Diese Vorgangsweise ist anwendbar, sofern die Volumsverzerrung klein ist. Ergebnisse für die endliche Verdrehung werden angegeben, wobei entweder die Längenänderung unterbunden ist oder die Länge sich ändern kann, so daß die resultierende Längskraft verschwindet.

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Notation F deformation gradient tensor - B=FF ^T left Cauchy-Green tensor - I identity matrix - I ₁=trB first invariant ofB - I ₂=1/2[(trB)²–trB ²] second invariant ofB - I ₃=detB third invariant ofB - W strain energy per unit volume of unstrained stateB ₀ - gradient operator for stateB - u small displacement from stateB - H=u small displacement gradient based on stateB - T true stress tensor based on stateB - f body force vector - T additional stress - =T+T true stress for final configuration - elastic constant equivalent to shear modulus for small deformation - elastic constant equivalent to Poisson's ratio for small deformation - r, ,z cylindrical polar coordinates - _r , , _z , _z physical components of stress tensor for cylindrical polar coordinates With 3 Figures     

Circular inhomogeneity and two concentric symmetric circular arc cracks problem in an infinite isotropic elastic plate under tension

A homogeneous infinite isotropic elastic plate contains two symmetrical circular arc cracks of equal radii and a concentric inhomogeneity. The radius of the inhomogeneity is less than that of the circular arc cracks. The plate is subjected to a traction at infinity. The stresses are found within the circular region bounded by the circular arcs, including the inhomogeneity. The problem is solved as a two-dimensional using the complex variable technique. Some numerical results are given to show the effects of the inhomogeneity.     

Transient heat conduction in a medium with multiple circular cavities and inhomogeneities

A two‐dimensional transient heat conduction problem of multiple interacting circular inhomogeneities, cavities and point sources is considered. In general, a non‐perfect contact at the matrix/inhomogeneity interfaces is assumed, with the heat flux through the interface proportional to the temperature jump. The approach is based on the use of the general solutions to the problems of a single cavity and an inhomogeneity and superposition. Application of the Laplace transform and the so‐called addition theorem results in an analytical transformed solution. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. Several numerical examples are given to demonstrate the accuracy and the efficiency of the method. The approximation error decreases exponentially with the number of the degrees of freedom in the problem. A comparison of the companion two‐ and three‐dimensional problems demonstrates the effect of the dimensionality. Copyright © 2009 John Wiley & Sons, Ltd.     

A numerical procedure for multiple circular holes and elastic inclusions in a finite domain with a circular boundary

This paper describes a numerical procedure for solving two-dimensional elastostatics problems with multiple circular holes and elastic inclusions in a finite domain with a circular boundary. The inclusions may have arbitrary elastic properties, different from those of the matrix, and the holes may be traction free or loaded with uniform normal pressure. The loading can be applied on all or part of the finite external boundary. Complex potentials are expressed in the form of integrals of the tractions and displacements on the boundaries. The unknown boundary tractions and displacements are approximated by truncated complex Fourier series. A linear algebraic system is obtained by using Taylor series expansion without boundary discretization. The matrix of the linear system has diagonal submatrices on its diagonal, which allows the system to be effectively solved by using a block Gauss-Seidel iterative algorithm.