Somigliana Formula Research Articles

Analysis for complex plane cracks in 1D orthorhombic quasicrystals using the singular integral equation method

A singular integral equation method is proposed to analyze the complex plane cracks in one-dimensional (1D) orthorhombic quasicrystals. Using the Somigliana formula, the singular integral equations of the curved crack are derived. Based on the general situation of the curved crack, the singular integral equations of the inclined crack and the arc crack are given. Then the analytical solutions for the singular phonon and phason stresses near the tips of the inclined and the arc crack are obtained. Gauss-Chebyshev quadrature method is introduced to calculate the singular integral equations, and a numerical algorithm for solving the stress intensity factor (SIF) is proposed. Numerical solutions for the phonon and phason SIFs of some examples are solved and discussed.

Application of GOCE Satellite Gravimetric Data For Mineral Exploration

Gravity is directly proportional to density thereby making possible the use of measurable gravity in mapping the density variations in the earth’s interior. The use of density as a means of mineral exploration is cost-effective for wider area coverage. GOCE gravity data, collected by ESA’s GOCE satellite at 255km altitude and 10km intervals leading to the acquisition of multiple gravity points over the globe made possible the application of gravimetry in mineral exploration. Gravity disturbance data sourced from BGI- was corrected with WGM2012 corrections computed using EGM2008 geoid and ETOPO1 models, with reference gravity computed using the Somigliana formula. The gravity disturbances were mapped over Zambia using Surfer and Q-GIS, with over 600 control points of known mineral occurrences plotted together with other surface features like roads, rivers, railways, etc. By relating the control points to the varying gravity disturbances using the triple integral principle, a cautious analysis led to the geological classification of the gravity disturbances which essentially involved mapping predominant mineral occurrences across different parts of Zambia. During ground truthing, it was observed that the results within a particular area of interest on the classified map, and those obtained using four (4) different metal detectors as well as another remote sensing method tallied. The metal detectors used were the GR-100MINI, AKS, AKS plus 3D, and Garrett Ace 400, each with its own characteristics. From the map outputs, the results showed that GOCE data can be used for geological classification and delineation of terrain types. The delineation of terrain types on the classified map output matched that on existing geological maps and also offered the delineation of sub-terrain types.

Somigliana formula in quaternion analysis

In this paper, we introduce a Somigliana formula in quaternion analysis. Solutions of the Lamé–Navier system in the forms of the Grigor’ev solution, the Papkovich–Neuber solution and the Ter-Mkrtychan–Naghdi–Hsu solution are included in this Somigliana formula. In application, a new generalized Kolosov–Muskhelishvili formula is constructed.

Normal gravity field in relativistic geodesy

Modern geodesy is subject to a dramatic change from the Newtonian paradigm to Einstein's theory of general relativity. This is motivated by the ongoing advance in development of quantum sensors for applications in geodesy including quantum gravimeters and gradientometers, atomic clocks and fiber optics for making ultra-precise measurements of the geoid and multipolar structure of the Earth's gravitational field. At the same time, VLBI, SLR, and GNSS have achieved an unprecedented level of accuracy in measuring coordinates of the reference points of the ITRF and the world height system. The main geodetic reference standard is a normal gravity field represented in the Newtonian gravity by the field of a Maclaurin ellipsoid. The present paper extends the concept of the normal gravity field to the realm of general relativity. We focus our attention on the calculation of the first post-Newtonian approximation of the normal field that is sufficient for applications. We show that in general relativity the level surface of the uniformly rotating fluid is no longer described by the Maclaurin ellipsoid but is an axisymmetric spheroid of the forth order. We parametrize the mass density distribution and derive the post-Newtonian normal gravity field of the rotating spheroid which is given in a closed form by a finite number of the ellipsoidal harmonics. We employ transformation from the ellipsoidal to spherical coordinates to deduce the post-Newtonian multipolar expansion of the metric tensor given in terms of scalar and vector gravitational potentials of the rotating spheroid. We compare these expansions with that of the normal gravity field generated by the Kerr metric and demonstrate that the Kerr metric has a fairly limited application in relativistic geodesy. Finally, we derive the post-Newtonian generalization of the Somigliana formula for the gravity field on the reference ellipsoid.

Analysis of two intersecting three-dimensional cracks by a BIEM

Hypersingular integral equations for two intersecting three-dimensional cracks are derived by using Somigliana formula, in which the unknown functions are the displacement discontinuities on the crack-surfaces. The singularity of the unknowns at the crack-front is analyzed by the analytical asymptotic method, and the analytical solutions of the singular stresses near the crack-front are given. Numerical solutions of the stress intensity factors for some examples are presented and discussed.

A time domain direct boundary integral method for a viscoelastic plane with circular holes and elastic inclusions

This paper considers the problem of an infinite, isotropic viscoelastic plane containing an arbitrary number of randomly distributed, non-overlapping circular holes and isotropic elastic inclusions. The holes and inclusions are of arbitrary size. All inclusions are assumed to be perfectly bonded to the material matrix but the elastic properties of the inclusions can be different from one another. The Kelvin model is employed to simulate the viscoelastic plane. The numerical approach combines a direct boundary integral method for a similar problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described in [Crouch SL, Mogilevskaya SG. On the use of Somigliana's formula and Fourier series for elasticity problems with circular boundaries. Int J Numer Methods Eng 2003;58:537–578], and a time-marching strategy for viscoelastic material analysis described in [Mesquita AD, Coda HB, Boundary integral equation method for general viscoelastic analysis. Int J Solids Struct 2002;39:2643–2664]. Several numerical examples are given to verify the approach. For benchmark problems with one inclusion, results are compared with the analytical solution obtained using the correspondence principle and analytical Laplace transform inversion. For an example with two holes and two inclusions, results are compared with numerical solutions obtained by commercial finite element software—ANSYS. Benchmark results for a more complicated example with 25 inclusions are also given.

On the use of Somigliana's formula and Fourier series for elasticity problems with circular boundaries

AbstractThis paper considers the problem of an infinite, isotropic elastic plane containing an arbitrary number of non‐overlapping circular holes and isotropic elastic inclusions. The holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can, if desired, be different. The analysis is based on the two‐dimensional version of Somigliana's formula, which gives the displacements at a point inside a region V in terms of integrals of the tractions and displacements over the boundary S of this region. We take V to be the infinite plane, and S to be an arbitrary number of circular holes within this plane. Any (or all) of the holes can contain an elastic inclusion, and we assume for simplicity that all inclusions are perfectly bonded to the material matrix. The displacements and tractions on each circular boundary are represented as truncated Fourier series, and all of the integrals involved in Somigliana's formula are evaluated analytically. An iterative solution algorithm is used to solve the resulting system of linear algebraic equations. Several examples are given to demonstrate the accuracy and efficiency of the numerical method. Copyright © 2003 John Wiley & Sons, Ltd.

A new boundary integral formulation for plane elastic bodies containing cracks and holes

This paper presents a new boundary integral formulation for a plane elastic body containing an arbitrary number of cracks and holes. The body is assumed to be linear elastic and isotropic, but can be of either finite or infinite extend. The cracks inside the body can be either internal or edge crack, and either straight or curvilinear; and the holes can be of arbitrary number and shape. Starting from Somigliana formula, we obtain a system of boundary integral equations by applying integration by parts. In complex variables notation, the stress and displacement components can be expressed in terms of Muskhelishvilis analytic functions, which are in turn written as functions of boundary traction and displacement data in the form of Cauchy integral. The complex boundary integral equations for traction involve only singularity of order 1/ r, where r is the distance measured from the singular boundary points, and no hypersingular terms appear. This new boundary integral formulation provides an effective basis in solving problems both analytically and numerically. To illustrate the validity of our new integral formulation, a number of classical problems are re-examined analytically using the present formulation: (i) an infinite body containing a circular hole subject to far field biaxial stress, internal pressure, and a point force on the holes boundary respectively; and (ii) an infinite body containing a circular-arc crack under remote uniaxial tension. To illustrate the applicability of the present formulation for boundary element method analysis, two numerical examples for the interactions between two collinear cracks are considered and the results agree well with the existing solutions by Chandra et al. (1995) for the case of finite rectangular plates and with Isida (cited in p. 195 of Murakami, 1987) for the case of infinite plates.

Dynamic analogues of Somigliana's formula for unsteady dynamics of elastic media with an arbitrary degree of anisotropy

The theory of generalized functions is used to obtain dynamic analogues of Somigliana's formula for the unsteady dynamics of linear elastic homogeneous anisotropic media. Regular integral representations of these analogues are constructed for the solutions of boundary-value problems of the theory of elasticity with homogeneous initial data. Analytic formulae are proposed for the kernels of the integral equations in the case of plane strain for orthotropic media.

Finite-part integral and boundary element method to solve embedded planar crack problems

Using the Somigliana formula and concepts of the finite-part integral, a set of hypersingular integral equations to solve the arbitrary flat crack in three-dimensional elasticity is derived, then its numerical method is proposed by combining the finite-part integral method with the boundary element method. In order to verify the method, several numerical examples are carried out. The results of the displacement discontinuities of the crack surface and the stress intensity factors at the crack front are in good agreement with the theoretical solutions.

Finite-part integral and boundary element method to solve flat crack problems

Using the Somigliana formula and the concepts of finite-part integral, a set of hypersingular integral equations to solve the arbitrary flat crack in three-dimensional elasticity is derived and its numerical method is then proposed by combining the finitepart integral method with boundary element method. In order to verify the method, several numerical examples are carried out. The results of the displacement discontinuities of the crack surface and the stress intensity factors at the crack front are in good agrement with the theoretical solutions.

Fundamental solutions in an elastic space in the case of moving loads

Green's matrix is constructed for a point force moving along an axis in an isotropic elastic space at sub-, super- and transonic speeds. The theory of generalized functions is used to establish an analogue of Somigliana's formula which enables one to construct boundary integral equations (BIEs) for solving boundary-value problems with moving loads.

POTENTIAL THEORY FOR QUASI-STATIC PROBLEMS OF UNCOUPLED THERMOELASTICITY

Abstract This article concerns the derivation of a new potential theory for quasi-static problems of uncoupled thermoelasticity. The theory is based on a complete system of differential equations for a quasi-static problem of uncoupled thermoelasticity treated as a non-self-conjugate differential operator equation. To formulate a reciprocity theorem (Green's second identity), the conjugate system of differential equations is used. The reciprocity theorem is then employed to obtain a Somigliana-type formula. The boundary and initial properties of potentials in the Somigliana formula are studied for a homogeneous isotropic medium, and boundary integral equations for basic initial-boundary value problems are obtained.

Somigliana formula and the completeness of Papkovich-Neuber and Boussinesq-Galerkin solutions in elasticity

Somigliana formula and the completeness of Papkovich-Neuber and Boussinesq-Galerkin solutions in elasticity

Penny-shaped cracks

The Somigliana formula is used to reduce an arbitrary elastic crack problem to a system of three integral equations for the components of displacement discontinuity. 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. In particular integral formulae are obtained for the stresses on the plane of the crack beyond the crack-tip, and hence for the stress intensity factors. The special case of uni-directional shear traction on the crack is examined.