Showing papers on "Displacement field published in 1996"
TL;DR: In this paper, the authors present an analysis of strong discontinuities in inelastic solids at finite strains, characterized by a discontinuous displacement field, which makes physical and mathematical sense if the softening modulus is reinterpreted as a singular distribution.
373 citations
TL;DR: The cytoskeletal activity of motile or adherent cells is frequently seen to induce detectable displacements of sufficiently compliant substrata to derive a Bayesian method for converting noisy measurements of substratum displacement into "images" of the actual traction forces exerted by adherent or locomoting cells.
297 citations
TL;DR: It is demonstrated that the topography is separable from the surface displacement field when a sequence of radar images are available and a pure displacement field can be obtained by removal of the topographic contribution to the interferometric phase at each pixel.
Abstract: Both topography and motion information are present in repeat pass ERS-1 interferograms over ice sheets. The authors demonstrate that the topography is separable from the surface displacement field when a sequence of radar images are available. If the velocity field is constant over the time span of observation, the topography can be derived from differential interferograms formed from sequential observations. With this measurement, a pure displacement field can then be obtained by removal of the topographic contribution to the interferometric phase at each pixel. Further, they discuss how the vertical and horizontal components of displacement affect the interferometrically-derived motion field. They illustrate their approach with four successive (3-day repeat) ERS-1 images of a flow feature in northeastern Greenland.
186 citations
TL;DR: In this paper, a combination of pairs of SAR images acquired by the European ERS-1 satellite maps deformation fields associated with two phenomena, both of small spatial extension and located in SE France: the one is rapid terrain deformation caused by a landslide near the city of Saint Etienne de Tinee, and the other is slower subsidence caused by underground coal mining near Gardanne.
Abstract: Interferometric combination of pairs of SAR images acquired by the European ERS-1 satellite maps deformation fields associated with two phenomena, both of small spatial extension and located in SE France: the one is rapid terrain deformation caused by a landslide near the city of Saint Etienne de Tinee, and the other is slower subsidence caused by underground coal mining near Gardanne. Unlike interferometric measurement of wide-field deformation, atmospheric propagation heterogeneity is not an accuracy-limiting factor. Although the radar data confirm prior knowledge concerning the landslide, such an application of SAR interferometry appears difficult under normal conditions of observation using current spaceborne radar systems. The study of soil subsidence, however, can be generalized and improves prior knowledge of the displacement field, which has here been modeled assuming elastic deformation in a half-space from several sources. The two examples help to understand the limits of the interferometric technique.
169 citations
TL;DR: In this article, a nonlinear continuum analysis is developed to show that stable, steady-state crack motion is limited not only by the macroscopic Rayleigh wave speed as asserted by the established theory of dynamic fracture, but also by a local wave speed governed by the elastic response near the crack tip.
Abstract: A nonlinear continuum analysis is developed to show that stable, steady-state crack motion is limited not only by the macroscopic Rayleigh wave speed as asserted by the established theory of dynamic fracture, but also by a local wave speed governed by the elastic response near the crack tip. The local limiting speed ensures that a subsonic steady-state field can be established in highly nonlinear material regions prior to rupture. A two-dimensional triangular lattice with nearest-neighbor interatomic bonding is studied as a model nonlinear elastic solid that is isotropic under infinitesimal strains, but becomes anisotropic and nonlinear when the lattice is heavily stretched. The local limiting speed is determined by considering the most critical state of deformation on the verge of bond rupture. If the critical state is assumed to be under equibiaxial stressing, the local limiting speed is found to be v 1 = c s σ max μ , where cs is the macroscopic shear wave speed, μ is the shear modulus and σmax is the equibiaxial cohesive strength of the solid (i.e. the maximum equibiaxial tensile stress that a flawless solid can stand without spontaneous rupture). The generality of this result is discussed by relaxing the restrictions in the model problem. It is also shown that lattice dispersion in front of a crack tip can further reduce the speed of bond-breaking stress waves with wavelength on the order of a few atomic spacings. This study lends further support for a viewpoint previously discussed by the author that high speed dynamic fracture involves a competition between a high inertia local crack-tip field and the surrounding low inertia apparent crack field. Motivated by recent molecular dynamics simulations of crack propagation in a 6–12 Lenard-Jones lattice, a variational principle for steady-state deformation is used together with a conjugate gradient minimization algorithm to compute atomistic responses near the tip of a crack moving with constant speed in a similar Lenard-Jones lattice. The computation is performed over a block which moves with the crack and is subjected along the boundary to a low inertia displacement field based on existing solutions for cracks moving in linear elastic solids. The critical velocity at the onset of local crack branching is found to be 0.30cs, in almost exact agreement with the earlier molecular dynamics study. In this case, the local limiting speed is calculated to be v1 = 0.37cs, which is 20% larger than the observed value. This difference can be attributed to the effects of local lattice dispersion. The results are fully supportive of the notion that global-local inertia competition is a key to understanding dynamic fracture instabilities.
144 citations
TL;DR: In this paper, a C0continuous finite element model has been developed for the free vibration analysis of laminated coposite plates, together with a higher order shear deformation theory (HSDT) to account for the parabolic variation of transverse shear stresses through the thickness and linear variation of the normal stresses.
131 citations
TL;DR: In this article, the authors present a theory to compute strains from the displacements in a discrete set of points and is particularly useful in the post-processing of experimentally measured displacement fields.
120 citations
TL;DR: In this paper, a double-ended crack of finite size embedded in a cylindrical simulation cell and fixed boundary conditions were prescribed along the periphery of the cell, whereas periodic boundary conditions are imposed parallel to the crack front.
Abstract: The behaviour of mode I cracks in -Fe is investigated using molecular statics computer simulation methods with an EAM potential. A double-ended crack of finite size embedded in a cylindrical simulation cell and fixed boundary conditions are prescribed along the periphery of the cell, whereas periodic boundary conditions are imposed parallel to the crack front. The displacement field of the finite crack is represented by that of an equivalent pile-up of opening dislocations distributed in a manner consistent with the anisotropy of the crystal and traction-free conditions of the crack faces. The crack lies on the plane and the crack front is located along , or directions. The crack tip response is rationalized in terms of the surface energy of the cleavage plane and the unstable stacking energies of the slip planes emanating from the crack front.
100 citations
TL;DR: In this paper, a displacement-based variable kinematic global-local finite element model is developed using hierarchical, multiple assumed displacement fields at two different levels: (1) at the element level, and (2) at mesh level.
Abstract: A displacement-based variable kinematic global–local finite element model is developed using hierarchical, multiple assumed displacement fields at two different levels: (1) at the element level, and (2) at the mesh level. The displacement field hierarchy contains both a conventional plate expansion (2-D) and a full layerwise (3-D) expansion. Depending on the accuracy desired, the variable kinematic element can use various terms from the composite displacement field, thus creating a hierarchy of different elements having a wide range of kinematic complexity and representing a number of different mathematical models. The VKFE is then combined with the mesh superposition technique to further increase the computational efficiency and robustness of the computational algorithm. These models are used to analyse a number of laminated composite plate problems that contain localized subregions where significant 3-D stress fields exist (e.g. free-edge effects).
93 citations
TL;DR: In this paper, the authors present two fully equivalent and regular forms of the hypersingular Somigliana stress identity in elasticity that are appropriate for problems in which the displacement field (and resulting stresses) is C 1,α continuous.
Abstract: The paper presents two fully equivalent and regular forms of the hypersingular Somigliana stress identity in elasticity that are appropriate for problems in which the displacement field (and resulting stresses) is C1,α continuous. Each form is found as the result of a single decomposition process on the kernels of the Somigliana stress identity in three dimensions. The results show that the use of a simple stress state for regularization arises in a direct manner from the Somigliana stress identity, just as the use of a constant displacement state regularization arose naturally for the Somigliana displacement identity. The results also show that the same construction leads naturally to a finite part form of the same identity. While various indirect constructions of the equivalents to these findings are published, none of the earlier forms address the fundamental issue of the usual discontinuities of boundary data in the hypersingular Somigliana stress identity that arise at corners and edges. These new findings specifically focus on the corner problem and establish that the previous requirements for continuity on the densities in the hypersingular Somigliana stress identity are replaced by a sole requirement on displacement field continuity.
The resulting regularized and finite part forms of the Somigliana stress identity leads to a regularized form of the stress boundary integral equation (stress-BIE). The regularized stress-BIE is shown to properly allow piecewise discontinuity of the boundary data subject only to C1,α continuity of the underlying displacement field. The importance of the findings is in their application to boundary element modeling of the hypersingular problem. The piecewise discontinuity derivation for corners is found to provide a rigorous and non-singular basis for collocation of the discontinuous boundary data for both the regularized and finite part forms of the stress-BIE. The boundary stress solution for both forms is found to be an average of the computed stresses at collocation points at the vertices of boundary element meshes. Collocation at these points is shown to be without any unbounded terms in the formulation thereby eliminating the use of non-conforming elements for the hypersingular equations. The analytical findings in this paper confirm the correct use of both regularized and finite part forms of the stress-BIE that have been the basis of boundary element analysis previously published by the first author of the current paper.
84 citations
TL;DR: In this paper, a physically motivated approach to modeling the dynamic behavior of viscoelastic structures using augmenting thermodynamic fields was presented. But instead of addressing physical damping mechanisms directly, as in the earlier approach, their effects on the displacement field were considered.
Abstract: A physically motivated approach to modeling the dynamic behavior of viscoelastic structures using augmenting thermodynamic fields was previously reported. Anelastic displacement fields, special kinds of augmenting thermodynamic fields, are now introduced. Instead of addressing physical damping mechanisms directly, as in the earlier approach, their effects on the displacement field are considered. In this approach, the total displacement field comprises two parts: 1) an elastic part and 2) an anelastic part. The material constitutive equations are developed, as well as the governing differential equations and boundary conditions for a one-dimensional structural member, and are compared to the results developed previously using the augmented thermodynamic fields approach. In addition, a physical interpretation of some of the quantities involved is advanced in terms of a classical mechanical analogy. Because the anelastic displacement field(s) and the total displacement field may be treated similarly in analytical or numerical study, the key practical benefit of this anelastic displacement fields approach is that it leads to the straightforward development of time-domain finite element models. Modal analyses and frequency response analyses have been implemented using the matrix manipulation capabilities of a commercial finite element code.
TL;DR: In this article, a finite volume formulation for discretizing and analysing plane elastostatic problems is described, where equilibrium equations which relate the displacements at the center of a general quadrilateral cell to those in neighbouring cells are developed.
Abstract: A finite volume formulation for discretizing and analysing plane elastostatic problems is described. Equilibrium equations which relate the displacements at the centre of a general quadrilateral cell to those in neighbouring cells are developed. After the application of suitable boundary conditions, an iterative method is employed to solve the resulting system of simultaneous equations and produce the displacement field, from which the strain and stress fields are derived subsequently. Stress distributions for a test problem, an elliptic plate pierced by an elliptic hole and loaded on the outer boundary, are determined for meshes of increasing refinement. The distributions are compared with those determined using triangular and quadrilateral finite elements and the analytical solution.
TL;DR: In this article, a higher-order shear deformation theory is presented for vibration analysis of thick, doubly curved shallow shells, where an orthogonal curvilinear coordinate system is used to arrive at the strain components.
Abstract: A higher-order shear deformation 'theory is presented for vibration analysis of thick, doubly curved shallow shells. An orthogonal curvilinear coordinate system is em- ployed to arrive at the strain components. A third-order displacement field in trans- verse coordinate is adopted. Though no transverse normal stress is assumed, the theory accounts for cubic distribution of the transverse shear strains through the shell thickness in contrast with existing parabolic shear distribution. The unsymmetric shear distribution is a physical consequence of the presence of shell curvatures where the stress and strain of a point above the mid-surface are different from its counterpart below the mid-surface. Imposing the vanishing of transverse shear strains on top and bottom surfaces, the rotation field is reduced from a six-degree to a two-degree system. The discrepancy between the existing and the present theories is highlighted.
TL;DR: In this paper, a theoretical model for the interaction of frequency components in arbitrary pulsed elastic waves during one-dimensional propagation in an infinite medium with extreme nonlinear response is presented.
Abstract: A theoretical model is presented that describes the interaction of frequency components in arbitrary pulsed elastic waves during one‐dimensional propagation in an infinite medium with extreme nonlinear response. The model is based on one‐dimensional Green’s function theory in combination with a perturbation method, as has been developed for a general source function by McCall [J. Geophys. Res. 99 (B2), 2591–2600 (Feb. 1994)]. A polynomial expansion of the equation of state is used in which four orders of nonlinearity in the moduli are accounted for. The nonlinear wave equation is solved for the displacement field at distance x from a symmetric ‘‘breathing’’ source with arbitrary Fourier spectrum imbedded in an infinite medium. The perturbation expression corresponds to a higher‐order equivalent of the Burgers’ equation solution for velocity fields in solids. The solution is implemented numerically in an iterative procedure which allows one to include an arbitrary attenuation function. Energy conservation is investigated in the absence of (linear) attenuation, and the notion of a hybrid (linear and nonlinear) dissipation is illustrated. Examples are provided showing the effect of each term in the perturbation solution on the spectral content of the waveform. Finally, the possibility of creating a parametric array for seismic exploration is briefly considered from a theoretical point of view.
TL;DR: In this article, the authors used the Airy stress eigenfunction expansion of Williams [1] to obtain simple expressions for the angular variations of the stress and displacement fields for n-material wedges and junctions subjected to inplane loading.
Abstract: The Airy stress eigenfunction expansion of Williams [1] has been used to obtain simple expressions for the angular variations of the stress and displacement fields for n-material wedges and junctions subjected to inplane loading. This formulation applies to real and complex roots, as well as the special transition case giving rise to r
−ω singular behavior. The asymptotic behavior of the general problem is similar to that of the bi-material interface crack. In the case of real roots, the stress and displacement expressions can be determined to within a multiplicative real constant (amplification), while for the complex case, the fields are determined to within a multiplicative complex constant (amplification plus rotation). Because of the rotation in the complex case, there are an infinite number of equivalent ways to express the angular variations (eigenfunctions) of the stress and displacement fields. Therefore, the fields are standardized in terms of ‘generalized stress intensity factors’ that are consistent with the bi-material interface crack and the homogeneous crack problems. As in the bi-material crack problem, for the complex case there are two stress intensity factors for each admissible order of the stress singularity. For specific n-material wedges and junctions, a small variation of material properties and/or geometry can change the eigenvalues from a pair of complex conjugate roots to two distinct real roots or vice-versa. An r
−ω singularity associated with a nonseparable solution in υ and θ exists at this point of bifurcation. Such behavior requires an adjustment in the standard eigenfunction approach to insure bounded stress intensity factors. The proper form of the solution is given both at and near this special material combination, and the smooth transition of the eigenfunctions as the roots change from real to complex is demonstrated in the results. Additional eigenfunction results are provided for particular cases of 2 and 3-material wedges and junctions.
TL;DR: In this paper, a finite element formulation is developed to determine the order and angular variation of singular stress states due to material and geometric discontinuities in anisotropic materials, and the formulation applies to any two-dimensional geometry that is prismatic in the third direction and has three-dimensional displacement fields.
TL;DR: In this article, the authors derived spherical harmonic expressions for the shear and tensile dislocations, which can be expressed by four independent solutions: a vertical strike slip, a vertical dip slip and a tensile opening in a horizontal plane.
Abstract: We define dislocation Love numbers [h nm ij , l nm ij , k nm ij , l nm t , ij ] and Green's functions to describe the elastic deformation of the Earth caused by a point dislocation and study the coseismic displacements caused in a radially heterogeneous spherical Earth model. We derive spherical harmonic expressions for the shear and tensile dislocations, which can be expressed by four independent solutions : a vertical strike slip, a vertical dip slip, a tensile opening in a horizontal plane, and a tensile opening in a vertical plane. We carry out calculations with a radially heterogeneous Earth model (1066A). The results indicate that the dominating deformations appear in the near field and attenuate rapidly as the epicentral distance increases. The shallower the point source, the larger the displacements. Both the Earth's curvature and vertical layering have considerable effects on the deformation fields. Especially the vertical layering can cause a 10% difference at the epicentral distance of 0.1°. As an illustration, we calculate the theoretical displacements caused by the 1964 Alaska earthquake (m w = 9.2) and compare the results with the observed vertical displacements at 10 stations. The results of the near field show that the vertical displacement can reach some meters. The far-field displacements are also significant. For example, the horizontal displacements (u ψ ) can be as large as 1 cm at the epicentral distance of 30°, 0.5 cm at about 40°, magnitudes detectable by modern instrument, such as satellite laser ranging (SLR), very long baseline interferometry (VLBI) or Global Positioning System (GPS). Globally, the displacement (u r ) caused by the earthquake is larger than 0.25 mm.
TL;DR: In this paper, a higher-order shell theory for symmetric laminated composites is presented, where the in-plane displacement fields are obtained by superimposing a globally cubic varying displacement field on a zigzag linearly varying one.
15 Apr 1996
TL;DR: This framework is based on the use of a non quadratic regularization technique in optical flow computation that preserves flow discontinuities and is used for processing oceanographic and atmospheric image sequences and presents an alternative to the very complex physical modelling techniques.
Abstract: We present in this paper a motion computation and interpretation framework for oceanographic satellite images. This framework is based on the use of a non quadratic regularization technique in optical flow computation that preserves flow discontinuities. We also show that using an appropriate tessellation of the image according to an estimate of the motion field can improve optical flow accuracy and yields more reliable flows. This method defines a non uniform multiresolution scheme that refines mesh resolution only in the neighborhood of moving structures. The second part of the paper deals with the interpretation of the obtained displacement field. We use a phase portrait model with a new formulation of the approximation of an oriented flow field. This allows us to consider arbitrary polynomial phase portrait models for characterizing salient flow features. This new framework is used for processing oceanographic and atmospheric image sequences and presents an alternative to the very complex physical modelling techniques.
TL;DR: In this paper, the authors derived continuity conditions for a fault modelled as a plane with isolated areas of slip, such that their overall effect is that of a distribution of circular cracks; discontinuities in both normal and tangential components of displacement are allowed, depending on internal conditions.
Abstract: SUMMARY Continuity conditions are derived for a fault modelled as a plane with isolated areas of slip. These slip areas are, for simplicity, taken to be such that their overall effect is that of a distribution of circular cracks; discontinuities in both normal and tangential components of displacement are allowed, depending on the internal conditions. Dry (gas-filled), partial or saturated liquid fill, or a fill of a weak visco-elastic solid are possible within the theory. The results are given in terms of the mean wave, which, at wavelengths long compared with the scale-lengths of the fault structure, is an accurate approximation to the displacement field. The continuity conditions that arise under this scheme are identical to those for a thin layer of visco-elastic material. However, unlike earlier, more empirical models of an ‘averaged’ fault, the parameters involved are directly related to the fault structure and include crack-crack interactions. It is
TL;DR: An anisotropic form is derived for the elastic field associated with a surface line force dipole using a two-dimensional surface Green tensor for a cubic elastic half-space within the Stroh formalism.
Abstract: Atomistic computer simulations and anisotropic elastic theory are employed to determine the elastic fields of surface steps and vicinal surfaces. The displacement field of and interaction energies between {l_angle}100{r_angle} steps on a {l_brace}001{r_brace} surface of Ni and Au are determined using atomistic simulations and embedded-atom method potentials. The step-step interaction energy found from the simulations is consistent with a surface line force dipole elastic model of a step. We derive an anisotropic form for the elastic field associated with a surface line force dipole using a two-dimensional surface Green tensor for a cubic elastic half-space within the Stroh formalism. Both the displacement fields and step-step interaction energy predicted by the theory are shown to be in excellent agreement with the simulations. The symmetry of the step displacement field is used to determine analytically the relative values of the components of the surface force dipole vector. {copyright} {ital 1996 The American Physical Society.}
TL;DR: In this paper, a clearly consistent finite element formulation for spatial stability analysis of thin-walled space frames is presented by applying linearized virtual work principle and introducing Vlasov's assumption.
Abstract: A clearly consistent finite element formulation for spatial stability analysis of thin-walled space frames is presented by applying linearized virtual work principle and introducing Vlasov's assumption. The improved displacement field for unsymmetric thin-walled cross-sections is introduced based on inclusion of second-order terms of finite rotations, and the potential energy corresponding to the semitangential moments is consistently derived. In the present formulation, displacement parameters of axial and bending deformations are defined at the centroid axis and parameters of lateral and torsional deformations at the shear centre axis, and all bending-torsional coupling effects due to unsymmetric cross-sections are taken into account. For finite element analysis, cubic Hermitian polynomials for the flexural beam with four types of end conditions are utilized as shape functions of Hermitian space frame element. Also, load correction stiffness matrices for off-axis point loadings are derived based on the second-order rotation terms. Finite element solutions for the spatial buckling analysis of thin-walled space frames are compared with available solutions and other researcher's results.
TL;DR: In this paper, an analytical model for laminated composite strips exhibiting extension-twist coupling is pre-sented and extended to include finite pretwist, and closed-form expressions relating applied extension to twisting rotation are obtained and the contribution of axial force to the twisting moment is isolated.
Abstract: An analytical model for laminated composite strips exhibiting extension-twist coupling is pre sented. The analysis is developed first for flat laminates and subsequently extended to include finite pretwist. The displacement field is developed in three steps, each accounting for a kinematic contribution. A finite rigid body twisting rotation is considered first. This is subsequently modified to include Saint-V6nant's type warping where the transverse normal and out-of-plane shear strains are neglected. Finally, inplane extension, shear, bending, and twisting curvatures are accounted for by superimposing a classical-type small-displacement field. Closed-form expressions relating applied extension to twisting rotation are obtained and the contribution of axial force to the twisting moment is isolated. Three approximate models are derived and the influence of the free edge conditions and Saint-V6nant's assumptions are assessed. Based on this assessment, a simple two-parameter model accounting for the axial force contribution to the twisting moment is proposed. Comparisons of analytical predictions with a finite-element simulations for both flat and pretwisted laminates illustrate the accuracy of the developed models. A set of pretwisted laminated composite strips made of a graphite/cyanate material system is manufactured and tested. A custom-made apparatus designed to allow the laminate to twist freely under axial loading is used to measure the twist angle associated with applied axial force. Test results depict the nonlinear axial force-twist behavior and the analytical predictions are in close agreement with test data. z.w
TL;DR: In this paper, a higher-order shear deformation theory is developed for accurately evaluating the transverse shear effects in delamination buckling and postbuckling of cylindrical shells under axial compression.
Abstract: A higher-order shear deformation theory is developed for accurately evaluating the transverse shear effects in delamination buckling and postbuckling of cylindrical shells under axial compression The theory assures an accurate description of displacement field and the satisfaction of stress-free boundary conditions for the delamination problem The governing differential equations of the present theory are obtained by applying the principle of virtual displacement The Rayleigh-Ritz method is used to solve both linear and nonlinear equations by assuming a double trigonometric series for the displacements Both linearized buckling analysis and nonlinear postbuckling analysis are performed for axially compressed cylindrical shells with clamped ends Comparisons made with the classical laminate theory and a first-order theory show significant deviations
TL;DR: In this article, a refined first-order shear-deformation theory is proposed and used to solve the plane strain bending; problem of both homogeneous plates and symmetric cross-ply laminated plates.
TL;DR: In this article, a generalization of the classical von Mises material is proposed; both the derivation of the model and the numerical treatment of the integration problem are discussed; the results turn out to be independent of the mesh spacing and the evolution laws for the internal variables can be derived from the postulate of maximum dissipation.
TL;DR: In this paper, a hierarchical model for thin elastic bodies is presented in which the variations of the displacement field through the thickness can be represented as a sequence of apolynomials, and it is found that the modeling error decreases monotonically as the order of the thickness polynomials increases and has a uniform convergence rate with respect to the thickness.
TL;DR: In this paper, a finite element for the study of the axial, torsional and flexural dynamic behavior of a rotating array of blades taking into account the gyroscopic effect and the centrifugal loadings was developed.
TL;DR: In this paper, the authors used the complex hypersingular boundary integral equation (CHBIE) for solving a wide range of plane elasticity problems, which is based on the use of the CHBIE for blocky bodies with cracks and holes.
Abstract: The effective numerical algorithm to solve a wide range of plane elasticity problems is presented. The method is based on the use of the complex hypersingular boundary integral equation (CHBIE) for blocky
The terminology refers to multiregions of interacting elastic bodies with generalized interfaces ranging from fixed to having displacement discontinuities. The terminology derives from Linkov (1983) and was associated with the hypersingular formulation in Linkov and Mogilevskaya (1991).
systems and bodies with cracks and holes. The BEM technique is employed to solve this equation. The unknown functions (displacement discontinuities (DD) or tractions) are approximated by Lagrange polynomials of the arbitrary degree. For the tip elements the asymptotics for the DD are taken into account. The boundaries of the blocks, cracks and holes are approximated by the arcs of the circles and the straight elements. In this case all the integrals (hypersingular, singular and regular) involved in this equation are evaluated in a closed form. Numerical results are given and compared either to the ones obtained by the other authors or to analytical solutions.
TL;DR: In this paper, a finite-volume procedure for determining the displacement fields and elastic stress distributions within structures that have axisymmetric geometries is presented, where an iterative technique is employed to solve these equations and provide the displacement field from which the strain and stress fields can be found.