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

Crouch, Steven L.; Mogilevskaya, Sofia G.

doi:10.1002/nme.789

Cited by 28 publications

(27 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This fact provides the means to verify the solution for the stresses obtained with our approach. To do so we performed the computation and compared the results for the stresses to those for the elastic problems given in [Wang et al 2003a] (the latter results have been verified with the benchmark results obtained earlier by [Ling 1948] and [Haddon 1967]). We achieved the same accuracy as reported in [Wang et al 2003a].…”

Section: Examplesmentioning

confidence: 75%

Semi-analytical solution for a viscoelastic plane containing multiple circular holes

Huang

Mogilevskaya

Crouch

2006

JOMMS

View full text Add to dashboard Cite

The paper considers the problem of an infinite, homogeneous, isotropic viscoelastic plane containing multiple circular holes. Constant or time-dependent loading is applied at infinity or on the boundaries of the holes. The sizes and locations of the holes are arbitrary provided they do not overlap. The solution of the problem is based on the use of the correspondence principle, and the governing equation in the Laplace domain is a complex hypersingular boundary integral equation written in terms of the unknown transformed displacements at the boundaries of the holes. The main feature of this equation is that the material parameters are only involved as multipliers for the terms other than the integrals of transformed displacements. The unknown transformed displacements are approximated by truncated complex Fourier series with coefficients dependent on the transform parameter. A system of linear algebraic equations is formulated using Taylor series expansion for determining these coefficients. The viscoelastic stresses and displacements are calculated through the viscoelastic analogs of Kolosov-Muskhelishvili potentials, and an inverse Laplace transform is used to provide the time domain solution. All the operations (space integration, Laplace transform and its inversion) are performed analytically. The method described in the paper enables the consideration of a variety of viscoelastic models. For the sake of illustration, examples are given for the cases where the viscoelastic solid responds as (i) a Boltzmann model in shear and elastically in dilatation, (ii) a Boltzmann model in both shear and dilatation, and (iii) a Burgers model in shear and elastically in dilatation. The accuracy and efficiency of the approach are demonstrated by comparing selected results with the solutions obtained by the finite element method (ANSYS) and the time stepping boundary element approach.

show abstract

Section: Examplesmentioning

confidence: 75%

Semi-analytical solution for a viscoelastic plane containing multiple circular holes

Huang

Mogilevskaya

Crouch

2006

JOMMS

View full text Add to dashboard Cite

show abstract

“…This fact provides the means to verify the solution for the stresses obtained with our approach. To do so we performed the computation and compared the results for the stresses to those for the elastic problems given in [Wang et al 2003a] (the latter results have been verified with the benchmark results obtained earlier by [Ling 1948] and[Haddon 1967]). We achieved the same accuracy as reported in [Wang et al 2003a].…”

Section: Examplesmentioning

confidence: 71%

“…However, the 'elastic constants' are functions of the transform parameter s, as are the transformed boundary conditions for the problem. By solving the corresponding elastic problem and taking the inverse Laplace transform, the time-dependent solution is found [Lee 1955;Findly et al 1989].…”

Section: Correspondence Principlementioning

confidence: 99%

Untitled

2006

JOMMS

View full text Add to dashboard Cite

This paper presents a methodology for detecting rotor imbalances, such as mass imbalance and crack-induced imbalance, using shaft synchronous vibrations. A vibration detection algorithm is derived based on the first order synchronous vibration response. A detection system was integrated by using state-of-the-art commercial analysis equipment. A laboratory rotor test rig with controlled mass imbalances was used to verify the integrated system. The system was then deployed to an engine subassembly test setup. Four specimens were used in the subassembly test and the test results are reported in the final section.

show abstract

“…Mogilevskaya and Crouch [11] have solved the problem of an infinite plane containing arbitrary number of circular inclusions based on the complex singular integral equation. Later, they [12] utilized Somigliana's formula and Fourier series for elasticity problems with circular boundaries. In their analysis procedure, the unknown tractions are approximated by using the complex Fourier series.…”

Section: Introductionmentioning

confidence: 99%

Torsional rigidity of a circular bar with multiple circular inclusions using the null-field integral approach

Chen

Lee

2009

Comput Mech

View full text Add to dashboard Cite

In this paper, a systematic approach is proposed to calculate the torsional rigidity of a circular bar containing multiple circular inclusions. To fully capture the circular geometries, the kernel function is expanded to the degenerate form and the boundary density is expressed into Fourier series. The approach is seen as a semi-analytical manner since error purely attributes to the truncation of Fourier series. By collocating the null-field point exactly on the real boundary and matching the boundary condition, a linear algebraic system is obtained. After obtaining the unknown Fourier coefficients, the solution can be obtained by using the integral representation. Finally, torsion problems are revisited to check the validity of our method. Torsional rigidities for a circular bar with an eccentric inclusion are compared well with the exact solution, BEM data and the Tang's results. Convergence study shows that only a few number of Fourier series terms can yield acceptable results. The torsional rigidities of two limiting case of cavity and rigid inclusion are also obtained using the present approach. Five gains of well-posed model, singularity free, free of boundary-layer effect, exponential convergence and mesh-free approach are achieved. A general-purpose program was developed to determine the torsional rigidity for a circular bar with arbitrary number, radii, positions and shear moduli of circular inclusions.

show abstract

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

Cited by 28 publications

References 32 publications

Semi-analytical solution for a viscoelastic plane containing multiple circular holes

Semi-analytical solution for a viscoelastic plane containing multiple circular holes

Untitled

Torsional rigidity of a circular bar with multiple circular inclusions using the null-field integral approach

Contact Info

Product

Resources

About