The Elastic Field Caused by a Circular Cylindrical Inclusion—Part II: Inside the Region x12+x22>a2,−∞

1995 ◽  
Vol 62 (3) ◽  
pp. 585-589 ◽  
Author(s):  
Linzhi Wu ◽  
Shanyi Y. Du
Keyword(s):  
Stress Field ◽  
Elastic Medium ◽  
Analytical Solutions ◽  
Elastic Field ◽  
Numerical Results ◽  
Elliptic Integrals ◽  
Cylindrical Inclusion ◽  
Stress Fields ◽  
Complete Elliptic Integrals

Analytical solutions are presented for the displacement and stress fields caused by a circular cylindrical inclusion with arbitrary uniform eigenstrains in an infinite elastic medium. The expressions obtained and those presented in Part I constitute the solutions of the whole elastic field, −∞<x1,x2,x3<∞. In the present paper, it is found that the analytical solutions within the region x12+x22>a2,−∞<x3<∞ can also be expressed as functions of the complete elliptic integrals of the first, second, and third kind. When the length of inclusion tends towards the limit (infinity), the present solutions agree with Eshelby’s results. Finally, numerical results are shown for the stress field.

The Elastic Field Caused by a Circular Cylindrical Inclusion—Part I: Inside the Region x12+x22

1995 ◽  
Vol 62 (3) ◽  
pp. 579-584 ◽  
Author(s):  
Linzhi Wu ◽  
Shanyi Du
Keyword(s):  
Analytical Solutions ◽  
Elastic Field ◽  
Companion Paper ◽  
Elliptic Integrals ◽  
Cylindrical Inclusion ◽  
Stress Fields ◽  
Elastic Fields ◽  
Complete Elliptic Integrals ◽  
Nonsingular Surface ◽  
Surface Integrals

The displacement and stress fields caused by uniform eigenstrains in a circular cylindrical inclusion are analyzed inside the region x12+x22<a2,−∞<x3<∞ and are given in terms of nonsingular surface integrals. Analytical solutions can be expressed as functions of the complete elliptic integrals of the first, second and third kind. The corresponding elastic fields in the region x12+x22>a2,−∞<x3<∞ are solved by using the same technique (by Green’s functions) in the companion paper (Part II).

The Elastic Field in a Half-Space With a Circular Cylindrical Inclusion

1996 ◽  
Vol 63 (4) ◽  
pp. 925-932 ◽  
Author(s):  
L. Z. Wu ◽  
S. Y. Du
Keyword(s):  
Half Space ◽  
Elastic Field ◽  
Elastic Half Space ◽  
Elliptic Integrals ◽  
Cylindrical Inclusion ◽  
Stress Fields ◽  
Function Technique ◽  
Explicit Solutions ◽  
Elastic Fields ◽  
Complete Elliptic Integrals

The problem of a circular cylindrical inclusion with uniform eigenstrain in an elastic half-space is studied by using the Green’s function technique. Explicit solutions are obtained for the displacement and stress fields. It is shown that the present elastic fields can be expressed as functions of the complete elliptic integrals of the first, second, and third kind. Finally, numerical results are shown for the displacement and stress fields.

Analytical Solutions of Stresses in Functionally Graded Circular Hollow Cylinder with Finite Length

2004 ◽  
Vol 261-263 ◽  
pp. 651-656 ◽  
Author(s):  
Z.S. Shao ◽  
L.F. Fan ◽  
Tie Jun Wang
Keyword(s):  
Young’S Modulus ◽  
Hollow Cylinder ◽  
Finite Length ◽  
Analytical Solutions ◽  
Radial Direction ◽  
Young's Modulus ◽  
Functionally Graded ◽  
Numerical Results ◽  
Stress Fields ◽  
Simply Supported

Analytical solutions of stress fields in functionally graded circular hollow cylinder with finite length subjected to axisymmetric pressure loadings on inner and outer surfaces are presented in this paper. The cylinder is simply supported at its two ends. Young's modulus of the material is assumed to vary continuously in radial direction of the cylinder. Moreover, numerical results of stresses in functionally graded circular hollow cylinder are appeared.

The determination of the elastic field of an ellipsoidal inclusion, and related problems

1957 ◽  
Vol 241 (1226) ◽  
pp. 376-396 ◽  
Keyword(s):  
Stress Field ◽  
Applied Stress ◽  
Elastic Constants ◽  
Elastic Solid ◽  
Elastic Field ◽  
Infinite Medium ◽  
Elliptic Integrals ◽  
Homogeneous Deformation ◽  
Spontaneous Change

It is supposed that a region within an isotropic elastic solid undergoes a spontaneous change of form which, if the surrounding material were absent, would be some prescribed homogeneous deformation. Because of the presence of the surrounding material stresses will be present both inside and outside the region. The resulting elastic field may be found very simply with the help of a sequence of imaginary cutting, straining and welding operations. In particular, if the region is an ellipsoid the strain inside it is uniform and may be expressed in terms of tabu­lated elliptic integrals. In this case a further problem may be solved. An ellipsoidal region in an infinite medium has elastic constants different from those of the rest of the material; how does the presence of this inhomogeneity disturb an applied stress-field uniform at large distances? It is shown that to answer several questions of physical or engineering interest it is necessary to know only the relatively simple elastic field inside the ellipsoid.

Analysis of Stress-Field Variations Expected on Subsurface Faults and Discontinuities in the Vicinity of Hydraulic Fracturing

2015 ◽  
Vol 19 (01) ◽  
pp. 054-069 ◽  
Author(s):  
Qian Gao ◽  
Yueming Cheng ◽  
Ebrahim Fathi ◽  
Samuel Ameri
Keyword(s):  
Stress Field ◽  
Analytical Solutions ◽  
Numerical Models ◽  
In Situ Stress ◽  
Strike Slip ◽  
Stress Fields ◽  
Hydraulic Fractures ◽  
Slip Tendency ◽  
The Stability

Summary In this study, a local stability evaluation method, slip-tendency analysis, is proposed on the basis of the Coulomb criterion to investigate the effects of hydraulic fracturing on stress-field variations and possibility of fault reactivation. The effects of net pressure and in-situ stress fields on the stability of faults are investigated in two typical faulting environments (i.e., normal and strike-slip faults). A 3D numerical model developed on the basis of the finite-element method (FEM) is also adopted to better understand the stability states around pressurized hydraulic fractures. The orientation and relative magnitudes of in-situ stress fields differ under different faulting environments, which, in turn, control the direction of fracture propagation and its geometry. It is found that the general patterns of slip-tendency distributions around pressurized hydraulic fractures are similar under different in-situ stress fields. Providing the normal and strike-slip faults with a same initial slip-tendency, the normal faulting environment demonstrates larger variations in slip-tendency than the strike-slip faulting environment. The comparison between analytical and numerical solutions indicates an excellent agreement was achieved, which certifies the validity of the proposed numerical models in complex situations. Numerical models and analytical solutions confirm the presence of both unstable and stable regions around the pressurized fractures. Fault stability during hydraulic operation depends on the position of faults with respect to the hydraulic fractures. The critical angle and distance between fault and hydraulic fracture in analytical solutions are identified when a region transits from stable to unstable status. For faults and discontinuities with an angle larger than 40° (i.e., with respect to horizontal direction) and distances less than 2.5 times the height of the fracture (i.e., from the center of pressurized fracture), the slip-tendency is greater than the initial value, indicating that the discontinuities within this zone are unstable and have the potential to slip. The developed model predicts that the unstable regions extend from fracture tips in both lateral and vertical directions. This generates relatively planar-distributed microseismic events, which are well-demonstrated in monitored field events. It was shown that the slippage of underground faults and other discontinuities could improve the fluid flow and transport by increasing the apparent permeability of the reservoir, such that the unstable regions could be recognized as permeability-improved zones.

The Elastic Field in a Half Space Due to Ellipsoidal Inclusions With Uniform Dilatational Eigenstrains

1979 ◽  
Vol 46 (3) ◽  
pp. 568-572 ◽  
Author(s):  
K. Seo ◽  
T. Mura
Keyword(s):  
Free Surface ◽  
Stress Field ◽  
Half Space ◽  
Elastic Stress ◽  
Elastic Field ◽  
Ellipsoidal Inclusion ◽  
Elliptic Integrals ◽  
Numerical Examples ◽  
Elastic Stress Field ◽  
Derivatives Of

The elastic stress field caused by an ellipsoidal inclusion with uniform dilatational eigenstrains is investigated for a semi-infinite solid. The result is expressed in terms of derivatives of elliptic integrals. Numerical examples are given to show the effect of the free surface with parameters of depth from the surface and shapes of the inclusion.

scholarly journals Master integrals for bipartite cuts of three-loop photon self energy

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
R. N. Lee ◽  
A. I. Onishchenko
Keyword(s):  
Spectral Density ◽  
High Energy ◽  
Elliptic Integrals ◽  
Iterated Integrals ◽  
Self Energy ◽  
Complete Elliptic Integrals ◽  
The One

Abstract We calculate the master integrals for bipartite cuts of the three-loop propagator QED diagrams. These master integrals determine the spectral density of the photon self energy. Our results are expressed in terms of the iterated integrals, which, apart from the 4m cut (the cut of 4 massive lines), reduce to Goncharov’s polylogarithms. The master integrals for 4m cut have been calculated in our previous paper in terms of the one-fold integrals of harmonic polylogarithms and complete elliptic integrals. We provide the threshold and high-energy asymptotics of the master integrals found, including those for 4m cut.

scholarly journals Analytical Investigation of Viscoelastic Stagnation-Point Flows with Regard to Their Singularity

2021 ◽  
Vol 11 (15) ◽  
pp. 6931
Author(s):  
Jie Liu ◽  
Martin Oberlack ◽  
Yongqi Wang
Keyword(s):  
Stress Distribution ◽  
Stress Field ◽  
Stagnation Point ◽  
Analytical Solutions ◽  
Wide Spectrum ◽  
Momentum Conservation ◽  
Stagnation Point Flow ◽  
Model Parameters ◽  
Viscoelastic Models ◽  
Special Cases

Singularities in the stress field of the stagnation-point flow of a viscoelastic fluid have been studied for various viscoelastic constitutive models. Analyzing the analytical solutions of these models is the most effective way to study this problem. In this paper, exact analytical solutions of two-dimensional steady wall-free stagnation-point flows for the generic Oldroyd 8-constant model are obtained for the stress field using different material parameter relations. For all solutions, compatibility with the conservation of momentum is considered in our analysis. The resulting solutions usually contain arbitrary functions, whose choice has a crucial effect on the stress distribution. The corresponding singularities are discussed in detail according to the choices of the arbitrary functions. The results can be used to analyze the stress distribution and singularity behavior of a wide spectrum of viscoelastic models derived from the Oldroyd 8-constant model. Many previous results obtained for simple viscoelastic models are reproduced as special cases. Some previous conclusions are amended and new conclusions are drawn. In particular, we find that all models have singularities near the stagnation point and most of them can be avoided by appropriately choosing the model parameters and free functions. In addition, the analytical solution for the stress tensor of a near-wall stagnation-point flow for the Oldroyd-B model is also obtained. Its compatibility with the momentum conservation is discussed and the parameters are identified, which allow for a non-singular solution.

scholarly journals Study on the nonlinear vibration of embedded carbon nanotube via the Hamiltonian-based method

2021 ◽  
pp. 146134842110327
Author(s):  
Kang-Jia Wang ◽  
Guo-Dong Wang
Keyword(s):  
Carbon Nanotubes ◽  
Carbon Nanotube ◽  
Nonlinear Vibration ◽  
Elastic Medium ◽  
Numerical Results ◽  
Frequency Property ◽  
New Novel ◽  
Novel Method

This article mainly studies the vibration of the carbon nanotubes embedded in elastic medium. A new novel method called the Hamiltonian-based method is applied to determine the frequency property of the nonlinear vibration. Finally, the effectiveness and reliability of the proposed method is verified through the numerical results. The obtained results in this work are expected to be helpful for the study of the nonlinear vibration.

Seismogram synthesis for multi-layered media with irregular interfaces by global generalized reflection/transmission matrices method. II. Applications for 2D SH case

1995 ◽  
Vol 85 (4) ◽  
pp. 1094-1106
Author(s):  
Xiaofei Chen
Keyword(s):  
Present Article ◽  
Analytical Solutions ◽  
Scattering Problem ◽  
Layered Media ◽  
Synthetic Seismograms ◽  
Numerical Results ◽  
New Method ◽  
New Formulation ◽  
Series Study

Abstract As the second part of a series study attempting to present a new method of seismogram synthesis for the irregular multi-layered media problems, the present article is devoted to discussing the aspects of the implementation of our new formulation developed earlier in part I of this series study (Chen, 1990). In this article, we have verified the validity of the formulation by comparing our numerical results with the existing analytical solutions for the scattering problem of a semi-circular canyon, and have shown its applicability by computing the synthetic seismograms for several selected irregular multi-layered media cases. Finally, applying our algorithm to the Whittier-Narrows earthquake of 1987, we have successfully interpreted the observed records.

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