scholarly journals Cones of localized shear strain in incompressible elasticity with prestress: Green's function and integral representations

2014 ◽  
Vol 470 (2169) ◽  
pp. 20140423 ◽  
Author(s):  
L. P. Argani ◽  
D. Bigoni ◽  
D. Capuani ◽  
N. V. Movchan
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Ad Hoc ◽  
Shear Failure ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Integral Representations ◽  
Element Formulation ◽  
Infinite Body ◽  
Incompressible Elasticity

The infinite-body three-dimensional Green's function set (for incremental displacement and mean stress) is derived for the incremental deformation of a uniformly strained incompressible, nonlinear elastic body. Particular cases of the developed formulation are the Mooney–Rivlin elasticity and the J 2 -deformation theory of plasticity. These Green's functions are used to develop a boundary integral equation framework, by introducing an ad hoc potential, which paves the way for a boundary element formulation of three-dimensional problems of incremental elasticity. Results are used to investigate the behaviour of a material deformed near the limit of ellipticity and to reveal patterns of shear failure. In fact, within the investigated three-dimensional framework, localized deformations emanating from a perturbation are shown to be organized in conical geometries rather than in planar bands, so that failure is predicted to develop through curved and thin surfaces of intense shearing, as can for instance be observed in the cup–cone rupture of ductile metal bars.

scholarly journals The dynamics of folding instability in a constrained Cosserat medium

2017 ◽  
Vol 375 (2093) ◽  
pp. 20160159 ◽  
Author(s):  
Panos A. Gourgiotis ◽  
Davide Bigoni
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Three Dimensional ◽  
Deformation Pattern ◽  
Infinite Body ◽  
Generalized Continua ◽  
Cosserat Medium ◽  
Time Harmonic ◽  
Propagating Waves ◽  
Cosserat Continua

Different from Cauchy elastic materials, generalized continua, and in particular constrained Cosserat materials, can be designed to possess extreme (near a failure of ellipticity) orthotropy properties and in this way to model folding in a three-dimensional solid. Following this approach, folding, which is a narrow zone of highly localized bending, spontaneously emerges as a deformation pattern occurring in a strongly anisotropic solid. How this peculiar pattern interacts with wave propagation in the time-harmonic domain is revealed through the derivation of an antiplane, infinite-body Green’s function, which opens the way to integral techniques for anisotropic constrained Cosserat continua. Viewed as a perturbing agent, the Green’s function shows that folding, emerging near a steadily pulsating source in the limit of failure of ellipticity, is transformed into a disturbance with wavefronts parallel to the folding itself. The results of the presented study introduce the possibility of exploiting constrained Cosserat solids for propagating waves in materials displaying origami patterns of deformation. This article is part of the themed issue ‘Patterning through instabilities in complex media: theory and applications.’

Alternative Methods of Evaluating Green’s Function in Three-Dimensional Ship-Wave Problems

1977 ◽  
Vol 21 (02) ◽  
pp. 89-93
Author(s):  
Grant E. Hearn
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Three Dimensional ◽  
Integral Representations ◽  
Alternative Methods ◽  
Ship Wave ◽  
Integral Forms ◽  
Finite Integral ◽  
Efficiency Comparison ◽  
Wave Problems

This paper introduces new alternative forms of Green's function, and its derivatives, suitable for the calculations associated with three-dimensional ship-wave problems. Rather than infinite-range integral forms, new finite integral representations of Green's function for harmonic motions are derived for the "single-body" and "double-body" descriptions of the interactive problem. The usefulness of these finite integral descriptions is illustrated by carrying out a computing efficiency comparison with other alternative, but mathematically equivalent, descriptions of the Green's function. The calculations carried out indicate that the proposed forms of the Green function are numerically more efficient than the standard forms without affecting accuracy.

scholarly journals Unified double- and single-sided homogeneous Green’s function representations

2016 ◽  
Vol 472 (2190) ◽  
pp. 20160162 ◽  
Author(s):  
Kees Wapenaar ◽  
Joost van der Neut ◽  
Evert Slob
Keyword(s):  
Green’S Function ◽  
Multiple Scattering ◽  
Green's Function ◽  
Time Reversal ◽  
Wave Theory ◽  
Boundary Integral ◽  
Open Boundary ◽  
Quantum Mechanical ◽  
Holographic Imaging ◽  
Scattering Time

In wave theory, the homogeneous Green’s function consists of the impulse response to a point source, minus its time-reversal. It can be represented by a closed boundary integral. In many practical situations, the closed boundary integral needs to be approximated by an open boundary integral because the medium of interest is often accessible from one side only. The inherent approximations are acceptable as long as the effects of multiple scattering are negligible. However, in case of strongly inhomogeneous media, the effects of multiple scattering can be severe. We derive double- and single-sided homogeneous Green’s function representations. The single-sided representation applies to situations where the medium can be accessed from one side only. It correctly handles multiple scattering. It employs a focusing function instead of the backward propagating Green’s function in the classical (double-sided) representation. When reflection measurements are available at the accessible boundary of the medium, the focusing function can be retrieved from these measurements. Throughout the paper, we use a unified notation which applies to acoustic, quantum-mechanical, electromagnetic and elastodynamic waves. We foresee many interesting applications of the unified single-sided homogeneous Green’s function representation in holographic imaging and inverse scattering, time-reversed wave field propagation and interferometric Green’s function retrieval.

Theory of Electron Diffraction by Crystals

1967 ◽  
Vol 22 (4) ◽  
pp. 422-431 ◽  
Author(s):  
Kyozaburo Kambe
Keyword(s):  
Integral Equation ◽  
Electron Diffraction ◽  
General Theory ◽  
Green’S Function ◽  
Green's Function ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Two Dimensions ◽  
The Third ◽  
Third Dimension

A general theory of electron diffraction by crystals is developed. The crystals are assumed to be infinitely extended in two dimensions and finite in the third dimension. For the scattering problem by this structure two-dimensionally expanded forms of GREEN’S function and integral equation are at first derived, and combined in single three-dimensional forms. EWALD’S method is applied to sum up the series for GREEN’S function.

Formulation of Three-Dimensional Green’s Function and its Application to Fatigue Life Evaluation of Pressurizer

2006 ◽  
Vol 324-325 ◽  
pp. 387-390
Author(s):  
Yoon Suk Chang ◽  
Shin Beom Choi ◽  
Jae Boong Choi ◽  
Young Jin Kim ◽  
Myung Jo Jhung ◽  
...  
Keyword(s):  
Finite Element ◽  
Fatigue Life ◽  
Green’S Function ◽  
Green's Function ◽  
Three Dimensional ◽  
Stress Transfer ◽  
Life Assessment ◽  
Element Analysis ◽  
Fatigue Evaluation ◽  
Effective Assessment

Major nuclear components have been designed by conservative codes to prevent unanticipated fatigue failure. However, more realistic and effective assessment is necessary in proof of continued operation beyond the design life. In the present paper, three-dimensional stress and fatigue evaluation is carried out for pressurizer employing complex full geometry itself instead of conventional discrete subcomponents. For this purpose, temperature and mechanical stress transfer Green’s functions are derived from finite element analyses and applied to critical locations of pressurizer. In accordance with comparison of resulting stresses obtained from the Green’s function and detailed finite element analysis, suitability of the specific Green’s function is investigated. Finally, prototype of fatigue life assessment results is provided along with relevant ongoing activities.

Green’s function–based surrogate model for windfields using limited samples

2017 ◽  
Vol 42 (3) ◽  
pp. 164-176 ◽  
Author(s):  
Joshua Paul Marshall ◽  
Joseph David Richardson ◽  
Carlos Jose Montalvo
Keyword(s):  
Green’S Function ◽  
Wind Velocity ◽  
Green's Function ◽  
Surrogate Model ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Fast Method ◽  
Three Dimensional Model ◽  
Wind Velocities ◽  
Wind Measurements

There exists many applications for which wind-velocity is desired over a three-dimensional space. The vector field associated with these wind velocities is known as a “windfield” or “velocity-windfield.” The present work provides a fast method to characterize windfields. The approach uses the free-space Green’s function for potential theory as an inexpensive surrogate model in lieu of either complicated physics-based models or other types of surrogate models, both of which require volumetric discretizations for the three-dimensional case. Using the gradient of the third Green’s identity, the wind-velocity in the interior of a domain is entirely characterized by a surface discretization while still providing a three-dimensional model. The unknown densities on the surface are determined from enforcement of the interior form of the identity at arbitrary points coinciding with wind measurements taken by unmanned aerial vehicles. Numerical results support the feasibility of the method.

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