511 Analytical Development of Three-dimensional Thermoelastic Problem for a Nonhomogeneous Infinite Body with an External Crack

Abstract This paper contains an exact closed solution for the stress distribution around a cavity in the shape of an ellipsoid of revolution in an infinite elastic body which is otherwise in an arbitrary uniform plane state of stress perpendicular to the axis of revolution of the cavity. The solution is based upon an extension to orthogonal curvilinear co-ordinates of the classical three-function approach to three-dimensional problems of the theory of elasticity. The technically important features of the ensuing stress concentration are discussed in detail.

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Generalized Thermoelastic Problem of an Infinite Body with a Spherical Cavity under Dual-Phase-Lags

Прикладная механика и техническая физика ◽

10.15372/pmtf20160409 ◽

2016 ◽

Keyword(s):

Spherical Cavity ◽

Thermoelastic Problem ◽

Dual Phase ◽

Infinite Body ◽

Phase Lags ◽

Generalized Thermoelastic

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

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2014.0423 ◽

2014 ◽

Vol 470 (2169) ◽

pp. 20140423 ◽

Cited By ~ 8

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.

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The dynamics of folding instability in a constrained Cosserat medium

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2016.0159 ◽

2017 ◽

Vol 375 (2093) ◽

pp. 20160159 ◽

Cited By ~ 6

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.’

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