On Eshelby’s inclusion problem in nonlinear anisotropic elasticity

2021 ◽  
pp. 2150002
Author(s):  
Arash Yavari
Keyword(s):  
Spherical Inclusion ◽  
Anisotropic Elasticity ◽  
Inclusion Problem ◽  
Elastic Solids ◽  
Spherical Ball ◽  
Solid Circular Cylinder ◽  
Stress And Deformation ◽  
Nonlinear Elastic ◽  
Radially Inhomogeneous ◽  
Isotropic Solids

In this paper, the recent literature of finite eignestrains in nonlinear elastic solids is reviewed, and Eshelby’s inclusion problem at finite strains is revisited. The subtleties of the analysis of combinations of finite eigenstrains for the example of combined finite radial, azimuthal, axial and twist eigenstrains in a finite circular cylindrical bar are discussed. The stress field of a spherical inclusion with uniform pure dilatational eigenstrain in a radially-inhomogeneous spherical ball made of arbitrary incompressible isotropic solids is analyzed. The same problem for a finite circular cylindrical bar is revisited. The stress and deformation fields of an orthotropic incompressible solid circular cylinder with distributed eigentwists are analyzed.

scholarly journals Nonlinear elastic inclusions in isotropic solids

2013 ◽  
Vol 469 (2160) ◽  
pp. 20130415 ◽  
Author(s):  
Arash Yavari ◽  
Alain Goriely
Keyword(s):  
Residual Stress ◽  
Spherical Inclusion ◽  
The Body ◽  
Model Systems ◽  
Stress Fields ◽  
Elastic Solids ◽  
Geometric Framework ◽  
Spherical Ball ◽  
Nonlinear Elastic ◽  
Nonlinear Solids

We introduce a geometric framework to calculate the residual stress fields and deformations of nonlinear solids with inclusions and eigenstrains. Inclusions are regions in a body with different reference configurations from the body itself and can be described by distributed eigenstrains. Geometrically, the eigenstrains define a Riemannian 3-manifold in which the body is stress-free by construction. The problem of residual stress calculation is then reduced to finding a mapping from the Riemannian material manifold to the ambient Euclidean space. Using this construction, we find the residual stress fields of three model systems with spherical and cylindrical symmetries in both incompressible and compressible isotropic elastic solids. In particular, we consider a finite spherical ball with a spherical inclusion with uniform pure dilatational eigenstrain and we show that the stress in the inclusion is uniform and hydrostatic. We also show how singularities in the stress distribution emerge as a consequence of a mismatch between radial and circumferential eigenstrains at the centre of a sphere or the axis of a cylinder.

Universal deformations in inhomogeneous isotropic nonlinear elastic solids

2021 ◽  
Vol 477 (2253) ◽  
Author(s):  
Arash Yavari
Keyword(s):  
Strain Energy ◽  
Elastic Solid ◽  
Necessary Condition ◽  
Density Functions ◽  
Elastic Solids ◽  
Universal Deformations ◽  
Green Strain ◽  
The Right ◽  
Nonlinear Elastic ◽  
Isotropic Solids

Universal (controllable) deformations of an elastic solid are those deformations that can be maintained for all possible strain-energy density functions and suitable boundary tractions. Universal deformations have played a central role in nonlinear elasticity and anelasticity. However, their classification has been mostly established for homogeneous isotropic solids following the seminal works of Ericksen. In this article, we extend Ericksen’s analysis of universal deformations to inhomogeneous compressible and incompressible isotropic solids. We show that a necessary condition for the known universal deformations of homogeneous isotropic solids to be universal for inhomogeneous solids is that inhomogeneities respect the symmetries of the deformations. Symmetries of a deformation are encoded in the symmetries of its pulled-back metric (the right Cauchy–Green strain). We show that this necessary condition is sufficient as well for all the known families of universal deformations except for Family 5.

Stress Concentrations in Three-Dimensional Electrostriction

1967 ◽  
Vol 34 (2) ◽  
pp. 431-436 ◽  
Author(s):  
T. E. Smith
Keyword(s):  
Displacement Vector ◽  
Spherical Inclusion ◽  
Crack Problem ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Uniform Electric Field ◽  
Inclusion Problem ◽  
Stress Concentrations ◽  
Displacement Potential ◽  
Penny Shaped Crack

Using the techniques employed in developing a Papkovich-Neuber representation for the displacement vector in classical elasticity, a particular integral of the kinematical equations of equilibrium for the uncoupled theory of electrostriction is developed. The particular integral is utilized in conjunction with the displacement potential function approach to problems of the theory of elasticity to obtain closed-form solutions of several stress concentration problems for elastic dielectrics. Under a prescribed uniform electric field at infinity, the problems of an infinite elastic dielectric having first a spherical cavity and then a rigid spherical inclusion are solved. The rigid spheroidal inclusion problem and the penny-shaped crack problem are also solved for the case where the prescribed field is parallel to their axes of revolution.

Synthesis and elastic and mechanical properties of Cr2GeC

2008 ◽  
Vol 23 (8) ◽  
pp. 2157-2165 ◽  
Author(s):  
Shahram Amini ◽  
Aiguo Zhou ◽  
Surojit Gupta ◽  
Andrew DeVillier ◽  
Peter Finkel ◽  
...  
Keyword(s):  
Mechanical Energy ◽  
Kink Band ◽  
Max Phase ◽  
Elastic Solids ◽  
Max Phases ◽  
Kink Bands ◽  
Young’S Moduli ◽  
Microscale Model ◽  
Energy Dissipated ◽  
Nonlinear Elastic

Herein we report on the synthesis and characterization of Cr2GeC, a member of the so-called Mn+1AXn (MAX) phase family of layered machinable carbides and nitrides. Polycrystalline samples were synthesized by hot pressing pure Cr, Ge, and C powders at 1350 °C at ∼45 MPa for 6 h. No peaks other than those associated with Cr2GeC and Cr2O3, in the form of eskolaite, were observed in the x-ray diffraction spectra. The samples were readily machinable and fully dense. The steady-state Vickers hardness was 2.5 ± 0.1 GPa. The Young’s moduli measured in compression and by ultrasound were 200 ± 10 and 245 ± 3 GPa, respectively; the shear modulus and Poisson’s ratio deduced from the ultrasound results were 80 GPa and 0.29, respectively. The ultimate compressive strength for a ∼20 μm grain size sample was 770 ± 30 MPa. Samples compressively loaded from 300 to ∼570 MPa exhibited nonlinear, fully reversible, reproducible, closed hysteretic loops that dissipated ∼20% of the mechanical energy, a characteristic of the MAX phases, in particular, and kinking nonlinear elastic solids, in general. The energy dissipated is presumably due to the formation and annihilation of incipient kink bands. The critical resolved shear stress of the basal plane dislocations—estimated from our microscale model—is ∼22 MPa. The incipient kink band and reversible dislocation densities, at the maximum stress of 568 MPa, are estimated to be 1.2 × 10−2 μm−3 and 1.0 × 1010 cm−2, respectively.

scholarly journals Plane-strain waves in nonlinear elastic solids with softening

Wave Motion ◽  
2019 ◽  
Vol 89 ◽  
pp. 65-78 ◽  
Author(s):  
Harold Berjamin ◽  
Bruno Lombard ◽  
Guillaume Chiavassa ◽  
Nicolas Favrie
Keyword(s):  
Plane Strain ◽  
Elastic Solids ◽  
Strain Waves ◽  
Nonlinear Elastic

scholarly journals Analysis of Stress Deformation Characteristics of Composite Geomembrane Rock-Fill Dam with Core Wall

2013 ◽  
Vol 470 ◽  
pp. 962-965
Author(s):  
Dong Yan Ding ◽  
Jian Min Ren
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Element Model ◽  
Elastic Model ◽  
Deformation Characteristics ◽  
Distribution Rule ◽  
Dimensional Finite Element ◽  
Stress And Deformation ◽  
Three Dimensional Finite Element ◽  
Nonlinear Elastic

The Chengzigou hydropower station of composite geomembrane rockfill dam as an example of the dam body and the composite geotechnical membrane stress and deformation characteristics are used nonlinear elastic model - Duncan EB model establish three-dimensional finite element model of rockfill,by using the large finite element softwareFLAC3D,whice provided geogrid element to simulate lexible geomembrane shear interaction with soil.The stress and deformation of the dam and the composite geomembrane is calculated under two conditionscompletion period and impoundment period.And analyze the change of the stress and strain distribution rule,whice will provide the basis for the design of the geomembrane.

Direct observation of nonlinear acoustoelastic hysteresis in kinking nonlinear elastic solids

2009 ◽  
Vol 94 (24) ◽  
pp. 241904 ◽  
Author(s):  
P. Finkel ◽  
A. G. Zhou ◽  
S. Basu ◽  
O. Yeheskel ◽  
M. W. Barsoum
Keyword(s):  
Direct Observation ◽  
Elastic Solids ◽  
Nonlinear Elastic ◽  
Kinking Nonlinear Elastic Solids

Three-Dimensional Nonlinear Finite Element Analysis on the Concrete Face Cock-Fill Dam of TianChi Upper Reservoir

2013 ◽  
Vol 838-841 ◽  
pp. 1763-1767
Author(s):  
Shuang Mei Chang ◽  
Wen She He ◽  
Yu Qiang Cheng ◽  
Su Min Zhao
Keyword(s):  
Finite Element ◽  
Water Pressure ◽  
Three Dimensional ◽  
Elastic Model ◽  
Finite Element Software ◽  
Dimensional Finite Element ◽  
Stress And Deformation ◽  
Three Dimensional Finite Element ◽  
Nonlinear Elastic ◽  
Concrete Face

Taking the concrete face cock-fill dam upper reservoir of Tianchi as an example, the stress and deformation characteristics of concrete face rock-fill dam are studied in-depth in this paper. The article builds a fine three-dimensional finite element model of Tianchi upper reservoir by a nonlinear elastic model of the finite element software ADINA. The stress and deformation of the two conditions under completion and storage for the dam are calculated ,which will be analyzed to obtain stress - strain distribution of the dam in two conditions, comparing dam stress and deformation before and after impoundment to get impact of the water pressure on the dam stress and deformation: comparing after impoundment and completion , the dam water level displacement of upstream side from role of horizontal water pressure will increase , the dam upstream offsets to downstream , but the offset is little ; Due to dam is affected by vertical hydrostatic pressure and uplift pressure after impoundment , the dam settlement is slightly less than the completion in storage. KEY WORDS: Tianchi upper reservoir, The concrete face cock-fill dam, Three-dimensional finite element, Nonlinear elastic model, Analysis of stress and deformation

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