The elastic field outside an ellipsoidal inclusion

1959 ◽  
Vol 252 (1271) ◽  
pp. 561-569 ◽  
Keyword(s):  
Velocity Field ◽  
Viscous Fluid ◽  
Elastic Constants ◽  
Strain Energy ◽  
Elastic Field ◽  
Ellipsoidal Inclusion ◽  
Harmonic Potential ◽  
Surface Distribution ◽  
Shear Transformation ◽  
General Method

The results of an earlier paper are extended. The elastic field outside an inclusion or inhomogeneity is treated in greater detail. For a general inclusion the harmonic potential of a certain surface distribution may be used in place of the biharmonic potential used previously. The elastic field outside an ellipsoidal inclusion or inhomogeneity may be expressed entirely in terms of the harmonic potential of a solid ellipsoid. The solution gives incidentally the velocity field about an ellipsoid which is deforming homogeneously in a viscous fluid. An expression given previously for the strain energy of an ellipsoidal region which has undergone a shear transformation is generalized to the case where the region has elastic constants different from those of its surroundings. The Appendix outlines a general method of calculating biharmonic potentials.

scholarly journals Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Mircea Bîrsan
Keyword(s):  
Energy Density ◽  
Constitutive Model ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Shell Theory ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Classical Shell Theory ◽  
Three Dimensional Elasticity ◽  
General Method

AbstractIn this paper, we present a general method to derive the explicit constitutive relations for isotropic elastic 6-parameter shells made from a Cosserat material. The dimensional reduction procedure extends the methods of the classical shell theory to the case of Cosserat shells. Starting from the three-dimensional Cosserat parent model, we perform the integration over the thickness and obtain a consistent shell model of order $$ O(h^5) $$ O ( h 5 ) with respect to the shell thickness h. We derive the explicit form of the strain energy density for 6-parameter (Cosserat) shells, in which the constitutive coefficients are expressed in terms of the three-dimensional elasticity constants and depend on the initial curvature of the shell. The obtained form of the shell strain energy density is compared with other previous variants from the literature, and the advantages of our constitutive model are discussed.

Wave Motion of a Compressible Viscous Fluid Contained in a Cylindrical Shell

1993 ◽  
Vol 115 (3) ◽  
pp. 302-312 ◽  
Author(s):  
J. H. Terhune ◽  
K. Karim-Panahi
Keyword(s):  
Velocity Field ◽  
Viscous Fluid ◽  
Imaginary Part ◽  
Fluid Velocity ◽  
Wave Number ◽  
Mode Shapes ◽  
Infinite Length ◽  
Complex Frequency ◽  
Compressible Viscous Fluid ◽  
Fluid Velocity Field

The free vibration of cylindrical shells filled with a compressible viscous fluid has been studied by numerous workers using the linearized Navier-Stokes equations, the fluid continuity equation, and Flu¨gge ’s equations of motion for thin shells. It happens that solutions can be obtained for which the interface conditions at the shell surface are satisfied. Formally, a characteristic equation for the system eigenvalues can be written down, and solutions are usually obtained numerically providing some insight into the physical mechanisms. In this paper, we modify the usual approach to this problem, use a more rigorous mathematical solution and limit the discussion to a single thin shell of infinite length and finite radius, totally filled with a viscous, compressible fluid. It is shown that separable solutions are obtained only in a particular gage, defined by the divergence of the fluid velocity vector potential, and the solutions are unique to that gage. The complex frequency dependence for the transverse component of the fluid velocity field is shown to be a result of surface interaction between the compressional and vortex motions in the fluid and that this motion is confined to the boundary layer near the surface. Numerical results are obtained for the first few wave modes of a large shell, which illustrate the general approach to the solution. The axial wave number is complex for wave propagation, the imaginary part being the spatial attenuation coefficient. The frequency is also complex, the imaginary part of which is the temporal damping coefficient. The wave phase velocity is related to the real part of the axial wave number and turns out to be independent of frequency, with numerical value lying between the sonic velocities in the fluid and the shell. The frequency dependencies of these parameters and fluid velocity field mode shapes are computed for a typical case and displayed in non-dimensional graphs.

On the Torsional Oscillations of a Solid Sphere in a Viscous Fluid

1956 ◽  
Vol 23 (4) ◽  
pp. 601-605
Author(s):  
G. F. Carrier ◽  
R. C. Di Prima
Keyword(s):  
Velocity Field ◽  
Viscous Fluid ◽  
Angular Displacement ◽  
Solid Sphere ◽  
Circumferential Velocity ◽  
Torsional Oscillations ◽  
Viscosity Measurements ◽  
First Approximation ◽  
Viscous Torque ◽  
Solid Bodies

Abstract Most treatments of the torsional oscillations of solid bodies assume that the velocity field is circumferential. In this paper the motion in planes containing the axis of oscillation is also considered. An expansion in terms of the angular displacement ϵ (assumed small) is made. The first approximation to the circumferential velocity is computed, and then used in computing the first approximation to the pumping motion. This is used to compute the correction to the circumferential velocity and, in particular, the correction to the viscous torque. For the range of parameters considered it is found that the correction to the torque is of the order of 0.04ϵ2|N0|, where N0 is the classical viscous torque. This problem is of interest in practical viscosity measurements.

Elastic Strain Energy of Dislocations in Ice

1971 ◽  
Vol 49 (16) ◽  
pp. 2181-2186 ◽  
Author(s):  
W. R. Tyson
Keyword(s):  
Elastic Constants ◽  
Strain Field ◽  
Elastic Strain ◽  
Strain Energy ◽  
Elastic Strain Energy ◽  
High Mobility ◽  
Anisotropic Elasticity ◽  
Hexagonal Ice ◽  
Complete Set ◽  
Elastic Strain Field

The energy stored in the elastic strain field of dislocations in hexagonal ice is calculated using anisotropic elasticity and the most complete set of elastic constants available. Ice is elastically fairly isotropic, and it is proposed that the high mobility of dislocations on the basal plane is due to dissociation of perfect dislocations on this plane.

Dynamical Limit of Compliant Lever Mechanisms

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
Michele Bonaldi ◽  
Mario Saraceni ◽  
Enrico Serra
Keyword(s):  
Finite Element ◽  
Multiobjective Optimization ◽  
Elastic Constants ◽  
Upper Bound ◽  
Strain Energy ◽  
Mechanical Energy ◽  
Element Model ◽  
Positive Definite ◽  
Optimization Analysis ◽  
Energy Conservation Principle

The application of the mechanical energy conservation principle sets a dynamical limit to the performances of compliant lever mechanisms endowed with a positive definite strain energy. The limit applies to every linear compliant lever and is given as an upper bound on the product between the static effective gain of the device and its bandwidth. The relevant parameters of this relation are determined only by the structures surrounding the device and not by its design. This result is obtained on the basis of a linear two-port model, with coefficients determined by the static elastic constants of the device. The model and the dynamical limit are validated by multiobjective optimization analysis interfaced with a finite element model of a practical mechanism.

scholarly journals Further Considerations of the Elastic Inclusion Problem

1964 ◽  
Vol 14 (1) ◽  
pp. 61-70 ◽  
Author(s):  
R. J. Knops
Keyword(s):  
Elastic Inclusion ◽  
Elastic Field ◽  
Ellipsoidal Inclusion ◽  
Inclusion Problem ◽  
Infinite Matrix ◽  
Mean Values ◽  
Shear Moduli ◽  
Weakly Interacting

SummaryAn equation is derived for the strains of an arbitrary elastic field in an infinite matrix perturbed by several inclusions. The equation is solved exactly when the shear moduli of the inclusions and matrix are identical, and also when only a single ellipsoidal inclusion perturbs a field uniform at infinity. Mean-values of the strains are then calculated for non-uniform fields perturbed either by an ellipsoid or by a system of weakly-interacting spheres.

scholarly journals On Mechanical Properties of Metallic Glass and its Liquid Vitrification Characteristics

2008 ◽  
Vol 41-42 ◽  
pp. 247-252
Author(s):  
Min Qiang Jiang ◽  
L.H. Dai
Keyword(s):  
Elastic Constants ◽  
Metallic Glasses ◽  
Critical Strain ◽  
Molar Mass ◽  
Characteristic Size ◽  
Transformation Zone ◽  
Shear Transformation ◽  
Shear Transformation Zone ◽  
Solid Glass ◽  
Characteristic Volume

A systematic survey of the available data such as elastic constants, density, molar mass, and glass transition temperature of 45 metallic glasses is conducted. It is found that a critical strain controlling the onset of plastic deformation is material-independent. However, the correlation between elastic constants of solid glass and vitrification characteristics of its liquid does not follow a simple linear relation, and a characteristic volume, viz. molar volume, maybe relating to the characteristic size of a shear transformation zone (STZ), should be involved.

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