Plane Deformations of an Inhomogeneity–Matrix System Incorporating a Compressible Liquid Inhomogeneity and Complete Gurtin–Murdoch Interface Model

2018 ◽  
Vol 85 (12) ◽  
Author(s):  
Ming Dai ◽  
Min Li ◽  
Peter Schiavone
Keyword(s):  
Analytic Solution ◽  
Elastic Solid ◽  
Compressible Liquid ◽  
External Loading ◽  
Interface Model ◽  
Interface Tension ◽  
Soft Solids ◽  
Remote Loading ◽  
Interface Elasticity ◽  
Plane Deformations

We consider the plane deformations of an infinite elastic solid containing an arbitrarily shaped compressible liquid inhomogeneity in the presence of uniform remote in-plane loading. The effects of residual interface tension and interface elasticity are incorporated into the model of deformation via the complete Gurtin–Murdoch (G–M) interface model. The corresponding boundary value problem is reformulated and analyzed in the complex plane. A concise analytical solution describing the entire stress field in the surrounding solid is found in the particular case involving a circular inhomogeneity. Numerical examples are presented to illustrate the analytic solution when the uniform remote loading takes the form of a uniaxial compression. It is shown that using the simplified G–M interface model instead of the complete version may lead to significant errors in predicting the external loading-induced stress concentration in gel-like soft solids containing submicro- (or smaller) liquid inhomogeneities.

Propagation of Longitudinal Waves in Circularly Cylindrical Bone Elements

1971 ◽  
Vol 38 (3) ◽  
pp. 578-584 ◽  
Author(s):  
J. L. Nowinski ◽  
C. F. Davis
Keyword(s):  
Linear Equations ◽  
Elastic Solid ◽  
Variable Coefficients ◽  
External Loading ◽  
Regular Singular Point ◽  
Two Phase ◽  
Longitudinal Waves ◽  
Living Bone ◽  
Poroelastic Material ◽  
Biot’S Equations

Two-phase poroelastic material is taken as a model of the living bone in the sense that the osseous tissue is treated as a linear isotropic perfectly elastic solid, and the fluid substances filling the pores as a perfect fluid. Using Biot’s equations, derived in his consolidation theory, four coupled governing differential equations for the propagation of harmonic longitudinal waves in circularly cylindrical bars of poroelastic material are derived. A longer manipulation reduces the task of solution to a single ordinary differential equation with variable coefficients and a regular singular point. The equation is solved by Frobenius’ method. Three boundary conditions on the curved surface of the bar, expressing the absence of external loading and the permeability of the surface, supply a system of three linear equations in three unknown coefficients. A nontrivial solution of the system gives two phase velocities of propagation of longitudinal waves in agreement with the finding of Biot for an infinite medium. A simplification to the purely elastic case yields the elementary classical result for the longitudinal waves.

Semi-Analytic Iterative Solution for the Adhesive Contact Between a Micro-Indenter and a Graded Elasticity Coating

2014 ◽  
Author(s):  
Stewart J. Chidlow ◽  
William W. F. Chong ◽  
Mircea Teodorescu
Keyword(s):  
Stress Field ◽  
Analytic Solution ◽  
Superposition Principle ◽  
Elastic Solid ◽  
Iterative Solution ◽  
Normal Pressure ◽  
Adhesive Contact ◽  
Collocation Methods ◽  
Subsurface Stress ◽  
The Fourier Transform

This paper proposes a hybrid (semi-analytic) solution for determining the contact footprint and subsurface stress field in a two-dimensional adhesive problem involving a multi-layered elastic solid loaded normally by a rigid indenter. The subsurface stress field is determined using a semi-analytic solution and the footprint using a fast converging iterative algorithm. The solid to be indented consists of a graded elasticity coating with exponential increase of decay of its shear modulus bonded on a homogeneously elastic substrate. By applying the Fourier Transform to the governing boundary value problem, we formulate expressions for the stresses and displacements induced by the application of line forces acting both normally and tangentially at the origin. The superposition principle is then used to generalize these expressions to the case of distributed normal pressure acting on the solid surface. A pair of coupled integral equations are further derived for the parabolic stamp problem which are easily solved using collocation methods.

scholarly journals The symmetry and loading-independency of multiple inclusions enclosing uniform stresses in an infinite elastic plane

2020 ◽  
Vol 41 (10) ◽  
pp. 1493-1496
Author(s):  
Ming Dai ◽  
P. Schiavone
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Particulate Composite ◽  
Specific Form ◽  
Internal Stresses ◽  
Elastic Plane ◽  
Plane Shear ◽  
The Matrix ◽  
Remote Loading ◽  
Plane Deformations

Abstract The identification of multiple interacting inclusions with uniform internal stresses in an infinite elastic matrix subjected to a uniform remote loading is of fundamental importance in the mechanics and design of particulate composite materials. In anti-plane shear and plane deformations, certain sufficient conditions have been established in the literature which guarantee uniform internal stresses inside multiple interacting inclusions displaying various symmetries when the matrix is subjected to specific uniform remote loading. Correspondingly, sufficient conditions which allow for the design of multiple interacting inclusions independent of any specific form of (uniform) remote loading have also been established. In this paper, we demonstrate rigorously that, in all cases, these sufficient conditions are also necessary conditions and indeed allow for the identification of all possible collections of such inclusions.

Uniqueness of Neutral Elastic Circular Nano-Inhomogeneities in Antiplane Shear and Plane Deformations

2016 ◽  
Vol 83 (10) ◽  
Author(s):  
Ming Dai ◽  
Peter Schiavone ◽  
Cun-Fa Gao
Keyword(s):  
Foreign Body ◽  
Stress Field ◽  
Antiplane Shear ◽  
Interface Effect ◽  
Interface Condition ◽  
External Loading ◽  
Surrounding Matrix ◽  
The Matrix ◽  
Interface Parameters ◽  
Plane Deformations

In elasticity theory, a neutral inhomogeneity is defined as a foreign body which can be introduced into a host solid without disturbing the stress field in the solid. The existence of circular neutral elastic nano-inhomogeneities has been established for both antiplane shear and plane deformations when the interface effect is described by constant interface parameters, and the surrounding matrix is subjected to uniform external loading. It is of interest to determine whether noncircular neutral nano-inhomogeneities can be constructed under the same conditions. In fact, we prove that only the circular elastic nano-inhomogeneity can achieve neutrality under these conditions with the radius of the inhomogeneity determined by the corresponding (constant) interface parameters and bulk elastic constants. In particular, in the case of plane deformations, the (uniform) external loading imposed on the matrix must be hydrostatic in order for the corresponding circular nano-inhomogeneity to achieve neutrality. Moreover, we find that, even when we relax the interface condition to allow for a nonuniform interface effect (described by variable interface parameters), in the case of plane deformations, only the elliptical nano-inhomogeneity can achieve neutrality.

Harmonic Shapes in Finite Elasticity Under Nonuniform Loading

2004 ◽  
Vol 72 (5) ◽  
pp. 691-694 ◽  
Author(s):  
G. F. Wang ◽  
P. Schiavone ◽  
C.-Q. Ru
Keyword(s):  
Stress Field ◽  
Finite Elasticity ◽  
Normal Stresses ◽  
Finite Plane ◽  
Hyperelastic Materials ◽  
Nonuniform Loading ◽  
Final Stress ◽  
Remote Loading ◽  
Finite Plane Deformations ◽  
Plane Deformations

We investigate the classic (inverse) problem concerned with the design of so-called harmonic shapes for an elastic material undergoing finite plane deformations. In particular, we show how to identify such shapes for a particular class of compressible hyperelastic materials of harmonic type. The “harmonic condition,” in which the sum of the normal stresses in the original stress field remains unchanged everywhere after the introduction of the harmonic hole or inclusion, is imposed on the final stress field. Using complex variable techniques, we identify particular harmonic shapes arising when the material is subjected nonuniform (remote) loading and discuss conditions for the existence of such shapes.

Analytical Solution for Size-Dependent Elastic Field of a Nanoscale Circular Inhomogeneity

2006 ◽  
Vol 74 (3) ◽  
pp. 568-574 ◽  
Author(s):  
L. Tian ◽  
R. K. N. D. Rajapakse
Keyword(s):  
Analytical Solution ◽  
Elastic Field ◽  
Infinite Matrix ◽  
Surface Residual Stress ◽  
Residual Surface Stress ◽  
Stress Effects ◽  
Size Dependent ◽  
Closed Form Analytical Solution ◽  
Remote Loading ◽  
Interface Elasticity

Two-dimensional elastic field of a nanoscale circular hole/inhomogeneity in an infinite matrix under arbitrary remote loading and a uniform eigenstrain in the inhomogeneity is investigated. The Gurtin–Murdoch surface/interface elasticity model is applied to take into account the surface/interface stress effects. A closed-form analytical solution is obtained by using the complex potential function method of Muskhelishvili. Selected numerical results are presented to investigate the size dependency of the elastic field and the effects of surface elastic moduli and residual surface stress. Stress state is found to depend on the radius of the inhomogeneity/hole, surface elastic constants, surface residual stress, and magnitude of far-field loading.

EFFECTS OF A LIQUID CORE ON THE PROPAGATION OF SEISMIC WAVES

1959 ◽  
Vol 37 (2) ◽  
pp. 109-128 ◽  
Author(s):  
George Duwalo ◽  
J. A. Jacobs
Keyword(s):  
Seismic Waves ◽  
Wave Length ◽  
Liquid Core ◽  
Elastic Solid ◽  
Compressible Liquid ◽  
Diffracted Waves ◽  
Viscous Compressible Liquid ◽  
Method Of Residues ◽  
Propagation Of Elastic Waves ◽  
The Waves

Effects of a spherical cavity in an infinite, homogeneous, isotropic elastic solid, containing non-viscous compressible liquid, on the propagation of elastic waves are investigated mathematically. The waves emitted by a simple harmonic point source in the solid are of the types known as SH and P in seismology. The discussion is restricted to the case ka » 1 (ka = 2 π cavity radius/wave length). Series solutions are transformed into contour integrals by Watson's method. Evaluation of these by the method of residues results in expressions describing the P and S components of the diffracted waves.

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