The Elastic Inclusion With a Sliding Interface

1984 ◽  
Vol 51 (2) ◽  
pp. 308-310 ◽  
Author(s):  
T. Mura ◽  
R. Furuhashi
Keyword(s):  
Stress Field ◽  
Shear Deformation ◽  
Spherical Inclusion ◽  
Elastic Inclusion ◽  
Ellipsoidal Inclusion ◽  
Spheroidal Inclusion ◽  
Sliding Interface ◽  
Shear Traction ◽  
The Matrix ◽  
Invariant Transformation

It is found that when an ellipsoidal inclusion undergoes a shear eigenstrain and the inclusion is free to slip along the interface, the stress field vanishes everywhere in the inclusion and the matrix. It is assumed in the analysis that the inclusion interface cannot sustain any shear traction. There exists a shear deformation that transforms an ellipsoid into the identical ellipsoid without changing its orientation (ellipsoid invariant transformation). This is not true, however, for a spheroidal inclusion. The amount of slip and the associated stress field are calculated for a spherical inclusion for a given uniform eigenstrain εij*.

A nanosized rigid spherical inclusion with interfacial diffusion and sliding

2017 ◽  
Vol 23 (7) ◽  
pp. 1049-1060
Author(s):  
Xu Wang
Keyword(s):  
Steady State ◽  
Stress Field ◽  
Normal Stress ◽  
Spherical Inclusion ◽  
Surface Elasticity ◽  
Relaxation Times ◽  
Imperfect Interface ◽  
Time Dependent ◽  
Interfacial Diffusion ◽  
The Matrix

We examine the time-dependent deformations around a nanosized rigid spherical inclusion in an infinite elastic matrix under uniaxial tension at infinity. The elastic matrix is first endowed with separate Gurtin–Murdoch surface elasticity. Furthermore, interfacial diffusion and sliding both occur on the inclusion–matrix interface. Closed-form expressions of the time-dependent displacements and stresses in the matrix are derived by using Papkovich–Neuber displacement potentials. A concise and elegant expression of the steady-state normal stress on the surface of the inclusion is also obtained. It is seen that the displacements and stresses in the matrix evolve with two relaxation times which are reliant on three size-dependent parameters, one from surface elasticity and the other two from interfacial diffusion and sliding. Numerical results are presented to demonstrate the influence of surface elasticity on the relaxation times and on the stress distribution near the inclusion. It is observed that the surface elasticity can alter the nature of the steady state normal stress on the surface of the inclusion from tension to compression. When the radius of the inclusion is not greater than the ratio of residual surface tension to remote tension, the steady state normal stress on the surface of the inclusion is always compressive. The related problem of a nanosized rigid spherical inclusion with a spring-type imperfect interface is also solved. We find that it is feasible to design a neutral spherical inclusion that does not disturb a prescribed uniform uniaxial stress field or even any uniform stress field outside the inclusion through a judicious choice of the four imperfect interface parameters.

Eshelby Tensor for an Elastic Inclusion With Slightly Weakened Interface

1993 ◽  
Vol 60 (4) ◽  
pp. 1048-1050 ◽  
Author(s):  
Jianmin Qu
Keyword(s):  
Integral Representation ◽  
Matrix Material ◽  
Elastic Inclusion ◽  
Ellipsoidal Inclusion ◽  
Eshelby Tensor ◽  
Matrix Interface ◽  
Infinite Extent ◽  
The Matrix ◽  
Perfect Bonding ◽  
Inclusion Matrix

The Eshelby tensor for an ellipsoidal inclusion in an elastic matrix of infinite extent is considered in this paper. Instead of assuming perfect bonding between the inclusion and the matrix, the interface between the inclusion and the matrix is modeled by a spring layer of vanishing thickness. The inclusion- matrix interface is said to be slightly weakened if the compliance of the spring layer is much smaller than that of the matrix material. By virtue of the Betti-Rayleigh reciprocal identity in linear elasticity, an integral representation for the displacement field due to an elastic inclusion with a spring layer interface is derived. Explicit expressions of the Eshelby tensor for an ellipsoidal inclusion with slightly weakened interface are obtained through an iteration procedure developed from the integral representation.

Cosserat Modeling of Size Effects in Crystalline Solids

2000 ◽  
Vol 653 ◽  
Author(s):  
Samuel Forest
Keyword(s):  
Stress State ◽  
Size Effects ◽  
Elastic Inclusion ◽  
Characteristic Size ◽  
Infinite Matrix ◽  
Crystalline Solids ◽  
Generalized Continua ◽  
Absolute Size ◽  
Kink Bands ◽  
The Matrix

AbstractThe mechanics of generalized continua provides an efficient way of introducing intrinsic length scales into continuum models of materials. A Cosserat framework is presented here to descrine the mechanical behavior of crystalline solids. The first application deals with the problem of the stress field at a crak tip in Cosserat single crystals. It is shown that the strain localization patterns developping at the crack tip differ from the classical picture : the Cosserat continuum acts as a bifurcation mode selector, whereby kink bands arising in the classical framework disappear in generalized single crystal plasticity. The problem of a Cosserat elastic inclusion embedded in an infinite matrix is then considered to show that the stress state inside the inclusion depends on its absolute size lc. Two saturation regimes are observed : when the size R of the inclusion is much larger than a characteristic size of the medium, the classical Eshelby solution is recovered. When R is much small than the inclusion, a much higher stress is reached (for an inclusion stiffer than the matrix) that does not depend on the size any more. There is a transition regime for which the stress state is not homogeneous inside the inclusion. Similar regimes are obtained in the study of grain size effects in polycrystalline aggregates of Cosserat grains.

A Spherical Inclusion with Inhomogeneous Interface in Conduction

2003 ◽  
Vol 19 (1) ◽  
pp. 1-8
Author(s):  
T. Chen ◽  
C. H. Hsieh ◽  
P. C. Chuang
Keyword(s):  
Infinite Number ◽  
Effective Conductivity ◽  
Spherical Inclusion ◽  
Cone Angle ◽  
Imperfect Interface ◽  
Infinite Matrix ◽  
Algebraic Equations ◽  
Legendre Series ◽  
Linear Set ◽  
The Matrix

ABSTRACTA series solution is presented for a spherical inclusion embedded in an infinite matrix under a remotely applied uniform intensity. Particularly, the interface between the inclusion and the matrix is considered to be inhomegeneously bonded. We examine the axisymmetric case in which the interface parameter varies with the cone angle θ. Two kinds of imperfect interfaces are considered: an imperfect interface which models a thin interphase of low conductivity and an imperfect interface which models a thin interphase of high conductivity. We show that, by expanding the solutions of terms of Legendre polynomials, the field solution is governed by a linear set of algebraic equations with an infinite number of unknowns. The key step of the formulation relies on algebraic identities between coefficients of products of Legendre series. Some numerical illustrations are presented to show the correctness of the presented procedures. Further, solutions of the boundary-value problem are employed to estimate the effective conductivity tensor of a composite consisting of dispersions of spherical inclusions with equal size. The effective conductivity solely depends on one particular constant among an infinite number of unknowns.

An Edge Dislocation in a Three-Phase Composite Cylinder Model With a Sliding Interface

2002 ◽  
Vol 69 (4) ◽  
pp. 527-538 ◽  
Author(s):  
X. Wang ◽  
Y.-p. Shen
Keyword(s):  
Edge Dislocation ◽  
Arbitrary Point ◽  
Circular Inclusion ◽  
Composite Cylinder ◽  
Phase Form ◽  
Sliding Interface ◽  
Three Phase ◽  
Cylinder Model ◽  
The Matrix ◽  
Perfect Interface

An exact elastic solution is derived in a decoupled manner for the interaction problem between an edge dislocation and a three-phase circular inclusion with circumferentially homogeneous sliding interface. In the three-phase composite cylinder model, the inner inclusion and the intermediate matrix phase form a circumferentially homogeneous sliding interface, while the matrix and the outer composite phase form a perfect interface. An edge dislocation acts at an arbitrary point in the intermediate matrix. This three-phase cylinder model can simultaneously take into account the damage taking place in the circumferential direction at the inclusion-matrix interface and the interaction effect between the inclusions. As an application, we then investigate a crack interacting with the slipping interface.

An elastic harmonic inclusion near a rigid harmonic inclusion loaded by a couple

2019 ◽  
Vol 84 (3) ◽  
pp. 555-566
Author(s):  
Xu Wang ◽  
Liang Chen ◽  
Peter Schiavone
Keyword(s):  
Rigid Inclusion ◽  
Elastic Inclusion ◽  
Complex Loading ◽  
Solution Procedure ◽  
External Loading ◽  
Surrounding Matrix ◽  
The Matrix ◽  
Mapping Techniques ◽  
Loading Parameter ◽  
Harmonic Inclusion

AbstractWe use conformal mapping techniques to solve the inverse problem concerned with an elastic non-elliptical harmonic inclusion in the vicinity of a rigid non-elliptical harmonic inclusion loaded by a couple when the surrounding matrix is subjected to remote uniform stresses. Both a size-independent complex loading parameter and a size-dependent real loading parameter are introduced as part of the solution procedure. The stress field inside the elastic inclusion is uniform and hydrostatic; the interfacial normal and tangential stresses as well as the hoop stress on the matrix side are uniform along each one of the two inclusion–matrix interfaces. The tangential stress along the interface of the elastic inclusion (free of external loading) vanishes, whereas that along the interface of the rigid inclusion (loaded by the couple) does not. A novel method is proposed to determine the area of the rigid inclusion.

The Stress Field Induced by Normal Contact Between Dissimilar Spheres

1987 ◽  
Vol 54 (1) ◽  
pp. 8-14 ◽  
Author(s):  
D. A. Hills ◽  
A. Sackfield
Keyword(s):  
Stress Field ◽  
Elastic Constants ◽  
Vertical Displacement ◽  
Partial Slip ◽  
Normal Contact ◽  
Shear Traction

The stress field induced by the mutual compression of two spheres having dissimilar elastic constants is deduced. Solutions are found assuming full slip, full stick, and partial slip at the interface, but coupling between the shear traction and vertical displacement of the surfaces is neglected. It is shown that significant modifications to the Hertzian stress field occur at the surface, and that these decay rapidly with depth.

scholarly journals Translation of a Coated Rigid Spherical Inclusion in an Elastic Matrix: Exact Solution, and Implications for Mechanobiology

2019 ◽  
Vol 86 (5) ◽  
Author(s):  
Xin Chen ◽  
Moxiao Li ◽  
Shaobao Liu ◽  
Fusheng Liu ◽  
Guy M. Genin ◽  
...  
Keyword(s):  
Mechanical Properties ◽  
Exact Solution ◽  
Analytical Solution ◽  
Spherical Inclusion ◽  
Matrix Material ◽  
Elastic Matrix ◽  
Protein Ligands ◽  
First Case ◽  
The Matrix ◽  
Matrix Modulus

The displacement of relatively rigid beads within a relatively compliant, elastic matrix can be used to measure the mechanical properties of the matrix. For example, in mechanobiological studies, magnetic or reflective beads can be displaced with a known external force to estimate the matrix modulus. Although such beads are generally rigid compared to the matrix, the material surrounding the beads typically differs from the matrix in one or two ways. The first case, as is common in mechanobiological experimentation, is the situation in which the bead must be coated with materials such as protein ligands that enable adhesion to the matrix. These layers typically differ in stiffness relative to the matrix material. The second case, common for uncoated beads, is the situation in which the beads disrupt the structure of the hydrogel or polymer, leading to a region of enhanced or reduced stiffness in the neighborhood of the bead. To address both cases, we developed the first analytical solution of the problem of translation of a coated, rigid spherical inclusion displaced within an isotropic elastic matrix by a remotely applied force. The solution is applicable to cases of arbitrary coating stiffness and size of the coating. We conclude by discussing applications of the solution to mechanobiology.

A New Method to Measure the Microresidual Stress-Field Around Inclusions

1968 ◽  
Vol 90 (4) ◽  
pp. 620-622 ◽  
Author(s):  
L. B. Gulbransen ◽  
S. K. Chatterjee
Keyword(s):  
Free Energy ◽  
Domain Wall ◽  
Stress Field ◽  
Domain Walls ◽  
Manganese Sulfide ◽  
Ferromagnetic Material ◽  
New Method ◽  
The Matrix

A brief review of various theories of interaction of inclusions and domain walls in a ferromagnetic material is presented. A postulate concerning the modification of domain patterns by inclusions, based on theory, is described and the various free-energy contributions of the inclusion to the matrix in the vicinity of the inclusion are discussed. Examples of domain wall bending in ingot iron by manganese sulfide inclusions are shown to agree with the postulated model of interaction of a stress field around the inclusion and the domain wall.

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.

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