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.

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.

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.

Algebraic Multigrid Preconditioning for Finite Element Solution of Inhomogeneous Elastic Inclusion Problems in Articular Cartilage

2011 ◽  
Vol 3 (6) ◽  
pp. 729-744
Author(s):  
Zhengzheng Hu ◽  
Mansoor A Haider
Keyword(s):  
Finite Element ◽  
Articular Cartilage ◽  
Analytical Solution ◽  
Conjugate Gradient ◽  
Numerical Solutions ◽  
Elastic Inclusion ◽  
Algebraic Multigrid ◽  
Inclusion Problem ◽  
Inclusion Problems ◽  
Cpu Time

AbstractIn studying biomechanical deformation in articular cartilage, the presence of cells (chondrocytes) necessitates the consideration of inhomogeneous elasticity problems in which cells are idealized as soft inclusions within a stiff extracellular matrix. An analytical solution of a soft inclusion problem is derived and used to evaluate iterative numerical solutions of the associated linear algebraic system based on discretization via the finite element method, and use of an iterative conjugate gradient method with algebraic multigrid preconditioning (AMG-PCG). Accuracy and efficiency of the AMG-PCG algorithm is compared to two other conjugate gradient algorithms with diagonal preconditioning (DS-PCG) or a modified incomplete LU decomposition (Euclid-PCG) based on comparison to the analytical solution. While all three algorithms are shown to be accurate, the AMG-PCG algorithm is demonstrated to provide significant savings in CPU time as the number of nodal unknowns is increased. In contrast to the other two algorithms, the AMG-PCG algorithm also exhibits little sensitivity of CPU time and number of iterations to variations in material properties that are known to significantly affect model variables. Results demonstrate the benefits of algebraic multigrid preconditioners for the iterative solution of assembled linear systems based on finite element modeling of soft elastic inclusion problems and may be particularly advantageous for large scale problems with many nodal unknowns.

Elliptic inclusions in a stressed matrix

1963 ◽  
Vol 59 (4) ◽  
pp. 821-832 ◽  
Author(s):  
R. D. Bhargava ◽  
H. C. Radhakrishna
Keyword(s):  
Minimum Energy ◽  
Elastic Field ◽  
Theory Of Elasticity ◽  
Inclusion Problem ◽  
Elliptic Inclusion ◽  
Linear Elasticity Theory ◽  
Energy Principle ◽  
Uniform Tension ◽  
Minimum Energy Principle ◽  
The Matrix

AbstractThis paper treats an extension of the problem considered by the authors in a recent paper (1). The minimum energy principle of the classical theory of elasticity was used in the above paper for evaluating the elastic field when an elliptic region (the inclusion, which could be of a material different from the rest) undergoes spontaneous dimensional change in an otherwise unstrained infinite medium (the matrix). By modification of this method, it has been possible to deal with the case when the inclusion is spherical or circular and the matrix is under uniform tension at infinity (2). The present paper deals with the much more general case when the matrix is under tension, at infinity, inclined at any angle to the major axis of the elliptic inclusion. The solution has been possible by the combination of the complex variable method coupled with minimum energy principle and superposition methods of linear elasticity theory. As a consequence we immediately derive almost without further calculation many particular cases, viz. (i) the inclusion problem in a matrix under axial tension parallel to either of the axes, (ii) under all round uniform tension (or pressure) etc. It is obvious that the results for the respective cases of a circular inclusion can be deduced from these results.It also solves the problem of composite sections under external forces at infinity because of the complete freedom in choosing the elastic constant of the inclusion which can be different from that of the matrix. As a corollary, it solves the problem of a cavity under stress at infinity.

Quantitative stress determination in SiC reinforced Al2O3

1990 ◽  
Vol 48 (4) ◽  
pp. 190-191
Author(s):  
Bernhard Caspers ◽  
Werner Mader ◽  
Peter Angelini
Keyword(s):  
Strain Analysis ◽  
Elastic Field ◽  
Elastic Interaction ◽  
Ceramic Composites ◽  
Infinite Matrix ◽  
Free Surfaces ◽  
Elastic Boundary ◽  
The Matrix ◽  
Quantitative Stress ◽  
Elastic Boundary Conditions

Whisker and fiber reinforced ceramic composites offer the potential for increased fracture toughness and fracture strength. However, differences in thermal expansion between whisker and matrix generate residual stresses which may affect the mechanical properties of such composites. The microscopic stress can be determined by analyzing the strain contrast around SiC whiskers quantitatively, similar to the strain analysis performed on ZrO2 inclusions in AI2O3. Previously obtained experimental TEM results on strain contrast around SiC whiskers embedded in Al2O3 were only qualitatively. In the present study the experimental observations are quantitatively evaluated.Contrast simulation requires the knowledge of the matrix displacement field in the vicinity of the whisker. Due to two force-free surfaces adjacent to the inclusion, Eshelby’s method for an inclusion in an infinite matrix can not be used for the evaluation of the elastic field in a TEM foil. Therefore, the Green’s function method was applied to fulfill the elastic boundary conditions for both surfaces neglecting only the elastic interaction between the surfaces.

Elastic inclusions and inhomogeneities in transversely isotropic solids

1994 ◽  
Vol 444 (1920) ◽  
pp. 239-252 ◽  
Keyword(s):  
Closed Form ◽  
Elastic Field ◽  
Transversely Isotropic ◽  
Potential Functions ◽  
Closed Form Expression ◽  
Inclusion Problem ◽  
Stress Vector ◽  
Equivalent Inclusion Method ◽  
Transversely Isotropic Solids ◽  
Isotropic Solids

A method that introduces a new stress vector function ( the hexagonal stress vector ) is applied to obtain, in closed form, the elastic fields due to an inclusion in transversely isotropic solids. The solution is an extension of Eshelby’s solution for an ellipsoidal inclusion in isotropic solids. The Green’s functions for double forces and double forces with moment are derived and these are then used to solve the inclusion problem. The elastic field inside the inclusion is expressed in terms of the newtonian and biharmonic potential functions, which are similar to those needed for the solution in isotropic solids. Two more harmonic potential functions are introduced to express the solution outside the inclusion. The constrained strain inside the inclusion is uniform and the relation between the constrained strain and the misfit strain has the same characteristics as those of the Eshelby tensor for isotropic solids, namely, the coefficients coupling an extension to a shear or one shear to another are zero. The explicit closed form expression of this tensor is given. The inhomogeneity problem is then solved by using Eshelby’s equivalent inclusion method. The solution for the thermoelastic displacements due to thermal inhomogeneities is also presented.

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