The Inversion and Kelvin's Transformation in Plane Thermoelasticity with Circular or Straight Boundaries

The problem of a circular elastic inclusion perfectly bonded to a matrix of infinite extent and subjected to arbitrarily thermal loading has been solved explicitly in terms of the corresponding homogeneous problem based on the inversion and Kelvin's transformation. It is to be noted that the relations established in this paper between the stress functions are algebraic and do not involve integration or solution of some other equations. Furthermore, the transformation leading from the solution for the homogeneous problem to that for the heterogeneous one is very simple, algebraic and universal in the sense of being independent of loading considered. The case of two bonded half-planes is obtained as a limiting case.

Sensitivity of the macroscopic response of elastic microstructures to the insertion of inclusions

This paper proposes an exact analytical formula for the topological sensitivity of the macroscopic response of elastic microstructures to the insertion of circular inclusions. The macroscopic response is assumed to be predicted by a well-established multi-scale constitutive theory where the macroscopic strain and stress tensors are defined as volume averages of their microscopic counterpart fields over a representative volume element (RVE) of material. The proposed formula—a symmetric fourth-order tensor field over the RVE domain—is a topological derivative which measures how the macroscopic elasticity tensor changes when an infinitesimal circular elastic inclusion is introduced within the RVE. In the limits, when the inclusion/matrix phase contrast ratio tends to zero and infinity, the sensitivities to the insertion of a hole and a rigid inclusion, respectively, are rigorously obtained. The derivation relies on the topological asymptotic analysis of the predicted macroscopic elasticity and is presented in detail. The derived fundamental formula is of interest to many areas of applied and computational mechanics. To illustrate its potential applicability, a simple finite element-based example is presented where the topological derivative information is used to automatically generate a bi-material microstructure to meet pre-specified macroscopic properties.

On Partially Debonded Circular Inclusions in Finite Plane Elastostatics of Harmonic Materials

We investigate a partially debonded circular elastic inclusion embedded in a particular class of harmonic materials by using the complex variable method under finite plane-strain deformations. A complete (or full-field) solution is derived. It is observed that the stresses in general exhibit oscillatory singularities near the two tips of the arc shaped interface crack. Particularly the traditional inverse square root singularity for stresses is observed when the asymptotic behavior of the harmonic materials obeys a constitutive restriction proposed by Knowles and Sternberg (1975, “On the Singularity Induced by Certain Mixed Boundary Conditions in Linearized and Nonlinear Elastostatics,” Int. J. Solids Struct., 11, pp. 1173–1201). It is also found that the number of admissible states under finite plane deformations for given external loads can be two, one, or even zero.

Stresses in a Spherical Shell With a Circular Elastic Inclusion

The problem of a circular elastic inclusion in a thin pressurized spherical shell is considered. Using Reissner’s differential equations governing the behavior of a thin shallow spherical shell, the solutions for the two regions are obtained in terms of Bessel and Hankel functions. Particular cases of a rigid circular inclusion free to move with the shell and a clamped rigid circular inclusion are also considered. Results are presented in nondimensional form which will greatly facilitate their use in the design of spherical shells containing a rigid or an elastic inclusion.

An Arc Crack Around a Circular Elastic Inclusion

The problem of a circular elastic inclusion bonded to a different elastic material except over an arc crack is considered. It is found that the solution yields an oscillation phenomenon near the ends of the crack of a form found previously by several authors, indicating that linear elasticity may not be used to predict the stresses and displacements in these regions.