Despite extensive study of the Eshelby's problem for inclusions of simple shape, little effort has been made to inclusions of arbitrary shape. In this paper, with aid of the techniques of analytical continuation and conformal mapping, a novel method is presented to obtain analytic solution for the Eshelby's problem of an inclusion of arbitrary shape in a plane or a half-plane. The boundary of the inclusion is characterized by a conformal mapping which maps the exterior of the inclusion onto the exterior of the unit circle. However, the boundary value problem is studied in the physical plane rather than in the image plane. The conformal mapping is used to construct an auxiliary function with which the technique of analytic continuation can be applied to the inclusion of arbitrary shape. The solution obtained by the present method is exact, provided that the expansion of the mapping function includes only a finite number of terms. On the other hand, if the exact mapping function includes infinite terms, a truncated polynomial mapping function should be used and then the method gives an approximate solution. In particular, this method leads to simple elementary expressions for the internal stresses within the inclusion in an entire plane. Several examples of practical interest are discussed to illustrate the method and its efficiency. Compared to other existing approaches for the two-dimensional Eshelby's problem, the present method is remarked by its elementary characters and applicability to inclusions of arbitrary shape in a plane or a half-plane.

Analytic Solution for Eshelby's Problem of an Inclusion of Arbitrary Shape in a Plane or Half-Plane

Time-dependent changes in myosin heavy chain (MHC) isoform expression were investigated in rat soleus muscle unloaded by hindlimb suspension. Changes at the mRNA level were measured by RT-PCR and c...

https://www.physiology.org/doi/pdf/10.1152/ajpcell.1999.277.6.C1044

Time-dependent changes in myosin heavy chain mRNA and protein isoforms in unloaded soleus muscle of rat.

To investigate the plasticity of slow and fast muscles undergoing slow-to-fast transition, rat soleus (SOL), gastrocnemius (GAS), and extensor digitorum longus (EDL) muscles were exposed for 14 day...

https://www.physiology.org/doi/pdf/10.1152/ajpcell.2000.279.5.C1558

Effects of unweighting and clenbuterol on myosin light and heavy chains in fast and slow muscles of rat

Thermal stress‐induced voiding in narrow aluminum‐based metallizations used as interconnects in microelectronic circuits has recently become a serious reliability concern. Room‐temperature stress relaxation and associated physical phenomena in passivated and unpassivated aluminum‐based metallizations, subsequent to exposure to high temperatures, are analyzed based both on theoretically estimated and experimentally determined thermal stresses. It is shown that stress relaxation at longer times involves mainly dislocation climb, while short‐term relaxation during cool down from higher temperatures, and immediately thereafter, involves significant dislocation glide. Void growth, frequently observed in passivated metallizations, provides a new source of atoms to feed stress relaxation by the same processes as in the absence of voiding.

Mechanisms of thermal stress relaxation and stress‐induced voiding in narrow aluminum‐based metallizations

Identification of O-linked N-Acetylglucosamine Proteins in Rat Skeletal Muscle Using Two-dimensional Gel Electrophoresis and Mass Spectrometry

Thermal stress‐induced void formation in aluminum interconnect has become a major reliability problem in the usage of very large scale integration circuits. The purpose of this work is to analytically evaluate stresses in the Al line. By applying Eshelby’s method in micromechanics, the stresses in the Al line were estimated analytically as a function of the aspect ratio of the Al line cross section. The yielding criteria in plasticity were applied to examine whether the calculated stresses can induce relaxation by plastic deformation. The analytically calculated results were compared with previous results of numerical calculation and experimental observation.

Stress distribution in an aluminum interconnect of very large scale integration

Using the previously obtained stress distributions in an Al line after relaxation by plastic deformation, another possible relaxation process by diffusion was analyzed. Even after this relaxation occurs, some stresses still remain in the Al line. If these remaining stresses are large enough, they can be responsible for the growth of voids causing line failure. Using a theory for diffusional growth of grain‐boundary voids, the time to failure of the Al line was estimated analytically. The previous and present papers together constitute a full analysis of the so‐called ‘‘stress migration’’ phenomenon.

Masaharu Kato

Papers

Stress distribution in an aluminum interconnect of very large scale integration

Diffusional relaxation and void growth in an aluminum interconnect of very large scale integration