The Mathematica Book
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Common characteristic of the structure of the crystalic or amorphous materials is the existence of micro-cracks. Reasons of the existence of micro-cracks are (a) the chemical composition of the material, (b) the kinetic mechanisms of the atoms or grains or atomic chains or dislocations or other structural elements during the solidification or forming or loading of a macroscopic material part. The dramatic effect of sub-critical crack growth on the mechanical integrity of structures was clearly brought to light by the often structural failures. As the micro-crack growth is the result of atomic bonds de-boding, the understanding of mechanisms and analysis of cracks is a useful tool for the chemistry to improve the chemical composition and the resulted micro-structure of materials in order to achieve better macroscopic fracture toughness. Due to the progress of the nanomechanics and the microelectronics, it is recognized today that the existence and growth of micro-cracks in the interface between thin layers such as thin films in electronic devices, sensors and actuators in smart materials or ceramic coating in machine components for wear or corrosion resistance, cause changes in the prescribed material properties and reduction of the material quality. In the present work, a mathematical analysis to determine the stress concentration areas in order strong particles to be inserted there (using chemical methods) for the material strengthening is attempted. To this scope, the equations describing the stress fields around interfacial micro-cracks in materials under internal singular loading sources are formulated. Numerical examples are included.