Inclusions With and Without Free Surfaces in a Plane Strain
The stress field and strain energy of inclusions with and without free surfaces in a plane strain condition are studied via an atomistic approach, in which a triangular lattice is considered with linear atomic interactions. The result of a polygonal inclusion in an infinite matrix shows that the stress concentrations at the polygonal vertices increase as the ratio of inclusion shear modulus to matrix shear modulus, µ*/µ, increases. The strain energy is nearly proportional to µ*/(2µ* + µ), which is the proportionality constant for an elliptic inclusion with a dilatational eigenstrain. For an undergrowth whose one side is free of traction, a maximum principal stress analysis reveals stress concentrations in a tensile mode at its traction-free side, indicating a possibility of crack initiation for a brittle undergrowth. For an overgrowth whose three sides are free of traction, a number of interesting results are found. Unlike an undergrowth or inclusion, the stress is mostly confined in the neighborhood of the overgrowth—substrate interface. For a positive eigenstrain, the stress field starts with a strong compressive state at the interface area and diminishes as the distance from the interface increases. However, nearby the substrate, the traction-free perimter of an overgrowth is also found to be in a tensile stress mode. Quite different from the case of inclusions and undergrowths is that the total strain energy associated with an overgrowth approaches an asymptotic value as its thickness increases. That is, the strain energy density of an overgrowth is found to depend on its shape and also on its size.
KeywordsStress Field Strain Energy Density Triangular Lattice Maximum Principal Stress Circular Inclusion
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