Thermal-stress induced phenomena in two-component material: Part II

Abstract

The paper deals with analytical models of the elastic energy gradient W sq representing an energy barrier. The energy barrier is a surface integral of the elastic energy density w q . The elastic energy density is induced by thermal stresses acting in an isotropic spherical particle (q = p) with the radius R and in a cubic cell of an isotropic matrix (q = m). The spherical particle and the matrix are components of a multi-particle-matrix system representing a model system applicable to a real two-component material of a precipitation-matrix type. The multi-particle-matrix system thus consists of periodically distributed isotropic spherical particles and an isotropic infinite matrix. The infinite matrix is imaginarily divided into identical cubic cells with a central spherical particle in each of the cubic cells. The dimension d of the cubic cell then corresponds to an inter-particle distance. The parameters R, d along with the particle volume fraction v = v(R, d) as a function of R, d represent microstructural characteristics of a real two-component material. The thermal stresses are investigated within the cubic cell, and accordingly are functions of the microstructural characteristics. The thermal stresses originate during a cooling process as a consequence of the difference α m − α p in thermal expansion coefficients between the matrix and the particle, α m and α p , respectively. The energy barrier W sq is used for the determination of the thermal-stress induced strengthening σ q . The strengthening represents resistance against compressive or tensile mechanical loading for α m − α p > 0 or α m − α p < 0, respectively.

Thermal stresses in two- and three-component anisotropic materials

“…Formulae for w q can be incorporated into those for the "curve" and "surface" energy density, W cq and W sq , determined in Refs. [2,3], respectively. On the one hand, [2,3] deal with thermal-stress induced phenomena in the two-component materials isotropic components.…”

“…[2,3], respectively. On the one hand, [2,3] deal with thermal-stress induced phenomena in the two-component materials isotropic components. On the other hand, the formulae for W cq and W sq are valid for isotropic as well as anisotropic components.…”

“…In case of [3], both mathematical procedures are applicable to such two-component materials which exhibit an elastic, elastic-plastic and plastic stress-strain state. Additionally, the mathematical procedures presented in [2,3] exhibits a general character, i.e.…”

“…Additionally, the mathematical procedures presented in [2,3] exhibits a general character, i.e. the energy density w q included in W cq and W sq can be induced by different types of stresses in the spherical particle and cell matrix.…”

“…Reference [3], concerning two-component materials of the precipitate-matrix type, presents mathematical procedures which are based on the "surface" energy density W sq .W sq represents an integral of w q over a suitable surface in the spherical particle and cell matrix. These mathematical procedures are then used for the determination of (3) An energy barrier in the cubic cell which is induced as a consequence of the presence of the spherical particle in the cubic cell, (4) The micro-strengthening σ sq = σ sq (R 1 , v) in the spherical particle (q = p) and cell matrix (q = m) and consequently the macro-strengthening σ sc = σ sc (R 1 , v) in the cubic cell.…”