Thermal-stress induced phenomena in two-component material: part I

Abstract

The paper deals with analytical fracture mechanics to consider elastic thermal stresses acting in an isotropic multi-particle-matrix system. The multi-particle-matrix system consists of periodically distributed spherical particles in an infinite matrix. 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 multi-particle-matrix system thus represents a model system applicable to a real two-component material of a precipitation-matrix type. The infinite matrix is imaginarily divided into identical cubic cells. Each of the cubic cells with the dimension d contains a central spherical particle with the radius R, where d thus corresponds to 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 twocomponent material. The thermal stresses are investigated within the cubic cell, and accordingly are functions of the microstructural characteristics. The analytical fracture mechanics includes an analytical analysis of the crack initiation and consequently the crack propagation both considered for the spherical particle (q = p) and the cell matrix (q = m). The analytical analysis is based on the determination of the curve integral W cq of the thermal-stress induced elastic energy density w q . The crack initiation is represented by the determination of the critical particle radius R qc = R qc (v). Formulae for R qc are valid for any two-component material of a precipitate-matrix type. The crack propagation for R > R qc is represented by the determination of the function f q describing a shape of the crack in a plane perpendicular L. Ceniga (B) to a plane of the crack propagation. The functions f p and f m are valid for an ideal-brittle particle and an ideal-brittle matrix, i.e. for the multi-particle-matrix system consisted of ceramic particles and ceramic matrix, 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.…”

“…This determination is then based on analytical results of this paper and Refs. [2,3]. (4) The determination of the radial stress p 2 = p 2 (ϕ, ν, R 1 , t) [see Eq.…”

“…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 [2], concerning two-component materials of the precipitate-matrix type, presents mathematical procedures which are based on the "curve" energy density W cq .W cq represents an integral of the energy density w q along a suitable curve in the spherical particle (q = p) and cell matrix (q = m). These mathematical procedures are then used for the determination of (1) The critical particle radius R 1qc which is a reason of a crack initiation in the spherical particle or the cell matrix.…”