Based on the fundamental equations of the mechanics of solid continuum, the paper employs an analytical model for determination of elastic thermal stresses in isotropic continuum represented by periodically distributed spherical particles with different distributions in an infinite matrix, imaginarily divided into identical cells with dimensions equal to inter-particle distances, containing a central spherical particle with or without a spherical envelope on the particle surface. Consequently, the multi-particle-(envelope)-matrix system, as a model system regarding the analytical modelling, is applicable to four types of multi-phase materials. As functions of the particle volume fraction v, the inter-particle distances d 1 , d 2 , d 3 along three mutually perpendicular axes, and the particle and envelope radii, R 1 and R 2 , respectively, the thermal stresses within the cell, are originated during a cooling process as a consequence of the difference in thermal expansion coefficients of phases represented by the matrix, envelope and particle. Analytical-(experimental)-computational lifetime prediction methods for multi-phase materials are proposed, which can be used in engineering with appropriate values of parameters of real multi-phase materials.

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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.

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.