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摘要
The paper presents four topics dealing with phenomena induced by elastic thermal stresses acting in an isotropic multi-particle-matrix system to represent a model system applicable to real multi-phase materials of a precipitation-matrix type. The isotropic multi-particle-matrix system consists of periodically distributed spherical particles in an infinite matrix to be imaginarily divided into cubic cells containing a central spherical particle. Formulae for the thermal stresses to be investigated within the cubic cell represents functions of the particle volume fraction v and the particle radius R. The thermal stresses originate during a cooling process as a consequence of the difference alpha(m) - alpha(p) in the thermal expansion coefficients alpha(m) and alpha(p) of the matrix and the particle, respectively. Additionally, such temperature range is considered within which the multi-particle-matrix system exhibits elastic deformations regarding the yield stress and the particle-matrix boundary adhesion strength. Analytical fracture mechanics to represent the first topic of this paper results from the determination of a curve integral of the thermal-stress induced elastic energy density. The curve elastic energy density results in the determination of the critical particle radii R(pc) and R(mc) as reasons of the crack initiation in the spherical particle and the matrix for alpha(m) - alpha(p) < 0 and alpha(m) - alpha(p) > 0, respectively. Consequently, the crack propagation to follow the crack initiation is a consequence of the particle radius R > R(qc) (q = p , m). Finally, a shape of the crack in a plane perpendicular to the direction of the crack propagation in the particle ( q = p) and the matrix ( q = p) is described by the function f(q) related to the ideal-brittle components. With regard to the crack initiation, the analytical determination of the radius Rqc is considered for any multi-phase materials of a precipitation-matrix type. With regard to the crack propagation, the analytical determination of the function fq along with an analysis concerning the crack dimension and directions of the crack propagation is considered for ceramic multi-phase materials which are generally assumed to be ideal-brittle. The thermal stresses induce resistance against compressive or tensile mechanical loading for alpha(m) - alpha(p) > 0 or alpha(m) - alpha(p) < 0, respectively. The analytical determination of the resistance results from the elastic energy gradient to represent the second topic of this paper. Derived by two equivalent mathematical techniques, the gradient within the cubic cell is defined as a surface integral of the thermal-stress induced elastic energy density. Consequently the 'surface' elastic energy density results in the analytical determination of the system strengthening to represent the third topic of this paper. Representing the fourth topic of this paper, an analytical model of stresses originating in isotropic crystalline lattices are derived. The stresses in the lattices are a consequence of the presence of a central substitutive atom. Additionally, elastic energy, induced by the substitute atom and accumulated in the lattices, is also derived. Finally, readers can substitute numerical values of parameters of real multi-phase materials into the presented formulae.
- 出版日期2008
