Some new problems in the micromechanics

Abstract

Main fundamental subjects in the micromechanics are mathematical and mechanical formulations for inclusions and dislocations as shown in Mura [T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff, Dordrecht, 1987]. Literature surveying inclusion problems, further mathematical development and applications of the inclusion theory to material transformation and composite materials were reported in [T. Mura, Appl. Mech. Rev. 41 (1988) 15–20] and [T. Mura, H.M. Shodja, Y. Hirose, Appl. Mech. Rev. 49 (1996) 118–127]. This paper proposes some new problems associated with the fundamental mathematical mechanical formulations of the inclusion. If an important old problem has not been solved yet, the old problem is a new frontier. Eshelby conjecture states that the ellipsoidal inclusions in an isotropic elastic medium are the only kind inclusions with the remarkable property that the stress field inside the inclusion is uniform if the prescribed eigenstrain in uniform [J.D. Eshelby, Proc. R. Soc. London, Ser. A. 46 (1957) 376–396; J.D. Eshelby, Elastic inclusions and inhomogeneities, in: I.N. Snedon, R. Hill (Eds.), Progress in Solid Mechanics, vol. 2, North Holland, Amsterdam, 1961, pp. 89–140]. Recently, Mura [T. Mura, Mech. Res. Commun. 24 (1997) 473–482] found that a m-pointed polygonal inclusion subjected to the uniform eigenstrain would also produce the uniform stress field inside the inclusion, if m is an odd number. Contradictory to this result, several papers have been reported as well, for instance, Rodin [G.J. Rodin, J. Mech. Phys. Solids 44 (1996) 473–482], Nozaki and Taya [H. Nozaki, M. Taya, J. Appl. Mech. (1997) 495–502], Markenscoff [X. Markenscoff, J. Mech. Phys. Solids (1998)], and Shozu et al. [W. Shozu, K. Hirashima, Y. Hirose, Jpn. Soc. Mech. Eng. (1996) 1643–1648]. The so-called mechanics of materials should study physical properties of materials more than the elastic moduli. A new method is proposed to satisfy the need mentioned above. The Schrödinger equation is used in determination of eigenstrain [T. Mura, The eigenstrain method applied to fracture and fatigue mechanics, in: G.C. Sih, H. Nishitani, T. Ishibara (Eds.), Role of Fracture Mechanics in Modern Technology, 1987, pp. 145–152]. The semi-infinite chain of point atoms bonded longitudinally by linear bendable elements and linear stretchable elements with spacing, a 0, zero-stress separation, b 0, and spring constants, α and β, employed by Kitamura et al. [K. Kitamura, I.L. Marksimov, K. Nishioka, Phil. Mag. Lett. 75 (6) (1997) 343–350], can be considered in terms of the eigendistortion to explain the effect of temperature on the crack-lattice trapping (CLT) phenomenon.