Theoretical and numerical analysis on 3-dimensional brittle fracture

In this paper, we propose a mathematical formulation of 3-dimensional quasi-static brittle fracture under varying loads. We give a precise formulation of internal cracking and surface cracking in 3D elastic bodies. In Sections 2 and 3, we provide geometrical results and results on Sobolev spaces. Most criteria in fracture mechanics are based on Griffith's energy balance theory, which is explained briefly in Section 5. GJ-integral is proposed by the author (1981), which is a generalization of J-integral widely used in 2D fracture problems. GJ-integral expresses the variation of energies with respect to crack extensions and relates to Griffith's energy balance theory. Under varying loads, we cannot use Griffith's energy balance theory directly, however GJ-integral is applicable to such cases too. For practical use, we must study the combination with numerical calculation. In the last section, we give an error estimate for finite element approximation of GJ-integral.