Since Mandelbrot proposed the concept of fractal in 1970s’, fractal has been applied in various areas such as science, economics, cultures and arts because of the universality of fractal phenomena. It provides a new analytical tool to reveal the complexity of the real world. Nowadays the calculus in a fractal space becomes a hot topic in the world. Based on the established definitions of fractal derivative and fractal integral, the fundamental theorems of fractal derivatives and fractal integrals are investigated in detail. The fractal double integral and fractal triple integral are discussed and the variational principle in fractal space has been proved. Moreover, a local fractional continuous random variable is proposed. The local fractional Hamilton operator is introduced in a fractal orthogonal system and the interaction between the local fractional Hamilton operator and fractal fields is discussed. Then the Green’s theorem, the Stokes’ theorem and the Gauss’s theorem are proved in a fractal space. In regard to mechanical applications of fractal mathematics, by applying the fractal calculus and the theory of fractal field, the governing equations in fractal elasticity are studied and principle of virtual work in fractal elasticity is proved; Applying the local fractional extreme value theorem in fractal space, the criterion of fractal Griffith’s crack propagation dependent on crack size is investigated. J-integral in fractal space is introduced and, using fractal Green’s theorem, the conservation of the J-integral is proved in detail. With the help of fractal statistical method, the size-scale statistical strength of brittle material is discussed in fractal medium.

- fractal calculus,
- virtual work principle,
- fractal Griffith’s crack,
- J-integral

- Analysis,
- Control Theory,
- Dynamic Systems,
- Engineering Physics,
- Non-linear Dynamics,
- Numerical Analysis and Computation,
- Ordinary Differential Equations and Applied Dynamics,
- Other Applied Mathematics,
- Other Mathematics,
- Other Physical Sciences and Mathematics,
- Partial Differential Equations and
- Theory and Algorithms