Elastic complex analysis and its applications in fracture mechanics

The key point in this paper is the introduction of elastic analytic functions. An elastic analytic function is a function of the form u: C4C2 which is di€erentiable and satis®es equations which are analogous to the Cauchy± Riemann equations of traditional complex analysis such that the following conditions hold: ®rst the real and imaginary part of the ®rst complex component of u satisfy the Navier equation of plane elasticity, and second, the derivative along a line of the real and imaginary part of the second complex component of u is proportional to the applied tractions along that line. Algebraical operations have been de®ned on elastic analytic functions such that they constitute a commutative algebra over the real ®eld and a module over the set of analytic functions. Next, a derivative and an integral of elastic analytic functions are introduced such that they behave in a similar way to complex di€erentiation and integration of analytic functions, in particular we have properties such as: the integral of an elastic analytic function around a contour is zero, a Cauchy-like integral formula and Plemelj-like formulae. These properties can be very useful in tackling problems of plane elasticity involving cracks through the boundary element method. It is also proved that path independent integrals in plane elasticity that are derived from Noetherʹs theorem, whose integrand only depends on the position and gradient of displacements, can be written as the integral of an elastic analytic function