Application of numerical methods for integrating differential equations of crack evolution models to the range of constant tension
Lucas Gimenis MOURA1; Cláudio Ávila1; Antônio Kozlik Junior1; Waldir Mariano Machado Junior1
1 Federal Technological University of Paraná
doi:10.20906/CPS/CILAMCE2015-0357
Resumo
A realistic approach to structures and components used in engineering should consider the existence of cracks. The presence of cracks in structures, or mechanical components, is usually associated with the existence of the phenomenon of fatigue. The propagation of a crack is strongly influenced by the state of stress in its vicinity. In this work were obtained approximated numerical solutions through the integration of differential equations describing the crack evolution laws of Paris-Erdogan type, Forman, Priddle and McEvily. These crack evolution laws can be formulated with the use of numerical methods of Euler, implicit and explicit, and explicit fourth-order Runge-Kutta (RK4). The equation presented by Paris sets an initial value problem (IVP), defined by a nonlinear and autonomous ordinary differential equation (ODE) of first order. More generally, the equation can be placed as a Cauchy problem that consists in determining the paths (functions) that satisfy the differential equation, passing through the point set in the initial condition ODE is separable, so the direct integrating method can be used for solutions to the equation. Nonetheless, generally, the definition of the function "geometric correction factor" prevents the explicit determination of function "crack size". Thus, it is possible to obtain numerical solution for the ODE, only, through the use of numerical methods. The use of a computing environment to assist the use of numerical methods makes it possible to generate graphics that show the evolution of a determined crack as the number of cycles increases. Small differences between the results of different methods are expected because each method works with a different degree of accuracy.
Palavras-chave: Paris-Erdogan's Law; Linear elastic fracture mechanics; Initial Value Problem