Application of numerical methods to dislocation modeling of creep
Three computer simulations of dislocation mechanics models of fundamental aspects of creep predict certain creep behaviors of solids. The first two simulate the dynamics of infinite straight dislocations in two models of creep proposed by Weertmann as the solution of a set of coupled ordinary differential equations describing the x and y velocities of the dislocations as functions of their locations, applied stress, and glide and climb mobilities. Dual-order, error-estimating, adaptive stepsize Runge-Kutta methods of Verner and of Fehlberg are used to solve the systems of ODE's. Predictions for the behavior of the stress exponent as a function of stress and model geometric parameters differ from predictions obtained using original models. Stress exponents on the order of 5 to 6 are found to occur without assigning a stress dependence to microstructural parameters. The third simulation is of a subgrain (tilt) boundary modeled as a set of straight parallel vertically aligned edge dislocations which bow out under applied shear stress in a direction perpendicular to the plane of the boundary. The curved dislocations of the boundary are modeled as assemblies of finite straight segments using the expressions of Hirth and Lothe for stress fields of such segments. Equilibrium configurations of the boundary under increasing applied stress are found by minimizing the sum of the squares of the glide shear stress at each segment endpoint over the boundary configuration. Minimization is done with a multi-dimensional quasi-Newton secant method which employs the rank-1 update of Broyden. Stress is incremented until no equilibrium configuration can be found in order to find the maximum stress at which the boundary is stable (Orowan stress for the boundary). Stresses exerted within the subgrain by the stable boundary configurations are calculated, and this backstress is related to the magnitude of the applied stress. Backstress within the subgrain is concluded to be nearly linear in the applied stress, but may increase or decrease with applied stress at differing subgrain boundary tilt angles.
Engineering, Materials science, Metallurgy, Mechanical engineering
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