Statistical Method for Tracing Hydraulic Fracture Front Without Evaluation of the Normal

In numerical simulation of hydraulic fracture propagation, tangent component of the fluid velocity generally considered to be neglected near the crack front. Then Reynolds transport theorem yields that the limit of the particle velocity coincides with the vector of the front propagation speed. We use this fact in combination with the Poiseuille-type equation, which implies that the particle velocity is always collinear to pressure gradient. We show that this specific feature of the hydraulic fracture problem may serve to simplify tracing the front propagation. The latter may be traced without explicit evaluation of the normal to the front, which is needed in conventional applications of the theory of propagating interfaces. Numerical experiments confirm that, despite huge errors in pressure and even greater errors in its gradient, the propagation speed, statistically averaged over a distance of a mesh size, is found quite accurate. We conclude that suggested method may simplify numerical simulation of hydraulic fractures driven by Newtonian and non-Newtonian fluids.

A NEW ALGORITHM FOR COMPUTING ONE-DIMENSIONAL STABLE AND UNSTABLE MANIFOLDS OF MAPS

A new algorithm is presented to compute one-dimensional stable and unstable manifolds of fixed points for both two-dimensional and higher dimensional diffeomorphism maps. When computing the stable manifold, the algorithm does not require the explicit expression of the inverse map. The global manifold is grown from a local manifold and one point is added at each step. The new point is located with a "prediction and correction" scheme, which avoids searching the computed part of the manifold with a bisection method and accelerates the searching process. By using the fact that the Jacobian transports derivatives along the orbit of the manifold, the tangent component of the manifold is determined and a new accuracy criterion is proposed to check whether the new point that defines the manifold is acceptable. The performance of the algorithm is demonstrated with several numerical examples.

Fatigue Property of Steel Q235 Based on Metal Magnetic Memory Testing

Fatigue damage is the most common kind of failure for metal materials. Traditional non-destructive testing (NDT) methods fail to predict early fatigue damage. Magnetic Metal Memory (MMM) Testing, a new NDT technology, has the potential of fatigue damage evaluation. This paper aims to investigate fatigue mechanical property of ferromagnetic material with the MMM method. Steel Q235 samples with different depth and breadth artificial crack are tested in rotary bending fatigue machinery. The MMM fatigue features are as follows: the tangent component Hp(x) of magnetic induction intensity is gradually increasing up to the peak value while the fatigue crack is initiating and growing; after that peak, Hp(x) is descending with the crack deepening. Furthermore, the features have been studied theoretically.