Análisis asintótico y aproximación de Padé en problemas de propagacion de grietas con difusión controlada

• Balueva, Alla V. ^[2] ; Germanovich, Leonid N. ^[1]
  1. [1] Georgia Institute of Technology

     Georgia Institute of Technology

     Estados Unidos

     
  2. [2] Gainesville State College, Mathematics Department
• Localización: Revista de Matemática: Teoría y Aplicaciones, ISSN 2215-3373, ISSN-e 2215-3373, Vol. 19, Nº. 2, 2012
• Idioma: español
• DOI: 10.15517/rmta.v19i2.1329
• Títulos paralelos:
  • Asymptotical analysis and Padé approximation in problems on diffusion-controlled cracks propagation
• Enlaces
  • Texto completo
• Resumen
  • español

    En este trabajo, consideramos la fractura de difución controlada axisimétrica en un espacio infinito, y en el semiespacio. Un ejemplo importante del crecimiento de una fractura de difusión controlada es dado por el hidrógeno inducido en agrietamiento. En metales, el hidrógeno es típicamente disuelto en forma de protones. Cuando los protones alcanzan la superficie de la grieta, se recombinan con electrones y forman hidrógeno molecular en la cavidad de la grieta. Entonces, la fractura puede propagar aún en ausencia de cualquier carga externa, esto es, sólo bajo presión excesiva de gas hidrógeno acumulado dentro de la grieta.Nuestros resultados muestran que en la aproximación asintótica a largo plazo (basada en la solución cuasiestática), la delaminación de difusión controlada propaga con velocidad constante. Nosotros determinamos una concentración crítica ́máxima que limita el uso de la solución cuasiestática. Una solución transitoria, que representa una aproximación asintótica de corto plazo, es usada cuando la concentración del gas excede la concentración crítica. Entonces apareamos estos dos casos usando el método de aproximaciones de Padé y presentamos soluciones en forma cerrada tanto para propagación de grietas de difusión controlada internas como cercanas a la superficie, en diferentes escalas de tiempo.Palabras clave: difusión, propagación de grietas, análisis asintótico, aproximación de Padé.Mathematics Subject Classification: 74A45, 74N25, 41A21.

  • English

    In this work, we consider the diffusion-controlled axisymmetric fracture in an infinite space, and half-space. An important example of diffusion-controlled fracture growth is given by hydrogen induced cracking. In metals, hydrogen is typically dissolved in the proton form. When protons reach the crack surface, they recombine with electrons and form molecular hydrogen in the crack cavity. Then, the fracture can propagate even in the absence of any external loading, that is, only under the excessive pressure of gas hydrogen accumulated inside the crack.Our results show that in the long-time asymptotic approximation (based on the quasi-static solution), the diffusion-controlled delamination propagates with constant velocity. We determine a maximum critical concentration that limits the use of the quasi-static solution. A transient solution, representing a short time asymptotic approximation, is used when the concentration of gas exceeds the critical concentration. We then match these two end-member cases by using the method of Padé approximations and present closed-form solutions for both internal and near-surface diffusion-controlled crack propagation at different time scales.Keywords: diffusion, crack propagation, asymptotic analysis, Padé approximation.Mathematics Subject Classification: 74A45, 74N25, 41A21.

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