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Mechanical analysis of the interaction between a semi-infinite Griffith crack and its coplanar plastic zone

Michot, Gérard

European journal of mechanics, A, Solids, 2023-07, Vol.100, p.104982, Article 104982 [Periódico revisado por pares]

Elsevier Masson SAS

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  • Título:
    Mechanical analysis of the interaction between a semi-infinite Griffith crack and its coplanar plastic zone
  • Autor: Michot, Gérard
  • Assuntos: Continuous dislocation distribution ; Crack ; Dislocation nucleation ; Dislocations ; Image distribution ; Pile-up of dislocations ; Plastic zone ; Shielding ; Stress analysis
  • É parte de: European journal of mechanics, A, Solids, 2023-07, Vol.100, p.104982, Article 104982
  • Descrição: This paper presents the complete analytical solution of the equilibrium conditions between a dislocation pile-up and a loaded crack. Contrary to previous models, the image distributions (forces) are included in this calculation. Their influence on the effective stress intensity factor, on the dislocation nucleation conditions and on the criteria for crack propagation are underlined. In semi-brittle materials the size of the dislocation free zone is controlled by material characteristics, as expected from the literature. However, in ductile materials its size is controlled by a mathematical limitation resulting from introduction of image forces. If plastic deformation takes place, the Griffith criteria must be modified by adding additional work required to separate dislocations from their images to the surface energy and the plastic work done to develop the plastic zone. A detailed analysis of the stress distribution inside and outside the pile-up is presented. Within the plastic zone the total stress is not constant while outside and close to the limits a square root variation with distance is noticed. Cet article présente la solution analytique complète des conditions d’équilibre d’un empilement de dislocations en présence d’une fissure chargée. Les effets des distributions (forces) image sont inclus, contrairement aux modèles disponibles dans la littérature. Leur influence sur le facteur d’intensité des contraintes effectif, sur les conditions de nucléation des dislocations et sur le critère de propagation de la fissure est soulignée. L’hypothèse d’une enclave élastique dont la taille est fixée par les caractéristiques physiques de la matière n’est valable que pour les matériaux semifragiles ; elle est limitée par une condition mathématique induite par les forces images dans le cas des matériaux ductiles. En présence de déformation plastique, le critère de Griffith doit être modifié par l’adjonction à l’énergie de surface et au travail plastique d’un travail additionnel effectué lors de la séparation des dislocations et de leurs images. Une analyse détaillée de la répartition des contraintes dans et hors de l’empilement est présentée. A l’intérieur de la zone plastique la contrainte totale n’est pas constante. A l’extérieur, à proximité de ses limites, elle varie comme la racine carrée de la distance This paper presents the complete solution of the distribution function which describes the equilibrium of N screw dislocations ahead and coplanar with a semi-infinite Griffith crack, extending from A1 to A2, loaded under a stress intensity facor K, and taking into account their image distribution within the crack.•K and N values must be given in order to determine the two unknowns A1 and A2 (or p2 = 1- A1/A2 and A2).•There is a maximum value, pc of p, beyond which the equilibrium equation (2) is no longer verified, introducing a mathematical limitation.•For this value the effective Ke, is minimum, shielding is maximum, the PZ cannot accommodate any more dislocation; in that situation, all the unknown parameters (a1, a2, and N) are fixed by the K value alone.•If the threshold value for dislocation nucleation, Kn, is larger than Kemin then the DFZ size is controlled by material characteristics as proposed by Chang et Ohr, 1981.•If not, the DFZ size is controlled by the mathematical limitation induced by the image term in (2), and is always finite. This result differs substantially from the BCS model and the conclusions of Chang and Ohr.•The Griffith criteria can be replaced by an effective surface energy, which includes the true surface energy, the plastic work spent to build the PZ and an additional work required to separate dislocations from their images.•A r −1/2 stress singularity is noticed close to the crack tip, whose amplitude is Ke.•A square root variation with distance of total stresses is observed outside the PZ, close to A1 and A2, as expected for a free limits pile-up.•The total stress is not constant within the PZ and varies as:∑t=τ(1+A0/X)where A0/X, comes directly from the image term; the total stress field ahead of the crack tip is continuous.
  • Editor: Elsevier Masson SAS
  • Idioma: Inglês
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  • Realização: ABCD-USP USP   Logos de Redes Sociais: Freepik

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