Crack Tip Interpolation, Revisited

Abstract

Abstract.It is well known that the near tip displacement field on a crack surface can be represented in a power series in the variable √ r, where r is the distance to the tip. It is shown herein that the coefficients of the linear terms on the two sides of the crack are equal. Equivalently, the linear term in the crack opening displacement vanishes. The proof is a completely general argument, valid for an arbitrary (e.g., multiple, nonplanar) crack configuration and applied boundary conditions. Moreover, the argument holds for other equations, such as Laplace. A limit procedure for calculating the surface stress in the form of a hypersingular boundary integral equation is employed to enforce the boundary conditions along the crack faces. Evaluation of the finite surface stress and examination of potentially singular terms lead to the result. Inclusion of this constraint in numerical calculations should result in a more accurate approximation of the displacement and stress fields in the tip region, and thus a more accurate evaluation of stress intensity factors.

Interactions between multiple rigid lamellae in a ductile metal matrix: Shear band magnification and attenuation in localization patterns

“…In order to provide the explicit expression for the linear system given by equations ( 39) and (40), it is instrumental to introduce the numbering of the nodes belonging to each inclusion, so that the -th inclusion node along the -th inclusion is defined by coordinates ̂ ( )[ ] 1 , ( ) 2 ( = 1, ..., ( ) , being ( ) the number of nodes of the -th inclusion). Defining…”

“…Although boundary integral approaches are particularly indicated and in wide use for problems involving singularities [8,18,33,36,40], finite elements permit large simulations, involving distributions of hundreds of inclusions, which can be investigated during an evolutive strain path in a fully elasto-damaging material [22,23]. From this point of view, finite elements appear to be superior to boundary integral techniques.…”

The dynamics of a shear band

“…Two incident waves with wavenumber Ωl/c 1 = 1 are considered, one orthogonal to the shear band (with inclination β = θ 0 + π/2) and the other aligned parallel to the x 1 −axis (with inclination β = 0). These inclinations of propagation represent the directions along which the wave velocity c assumes the minimum and maximum values respectively, see equation (37).…”

“…is the nodal value of the incremental displacement jump (Figure 2). The square root variation is adopted to take into account the singularity at the shear band tip, as is usual for the crack tip problem [37,44,45,48]. When ŷ = x(p) (p = 1, .…”

Displacement Correlation Technique for Interface Crack by FEM

“…Using the displacement correlation technique [2,4,5] the mode I and II stress intensity factor is given by;…”