SUMMARYA methodology for solving three-dimensional crack problems with geometries that are independent of the mesh is described. The method is based on the extended ÿnite element method, in which the crack discontinuity is introduced as a Heaviside step function via a partition of unity. In addition, branch functions are introduced for all elements containing the crack front. The branch functions include asymptotic near-tip ÿelds that improve the accuracy of the method. The crack geometry is described by two signed distance functions, which in turn can be deÿned by nodal values. Consequently, no explicit representation of the crack is needed. Examples for three-dimensional elastostatic problems are given and compared to analytic and benchmark solutions. The method is readily extendable to inelastic fracture problems.
SUMMARYWe present a level set method for treating the growth of non-planar three-dimensional cracks. The crack is deÿned by two almost-orthogonal level sets (signed distance functions). One of them describes the crack as a two-dimensional surface in a three-dimensional space, and the second is used to describe the one-dimensional crack front, which is the intersection of the two level sets. A Hamilton-Jacobi equation is used to update the level sets. A velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface. The technique is coupled with the extended ÿnite element method which approximates the displacement ÿeld with a discontinuous partition of unity. This displacement ÿeld is constructed directly in terms of the level sets, so the discretization by ÿnite elements requires no explicit representation of the crack surface. Numerical experiments show the robustness of the method, both in accuracy and in treating cracks with signiÿcant changes in topology.
In order to model brittle fracture, we have implemented a two and three dimensional phase-field method in the commercial finite element code Abaqus/Standard. The method is based on the rate-independent variational principle of diffuse fracture. The phase-field is a scalar variable between 0 and 1 which connects broken and unbroken regions. If its value reaches one the material is fully broken, thus both its stiffness and stress are reduced to zero. The elastic displacement and the fracture problem are decoupled and solved separately as a staggered solution. The approach does not need predefined cracks and it can simulate curvilinear fracture paths, branching and even crack coalescence. Several examples are provided to explain the advantages and disadvantages of the method. The provided source codes and the tutorials make it easy for practicing engineers and scientists to model diffuse crack propagation in a familiar computational environment.
SUMMARYThis paper proposes a generalization of the eXtended finite element method (X-FEM) to model dynamic fracture and time-dependent problems from a more general point of view, and gives a proof of the stability of the numerical scheme in the linear case. First, we study the stability conditions of Newmark-type schemes for problems with evolving discretizations. We prove that the proposed enrichment strategy satisfies these conditions and also ensures energy conservation. Using this approach, as the crack propagates, the enrichment can evolve with no occurrence of instability or uncontrolled energy transfer. Then, we present a technique based on Lagrangian conservation for the estimation of dynamic stress intensity factors for arbitrary 2D cracks. The results presented for several applications are accurate for stationary or moving cracks.
A full three dimensional study of a fatigue crack in cast iron is presented. This analysis involves combining various tools, namely, Synchrotron X-ray microtomography
Preprint submitted to Elsevier 4 October 2009of an in situ experiment, image acquisition and treatment, 3D volume correlation to measure 3D displacement fields, extraction of the crack geometry, extended digital image correlation to account for the crack displacement discontinuity, crack modeling in an elastic material exploiting the actual crack geometry, and finally estimation of stress intensity factors. All these different tasks are based on specific multiscale approaches.
3D tomographic images of a nodular graphite cast iron obtained using a laboratory X-ray source were used to analyze the opening of a fatigue crack during in-situ mechanical loading. Direct image analysis and digital image correlation are utilized to obtain the 3D morphology and front location of the crack as well as the displacement fields in the bulk of the specimen. From DIC results, it is possible to extract the Crack Opening Displacement (COD) map in the whole sample cross-section and to compute Stress Intensity Factors (SIF) all along the crack front even for COD values that are less than the image resolution. The comparison of COD maps with local values of SIF enabled for an estimation of the opening SIF, K op , equal to 6 MPa√m.
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