13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 2. Modelling 2.1 Geometry To determine the fracture toughness standardized test geometries are mainly used in accordance with ASTM standards, such as the three-point bending test (3PB) or the compact tensile specimen (CT). However, it should be noted that the stress intensity factor K depends on the specimen geometry as well as the respective crack opening mode. Mode I, the opening mode, represent the most important type of crack opening which is characterized by a tensile stress normal to the crack plane. By considering the critical case under mode I and plane strain conditions, the fracture toughness KIc can be determined at the beginning of the crack extension according to ASTM standard E399-90 [7]. As in the present work fracture is studied at the micro scale, the rules of the ASTM standard do not hold anymore. Therefore a new geometry had to be developed. To ensure efficient and rapid manufacturing of the tungsten micro cantilevers, a geometry based on the specifications of the standard ASTM sample is chosen. This geometry is shown in Fig. 1. Its width W is 55 µm with a thickness B of 28 µm and a crack length a of 15 µm. The proportions between width W, thickness B and crack length a are identical to the ones of the standard ASTM samples. This is to ensure the relationship between macro and micro scale. The developed sample geometry allows the analysis of micro-specific effects of notches and multiaxial load conditions. Due to the miniaturized geometry size-effects occur, which is caused by proportions of the plastic zone in front of the crack tip as well as the changed sample ratio of surface to volume. Therefore it is not possible to transfer known macroscopic material properties into the micro scale. Based on this fact an experimental programme is carried out (N. J. Schmitt) and combined with a numerical analysis to determine the necessary characteristics at the micro scale (100 to 300 µm). 2.2 Finite element model The notched micro cantilever is represented by a three dimensional finite element (FE) model shown in Fig. 2. Due to the symmetry only one half of the specimen is modeled and meshed with 8-node brick elements with linear function (C3D8). Besides the symmetry conditions fixed boundary conditions are applied at the right end (see Fig. 2). The indenter is modeled as rigid body. Its movement in z-direction is prescribed while the indenter cannot move in x-direction (lateral direction) and in y-direction. The developed model is implemented in the finite element code ABAQUS [8]. In the first simulations purely linear elasticity is applied as constitutive law. In later simulations crystal plasticity is added. 2.3 Crystal plasticity as constitutive law As plastic behavior can be observed at the crack tip, crystal plasticity is implemented in the FE model as a plastic constitutive law as it allows specifying the crystal orientation of the tungsten single crystal. The theory of crystal plasticity is based on the assumption that plastic deformation (crystalline slip) results as the sum of all activated slip systems. Schmid (1931) [9] found, that the resolved shear stress onto a crystallographic plane leads to plastic slip, if stress reaches a critical value. This resolved stress on a slip system, which is also called the Schmid stress is assumed for this constitutive law as the only driving force for slip. The exact theory was formulated by Hill and Rice (1972) [10]. First FE studies of single crystals have been carried by Peirce, Asaro and Needleman (1982) [11]. The rate dependent plastic constitutive law was formulated by Asaro [12] and written by Huang [13] as a user-material subroutine UMAT. This UMAT is used in the presented simulations. It allows investigating the influence of the crystal orientation on the stress intensity factor. Two slip system families are taken into account, namely the {110}<111> and the {112}<111>.
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