서강대학교

초록/요약

Indentation is a versatile, fast, inexpensive and therefore often-applied method for material property evaluation. However, material deformation below the indenter is complex and analytical models are based on quite a number of assumptions. For the investigation of problems of such complexity, the (extended) finite element method [(X)FEM] represents an ideal tool. In this work, we focus on two parameters that are crucial for design, (i) residual stresses (RS) and (ii) fracture toughness Kc. In Part I, we develop a method for estimating in-plane biaxial RS. Despite abundant research, no universal method has been developed that encompasses tensile and compressive biaxial RS. The Knoop indenter with its large aspect-ratio proves to be a good choice here because the curvature of the load-depth (P-h) curve, C, is sensitive not only to the magnitude but also to the orientation of RS. Having obtained maximum and minimum C values for a wide range of materials, we find that the RS ratio can be estimated from the C ratio independent of magnitude and sign of RS. Following Lee et al. (2010), C values for the non-equibiaxial RS case are converted to equivalent C values for two equibiaxial RS cases. The magnitude of RS is then determined from the converted equibiaxial RS cases. An algorithm is developed that allows the extraction of biaxial RS components from C. Finally the method is validated by comparison with results from indentation of bended cross-shaped steel specimens. Sharp indentation of brittle materials is accompanied by radial-median cracking. The length of these cracks c can be used to derive the fracture toughness. Knoop indentation has the merit that only one large crack is produced. This and the shallow plastic zone make it the preferred choice for crack growth experiments, which however require information on the crack aspect-ratio rho. In Part II, we establish a way to determine Kc by Knoop indentation and depth cz of so-induced cracks. This work represents the first numerical treatment of Knoop indentation cracking with XFEM. Results show that the point-load assumption holds for sufficiently high loads, which means that the equation for Kc by Lawn et al. (1980a) (LEM equation) is applicable. During loading the crack depth cz is found to evolve according to h proportional to cz^3/4. For well-developed cracks, rho (≡ cz/c) is load-independent. Through comparison with experimental results the XFE model is shown to predict crack evolution properly. Based on parameter studies, we establish mapping functions for determining Kc and rho from load Pmax, c and material properties. It is demonstrated that the mapping functions work well. Part III addresses the evaluation of equibiaxial RS in brittle materials by Vickers indentation fracture. Previous analytical models were established on the assumptions that (i) the crack is of semi-circular shape and (ii) that the shape is not affected by RS. A new analytical model that accounts for the crack shape and its change is presented. XFE results reveal that tensile and compressive RS are fundamentally different; the crack shape is generally not semi-circular and affected by RS. Parameter studies are performed to calibrate the proposed analytical model. Comparison of results calculated by the analytical models with XFE results reveals the inaccuracy inherent in the previous analytical models, namely the neglect of (the change of) the crack aspect-ratio, in particular for tensile RS. These models should therefore be treated with caution and, if at all, only used for compressive RS. The new model, on the other hand, gives a much more accurate description of the RS but requires the crack depth. In his work on indentation cracking, Hyun (2011) treated monocrystalline silicon as an isotropic material, despite its inherent anisotropy. In Part IV, we therefore propose a (simplified) material model that allows the simulation of anisotropic cracking in Vickers indentation of silicon. As cracking takes place exclusively in the elastic zone, anisotropy is confined to the elastic domain; the complex plastic deformation is considered through an effective yield strength, into which diverse kinds of irreversible deformations are merged. Numerically obtained hardness values agree well with literature values. The LEM equation is modified to account for the anisotropy of cubic materials. The variation of the crack size for diverse orientations on Si(001), Si(110) and Si(111) is analyzed. Numerical results are found to be in a generally good agreement with the experimental results by Ebrahimi and Kalwani (1999). Considering that cubic materials are quite similar in their anisotropy, the model can be applied to other similarly structured materials such as germanium.