Investigation on Indentation Cracking-Based Approaches for Residual Stress Evaluation

Abstract

Vickers indentation fracture can be used to estimate equibiaxial residual stresses (RS) in brittle materials. Previous, conceptually-equal, analytical models were established on the assumptions that (i) the crack be of a semi-circular shape and (ii) that the shape not be affected by RS. A generalized analytical model that accounts for the crack shape and its change is presented. To assess these analytical models and to gain detailed insight into the crack evolution, an extended finite element (XFE) model is established. XFE analysis results show that the crack shape is generally not semi-circular and affected by RS and that tensile and compressive RS have different effects on the crack evolution. Parameter studies are performed to calibrate the generalized analytical model. Comparison of the 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. Previous models should therefore be treated with caution and, if at all, used only for compressive RS. The generalized model, on the other hand, gives a more accurate description of the RS, but requires the crack depth.

Keywords: extended finite element analysis; fracture toughness; indentation fracture; residual stress.

Figure 1

Semi-elliptical surface crack in a semi-infinite medium subject to remote tensile stress σ^∞ (top) and the definition of the parametric angle (bottom) (following Anderson [41]).

Figure 2

Residual force P_R superimposes on wedging forces P_⏊, both acting normal to the crack surface A_crack.

Figure 3

Change of shape factor Y with ω for (a) diverse ρ (Equation (2)) and (b) with ρ at Points A and B; for ρ = 0.826, Y has equal values at A and B.

Figure 4

Quarter FE model for evaluation of residual stresses (RS) in brittle materials by Vickers indentation cracking.

Figure 5

Relative change of Kick’s law coefficient C/C_o, crack length c/c_o, crack depth c_z/c_zo and crack aspect-ratio ρ/ρ_o with σ_R^FE (FE input).

Figure 6

Influence of compressive and tensile RS on crack evolution at load reversal (semi-transparent lines) and after unloading.

Figure 7

Change of χ^app (a); χ_z^app and ρ (b) with h_max for σ_R = 0.1, 0 and −0.1 GPa.

Figure 8

Coefficient k in Equation (19) vs. σ_R.

Figure 9

Normalized crack ratio ρ/ρ_o for all combinations of E = {200, 600} GPa, ν = {0.1, 0.3} and σ_y = {3, 5, 8} GPa (note that not for all materials have radial-median cracks formed).

Figure 10

Mean average values of k/χ over the whole range of equibiaxial RS states vs. E; h_max = 1.5 (a) and 2.0 µm (b); data points for ν = 0.1 and 0.3 are plotted slightly left and right, respectively, of their real location for better visibility.

Figure 11

Comparison of RS calculated by Equation (19) (σ_R^Eq) with FE input values (σ_R^FE) for materials with different Γ; E = 200 GPa, ν = 0.3, σ_y = 5 GPa; h_max = 1.5 (a) and 2.0 µm (b) and E = 600 GPa, ν = 0.1, σ_y = 8 GPa, h_max = 1.5 µm (c).

Figure 12

Deviations (=(σ_R^Eq – σ_R^FE)/σ_R^FE) of results calculated by ‘simple’ (a); corrected ‘simple’ (b) and ‘generalized’ analytical models (c) from FE input RS.