The equivalence of axisymmetric indentation model for three-dimensional indentation hardness

  • PDF / 937,280 Bytes
  • 8 Pages / 584.957 x 782.986 pts Page_size
  • 12 Downloads / 258 Views

DOWNLOAD

REPORT


Y. Huanga) Department of Civil and Environmental Engineering and Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208

J. Xiao Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208

K.C. Hwang Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China (Received 30 July 2008; accepted 3 October 2008)

Nix and Gao [J. Mech. Phys. Solids 46, 411 (1998)] established an important relation between the microindentation hardness and indentation depth for axisymmetric indenters. We use the conventional theory of mechanism-based strain gradient plasticity [Y. Huang et al., Int. J. Plast. 20, 753 (2004)] established from the Taylor dislocation model [G.I. Taylor, Proc. R. Soc. London A 145, 362 (1934); G.I. Taylor, J. Inst. Met. 62, 307 (1938)] to study the Berkovich and other triangular pyramid indenters. The three-dimensional finite element analysis shows that the widely used equivalence of equal base area leads to significant errors, particularly in microindentation. A new equivalence of equal angle is proposed for triangular pyramid indenters, and it has been validated for a large range of indenter angles and indentation depths.

I. INTRODUCTION

The microindentation hardness of metallic materials typically increases by a factor of two as the indentation depth decreases to submicrometers; i.e., “smaller is harder.”1–5 The classical plasticity theories do not possess any intrinsic material length and cannot explain this indentation size effect. The strain gradient plasticity theories have been developed6–11 to extend the classical plasticity theories down to the submicrometer scale. They have been applied to study the indentation size effect in bulk materials12–16 and thin films on substrates,17,18 as well as the indenter shape and angle.19–24 Based on the Taylor dislocation model25,26 and a model of geometrically necessary dislocations underneath a sharp, conical indenter, Nix and Gao27 established the following simple relation between the micro-indentation hardness H and depth h  2 H h ; ð1Þ ¼1þ H0 h where H0 is the macroindentation hardness for a large indentation depth, a)

Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2009.0095

776

http://journals.cambridge.org

 2 27M2 ba2 m h ¼ 2 tan2 y H0 

J. Mater. Res., Vol. 24, No. 3, Mar 2009 Downloaded: 31 Jan 2015

;

ð2Þ

is a characteristic length that depends on both the angle y of the conical indenter (Fig. 1) and material [Burgers vector b, shear modulus m, empirical coefficient a around 0.3, M = 3.06 for face-centered cubic (fcc) metals as well as for body-centered cubic (bcc) metals that slip on {110} planes]. Equation (1) predicts a linear relation between the square of microindentation hardness H2 and the reciprocal of indentation depth 1/h, which agrees with the experimental data for single-crystal and polycrystalline copper5 and single-crystal silver.10 The microindentation model in Eq. (1) has been extended to spherical

Data Loading...