XXIII ICTAM, 19–24 August 2012, Beijing, China ANALYSIS OF INDENTATION SIZE EFFECTS UNDER STRAIN GRADIENT VISCOPLASTICITY Suman Guha *a , Sandeep Sangal * & Sumit Basu ** * Department of Materials Science and Engineering, Indian Institute of Technology Kanpur, Kanpur, 208016, India ** Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, 208016, India Summary Indentation size effects have been observed to exist extensively in several micro/nano-indentation experiments. Different strain gradient plasticity model has been proposed by several researchers to capture the observed size effects. Nix and Gao’s gradient plasticity simulations has been found to comply well with the experimental observations mostly at mircon scales. In this article a large deformation finite element simulations have been carried out to analyze indentation of a half space by an wedge shaped indenter based on Fleck-Hutchinson’s gradient plasticity formulation. The behavior of the size effects in indentation has been studied based on the varying mechanical properties and the results are found to be agreeing well with the experimental trends. INTRODUCTION It has been well established that materials exhibit more resistance to deformation at small scale, which is commonly known as size effect or scale effect. Size effects have been observed to exist predominantly in most of the micro/nano- indentation hardness measurement. In case of most of the microindentation experiments it has been observed that the square of the hardness generally follows a linear trend with the inverse of the indentation depth. Nix and Gao proposed a gradient plasticity theory, based on the concepts of Geometrically Necessary Dislocations (GNDs) which is able to capture the observed scaling effect[3]. The fundamental basis of this theory lies on the Taylors dislocation model for flow stress: τ = αµb √ ρ t , where τ is the flow stress in shear and ρ t = ρ s + ρ g , where ρ t is the total dislocation density and ρ s and ρ g are the densities of statistically stored dislocations and geometrically necessary dislocations respectively. µ is the shear modulus and b is the Burger’s vector. α is an empirical constant. However the observed linear variation in indentation can be predicted only for the indenters of particular geometry[6] (e.g. conical, wedge shaped, Berkovich etc.) and only when the total dislocation density is assumed to be given by the direct sum of ρ s and ρ g . It has also been reported that in case of nano-indentation the observed linearity is lost for various materials[4]. Fleck-Hutchinson’s (FH) higher order gradient plasticity theory is also capable of capturing the size effect. However the FH model incorporates a relationship which assumes a harmonic mean of ρ s and ρ g to derive ρ t . Accordingly the above mentioned linear variation is not reproduced for indentation simulations, if FH model is used. From various indentation hardness data summarized in [4] it is observed that materials categorized according to their mechanical property exhibit a common trend. It has been identified that the hardness trends vary mainly with the yield strength, the hardening parameter and the characteristic length scale (l ∗ ) of the material. In this work, 2D large deformation finite element simulations have been carried out using FH gradient plasticity theory to observe the effect of the materials yield strength, strain hardening parameter and l ∗ on the hardness trends for indentation of a half space by an wedge shaped indenter. FINITE ELEMENT FORMULATION In order to formulate the finite element equations the virtual work principle for strain gradient material is obtained by assuming the existence of a higher order traction which induces a higher order stress in the material as[2]: ∫ V (σ ij δϵ e ij + Qδϵ p + τ i δϵ p ,i )dV = ∫ S (T i δu i + tδϵ p )dS, (1) where τ i is the higher order stress work conjugate to the plastic strain gradient ϵ p ,i , Q is the scalar microstress, work conjugate to the equivalent plastic strain ϵ p and t is the higher order traction. In the absence of strain gradient (1) reduces to the virtual work principle for conventional plasticity and Q = σ e , where σ e is the equivalent stress defined in the conventional manner as σ e = √ σ ′ ij σ ′ ij and σ ′ ij = σ ij - 1 3 δ ij σ kk . The incremental form of the virtual work principle is derived for large deformation and in an updated lagrangian framework according to [5] using the uniaxial viscoplastic stress-strain law σ c = σ 0 (1 + E p ϵ 0 ) 1/n ( ˙ E p / ˙ ϵ 0 ) m , (2) where m is the strain rate sensitivity and ˙ ϵ 0 is the reference strain rate. ˙ E p is the effective plastic strain rate defined as ˙ E 2 p =˙ ϵ p 2 + l 2 ∗ ˙ ϵ p ,i ˙ ϵ p ,i , (3) where l ∗ is the characteristic length scale. The present formulation is different from the conventional finite deformation based elasto-plasticity in a manner that both the velocity and plastic strain rate have been used as nodal variables in 6 noded triangular elements. Finally a coupled FE equations are derived according to [1] and solved using a 2D finite element code. a Corresponding author. E-mail: gsuman@iitk.ac.in.