Relationship Between Rockwell C Hardness and Inelastic Material Constants

This paper describes the relationship between Rockwell C hardness and elastic-plastic material constants by using finite element analyses. Finite element Rockwell C hardness analyses were carried out to examine the effects of friction coefficient and elastic-plastic material constants on the hardness. The friction coefficient and Young’s modulus have no influence on the hardness but the inelastic materials constants, yield stress and strain hardening coefficient and exponent, have a significant influence on the hardness. A new equation for predicting the hardness was proposed as a function of yield stress and strain hardening coefficient and exponent. The equation evaluated the hardness within a ± 5% difference for all the finite element and experimental results.


INTRODUCTION
Hardness testing is one of the most frequent to material testings and is understood as a testing to measure the resistance to compressive permanent deformation. However, the physical meaning of hardness has been an open question. Especially, inelastic material contains are supposed to have some connection but a quantitative relationship has not been well defined. Experimental studies tried to obtain the quantitative relationship between the materials constants and hardness using many different materials that have different tensile properties, but they did not succeed because every material constant changes when using a different material [1]. To develop a quantitative relationship, an analytical method of changing only a specific material constant is needed keeping the other constants unchanged. Recent development finite element (FE) contact analysis enables this . The authors applied this approach to Brinell hardness and proposed a quantitative relationship [3]. Rockwell C hardness (HRC), however, is a representative hardness testing and a quantitative equation has been also needed. This paper studies the relationship between Rockwell C hardness and the elastic-plastic material constants using finite element analyses. Finite element contact analyses were made by varying yield stress and strain hardening coefficient and exponent. Based on the quantitative relationship between HRC and those material constants, an equation predicting the hardness from the material constants well be proposed. The accuracy of the equation well be discussed by comparing the hardness between the prediction, FE analysis and experiments. Figure 1 shows the FE meshes used for the analysis. An axi-symmetric 2D isoparametric 4-nodes element was used to obtain the relationship between HRC and the inelastic material constants, Figure 1 (a). The same mesh was used for analyzing the critical thickness for the hardness testing. An isoparametric 3D 8-nodes element was used for the analysis of the critical distance from specimen edge, Figure 2 (b). A quarter part of a whole model was meshed in 3D analysis from the symmetry of the model. The circular conical indenter with a flank angle of 120 degrees and 0.2 mm tip radius was modeled to a rigid body in both meshes based on the previous result that the elastic deformation of the indenter has no effect on Brinell hardness [3]. The numbers of nodes and elements are 773 and 727 in the 2D mesh, and those of nodes and elements are 3785 and 2136 in the 3D mesh, respectively. Thickness, t, in Figure 1 (a) was changed from 0.156 mm to 5.00 mm and the edge distance, w, in Figure 1  In FE analysis, load of 98 N was initially applied, increased to the maximum value of 1471 N and decreased to 98 N to simulate HRC testing. The relative indenter displacement ( ) between 98 N in loading stage and 98 N in unloading stage was analyzed to obtain HRC values. HRC values were calculated by .

RESULTS AND DISCUTION Finite Element Model
MARC K6 was used as a FE code and MENTAT II as pre and post processors. True stress -logarithmic strain relationships were used in the FE calculation with large deformation, update and finite strain options [4]. Contact was judged when the indenter approached to the specimen in a distance less than . The Coulomb friction below was employed to express a friction force in the analysis.
where is the friction force, F µ the friction coefficient and the normal force at a contact point. W

Effect of inelastic material constants on HRC
Effects of the friction coefficient and Young's modulus on HRC Figure 2 shows an example of force -indenter displacement curve. The curve is approximated with a quadratic function as , where c b a P + + = δ δ 2 P is the load applied to the indenter and δ is the indenter displacement. In the Brinell analysis with a ball indenter, the curve was not quadratic but a linear relationship was found in the initial loading stage [2]. These results suggest that the type of indenter influences the shape of load -indenter curve.
δ 1471 Indenter displacement δ , mm This study assumes the power law inelastic constitutive equation expressed below where ε p , σ, σ y , A and n are plastic strain, flow stress, yield stress and work hardening coefficient and exponent, respectively. These material constants of TAB6400 titanium alloy, INCONEL 718 nickel base superalloy and SCM 430 ferritic alloy are tabulated in Table 1   The variation of the hardness with yield stress is plotted in Figure 3 for E=206 GPa, A=1000MPa and n=0.3. The indenter displacements decrease with increasing yield stress and HRC increases accordingly. Yield stress has a significant effect on the hardness.  Quantitative formulation of HRC with inelastic material constants As shown in Figure 5, the relative displacement can be equated as a linear logarithmic function of n, so the relationship is expressed as  Table 1. All the hardnesses including the experimental results are correlated within a narrow scatter band of a factor of 1.05. In the correlation in Figure 6, the inelastic material constants are ranged from 588 to 3727 MPa for y σ , from 800 to 1440 MPa for A and 0.2 to 0.6 for n.
These ranges mostly cover the actually existing materials suitable for HRC testing. Thus, Eq.(6) is an appropriate equation for predicting HRC from the inelastic material constants for a wide range of matrials. Eq.(6) also indicates that how the inelastic material constants, yield stress and strain hardening coefficient and exponent, influence HRC value quantitatively.