Material characterization by indentation

Transcription

1 Material characterization by indentation Erland Nordin Project Work in Contact Mechanics Introduction Contact between a sphere and a plane is a classical and relatively simple problem due to the solutions by Hertz, published There are four simplifying assumptions: [1] The surfaces must be continuous and non-conforming. The strains must be small. Each solid could be considered as elastic half-spaces. The surfaces should be frictionless. The first three assumptions all means that the contact radius should be small in comparison with the other dimensions of the problem, particularly the radius of the sphere. Considering the problem of a rigid sphere indenting an elastic half-space the contact radius can be calculated with equation 1. a = ( ) 1 3P R 3 4E (1) Here P is the force pressing the sphere into the elastic plane, R the radius of the sphere and E is the effective elastic modulus (se equ. 4) of the half-space. Equation 2 calculates the indentation depth δ for a rigid ball against an elastic half-space. For two elastic bodies δ will be the approach of distant points on each body. ( ) 1 δ = a2 9P 2 3 R = 16RE 2 (2) The maximum contact pressure is denoted p 0 (equ. 3) and is related to the mean contact pressure by p m = 2 3 p 0 = P πa 2. p 0 = 3P 2πa 2 = ( 6P E 2 π 3 R 2 ) 1 3 (3) 1

2 If the indenting ball is not rigid but elastic the effective elastic modulus is calculated from the two bodies elastic properties using equation 4. For the case when the ball is rigid E 1 or E 2 is infinite and the corresponding term cancel. 1 E = 1 ν2 1 E ν2 2 E 2 (4) If two spheres are pressed together an equivalent radius is calculated with equation 5. The case of a rigid sphere against an elastic half-space considered above is a special case where the radius of the half-plane is infinite and the equivalent radius is thus the same as the radius of the rigid sphere. 1 R = 1 R R 2 (5) The surface displacements and stresses due to the Hertzian pressure can be calculated when the maximum pressure p 0 and contact radius a has been calculated using the previous equations.the displacements in radial direction is calculated by equation 6. ū r (r) = (1 2ν)(1 + ν) a 2 3E r p 0 ] [1 (1 r2 a 2 ) 3 2 r a (6) (1 2ν)(1 + ν) a 2 = p 0 r > a 3E r Here E and ν are the elastic properties for the body of interest. The radius r is the distance from the center of the contact to the point of interest along the surface. The normal displacements is calculated with equation 7. ū z (r) = 1 ν2 E = 1 ν2 E πp 0 4a (2a2 r 2 ) r a (7) [ p 0 (2a 2 r 2 ) arcsin( a 2a r ) + a ) 1 ] (1 r2 a2 2 r r 2 r > a The stresses on the surface is first divided into points lying inside the contact radius: σ r = 1 2ν ( ) ] ) 1 a 2 [1 p 0 3 r 2 (1 r2 a 2 ) 3 2 (1 r2 2 a 2 (8a) σ θ = 1 2ν ( ) ] ) 1 a 2 [1 p 0 3 r 2 (1 r2 a 2 ) 3 2 2ν (1 r2 2 a 2 (8b) ) 1 σ z = (1 r2 2 p 0 a 2 (8c) The last equation, 8c, is of course the Hertzian pressure but with a minus sign. Outside the contact zone, but still restricted to the surface, the stresses are: σ r p 0 = σ θ p 0 = (1 2ν) a2 3r 2 (9) σ z = 0 2

3 Inside the material, beneath the surface, the stress equations are restricted to be along the centerline beneath the contact. Letting the z-axis start at the center of the contact at the surface and be directed into the body of interrest the stresses are: σ r = σ θ = (1 + ν) [1 z ] p 0 p 0 a arctan(a z ) + 1 ) 1 (1 + z2 2 a 2 (10a) ) 1 σ z = (1 + z2 p 0 a 2 (10b) The σ r, σ θ and σ z stresses along the z-axis are principal stresses. The principal shear stress τ 1 can therefore be calculated as τ 1 = 1 2 σ z σ θ (11) Similarly the octahedral shear stress is calculated with equation 12 and τ oct = 1 3 [ (σr σ θ ) 2 + (σ θ σ z ) 2 + (σ r σ z ) 2] 1 2 (12) = {σ r = σ θ } = 1 3 [ (σθ σ z ) 2 + (σ r σ z ) 2] 1 2 the von Mises equivalent stress is then according to equation 13. σ e = 1 2 [ (σr σ θ ) 2 + (σ θ σ z ) 2 + (σ r σ z ) 2] 1 2 (13) = {σ r = σ θ } = 1 2 [ (σθ σ z ) 2 + (σ r σ z ) 2] 1 2 Figure 1 shows how the surface stresses look if the stresses are normalized with the maximum pressure p 0 and the radial distance is normalized with the contact radius a. The only tensile stress is the radial stress s r outside the contact. Figure 2 shows the corresponding stresses along the centerline beneath the contact. Assuming Poisson s ratio ν = 0.3 the maximum shear stress will be τ 1,max = 0.31p 0, the maximum octrahedral shear stress is τ oct = 0.29p 0 and maximum von Mises is σ e = 0.62p 0. All occurs at a depth of z = 0.48a.Therefore initial plasticity will start at a depth of about half the contact radius under the surface. Figure 3 shows a comparison of the stresses and displacements between a load that is just below start of plasticity (elastic region) with a load after plasticity had started (corresponding to a max equivalent plastic strain of 0.5%). Figure 4 shows the amount of equivalent plastic strain corresponding to the chosen plastic region. The plastic deformation has not reached the surface and is still constrained by elastic material. Although the relatively large plastic deformation beneath the surface the difference in surface displacements for a completely elastic deformation at the same load is very small. Figure 5 shows the force-indentation depth response from an finite element model and analytically. There is practically no difference in measured force between the elastic and 3

4 Figure 1: Normalized stresses on surface. stress (sth) and surface normal stress (sz). Radial stress (sr), circumferential Figure 2: Normalized stresses along centerline beneath centrum of contact. Radial stress (sr), circumferential stress (sth), stress along centeraxis beneath contact (sz), maximum shear stress (tau1), octahedral shear stress (octshear) and von Mises equivalent stress. 4

5 plastic solution below 3 4 times the indentation depth that cause first yielding. Figure 6 shows the mean pressure normalized with yield stress plotted versus the Johnson parameter. The Johnson parameter is indentation size normalized with ball radius times the ratio of elastic modulus to the yield stress. It can also here be seen that there is very little difference a good bit above the yield point before an elastic solution diverges from the plastic results. It is therefore very difficult to detect the yield point by practical indentation experiments. The initial suggestion by Hertz that the hardness should be defined as the contact pressure at initial yield is not used today for that reason. Instead the Vickers pyramid for example gives the hardness at approximately 8% strain and Brinell indentations are always presented with the corresponding ball material/diameter and load. 2 Material parameters for aluminum In this section an aluminum piece of unknown material properties is investigated. The goal is to determine elastic modulus and the stress-strain curve using indentation techniques. 2.1 Vickers indentation The Vickers indenter is a diamond pyramid which creates a self-similar stress field. This means that the stress field looks the same regardless of the scale of the indentation. This applies as long as the indent size is significant larger than the grain size of the material so that discrete grains are averaged out [2]. Figure 7 shows an example of a Vickers indentation on the aluminum used. The force used to press the diamond pyramid down into the material was 500 gf, i.e. the same force a 500 g weight would press down with due to gravity. The two diagonal measurements are averaged and used to calculate the hardness number with equation 14. P is the force in kgf, L is the mean of two diagonal measurements on the indent in mm and θ is the angle between the diamond tips faces (θ = 136 ). In order to check if there were any size dependences for this aluminum several measurements at different forces were made. Figure 8 shows how different Vickers indents compare to each other when different forces are used. The measured diagonals of the indents and the corresponding hardness is presented in Table 1. There were only natural scatter in the results and as a final Vickers hardness number for this aluminum an average was used, which gives 189 HV. 2.2 Brinell indentation HV = 2P sin( θ 2 ) L 2 (14) Using the Vickers indenter can give an easily measured hardness but due to the self-similar feature the result correspond to only one point in a stress-strain 5

6 (a) Surface stress elastic region (b) Surface stress plastic region (c) Bulk stress elastic region (d) Bulk stress plastic region (e) Surface displacements elastic region (f) Surface displacements plastic region Figure 3: Comparison of stresses and surface displacements in the elastic region (left column) and plastic region (right colummn). The plastic region had an maximum equivalent plastic strain of 0.5%. Dashed lines are analytical results and point results are from a finite element model. 6

7 Figure 4: Finite element results for equivalent plastic strain for plastic region. Table 1: Vickers indentation with varying force. All measurements except 2 kgf was measured with a 40x objective. Force L1 [um] L2 [um] HV 25 gf gf gf gf gf gf kgf kgf (10x objective)

8 Figure 5: Force versus indentation depth for a rigid ball against a steel plate Figure 6: Hardness versus Johnson parameter. 8

9 Figure 7: Vickers indentation with 0.5 kgf force. Figure 8: Vickers indentation with different forces. Left 2 kgf, upper middle 50 gf, lower middle 25 gf and to the right 500 gf. 9

10 curve, namely at approximately 8% strain. The Brinell indentation method use a sphere as the indent body instead of a sharp pyramid. The indent for a sphere will not be self-similar and while that might be a draw-back it can also be used to measure the stress-strain curve at different equivalent strains. The Brinell measurements was made on a Wolpert machine that had an indent sphere with a diameter of 2.5 mm. Table 2 shows the Brinell measurements made along with the calculated Brinell hardness number, calculated by equation 15. HB = πd 2 P ( (15) D D2 d2) Here P is the force on the ball in kgf, D is the ball diameter in mm and d is the remaining indent diameter in mm. Table 2: Brinell indentation with varying force. Force [kgf] d [mm] HB Approximating stress-strain curve The stress-strain curve can be approximated from hardness values by converting hardness to a corresponding flow stress. For a Vickers indentation the calculated flow stress will always correspond to a strain of 8% because of the self-similar stress-field. For a Brinell (sphere) indentation a representative strain is approximated which depend on the relative size of the indent. Using a power law model for the stress-strain relationship, (equation 16) σ(ɛ pl ) = κɛ m pl (16) Tabor [3] found that the relation, ( a ) m H = C 1 κ C 2 (17) D is valid when the indentation is in the fully plastic region. Tabor determined the constants to: C 1 = 2.8 (18) C 2 = 0.4 (19) The exponent m and κ are material parameters. H is the hardness defined as the indentation force divided by the projected area. Therefore this is not 10

11 the same hardness as reported from Vickers och Brinell indentations which use actual contact area. For the Brinell indentation the correct hardness value to use is: and for Vicker indentation the conversion is: H B = P g πa 2 (20) H V = HV g sin( θ 2 ) = 2P g L 2 (21) The conversion from kgf used in the Brinell and Vickers hardnesss to force is performed by multiplying with a standardized gravitational constant (g = ). Comparing equations 16 and 17 it is recognized that the representative strain for a sphere indentation is approximated as: ɛ R = 0.4 a D (22) and that the flow stress is approximated as: σ = H 2.8 (23) For a Vickers indentation the representative strain is always 8% and the flow stress is also approximated with equation 23. Figure 9 shows the Brinell and Vickers indents converted to corresponding strains and stresses. Equation 16 is fitted to the Brinell measurements and shown as the fitted curve. The values of the material parameters for this case is: κ = 1939 MPa m = 0.35 Using equation 16 outside the interval for the measured points, strains 0.04 to 0.08 in this case, should of cause be made with caution. 2.4 Measuring elastic modulus To estimate the elastic modulus a ball with 12.5 mm diameter was pressed into the aluminum block while the force and displacement was recorded. The ball was loaded up to 1900 N, held for stable for 8 s and then unloaded at a quasi-static rate. When a ball is pressed beyond the elastic limit the pressure distribution changes from the hertzian to a more flat distribution. The max load is therefore chosen to give a clear and large indent. The indent mark in this case is shown in Figure 10. The indent radius is 0.67 mm and therefore about 10% of the ball radius. Since the pressure distribution at max load is approximately flat and the unloading is elastic a relation between the load and displacement can be calculated by the flat punch equation 24. δ = πp 0 a 1 ν2 E = P 2aE (24) 11

12 Figure 9: Stress-strain points calculated from the Brinell and Vickers indentations. Solving for the effective elastic modulus, changing to an incremental version and inserting a constant that will be determined by FEM-simulations we get E = 1 c 1 Aproj P δ (25) where A proj = πa 2 is the projected indent area. The slope of the curve at unloading is changing so P/ δ is evaluated at a chosen position, close to the start of unloading. To determine the constant an fem-simulation of the process is made. As indenter a sphere with the same diameter and elastic properties, E = 206 GPa and ν = 0.3, as in the experiment is chosen. The target is chosen to have E = 70GP a and ν = 0.3 and an ideal-plastic material with yield stress 714 MPa (compare with Figure 9). No friction is included in the analysis. Figure 11 shows the simulated force versus indentation depth. The thick line part is the region used to evaluate the slope. In the simulation the contact diameter is mm at 1900 N force which is quite similar to the measured indent diameter of 1.34 mm. Using equation 25, with the known elastic modulus, the constant is determined to c = Figure 12 shows the measured force versus displacement. In the measurement the machine compliance has been accounted for so the displacement should correspond to the indentation depth. The thick red part of the line is the part where the slope is evaluated. To avoid some dynamic effects and letting the 12

13 ball settle before unloading a wait time of 8 s is used before unloading starts. The force versus time procedure is shown in Figure 13. The evaluated elastic modulus of the aluminum, assuming Poisson s ratio to be ν = 0.3, is E = 84 GPa. This is a reasonable value compared to the elastic modulus of aluminum of 70 GPa that is usually assumed. Figure 10: Indent caused by a 12.5 mm diameter ball pressed into the aluminum with 1900 N. The indent diameter is 1.34 mm. 3 Conclusions In the first part of this report it was shown that the measurable properties (forces, displacements) in en indent test is very little influenced by the start if yielding. This is because the yielding starts beneath the surface and is constrained by elastic material around it. The second part deals with how an unknown material can be characterized by indentation methods. The stress-curve can be approximated with Brinell and Vickers indentation techniques and both gave similar result. At 8% representative strain the flow stress was 714 MPa which is quite large for aluminum. The elastic modulus could be reasonable well measured with a ball indentation if force and displacement is continuously measured at unloading. The resulting 84 GPa is a little bit higher than what is usually used for aluminum but still a good value for a relatively simple experiment. 13

14 Figure 11: Force versus displacement for simulated ball indentation. Figure 12: Force versus displacement for ball indenting aluminum plate. 14

15 Figure 13: Force versus time for the ball indentation process. References [1] Johnson, K.L., Contact mechanics, Cambridge University Press, 1985 [2] Elmustafa, A.A., Eastman, J.A., Rittner, M.N., Weertman, J.R., Stone, D.S., Indentation size effect: Large grained aluminum versus nanocrystalline aluminum-zirconium alloys, Scripta Materialia 43 (2000), [3] Tabor, D., The Hardness of Metals, Oxford University Press,