Measurement and modelling of contact stiffness

Transcription

1 Measurement and modelling of contact stiffness D. Nowell, University of Oxford, UK Joints workshop, Chicago, 16/8/12

2 Difficulties in modelling contacts In general, the normal and tangential stiffnesses need to be experimentally measured, along with the friction coefficient These properties may change with time (e.g. as the contact wears, with position, and with load) Progress is needed towards a model of interface behaviour, which is based on more fundamental properties (material properties, surface geometry etc). We also need to understand how to incorporate the interface behaviour into global (FE) models of the system

3 Measurement of Contact behaviour Oxford and Imperial rigs 8 mm 2 flat and rounded contact 1Hz Frequency.6mm sliding distance Displacement measurement by remote LVDT or digital image correlation 1 mm 2 flat on flat contact ~1Hz Frequency 3m sliding distance Displacement measurement integration of LDV measurements

4 Measured and idealised hysteresis loops 15 1 LVDT DIC_specimen DIC_relative 5 Force (kn) Displacement (mm) Tangential Force P Idealised loop is characterised by contact stiffness, k and friction coefficient, These can be reasonably representative of real loops k Displacement

5 Variation of contact stiffness with measurement location 2 18 Ti rough experiment Contact stiffness (kn/mm) N=5 N=1 N= Distance between two measured points (mm) d

6 Predicted (FE) variation of tangential stiffness 3 δ(t) =.1Sin(2πt) mm 25 2 FE DIC, N=5, (M.E. Kartal) Compliant Elastic Layer E = 2.29 GPa, ν =.32 Individual Layer Thickness: 5 μm (Total: 1 μm) Kt (kn/mm) 15 1 μm p 1 5 Bulk material: Ti-6Al-4V E = 115 GPa, ν = Distance from Interface (μm) FE predictions of stiffness based on smooth contact are much higher than experimental measurements

7 qr () q i r 1 ai 2 Modelling - basic assumptions To develop a model for contact stiffness, we need consider surface roughness Initial tangential loading is likely to be predominantly elastic Consider a rough elastic surface in contact with a smooth rigid one. This puts all the elasticity and roughness on one surface and is easier to deal with At light loads, asperity contacts will be relatively widely-spaced and may be modelled as 2 Hertzian r p( r) p 1 a

8 qr () q i r 1 ai 2 Formulation When tangentially loaded, all contacts will initially be stuck, so the shear traction at each contact will be given by qi qr () r 1 ai 2 Mindlin gives the compliance for this traction distribution as (1 )(2 ) i Qi 8ai G 4ai E From this, the Greenwood/Williamson approach can be used to derive an expression for tangential stiffness

9 Result The approach leads to Note that this is independent of Young s modulus This is consistent with the results of Berthoud and Baumberger (1997), who found limited effect of modulus and P Where is a length scale of the order of microns (i.e. similar to ) Normalisation by area gives

10 Area effect 16 Effect of contact area on tangential contact stiffness for 7 MPa average pressure Tangential Contact stiffness (kn/mm) N 7 N 4 N 217 N 452 Larger area N 8 Larger area N 2 Larger area N Distance between measured points (mm) Before normalising After normalising Experiments carried out with different contact area do suggest that stiffness is approximately proportional to apparent area of contact This is because almost all of the compliance is in the surface layer

11 Effect of normal load Distance between measured points.18 mm Effect of normal pressure on tangential contact stiffness, N=2-25 cycles

12 Comparison with numerical model As part of our joint project with Imperial College, Medina has produced a numerical model of rough elastic contact Comparison shows good agreement at low loads, but reduced stiffness in numerical model at higher loads Effect is almost certainly caused by asperity interaction Similar effect noted for normal contact by Ciavarella et al (28) Tangential Stiffness (N/μm) Model B+B G-V OXF Load Numerical Simulation Pressure (MPa) 1 k k 1 smooth k 1 interface

13 Comparison with ultrasound measurements Recent work in collaboration with Sheffield Univ has compared stiffness measured with DIC with that using ultrasound Note that (in this case) initial value is very similar, but variation with Q is very different Ultrasound is measuring an unloading stiffness

14 Ultrasound measurement Ultrasound measures an unloading stiffness: In the case of normal stiffness, there is a similar effect, in this case there is an increase of stiffness with (normal) load and growth of the real contact area

15 Perspectives Tangential stiffness models should almost certainly include a dependence on normal load. What models are appropriate How can we improve the models we have? How do we capture time dependence? Measurement of stiffness in real contacts is not straightforward. There is a need for reconciliation between different techniques. We cannot model what we cannot measure. Modelling friction is far more challenging than contact stiffness More multiphysics in this problem Once again, time dependence is an issue We need better models for wear

16 Part 2: FRICTIONAL SHAKEDOWN 16

17 Punch on Half Plane with Tension 2a p Normal Load P = 2ap Bulk Load T = 2t Pad 121º 121º Grip Fixed end t Specimen Axis of symmetry Centreline

18 Simplified Model: Load vs. f Map / p Corner Leading Edge Partial Slip (PS) Edge Partial Slip f =.296 Incipient Interior Partial Slip Interior Trailing Edge PS Example Calculation Point Incipient Interior Partial Interior Slip Leading Edge PS Separation Fully Adhered f crit for / p > <([ T/t] / [P/a])<1 vs. COF f crit for / p < -1.9<([ T/t] / [P/a])< vs. COF f

19 Simplified Model: Load vs. f Map Probing shakedown / p Interior Cyclic Slip (PS) Possible Frictional Shakedown Melan s equivalent theorem Separation Corner Leading Edge Edge Partial Slip Slip (PS) Probing Point Fully Adhered f crit for / p > <([ T/t] / [P/a])<1 vs. COF.4 f Melan Frictional Shakedown Limit Melan f = f

20 Simplified Model: Load trajectories Step: LOAD Normal Load, P Bulk Pre-load, T Case 1 Case 2 Case 4 Case 3 Bulk Load, T Case 3 Case 4 Case 2 Case 1 TIME Melan s Theorem predicts shakedown for ANY and all residual stresses Input Residual Stress Cyclical Loading

21 q(x)/(f xy f yy p(x)) ) 1.8 Loading Sequence: History - 4 Cases q(x)/(f xy f yy ) p(x)) Case 1 Case x/a x/a q(x)/(f xy f yy p(x)) ) Case 4 Step2 Step3 Step4 Step5 Step6 Case Case 2 Case q(x)/(f xy f yy ) p(x)) Case 3 Case Step2 Step3 Step4 Step5 Step x/a x/a -1 Step2 Step3 Step4 Step5 Step6-1 Step2 Step3 Step4 Step5 Step6

22 Case 2 (no pre-stress) Does it shakedown? Case 2 Slip 5E-1 4E-1 3E-1 2E-1 Transient Step 4 Step 5 Step 6 1E-1 Steady State x/a

23 Simplified Model: Load vs. f Map Probing shakedown / p 2.4 Interior Cyclic Slip (PS) Possible Frictional Shakedown Melan s equivalent theorem 2 Probing Point Separation Corner Leading Edge Edge Partial Slip Slip (PS) Fully Adhered f crit for / p > <([ T/t] / [P/a])<1 vs. COF Cyclic Slip should occur above the threshold predicted by Melan s Theorem.4 f =.296 f Melan Frictional Shakedown Limit Melan f

24 Loading Sequence: Interior (?) Cyclic Slip LOAD Normal Load, P xy /( f yy ) q(x)/(f p(x)) Bulk Load, T TIME x/a Step2 Step3 Step4 Step5 Step6

25 Loading Sequence: Interior (?) Cyclic Slip Slip LOAD Normal Load, P 1.6E-8 1.4E-8 Transient Bulk Load, T 1.2E-8 1E-8 8E-9 6E-9 TIME 4E-9 2E-9-2E-9 Steady State x/a Step 2 Step 3 Step 4 Step 5 Step 6

26 Simplified Model: Load vs. f Map Probing shakedown Loading scenarios designed to lock-in different sets of residual stresses / p Separation, K I o > Interior Cyclic Slip (PS) Bulk Preload, T Case 1 Possible Frictional Shakedown Melan s equivalent theorem Case 2 Case 3 Case 4 LOAD K Io = Normal Load, P Bulk Load, T TIME Corner Leading Edge Partial Slip (PS) Probing Point Fully Adhered f crit for / p > <([ T/t] / [P/a])<1 vs. COF Melan s theorem predicts frictional shakedown for all 4 cases.4 f Melan Frictional Shakedown f = f

27 Loading Sequence: Step 6 xy /( f yy ) Interior Trailing Edge (PS and no shakedown) Fully adhered Slipping from the edge (a long way in) Fully adhered x/a Case 1 - Step 6 Case 2 - Step 6 Case 3 - Step 6 Case 4 - Step 6 Bulk Preload, T Case 1 Case 2 Case 3 Case 4 LOAD Normal Load, P Bulk Load, T TIME