Issue 60

D.-E. Semsoum et alii, Frattura ed Integrità Strutturale, 60 (2022) 407-415; DOI: 10.3221/IGF-ESIS.60.28

    m

f P B h h

(8)

where B and m are smoothing parameters of the power law, and h f is the final depth after total unloading of the indenter. Under these conditions, the slope S is found by taking the derivative of this function at its deepest point:        1 max m max f h h S dP mB h h dh (9)

Hence, the contact stiffness is expressed as follows:

 4

S

E A

(10)

r

c



where E r is the reduced (mixed) Young's modulus, given by:

2

2





 i

 s

1

1

1





(11)

r i s E E E

with E s and ν s are respectively the Young modulus and Poisson's ratio of the indented sample and E i and ν i are those of the penetrator.

 / M H P S SPECIFIC TO THE PILE - UP

M ODELING OF THE ANALYTICAL EXPRESSION

T

ransformation of Eqn. (3) according to the model of Bull and Page [12] gives:



2

  

  

P



H

1

m





(12)





2

2 cH E



 h h m

r

0

The classic Martens hardness, H M , expresses the ratio of the ultimate indentation load to the maximum projected area with the imposed tip defect correction as follows:

P

P

  m

m

H

(13)

M





2

 A h h 26.43

m

0

Hence, we express the Martens hardness as a function of the contact hardness and the reduced modulus for the pile-up mode by combining Eqs. (12) and (13). We obtain:



2

  

  



H

1

1





H

(14)

M

26.43

2 cH E



r

From Joslin and Oliver's relationship [13]:

  P H S E 2 4 m

2 IT r

(15)

The ratio of hardness to the square of the modulus is expressed as:

410