PSI - Issue 10

I.D. Gavardinas et al. / Procedia Structural Integrity 10 (2018) 18–24 I.D. Gavardinas and A.E. Giannakopoulos / Structural Integrity Procedia 00 (2018) 000 – 000

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4

only E 0 . The experimental setup is illustrated in Fig.2, in which a commercial vascular bovine patch without prestretch is indented by an M-type Shore durometer. Reporting parameters of prestretched bovine samples is out of the scope of this work and will be reported in due time. Our attention focuses on the indentation of a hyperelastic substrate by a rigid indenter whose spherical tip of radius R merges smoothly (tangentially) with the cone body of half apical angle θ giving a transition radius b=Rcos θ , as shown in Fig.3.

P θ θ

Bovine pa tch

steel

h

b b R

t

α

α

. R mm  0 1

o





. 15 0 09659 1 2 23 . mm

b

mm



t



o T C 

Fig. 3. The indentation configuration.

The present methodology pertains to quasi-static testing, whereas prolonged indentation tests would require a viscoelastic analysis; see for example Mattice et al. (2006). The loading rate was controlled by the durometer device, as suggested by the ASTM D2240 standard. The corresponding linear elasticity contact problem (in the absence of friction) has been addressed analytically by Briscoe et al. (1994). Based on their results, we can formulate an

indentation procedure in order to obtain the elastic modulus E 0 as follows. The indentation depth h relates to the contact radius a , according to:



   

1/2

a b             

  

2

h a

b a        

a b

 

1



arcsin

cos

1   









(2)

tan 2  

The required vertical load P relates to the initial elastic modulus E 0 :

1/2    



2

2

3

  

2 2 (1 ) 1  

2 ( / ) 2 a b 

P v

a h a b a b   

a b      

1

1

     

     

1

cos

cos











(3)

 

o E b

2

2

6

2 tan 6 





The indentor is steel and is practically treated as rigid in comparison to the soft tissues. An M-type Shore durometer according to ASTM D2240-15 standard (2015) was used. The indentor’s geometry shares the properties: θ = 150 o , R =0.1 mm, ν =0.5 and b =0.9667 mm. Eqs.(2) and (3) were found to hold for all incompressible hyperelastic materials (ν=0.5 and E=E 0 ). They can be used in order to construct Table 1, which in turn can be used to obtain the initial modulus E = E 0 (regardless of the energy density function). This interesting result is in line with the recent work by Zisis et al. (2015) and references therein. All specimens were folded so that the final thickness, t , is four times the thickness of the patch ( t = 1.8 mm) so as to minimize the effect of the rigid stage that supports the specimens. Four samples for each bovine patch were tested. Each sample was indented four to six times at different locations on the sample surface.