PSI - Issue 41

Christos F. Markides et al. / Procedia Structural Integrity 41 (2022) 351–360 Christos F. Markides et al. / Structural Integrity Procedia 00 (2019) 000 – 000

353

3

P frame

y

Platen Stamp

P frame

x

Stamp Ο

t

L

– L

FBD

FBD

R

a

Stamp Platen

Fig. 2. FBDT with interposed stamps.

Fig. 3. FBD - stamp contact region.

This configuration corresponds to the case of two indenters (the stamps) which are placed in between the platens and the FBD (Fig.2). The case when platens exceeding in length the flat edges of the FBD are in direct contact with the FBD (Fig.1a) will be dealt with in a future study. 2. Theoretical analysis 2.1 The stress distributions to be used for the boundary conditions on the flat edge of the FBD Assume that the FBD occupies the lower half plane, the stamp is flat, its length is equal to that of the flat edge of the FBD, and, finally that the stamp is much stiffer than the FBD (Fig.3). Let the stamp be compressed against the FBD with an overall load P frame in the presence of friction, compelling the material points of the flat edge of the FBD to remain stuck with the stamp. The compressive and frictional stresses developed on the flat edge of the FBD are then provided by the Muskhelishvili‟s related contact problem as (Muskhelishvili 1963):

P



1

L x L x L x L x  









 

frame

cos log log 2   





y

 



2 t L x P t L x   2 frame

2



(1)

1









 

sin log log 2   





xy

 



2



where ± L are the end points of both the stamp and FBD‟s flat edge, i.e., – L ≤ x ≤ L , t is FBD‟s thickness, and κ equals 3 – 4 ν or (3 – ν )/(1+ ν ) for plane strain or generalized plane stress, respectively ( ν is Poisson‟s ratio). The distributions of the boundary stresses on the flat edge of the FBD due to Eqs.(1) are plotted in Fig.4, assuming an FBD of thickness t =10 mm, radius R =50 mm, ν =0.36 and α =11 o , so that L = R sin α =9.54 mm. Moreover, without loss of gen erality, assume that P frame =36 kN, and that generalized plane stress conditions prevail. The contact stresses plotted in Fig.4 will be now used for writing down the boundary conditions in the 1 st fundamental problem for the isolated FBD. 2.2 From the complete disc, to the 1 st fundamental problem for the FBD The chord – LL of the complete disc will stand as the flat edge of the FBD. In this context, – LL is first divided into a total number of 2 n +1 infinitesimal portions (Fig.5). In the middle of each one of these portions an infinitesimal hole is considered, containing (and in the limit coinciding with) the point Z j , j =1,…, n (countered from right to left in the first quadrant). The point Z m , after Z n in the previous sense, corresponds to the central infinitesimal hole at the inter section of – LL with the y -axis. A pair of forces ( P j , F j ) (i.e., a compressive and a frictional one), is applied to each point Z j (at Z m , F m =0). P j and F j are forces per unit thickness, obtained directly from the respective stress distributions of Eqs.(1). In this context their magnitudes can be written as follows: