PSI - Issue 2_B

Tuncay YALÇINKAYA et al. / Procedia Structural Integrity 2 (2016) 1716–1723 Tuncay Yalc¸inkaya and Alan Cocks / Structural Integrity Procedia 00 (2016) 000–000

1718

3

Fig. 2. Loading of the representative cell.

assume that all cells remain the same. Also assume that deformation is concentrated in these zones, i.e. there is an axial macroscopic strain, however the in plane macroscopic strain is zero, i.e. if u is the radial displacement in the plane, u = 0 at r = l . Initially we consider the response to transverse traction T n (see Fig. 2). The initial volume of a pore is π a 2 0 h 0 . The representative cell is fully constrained so that l remains constant. Through the incompressibility condition in the matrix material, ˙ ε r + ˙ ε θ + ˙ ε z = 0 and using the small strain-displacement relations in pola r coordinates ˙ ε z = ˙ h / h , ˙ ε θ = ˙ u / r , ˙ ε r = ˙ u / dr and the boundary conditions ( ˙ u = 0 at r = l ) we get ˙ u = ( ˙ ε z r / 2) / ( l 2 / r 2 − 1) and other strain rate components becomes

l 2 r 2 −

1) and ˙ ε r = − ( ˙ ε θ + ˙ ε z ) = − ˙ ε z 2 ( l 2 r 2 + 1)

˙ u r =

˙ ε z 2

(

˙ ε θ =

(1)

Next we apply the upper bound for a perfectly plastic material, where work done by the limit load is smaller or equal to the integral of internal energy dissipation of the e ff ective strain rate ˙ ε e at yield stress

l

n ˙ δ n ≤

π l 2 T

˙ ε e σ y 2 π rhdr

(2)

a

Using (1) the e ff ective strain rate could be written as

˙ ε e = ˙ ε z 1 +

l 4 3 r 4

(3)

Substituting (3) into inequality (2) and using ˙ δ n = ˙ h and ˙ ε z = ˙ h / h gives

1 f 1 + (1 / 3 v 2 ) dv where v = r 2 / l 2 and dv = 2 r / l 2 dr

T n ≤ σ y

(4)