PSI - Issue 7

D. Regazzi et al. / Procedia Structural Integrity 7 (2017) 399–406 Regazzi et al. / Structural Integrity Procedia 00 (2017) 000–000

402

4

(1979):

2 3 C k n d p − 2

3 γ k α k d p

d α k =

(8)

αα = N

k = 1 α k

The non-proportional hardening induced by the rolling contact stress is taken into account by defining a variation of the shear yield stress k as proposed by Tanaka (1994):

d k d p = b 0 ( k T − k ) k T = k 0 exp N p A

(9)

with the introduction of two additional material parameters, N p and b 0 , and the Tanaka’s parameter A used to quantify the amount of non-proportional hardening starting from the definition of a fourth order tensor which is a function of the plastic strain and the direction of the plastic strain increment. Considering Eq. 7 and writing Eq. 1 in terms of deviatoric stress, it is possible to rewrite the yield surface as: S el − α ∗ : S el − α ∗ − 2 k 2 = 0 (10)

where α ∗ is a new second tensorial internal variable:

α ∗ = ( α − ρ )

(11)

The modified backstress, α ∗ , is used to relocate the yield surface to a location in stress space depending on the residual stress:

C k n −

M k = 1   2 3

γ k α k   − K ρ n

d α ∗ d p =

2 3

(12)

By imposing the consistency condition during the load passage:

S el + d S el − ( α ∗ + d α ∗ ) : S el + d S el − ( α ∗ + d α ∗ ) − 2 ( k + d k ) 2 = 0

(13)