Issue 28

M.Malnati, Frattura ed Integrità Strutturale, 28 (2014) 12-18 ; DOI: 10.3221/IGF-ESIS.28.02 14 d X C = 1 2 H(  P ) <d s : n > n (5.a) d  y = 1 2 H(  P ) 2 3 <d s : n > (5.b) where: H(x) = 1 if x≥ 0, H(x) = 0 if x< 0 (6) <x>= xH(x) i.e. <x>= x if x≥ 0, <x>= 0 if x< 0 (7)   P ( s; X C ,  y ) = f P ( s – X C ) -  y (8) and n is the local outward vector normal to the surface in the deviatoric stress space, having unit length in the following sense: n : n = 1 (9) The symbol : in Eqs. (5.a), (5.b) and (9) designates the following scalar product in the stress space, between two tensors s A , s B : s A : s B = 3 Aij Bij i,j 1 s s   (10) SinceVonMises stress is used, it is possible - as done in [3] - to give to n the following explicit form: n = 3 2 ( s – X C ) / f P ( s – X C ) (11) If at a given instant the same maximum  y belongs to two or more yield surfaces contemporarily then the one to be hardened according to the Eqs. (5.a) and (5.b) is chosen conservatively as the one having the maximum value of the scalar product d s : n . Additionally, if two or more surfaces have not only the same maximum  y but also the same maximum d s : n , then the choice for the one to be hardened according to the Eqs. (5.a) and (5.b) becomes physically arbitrary: we simply choose the onewhich hadbeenpreviously created firstly. The other surfaces that are hardened at a given instant - because the stress point lies on the surface and is moving towards the outside - simplymove rigidlywith a pure kinematic hardening (  y remain constant) defined by: d X C =H(  P ) <d s : n > n (12.a) d  y = 0 (12.b) Actually only the yield surface selected by the criteria illustrated above is considered to be plastically active and it is called “active surface”. The other hardened surfaces having a pure kinematic hardening expressed by the Eqs. (12.a) and (12.b) are not directly considered in the current global computation of the fatigue damage, but their updated position will have an influence on the subsequent response in fatigue. They are designated as “transported surfaces”. A third category is given by those surfaces that are not moving at a current instant: they are denoted as “resting surfaces” since the stress pointmoves internally. The presence of the transported and resting yield surfaces represents a memory effect in the fatigue process, since they can be activated in the subsequent stress history. As already remarked above, all the existing surfaces do not form obligatorily a set of nested surfaces but they can intersect each other.  An increment of plastic deformation at the grain scale can be by hypothesis created at each instant under a deviatoric stress variation. To take into account this, if the stress point ismoving in such a way that none of the already existing yield surfaces are hardened then a new surface is created starting from the size  y = 0 and obeying to the hardening law for the active surface expressed by Eqs. (5.a) and (5.b). Of course a first surface is immediately created at the start of the stress history. A problem arises when a surface is created: its normal n is not defined since the surface is collapsed in a point (  y = 0). This problem is bypassed by defining, at the instant where a yield surface is created, a normal drivenby the stress variationd s :