Issue 28

M.Malnati, Frattura ed Integrità Strutturale, 28 (2014) 12-18 ; DOI: 10.3221/IGF-ESIS.28.02 16 E FFECTOFTHEHYDROSTATICPRESSURE t is necessary to take into account the effect of the hydrostatic pressure, since the deviatoric fatigue equivalent stress  EQ calculated on the fatigue cycles as defined in the previous paragraph cannot entirely describe the fatigue damage process. This is due to the fact that the function f P describing the yield surfaces depends on the deviatoric part only of the stress tensor. Several possibilities arise. It is chosen here to give to the hydrostatic pressure effects a formulation inspired by the classical hypothesis of Sines (see e.g. [1, 4]) inwhich the average hydrostatic pressure during a load cycle is used. But it is important to remark that this hypothesis is not the only possible in the frame of thismethodology: by introducing it, we donot loose generality regarding the possibility to use othermechanisms of hydrostatic pressure influence. Let us consider firstly that during the advancement of the stress history the deviatoric fatigue equivalent stress  EQ for a given stress cycle ‘j’ is amonotonic non-decreasing function going from 0 at the initial instant to its final maximum value  EQ,j Thus the evolution of the hydrostatic pressure H p can be regarded as a function of  EQ,j , so that it is possible to calculate for the considered stress cycle ‘j’ the average value H,j p in the followingway: EQ,j τ H,j H EQ,j EQ,j 0 EQ,j 1 p p (τ )dτ τ     (15) Let us remark explicitly that H,j p is different for each stress cycle. The quantity H,j p is then used in combinationwith  EQ,j inorder to evaluate the fatigue damage of the cycle itself. To this scope in the frame of the present method it is required to have for a givenmaterial a relationshipwhich provides, for an uniaxial tension/compression fatigue problem under Constant Amplitude (CA) cycles, the fatigue equivalent stress as a function of the stress amplitude S a andof the average stress S , as follows: S EQ = S EQ (S a , S ) (16) When such an equation is not available for a material, then it is necessary to do a hypothesis to describe the influence of the average stress for uniaxial CA cycles. An “equivalent fatigue stress”  EQ,j for each of the yield surfaces is then calculated by applying the same law of Eq. (16) on the calculated  EQ,j and 3 H,j p :  EQ,j = S EQ (  EQ,j , 3 H,j p ) (17) This can be better illustrated by a simple example: in the “MMPDS” handbook [15] the expression of the equivalent stress for uniaxial CA cycles is givenby the following formula: S EQ = S MAX (1-R) q (18) whereR is the cycle ratioR= S MIN /S MAX . Conformingly to theEq. (16) this equation canbe rewritten in terms of S a and S : S EQ = 1 q a S S 2        S a q (19) The applicationofEq. (17) gives for a given stress cycle ‘j’:   EQ,j = 1 q EQ,j H,j τ 3p 2          EQ,j q (20) The so-calculated equivalent fatigue stress  EQ,j can be then entered directly in a S EQ -N curve available for a material, in order to calculate the damagemade by a single stress cycle. However, an important remark must be made: in the evaluation of their damage, the yield surfaces obtained by the procedure described abovemust be considered as half-cycles (or stress-reversals) and not as entire cycles. This can bewell I