7/28/2019 Histeresis in Fatigue

1/2

Histeresis in Fatigue

If the yield stress is exceeded at notches in a structure, hysteresis loops of various sizes will be traversedat points around the notches. Providing the load is not too large, shakedown is achieved, in which theloops stabilise at all points, after a small number of cycles. But sustained repetition of this load of typically, between 102 and 107 cycles, will have to consider the separate and longer term phenomenonof fatigue damage.Aside from Youngs Modulus and Poissons Ratio, the material properties required for a shakedownanalysis are the yield stress and the post-yield stiffness. Fatigue behaviour is described by furthermaterial constants, but these are not required in the main FEA solution phase.In fatigue, loads are cyclic and fatigue damage at a point is strongly related to the range of stressencountered during a given load cycle, rather than the peak stress due to the extreme of the cycle. Eachstress cycle will damage the material by a small amount, eventually causing cracks to form. Generally, noaccount is made of the shape of each individual peak to peak load (and corresponding stress) and, for asequence of different load ranges, no account is made of the order in which the loads are imposed.Figure 1 shows common terminology used to describe load or stress cycles.

It will be obvious that although the range of stress might be the most important determinant of fatiguedamage, the mean stress value must have some effect. This can be conveniently considered separatelyhowever; the main effort in a typical fatigue analysis is expended in isolating the discrete stress rangeswithin a complex load cycle, with each cycles mean stress accounted for separately.The fatigue life of ferrous and some aluminium alloys can be predicted based on the followingassumptions. For other material including cast irons and other aluminium alloys, at least some of theseassumptions are not valid, so fatigue lifing may not always be viable for such materials:Life is defined by material properties, that is, quantities that are assumed constant for a given material.These are obtained from cyclic loading or straining of specimens of the said material. There is a greateramount of scatter of fatigue life data from material specimens than of say, stiffness. Such scatter may be

accounted for by taking the mean minus two standard deviation life or other statistical approaches.For a fixed, repeatedly applied load cycle, the hysteresis stress-strain curve at any point will not change,until failure begins. This is termed cyclic stability. In reality, hardening or softening may occur initially, soa specimen hysteresis loop changes shape, but this stabilises after a few cycles and is one aspect of shakedown behaviour, already described.The memory effect underpins the range counting techniques used in fatigue analysis. This implies thatany complex sequence of load cycles will give rise to several complete (closed) hysteresis loopscorresponding to each peak to peak excursion. Thus, the damage due to each loop is related only to the

7/28/2019 Histeresis in Fatigue

2/2

size of that loop, independently of the other loops or their relative position. This is explained in moredetail below.The hysteresis loops described in previous articles used straight line portions, indicative of the typicalbilinear stiffness representations used in FEA. Real hysteresis loops are curved, as shown in Figure 3.This figure shows the hysteresis loop B-C-B enclosed within the loop O-A-D-O.

Figure 2 shows a load cycle in which several stress ranges could possibly be considered in a damagecalculation, such as O-A, C-D and D-E. But the actual ranges chosen for the damage calculation will bethose that give rise to closed hysteresis loops. At point B in Figure 2, the load reversal causes a newhysteresis loop to be started in Figure 3. At some point between C and D in Figure 2, the B-C-B loop inFigure 3 will be closed. The stress strain curve does not follow the continuation of the CB curve, showndotted, but instead resumes the continuation of the A-B loop, until the load at D is reached. This is thememory effect. Hence the ranges to consider in the ensuing fatigue calculation are A-D and B-C.