Wang & Brown method

that maximized the local loads over the whole load history in order to describe better the continuous increase of damage during the loading with Palmgren-Miner rule:

The plane with maximum shear strain range Δgmax is searched - the original proposal belongs to MSSR concepts. The normal strain excursion during each shear strain half-cycle related to that plane is marked as an effective normal strain eeff. The mean stress effect is included in the Basquin part of the e-N curve in an analogy to Landgraf parameter. It was added to the original criteria at another paper [WB96]. Note the use of elastic Poisson's ν ratio in the Basquin part of the e-N curve and of plastic Poisson's ration νp in the Manson-Coffin curve. There is only one strange parameter in the formula - the multiaxial parameter S. Wang a Brown refer on its value lying in the range 1.0-2.4 for the group of materials they tested.

Kim, Park and Lee in [KPL99] tested another variant (let it be called KPL in the further text) of retrieval of effective normal strain. They related the maximum excursion of normal strain (i.e. effective normal strain eeff) to a whole cycle. Comparison of both variants lead them to the conclusion to prefer this new variant. Both variants are implemented in PragTic.

There is another important choice in the computational scheme. In addition to the originally used MSSR scheme, a possibility to switch to MD concept was added.

Up to now, the information concerned the original Wang and Brown method as published in [WB93] - the method is marked as WB 1993 in the Calculation Methods window. The authors do not comment the decomposition of the load history to separate cycles except for the fact that the shear strain should be the decisive variable for the rain-flow decomposition. The common rain-flow scheme with cycles separation according to the course of resolved shear stress (i.e. the shear stress on a specific plane and in a specific direction). There are two more options. One of them is the cycle retrieval according to normal strain.

The last option is as the decomposition Wang & Brown v.'96. This is a special computational scheme that was published in [WB96] and can be described in several steps:

2) The maximum value throughout the load history is picked (and designated as point A).

3) Relative strain tensors computed as a difference between actual strain tensor and the strain tensor in point A are computed.

6) Points A and B form one half-cycle, the point B splits the load history into two parts. The procedure with steps 1-6 is repeated for each of those two parts.

This scheme was programmed into PragTic. It should be usable universally in conjunction with any other fatigue analysis method, but currently it is usable only for the Wang and Brown method. The current version in PragTic is implemented as a standalone method WB 1996. Be aware, that the possibility to choose Wang & Brown v.'96 in the Decomposition dialogue is currently only fictitious for WB 1993 method and the rain-flow decomposition of resolved shear strain variable is used instead.

Note: The original proposal in [WB96] expects that the MSSR concept should be applied on each retrieved load cycle separately. The damage is thus maximized and Wang and Brown themselves mark this solution as a conservative border. The method is not implemented in PragTic in this mode, but the MSSR or MD preference is applied on the whole load history (i.e. to all retrieved cycles), which means that there is only one critical plane for whole life in the examined point.

Mark	Unit	PragTic variable	Meaning
eeff	[-]		effective normal strain excursion acc. to original [WB93] or KPL modified [KPL99] variant
Δgmax	[-]		maximum shear strain range on the critical plane
Δe	[-]		normal strain range on the critical plane
Nm	[MPa]		mean normal stress on the critical plane
E	[MPa]	E	tensile modulus
	[MPa]	SIG_F	fatigue strength coefficient
	[-]	EPS_F	fatigue ductility coefficient
b	[-]	EXP_B	fatigue strength exponent
c	[-]	EXP_C	fatigue ductility exponent
S	[-]	S_WB	Wang & Brown multiaxial parameter
N	[-]		number of cycles to crack initiation
ν	[-]	NU	elastic Poisson's ratio
νp	[-]	NU_PL	plastic Poisson's ratio

Note: The Neuber-like methods allowing input of elastic stresses and strains into the fatigue damage calculation for multiaxial solution are not implemented in PragTic. The only allowed input thus is the input of transient analysis, where the elastic-plastic constitutive rules were applied in the non-linear elastic-plastic FE-solution.

Solution option - their actual set depends on the method and decomposition method chosen

- Shear component description <0~MCCM, 1~LCM, 2~by normal line> - fill in zero, please - the MCCM solution is used as default currently. Any other choice will not be reflected in the computation.

- Normal strain amp. <0~on shear half-cycle, 1~on whole cycle> - the option concerns the choice between the original retrieval of effective normal strain and the KPL variant - see here.

- Zero deviation - this value describes the maximum distance of two points on the shear load path that are claimed to be coincident. Recommended value: 1e-8

- Minimum damage – a reasonable value of damage allowing the Newton-Raphson iterative algorithm to get to the final and real value is expected. Recommended values: 1e-15 - 1e-20

E	[MPa]	tensile modulus
NU	[-]	Poisson’s ratio
SIG_F	[MPa]	fatigue strength coefficient
EPS_F	[-]	fatigue ductility coefficient
EXP_B	[-]	fatigue strength exponent
EXP_C	[-]	fatigue ductility exponent
S_WB	[-]	Wang & Brown multiaxial parameter
NU_PL	[-]	plastic Poisson's ratio