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The decomposition of solenoidal fields into a poloidal and a toroidal component is studied when the underlying domain Ω is a three-dimensional ball or spherical shell. The old result of Backus [4] is extended in such a way that a formulation in terms of projections operating on the subspace of solenoidal fields in L P (Q) 3 is attained. Therefore the given result is applicable to the solenoidal part of the Helmholtz decomposition. In addition, Sobolev estimates for the poloidal and toroidal part are supplied.

The plasma immersion ion implantation treatment (PIII) acts to increase mechanical resistance with solid solution formation, new phases and defects inclusion, besides chemical and residual stress profile modification. Ti-6Al-4V alloy presents poor tribological properties and high affinity with interstitial elements, such as nitrogen and oxygen, this makes it more reactive at high temperatures. This paper aims to study Ti-6Al-4V alloy fatigue behavior subjected to nitrogen addition by plasma immersion ion implantation. It was investigated the frequency parameter influence on fatigue resistance. Ti-6Al-4V alloy was PIII treated with voltage equal to 9.5 kV, frequencies varying between 1000 and 1500 Hz and submitted to axial fatigue tests. Axial fatigue tests were performed, at room temperature and R = 0.1. Ti-6Al-4V alloy fatigue results were supported by Weibull statistics analysis. Ti-6Al-4V alloy microstructural analysis showed equiaxed α + β grains. Weibull analysis at untreated condition presented m values greater than 1, indicating reliability and uniformity. For a lifetime of 10 7 cycles, fatigue resistance was equal to 829 MPa for untreated condition, 644, 767 and 417 MPa, for f = 1000, 1200 and 1500 Hz, respectively. The nitrogen-based compounds were detected only at the condition where f = 1200 Hz. Thus, the combination of PIII treatment parameters, when f = 1200 Hz, hindered crack nucleation and increasing fatigue resistance of treated Ti-6Al-4V alloy when compared with the other two treatment conditions.

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