Hydrogen Diffusion in Welds
Hydrogen cracking is an important cause of failures in many welded structures. Given a sufficiently sensitive microstructure, a sufficient hydrogen concentration and a high enough tensile stress state, hydrogen cracking, sometimes called cold cracking, is likely to occur. Hydrogen cracking usually involves delays of minutes to years. If cellulosic electrodes are used, it is usual to assume that the weld pool is saturated with hydrogen. The solubility of hydrogen is highest in the weld pool and next highest in austenite and lowest in ferrite or martensite.
Therefore a CWM model that predicts hydrogen cracking must compute the diffusion of hydrogen in a transient temperature, microstructure and stress field. Hydrogen diffuses toward high hydrostatic stresses, i. e., tension. This is particularly important at low temperatures where stresses are higher and hydrogen diffusivity lower. In a weld, the diffusivity of hydrogen varies by several orders of magnitude with temperature and with phase changes. In addition, it requires a model for crack initiation and propagation. The simplest models simply assume the risk of cracking is too high if the hydrogen level exceeds a certain threshold level and the microstructure is sensitive to hydrogen cracking.
The hydrogen in a welded joint has three sources: from base metal and filling metals, from other consumables and from ambient air. The fraction of hydrogen from base metal and filling metal is essentially negligible because of their low hydrogen levels.
Microscopically, plastic deformations, dislocations, micro-voids, various inclusions, etc. may exist in a welded joint dispersively and vary from here to there. The activation coefficient will thus be greatly different from a “sound crystal”. As a result, the diffusion does not follow Fick’s law .
Recent work by Olson et al [30-32] clearly demonstrates the need for coupled weld metal and heat affected zone (WM/HAZ) approaches for dealing with the general problem of НІС, especially in regard to the high strength, low carbon materials. This is a consequence of the fact that the transformation temperature of the WM can be either higher or lower than the base material depending on relative filler and base metal compositions. It is said that the location of hydrogen cracks depends to a great extent on the relative martensite start temperatures of the two zones. If the martensite start temperature Ms of the WM is higher than of the HAZ, hydrogen accumulates in the coarse grained region. Austenite acts as something of a barrier to hydrogen movement into the HAZ, which accumulates in the boundary area causing the НІС problem at this location. On the other hand if the Ms of the WM is lower than that of the HAZ, there is less hydrogen accumulation in the sensitive boundary region and НІС is more likely in the WM. Bibby et al  propose an index based on the WM/HAZ difference in martensite start temperatures as follows:
AM = M - Ms (6-8)
5 SWM SIIAZ x '
M, =521-350С -14.30 -17.5M - 28.9 Mn
-37.65/ - 29.5M? -1.19Cr. Ni + 23.1(0 + Mo).С
Ms = 521 -350C-13.6Cr-16.6M-25.1Mn
wu (6-10) - 30.15/ - 40AMo - 40Al -1 .OlCr. Ni + 21.9(Cr + 0.1ЪМо).С
ForAAfy >0, the fracture is expected to occur in the HAZ and
vice versa for AMS < 0 .
The driving force of hydrogen diffusion is the gradient of chemical potential, that is:
F = - VjU (6-11)
where Ц (cal/mol) is chemical potential, and:
where /u° is the chemical potential when a=l, namely standard chemical potential; a is the activity, R is the gas constant and T is the absolute temperature.
Generally the physical-chemical behavior of hydrogen during diffusion is represented by the activation coefficient у and the product of the у by the hydrogen concentration c, i. e., the so called activity a. That is a= у с.
Microstructural defects, such as grain-boundaries, dislocations and voids can be regarded as hydrogen “traps”. When hydrogen diffuses and passes through them, it will be trapped partially. The “trapped” hydrogen does not participate in diffusion and only the “effective, diffusible” hydrogen, as the diffusing mass, must be considered in the diffusion equation. The activity reflects the very difference between the hydrogen concentration in the considered medium with and without these defects. The activity a may be called also “corrected hydrogen concentration” or “effective diffusible hydrogen concentration”, .
Under a driving force F, the hydrogen moves with average velocity:
v = — F (6-13)
Then the hydrogen flux, (mol/m2s):
J=vc = -^-SZjU (6-14)
RT v '
The hydrogen diffusivity has the general form £) = £)0 exp [——],
m2/s. The hydrogen diffusivity in austenite and ferrite as function of temperature :
Z) = l. lxl(T6exp(- 41600 ) (6-15)
Z) = 0.22 x 10 exp(----------------- ) (6-16)
By the law of conservation mass, if there is a hydrogen-source with an intensity of QH in certain infinitesimal element, the hydrogen concentration, c, should satisfy the following integral equation; n is the unit vector outwards and perpendicular to the surface:
J^<i£2+ J J-nds - ^QHd£l (6-17)
fi ® 92Q Q
From Gauss-Green integration law, the above Equation can be rewritten as:
^ + V-J-QH=0 (6-18)
The equation above is proposed by Zhang et al [3 and 24] as the general equation for hydrogen diffusion.
When hydrogen diffuses into a metal containing a stress field o, there is an increase in the volume A V = Vhn, where Vh is the volume
increase due to one hydrogen atom or one mole of hydrogen atoms
and n is the number of hydrogen atoms or number of moles of
hydrogen added. The potential from the stress field that causes a hydrogen flux is, :
У(о-) = >/„(у+~£) (6-19)
The flux associated with the stress field is:
The partial molar volume of hydrogen in ferrite, , is
V = 2.5x1 0“6 (m3/gram-atom), and in austenite, , is
V = 1.75xl0“6 (m3/gram-atom)
The evolution of hydrogen content as a function of temperature in a stress field is:
c-[V-DVc +V-( 4ys)]-QH =0
If с = 0 or V у/ = 0 on a boundary, then the hydrogen flux J on this boundary is zero. If сф OorV^^O on a boundary, then the hydrogen flux on this boundary J is not zero. Sofronis et al  presents a solution for the hydrogen diffusion near a crack tip.
Streitberger and Koch have given a full account of the numerical solution of the diffusion equation with a stress term for a sharp crack tip under different loading conditions. The two-dimensional time - dependent drift-diffusion equation for a crack tip under mixed-mode loading and with tip as an ideal sink for solute atoms is solved by a finite difference method .
In all theoretical models it is assumed that the crack growth rate depends in a crucial way on the segregation rate of the embrittling solute to the crack tip region, whereas the segregation rate is governed by the concentration field с = c(r, t) of the solute obeying the diffusion equation:
— = DAc + — (VE)Vc +—cAE dt kT kT
where D is the bulk or grain boundary diffusivity, T the absolute temperature, к the Boltzmann constant and E = E(r) the relevant interaction energy between the external stress field and the impurity.
In an alternative model of dynamic embrittlement the equation (622) is considered for the somewhat simpler problem of the onedimensional diffusion along a grain boundary ahead of a plastic crack tip. Only for the specific problem of transient hydrogen transport near a blunting crack has a full numerical analysis of the diffusion initial-boundary-value problem in conjunction with the
elastic-plastic boundary value problem been carried out using finite element procedures .
With the assumption that the solute flow in the vicinity of the crack tip is dominated by the strong and long-ranged elastic interaction field and that the flow arising from random diffusion processes can be ignored at least for small time, the equation (6-22) for the harmonic field takes the explicit form following, [25 and 27]: де AD. 6 Эс в 1 Эс
A = (—)m(} + v)KAr - я <6 <x, r «а (6-23)
where К - <Tj (ж?)12 is the stress intensity factor.
The complete drift-diffusion equation (6-22), which governs the migration of point defects around the crack tip under the action of the harmonic interaction potential (6-23), takes the form:
Эс Э 2c 1 Эс 1 Э2с
dt dr2 r dr г2 дв2
Qsinia) . . ,6 .Эс.6 . 1 Эсч
_ с _ г - Dt where с - —,г =—,t = —г-are the scaled variables and the
c0 R R2
parameter Q is defined by:
R is an additional length scale that defines a large cylindrical region around the crack tip where the numerical solution of equation (6-24) is performed. The initial and boundary conditions are respectively: c(t, r =1,6) = 1
_ - _ I (6-26)
c(t, r =a,6) = 0
Furthermore, at the crack faces (6 = ±K, r > a) the tangential component of the solute flux density is required to be zero:
je = — + —j sin a cos(— - a) = 0 for 6 = ±n (6-27)
Эс Qc. .6
— + —sin a cos(— д6 r 2
The zero-flux boundary condition (6-27) is required to model impermeable crack faces.