Analytical approach to DHC description in zirconium alloys

Abstract

A new model for the threshold stress intensity factor (K_IH) for delayed hydride cracking (DHC) has been developed that account for the crack tip stress shielding by hydrides and plastic strains. The article reviews existing approaches to the evaluation of K_IH and key features that govern the DHC process: hydride shape and stress field at the crack tip. The evaluation and validation of hydride length and thickness at the crack tip have been carried out. The new parametric model for K_IH is based on a direct account for stress concentrator shielding by plastic zone and hydrides at the crack tip. The model results are in good agreement with experimental data.

Appendix

Let us consider a cylindrically symmetric diffusion task with a point sink near the crack tip. Diffusion flow is determined by the gradient of chemical potential $\mu = RT\ln C + \sigma v_{{\text{H}}}$:

$$ J = C\frac{D}{RT}\frac{d\mu }{{dr}} = D\frac{\partial C}{{\partial r}} - v_{{\text{H}}} C\frac{D}{RT}\frac{K}{2}r^{{ - {3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}} $$

(27)

It was assumed that mechanical stresses decrease as $\sigma = {K \mathord{\left/ {\vphantom {K {\sqrt r }}} \right. \kern-\nulldelimiterspace} {\sqrt r }}$ with distance r to the crack tip. The hydrogen concentration at stationary conditions must satisfy the flow conservation:

$$ divJ = \frac{1}{r}\frac{\partial }{\partial r}\left( {Jr} \right) = 0 $$

(28)

Substitute Eqs. 27) to (28):

$$ Dr\frac{\partial C}{{\partial r}} - C\frac{{v_{{\text{H}}} D}}{RT}\frac{K}{2}r^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} = Const $$

(29)

The solution of the homogeneous equation of type Eq. (29) is a function $C = \exp \left( { - {{Kv_{{\text{H}}} r^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} } \mathord{\left/ {\vphantom {{Kv_{{\text{H}}} r^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} } {RT}}} \right. \kern-\nulldelimiterspace} {RT}}} \right)$ and the full solution of the Eq. (29) is:

$$ C = Z\left( r \right)\exp \left( { - \frac{{Kv_{{\text{H}}} }}{RT}r^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} } \right) $$

(30)

Substituting Eq. (30) in (29):

$$ \frac{{{\text{d}}Z}}{{{\text{d}}r}} = \frac{Const}{{Dr}}\exp \left( {\frac{{Kv_{{\text{H}}} }}{RT}r^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} } \right) $$

(31)

Next, we make the variable change y = r^−1/2, − 2dy/y = dr/r:

$$ Z = - \frac{2Const}{D}\int {\exp \left( {\frac{{Kv_{{\text{H}}} }}{RT}y} \right)\frac{{{\text{d}}y}}{y}} = - \frac{2Const}{D}\left( {\ln r^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} + \sum\limits_{i = 1}^{\infty } {\frac{{\left( {\frac{{Kv_{{\text{H}}} }}{RT}r^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} } \right)^{i} }}{i \cdot i!}} + Const2} \right) $$

(32)

It should be noted that the stationary solution of hydrogen profile [Eqs. (30) and (32)] has no limit at r → ∞. The constants Const and Const2 in Eq. (32) can be found from the boundary conditions: the hydrogen concentration is equal to the C^TSSP near the crack tip at r₀ and concentration in the metal matrix C₀ at a large distance R₀ (say, as much as the crack length):

$$ \left\{ \begin{gathered} \left. C \right|_{{r = r_{0} }} = C^{TSSP} \hfill \\ \left. C \right|_{{r = R_{0} }} = C_{0} \hfill \\ \end{gathered} \right. $$

(33)

From Eqs. (30), (32) and (33):

$$ C^{\prime} = \frac{Const}{{DC^{TSSP} }} = \frac{{\exp \left( {\frac{{Kv_{{\text{H}}} }}{RT}r_{0}^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} } \right) - \frac{{C_{0} }}{{C^{TSSP} }}\exp \left( {\frac{{Kv_{{\text{H}}} }}{RT}R_{0}^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} } \right)}}{{2\left( {\ln \left( {{{r_{0} } \mathord{\left/ {\vphantom {{r_{0} } {R_{0} }}} \right. \kern-\nulldelimiterspace} {R_{0} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} + \sum\limits_{i = 1}^{\infty } {\frac{{\left( {\frac{{Kv_{{\text{H}}} }}{RT}R_{0}^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} } \right)^{i} - \left( {\frac{{Kv_{{\text{H}}} }}{RT}r_{0}^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} } \right)^{i} }}{i \cdot i!}} } \right)}} $$

(34)

Unitless parameter C’ is a weak function of two variables: (R₀/r₀)^1/2 and Kv_Hr₀^−1/2/RT. C’ plotted in Fig. 7 as a function on (R₀/r₀)^1/2 at three values of Kv_Hr₀^−1/2/RT: 10^–7, 10^–6 (for r₀ = 1 μm and T = 250 °C), and 10^–5; C⁰/C^TSSP was chosen equal to 1.1.

Fig. 7

Values of parameter C’. Value of Kv_Hr₀^−1/2/RT: solid line—10^–7 and 10^–6 (indistinguishable), dotted line—10^–5

For order-of-magnitude estimation the truncation radii r₀ = 1 μm, R₀ = 1 mm were chosen. Hydrogen flow from Eq. (28):

$$ J = DC^{\prime} \cdot \frac{{C^{{{\text{TSSP}}}} }}{r} $$

(35)

Now consider hydride size with the assumption that the hydride at the crack tip is a plate parallel to the crack with length L and thickness h. If all of the hydrogen transferred by the diffusion flux in a stationary approximation [Eq. (35)] precipitated in the form of hydrides, then:

$$ Lh = 2\pi r \cdot J\left( r \right)\Omega_{{\text{H}}} \cdot t = 2\pi \cdot DC^{\prime}C^{{{\text{TSSP}}}} \cdot \Omega_{{\text{H}}} \cdot t $$

(36)

Ω_H—the volume of a hydrogen atom in the hydride, Ω_H = v_H/N_A, where N_A—Avogadro constant.