An analytical study on cracking directions and damage in thermal fatigue crazing subjected to variable amplitude loadings

Abstract

In this paper, the effects of biaxial mean stress, mainly contributed by the weld residual stress, and thermal loading conditions on cracking directions and damage in high cycle thermal fatigue crazing subjected to variable amplitude loadings are investigated by a combined analytical and computational approach. The cracking directions are related to the orientation of the critical plane defined by the maximum damage. Analytical solutions of the critical plane orientation under constant amplitude biaxial tension/compression loadings are first derived and then employed to study the effects of biaxial mean stress on the critical plane orientation. The critical plane orientation appears to strongly depend on the dominant direction defined by the larger maximum stress. The developed analytical solution of the critical plane orientation and the analytical solution of the thermal stress from the literature are employed to study the effects of thermal loading conditions on the critical plane orientation. The critical plane orientation does not seem to significantly depend on the frequency, the amplitude and the mean value of the fluid temperature fluctuations, and the heat transfer film coefficient between the fluid and the pipe wall. The critical plane orientation under variable amplitude loadings is also studied, and an approximate solution is proposed for convenient engineering applications. The critical plane orientations are used to partially explain the observed cracking directions in the high cycle thermal fatigue crazing in the old residual heat removal system of a nuclear power plant. Finally, the effects of biaxial mean stress and thermal loading conditions on the fatigue crack initiation life are discussed.

Appendix

The mathematical derivation process to obtain the analytical solution of the critical plane orientation consists of the following steps:

• (1) Consider the stress tensor ${\varvec{\upsigma}}$ which represents the biaxial tension/compression stress state in the elementary cube as

  $$ {\varvec{\upsigma}}\left( t \right) = \left[ {\begin{array}{*{20}c} {\sigma_{x} \left( t \right)} & 0 & 0 \\ 0 & {\sigma_{y} \left( t \right)} & 0 \\ 0 & 0 & 0 \\ \end{array} } \right] $$

  (24)

• (2) Calculate the stress vector $\overrightarrow {F}$ acting on the candidate plane Δ of the unit normal vector $\vec{n}$ as

  $$ \overrightarrow {F} \left( {n_{x} ,n_{y} ,t} \right) = {\varvec{\upsigma}} \cdot \overrightarrow {n} = \left[ {\begin{array}{*{20}c} {\sigma_{x} \left( t \right)n_{x} } \\ {\sigma_{y} \left( t \right)n_{y} } \\ 0 \\ \end{array} } \right] $$

  (25)

• (3) Project the stress vector $\overrightarrow {F}$ into a normal stress $\overrightarrow {N}$ and a shear stress $\overrightarrow {\tau }$ acting on the candidate plane Δ as

  $$ \overrightarrow {N} \left( {n_{x} ,n_{y} ,n_{z} ,t} \right) = \left( {\left( {\overrightarrow {F} } \right)^{\text{T}} \cdot \overrightarrow {n} } \right)\overrightarrow {n} = \left( {\sigma_{x} \left( t \right)n_{{_{x} }}^{2} + \sigma_{y} \left( t \right)n_{{_{y} }}^{2} } \right)\left[ {\begin{array}{*{20}c} {n_{x} } \\ {n_{y} } \\ {n_{z} } \\ \end{array} } \right] $$

  (26)
  $$ \overrightarrow {\tau } \left( {n_{x} ,n_{y} ,n_{z} ,t} \right) = \overrightarrow {F} - \overrightarrow {N} = \left[ {\begin{array}{*{20}c} {\sigma_{x} \left( t \right)n_{x} - \sigma_{x} \left( t \right)n_{{_{x} }}^{3} - \sigma_{y} \left( t \right)n_{x} n_{{_{y} }}^{2} } \\ {\sigma_{y} \left( t \right)n_{y} - \sigma_{x} \left( t \right)n_{{_{x} }}^{2} n_{y} - \sigma_{y} \left( t \right)n_{{_{y} }}^{3} } \\ { - \sigma_{x} \left( t \right)n_{{_{x} }}^{2} n_{z} - \sigma_{y} \left( t \right)n_{{_{y} }}^{2} n_{z} } \\ \end{array} } \right] $$

  (27)

• (4) Calculate the maximum normal stress $N_{\hbox{max} }$ and the shear stress range $\Delta \tau$ acting on the candidate plane Δ as

  $$ N_{\hbox{max} } \left( {n_{x} ,n_{y} } \right) = \mathop {\hbox{max} }\limits_{{t \in [0,t_{0} ]}} \left\| {\overrightarrow {N} \left( {n_{x} ,n_{y} ,n_{z} ,t} \right)} \right\| = \sigma_{x,\hbox{max} } n_{{_{x} }}^{2} + \sigma_{y,\hbox{max} } n_{{_{y} }}^{2} $$

  (28)
  $$ \begin{gathered} \Delta \tau \left( {n_{x} ,n_{y} } \right) = \mathop {\hbox{max} }\limits_{{i,j \in [0,t_{0} ]}} \left\| {\overrightarrow {\tau } \left( {n_{x} ,n_{y} ,n_{z} ,t = i} \right) - \overrightarrow {\tau } \left( {n_{x} ,n_{y} ,n_{z} ,t = j} \right)} \right\| \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = 2\sigma_{a} \sqrt {n_{{_{x} }}^{2} + n_{{_{y} }}^{2} - \left( {n_{{_{x} }}^{2} + n_{{_{y} }}^{2} } \right)^{2} } \; \hfill \\ \end{gathered} $$

  (29)

  Similarly, the strain vector can also be projected on the candidate plane. Note that an elastic material model is used. The shear strain range $\Delta \gamma$ can be expressed as

  $$ \Delta \gamma \left( {n_{x} ,n_{y} } \right) = \frac{{2\left( {1 + \nu } \right)\sigma_{a} }}{E}\sqrt {n_{{_{x} }}^{2} + n_{{_{y} }}^{2} - \left( {n_{{_{x} }}^{2} + n_{{_{y} }}^{2} } \right)^{2} } $$

  (30)

• (5) Calculate the equivalent parameter σ _eq of the modified Matake’s criterion and ε _eq of the modified Fatemi–Socie’s criterion. Note that $\sigma_{x,\hbox{max} } = \sigma_{a} \left( {\alpha_{x} + 1} \right)$ and $\sigma_{y,\hbox{max} } = \sigma_{a} \left( {\alpha_{y} + 1} \right)$. By substituting Eqs. (28) and (29) into Eq. (3), the equivalent parameter of the modified Matake’s criterion can be written as

  $$ \sigma_{\rm eq} \left( {n_{x} ,n_{y} } \right) = \sigma_{a} \left\{ {\sqrt {n_{x}^{2} + n_{y}^{2} - \left( {n_{x}^{2} + n_{y}^{2} } \right)^{2} } + a \cdot \hbox{max} \left\{ {0,\left[ {\left( {1 + \alpha_{x} } \right)n_{x}^{2} + \left( {1 + \alpha_{y} } \right)n_{y}^{2} } \right]} \right\}} \right\} $$

  (31)

  Similarly, the equivalent parameter of the modified Fatemi–Socie’s criterion can be written as

  $$ \varepsilon_{\rm eq} \left( {n_{x} ,n_{y} } \right) = \frac{{\sigma_{a} }}{E}\left( {1 + \nu } \right)\sqrt {n_{x}^{2} + n_{y}^{2} - \left( {n_{x}^{2} + n_{y}^{2} } \right)^{2} } \left\{ {1 + k\sigma_{a} \frac{{\hbox{max} \left\{ {0,\left[ {\left( {1 + \alpha_{x} } \right)n_{x}^{2} + \left( {1 + \alpha_{y} } \right)n_{y}^{2} } \right]} \right\}}}{{S_{y} }}} \right\} $$

  (32)

• (6) Finally, calculate the critical plane orientation by finding the maximum value of the equivalent parameter $\sigma_{\rm eq} \,(n_{x} ,\,n_{y} )$ or ε _eq (n _x , n _y ) which is a two-variable function. The analytical solution of the critical plane orientation is then obtained by solving the following equation system:

  $$ \left\{ {\begin{array}{*{20}l} {\frac{{\partial \sigma_{\rm eq} }}{{\partial n_{x} }} = 0} \hfill \\ {\frac{{\partial \sigma_{\rm eq} }}{{\partial n_{y} }} = 0} \hfill \\ {\left( {\frac{{\partial^{2} \sigma_{\rm eq} }}{{\partial n_{x} \partial n_{y} }}} \right)^{2} - \frac{{\partial^{2} \sigma_{\rm eq} }}{{\partial n_{x}^{2} }} \cdot \frac{{\partial^{2} \sigma_{\rm eq} }}{{\partial n_{y}^{2} }} < 0} \hfill \\ {\frac{{\partial^{2} \sigma_{\rm eq} }}{{\partial n_{x}^{2} }} < 0} \hfill \\ \end{array} } \right.\,{\text{or}}\,\left\{ {\begin{array}{*{20}l} {\frac{{\partial \varepsilon_{\rm eq} }}{{\partial n_{x} }} = 0} \hfill \\ {\frac{{\partial \varepsilon_{\rm eq} }}{{\partial n_{y} }} = 0} \hfill \\ {\left( {\frac{{\partial^{2} \varepsilon_{\rm eq} }}{{\partial n_{x} \partial n_{y} }}} \right)^{2} - \frac{{\partial^{2} \varepsilon_{\rm eq} }}{{\partial n_{x}^{2} }} \cdot \frac{{\partial^{2} \varepsilon_{\rm eq} }}{{\partial n_{y}^{2} }} < 0} \hfill \\ {\frac{{\partial^{2} \varepsilon_{\rm eq} }}{{\partial n_{x}^{2} }} < 0} \hfill \\ \end{array} } \right. $$

  (33)

Due to symmetry, only $n_{x} ,\;n_{y} ,\;n_{z} \ge 0$ are considered. Also, $\alpha_{x}$ and $\alpha_{y}$ are taken to be larger than −1 for the convenience of derivation. The analytical solution of the critical plane orientation, which is represented by the critical angles $\varphi_{x}$ and $\varphi_{y}$, is summarized in Table 1. Note that $\varphi_{z}$ is also presented for convenience.

In the table, the auxiliary functions C _M (α _i ) for the modified Matake’s criterion and C _F (α _i ) for the modified Fatemi–Socie’s criterion are defined as

$$ C_{\rm M} \left( {\alpha_{i} } \right) = \sqrt {\frac{{1 + \sqrt {1 - \frac{1}{{a^{2} (1 + \alpha_{i} )^{2} + 1}}} }}{2}} $$

(34)

$$ C_{\rm F} (\alpha_{i} ) = \sqrt {\frac{{3 - \frac{{2S_{y} }}{{k\sigma_{a} \left( {1 + \alpha_{i} } \right)}} + \sqrt {\left( {1 + \frac{{2S_{y} }}{{k\sigma_{a} \left( {1 + \alpha_{i} } \right)}}} \right)^{2} + 8} }}{8}} $$

(35)

where the subscript i = x, y represent the x and y directions, respectively.