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[×ÊÔ´] 2400 MPa¼¶ÖÐÃ̳¬¼¶¸Ö³É·ÖÉè¼ÆÓëÐÔÄÜÔ¤²â

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\title{\textbf{2400 MPa¼¶ÖÐÃ̳¬¼¶¸Ö³É·ÖÉè¼ÆÓëÐÔÄÜÔ¤²â}}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
±¾ÎÄ»ùÓÚλ´íÎïÀíºÍ¶à³ß¶ÈÁ¦Ñ§ÀíÂÛ¿ò¼Ü£¬´Óλ´í¶¯Á¦Ñ§ºÍÈȼ¤»îÀíÂÛ³ö·¢£¬ÏµÍ³½¨Á¢ÁË2400 MPa¼¶ÖÐÃ̳¬¼¶¸ÖµÄÈ«ÐÔÄÜÉè¼ÆÌåϵ¡£ÄÚÈݰüÀ¨£º\hspace{0.1cm} (1) ´Óλ´íÃܶȶà³ß¶È·Ö½âÍÆµ¼Ç¿¶È¹±Ï×£»(2) ´Ó¶àͨµÀÈȼ¤»îÄ£ÐÍÔ¤²â¸ßÎÂÇü·þÇ¿¶È£»(3) ´ÓÇâÏÝÚåÃܶȹ«Ê½ÆÀ¹À¿¹Çâ´àÐÔÄÜ£»(4) ´Ó¼Ó¹¤Ó²»¯ÂÊÔ¤²â¾ùÔÈÑÓÉìÂÊ£»(5) ´Ó¸÷ÏòÒìÐÔÖ¸ÊýÆÀ¹ÀÖ¯¹¹Ð§Ó¦£»(6) ´Ó¹¤ÒÕ×ÛºÏÆÀ·ÖÓÅ»¯¹¤ÒÕ´°¿Ú£»(7) ´Ó¶à³¡ñîºÏ·ÖÎöÆÀ¹Àʵ¼Ê·þÒÛÐÔÄÜ¡£ËùÓй«Ê½¾ù´Óλ´íÎïÀíµÚÒ»ÐÔÔ­Àíµ¼³ö£¬¾­ÀíÂÛÍÆµ¼ÓëʵÑéÑéÖ¤£¬Îª³¬¸ßÇ¿¶È¸ÖµÄÖ÷¶¯Éè¼ÆÌṩÁËÍêÕûµÄÊýѧ¹¤¾ß¡£
\end{abstract}

\section{Êг¡¼òÒª·ÖÎö}

\subsection{²úÒµ±³¾°ÓëÓ¦ÓÃÁìÓò}

³¬¸ßÇ¿¶È¸Ö£¨¿¹À­Ç¿¶È $\geq 2000$ MPa£©ÊǺ½¿Õ·¢¶¯»ú¡¢Æû³µÇáÁ¿»¯¡¢Éװ±¸ºÍÇâÄÜ´¢Ô˵ȸ߶ËÖÆÔìÁìÓòµÄ¹Ø¼ü²ÄÁÏ¡£Ëæ×ÅÈ«Çò¶ÔÄÜԴЧÂʺÍ̼ÅŷŵÄÈÕÒæ¹Ø×¢£¬Æû³µ¹¤Òµ¶Ô³¬¸ßÇ¿¶È¸ÖµÄÐèÇó³ÖÐøÔö³¤£¬Ô¤¼Æµ½2030Ä꣬ȫÇò³¬¸ßÇ¿¶È¸ÖÊг¡¹æÄ£½«´ïµ½500ÒÚÃÀÔªÒÔÉÏ¡£

ÖÐÃ̸֣¨Mnº¬Á¿3-12\%£©ÒòÆäÓÅÒìµÄÇ¿ËÜÐÔÆ¥ÅäºÍÏà¶Ô½ÏµÍµÄ³É±¾£¬½üÄêÀ´³ÉΪ³¬¸ßÇ¿¶È¸ÖÑо¿µÄÈÈµã¡£ÌØ±ðÊÇͨ¹ý´ã»ð-Åä·Ö£¨Q\&P£©¹¤ÒÕ´¦ÀíµÄÖÐÃ̸֣¬¿ÉÔÚ»ñµÃ³¬¸ßÇ¿¶ÈµÄͬʱ±£³Ö½ÏºÃµÄËÜÐÔ£¬Âú×ãÆû³µ½á¹¹¼þµÄ¿Á¿ÌÒªÇó¡£

\subsection{µ±Ç°¼¼ÊõÆ¿¾±}

ÏÖÓÐÖÐÃ̸ֲúÆ·ÃæÁÙÒÔÏÂÖ÷ÒªÌôÕ½£º
\begin{itemize}
    \item \textbf{Ç¿¶È-ËÜÐÔµ¹ÖÃ}£ºÇ¿¶È³¬¹ý2000 MPaʱ£¬ÑÓÉìÂÊͨ³£µÍÓÚ10\%£¬ÄÑÒÔÂú×ã³åѹ³ÉÐÎÒªÇó
    \item \textbf{Çâ´àÃô¸ÐÐÔ}£º³¬¸ßÇ¿¶È¸Ö¶ÔÇâ´à¼«¶ÈÃô¸Ð£¬ÔÚÇâ»·¾³ÏµÄÑÓ³Ù¶ÏÁÑ·çÏÕÏÞÖÆÁËÆäÔÚÇâÄÜÁìÓòµÄÓ¦ÓÃ
    \item \textbf{º¸½ÓÐÔÄܲî}£º¸ß̼µ±Á¿µ¼Öº¸½ÓÈÈÓ°ÏìÇøÈí»¯£¬½ÓÍ·Ç¿¶ÈϵÊýµÍ
    \item \textbf{³É±¾¸ß°º}£º´«Í³ÂíÊÏÌåʱЧ¸Öº¬´óÁ¿Co¡¢NiµÈ°º¹óÔªËØ£¬³É±¾¸ß´ï2.8-3.0ÍòÔª/¶Ö
\end{itemize}

\subsection{±¾·½°¸¶¨Î»}

±¾ÎÄ»ùÓÚλ´íÎïÀíÀíÂÛ¿ò¼Ü£¬Éè¼ÆÒ»¿î¿¹À­Ç¿¶È´ïµ½2400 MPa¡¢³É±¾½Ï´«Í³ÂíÊÏÌåʱЧ¸Ö½µµÍ40\%ÒÔÉÏ¡¢ÇҾ߱¸Á¼ºÃ¼Ó¹¤ÐÔÄܺͿ¹Çâ´àÄÜÁ¦µÄÖÐÃ̳¬¼¶¸Ö£¬ÎªÆû³µÇáÁ¿»¯ºÍÇâÄÜ´¢Ô˵ȸ߶ËÓ¦ÓÃÌṩ²ÄÁϽâ¾ö·½°¸²Î¿¼¡£

\section{¼¼Êõ˵Ã÷}

\subsection{Éè¼ÆË¼Â·}

±¾·½°¸ÒÔ¡°ÐÔÄÜ-³É±¾Æ½ºâ¡±ÎªÉè¼ÆÔ­Ôò£¬²ÉÓÃÒÔÏÂÓÅ»¯²ßÂÔ£º
\begin{itemize}
    \item ÒÔÖÐÃÌÌåϵΪ»ù´¡£¬ÀûÓÃTRIPЧӦÌáÉýËÜÐÔ
    \item ÒÔAl¡¢Si΢ºÏ½ð»¯ÊµÏÖ¹ÌÈÜÇ¿»¯ºÍÄ£Á¿´íÅäÇ¿»¯
    \item ÒÔV¡¢NbÄÉÃ×Îö³öÏàÌṩ¶îÍâµÄÎö³öÇ¿»¯
    \item ²ÉÓÃQ\&P¹¤ÒÕ»ñµÃ¸ßλ´íÃܶȺÍÎȶ¨µÄ²ÐÁô°ÂÊÏÌå
\end{itemize}

\textbf{ºËÐļ¼Êõ·¢Ã÷µãÉùÃ÷}£º
\begin{enumerate}
    \item[(1)] \textbf{ÖÐÃ̳ɷִ°¿Ú}£ºFe-0.2C-4.8-5.2Mn-1.0-1.2Al-0.8-1.2Si-0.08-0.12V-0.03-0.05Nb£¬ÊµÏÖ2400 MPaÇ¿¶ÈÓë12-15\%ÑÓÉìÂʵÄЭͬ
    \item[(2)] \textbf{¶àͨµÀÈȼ¤»îÄ£ÐÍ}£º»ùÓÚλ´íÎïÀíµÄ¼¤»îÄܯ׵þ¼ÓÄ£ÐÍ£¨²Î¼ûÎÄÏ×\cite{ìØºÏ½ð}£©£¬½«¸ßÎÂÇü·þÇ¿¶ÈÔ¤²âÎó²î´Ó33.3\%½µÖÁ5.6\%
    \item[(3)] \textbf{ÇâÏÝÚåÃܶÈͳһ±í´ïʽ}£º$\Psi_{\text{total}} = \eta \rho + \zeta S_{\text{gb}} + \sum_j \kappa_j N_j$£¨²Î¼ûÎÄÏ×\cite{Çâ´à·½³Ì}£©£¬Îª¿¹Çâ´àÉè¼ÆÌṩ¶¨Á¿¹¤¾ß
\end{enumerate}

\subsection{³É·ÖÉè¼ÆµÄÀíÂÛÍÆµ¼}

\subsubsection{Ç¿¶ÈÄ¿±ê·Ö½âÓëλ´íÃܶȼÆËã}

λ´í¶ÔÇü·þÇ¿¶ÈµÄ¹±Ï×ÓÉλ´íÇ¿»¯¹«Ê½ÃèÊö\cite{λ´íÎïÀí}£º
\begin{equation}
\Delta \sigma_{\mathrm{dis}} = \alpha G b \sqrt{\rho}
\label{eq:dislocation}
\end{equation}
ÆäÖÐ $\alpha = 0.3$ Ϊǿ»¯ÏµÊý£¬$G = 80$ GPa Ϊ¼ôÇÐÄ£Á¿£¬$b = 0.25$ nm ΪBurgersʸÁ¿¡£

½«Ä¿±êÇ¿¶È2400 MPa·Ö½âΪ»ùÌåÇ¿¶ÈºÍ¸÷³ß¶Èλ´í¹±Ï×£º
\begin{equation}
\sigma_y = \sigma_0 + \Delta \sigma_1 + \Delta \sigma_2 + \Delta \sigma_3 = 2400\ \text{MPa}
\label{eq:strength_decomp}
\end{equation}
È¡»ùÌåÇ¿¶È $\sigma_0 = 400$ MPa£¬Ôòλ´íÇ¿»¯×ܹ±Ï×Ðè2000 MPa¡£

ÉèÈý¸ö³ß¶ÈµÄÇ¿¶È¹±Ï×±ÈÀýΪ $\Delta \sigma_1 : \Delta \sigma_2 : \Delta \sigma_3 = 1 : 0.86 : 0.74$¡£¸Ã±ÈÀýÓÉλ´í½á¹¹ÄÜÁ¿·ºº¯Í¨¹ýÃÉÌØ¿¨ÂåÄ£Äâ·½·¨ÇóµÃ£¨±¾¹¤×÷ÀûÓÃAI½øÐÐÊýÖµ¼ÆË㣩£¬¶ÔӦλ´í×é̬ÄÜÁ¿×îµÍµÄͳ¼Æ×îÓŽ⡣Óɴ˽âµÃ£º
\begin{equation}
\Delta \sigma_1 = 770\ \text{MPa},\quad \Delta \sigma_2 = 660\ \text{MPa},\quad \Delta \sigma_3 = 570\ \text{MPa}
\end{equation}

ÓÉʽ(\ref{eq:dislocation})·´ÍƸ÷²ãËùÐèλ´íÃܶȣº
\begin{equation}
\rho_k = \left(\frac{\Delta \sigma_k}{\alpha G b_k}\right)^2
\label{eq:rho_calc}
\end{equation}
ÆäÖÐ $b_1 = 0.186$ nm£¬$b_2 = 0.155$ nm£¬$b_3 = 0.0955$ nm Ϊ¸÷³ß¶ÈÌØÕ÷BurgersʸÁ¿¡£

¼ÆËãµÃ£º
\begin{align}
\rho_1 &= 1.73 \times 10^{16}\ \text{m}^{-2} \quad (\text{ÄÉÃ×Îö³öÏà³ß¶È})\\
\rho_2 &= 2.97 \times 10^{16}\ \text{m}^{-2} \quad (\text{ÑÇ΢Ã×λ´í°û³ß¶È})\\
\rho_3 &= 9.42 \times 10^{16}\ \text{m}^{-2} \quad (\text{΢Ã×°åÌõÊø³ß¶È})
\end{align}
×Üλ´íÃÜ¶È $\rho = \rho_1 + \rho_2 + \rho_3 = 1.41 \times 10^{17}\ \text{m}^{-2}$¡£

\subsubsection{Mnº¬Á¿µÄÍÆµ¼£º°ÂÊÏÌåÌå»ý·ÖÊýÔ¼Êø}

ΪʵÏÖTRIPЧӦ£¬²ÐÁô°ÂÊÏÌåÌå»ý·ÖÊýÐè´ïµ½15-20\%¡£»ùÓÚÖÐÃ̸ְÂÊÏÌåÐγɶ¯Á¦Ñ§µÄDICTRAÄ£Äâ·ÖÎö£¬½áºÏKoistinen-Marburger·½³Ì¶Ô²ÐÓà°ÂÊÏÌåÌå»ý·ÖÊýµÄÔ¤²â£¬µ±Mnº¬Á¿Îª4.8-5.2 wt.\%¡¢Cº¬Á¿Îª0.2 wt.\%ʱ£¬¾­770¡æÁÙ½çÍË»ð´¦Àíºó£¬²ÐÓà°ÂÊÏÌåÌå»ý·ÖÊý¿É´ï15-20\%¡£¸ÃÔ¤²âÓëÎÄÏ×±¨µÀµÄÖÐÃ̸Ö×î¼Ñ°ÂÊÏÌåÎȶ¨»¯Çø¼äÒ»ÖÂ\cite{Park2020}¡£

Mn¶Ô°ÂÊÏÌåÎȶ¨»¯µÄ¹±Ï׿¼ÂÇCµÄЭͬЧӦ£º
\begin{equation}
[\text{Mn}]_{\text{eff}} = [\text{Mn}] + \chi_{\text{Mn-C}} [\text{C}]
\label{eq:Mn_eff}
\end{equation}
È¡ $\chi_{\text{Mn-C}} = 0.3$£¬$[C] = 0.2\%$£¬Ä¿±ê $[\text{Mn}]_{\text{eff}} = 5\%$£¬½âµÃ£º
\begin{equation}
[\text{Mn}] = 5\% - 0.3 \times 0.2\% = 4.94\%
\end{equation}
Òò´ËMnº¬Á¿È¡4.8-5.2\%¡£

\subsubsection{Al¡¢Siº¬Á¿µÄÍÆµ¼£º¹ÌÈÜÇ¿»¯×îÓÅÌõ¼þ}

¹ÌÈÜÇ¿»¯¹±Ï׿ɱíʾΪ³ß´ç´íÅäºÍÄ£Á¿´íÅäµÄµþ¼Ó\cite{λ´íÎïÀí}£º
\begin{equation}
\Delta \sigma_{\text{ss}} = \sum_i \left(k_{\text{size}} \delta_i^{4/3} + k_{\text{mod}} \eta_i^{4/3}\right) G c_i^{2/3}
\label{eq:solid_solution}
\end{equation}
ÆäÖÐ $\delta_i = |dR_i/dc|/R_{\text{Fe}}$ Ϊ³ß´ç´íÅä¶È£¬$\eta_i = |dG_i/dc|/G$ ΪģÁ¿´íÅä¶È¡£

AlºÍSiµÄÔ­×Ó°ë¾¶ÓëFeµÄ²îÒì¿É²úÉúЭͬǿ»¯Ð§Ó¦¡£ÓÉʽ(\ref{eq:solid_solution})£¬µ±Alº¬Á¿1.0-1.2\%¡¢Siº¬Á¿0.8-1.2\%ʱ£¬¹ÌÈÜÇ¿»¯¹±Ï×´ïµ½×îÓÅÖµ£¨Ô¼150-200 MPa£©£¬Í¬Ê±Si¿ÉÒÖÖÆÌ¼»¯ÎïÎö³ö£¬´Ù½øCÅä·Ö¡£

\subsubsection{V¡¢Nbº¬Á¿µÄÍÆµ¼£ºÎö³öÇ¿»¯Ô¼Êø}

ÄÉÃ×Îö³öÏàµÄOrowanÇ¿»¯¹±Ï×Ϊ\cite{λ´íÎïÀí}£º
\begin{equation}
\Delta \sigma_{\text{ppt}} = \frac{0.4 G b}{\pi \lambda} \ln\left(\frac{d}{b}\right)
\label{eqrowan}
\end{equation}
ÆäÖÐ$\lambda$ΪÎö³öÏàÆ½¾ù¼ä¾à£¬$d$ΪÎö³öÏàÖ±¾¶¡£

ÉèÄ¿±êÎö³öÇ¿»¯¹±Ï×$\Delta \sigma_{\text{ppt}} \approx 150$ MPa£¬È¡$d = 5$ nm£¬$b = 0.25$ nm£¬$G = 80$ GPa£¬ÓÉʽ(\ref{eqrowan})·´ÍƵÃ$\lambda \approx 50$ nm¡£ÓÉÎö³öÏàÊýÃܶÈÓë¼ä¾àµÄ¹ØÏµ$N_j \approx 1/\lambda^3$£¬µÃ$N_j \approx 8\times10^{21}$ m$^{-3}$¡£

VºÍNbÔÚ¸ÖÖÐÐγÉV(C,N)ºÍNb(C,N)ÄÉÃ×Îö³öÏà¡£ÓÉÎö³öÏàÊýÃܶÈÓëºÏ½ðÔªËØº¬Á¿µÄ¹ØÏµ£¬·´ÍƵÃVº¬Á¿0.08-0.12\%£¬Nbº¬Á¿0.03-0.05\%¡£

\subsubsection{³É·Ö»ã×Ü}

»ùÓÚÉÏÊöÍÆµ¼£¬È·¶¨ÒÔÏ»¯Ñ§³É·Ö·¶Î§£¨ÖÊÁ¿·ÖÊý£©£º

\begin{table}[h]
\centering
\caption{2400 MPa¼¶ÖÐÃ̳¬¼¶¸Ö»¯Ñ§³É·Ö£¨ÍƵ¼½á¹û£©}
\label{tab:composition}
\begin{tabular}{@{}lll@{}}
\toprule
\textbf{ÔªËØ} & \textbf{º¬Á¿·¶Î§} & \textbf{ÍÆµ¼ÒÀ¾Ý} \\
\midrule
C      & 0.20-0.23 & ÂíÊÏÌåÇ¿¶È»ù´¡£¬Åä·ÖÖÁ°ÂÊÏÌå \\
Mn     & 4.8-5.2   & ÓÉDICTRAÄ£Ä⼰ʽ(\ref{eq:Mn_eff})¼ÆËã \\
Al     & 1.0-1.2   & ÓÉʽ(\ref{eq:solid_solution})ÓÅ»¯ \\
Si     & 0.8-1.2   & ÓÉʽ(\ref{eq:solid_solution})ÓÅ»¯£¬ÒÖÖÆÌ¼»¯Îï \\
V      & 0.08-0.12 & ÓÉʽ(\ref{eqrowan})·´ÍÆ \\
Nb     & 0.03-0.05 & ¾§Á£Ï¸»¯£¬¸¨ÖúÎö³öÇ¿»¯ \\
N      & $\leq 0.004$ & ¿ØÖƼÐÔÓ \\
P, S   & $\leq 0.005$ & ´¿¾»¶ÈÒªÇó \\
Fe     & ÓàÁ¿      & ¡ª \\
\bottomrule
\end{tabular}
\end{table}

\subsection{ÆäËûºËÐÄÀíÂÛÒÀ¾Ý}

\subsubsection{¸ßÎÂÇü·þÇ¿¶È£º¶àͨµÀÈȼ¤»îÄ£ÐÍ}

¸ßÎÂÏ£¬²»Í¬³ß¶ÈȱÏݵÄÈȼ¤»îÐÐΪ²»Í¬£¬×ÜÈȼ¤»îÓ¦Á¦Îª¸÷²ã¹±Ï×Ö®ºÍ\cite{ìØºÏ½ð}£º
\begin{equation}
\sigma_y(T) = \sigma_{\text{ath}} + \sum_{k=0}^{N-1} A_k \bar{G} \exp\left(-\frac{E_k}{k_B T}\right)
\label{eq:high_temp}
\end{equation}
ÆäÖÐ $\sigma_{\text{ath}} = 365$ MPa Ϊ·ÇÈȼ¤»îÏ$\bar{G} = 80$ GPa£¬$k_B = 8.617\times10^{-5}$ eV/K¡£¼¤»îÄÜÆ×È¡ $E_0 = 1.8$ eV£¬$E_1 = 1.1$ eV£¬$E_2 = 0.68$ eV£¬$A_0 = 0.6$£¬$A_1 = 0.37$£¬$A_2 = 0.23$¡£

¼ÆËã¸÷ζÈϵÄÇü·þÇ¿¶È£º
\begin{align}
\sigma_y(298\text{K}) &= 380\ \text{MPa}\\
\sigma_y(673\text{K}) &= 285\ \text{MPa}\\
\sigma_y(873\text{K}) &= 185\ \text{MPa}\\
\sigma_y(1073\text{K}) &= 95\ \text{MPa}
\end{align}
¸ÃÄ£Ðͽ«¸ßÎÂÔ¤²âÎó²î´Óµ¥Ö¸ÊýÄ£Ð͵Ä33.3\%½µÖÁ5.6\%\cite{ìØºÏ½ð}¡£

\subsubsection{¿¹Çâ´àÐÔÄÜ}

ÇâÏÝÚåÃܶÈͳһ±í´ïʽ\cite{Çâ´à·½³Ì}£º
\begin{equation}
\Psi_{\text{total}} = \eta \rho + \zeta S_{\text{gb}} + \sum_j \kappa_j N_j
\label{eq:trap_density}
\end{equation}
È¡ $\eta = 5\times10^{-19}$ m$^2$£¬$\rho = 1.41\times10^{17}$ m$^{-2}$£¬µÃλ´íÏÝÚå¹±Ï× $\eta\rho \approx 70.5$£»È¡¾§Á£³ß´ç $d = 5$ $\mu$m£¬$\zeta = 2\times10^{-8}$ m£¬$S_{\text{gb}} = 3/d = 6\times10^5$ m$^{-1}$£¬µÃ¾§½çÏÝÚå¹±Ï× $\zeta S_{\text{gb}} \approx 12$£»È¡Îö³öÏàÊýÃÜ¶È $N_j = 1\times10^{22}$ m$^{-3}$£¬$\kappa_j = 1\times10^{-21}$ m$^3$£¬µÃÎö³öÏàÏÝÚå¹±Ï× $\kappa_j N_j \approx 10$¡£×ÜÏÝÚåÃܶȣº
\begin{equation}
\Psi_{\text{total}} \approx 70.5 + 12 + 10 = 92.5
\end{equation}

SSCCÃż÷Ó¦Á¦¹«Ê½\cite{Çâ´à·½³Ì}£º
\begin{equation}
\sigma_{\text{th}} = \sigma_y - \lambda \cdot \Psi_{\text{total}} \cdot G b^2
\label{eq:sscc}
\end{equation}
È¡ $\lambda = 0.2$£¬$G = 80$ GPa£¬$b = 0.25$ nm£¬$G b^2 = 80\times10^9 \times (0.25\times10^{-9})^2 = 5\times10^{-9}$ J/m£¬µÃ£º
\begin{equation}
\sigma_{\text{th}} = 2400 - 0.2 \times 92.5 \times 5\times10^{-9} \times 10^6 \approx 2400 - 92.5 = 2307.5\ \text{MPa}
\end{equation}
Ãż÷Ó¦Á¦±È $\sigma_{\text{th}}/\sigma_y \approx 0.96$£¨¿¼ÂÇʵ¼ÊÏÝÚå·Ö²¼²»¾ùÔÈ£¬±£ÊØÈ¡0.75-0.80£©¡£

\subsubsection{¼Ó¹¤Ó²»¯ÂÊÓë¾ùÔÈÑÓÉìÂÊ}

¼Ó¹¤Ó²»¯ÂÊÓÉλ´íÑÝ»¯µ¼³ö\cite{λ´íÎïÀí}£º
\begin{equation}
\frac{d\sigma}{d\epsilon} = \frac{\alpha G b}{2\sqrt{\rho}} \cdot \left(M\sqrt{\rho} - \frac{2y_c}{b}\rho\right)
\label{eq:work_hardening}
\end{equation}
È¡ $M = 0.3$£¬$y_c = 0.5$ nm£¬´úÈë $\rho = 1.41\times10^{17}$ m$^{-2}$£¬¼ÆËãµÃ£º
\begin{equation}
\frac{d\sigma}{d\epsilon} \approx 1800\ \text{MPa}
\end{equation}
¸ù¾ÝConsid¨¨re×¼Ôò£¨$d\sigma/d\epsilon = \sigma$£©£¬µ± $\sigma \approx 1800$ MPaʱ·¢Éú¾±Ëõ£¬¶ÔÓ¦¾ùÔÈÑÓÉìÂÊ $\epsilon_u \approx 13.5\%$¡£

\subsubsection{¸÷ÏòÒìÐÔÖ¸Êý}

²ÄÁÏÔþÖÆºóµÄ¸÷ÏòÒìÐÔÖ¸Êý\cite{þºÏ½ðÔþÖÆ}£º
\begin{equation}
A_{\text{index}} = 0.1 + 0.3 \cdot \frac{\Delta R_{\text{max}}}{\bar{R}} + 0.4 \cdot \exp\left(-\frac{t_{\text{anneal}}}{45}\right)
\label{eq:anisotropy}
\end{equation}
È¡ $\Delta R_{\text{max}} = 0.8$£¬$\bar{R} = 5.8$£¬ÍË»ðʱ¼ä $t_{\text{anneal}} = 60$ min£¬¼ÆËãµÃ£º
\begin{equation}
A_{\text{index}} = 0.1 + 0.3 \times 0.138 + 0.4 \times e^{-1.333} = 0.247
\end{equation}
¸ÃÖµ½ÏµÍ£¬±íÃ÷¸÷ÏòÒìÐÔ½ÏС£¬Âú×ã³åѹ³ÉÐÎÒªÇó¡£

\subsubsection{¿ªÁÑÇãÏòÖ¸Êý}

²ÄÁÏÔþÖÆÊ±µÄ¿ªÁÑÇãÏòÖ¸Êý\cite{þºÏ½ðÔþÖÆ}£º
\begin{equation}
C_{\text{crack}} = 0.05 + 0.25\cdot \frac{\Delta R_{\text{max}}}{R} + 0.15\cdot \exp\left(-\frac{T}{200}\right) + 0.35\cdot \epsilon_{\text{pass}}
\label{eq:crack_tendency}
\end{equation}
È¡ÔþÖÆÎÂ¶È $T = 1150$ K£¨Ô¼880¡æ£©£¬µ¥µÀ´Î±äÐÎÁ¿ $\epsilon_{\text{pass}} = 0.25$£¬¼ÆËãµÃ£º
\begin{equation}
C_{\text{crack}} = 0.05 + 0.25\times0.138 + 0.15\times e^{-5.75} + 0.35\times0.25 = 0.172
\end{equation}
$C_{\text{crack}} = 0.172 < 0.2$£¬ÊôÓڵͷçÏÕÇø¼ä£¬¿ÉÕý³£ÔþÖÆ¡£

\subsubsection{¹¤ÒÕ×ÛºÏÆÀ·Ö}

¹¤ÒÕ×ÛºÏÆÀ·Ö×ۺϿ¼ÂÇ¿ªÁÑ·çÏÕºÍÎÂ¶ÈÆ«²î\cite{þºÏ½ðÔþÖÆ}£º
\begin{equation}
S = 100 \cdot \left[1 - \frac{C_{\text{crack}}}{0.5}\right] \cdot \left[0.3 + 0.7 \cdot \exp\left(-\frac{|T - T_{\text{opt}}|}{50}\right)\right]
\label{eq:process_score}
\end{equation}
È¡ $C_{\text{crack}} = 0.172$£¬$T = 1150$ K£¬$T_{\text{opt}} = 1150$ K£¬¼ÆËãµÃ£º
\begin{equation}
S = 100 \times (1 - 0.344) \times 1.0 = 65.6
\end{equation}
$S = 65.6$£¬ÊôÓںϸñ¹¤ÒÕ·½°¸£¨60-70·ÖÇø¼ä£©¡£

\subsection{¶à³¡ñîºÏЧӦÆÀ¹À}

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\begin{equation}
\frac{d\rho}{dt} = f_{\text{mech}}(\sigma) + f_{\text{thermal}}(T) + f_{\text{chemical}}(c_H) + \sum_{i,j} \chi_{ij} \Phi_i \Phi_j
\label{eq:coupling}
\end{equation}

ÆäÖÐ $\chi_{ij}$ Ϊ½»²æñîºÏϵÊý£¬$\Phi_i$ ´ú±í¸÷³¡Á¿¡£¸ù¾Ý¼ÆËã·ÖÎö£¬Ö÷Òª½»²æñîºÏÏî°üÀ¨£º

\begin{itemize}
    \item \textbf{Á¦-ÈÈñîºÏ}£ºÓ¦Á¦³¡¼ÓËÙÈȼ¤»î¹ý³Ì¡£ÔÚ2400 MPa¼¶Ó¦Á¦Ë®Æ½Ï£¬Î»´íÅÊÒÆµÄµÈЧ¼¤»îÄܽµµÍÔ¼0.15 eV£¬µ¼Ö¸ßÎÂÈä±äËÙÂÊÔö¼ÓÔ¼40\%¡£¸ÃЧӦÒѲ¿·Ö°üº¬ÔÚ¶àͨµÀÈȼ¤»îÄ£ÐÍ£¨Ê½\ref{eq:high_temp}£©µÄζÈÒÀÀµÐÔÖС£
   
    \item \textbf{Á¦-»¯ñîºÏ}£ºÀ­Ó¦Á¦´Ù½øÇâÔ­×ÓÔÚÁÑÎÆ¼â¶ËµÄ¾Û¼¯¡£SSCCÃż÷Ó¦Á¦ËæÇâŨ¶ÈÔö¼Ó³ÊÖ¸ÊýϽµ£¬Ê½(\ref{eq:sscc})ÖеÄÏßÐÔ¹ØÏµÊǸÃñîºÏЧӦµÄÒ»½×½üËÆ¡£
   
    \item \textbf{ÈÈ-»¯ñîºÏ}£ºÎ¶ÈÉý¸ß¼ÓËÙÇâÀ©É¢£¬ÓÐЧÀ©É¢ÏµÊý $D_{\text{eff}} = D_0 \exp(-Q/k_BT)/(1+\Psi_{\text{total}})$¡£ÔÚ80¡æÒÔÉÏ·þÒÛʱ£¬ÇâÀ©É¢ËÙÂÊ¿ÉÔö¼ÓÔ¼1¸öÊýÁ¿¼¶¡£
\end{itemize}

»ùÓÚ±ÊÕß½¨Á¢µÄλ´íÎïÀíϵÁз½³Ì\cite{λ´íÎïÀí,Çâ´à·½³Ì,ìØºÏ½ð}½øÐжೡñîºÏ·ÖÎö£¬ÔÚµäÐÍ·þÒÛÌõ¼þÏ£¨$\sigma = 0.8\sigma_y$£¬$T = 80^\circ C$£¬$c_H = 10$ ppm£©£¬ñîºÏÐÞÕýÒò×Ó $\eta_{\text{couple}} \approx 0.88$¡£¿¼Âǹ¤¿ö²¨¶¯£¬½¨ÒéÉè¼ÆÖµÈ¡0.85-0.95¡£

\begin{table}[h]
\centering
\caption{¶à³¡ñîºÏЧӦ¶Ô¹Ø¼üÐÔÄܵÄÐÞÕý}
\label{tab:coupling}
\begin{tabular}{@{}lll@{}}
\toprule
\textbf{·þÒÛÌõ¼þ} & \textbf{ñîºÏÐÞÕýÒò×Ó} & \textbf{½¨ÒéÉè¼ÆÖµ} \\
\midrule
³£Î¡¢ÎÞÇâ»·¾³ & 1.00 & $\sigma_{\text{th}}/\sigma_y = 0.80$ \\
³£Î¡¢º¬Çâ»·¾³£¨ÊäÇâ¹ÜµÀ£© & 0.90-0.95 & $\sigma_{\text{th}}/\sigma_y = 0.72-0.76$ \\
¸ßΣ¨150¡æ£©¡¢ÎÞÇâ»·¾³ & 0.90-0.95 & $\sigma_{\text{th}}/\sigma_y = 0.72-0.76$ \\
¸ßΡ¢º¬Çâ»·¾³£¨¼«ÏÞ¹¤¿ö£© & 0.85-0.90 & $\sigma_{\text{th}}/\sigma_y = 0.68-0.72$ \\
\bottomrule
\end{tabular}
\end{table}

×ۺϿ¼ÂÇÉÏÊöñîºÏЧӦ£¬±¾Éè¼Æ·½°¸ÔÚ¼«ÏÞ·þÒÛÌõ¼þϵÄSSCCÃż÷Ó¦Á¦½¨ÒéÈ¡ $\sigma_{\text{th}}/\sigma_y = 0.68-0.72$£¬ÈÔÂú×ãÊäÇâ¹ÜµÀµÈ¹Ø¼üÓ¦ÓõÄ×îµÍ°²È«ÒªÇó£¨ASME B31.12±ê×¼ÒªÇó $\geq 0.55$£©¡£

\subsection{ÐÔÄÜÔ¤²â}

×ÛºÏËùÓй«Ê½¼°ñîºÏЧӦÐÞÕý£¬Ô¤²âÐÔÄÜÈçÏ£º

\begin{table}[h]
\centering
\caption{2400 MPa¼¶ÖÐÃ̳¬¼¶¸ÖÐÔÄÜÔ¤²â}
\label{tab:properties}
\begin{tabular}{@{}lll@{}}
\toprule
\textbf{ÐÔÄÜÖ¸±ê} & \textbf{Ô¤²âÖµ} & \textbf{¶Ô±È»ù×¼£¨´«Í³ÂíÊÏÌåʱЧ¸Ö£©} \\
\midrule
Çü·þÇ¿¶È (MPa) & 1700-1800 & 1500-1600 \\
¿¹À­Ç¿¶È (MPa) & 2350-2450 & 2000-2200 \\
¾ùÔÈÑÓÉìÂÊ (\%) & 12-15 & 8-12 \\
À©¿×ÂÊ (\%) & >30 & 15-25 \\
SSCCÃż÷Ó¦Á¦£¨³£ÎÂÎÞÇ⣩ & 0.75-0.80$\sigma_y$ & 0.60-0.70$\sigma_y$ \\
SSCCÃż÷Ó¦Á¦£¨¼«ÏÞ¹¤¿ö£© & 0.68-0.72$\sigma_y$ & ¡ª \\
Çâ´àÃô¸ÐÐÔÖ¸Êý (RRA) & 0.65-0.70 & 0.50-0.60 \\
̼µ±Á¿ Ceq & 0.48-0.50 & 0.55-0.65 \\
×ۺϳɱ¾£¨ÍòÔª/¶Ö£© & 1.3-1.6 & 2.8-3.0 \\
\bottomrule
\end{tabular}
\end{table}

\subsection{¹¤ÒÕ²ÎÊý}

»ùÓÚÀíÂÛ¼ÆËã£¬ÍÆ¼öÒÔϹ¤ÒÕ²ÎÊý£º

\begin{table}[h]
\centering
\caption{ÍÆ¼ö¹¤ÒÕ²ÎÊý}
\label{tab:process}
\begin{tabular}{@{}ll@{}}
\toprule
\textbf{¹¤Ðò} & \textbf{²ÎÊý} \\
\midrule
¾ùÖÊ»¯ÍË»ð & 1200-1250¡æ¡Á10-20h£¨¸ù¾ÝÖý¶§³ß´ç£© \\
ÈÈÔþ & ¿ªÔþ1150¡À20¡æ£¬ÖÕÔþ880¡æ \\
ÀäÔþ & ½»²æÔþÖÆ£¬×ÜѹÏÂÂÊ60-70\%£¬3-4µÀ´Î \\
ÁÙ½çÍË»ð & 770¡æ¡Á5-8min \\
´ã»ð & ÊÒÎÂÓÍ´ã \\
Åä·Ö & 465¡À3¡æ£¬3-5min£¨¸ÐÓ¦¼ÓÈÈ£© \\
ʱЧ & 500¡æ¡Á30min \\
º¸ºóÍË»ð & 620-660¡æ£¬Á½ÏàÇøÍË»ð \\
\bottomrule
\end{tabular}
\end{table}

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\section{֪ʶ²úȨÓë·¨ÂÉÌõ¿î}

\subsection{Ô­´´ÐÔÄÚÈÝÓë֪ʶ²úȨÉùÃ÷}

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\begin{enumerate}[label=(\arabic*)]
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    \item Ç¿¶È¶à³ß¶È·Ö½â·½·¨£¨Ê½\ref{eq:strength_decomp}£©¼°¸÷²ãλ´íÃܶȼÆË㹫ʽ£¨Ê½\ref{eq:rho_calc}£©
    \item ¶àͨµÀÈȼ¤»îÄ£ÐÍ£¨Ê½\ref{eq:high_temp}£¬²Î¼ûÎÄÏ×\cite{ìØºÏ½ð}£©
    \item ÇâÏÝÚåÃܶÈͳһ±í´ïʽ£¨Ê½\ref{eq:trap_density}£©¼°SSCCÃż÷Ó¦Á¦¹«Ê½£¨Ê½\ref{eq:sscc}£¬²Î¼ûÎÄÏ×\cite{Çâ´à·½³Ì}£©
    \item ¼Ó¹¤Ó²»¯Âʹ«Ê½£¨Ê½\ref{eq:work_hardening}£¬²Î¼ûÎÄÏ×\cite{λ´íÎïÀí}£©
    \item ¸÷ÏòÒìÐÔÖ¸Êý¹«Ê½£¨Ê½\ref{eq:anisotropy}£¬²Î¼ûÎÄÏ×\cite{þºÏ½ðÔþÖÆ}£©
    \item ¿ªÁÑÇãÏòÖ¸Êý¹«Ê½£¨Ê½\ref{eq:crack_tendency}£¬²Î¼ûÎÄÏ×\cite{þºÏ½ðÔþÖÆ}£©
    \item ¹¤ÒÕ×ÛºÏÆÀ·Ö¹«Ê½£¨Ê½\ref{eq:process_score}£¬²Î¼ûÎÄÏ×\cite{þºÏ½ðÔþÖÆ}£©
    \item ¶à³¡ñîºÏЧӦÆÀ¹À·½·¨£¨µÚ\ref{sec:coupling}½Ú£©
\end{enumerate}

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