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Chiziqli elastik statik tahlil (kirish)

Ushbu bo‘lim kichik deformatsiya nazariyasiga asoslangan elastik statik tahlil formulatsiyasini bayon qiladi. Kuchlanish–deformatsiya munosabati uchun chiziqli elastiklik qabul qilinadi. Bu bob chekli elementli strukturaviy tahlilning umumiy tuzilishini tushunishga kirish sifatida mo‘ljallangan va mustaqil yakunlangan bob ko‘rinishida tuzilgan.

Virtual ish prinsipining umumiy nazariyasi (joriy konfiguratsiya, boshlang‘ich konfiguratsiya va kichik deformatsiyaga keltirish) uchun Virtual ish prinsipi, chiziqli elastik konstitutiv qonun tafsilotlari uchun Chiziqli elastiklik, tensor va Voigt belgilashlari uchun Tensor belgilashlari va matematik asoslar, chekli deformatsiyaning umumiy formulatsiyasi uchun Harakat, deformatsiya va strain, nochiziqli masalalarni yechish usullari uchun esa Urinma qattiqlik matritsasi ga qarang.

Asosiy tenglamalar

Kichik deformatsiya va chiziqli elastiklik farazlari ostida qattiq jism mexanikasining chegaraviy masalasi muvozanat tenglamasi, mexanik chegaraviy shart va geometrik (asosiy) chegaraviy shartdan iborat (2.1.1-rasm):

\[\begin{equation} \nabla \cdot \boldsymbol{\sigma} + \overline{\boldsymbol{b}} = \boldsymbol{0} \quad \text{in} \ V \label{eq:2.1.1} \end{equation}\]
\[\begin{equation} \boldsymbol{\sigma} \cdot \boldsymbol{n} = \overline{\boldsymbol{t}} \quad \text{on} \ S_t \label{eq:2.1.2} \end{equation}\]
\[\begin{equation} \boldsymbol{u} = \overline{\boldsymbol{u}} \quad \text{on} \ S_u \label{eq:2.1.3} \end{equation}\]

Bu yerda \(\boldsymbol{\sigma}\) — Cauchy kuchlanishi, \(\overline{\boldsymbol{b}}\) — hajm birligiga to‘g‘ri keladigan hajmiy kuch, \(\overline{\boldsymbol{t}}\) — berilgan sirt tortish kuchi, \(\overline{\boldsymbol{u}}\) — berilgan siljish, \(S_t, S_u\) esa mos ravishda mexanik va geometrik chegaralardir.

Qattiq jism mexanikasining chegaraviy masalasi (kichik deformatsiya)

2.1.1-rasm Qattiq jism mexanikasining chegaraviy masalasi (kichik deformatsiya)

Simmetrik gradient operatori yordamida deformatsiya–siljish munosabati

\[\begin{equation} \boldsymbol{\varepsilon} = \nabla_S \boldsymbol{u} \label{eq:2.1.4} \end{equation}\]

ko‘rinishida yoziladi. Chiziqli elastik konstitutiv tenglama esa

\[\begin{equation} \boldsymbol{\sigma} = \boldsymbol{C} : \boldsymbol{\varepsilon} \label{eq:2.1.5} \end{equation}\]

bo‘lib, bu yerda \(\boldsymbol{C}\) — to‘rtinchi tartibli elastiklik tensoridir.

Virtual ish prinsipi

Virtual ish prinsipining umumiy shakllari (joriy konfiguratsiya, boshlang‘ich konfiguratsiya va kichik deformatsiyaga keltirish) Virtual ish prinsipi da jamlangan. Kichik deformatsiya va chiziqli elastiklik farazlarida kuchsiz shakl quyidagicha:

\[\begin{equation} \int_V \boldsymbol{\sigma} : \delta \boldsymbol{\varepsilon}\, dV = \int_{S_t} \overline{\boldsymbol{t}} \cdot \delta \boldsymbol{u}\, dS + \int_V \overline{\boldsymbol{b}} \cdot \delta \boldsymbol{u}\, dV \label{eq:2.1.6} \end{equation}\]
\[\begin{equation} \delta \boldsymbol{u} = \boldsymbol{0} \quad \text{on} \ S_u \label{eq:2.1.7} \end{equation}\]

Konstitutiv tenglama \eqref{eq:2.1.5} ni qo‘yib, Voigt belgilashida \(\hat{\sigma} = D\, \hat{\varepsilon}\) deb yozsak, diskretlashda bevosita ishlatiladigan shakl olinadi:

\[\begin{equation} \int_V \delta \hat{\varepsilon}^T\, D\, \hat{\varepsilon}\, dV = \int_{S_t} \delta \boldsymbol{u}^T\, \overline{\boldsymbol{t}}\, dS + \int_V \delta \boldsymbol{u}^T\, \overline{\boldsymbol{b}}\, dV \label{eq:2.1.10} \end{equation}\]

Bu yerda \(D\) — Chiziqli elastiklik da ta’riflangan elastiklik matritsasi. \eqref{eq:2.1.10} va \eqref{eq:2.1.7} tenglamalari quyida diskretlanadigan virtual ish prinsipini tashkil qiladi.

Diskretlash va global tenglamani yig‘ish

\( \eqref{eq:2.1.10} \) dagi virtual ish prinsipini chekli elementlar bo‘yicha diskretlasak,

\[\begin{equation} \sum_e \int_{V^e} \delta \hat{\varepsilon}^T\, D\, \hat{\varepsilon}\, dV = \sum_e \int_{S^e_t} \delta \boldsymbol{u}^T\, \overline{\boldsymbol{t}}\, dS + \sum_e \int_{V^e} \delta \boldsymbol{u}^T\, \overline{\boldsymbol{b}}\, dV \label{eq:2.1.11} \end{equation}\]

olinadi. Har bir elementda siljish maydoni elementni tashkil qiluvchi tugunlar siljishlari yordamida quyidagicha interpolatsiya qilinadi:

\[\begin{equation} \boldsymbol{u} = \sum^m_{i=1} N_i\, \boldsymbol{u}_i = \boldsymbol{N}\, \boldsymbol{U} \label{eq:2.1.12} \end{equation}\]

Shunda \(\eqref{eq:2.1.4}\) dan deformatsiya

\[\begin{equation} \hat{\varepsilon} = \boldsymbol{B}\, \boldsymbol{U} \label{eq:2.1.13} \end{equation}\]

ko‘rinishida beriladi. \(\eqref{eq:2.1.12}\) va \(\eqref{eq:2.1.13}\) ni \(\eqref{eq:2.1.11}\) ga qo‘yib,

\[\begin{equation} \sum_e \delta \boldsymbol{U}^T \left( \int_{V^e} \boldsymbol{B}^T D\, \boldsymbol{B}\, dV \right) \boldsymbol{U} = \sum_e \delta \boldsymbol{U}^T \int_{S_t^e} \boldsymbol{N}^T\, \overline{\boldsymbol{t}}\, dS + \sum_e \delta \boldsymbol{U}^T \int_{V^e} \boldsymbol{N}^T\, \overline{\boldsymbol{b}}\, dV \label{eq:2.1.14} \end{equation}\]

olinadi. \(\eqref{eq:2.1.14}\) tenglamani

\[\begin{equation} \delta \boldsymbol{U}^T\, \boldsymbol{K}\, \boldsymbol{U} = \delta \boldsymbol{U}^T\, \boldsymbol{F} \label{eq:2.1.15} \end{equation}\]

ko‘rinishida yozish mumkin. Bu yerda \(\eqref{eq:2.1.16}\) va \(\eqref{eq:2.1.17}\) bilan ta’riflangan matritsa va vektor komponentlari har bir chekli element uchun hisoblanib, superpozitsiya orqali yig‘iladi.

\[\begin{equation} \boldsymbol{K} = \sum_e \int_{V^e} \boldsymbol{B}^T\, D\, \boldsymbol{B}\, dV \label{eq:2.1.16} \end{equation}\]
\[\begin{equation} \boldsymbol{F} = \sum_e \left( \int_{S_t^e} \boldsymbol{N}^T\, \overline{\boldsymbol{t}}\, dS + \int_{V^e} \boldsymbol{N}^T\, \overline{\boldsymbol{b}}\, dV \right) \label{eq:2.1.17} \end{equation}\]

\(\eqref{eq:2.1.15}\) tenglama ixtiyoriy virtual siljish \(\delta \boldsymbol{U}\) uchun bajarilgani sababli,

\[\begin{equation} \boldsymbol{K}\, \boldsymbol{U} = \boldsymbol{F} \label{eq:2.1.18} \end{equation}\]

olinadi. Shu bilan birga, \(\eqref{eq:2.1.3}\) dagi siljish chegaraviy sharti

\[\begin{equation} \boldsymbol{U} = \overline{\boldsymbol{U}} \label{eq:2.1.19} \end{equation}\]

ko‘rinishida ifodalanadi. \(\eqref{eq:2.1.18}\) ni \(\eqref{eq:2.1.19}\) cheklov sharti ostida yechib, tugun siljishi \(\boldsymbol{U}\) aniqlanadi.

Bog‘liq bo‘limlar

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