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Ichki kuchlar virtual ishini diskretlashtirish

Inkremental tahlil doirasi da \(t + \Delta t\) vaqti uchun keltirilgan virtual ish tenglamasi tayanch konfiguratsiyani tanlashga qarab Updated Lagrange va Total Lagrange formulirovkalariga ajraladi. Ushbu bobda Shakl funksiyalari va chekli elementlar approksimatsiyasi hamda Shakl funksiyalarining fazoviy hosilalari da kiritilgan chekli element approksimatsiyasidan foydalanib, ikkala formulirovkadagi ichki kuchlar virtual ishi fazoda diskretlashtiriladi va element ichki kuch vektorlari \(\boldsymbol{q}^e\) (UL) hamda \(\boldsymbol{Q}^e\) (TL) olinadi.

\(e\) elementning \(\alpha = 1, \ldots, n_e\) tugunlari uchun ko‘chishlarni \(\boldsymbol{u}^e_\alpha\) deb belgilaymiz va element tugun ko‘chishlari vektorini \(\boldsymbol{u}^e = (\boldsymbol{u}^{eT}_1, \ldots, \boldsymbol{u}^{eT}_{n_e})^T\) tartibida tuzamiz. Virtual ko‘chish \(\delta \boldsymbol{u}^e\) ham xuddi shu tartibda aniqlanadi. Element ichidagi ko‘chish shakl funksiyalari yordamida \(\boldsymbol{u} = \sum_\alpha N_\alpha^e \boldsymbol{u}^e_\alpha\) ko‘rinishida interpolatsiya qilinadi.

Updated Lagrange formulirovkasidagi ichki kuchlar virtual ishi

Updated Lagrange formulirovkasida \(t\) vaqtdagi joriy konfiguratsiya \({}^{t}\Omega\) tayanch konfiguratsiya sifatida olinadi va ichki kuchlar virtual ishi Cauchy kuchlanishi \(\boldsymbol{\sigma}\) hamda Almansi nisbiy deformatsiyasining chiziqli qismi \(\boldsymbol{A}_{(L)}\) orqali quyidagicha yoziladi:

\[ \delta W^{\mathrm{int}} = \sum_e \int_{\Omega^e} \boldsymbol{\sigma} : \delta \boldsymbol{A}_{(L)}\, dv \]

\(\delta \boldsymbol{A}_{(L)}\) ning har bir komponenti joriy konfiguratsiya koordinatalari \(\boldsymbol{x}\) bo‘yicha shakl funksiyasi hosilalari \(\partial N_\alpha^e/\partial x_i\) va tugun virtual ko‘chishlari \(\delta u^e_{i\alpha}\) ning chiziqli kombinatsiyasi sifatida ifodalanadi. Voigt yozuvida bu quyidagicha jamlanadi:

\[ \delta \boldsymbol{A}_{(L)} = \boldsymbol{B}_L\, \delta \boldsymbol{u}^e, \qquad \boldsymbol{B}_L = [\boldsymbol{B}_{L1}, \ldots, \boldsymbol{B}_{Ln_e}] \]

Tugun bloki \(\boldsymbol{B}_{L\alpha}\) \(\partial N_\alpha^e/\partial x_i\) ni Voigt qoidasi bo‘yicha \(6 \times 3\) matritsa ko‘rinishida joylashtirish orqali tuziladi va \(\boldsymbol{B}_L\) UL formulirovkasining nisbiy deformatsiya–ko‘chish matritsasi bo‘ladi. Buni ichki kuchlar virtual ishiga qo‘yib, \(\delta \boldsymbol{u}^e\) ni ajratsak,

\[ \delta W^{\mathrm{int}} = \sum_e \delta \boldsymbol{u}^{eT}\, \boldsymbol{q}^e, \qquad \boldsymbol{q}^e = \int_{\Omega^e} \boldsymbol{B}_L^T \boldsymbol{\sigma}\, dv \]

\(\boldsymbol{q}^e\) uchun \(\boldsymbol{q}^e_\alpha = \int_{\Omega^e} \boldsymbol{B}_{L\alpha}^T \boldsymbol{\sigma}\, dv\) tugun bloki \(\Omega^e\) elementning \(\alpha\) tuguniga ta’sir etuvchi ichki kuchdir.

Total Lagrange formulirovkasidagi ichki kuchlar virtual ishi

Total Lagrange formulirovkasida tayanch konfiguratsiya \(\Omega_0\) olinadi va ichki kuchlar virtual ishi ikkinchi Piola-Kirchhoff kuchlanishi \(\boldsymbol{S}\) hamda Green-Lagrange nisbiy deformatsiyasi \(\boldsymbol{E}\) orqali quyidagicha yoziladi:

\[ \delta W^{\mathrm{int}} = \sum_e \int_{\Omega^e_0} \boldsymbol{S} : \delta \boldsymbol{E}\, dV \]

\(\delta \boldsymbol{E}\) variatsiyasi virtual ko‘chishga nisbatan chiziqli had va joriy ko‘chish gradienti \(\partial u_k/\partial X_j\) bilan ko‘paytmalarni o‘z ichiga olgan nochiziqli hadga ajratiladi:

\[ \delta E_{(L)ij} = \frac{1}{2}\left( \frac{\partial \delta u_i}{\partial X_j} + \frac{\partial \delta u_j}{\partial X_i} \right), \quad \delta E_{(NL)ij} = \frac{1}{2}\left( \frac{\partial \delta u_k}{\partial X_i}\, \frac{\partial u_k}{\partial X_j} + \frac{\partial u_k}{\partial X_i}\, \frac{\partial \delta u_k}{\partial X_j} \right). \]

Chiziqli had UL formulirovkasidagi kabi joylashtirish qoidasini \(\partial N_\alpha^e/\partial X_i\) ga qo‘llash orqali \(\boldsymbol{B}_{L\alpha}\) tugun bloki yordamida yoziladi:

\[ \delta \boldsymbol{E}_{(L)} = \boldsymbol{B}_L\, \delta \boldsymbol{u}^e \]

Tayanch konfiguratsiya boshqacha bo‘lgani sababli faqat tarkibiy hosilalar \(\partial N_\alpha^e/\partial x_i\) dan \(\partial N_\alpha^e/\partial X_i\) ga almashadi; belgi esa UL formulirovkasidagidek saqlanadi. Nochiziqli had joriy ko‘chish gradienti \(\partial u_k/\partial X_j\) va \(\partial N_\alpha^e/\partial X_i\) ko‘paytmalarini Voigt qoidasi bo‘yicha joylashtirib, \(\boldsymbol{B}_{NL\alpha}\) tugun blokini hosil qiladi:

\[ \delta \boldsymbol{E}_{(NL)} = \boldsymbol{B}_{NL}\, \delta \boldsymbol{u}^e, \qquad \boldsymbol{B}_{NL} = [\boldsymbol{B}_{NL1}, \ldots, \boldsymbol{B}_{NLn_e}] \]

Shunday qilib, \(\delta \boldsymbol{E} = (\boldsymbol{B}_L + \boldsymbol{B}_{NL})\, \delta \boldsymbol{u}^e\) bo‘ladi va \(\boldsymbol{B}_L + \boldsymbol{B}_{NL}\) TL formulirovkasining nisbiy deformatsiya–ko‘chish matritsasidir. Buni ichki kuchlar virtual ishiga qo‘ysak,

\[ \delta W^{\mathrm{int}} = \sum_e \delta \boldsymbol{u}^{eT}\, \boldsymbol{Q}^e, \qquad \boldsymbol{Q}^e = \int_{\Omega^e_0} (\boldsymbol{B}_L + \boldsymbol{B}_{NL})^T \boldsymbol{S}\, dV \]

\(\boldsymbol{Q}^e_\alpha\) tugun bloki \(\Omega^e_0\) elementning \(\alpha\) tuguniga ta’sir etuvchi ichki kuchdir.

UL/TL orasidagi moslik va hisoblash oqimi

Updated Lagrange va Total Lagrange formulirovkalaridagi element ichki kuch vektorlari quyidagicha mos keladi.

Band Updated Lagrange formulirovkasi Total Lagrange formulirovkasi
Tayanch konfiguratsiya Joriy konfiguratsiya \({}^{t}\Omega^e\) Tayanch konfiguratsiya \(\Omega^e_0\)
Kuchlanish tenzori Cauchy kuchlanishi \(\boldsymbol{\sigma}\) Ikkinchi PK kuchlanishi \(\boldsymbol{S}\)
Nisbiy deformatsiya variatsiyasi \(\delta \boldsymbol{A}_{(L)}\) \(\delta \boldsymbol{E} = \delta \boldsymbol{E}_{(L)} + \delta \boldsymbol{E}_{(NL)}\)
B matritsasi \(\boldsymbol{B}_L\) (\(\partial N/\partial x\)) \(\boldsymbol{B}_L + \boldsymbol{B}_{NL}\) (\(\partial N/\partial X\), \(\partial u/\partial X\))
Element ichki kuchi \(\boldsymbol{q}^e = \int_{\Omega^e} \boldsymbol{B}_L^T \boldsymbol{\sigma}\, dv\) \(\boldsymbol{Q}^e = \int_{\Omega^e_0} (\boldsymbol{B}_L + \boldsymbol{B}_{NL})^T \boldsymbol{S}\, dV\)

Har ikkisi bir xil tartibda hisoblanadi: shakl funksiyalarining fazoviy hosilalaridan \(\boldsymbol{B}_L\) tuziladi; TL formulirovkasida joriy ko‘chish gradientidan \(\boldsymbol{B}_{NL}\) tuzilib qo‘shiladi; kuchlanish (\(\boldsymbol{\sigma}\) yoki \(\boldsymbol{S}\)) konstitutiv qonunga muvofiq yangilanadi; integrallash nuqtalarida \(\boldsymbol{B}^T \boldsymbol{\sigma}\) yoki \((\boldsymbol{B}_L+\boldsymbol{B}_{NL})^T \boldsymbol{S}\) element sohasi bo‘yicha sonli integrallanadi (Sonli integrallash). Tayanch konfiguratsiyani almashtirish (tugun koordinatalari va \(\boldsymbol{B}\) matritsasini tuzish) hamda kuchlanish tenzorini almashtirishdan tashqari hisoblash umumiydir, shuning uchun FrontISTR ikkala formulirovka uchun ichki kuch hisoblarini umumiy quyi dasturlar orqali amalga oshiradi. Element ichki kuch vektorlari \(\boldsymbol{q}^e\) va \(\boldsymbol{Q}^e\) ni global ichki kuch vektoriga yig‘ish Tashqi kuchlarning virtual ishi va global tenglamalarni yig‘ish da ko‘rib chiqiladi.

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