Kuchlanish va saqlanish qonunlari¶

Ushbu bobda kuchlanish tensorlarining ta’rifi va o‘zaro bog‘lanishi, shuningdek massa, impuls va burchak impulsi saqlanish qonunlaridan kelib chiqadigan muvozanat tenglamalari hamda kuchlanish simmetriyasi jamlanadi. Uchta kuchlanish tensori — Cauchy kuchlanishi \(\boldsymbol{\sigma}\), birinchi Piola-Kirchhoff kuchlanishi \(\boldsymbol{P}\) va ikkinchi Piola-Kirchhoff kuchlanishi \(\boldsymbol{S}\) — sirt va kuch vektori uchun qaysi konfiguratsiya tanlanishiga qarab kiritiladi. Konfiguratsiya, harakat va deformatsiya gradienti ta’riflari uchun Harakat, deformatsiya va strain, belgilar ro‘yxati uchun Fizik miqdorlar belgilari ro‘yxati ga qarang.

Cauchy kuchlanish tensori¶

Joriy konfiguratsiyadagi \(\boldsymbol{x}\) nuqtada kichik sirtga (yuza \(d\Gamma\), tashqi birlik normal \(\boldsymbol{n}\)) ta’sir qiluvchi kuch \(d\boldsymbol{f}\) bo‘lsin. Kuchlanish vektori \(\boldsymbol{t}\) birlik yuzaga to‘g‘ri keladigan kuch sifatida \(d\boldsymbol{f} = \boldsymbol{t}\, d\Gamma\) bilan aniqlanadi. \(\boldsymbol{t}\) ham \(\boldsymbol{x}\) joylashuvga, ham \(\boldsymbol{n}\) normalga bog‘liq. Kichik tetraedrga impuls saqlanish qonunini qo‘llashdan \(\boldsymbol{t}\) ning \(\boldsymbol{n}\) ga nisbatan chiziqli ekani va ikkinchi tartibli \(\boldsymbol{\sigma}(\boldsymbol{x}, t)\) tensor mavjudligi kelib chiqadi, shundayki

\[ \boldsymbol{t} = \boldsymbol{\sigma} \boldsymbol{n}, \qquad t_i = \sigma_{ij} n_j \]

bu munosabat ixtiyoriy \(\boldsymbol{n}\) uchun bajariladi (Cauchy kuchlanish teoremasi). \(\boldsymbol{\sigma}\) Cauchy kuchlanish tensori deb ataladi va joriy konfiguratsiya geometriyasidagi birlik yuzaga nisbatan kuch bo‘lgani uchun haqiqiy kuchlanish deb ham yuritiladi. \(\sigma_{ij}\) komponenti joriy konfiguratsiyada \(x_j\) koordinata o‘qiga normal kichik sirtning birlik yuzasiga ta’sir qiluvchi kuchning \(x_i\) komponentini bildiradi.

Quyida ko‘rsatiladigan burchak impulsi saqlanish qonuniga ko‘ra Cauchy kuchlanishi simmetrik tensor

\[ \boldsymbol{\sigma} = \boldsymbol{\sigma}^T, \qquad \sigma_{ij} = \sigma_{ji} \]

bo‘lib, 3D da oltita mustaqil komponentga (\(\sigma_{11}, \sigma_{22}, \sigma_{33}, \sigma_{12}, \sigma_{23}, \sigma_{31}\)) ega. Voigt belgilanishidagi kuchlanish vektori aynan shu simmetriyaga asoslanadi (Tensor belgilanishi va matematik asoslar ga qarang). Cauchy kuchlanishi FrontISTR ning Updated Lagrange usulidagi geometrik nochiziqli tahlil va kichik deformatsiya tahlilida kuchlanish natijasi sifatida ishlatiladi.

Birinchi va ikkinchi Piola-Kirchhoff kuchlanishlari¶

Chekli deformatsiyada kuchlanishni referens konfiguratsiyadagi sirt va vektor orqali ifodalash ko‘pincha qulay bo‘ladi, shu sababli ikki turdagi Piola-Kirchhoff kuchlanish tensori kiritiladi. Referens konfiguratsiyadagi kichik sirt yuzi \(d\Gamma_0\), tashqi birlik normal \(\boldsymbol{N}\) bo‘lsin.

Birinchi Piola-Kirchhoff kuchlanishi (birinchi PK kuchlanishi, nominal kuchlanish) \(\boldsymbol{P}\) joriy konfiguratsiyadagi \(d\boldsymbol{f}\) kuchning referens konfiguratsiyadagi kichik sirtga ta’sirini ifodalovchi kuchlanish tensori sifatida

\[ d\boldsymbol{f} = \boldsymbol{P} \boldsymbol{N}\, d\Gamma_0 \]

bilan aniqlanadi. Nanson formulasi \(\boldsymbol{n}\, d\Gamma = J \boldsymbol{F}^{-T} \boldsymbol{N}\, d\Gamma_0\) va Cauchy kuchlanish teoremasidan Cauchy kuchlanishi bilan bog‘lanish

\[ \boldsymbol{P} = J\, \boldsymbol{\sigma} \boldsymbol{F}^{-T}, \qquad \boldsymbol{\sigma} = \frac{1}{J} \boldsymbol{P} \boldsymbol{F}^T \]

olinadi. Birinchi PK kuchlanishi odatda nosimmetrik tensordir.

Ikkinchi Piola-Kirchhoff kuchlanishi (ikkinchi PK kuchlanishi) \(\boldsymbol{S}\) joriy konfiguratsiyadagi \(d\boldsymbol{f}\) kuch \(\boldsymbol{F}^{-1}\) bilan referens konfiguratsiyaga qaytarilib, referens konfiguratsiyadagi kichik sirtga ta’sir ettirilgandagi kuchlanish tensori sifatida

\[ \boldsymbol{F}^{-1}\, d\boldsymbol{f} = \boldsymbol{S} \boldsymbol{N}\, d\Gamma_0 \]

bilan aniqlanadi. Kuch vektori ham, ta’sir sirti ham referens konfiguratsiya miqdorlari bilan ifodalanadi; tensor qattiq jism aylanishiga invariant va simmetrikdir. Birinchi PK kuchlanishi bilan bog‘lanish va Cauchy kuchlanishiga o‘tish formulalari

\[ \boldsymbol{P} = \boldsymbol{F} \boldsymbol{S}, \qquad \boldsymbol{\sigma} = \frac{1}{J} \boldsymbol{F} \boldsymbol{S} \boldsymbol{F}^T, \qquad \boldsymbol{S} = J\, \boldsymbol{F}^{-1} \boldsymbol{\sigma} \boldsymbol{F}^{-T} \]

dir. Ikkinchi PK kuchlanishi giperelastik strain energiyasi funksiyasi \(W(\boldsymbol{C})\) dan \(\boldsymbol{S} = 2\,\partial W / \partial \boldsymbol{C}\) sifatida kelib chiqadi va Total Lagrange usulida Green-Lagrange deformatsiyasi \(\boldsymbol{E}\) bilan energiyaviy konjugat juft \((\boldsymbol{S}, \boldsymbol{E})\) sifatida ishlatiladi.

Referens konfiguratsiya va simmetriya quyidagi jadvalda keltirilgan. Kichik deformatsiya limitida (\(\boldsymbol{F} \to \boldsymbol{I}\), \(J \to 1\)) uchala kuchlanish bir-biriga tenglashadi.

Kuchlanish tensori	Referens sirt	Kuch vektori	Simmetriya	Qo‘llanishi
Cauchy kuchlanishi \(\boldsymbol{\sigma}\)	Joriy konfiguratsiya \(d\Gamma, \boldsymbol{n}\)	Joriy konfiguratsiya \(d\boldsymbol{f}\)	Simmetrik	Updated Lagrange usuli va kichik deformatsiya tahlili
Birinchi PK kuchlanishi \(\boldsymbol{P}\)	Referens konfiguratsiya \(d\Gamma_0, \boldsymbol{N}\)	Joriy konfiguratsiya \(d\boldsymbol{f}\)	Odatda nosimmetrik	Referens konfiguratsiyadagi muvozanat tenglamasi
Ikkinchi PK kuchlanishi \(\boldsymbol{S}\)	Referens konfiguratsiya \(d\Gamma_0, \boldsymbol{N}\)	Referens konfiguratsiya \(\boldsymbol{F}^{-1} d\boldsymbol{f}\)	Simmetrik	Total Lagrange usuli va giperelastiklik

Massa va impuls saqlanish qonunlari hamda muvozanat tenglamasi¶

Joriy konfiguratsiyadagi massa zichligi \(\rho\), referens konfiguratsiyadagi massa zichligi \(\rho_0\) bo‘lsin. Massa saqlanish qonuni \(\int_{\Omega} \rho\, dv = \int_{\Omega_0} \rho_0\, dV\) hajm elementi \(dv = J\, dV\) orqali lokal ko‘rinish

\[ \rho_0 = J \rho \]

ga keladi.

Jism kuchi (birlik massaga) \(\boldsymbol{g}\), tezlanish \(\boldsymbol{a}\) bo‘lsin. Impuls saqlanish qonuniga (Eulerning birinchi harakat qonuni) Cauchy kuchlanish teoremasi va Gauss divergensiya teoremasini qo‘llasak, joriy konfiguratsiyadagi lokal muvozanat tenglamasi (harakat tenglamasi)

\[ \nabla_x \cdot \boldsymbol{\sigma} + \rho \boldsymbol{g} = \rho \boldsymbol{a}, \qquad \frac{\partial \sigma_{ij}}{\partial x_j} + \rho g_i = \rho a_i \]

olinadi. Nanson formulasi va \(\boldsymbol{P} = J \boldsymbol{\sigma} \boldsymbol{F}^{-T}\) yordamida referens konfiguratsiyadagi integralga o‘tkazilganda lokal shakl

\[ \nabla_X \cdot \boldsymbol{P} + \rho_0 \boldsymbol{g} = \rho_0 \boldsymbol{a}, \qquad \frac{\partial P_{ij}}{\partial X_j} + \rho_0 g_i = \rho_0 a_i \]

olinadi. Ikki shakl kuchlanish almashtirish qoidasi va \(\rho_0 = J\rho\) orqali ekvivalentdir. Inersiya hadi e’tiborga olinmasa

\[ \nabla_x \cdot \boldsymbol{\sigma} + \rho \boldsymbol{g} = \boldsymbol{0}, \qquad \nabla_X \cdot \boldsymbol{P} + \rho_0 \boldsymbol{g} = \boldsymbol{0} \]

statik muvozanat tenglamasi olinadi va u FrontISTR ning chiziqli hamda nochiziqli statik tahlillari uchun boshlang‘ich tenglamadir.

Burchak impulsi saqlanish qonuni va kuchlanish simmetriyasi¶

Burchak impulsi saqlanish qonunini (Eulerning ikkinchi harakat qonuni) impuls saqlanishidan olingan muvozanat tenglamasi bilan birlashtirsak, Cauchy kuchlanish tensorining simmetriyasi

\[ \boldsymbol{\sigma} = \boldsymbol{\sigma}^T, \qquad \sigma_{ij} = \sigma_{ji} \]

kelib chiqadi va 3D da u oltita mustaqil komponentdan iborat. \(\boldsymbol{\sigma} = J^{-1} \boldsymbol{F} \boldsymbol{S} \boldsymbol{F}^T\) almashtirish formulasining ikki tomonini transpozitsiya qilish va \(\boldsymbol{F}\) ning regularligidan ikkinchi PK kuchlanishi ham simmetrik

\[ \boldsymbol{S} = \boldsymbol{S}^T \]

bo‘lib, oltita mustaqil komponentga ega. Boshqa tomondan, birinchi PK kuchlanishi \(\boldsymbol{P} = \boldsymbol{F} \boldsymbol{S}\) odatda simmetrik emas; faqat \(\boldsymbol{P} \boldsymbol{F}^T = \boldsymbol{F} \boldsymbol{P}^T\) (ya’ni \(\boldsymbol{P} \boldsymbol{F}^T\) simmetrik) munosabati bajariladi va mustaqil komponentlar soni 9 ta.

Ushbu simmetriyalar kuchlanishni Voigt belgilanishida oltita komponentli vektorga aylantirish uchun asos bo‘ladi. Voigt qoidalari va material matritsasining tuzilishi uchun Tensor belgilanishi va matematik asoslar hamda Chiziqli elastiklik ga qarang.