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Fizik kattaliklar belgilarining ro‘yxati

Ushbu sahifada FrontISTR nazariya qo‘llanmasida ishlatiladigan fizik kattaliklar belgilari ro‘yxati keltiriladi. Belgilash qoidalari (vektor va tenzorlarni qalin shrift bilan yozish, Einstein yig‘indi qoidasi va Voigt yozuvi) uchun Tenzor belgilashi va matematik asoslar ga qarang.

Konfiguratsiyalar va koordinatalar tizimlari

Belgi Tavsif
\(\boldsymbol{X}\) Material nuqtaning tayanch (boshlang‘ich) konfiguratsiyadagi holat vektori (material koordinatalari)
\(\boldsymbol{x}\) Material nuqtaning joriy konfiguratsiyadagi holat vektori (fazoviy koordinatalar)
\(\phi(\boldsymbol{X}, t)\) Harakat akslantirishi: \(\boldsymbol{x} = \phi(\boldsymbol{X}, t)\)
\(\Omega_0\) Tayanch konfiguratsiyada jism egallagan soha
\(\Omega\) Joriy konfiguratsiyada jism egallagan soha
\(\Gamma_0\) \(\Omega_0\) ning chegarasi
\(\Gamma\) \(\Omega\) ning chegarasi
\(\Gamma_B, \Gamma_{0B}\) Geometrik (siljish) chegara: joriy/tayanch konfiguratsiyadagi \(\boldsymbol{u} = \bar{\boldsymbol{u}}\) belgilangan chegara
\(\Gamma_t, \Gamma_{0t}\) Mexanik chegara: joriy/tayanch konfiguratsiyadagi \(\boldsymbol{\sigma}\boldsymbol{n} = \bar{\boldsymbol{t}}\) (yoki \(\boldsymbol{P}\boldsymbol{N} = \bar{\boldsymbol{t}}\)) belgilangan chegara
\(\boldsymbol{n}\) Joriy konfiguratsiyadagi sirtning tashqariga yo‘nalgan birlik normal vektori
\(\boldsymbol{N}\) Tayanch konfiguratsiyadagi sirtning tashqariga yo‘nalgan birlik normal vektori
\(t\) Vaqt

Qoidа: tayanch konfiguratsiyadagi fizik kattaliklar katta harf yoki \(0\) pastki indeksi bilan, joriy konfiguratsiyadagi kattaliklar esa kichik harf bilan belgilanadi.

Vaqt va inkrementlar (inkremental tahlil)

Belgi Tavsif
\(\Delta t\) Vaqt inkrementi: \(\Delta t = t_{n+1} - t_n\)
\(^{t}(\cdot)\) \(t\) vaqtdagi fizik kattalik (chap yuqori indeks). Masalan, \(^{t}\boldsymbol{\sigma}\) — \(t\) vaqtdagi Cauchy kuchlanishi
\(^{t}\Omega\) \(t\) vaqtdagi joriy konfiguratsiya sohasi
\(^{t}\Gamma\) \(t\) vaqtdagi joriy konfiguratsiya chegarasi

Qoidа: inkremental tahlilda \(t\) vaqtgacha bo‘lgan holat ma’lum, \(t + \Delta t\) vaqtdagi holat esa noma’lum deb olinib yechiladi. Vaqtni aniq ko‘rsatish zarur bo‘lmasa, chap yuqori indeks tashlab ketiladi. Tayanch sifatida tayanch konfiguratsiya \(\Omega_0\) ni oladigan formulalash Total Lagrange usuli, inkrement boshidagi joriy konfiguratsiya \(^{t}\Omega\) ni oladigan formulalash esa Updated Lagrange usuli deb ataladi.

Siljish, tezlik, tezlanish va hajmiy kuch

Belgi Tavsif
\(\boldsymbol{u}\) Siljish vektori: \(\boldsymbol{u} = \boldsymbol{x} - \boldsymbol{X}\)
\(\bar{\boldsymbol{u}}\) Geometrik chegara \(\Gamma_B\) da belgilangan siljish
\(\delta \boldsymbol{u}\) Virtual siljish (sinov funksiyasi): \(\Gamma_B\) da \(\delta \boldsymbol{u} = \boldsymbol{0}\)
\(\boldsymbol{v}\) Tezlik vektori: \(\boldsymbol{v} = \dot{\boldsymbol{u}}\)
\(\boldsymbol{a}\) Tezlanish vektori: \(\boldsymbol{a} = \dot{\boldsymbol{v}}\)
\(\boldsymbol{g}\) Hajmiy kuch (birlik massaga)
\(\boldsymbol{t}\) Sirt kuchi vektori (birlik yuzaga): \(\boldsymbol{t} = \boldsymbol{\sigma}\boldsymbol{n}\)
\(\bar{\boldsymbol{t}}\) Mexanik chegara \(\Gamma_t\) da belgilangan sirt kuchi (Neumann ma’lumoti)

Differensial operatorlar

Belgi Tavsif
\(\dot{(\cdot)}\) Material vaqt hosilasi (bir xil material nuqtani kuzatib olingan vaqt hosilasi): \(\dot{A} \equiv DA/Dt\)
\(\nabla_X\) Tayanch konfiguratsiya gradienti (material koordinatalar \(\boldsymbol{X}\) bo‘yicha gradient): \(\nabla_X = \partial / \partial \boldsymbol{X}\)
\(\nabla_x\) Joriy konfiguratsiya gradienti (fazoviy koordinatalar \(\boldsymbol{x}\) bo‘yicha gradient): \(\nabla_x = \partial / \partial \boldsymbol{x}\)
\(\nabla_S\) Simmetrik gradient operatori: \(\nabla_S \boldsymbol{u} = \tfrac{1}{2}(\nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^T)\)

Izoh: \(\nabla_X\) va \(\nabla_x\) \(\nabla_X = \boldsymbol{F}^T \nabla_x\) orqali bog‘langan. Pastki indekssiz \(\nabla\) konfiguratsiya kontekstdan aniq bo‘lganda yoki kichik deformatsiyadagi kabi farq yo‘qolganda ishlatiladi.

Deformatsiya tenzorlari

Belgi Tavsif
\(\boldsymbol{F}\) Deformatsiya gradienti tenzori: \(F_{ij} = \partial x_i / \partial X_j\)
\(J\) Hajm o‘zgarishi nisbati: \(J = dv/dV = \det \boldsymbol{F}\)
\(\boldsymbol{C}\) O‘ng Cauchy-Green deformatsiya tenzori: \(\boldsymbol{C} = \boldsymbol{F}^T \boldsymbol{F}\)
\(\boldsymbol{b}\) Chap Cauchy-Green deformatsiya tenzori: \(\boldsymbol{b} = \boldsymbol{F}\boldsymbol{F}^T\)
\(\boldsymbol{L}\) Tezlik gradienti tenzori: \(L_{ij} = \partial v_i / \partial x_j = \dot{F}_{ik}F^{-1}_{kj}\)
\(\boldsymbol{D}\) Deformatsiya tezligi tenzori (\(\boldsymbol{L}\) ning simmetrik qismi): \(\boldsymbol{D} = \tfrac{1}{2}(\boldsymbol{L}+\boldsymbol{L}^T)\)
\(\boldsymbol{W}\) Spin tenzori (\(\boldsymbol{L}\) ning antisimmetrik qismi): \(\boldsymbol{W} = \tfrac{1}{2}(\boldsymbol{L}-\boldsymbol{L}^T)\)

Deformatsiya o‘lchovi tenzorlari

Belgi Tavsif
\(\boldsymbol{E}\) Green-Lagrange deformatsiya tenzori (tayanch konfiguratsiya): \(\boldsymbol{E} = \tfrac{1}{2}(\boldsymbol{C}-\boldsymbol{I})\)
\(\boldsymbol{e}\) Almansi deformatsiya tenzori (joriy konfiguratsiya): \(\boldsymbol{e} = \tfrac{1}{2}(\boldsymbol{I}-\boldsymbol{b}^{-1})\)
\(\boldsymbol{A}_{(L)}\) Almansi deformatsiya tenzorining chiziqli qismi (joriy konfiguratsiyadagi simmetrik siljish gradienti): \(\boldsymbol{A}_{(L)} = \tfrac{1}{2}(\nabla_x \boldsymbol{u} + (\nabla_x \boldsymbol{u})^T)\)
\(\boldsymbol{\varepsilon}\) Kichik deformatsiya tenzori (chiziqli yaqinlashuv): \(\varepsilon_{ij} = \tfrac{1}{2}(\partial u_i/\partial x_j + \partial u_j/\partial x_i)\)

Izoh: \(\boldsymbol{E}\) asosan Total Lagrange usulida, \(\boldsymbol{D}\) esa asosan Updated Lagrange usulida ishlatiladi.

Kuchlanish tenzorlari

Belgi Tavsif
\(\boldsymbol{\sigma}\) Cauchy kuchlanish tenzori (haqiqiy kuchlanish, joriy konfiguratsiya): \(d\boldsymbol{f} = \boldsymbol{\sigma}\boldsymbol{n}\,d\Gamma\)
\(\boldsymbol{P}\) Birinchi Piola-Kirchhoff kuchlanish tenzori (nominal kuchlanish): \(d\boldsymbol{f} = \boldsymbol{P}\boldsymbol{N}\,d\Gamma_0\)
\(\boldsymbol{S}\) Ikkinchi Piola-Kirchhoff kuchlanish tenzori (tayanch konfiguratsiya, simmetrik): \(\boldsymbol{F}^{-1}d\boldsymbol{f} = \boldsymbol{S}\boldsymbol{N}\,d\Gamma_0\)

Kuchlanish tenzorlari orasidagi o‘zgartirish munosabatlari:

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

Qoidа: Total Lagrange usulida \((\boldsymbol{S}, \boldsymbol{E})\), Updated Lagrange usulida esa \((\boldsymbol{\sigma}, \boldsymbol{D})\) juftligi ishlatiladi.

Zichlik va massa

Belgi Tavsif
\(\rho\) Joriy konfiguratsiyadagi massa zichligi
\(\rho_0\) Tayanch konfiguratsiyadagi massa zichligi

Massaning saqlanish qonuni: \(\rho_0 = J\rho\).

Material konstantalari (chiziqli elastiklik)

Belgi Tavsif
\(E\) Yung moduli
\(\nu\) Puasson koeffitsiyenti
\(\lambda\) Birinchi Lamé konstantasi: \(\lambda = E\nu / [(1+\nu)(1-2\nu)]\)
\(\mu\) Ikkinchi Lamé konstantasi (siljish moduli): \(\mu = E / [2(1+\nu)]\)
\(\boldsymbol{\mathsf{C}}\) Elastiklik tenzori (to‘rtinchi tartib): \(\boldsymbol{S} = \boldsymbol{\mathsf{C}}:\boldsymbol{E}\), komponentlari \(C_{ijkl}\)
\(D\) (yoki \(\hat{\tilde{C}}\)) Material matritsasi (Voigt yozuvi, 3D da \(6\times6\)): \(\hat{\sigma} = D\,\hat{\varepsilon}\)

Izotrop chiziqli elastik materialning elastiklik tenzori komponentlari:

\[ C_{ijkl} = \lambda\,\delta_{ij}\delta_{kl} + \mu\,(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}) \]

Giperelastik materiallar

Belgi Tavsif
\(W(\boldsymbol{C})\) Elastik potensial funksiya (deformatsiya energiyasi zichligi funksiyasi)
\(I_1, I_2, I_3\) O‘ng Cauchy-Green tenzori \(\boldsymbol{C}\) ning bosh invariantlari
\(\tilde{I}_1, \tilde{I}_2, \tilde{I}_3\) \(\boldsymbol{C}\) ning kamaytirilgan invariantlari (hajm o‘zgarishi ajratilgan)
\(C_1, C_2\) Mooney-Rivlin modelining material konstantalari
\(D\) Hajmiy elastiklikka tegishli material konstantasi

Elastoplastik materiallar

Belgi Tavsif
\(\boldsymbol{D}^e\) Deformatsiya tezligi tenzorining elastik komponenti
\(\boldsymbol{D}^p\) Deformatsiya tezligi tenzorining plastik komponenti
\(F(\boldsymbol{\sigma}, \kappa)\) Oqish funksiyasi
\(\kappa\) Plastik holatni ifodalovchi ichki o‘zgaruvchi (izotrop qotish o‘zgaruvchisi)
\(\lambda^p\) Plastik ko‘paytuvchi (plastik deformatsiya tezligi ko‘paytuvchisi): \(\lambda^p \geq 0\)
\(\Theta\) Plastik potensial (bog‘langan oqim qoidasida \(\Theta = F\))
\(\bar{\sigma}\) Ekvivalent kuchlanish (masalan, von Mises kuchlanishi)
\(\bar{\varepsilon}^p\) Ekvivalent plastik deformatsiya
\(H(\bar{\varepsilon}^p)\) Izotrop qotish funksiyasi

Komplementarlik sharti: \(\lambda^p F(\boldsymbol{\sigma}, \kappa) = 0\), \(\lambda^p \geq 0\), \(F \leq 0\).

Chekli elementlar usuli

Belgi Tavsif
\(\Omega^h, \Omega_0^h\) Chekli elementlarga bo‘lish orqali yaqinlashtirilgan joriy va tayanch konfiguratsiya sohalari: \(\Omega^h = \bigcup_e \Omega^e\)
\(\Omega^e, \Omega^e_0\) Elementning joriy va tayanch konfiguratsiyadagi sohasi
\(\Gamma^e_t, \Gamma^e_{0t}\) Element chegarasining mexanik chegaraga tegishli qismi
\(\boldsymbol{r}\) Tabiiy koordinatalar (elementning mahalliy koordinatalari)
\(\boldsymbol{r}_\alpha\) \(\alpha\) tugunga mos tabiiy koordinatalardagi nuqta
\(N_\alpha^e(\boldsymbol{r})\) \(e\) elementning \(\alpha\) tuguniga mos shakl funksiyasi
\(n_e\) Elementni tashkil etuvchi tugunlar soni
\(n_g\) Global tugunlarning umumiy soni
\(\boldsymbol{X}^e_\alpha, \boldsymbol{u}^e_\alpha\) Elementning \(\alpha\) tuguni koordinatasi va siljishi
\(\boldsymbol{X}^e, \boldsymbol{u}^e\) Element tugun koordinatalari va siljish vektorlari: \(\boldsymbol{X}^e = (\boldsymbol{X}^{eT}_1, \ldots, \boldsymbol{X}^{eT}_{n_e})^T\) va hokazo
\(\boldsymbol{X}^n, \boldsymbol{u}^n\) Global tugun koordinatalari va siljish vektorlari (tugun raqami, so‘ng erkinlik darajasi tartibida)
\(\boldsymbol{B}\) Deformatsiya–siljish munosabati matritsasi (B-matritsa)
\(\boldsymbol{N}_\alpha, \boldsymbol{N}\) Shakl funksiyalari matritsasi: \(\boldsymbol{N}_\alpha\) — \(\alpha\) tugunning \(N_\alpha^e\) shakl funksiyasi diagonalga joylangan \(d \times d\) blok, \(\boldsymbol{N} = [\boldsymbol{N}_1, \ldots, \boldsymbol{N}_{n_e}]\). \(\delta\boldsymbol{u} = \boldsymbol{N}\, \delta\boldsymbol{u}^e\)
\(\boldsymbol{K}^e\) Element qattiqlik matritsasi
\(\boldsymbol{K}^e_X, \boldsymbol{K}^e_x\) Element qattiqlik matritsasining integral osti hadlari (tayanch va joriy konfiguratsiyalarda ifodalangan): \(\boldsymbol{K}^e = \int_{\Omega^e_0} \boldsymbol{K}^e_X\, dV = \int_{\Omega^e} \boldsymbol{K}^e_x\, dv\)
\(\boldsymbol{Q}^e\) (TL usuli), \(\boldsymbol{q}^e\) (UL usuli) Element ichki kuch vektori
\(\boldsymbol{F}^e_\alpha, \boldsymbol{F}^e\) Elementning \(\alpha\) tuguniga ta’sir etuvchi tugun tashqi kuchi va element tugun tashqi kuch vektori: \(\boldsymbol{F}^e = (\boldsymbol{F}^{eT}_1, \ldots, \boldsymbol{F}^{eT}_{n_e})^T\)
\(\boldsymbol{Q}, \boldsymbol{F}\) Global ichki kuch va global tashqi kuch vektorlari: \(\boldsymbol{Q} = (\boldsymbol{Q}^T_1, \ldots, \boldsymbol{Q}^T_{n_g})^T\); \(\boldsymbol{F}\) ham xuddi shunday
\(\boldsymbol{Q}_{i_g}, \boldsymbol{F}_{i_g}\) Global \(i_g\) tugundagi tugun ichki va tashqi kuchlari (\(\boldsymbol{Q}, \boldsymbol{F}\) ning tugun bloklari)
\(\boldsymbol{K}\) Global tangensial qattiqlik matritsasi: \(\boldsymbol{K} = \partial \boldsymbol{Q}/\partial \boldsymbol{u}\). \(3\times 3\) blok \(\boldsymbol{K}_{i_gi_h}\) ni \(i_g\)-qator va \(i_h\)-ustunga joylashtirishdan hosil bo‘lgan matritsa
\(\boldsymbol{R}_i\) Newton-Raphson iteratsiyasidagi qoldiq vektori: \(\boldsymbol{R}_i = \boldsymbol{F} - \boldsymbol{Q}(\boldsymbol{u}_n + \Delta\boldsymbol{u})\)
\(d\boldsymbol{u}_i\) Newton-Raphson iteratsiyasidagi tuzatma (\(i\)-iteratsiyada \(\boldsymbol{K}_i\, d\boldsymbol{u}_i = \boldsymbol{R}_{i-1}\) chiziqli tenglamani yechib olinadi)
\(i_g\) Global tugun raqami (\(1 \leq i_g \leq n_g\))
\(\mathrm{gdx}(e, \alpha)\) \(e\) elementdagi mahalliy tugun raqami \(\alpha\) ga mos global tugun raqami: \(\mathrm{gdx}(e, \alpha) = i_g\)
\(\mathcal{E}(i_g)\) Global tugun raqami \(i_g\) ga mos \((e, \alpha)\) juftliklari to‘plami: \(\mathcal{E}(i_g) = \{ (e, \alpha) \mid \mathrm{gdx}(e, \alpha) = i_g \}\). Yig‘ish amallarida ishlatiladi
\(\mathcal{E}^2(i_g, i_h)\) Global tugun raqamlari jufti \((i_g, i_h)\) ga mos \((e, \alpha, \beta)\) uchliklari to‘plami; qattiqlik matritsasini yig‘ishda ishlatiladi
\(\alpha, \beta, \gamma, \ldots\) Elementni tashkil etuvchi tugunlar indekslari
\(i, j, k, l, \ldots\) Erkinlik darajalari indekslari (3D da \(1, 2, 3\))

Yuqori samarali va konstruktiv elementlar

Belgi Tavsif
\(\bar{\boldsymbol{B}}\) B-bar usulida hajmiy qismi tuzatilgan deformatsiya–siljish munosabati matritsasi (hajmiy komponent element markazida baholangan B-matritsa bilan almashtiriladi)
\(\bar{\boldsymbol{F}}\) F-bar usulida hajmiy qismi tuzatilgan deformatsiya gradienti: \(\bar{\boldsymbol{F}} = (J_0/J)^{1/3} \boldsymbol{F}\)
\(J_0\) Element markazi \(\boldsymbol{r}=\boldsymbol{0}\) da baholangan deformatsiya gradientining hajm nisbati: \(J_0 = \det \boldsymbol{F}(\boldsymbol{0})\)
\(\boldsymbol{\alpha}\) Mos kelmaydigan modali elementdagi ichki erkinlik darajalari vektori (element chegarasida uzluksizlikni majburlamaydigan qo‘shimcha siljish modalari koeffitsiyentlari)
\(M_k(\boldsymbol{r})\) Mos kelmaydigan modalar shakl funksiyalari: \(M_1=1-\xi^2\), \(M_2=1-\eta^2\), \(M_3=1-\zeta^2\)
\(h\) Qobiq element qalinligi
\(A\) Balka elementi kesimining yuzasi
\(I\) Balka elementi kesimining ikkinchi yuza momenti
\(G\) Siljish moduli: \(G = E/[2(1+\nu)]\)

Tegishli mavzular

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