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Kanuni ya kazi halisi

Kutokana na equilibrium equation na boundary conditions zilizotolewa katika Stress na conservation laws, tunatoa kanuni ya kazi halisi, ambayo ni weak form ya boundary-value problem ya continuum mechanics. FEM discretization huanzia kwenye weak form hii. Sura hii inaonyesha maumbo yote mawili: current configuration (kwa Cauchy stress na linear part ya Almansi strain) na reference configuration (kwa second Piola-Kirchhoff stress na Green-Lagrange strain), inaonyesha kuwa yanafanana, kisha inathibitisha kikomo cha small deformation.

Equilibrium equation na masharti ya mpaka

Tuchukulie body force kwa unit mass inayotenda kwenye continuum kuwa \(\boldsymbol{g}\), na kitu kinachokaa katika domain \(\Omega\) kwenye current configuration. Boundary \(\Gamma\) hugawanywa kuwa geometric boundary ambako displacement imewekwa kuwa \(\bar{\boldsymbol{u}}\), yaani \(\Gamma_B\), na mechanical boundary ambako surface traction imewekwa kuwa \(\bar{\boldsymbol{t}}\), yaani \(\Gamma_t\), huku \(\Gamma = \Gamma_B \cup \Gamma_t\) na \(\Gamma_B \cap \Gamma_t = \emptyset\). Kwa static problem, tukiondoa inertial term kutoka conservation of momentum iliyoonyeshwa katika Stress na conservation laws, tunapata

\[ \nabla_x \cdot \boldsymbol{\sigma} + \rho \boldsymbol{g} = \boldsymbol{0} \quad \text{in} \ \Omega \]

kama equilibrium equation. Boundary conditions ni

\[ \boldsymbol{\sigma} \boldsymbol{n} = \bar{\boldsymbol{t}} \quad \text{on} \ \Gamma_t \]
\[ \boldsymbol{u} = \bar{\boldsymbol{u}} \quad \text{on} \ \Gamma_B \]

Kuanzia hapa, kanuni ya kazi halisi inatolewa kama weak form ya equilibrium equation na mechanical boundary condition \(\boldsymbol{\sigma} \boldsymbol{n} = \bar{\boldsymbol{t}}\). Geometric boundary condition \(\boldsymbol{u} = \bar{\boldsymbol{u}}\) inaingizwa kupitia uchaguzi wa test function.

Weak form katika current configuration

Katika weak form, admissible space ya displacement isiyojulikana na test-function space hufafanuliwa mtawalia kama

\[ \mathcal{U} = \{ \boldsymbol{u} \in [H^1(\Omega)]^d \mid \boldsymbol{u} = \bar{\boldsymbol{u}} \ \text{on} \ \Gamma_B \} \]
\[ \mathcal{V} = \{ \delta \boldsymbol{u} \in [H^1(\Omega)]^d \mid \delta \boldsymbol{u} = \boldsymbol{0} \ \text{on} \ \Gamma_B \} \]

Hapa \(d\) ni spatial dimension, \(H^1(\Omega)\) ni Sobolev space yenye square-integrable weak derivatives hadi order ya kwanza, na \(\delta\) ni alama ya variation. Katika current-configuration representation, \(\Omega\) ni configuration baada ya mgeuko; katika numerical solution halisi, quantities hurudishwa kwenye reference configuration au intermediate configuration inayojulikana.

Tukizidisha equilibrium equation kwa weight \(\delta \boldsymbol{u} \in \mathcal{V}\) na kutumia divergence theorem ya Gauss pamoja na mechanical boundary condition, kanuni ya kazi halisi katika current configuration huwa

\[ \int_{\Omega} \boldsymbol{\sigma} : \delta \boldsymbol{A}_{(L)}\, dv = \int_{\Gamma_t} \delta \boldsymbol{u}^T \bar{\boldsymbol{t}}\, d\Gamma + \int_{\Omega} \delta \boldsymbol{u}^T \rho \boldsymbol{g}\, dv \]

Hapa \(\boldsymbol{A}_{(L)}\) ni linear part ya Almansi strain tensor, na

\[ \boldsymbol{A}_{(L)} = \frac{1}{2}\left( \nabla_x \boldsymbol{u} + (\nabla_x \boldsymbol{u})^T \right), \qquad A_{(L)ij} = \frac{1}{2}\left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \]

hufafanuliwa hivyo. Variation yake ni \(\delta \boldsymbol{A}_{(L)} = \tfrac{1}{2}(\nabla_x \delta \boldsymbol{u} + (\nabla_x \delta \boldsymbol{u})^T)\). Yaani, tunatafuta \(\boldsymbol{u} \in \mathcal{U}\) inayotimiza mlinganyo wa kazi halisi kwa \(\delta \boldsymbol{u} \in \mathcal{V}\) yoyote. Upande wa kushoto ni kazi halisi ya internal force, na upande wa kulia ni kazi halisi ya external force kutokana na prescribed surface traction na body force.

Kwa kuwa mlinganyo huu umeandikwa kwenye domain baada ya mgeuko (current configuration), katika solution halisi initial configuration \(\Omega_0\) (reference configuration) au intermediate configuration inayojulikana huchaguliwa upya kama reference, kisha mlinganyo huandikwa katika incremental form kabla ya kutatuliwa. Kwa uchaguzi maalum wa reference configuration (Total Lagrange / Updated Lagrange) na incremental decomposition, rejelea Mfumo wa uchanganuzi wa nyongeza.

Weak form katika initial configuration

Tuchukulie kitu kinachokaa katika domain \(\Omega_0\) kwenye reference configuration, na tugawanye boundary yake \(\Gamma_0\) kuwa \(\Gamma_{0B} \cup \Gamma_{0t}\). Tukirudisha current-configuration representation kwenye reference configuration, conjugate pair ya stress na strain huwa second Piola-Kirchhoff stress \(\boldsymbol{S}\) na Green-Lagrange strain \(\boldsymbol{E}\). Kanuni ya kazi halisi katika initial configuration huwa

\[ \int_{\Omega_0} \boldsymbol{S} : \delta \boldsymbol{E}\, dV = \int_{\Gamma_{0t}} \delta \boldsymbol{u}^T \bar{\boldsymbol{t}}\, d\Gamma_0 + \int_{\Omega_0} \delta \boldsymbol{u}^T \rho_0 \boldsymbol{g}\, dV \]

Hapa \(\rho_0\) ni mass density katika reference configuration, na kupitia conservation of mass \(\rho_0 = J\rho\) ni sawa na body-force representation katika current configuration.

Usawa wa current- na initial-configuration representations

Kazi halisi ya internal force katika representations zote mbili inalingana kupitia transformation kwa deformation gradient \(\boldsymbol{F}\) na volume ratio \(J = \det \boldsymbol{F}\). Yaani,

\[ \int_{\Omega_0} \boldsymbol{S} : \delta \boldsymbol{E}\, dV = \int_{\Omega} \boldsymbol{\sigma} : \delta \boldsymbol{A}_{(L)}\, dv \]

External-force terms pia ni sawa kupitia conservation of mass na transformation ya surface traction. Kwa hiyo milinganyo ya kazi halisi katika current na initial configurations ni kanuni ileile iliyoonyeshwa katika configurations tofauti. Solution inayorejelea reference configuration inalingana na Total Lagrange, na inayorejelea current configuration (configuration ya convergence ya awali) inalingana na Updated Lagrange.

Kikomo cha mgeuko mdogo

Chini ya small-deformation assumption \(\boldsymbol{F} \approx \boldsymbol{I}\) na \(J \approx 1\), tofauti kati ya current na reference configurations hutoweka, second PK stress hulingana na Cauchy stress (\(\boldsymbol{S} \to \boldsymbol{\sigma}\)), na Green-Lagrange strain pamoja na linear part ya Almansi strain zote hupungua kuwa small strain \(\boldsymbol{\varepsilon}\).

\[ \boldsymbol{\varepsilon} = \nabla_S \boldsymbol{u} = \frac{1}{2}\left( \nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^T \right), \qquad \varepsilon_{ij} = \frac{1}{2}\left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \]

Katika hali hii, kanuni ya kazi halisi hupungua kuwa weak form katika Cauchy stress \(\boldsymbol{\sigma}\) na small strain \(\boldsymbol{\varepsilon}\)

\[ \int_{\Omega} \boldsymbol{\sigma} : \delta \boldsymbol{\varepsilon}\, dV = \int_{\Gamma_t} \delta \boldsymbol{u}^T \bar{\boldsymbol{t}}\, d\Gamma + \int_{\Omega} \delta \boldsymbol{u}^T \rho \boldsymbol{g}\, dV \]
\[ \delta \boldsymbol{u} = \boldsymbol{0} \quad \text{on} \ \Gamma_B \]

Hii ndiyo weak form inayotumiwa moja kwa moja katika discretization ya small-deformation linear-elastic static analysis (Linear-elastic static analysis (utangulizi/appendix) huanzia hapa na kuonyesha uundaji wa element stiffness \(\boldsymbol{K}^e\) na assembly ya global equation \(\boldsymbol{K}\boldsymbol{U} = \boldsymbol{F}\)).

Tukibadilisha linear-elastic constitutive law \(\boldsymbol{\sigma} = \boldsymbol{\mathsf{C}} : \boldsymbol{\varepsilon}\) na kuandika \(\hat{\sigma} = D\, \hat{\varepsilon}\) kwa Voigt notation, weak form huwa

\[ \int_{\Omega} \delta \hat{\varepsilon}^T D\, \hat{\varepsilon}\, dV = \int_{\Gamma_t} \delta \boldsymbol{u}^T \bar{\boldsymbol{t}}\, d\Gamma + \int_{\Omega} \delta \boldsymbol{u}^T \rho \boldsymbol{g}\, dV \]

katika umbo hilo.

Vipengee vinavyohusiana

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