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Uchanganuzi Tuli wa Unyumbufu wa Mstari (Utangulizi)

Hapa tunaonyesha uundaji wa uchanganuzi tuli wa unyumbufu unaotegemea nadharia ya uharibifu mdogo. Unyumbufu wa mstari unadhaniwa katika uhusiano wa mkazo na strain. Sura hii ni utangulizi wa kuelewa picha ya jumla ya uchanganuzi wa miundo kwa mbinu ya elementi finiti, na imepangwa ili iweze kusomwa kama sura inayojitegemea.

Kwa nadharia ya jumla ya kanuni ya kazi pepe (umbo katika usanidi wa sasa, umbo katika usanidi wa awali, na upunguzaji hadi uharibifu mdogo), rejelea Kanuni ya kazi pepe; kwa maelezo ya sheria ya constitutive ya unyumbufu wa mstari, rejelea Unyumbufu wa mstari; kwa kanuni za nukuu za tensor na Voigt, rejelea Nukuu za tensor na misingi ya hisabati; kwa uundaji wa jumla wa uharibifu mkubwa, rejelea Mwendo, uharibifu na strain; na kwa mbinu za kutatua matatizo yasiyo ya mstari, rejelea Matriki ya ugumu wa tangent.

Milinganyo ya msingi

Chini ya dhana za uharibifu mdogo na unyumbufu wa mstari, tatizo la thamani za mipaka katika mekanika ya yabisi linajumuisha mlinganyo wa usawa, masharti ya mipaka ya kimitambo, na masharti ya mipaka ya kijiometri (masharti muhimu ya mipaka) (tazama Mchoro 2.1.1):

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

Hapa, \(\boldsymbol{\sigma}\) ni mkazo wa Cauchy, \(\overline{\boldsymbol{b}}\) ni nguvu ya mwili kwa kila ujazo wa kitengo, \(\overline{\boldsymbol{t}}\) ni traction ya uso iliyowekwa, \(\overline{\boldsymbol{u}}\) ni uhamisho uliowekwa, na \(S_t, S_u\) ni mipaka ya kimitambo na ya kijiometri.

Tatizo la thamani za mipaka katika mekanika ya yabisi (tatizo la uharibifu mdogo)

Mchoro 2.1.1 Tatizo la thamani za mipaka katika mekanika ya yabisi (tatizo la uharibifu mdogo)

Kwa kutumia opereta ya gradient linganifu, uhusiano wa strain na uhamisho ni

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

Mlinganyo wa constitutive wa unyumbufu wa mstari ni

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

ambapo \(\boldsymbol{C}\) ni tensor ya unyumbufu ya daraja la nne.

Kanuni ya kazi pepe

Maumbo ya jumla ya kanuni ya kazi pepe (umbo la usanidi wa sasa, umbo la usanidi wa awali, na upunguzaji hadi uharibifu mdogo) yamekusanywa katika Kanuni ya kazi pepe. Chini ya dhana za uharibifu mdogo na unyumbufu wa mstari, umbo dhaifu ni

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

Kwa kubadilisha mlinganyo wa constitutive \eqref{eq:2.1.5} na kuandika \(\hat{\sigma} = D\, \hat{\varepsilon}\) kwa nukuu ya Voigt, tunapata umbo linalotumiwa moja kwa moja katika udiscretishaji:

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

ambapo \(D\) ni matriki ya unyumbufu iliyoelezwa katika Unyumbufu wa mstari. Mlinganyo \eqref{eq:2.1.10} na mlinganyo \eqref{eq:2.1.7} ndio kanuni ya kazi pepe inayodiscretishwa hapa chini.

Udiscretishaji na uunganishaji wa mlinganyo wa jumla

Kwa kudiscretisha kanuni ya kazi pepe katika mlinganyo \( \eqref{eq:2.1.10} \) kwa kila elementi finiti, tunapata

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

Kwa kila elementi, uga wa uhamisho huingizwa kwa kutumia uhamisho wa nodi zinazounda elementi kama ifuatavyo.

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

Kisha strain hutolewa kwa kutumia mlinganyo \(\eqref{eq:2.1.4}\) kama ifuatavyo.

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

Kwa kubadilisha mlinganyo \(\eqref{eq:2.1.12}\) na \(\eqref{eq:2.1.13}\) katika mlinganyo \(\eqref{eq:2.1.11}\), tunapata

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

Mlinganyo \(\eqref{eq:2.1.14}\) unaweza kuandikwa kwa ufupi kama

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

Hapa, vipengele vya matriki na vekta vinavyofafanuliwa na milinganyo \(\eqref{eq:2.1.16}\) na \(\eqref{eq:2.1.17}\) vinaweza kukokotolewa kwa kila elementi finiti na kuunganishwa kwa kujumlisha.

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

Kwa kuwa mlinganyo \(\eqref{eq:2.1.15}\) unatumika kwa uhamisho wowote pepe \(\delta \boldsymbol{U}\), tunapata

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

Kwa upande mwingine, sharti la mpaka wa uhamisho katika mlinganyo \(\eqref{eq:2.1.3}\) huandikwa kama

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

Kwa kutatua mlinganyo \(\eqref{eq:2.1.18}\) chini ya sharti la kizuizi la mlinganyo \(\eqref{eq:2.1.19}\), uhamisho wa nodi \(\boldsymbol{U}\) unaweza kuamuliwa.

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