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Elastiko lineala Analisi estatikoa (Introduction)

Atal honek presents formulazioa -ren elastiko analisi estatikoa oinarrituta infinitesimal deformazio theory. Elastikotasun lineala suposatzen da stress-strain erlazio. Kapitulu honek da zuzenduta gisa introduction understanding orokor structure -ren finite-element egitura-analisia eta da organized gisa self-contained kapitulu.

orokor theory -ren lan birtualaren printzipioa (current-configuration forma, initial-configuration forma, eta murrizketa infinitesimal deformazio), ikus Lan birtualaren printzipioa; details -ren linear-elastic lege konstitutiboa, ikus Elastikotasun lineala; conventions -ren tentsore eta Voigt notazio, ikus Tensor Notation and Mathematical Foundations; orokor formulazioa -ren finitu deformazio, ikus Motion, Deformation, and Strain; eta soluzio metodoak ez-lineal problemak, ikus Zurruntasun tangentzialaren matrizea.

Oinarrizko Ekuazioak

Pean assumptions -ren infinitesimal deformazio eta elastikotasun lineala, boundary-value problema -n solido mekanika osatzen da -ren equilibrium ekuazio, mekaniko muga-baldintzak, eta geometriko muga-baldintzak (essential muga-baldintzak) (ikus Irudi 2.1.1):

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

Hemen, \(\boldsymbol{\sigma}\) da Cauchy-ren tentsioa, \(\overline{\boldsymbol{b}}\) da body indar bakoitzeko unit bolumen, \(\overline{\boldsymbol{t}}\) da agindutako gainazal traction, \(\overline{\boldsymbol{u}}\) da agindutako desplazamendu, eta \(S_t, S_u\) dira mekaniko eta geometriko boundaries.

Boundary-value problem in solid mechanics (infinitesimal-deformation problem)

Irudi 2.1.1 Boundary-value problema -n solido mekanika (infinitesimal-deformation problema)

Erabiliz symmetric gradient operator, strain-displacement erlazio da

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

linear-elastic ekuazio konstitutiboa da

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

non \(\boldsymbol{C}\) da fourth-order elastikotasuna tentsore.

Lan birtualaren printzipioa

orokor formak -ren lan birtualaren printzipioa (current-configuration forma, initial-configuration forma, eta murrizketa infinitesimal deformazio) dira laburbilduta -n Lan birtualaren printzipioa. Pean assumptions -ren infinitesimal deformazio eta elastikotasun lineala, weak forma da

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

Ordezkatuz ekuazio konstitutiboa \eqref{eq:2.1.5} eta writing \(\hat{\sigma} = D\, \hat{\varepsilon}\) -n Voigt notation ematen du forma erabilia zuzenean diskretizazio:

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

lortzen da, non \(D\) da elastiko matrize definitutako -n Elastikotasun lineala. Ekuazioak \eqref{eq:2.1.10} eta \eqref{eq:2.1.7} constitute lan birtualaren printzipioa discretized behean.

Diskretizazio eta Muntaketa -ren Global Ekuazio

Discretizing lan birtualaren printzipioa -n Eq.\( \eqref{eq:2.1.10} \) gainean finitu elementuak ematen du

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

bakoitz elementu, desplazamendu field da interpolated erabiliz desplazamenduak -ren nodoak composing elementu honela.

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

deformazio da orduan emandako erabiliz Eq.\(\eqref{eq:2.1.4}\) honela.

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

Ordezkatuz Eqs.\(\eqref{eq:2.1.12}\) eta \(\eqref{eq:2.1.13}\) ra Eq.\(\eqref{eq:2.1.11}\) ematen du

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

Ekuazio \(\eqref{eq:2.1.14}\) honela idatz daiteke

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

Hemen, osagaiak -ren matrize eta bektore definitutako bidez Eqs.\(\eqref{eq:2.1.16}\) eta \(\eqref{eq:2.1.17}\) daiteke izan kalkulatutako bakoitz elementu finitu eta assembled bidez superposition.

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

Delako Eq.\(\eqref{eq:2.1.15}\) holds arbitrary birtual desplazamendu \(\delta \boldsymbol{U}\), honako ekuazioa lortzen da.

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

Meanwhile, desplazamendu muga-baldintza -n Eq.\(\eqref{eq:2.1.3}\) honela adierazten da follows.

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

Bidez ebatziz Eq.\(\eqref{eq:2.1.18}\) baldintzapean murriztapen baldintza -n Eq.\(\eqref{eq:2.1.19}\), nodal desplazamendu \(\boldsymbol{U}\) daiteke izan zehaztuta.

Ikus Ere

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