From baeefbef83dd26d44b5f17b311bc2d6e071ca22c Mon Sep 17 00:00:00 2001 From: "Denis V. Dedkov" Date: Sun, 30 Mar 2014 14:03:25 +0600 Subject: [PATCH] Chapter 2 was refactored --- c2.tex | 102 +++++++++++++++++++++++---------------- disser.kilepr | 54 ++++++++++----------- fig/mesh/v1/matrix.png | Bin 122670 -> 120496 bytes stress_concentartors.tex | 2 +- tables.tex | 42 +--------------- 5 files changed, 90 insertions(+), 110 deletions(-) diff --git a/c2.tex b/c2.tex index 6299371..b7fa82c 100644 --- a/c2.tex +++ b/c2.tex @@ -3,13 +3,7 @@ В главе\insecondtext -\section{Математическая модель упруго-хрупкого поведения тканого композита с -поликристаллической матрицей} - -\section{Модели тканого композита с поликристаллической матрицей с периодическим -и квазипериодическим расположением волокон} - -\subsection{Численное решение краевой задачи упругости методом конечных +\section{Численное решение краевой задачи упругости методом конечных элементов} Краевая задача \eqref{eq:Eqvilibrium} -- \eqref{eq:Guck} с граничными условиями @@ -54,27 +48,49 @@ \label{fig:mesh:fibers} \end{figure} -Степень дискретизации выбиралась таким образом, чтобы чтобы полученные значения -структурных перемещений, деформаций и напряжений в слое тканого композита без -локальных дефектов и с несовершенствами ни качественно, ни количественно не -изменялись при уменьшении характерных размеров конечных элементов. +Для сопряжения конечно-элементных сеток армирующего каркаса и матрицы на этапе +дискретизации выделялись <<главная>> и <<подчиненная>> поверхности. На этапе +расчета узловые точки <<подчиненной>> поверхности (например, принадлежащие +матрице) проецировались на те ближайшие конечные элементы, грани которых +расположены на <<главной>> поверхности, и считались принадлежащими этим +элементам. Перемещения точек <подчиненной>> поверхности заменялись перемещениями +их проекций на элемент <<главной>> поверхности \cite{bib:code-aster:contact}. -Из таблицы~{\ref{tab:convergence}}, в которой показана зависимость максимальных -интенсивностей напряжений от количества конечных элементов, видно, что -расхождение между двумя последними строками не превышает $1\%$. Это говорит о -достаточной степени дискретизации модели. +Модуль Юнга $E_f = 280$~ГПа и коэффициент Пуассона $\nu_f = 0{,}20$ +волокон соответствовали данным работы \cite{bib:tarnapolsky}. +Упругие модули поликристаллической матрицы были выбраны следующими: $E_m += 0{,}28$~ГПа и коэффициент Пуассона $\nu_m = 0,40$. + +\section{Тестирование математической модели тканого композита с искривленными +волокнами} + +Для проверки корректности построения математической модели решалась задача по +определению напряженно-деформированного состояния при двухосном +равнокомпонентном деформировании слоя тканого композита с искривленными +волокнами для сеток с разным количеством конечных элементов и проводилось +сравнение значений интенсивностей напряжений $\sigma_I$ в точке, находящейся +в геометрическом центре слоя тканого композита с бездефектной идеальной +периодической структурой. Такие же задачи решались для модели слоя тканого +композита с дефектом в виде туннельной поры, для случаев когда полость, +возникающая в следствие дефекта доуплотняется материалом связующего или +остается незаполненной. + +Зависимость интенсивностей напряжений в точке, находящейся в центре слоя +тканного композита от количества конечных элементов показана в таблице +\ref{tab:convergence}. \begin{table}[ht!] - \caption{Зависимость максимальных интенсивностей напряжений от количества - \newline конечных элементов} + \caption{Зависимость интенсивностей напряжений от количества конечных +элементов} \begin{tabular}{|c|c||c|c||c|c|} \hline \multicolumn{2}{|p{5cm}||}{Идеальная периодическая структура}& - \multicolumn{2}{|p{5cm}||}{Тунельная пора}& - \multicolumn{2}{|p{5cm}|}{Туннельная пора, доуплотненная матрицей} \\ + \multicolumn{2}{|p{5cm}||}{Туннельная пора}& + \multicolumn{2}{|p{5cm}|}{Туннельная пора, доуплотненная материалом +связующего} \\ \hline - $C$ & $\sigma_{max}$ & $C$ & $\sigma_{max}$ & $C$ & $\sigma_{max}$ \\ + $C$ & $\sigma_{I}$ & $C$ & $\sigma_{I}$ & $C$ & $\sigma_{I}$ \\ \hline \hline 218 207 & 33.6 & 213 381 & 38.0 & 194 196 & 37.9 \\ @@ -89,6 +105,28 @@ \label{tab:convergence} \end{table} +Из таблицы видно, что расхождение между интенсивностями напряжений в двух +последних вычислительных экспериментах не превышает $1\%$, что может +свидетельствовать о достаточной степени дискретизации модели. + +Распределения интенсивностей напряжений в слое тканого композита с идеальной +периодической структурой, полученные в ходе решения задачи показаны на +рис.~\ref{fig:vmis_v1_s1}. + +\begin{figure}[ht] + \includegraphics[width=15cm]{vmis_v1_s1} + \caption{Поля интенсивности напряжений в слое тканого композита с идеальной +периодической структурой} + \label{fig:vmis_v1_s1} +\end{figure} + +Из рисунка видно, что распределение искомых полей в рассматриваемом случае +удовлетворяют условиям симметрии и периодичности геометрической модели и +приложенной внешней нагрузке, что говорит о корректно построенной +геометрической модели и корректности полученного численного решения. +Максимальных значений интенсивность напряжений достигает в местах наибольшей +кривизны волокон. + Параметры конечно-элементной сетки, удовлетворяющие условиям неизменности качественных и количественных характеристик для моделей с различными видами дефектов, а также для модели с идеальной периодической структурой представлены @@ -121,28 +159,8 @@ \label{tab:discr} \end{table} -Модуль Юнга $E_f = 280$~ГПа и коэффициент Пуассона $\nu_f = 0{,}20$ -волокон соответствовали данным работы \cite{bib:tarnapolsky}. -Упругие модули поликристаллической матрицы были выбраны следующими: $E_m -= 0{,}28$~ГПа и коэффициент Пуассона $\nu_m = 0,40$. - -Распределения интенсивностей напряжений в слое тканого композита с идеальной -периодической структурой, полученные в ходе решения задачи показаны на -рис.~\ref{fig:vmis_v1_s1}. - -\begin{figure}[ht] - \includegraphics[width=15cm]{vmis_v1_s1} - \caption{Поля интенсивности напряжений в слое тканого композита с идеальной -периодической структурой} - \label{fig:vmis_v1_s1} -\end{figure} - -Из рисунка видно, что распределение искомых полей в рассматриваемом случае -удовлетворяют условиям симметрии и периодичности геометрической модели и -приложенной внешней нагрузке, что говорит о корректно построенной -геометрической модели и корректности полученного численного решения. -Максимальных значений интенсивность напряжений достигает в местах наибольшей -кривизны волокон. +\section{Модели тканого композита с поликристаллической матрицей с периодическим +и квазипериодическим расположением волокон} \subsection{Коэффициенты концентрации напряжений} diff --git a/disser.kilepr b/disser.kilepr index 8816ca7..f3b7b74 100644 --- a/disser.kilepr +++ b/disser.kilepr @@ -4,7 +4,7 @@ img_extIsRegExp=false img_extensions=.eps .jpg .jpeg .png .pdf .ps .fig .gif kileprversion=2 kileversion=2.1.0 -lastDocument=bibliography.bib +lastDocument=stress_concentartors.tex masterDocument= name=disser pkg_extIsRegExp=false @@ -98,50 +98,50 @@ ReadWrite=true [item:bibliography.bib] archive=true -column=1 +column=0 encoding=UTF-8 highlight=BibTeX -line=203 +line=0 mode=BibTeX open=true order=3 [item:c1.tex] archive=true -column=0 +column=51 encoding=UTF-8 highlight=LaTeX -line=0 +line=520 mode=LaTeX open=true order=4 [item:c2.tex] archive=true -column=47 +column=72 encoding=UTF-8 highlight=LaTeX -line=380 +line=11 mode=LaTeX open=true order=5 [item:c3.tex] archive=true -column=73 +column=62 encoding=UTF-8 highlight=LaTeX -line=407 +line=457 mode=LaTeX open=true order=6 [item:common.tex] archive=true -column=66 +column=14 encoding=UTF-8 highlight=LaTeX -line=21 +line=162 mode=LaTeX open=true order=1 @@ -178,41 +178,41 @@ order=2 [item:stress_concentartors.tex] archive=true -column=15 +column=0 encoding=UTF-8 highlight=LaTeX -line=17 +line=36 mode=LaTeX open=true order=0 [view-settings,view=0,item:bibliography.bib] 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