From d28db9f1a2273fd4d9f5ee8e683a887068bac75b Mon Sep 17 00:00:00 2001 From: "Denis V. Dedkov" Date: Sat, 15 Sep 2012 20:59:45 +0600 Subject: [PATCH] Chapter 2 was finished --- bibliography.bib | 23 ++ c2.tex | 528 +++++++++++++++++++++++++- c3.tex | 49 --- disser.kilepr | 32 +- fig/concentrators/v1/scheme1/d1d2.png | Bin 0 -> 597813 bytes fig/concentrators/v1/scheme1/d3d6.png | Bin 0 -> 603215 bytes fig/concentrators/v1/scheme1/d4d7.png | Bin 0 -> 622411 bytes fig/concentrators/v1/scheme2/d1d2.png | Bin 0 -> 609202 bytes fig/concentrators/v1/scheme2/d3d6.png | Bin 0 -> 615778 bytes fig/concentrators/v1/scheme2/d4d7.png | Bin 0 -> 635287 bytes fig/concentrators/v1/scheme3/d1d2.png | Bin 0 -> 593727 bytes fig/concentrators/v1/scheme3/d3d6.png | Bin 0 -> 637563 bytes fig/concentrators/v1/scheme3/d4d7.png | Bin 0 -> 640493 bytes fig/elements.png | Bin 0 -> 11167 bytes fig/elements.svg | 237 ++++++++++++ fig/mesh/v1/fibers.png | Bin 0 -> 152976 bytes fig/mesh/v1/matrix.png | Bin 0 -> 122670 bytes fig/vmis_v1_s1.png | Bin 0 -> 339721 bytes fig/vmis_v1_s2.png | Bin 0 -> 318650 bytes fig/vmis_v1_s3.png | Bin 0 -> 271277 bytes stress_concentartors.tex | 2 + 21 files changed, 797 insertions(+), 74 deletions(-) create mode 100644 fig/concentrators/v1/scheme1/d1d2.png create mode 100644 fig/concentrators/v1/scheme1/d3d6.png create mode 100644 fig/concentrators/v1/scheme1/d4d7.png create mode 100644 fig/concentrators/v1/scheme2/d1d2.png create mode 100644 fig/concentrators/v1/scheme2/d3d6.png create mode 100644 fig/concentrators/v1/scheme2/d4d7.png create mode 100644 fig/concentrators/v1/scheme3/d1d2.png create mode 100644 fig/concentrators/v1/scheme3/d3d6.png create mode 100644 fig/concentrators/v1/scheme3/d4d7.png create mode 100644 fig/elements.png create mode 100644 fig/elements.svg create mode 100644 fig/mesh/v1/fibers.png create mode 100644 fig/mesh/v1/matrix.png create mode 100644 fig/vmis_v1_s1.png create mode 100644 fig/vmis_v1_s2.png create mode 100644 fig/vmis_v1_s3.png diff --git a/bibliography.bib b/bibliography.bib index ece7f7f..a3f2881 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -107,4 +107,27 @@ Algorithm. Backgrounds.}, author = {<<Ладуга>> Инженерные услуги}, year = {2012}, language = {russian} +} + +@ONLINE{bib:code-aster:common, + url = {http://www.code-aster.org/V2/UPLOAD/DOC/Presentation/2007_nafems.pdf}, + title = {Free Software for Computational Mechanics: EDF’s Choice}, + author = {Christophe Durand}, + year = {2007} +} + +@ONLINE{bib:code-aster:presentation, + url = +{http://www.code-aster.org/V2/UPLOAD/DOC/Presentation/2008_Manchester.pdf}, + title = {Analyse des Structures et Thermo-m\'{e}canique pour des \'{E}tudes +et des Recherches}, + year = {2007} +} + +@BOOK{bib:rossum, + author = {Г.~Россум and Ф.~Л.~Дж.~Дрейк and Д.~С.~Откидач}, + title = {Язык программирования Python}, + year = {2001}, + pages = {454}, + language = {russian} } \ No newline at end of file diff --git a/c2.tex b/c2.tex index 6252e23..e0dbd3d 100644 --- a/c2.tex +++ b/c2.tex @@ -42,7 +42,7 @@ NUclear REactor SIMulations), предназначенного для полно (рис.~\ref{fig:c2:regular}~б) \cite{bib:salome:geom, bib:salome:additional_geom, bib:laduga:geom}. -\begin{figure}[h] +\begin{figure}[ht] \centering \includegraphics[width=17cm]{geometry/v1/regular_all} \caption{Фрагмент слоя тканого композита с идеальной периодической структурой: @@ -62,7 +62,7 @@ bib:salome:additional_geom, bib:laduga:geom}. (рис.~\ref{fig:c2:two_fibers_break}), а также внутреннюю технологическую пору (рис.~\ref{fig:c2:pore}). -\begin{figure}[h] +\begin{figure}[ht] \centering \includegraphics[width=17cm]{geometry/v1/d1d2} \caption{Фрагмент тканого композита с пропуском нити без дополнительной @@ -70,7 +70,7 @@ bib:salome:additional_geom, bib:laduga:geom}. \label{fig:c2:fiber_skip} \end{figure} -\begin{figure}[h!] +\begin{figure}[ht!] \centering \includegraphics[width=17cm]{geometry/v1/d3d6} \caption{Фрагмент тканого композита с разрывом волокна основы без @@ -183,7 +183,269 @@ C_{ijkl}^{m} \left[ 1-\lambda({\bf r}) \right ] \right\} \label{eq:b_cond_free} \end{equation} -Заменяя граничные условия \ref{eq:b_cond} граничными условиями +\section{Модели тканого композита с поликристаллической матрицей с периодическим +и квазипериодическим расположением волокон} + +\subsection{Численное решение краевой задачи упругости методом конечных +элементов} + +Краевая задача \eqref{eq:Eqvilibrium} -- \eqref{eq:Guck} с граничными условиями +\eqref{eq:b_cond} -- \eqref{eq:b_cond_free} решается численно методом конечных +элементов, который является одним из наиболее эффективных методов решения задач +механики деформируемого твердого тела и расчета конструкций из тканых +композитов. + +Решать задачу будем с помощью некоммерческого пакета Code-Aster, входящего в +состав платформы SALOME-MECA. Этот пакет был разработан и сертифицирован +специально для французской энергетической отрасли и предназначен для задач +механики сплошных сред, термо- и гидродинамики, акустики и магнетизма, +выполнения расчетов для строительных конструкций и сооружений +\cite{bib:code-aster:common, bib:code-aster:presentation}. + +Дискретизация матрицы проводилась на 14-узловые тетраэдральные элементы +(рис.~\ref{fig:elements}~а), волокно разбивалось на 20-узловые гексаэдральные +элементы (рис.~\ref{fig:elements}~б). + +\begin{figure}[ht!] + \centering + \includegraphics[width=8cm]{elements} + \caption{Пример конечных элементов: а) тетраэдральный, б) гексаэдральный} + \label{fig:elements} +\end{figure} + +На рис.~\ref{fig:mesh:matrix} представлена конечно-элементная сетка фрагмента +матрицы слоя модельного тканого композита полотняного переплетения. +Конечно-элементная сетка волокон представлена на рис.~\ref{fig:mesh:fibers}. + +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. + +Из таблицы~{\ref{tab:convergence}}, в которой показана зависимость максимальных +интенсивностей напяжений от количества конечных элементов, видно, что +расхождение между двумя последними строками не превышает $1\%$. Это говорит о +достаточной степени дискретизации модели. + +\begin{figure}[ht!] + \centering + \includegraphics[width=17cm]{mesh/v1/matrix} + \caption{Пример дискретизации матрицы} + \label{fig:mesh:matrix} +\end{figure} + +\begin{figure}[ht!] + \centering + \includegraphics[width=17cm]{mesh/v1/fibers} + \caption{Пример дискретизации волокон} + \label{fig:mesh:fibers} +\end{figure} + +\begin{table}[ht!] + \caption{Зависимость максимальных интенсивностей напряжений от количества + \newline 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} \\ + \hline + $C$ & $\sigma_{max}$ & $C$ & $\sigma_{max}$ & $C$ & $\sigma_{max}$ \\ + \hline + \hline + 218 207 & 33.6 & 213 381 & 38.0 & 194 196 & 37.9 \\ + \hline + 271 644 & 32.0 & 261 695 & 36.2 & 241 932 & 36.0 \\ + \hline + 365 283 & 31.1 & 345 396 & 35.2 & 326 327 & 35.2 \\ + \hline + 427 855 & 31.2 & 402 304 & 35.4 & 382 954 & 35.3 \\ + \hline + \end{tabular} + \label{tab:convergence} +\end{table} + +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 +в таблице~\ref{tab:discr}. + +\begin{table}[ht!] + \caption{Параметры конечно-элементной сетки} + \begin{tabular}{|p{8.25cm}||>{\centering}p{3.45cm}|p{3.45cm}<{\centering}|} + \hline + & Тетраэдральные элементы & Гексаэдральные элементы \\ + \hline + \hline + Идеальная периодическая структура & 298 255 & 77 760 \\ + \hline + Тунельная пора & 285 664 & 69 984 \\ + \hline + Туннельная пора с доуплотнением & 266 314 & 69 984 \\ + \hline + Разрыв волокна основы & 285 466 & 75 168 \\ + \hline + Разрыв волокна основы с доуплотнением & 296 499 & 75 168 \\ + \hline + Разрыв волокон основы и утка & 279 276 & 72 576 \\ + \hline + Разрыв волокон основы и утка с доуплотнением & 276 175 & 72 576 \\ + \hline + Внутренняя технологическая пора & 287 934 & 77 760 \\ + \hline + \end{tabular} + \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 интесивностей 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 решенеия. +Максимальных значений интенсивность напряжений достигает в местах наибольшей +кривизны волокон. + +\subsection{Коэффициенты концентрации напряжений} + +Безразмерные коэффициенты $K_{\sigma_{ij}} = \sigma_{ij}({\bf r}) / +\sigma_{ij}^{per}({\bf r})$ вычислялись как отношение компонент тензора +напряжений в слое модельного тканого композита с локальным дефектом к +соответствующим компонентам в слое материала идеальной периодической структуры. + +Для расчета коэффициентов концентрации был написан пакет вспомогательных +программ с использованием языка программирования Python, который является +простым и, в то же время, мощным интерпретируемым объектно-ориентированным +языком программирования. Он предоставляет структуры данных высокого уровня, +имеет изящный синтаксис и использует динамический контроль типов, что делает +его деальным языком для быстрого написания различных приложений, работающих +на большинстве распространенных платформ \cite{bib:rossum}. + +Максимальные значения коэффициентов концентрации напряжений представлены в +таблице~\ref{tab:max_k_s1}: + +\begin{table}[ht] + \centering + \caption{Максимальные коэффициенты концентрации напряжений в слое тканого +композита} + \begin{tabular}{|p{8cm}||c|c|c|c|c|c|} + \hline + & $K_{\sigma_{11}}$ + & $K_{\sigma_{22}}$ + & $K_{\sigma_{33}}$ + & $K_{\sigma_{12}}$ + & $K_{\sigma_{13}}$ + & $K_{\sigma_{23}}$ \\ + \hline + \hline + Пропуск волокна основы + & $1{,}34$ & $2{,}11$ & $1{,}53$ & $1{,}36$ & ${\bf 2{,}50}$ & $1{,}42$ \\ + \hline + Пропуск волокна основы (доуплотнение) + & $1{,}28$ & $1{,}77$ & $1{,}31$ & $1{,}29$ & ${\bf 2{,}43}$ & $1{,}23$ \\ + \hline + \hline + Разрыв нити основы + & $1{,}29$ & $1{,}63$ & $1{,}30$ & $1{,}25$ & ${\bf 2{,}31}$ & $1{,}44$ \\ + \hline + Разрыв нити основы (доуплотнение) + & $1{,}26$ & $1{,}49$ & $1{,}27$ & $1{,}25$ & ${\bf 2{,}20}$ & $1{,}32$ \\ + \hline + \hline + Разрыв нитей основы и утка + & $1{,}50$ & $1{,}92$ & $1{,}56$ & $1{,}58$ & ${\bf 2{,}53}$ & $1{,}70$ \\ + \hline + Разрыв нитей основы и утка (доуплотнение) + & $1{,}35$ & $1{,}68$ & $1{,}41$ & $1{,}41$ & ${\bf 2{,}21}$ & $1{,}50$ \\ + \hline + \hline + Внутренняя пора + & $1{,}31$ & $1{,}93$ & $1{,}35$ & ${\bf 4{,}38}$ & $1{,}73$ & ${\bf 4{,}56}$\\ + \hline + \end{tabular} + \label{tab:max_k_s1} +\end{table} + +Как видно из таблицы, наибольший вклад в коэффициенты концентрации для всех +типов дефектов кроме внутренней технологической поры вносит касательная +составляющая тензора напряжений $\sigma_{13}$, её значение в модели с +дефектом более чем в $2$ раза превышает соответствующее значение в идеальной +периодической модели. В случае внутренней технологической поры значения +коэффициентов концентраций превышают $4$ и соответствуют касательным +составляющим тензора напряжений $\sigma_{12}$ и $\sigma_{23}$. + +На рис.~\ref{fig:k_d1d2_s1}~--~\ref{fig:k_d5_s1} показаны распределения +коэффициентов концентрации интенсивностей напряжений в слое тканого композита с +искривленными волокнами и поликристалической матрицей при наличии различных +типов технологических дефектов и с учётом дополнительной пропитки композита +материалом матрицы. Расположение областей, в которых интенсивность напряжений +достигает максимальных значений в местах, где искривленные волокна основы или +утка имеют наибольшую кривизну, строго периодично. Исключение составляют +области, расположенные вблизи локальных дефектов, где интенсивности напряжений +превышают соответствующие интенсивности напряжений определенное для композита +идеальной периодической структуры в $1{,}2$ раза для случаев разрыва волокна +основы и внутренней технологической поры, в $1{,}4$ раза для случая пропуска +волокна основы и в $1{,}5$ раз для одновременного разрыва волокон основы и +утка. При этом, в случае пропуска волокна основы или разрыва волокон основы и +утка, значение коэффициентов концентрации интенсивностей напряжений может быть +снижено до $1{,}3$ с помощью дополнительных операций доуплотнения +поликристаллической матрицы. + +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v1/scheme1/d1d2} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с пропуском волокна основы~(а) и с учётом +доуплотнения~(б)} + \label{fig:k_d1d2_s1} +\end{figure} + +\pagebreak + +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v1/scheme1/d3d6} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с разрывом волокна основы~(а) и с учётом +доуплотнения~(б)} + \label{fig:k_d3d6_s1} +\end{figure} + +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v1/scheme1/d4d7} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с разрывом волокон основы и утка~(а) и с учётом +доуплотнения~(б)} + \label{fig:k_d4d7_s1} +\end{figure} + +\pagebreak + +\begin{figure}[ht!] + \centering + \includegraphics[width=10cm]{concentrators/v1/scheme1/d5} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с внутренней технологической порой} + \label{fig:k_d5_s1} +\end{figure} + +\subsection{Коэффициенты концентрации напряжений при чистом сдвиге} + +Если в краевой задаче \eqref{eq:Eqvilibrium} -- \eqref{eq:Guck} заменить +граничные условия \ref{eq:b_cond} граничными условиями \begin{equation} \begin{array}{c} @@ -198,7 +460,126 @@ C_{ijkl}^{m} \left[ 1-\lambda({\bf r}) \right ] \right\} \label{eq:b_cond:s2} \end{equation} -\noindent получим задачу на чистый сдвиг, а при замене граничными условиями +\noindent получим задачу на чистый сдвиг, решив которую получим распределение +интенсивностей напряжений, показанных на рис.~\ref{fig:vmis_v1_s2}. + +\begin{figure}[ht] + \includegraphics[width=15cm]{vmis_v1_s2} + \caption{Поля интенсивности напряжений в слое тканого композита с идеальной +периодической структурой при чистом сдвиге} + \label{fig:vmis_v1_s2} +\end{figure} + +Максимальные значения коэффициентов концентрации напряжений в слое тканного +композита с искривленными волокнами и поликристаллической матрицей при наличии +различных технологических дефектов под воздействием сдвиговых +нагрузок представлены в таблице~\ref{tab:max_k_s2}: + +\begin{table}[ht!] + \centering + \caption{Максимальные коэффициенты концентрации напряжений в слое тканого +композита при чистом сдвиге} + \begin{tabular}{|p{8cm}||c|c|c|c|c|c|} + \hline + & $K_{\sigma_{11}}$ + & $K_{\sigma_{22}}$ + & $K_{\sigma_{33}}$ + & $K_{\sigma_{12}}$ + & $K_{\sigma_{13}}$ + & $K_{\sigma_{23}}$ \\ + \hline + \hline + Пропуск волокна основы + & $1{,}24$ & $1{,}31$ & $\bf2{,}30$ & $1{,}36$ & $\bf2{,}02$ & $1{,}52$ \\ + \hline + Пропуск волокна основы (доуплотнение) + & $1{,}22$ & $1{,}27$ & $\bf2{,}14$ & $1{,}30$ & $\bf2{,}13$ & $1{,}55$ \\ + \hline + \hline + Разрыв нити основы + & $1{,}25$ & $1{,}27$ & $\bf2{,}38$ & $1{,}29$ & $\bf1{,}97$ & $1{,}59$ \\ + \hline + Разрыв нити основы (доуплотнение) + & $1{,}23$ & $1{,}25$ & $\bf2{,}03$ & $1{,}35$ & $\bf1{,}89$ & $1{,}52$ \\ + \hline + \hline + Разрыв нитей основы и утка + & $1{,}60$ & $1{,}56$ & $\bf3{,}28$ & $1{,}95$ & $\bf2{,}42$ & $2{,}01$ \\ + \hline + Разрыв нитей основы и утка (доуплотнение) + & $1{,}48$ & $1{,}45$ & $\bf2{,}59$ & $1{,}76$ & $\bf2{,}17$ & $1{,}82$ \\ + \hline + \hline + Внутренняя пора + & $1{,}19$ & $1{,}28$ & $\bf4{,}90$ & $\bf4{,}80$ & $1{,}30$ & $\bf5{,}04$\\ + \hline + \end{tabular} + \label{tab:max_k_s2} +\end{table} + +Из таблицы видно, что в случае приложения сдвиговых нагрузок к +фрагменту композита с локальными технологическими дефектами максимальные +значения принимают коэффициенты концентрации касательной составляющей +$\sigma_{13}$ и нормальной составляющей $\sigma_{33}$ компонент тензор +напряжений. Для фрагмента с внутренней технологической порой максимальный вклад +в коэффициенты концентрации напряжений вносят касательные составляющие +$\sigma_{12}$ и $\sigma_{23}$ и нормальная составляющая $\sigma_{33}$ тензора +напряжений. + +На рис.~\ref{fig:k_d1d2_s2}~--~\ref{fig:k_d5_s2} показаны распределения +коэффициентов концентрации интенсивностей напряжений в слое тканого композита с +искривленными волокнами и поликристалической матрицей при наличии различных +типов технологических дефектов и с учётом дополнительной пропитки композита +материалом матрицы под воздействием сдвиговых нагрузок. Вблизи локальных +дефектов интенсивности напряжений превышают соответствующие интенсивности +напряжений определенное для композита идеальной периодической структуры в +$1{,}2$ раза при наличии внутренней технологической поры, в $1{,}3$ раза для +слчая пропуска или разрыва волокна основы и в $1{,}6$ раз для одновременного +разрыва волокон основы и утка. При этом, в случае разрыва волокна основы +или волокон основы и утка, значение коэффициентов концентрации интенсивностей +напряжений ожет быть снижено до $1{,}2$ и $1{,}5$ соответственно, с помощью +дополнительных операций доуплотнения поликристаллической матрицы. + +\pagebreak + +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v1/scheme2/d1d2} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с пропуском волокна основы~(а) и с учётом +доуплотнения~(б) при чистом сдвиге} + \label{fig:k_d1d2_s2} +\end{figure} + +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v1/scheme2/d3d6} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с разрывом волокна основы~(а) и с учётом +доуплотнения~(б) при чистом сдвиге} + \label{fig:k_d3d6_s2} +\end{figure} + +\pagebreak + +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v1/scheme2/d4d7} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с разрывом волокон основы и утка~(а) и с учётом +доуплотнения~(б) при чистом сдвиге} + \label{fig:k_d4d7_s2} +\end{figure} + +\begin{figure}[ht!] + \centering + \includegraphics[width=10cm]{concentrators/v1/scheme2/d5} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с внутренней технологической порой при чистом сдвиге} + \label{fig:k_d5_s2} +\end{figure} + +\subsection{Коэффициенты концентрации напряжений при одноосном растяжении} + +В случае, если граничные условия \ref{eq:b_cond} в краевой задаче +\eqref{eq:Eqvilibrium} -- \eqref{eq:Guck} будут принимать вид \begin{equation} \begin{array}{c} @@ -214,9 +595,138 @@ C_{ijkl}^{m} \left[ 1-\lambda({\bf r}) \right ] \right\} \end{equation} \noindent получим задачу на одноосное растяжение слоя тканого композита в -направлении, соответсвующем направлению утка. +направлении, соответствующем направлению утка. -\section{Модели тканого композита с поликристаллической матрицей с периодическим -и квазипериодическим расположением волокон} +Решив задачу \eqref{eq:Eqvilibrium} -- \eqref{eq:Guck} с граничными условиями +\ref{eq:b_cond_ideal} -- \ref{eq:b_cond_free} и \ref{eq:b_cond:s3} методом +конечных элементов, получим распределение интенсивности напряжений +(рис.~\ref{fig:vmis_v1_s3}) и максимальные значения коэффициентов концентрации +напряжений (таблица~\ref{tab:max_k_s3}). -\section{Выводы ко второй главе} \ No newline at end of file +\begin{figure}[ht] + \includegraphics[width=15cm]{vmis_v1_s3} + \caption{Поля интенсивности напряжений в слое тканого композита с идеальной +периодической структурой при одноосном растяжении} + \label{fig:vmis_v1_s3} +\end{figure} + +\begin{table}[ht!] + \centering + \caption{Максимальные коэффициенты концентрации напряжений в слое тканого +композита при одноосном растяжении} + \begin{tabular}{|p{8cm}||c|c|c|c|c|c|} + \hline + & $K_{\sigma_{11}}$ + & $K_{\sigma_{22}}$ + & $K_{\sigma_{33}}$ + & $K_{\sigma_{12}}$ + & $K_{\sigma_{13}}$ + & $K_{\sigma_{23}}$ \\ + \hline + \hline + Пропуск волокна основы + &$1{,}26$ & $1{,}39$ & $\bf2{,}14$ & $1{,}36$ & $\bf2{,}66$ & $\bf2{,}64$\\ + \hline + Пропуск волокна основы (доуплотнение) + &$1{,}24$ & $1{,}34$ & $\bf2{,}10$ & $1{,}29$ & $\bf2{,}75$ & $\bf3{,}00$\\ + \hline + \hline + Разрыв нити основы + &$1{,}26$ & $1{,}36$ & $\bf1{,}92$ & $1{,}27$ & $\bf2{,}50$ & $\bf2{,}01$\\ + \hline + Разрыв нити основы (доуплотнение) + &$1{,}24$ & $1{,}35$ & $\bf1{,}87$ & $1{,}35$ & $\bf2{,}41$ & $\bf2{,}81$\\ + \hline + \hline + Разрыв нитей основы и утка + &$1{,}43$ & $1{,}73$ & $\bf2{,}06$ & $1{,}46$ & $\bf2{,}66$ & $\bf2{,}17$\\ + \hline + Разрыв нитей основы и утка (доуплотнение) + &$1{,}31$ & $1{,}55$ & $\bf1{,}91$ & $1{,}32$ & $\bf2{,}45$ & $\bf2{,}91$\\ + \hline + \hline + Внутренняя пора + & $1{,}23$ & $1{,}39$ & $1{,}62$ & $\bf4{,}59$ & $1{,}40$ & $1{,}46$ \\ + \hline + \end{tabular} + \label{tab:max_k_s3} +\end{table} + +Из таблицы \ref{tab:max_k_s3} можно заметить, что наибольший вклад в +коэффициенты концентраций вносят касательные составляющие тензора напряжений +$\sigma_{13}$ и $\sigma_{23}$ и нормальная составляющая $\sigma_{33}$. +Исключение составляет случай, когда в слое тканого композита присутствует +внутренняя технологическая пора. В этом случае значение касательной +компоненты тензора напряжений превышает соответсвующее значение в и деальной +периодической структуре в $4{,}59$ раз. + +Распределение коэффициентов концентрации интенсивностей напряжений в слое +тканого композита с искривленными волокнами и поликристалической матрицей при +наличии различных типов технологических дефектов и с учётом дополнительной +пропитки композита материалом матрицы под воздействием сдвиговых нагрузок +представлены на рис.~\ref{fig:k_d1d2_s3}~--~\ref{fig:k_d5_s3}. + +\pagebreak + +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v1/scheme3/d1d2} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с пропуском волокна основы~(а) и с учётом +доуплотнения~(б) при одноосном растяжении} + \label{fig:k_d1d2_s3} +\end{figure} + +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v1/scheme3/d3d6} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с разрывом волокна основы~(а) и с учётом +доуплотнения~(б) при одноосном растяжении} + \label{fig:k_d3d6_s3} +\end{figure} + +\pagebreak + +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v1/scheme3/d4d7} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с разрывом волокон основы и утка~(а) и с учётом +доуплотнения~(б) при одноосном растяжении} + \label{fig:k_d4d7_s3} +\end{figure} + +\begin{figure}[ht!] + \centering + \includegraphics[width=10cm]{concentrators/v1/scheme3/d5} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с внутренней технологической порой при одноосном +растяжении} + \label{fig:k_d5_s3} +\end{figure} + +Из рисунков видно что вблизи локальных дефектов интенсивности напряжений +превышают соответствующие интенсивности напряжений определенное для композита +идеальной периодической структуры в $1{,}2$ раза при наличии внутренней +технологической поры или разрыва волокна основы, в $1{,}3$ раза для слчая +пропуска или разрыва волокна основы и в $1{,}4$ раз для одновременного +разрыва волокон основы и утка. При этом, в случае пропуска или разрыва волокна +основы, значение коэффициентов концентрации интенсивностей напряжений может быть +снижено до $1{,}2$ и $1{,}3$ соответственно, с помощью дополнительных операций +доуплотнения поликристаллической матрицы. + +\section*{Выводы ко второй главе} +\addcontentsline{toc}{section}{Выводы ко второй главе} + +\begin{enumerate} + \item Построена твердотельная модель фрагмента слоя тканого композита с +искривленными волокнами и поликристалической матрицей с идеальной +периодической структурой и локальными технологическими дефектами, такими как +пропуск волокна основы, разрыв волокна основы, разрыв волокон основы и утка и +внутренняя технологическая пора. + \item Получены численные решения краевых задач на двухосное растяжение в +плоскости слоя, чистый сдвиг и одноосное растяжение в направлении утка. + \item Вычислены безразмерные коэффициенты концентрации напряжений, вызванные +наличием локальных технологическх дефектов в виде пропуска волокна основы, +разрыва волокна основы, разрыва волокон основы и утка, а также внутренней +технологической поры. + \item Определены механизмы инициирующие разрушение матрицы. +\end{enumerate} diff --git a/c3.tex b/c3.tex index 761035d..86d44c1 100644 --- a/c3.tex +++ b/c3.tex @@ -42,16 +42,6 @@ f | \sigma_{nn} {\bf (r)} | \right ] |_{\Gamma_9^{-}}, \quad исключается соприкосновение волокон. Это герметичные полости, недоступные для материала матрицы, - - -Краевая задача \eqref{eq:Eqvilibrium}--\eqref{eq:Guck} с -граничными условиями \eqref{eq:b_cond}---\eqref{eq:b_cond_free} -решается численно методом конечных элементов в некоммерческом пакете -Code-Aster, входящим в платформу SALOME--MECA. Этот пакет был разработан -и сертифицирован специально для французской энергетической отрасли и -предназначен для задач механики сплошных сред, термо- и гидродинамики, акустики -и магнетизма, выполнения расчетов для строительных конструкций и сооружений. - \begin{figure}[!ht] \centering % \includegraphics[width=0.83\linewidth]{img/matrix} @@ -59,45 +49,6 @@ Code-Aster, входящим в платформу SALOME--MECA. Этот пак \label{fig:matrix} \end{figure} -Дискретизация фрагмента проводилась на 16-узловые тетераэдральные и -20-узловые гексаэдральные изопараметрические элементы. На -рис.~\ref{fig:matrix} представлена дискретизация области матрицы слоя -модельного тканого композита полотняного плетения. Степень -дискретизации выбиралась таким образом, чтобы сетка сгущалась в областях, -имеющих наибольшую кривизну и располагающихся вблизи поверхности контакта -нитей, а также в местах расположения внутренни технологических пор. Полученные -в результате численного решения 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. Этим условиям -удовлетворяют конечноэлементные сетки, параметры которых представлены -в табл.~\ref{tab:discr}. Значения, стоящие в числителе, соответствуют -случаю, когда каждая нить армирующего каркаса окружена гарантированным -слоем матрицы, а в знаменателе --- случаю, когда нити основы и утка имеют -общую поверхность контакта с трением. - -\begin{table}[htp] - \centering - \caption{Параметры конечноэлементной сетки} -\begin{tabular}{l||c|c} - \hline - & Тетраэдральные & Гексаэдральные \\ - & элементы & элементы \\ - \hline - \hline - Идеальная структура & $\frac{298~255} {405~480}$ & $\frac{77~760} {77~760}$ -\\ - \hline - Разрыв волокна основы & $\frac{285~466} {405~480}$ & $\frac{75~168} -{75~168}$ \\ - \hline - Разрыв волокон основы и утка & $\frac{279~276} {405~480}$ & $\frac{72~576} -{72~576}$ \\ - \hline - \end{tabular} - \label{tab:discr} -\end{table} - На рис.~\ref{fig:sigma} показаны распределения интенсивностей напряжений в искривленных нитях основы и утка при равнокомпонентном двухосном однородном деформировании слоя модельного тканого композита diff --git a/disser.kilepr 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