From a9796061988ad85a22e519b814a357b556d9e3b7 Mon Sep 17 00:00:00 2001 From: "Denis V. Dedkov" Date: Sun, 2 Sep 2012 18:27:46 +0600 Subject: [PATCH] Subsections 2.1.1 and 2.1.2 is finished --- bibliography.bib | 30 ++ c1.tex | 30 +- c2.tex | 243 ++++++++--------- c3.tex | 43 ++- fig/concentrators/v1/scheme1/d1.png | Bin 0 -> 272906 bytes fig/concentrators/v1/scheme1/d2.png | Bin 0 -> 251651 bytes fig/concentrators/v1/scheme1/d3.png | Bin 0 -> 278654 bytes fig/concentrators/v1/scheme1/d4.png | Bin 0 -> 275346 bytes fig/concentrators/v1/scheme1/d5.png | Bin 0 -> 260615 bytes fig/concentrators/v1/scheme1/d6.png | Bin 0 -> 255384 bytes fig/concentrators/v1/scheme1/d7.png | Bin 0 -> 271943 bytes fig/concentrators/v1/scheme2/d1.png | Bin 0 -> 272922 bytes fig/concentrators/v1/scheme2/d2.png | Bin 0 -> 251431 bytes fig/concentrators/v1/scheme2/d3.png | Bin 0 -> 278709 bytes fig/concentrators/v1/scheme2/d4.png | Bin 0 -> 275874 bytes fig/concentrators/v1/scheme2/d5.png | Bin 0 -> 263187 bytes fig/concentrators/v1/scheme2/d6.png | Bin 0 -> 254492 bytes 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create mode 100644 fig/geometry/v1/bc.png create mode 100644 fig/geometry/v1/d1d2.png create mode 100644 fig/geometry/v1/d3d6.png create mode 100644 fig/geometry/v1/d4d7.png create mode 100644 fig/geometry/v1/d5.png create mode 100644 fig/geometry/v1/regular.png create mode 100644 fig/geometry/v1/regular_all.png create mode 100644 fig/geometry/v1/regular_in_matrix.png create mode 100644 fig/geometry/v2/d1.png create mode 100644 fig/geometry/v2/d2.png create mode 100644 fig/geometry/v2/d3.png create mode 100644 fig/geometry/v2/d4.png create mode 100644 fig/geometry/v2/matrix.png create mode 100644 fig/geometry/v2/regular.png create mode 100644 fig/geometry/v2/regular_in_matrix.png diff --git a/bibliography.bib b/bibliography.bib index 0f84ed5..ece7f7f 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -77,4 +77,34 @@ title = {{[R5.03.50]} Discrete formulation of the contact-friction}, author = {Mickael ABBAS}, year = {2012} +} + +@ONLINE{bib:salome, + url = {http://salome-platform.org/user-section/salome-brochure}, + title = {SALOME 6. The Open Source integration platform for numerical +simulations}, + author = {Vincent Bergeaud and Vincent Lefebvre and \'{E}tinne Rossignon}, + year = {2012} +} + +@ONLINE{bib:salome:geom, + url = {http://docs.salome-platform.org/salome_6_5_0/gui/GEOM/index.html}, + title = {SALOME Geometry User's Guide}, + year = {2012} +} + +@ONLINE{bib:salome:additional_geom, + url = +{http://docs.salome-platform.org/salome_6_5_0/gui/GEOM/SALOME_BOA_PA.pdf}, + title = {General Fuse Algorithm, Partition Algorithm, Boolean Operations +Algorithm. Backgrounds.}, + year = {2010} +} + +@ONLINE{bib:laduga:geom, + url = {http://www.laduga.ru/salome/salome.shtml}, + title = {Документация Salome Geometry}, + author = {<<Ладуга>> Инженерные услуги}, + year = {2012}, + language = {russian} } \ No newline at end of file diff --git a/c1.tex b/c1.tex index 721249d..8d5096f 100644 --- a/c1.tex +++ b/c1.tex @@ -102,19 +102,19 @@ $900\dots1500^\circ\mathrm{C}$. На этой стадии продолжают В промышленности используют ткани, имеющие различные типы переплетения. Наиболее простым и широко применяемым является полотняное переплетение -(рис.~\ref{fig:schemas},~a), где каждая нить основы и утка проходит поочередно -сверху и снизу пересекающихся нитей. +(рис.~\ref{fig:c1:schemas},~a), где каждая нить основы и утка проходит +поочередно сверху и снизу пересекающихся нитей. \begin{figure}[h] \includegraphics[width=17cm]{all_structs} \caption{Схемы типов переплетения: а) полотняное, б) сатиновое, в) саржевое $2\times2$} - \label{fig:schemas} + \label{fig:c1:schemas} \end{figure} -Сатиновое переплетение (рис.~\ref{fig:schemas},~б) получают путем перекрытия +Сатиновое переплетение (рис.~\ref{fig:c1:schemas},~б) получают путем перекрытия одной нитью утка четырех нитей основы. При саржевом переплетении -(рис.~\ref{fig:schemas},~в) нити основы и утка проходят поочередно сверху и +(рис.~\ref{fig:c1:schemas},~в) нити основы и утка проходят поочередно сверху и снизу двух и четырех пересекающих их нитей. Ткани подразделяют по ширине: $40\dots75$~см. --- узкие, $75\dots100$~см. --- @@ -372,39 +372,39 @@ $6\dots100$~МПа при температуре $550\dots 650^\circ\mathrm{C}$. Дефекты связанные с отклонениями от расчетных параметров структуры возникают на этапе изготовления ткани. К числу таких дефектов можно отнести отклонения в напралении армирующих нитей, пропуски нитей в направлении армирования (рис. -\ref{fig:no_fiber}). +\ref{fig:c1:no_fiber}). На этапе формирования матрицы могут возникнуть дефекты связанные с отклонением от расчетного распределения плотности конечного материала, а так же нарушение структуры армирующей ткани, возникающие на подготовительных операциях. Также, в следствие нарушения технологического процесса на этом этапе могут возникнуть -трещины и внутренние поры (рис. \ref{fig:pore}). +трещины и внутренние поры (рис. \ref{fig:c1:pore}). -Разрывы волокон утка или основы (рис. \ref{fig:break}) могут возникать на каждом -из этапов: на этапе формирования ткани --- вследствии очень тесного размещения -нитей, в процессе сшивки слоев ткани при формировании конструкции, на этапе -формирования матрицы --- из-за внутренний напряжений, возникающих в материале во -время его изготовления. +Разрывы волокон утка или основы (рис. \ref{fig:c1:break}) могут возникать на +каждом из этапов: на этапе формирования ткани --- вследствии очень тесного +размещения нитей, в процессе сшивки слоев ткани при формировании конструкции, на +этапе формирования матрицы --- из-за внутренний напряжений, возникающих в +материале во время его изготовления. \begin{figure} \centering \includegraphics[width=12cm]{d3} \caption{Пропуск нити в направлении армирования} - \label{fig:no_fiber} + \label{fig:c1:no_fiber} \end{figure} \begin{figure} \centering \includegraphics[width=14cm]{d4} \caption{Внутренняя технологическая пора} - \label{fig:pore} + \label{fig:c1:pore} \end{figure} \begin{figure} \centering \includegraphics[width=17cm]{d1d2} \caption{Разрывы нитей утка (а) и нитей основы и утка (б)} - \label{fig:break} + \label{fig:c1:break} \end{figure} Кроме того, на каждом из этапов в тканый композит могут попасть посторонние diff --git a/c2.tex b/c2.tex index 6a7658e..12ac523 100644 --- a/c2.tex +++ b/c2.tex @@ -4,27 +4,24 @@ \section{Математическая модель упруго-хрупкого поведения тканого композита с поликристаллической матрицей} -Рассмотрим слой тканого композита с армирующим каркасом полотняного +\subsection{Геометрическая модель слоя тканого композита} + +Будем моделировать слой тканого композита с армирующим каркасом полотняного переплетения образованного волокнами круглого поперечного сечения постоянного диаметра $D$, толщина которого которого составляет $2,5 D$. Будем считать, что искривление нитей основы и утка ткани задается дугой окружности $a$ с центральным углом $\alpha = \pi \mathord{\left/ -{\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4 $ и прямой $b$ (рис. -\ref{fig:geometry}) \cite{bib:imankulova}. +{\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4 $ и прямой $b$ +(рис.~\ref{fig:c2:geometry}) \cite{bib:imankulova}. В силу малости деформаций +будем считать углы $\alpha$ неизменными при нагружении слоя. \begin{figure} + \centering + \includegraphics[width=17cm]{geom} \caption{Геометрия изгиба волокна} - \label{fig:geometry} + \label{fig:c2:geometry} \end{figure} -В процессе изготовления композита не удается исключить соприкосновения -нитей основы и утка. Поэтому будем предполагать, что искривленные -волокна, принадлежащие слою тканого композита с идеальной -периодической структурой, не всегда окружены гарантированным -слоем поликристаллической матрицы, в результате чего основа и уток -соприкасаются. Кроме того, в силу малости деформаций будем считать углы -$\alpha$ неизменными при нагружении слоя. - Построение геометрической модели слоя тканого композита будем проводить с помощью платформы для численного моделирования SALOME, которая представляет собой набор пре- и постпроцессинга. Первоначально задуманная как @@ -33,67 +30,96 @@ $\alpha$ неизменными при нагружении слоя. приложениях численного моделирования и САПР. Так, например, платформа SALOME используется как база для проекта NURESIM (European Platform for NUclear REactor SIMulations), предназначенного для полномасштабного -моделирования реакторов. +моделирования реакторов \cite{bib:salome}. -% На рис.~\ref{fig:defects}~а и б представлен фрагмент слоя тканого композита, -% армирующий каркас которого образован полотняным переплетением утка и основы -% (с коэффициентами армирования $\alpha_{1} = \alpha_{3} = 0,14$ -% соответственно). Здесь и далее оси $x_1$ и $x_3$ ортогональной декартовой -% системы координат принадлежат плоскости слоя. -% -% В рассматриваемом случае локальными концентраторами напряжений -% являются технологические поры, возникающие в областях, расположенных -% вблизи участков волокон с наибольшей кривизной (рис.~\ref{fig:pore}), и -% дефекты, связанные со случайными разрывами нитей утка -% (рис.~\ref{fig:defects},~а) или основы и утка (рис.~\ref{fig:defects},~б) -% в процессе прошивки слоев. Обратим внимание на то, что локальные разрывы -% нитей армирующего каркаса могут иметь место и в исходной ткани до -% прошивки. Образующаяся в результате полости имеют характерные -% размеры, соизмеримые с характерными размерами неоднородностей, не -% изменяют значительно интегральные коэффициенты армирования композита, -% могут оказаться заполненными материалом матрицы (при дополнительном уплотнении -% с последующей карбонизацией или доосаждением материала из газовой фазы) или -% оставаться незаполненными. -% -% \begin{figure} -% \begin{minipage}[h]{0.47\linewidth} -% % \center{\includegraphics[width=1\linewidth]{img/d1}} \\ а) -% \end{minipage} -% \hfill -% \begin{minipage}[h]{0.47\linewidth} -% % \center{\includegraphics[width=1\linewidth]{img/d2}} \\ б) -% \end{minipage} -% \caption{Локальные разрывы нитей слоя тканого композита} -% \label{fig:defects} -% \end{figure} -% -% \begin{figure} -% \centering -% % \includegraphics[width=0.77\linewidth]{img/pore} -% \caption{Внутренняя технологическая пора} -% \label{fig:pore} -% \end{figure} +С помощью операции экструзии вдоль кривой, показанной на +рис.~\ref{fig:c2:geometry}, формируется сегмент волокна, из которого, в свою +очередь, с помощью операций трансляции и зеркалирования формируется фрагмент +ткани (рис.~\ref{fig:c2:regular}~а). Матрица моделируется с помощью операции +вычитания из твердотельного прямоугольного параллилепипеда фрагмента ткани, +после чего матрица и фрагмент ткани совмещаются для получения твердотельной +модели тканого композита с поликристаллической матрицей +(рис.~\ref{fig:c2:regular}~б) \cite{bib:salome:geom, +bib:salome:additional_geom, bib:laduga:geom}. + +\begin{figure}[h] + \centering + \includegraphics[width=17cm]{geometry/v1/regular_all} + \caption{Фрагмент слоя тканого композита с идеальной периодической структурой: +а)~только волокна, б)~волокна, окруженные матрицей} + \label{fig:c2:regular} +\end{figure} + +Коэффициенты армирования моделируемого слоя тканого композита с +поликристаллической матрицей --- $\alpha_{1} = \alpha_{3} = 0{,}14$. Здесь и +далее оси $x_1$ и $x_3$ ортогональной декартовой системы координат принадлежат +плоскости слоя. + +Будем рассматривать дефекты, типичные для тканых композитов с +поликристаллической матрицей: пропуск нити основы +(рис.~\ref{fig:c2:fiber_skip}), разрыв волокна основы +(рис.~\ref{fig:c2:one_fiber_break}), разрыв волокон основы и утка +(рис.~\ref{fig:c2:two_fibers_break}), а также внутреннюю технологическую пору +(рис.~\ref{fig:c2:pore}). + +\begin{figure}[h] + \centering + \includegraphics[width=17cm]{geometry/v1/d1d2} + \caption{Фрагмент тканого композита с пропуском нити без дополнительной +пропитки (а) и с пропиткой (б)} + \label{fig:c2:fiber_skip} +\end{figure} + +\begin{figure}[h!] + \centering + \includegraphics[width=17cm]{geometry/v1/d3d6} + \caption{Фрагмент тканого композита с разрывом волокна основы без +дополнительной пропитки (а) и с пропиткой (б)} + \label{fig:c2:one_fiber_break} +\end{figure} + +\begin{figure}[ht!] + \centering + \includegraphics[width=17cm]{geometry/v1/d4d7} + \caption{Фрагмент тканого композита с разрывом волокон основы и утка без +дополнительной пропитки (а) и с пропиткой (б)} + \label{fig:c2:two_fibers_break} +\end{figure} + +\begin{figure}[ht!] + \centering + \includegraphics[width=10cm]{geometry/v1/d5} + \caption{Фрагмент тканого композита с внутренней технологической порой} + \label{fig:c2:pore} +\end{figure} + +Полости, образующиеся в результате разрывов нити основы, нитей основы или утка +или вызванные наличием внутренней технологической поры имеют характерные +размеры, соизмеримые с характерными размерами неоднородностей, не изменяют +значительно интегральные коэффициенты армирования композита. Полость, +образующаяся при пропуске волокна основы уменьшает коэффициент армирования +вдоль основы до $0{,}13$. При дополнительном уплотнении с последующей +карбонизацией или доосаждением матрицы из газовой фазы эти полости могут быть +заполнены материалом матрицы либо оставаться незаполненными. + +\subsection{Постановка краевой задачи теории упругости} Будем предполагать, для простоты, что волокна и матрица слоя модельного тканого композита изотропные, линейно упругие, не изменяющие геометрию, взаимное расположение и тип симметрии при нагружении. Тогда компоненты -тензора напряжений -$\sigma_{ij,j} ({\bf r})$ -удовлетворяют -уравнениям равновесия +тензора напряжений $\sigma_{ij,j} ({\bf r})$ удовлетворяют уравнениям равновесия \begin{equation} - \sigma_{ij,j} ({\bf r}) = 0,\label{eq:kov:Eqvilibrium} + \sigma_{ij,j} ({\bf r}) = 0,\label{eq:Eqvilibrium} \end{equation} \noindent а компоненты тензора малых деформаций $\varepsilon_{ij}$ связаны -с компонентами вектора перемещений $u_{i}$ геометрическими соотношениями -Коши +с компонентами вектора перемещений $u_{i}$ геометрическими соотношениями Коши \begin{equation} \varepsilon_{ij} ({\bf r}) = \frac{1}{2}\left[u_{i,j} ({\bf r}) + u_{j, i}({\bf r}) \right]. -\label{eq:kov:Koshi} +\label{eq:Koshi} \end{equation} Введем для описания геометрии слоя тканого композита единичную @@ -106,96 +132,55 @@ ${\bf r}$, которая принимает значение $1$, если то \sigma_{ij} ({\bf r}) = \left\{ C_{ijkl}^{f}\lambda({\bf r}) + C_{ijkl}^{m} \left[ 1-\lambda({\bf r}) \right ] \right\} \varepsilon_{kl}({\bf r}), -\label{eq:kov:Guck} +\label{eq:Guck} \end{equation} \noindent где верхними индексами $f$ и $m$ отмечены материальные коэффициенты, относящиеся к волокнам и матрице соответственно. -Краевая задача \eqref{eq:kov:Eqvilibrium}--\eqref{eq:kov:Guck} должна +Краевая задача \eqref{eq:Eqvilibrium}--\eqref{eq:Guck} должна быть дополнена граничными условиями -\begin{equation} u_1 {\bf (r)}|_{\Gamma_2} = u_1^0, \quad u_3 {\bf -(r)}|_{\Gamma_1} = -u_3^0, \label{eq:kov:b_cond} +\begin{equation} + \begin{array}{c} + u_1 {\bf (r)}|_{\Gamma_2} = u_1^0, \quad u_3 {\bf (r)}|_{\Gamma_1} = u_3^0, \\ + u_1 {\bf (r)}|_{\Gamma_4} = u_3 {\bf (r)}|_{\Gamma_3} = u_2 + {\bf (r)}|_{\Gamma_5} = u_2 {\bf (r)}|_{\Gamma_6} = 0, \\ + \sigma_{12} {\bf (r)}|_{\Gamma_4} = \sigma_{13} {\bf (r)}|_{\Gamma_4} = + \sigma_{13} {\bf (r)}|_{\Gamma_3} = \sigma_{23} {\bf (r)}|_{\Gamma_3} = 0, \\ + \sigma_{12} {\bf (r)}|_{\Gamma_5} = \sigma_{13} {\bf (r)}|_{\Gamma_5} = + \sigma_{12} {\bf (r)}|_{\Gamma_6} = \sigma_{13} {\bf (r)}|_{\Gamma_6} = 0, + \end{array} + \label{eq:b_cond} \end{equation} -$$ u_1 {\bf (r)}|_{\Gamma_4} = u_3 {\bf (r)}|_{\Gamma_3} = u_2 -{\bf (r)}|_{\Gamma_5} = u_2 {\bf (r)}|_{\Gamma_6} = 0, -$$ - -$$ \sigma_{12} {\bf (r)}|_{\Gamma_4} = \sigma_{13} {\bf (r)}|_{\Gamma_4} = - \sigma_{13} {\bf (r)}|_{\Gamma_3} = \sigma_{23} {\bf (r)}|_{\Gamma_3} = - 0, -$$ - -$$ \sigma_{12} {\bf (r)}|_{\Gamma_5} = \sigma_{13} {\bf (r)}|_{\Gamma_5} = - \sigma_{12} {\bf (r)}|_{\Gamma_6} = \sigma_{13} {\bf (r)}|_{\Gamma_6} = - 0, -$$ - -\noindent обеспечивающими заданное макрооднородное -равнокомпонентное деформирование в плоскости слоя и условиями -идеального сопряжения +\noindent обеспечивающими заданное макрооднородное равнокомпонентное +деформирование в плоскости слоя и условиями идеального сопряжения \begin{equation} -\left[\sigma_{ij} {\bf (r)} n_{j} \right] |_{\Gamma_7^{+}} = -\left[\sigma_{ij} {\bf (r)} n_{j} \right] |_{\Gamma_7^{-}}, \quad -\left[u_i {\bf (r)}\right]|_{\Gamma_7^{+}} = \left[u_i {\bf -(r)}\right]|_{\Gamma_7^{-}} \label{eq:kov:b_cond_ideal} + \left[\sigma_{ij} {\bf (r)} n_{j} \right] |_{\Gamma_7^{+}} = + \left[\sigma_{ij} {\bf (r)} n_{j} \right] |_{\Gamma_7^{-}}, \quad + \left[u_i {\bf (r)}\right]|_{\Gamma_7^{+}} = \left[u_i + {\bf(r)}\right]|_{\Gamma_7^{-}} + \label{eq:b_cond_ideal} \end{equation} -\noindent на границах раздела фаз $\Gamma_7$ (рис.~\ref{fig:b_cond}). +\noindent на границах раздела фаз $\Gamma_7$ (рис.~\ref{fig:c2:b_cond}). \begin{figure}[!ht] \centering -% \includegraphics[width=0.53\linewidth]{img/gu} - \caption{Фрагмент тканого композита с искривленными волокнами} - \label{fig:b_cond} + \includegraphics[width=12cm]{geometry/v1/bc} + \caption{Граничные условия краевой задачи теории упругости} + \label{fig:c2:b_cond} \end{figure} -В случае, если в модельном материале не исключается возможность контакта -нитей основы и утка, на соответствующих контактных поверхностях -$\Gamma_9$ (положение и геометрия которых считается заданными и неизменными -в процессе нагружения слоя) будем считать справедливыми условия контакта -с кулоновским трением. На $\Gamma_9$ следует задать 2 условия: - -\noindent если $\left[\sigma_{n\tau} {\bf (r)} -\right] |_{\Gamma_9^{+}} < \left[ {f | \sigma_{nn} {\bf (r)} |} -\right] |_{\Gamma_9^{-}}$, то +Полости, вызванные наличием локальных дефектов и незаполенные матрицей имеют +внутреннюю поверхность $\Gamma_8$, на которой отсутствуют ограничения на +перемещения, а сама поверхность свободна от напряжений: \begin{equation} -\left[\sigma_{n\tau} {\bf (r)} n_{n} \right] |_{\Gamma_9^{+}} = -\left[\sigma_{n\tau} {\bf (r)} n_{n} \right] |_{\Gamma_9^{-}}, \quad -\left[u_n {\bf (r)}\right]|_{\Gamma_9^{+}} = \left[u_n {\bf -(r)}\right]|_{\Gamma_9^{-}} , -\label{eq:kov:b_cond_Colomb_1} -\end{equation} - -\noindent а, если $\left[\sigma_{n\tau} {\bf (r)} \right] -|_{\Gamma_9^{+}} \geq \left[ {f | \sigma_{nn} {\bf (r)} |} \right ] -|_{\Gamma_9^{-}}$, то - -\begin{equation} -\left[\sigma_{n\tau} {\bf (r)} \right] |_{\Gamma_9^{+}} \geq \left[ -f | \sigma_{nn} {\bf (r)} | \right ] |_{\Gamma_9^{-}}, \quad -\left[u_n {\bf (r)}\right]|_{\Gamma_9^{+}} = \left[u_n {\bf -(r)}\right]|_{\Gamma_9^{-}} , \label{eq:kov:b_cond_Colomb_2} -\end{equation} - -\noindent где $f$ --- статический коэффициент трения, а индексы $n$ и $\tau$ ---- определяют направление внешней нормали и касательной к -поверхности $\Gamma_9$. - -Внутренние поры имеют место в слое композита в случае, если не -исключается соприкосновение волокон. Это герметичные полости, недоступные -для материала матрицы, имеют внутреннюю поверхность $\Gamma_8$, на -которой отсутствуют ограничения на перемещения, а сама поверхность свободна -от напряжений: - -\begin{equation} -\sigma_{ij} {\bf (r)} n_{j} |_{\Gamma_8} = 0. -\label{eq:kov:b_cond_free} + \sigma_{ij} {\bf (r)} n_{j} |_{\Gamma_8} = 0. + \label{eq:b_cond_free} \end{equation} \section{Модели тканого композита с поликристаллической матрицей с периодическим diff --git a/c3.tex b/c3.tex index 6b51b0d..761035d 100644 --- a/c3.tex +++ b/c3.tex @@ -5,8 +5,47 @@ композита с поликристаллической матрицей при наличии контакта с трением между волокнами} -Краевая задача \eqref{eq:kov:Eqvilibrium}--\eqref{eq:kov:Guck} с -граничными условиями \eqref{eq:kov:b_cond}---\eqref{eq:kov:b_cond_free} +В случае, если в модельном материале не исключается возможность контакта +нитей основы и утка, на соответствующих контактных поверхностях +$\Gamma_9$ (положение и геометрия которых считается заданными и неизменными +в процессе нагружения слоя) будем считать справедливыми условия контакта +с кулоновским трением. На $\Gamma_9$ следует задать 2 условия: + +\noindent если $\left[\sigma_{n\tau} {\bf (r)} +\right] |_{\Gamma_9^{+}} < \left[ {f | \sigma_{nn} {\bf (r)} |} +\right] |_{\Gamma_9^{-}}$, то + +\begin{equation} +\left[\sigma_{n\tau} {\bf (r)} n_{n} \right] |_{\Gamma_9^{+}} = +\left[\sigma_{n\tau} {\bf (r)} n_{n} \right] |_{\Gamma_9^{-}}, \quad +\left[u_n {\bf (r)}\right]|_{\Gamma_9^{+}} = \left[u_n {\bf +(r)}\right]|_{\Gamma_9^{-}} , +\label{eq:b_cond_Colomb_1} +\end{equation} + +\noindent а, если $\left[\sigma_{n\tau} {\bf (r)} \right] +|_{\Gamma_9^{+}} \geq \left[ {f | \sigma_{nn} {\bf (r)} |} \right ] +|_{\Gamma_9^{-}}$, то + +\begin{equation} +\left[\sigma_{n\tau} {\bf (r)} \right] |_{\Gamma_9^{+}} \geq \left[ +f | \sigma_{nn} {\bf (r)} | \right ] |_{\Gamma_9^{-}}, \quad +\left[u_n {\bf (r)}\right]|_{\Gamma_9^{+}} = \left[u_n {\bf +(r)}\right]|_{\Gamma_9^{-}} , \label{eq:b_cond_Colomb_2} +\end{equation} + +\noindent где $f$ --- статический коэффициент трения, а индексы $n$ и $\tau$ +--- определяют направление внешней нормали и касательной к +поверхности $\Gamma_9$. + +Внутренние поры имеют место в слое композита в случае, если не +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. 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