commit 0db55afee40570823758aa4b1aeac790a7516b24 Author: Denis V. Dedkov Date: Sun Jun 1 17:29:10 2014 +0600 Initial commit diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..7e652c1 --- /dev/null +++ b/.gitignore @@ -0,0 +1,12 @@ +*.aux +*.toc +*.log +*.nav +*~ +*.backup +*.snm +*.out +*.swp +.directory +*.pdf +*.dvi diff --git a/1spec.tex b/1spec.tex new file mode 100755 index 0000000..1930ea5 --- /dev/null +++ b/1spec.tex @@ -0,0 +1,591 @@ +\pagebreak +\section{Введение} + +Прогнозирование эффективных деформационных свойств и определение статистических характеристик случайных полей напряжений и деформаций в компонентах волокнистых и дисперсно-упрочненных композитов связаны с необходимостью решения стохастически нелинейных краевых задач, для построения приближенных решений которых (например, полного корреляционного приближения) требуются описывающие многочастичное взаимодействие в системе армирующих элементов моментные функции структурных модулей упругости второго, третьего, четвертого и пятого порядков. + +При построении приближенных решений нелинейных стохастических краевых задач используются различные (но очень часто не вполне обоснованные) гипотезы о характере многочастичного взаимодействия в ансамбле частиц армирующего наполнителя (например, предельная локальность) и аппроксимации центральных моментов случайного индикатора. Поэтому, во-первых, существует потребность в идентификации и «отбраковке» соответствующих статистических моделей механики структурно-неоднородных сред. Во-вторых, «традиционные» алгоритмы построения условных и безусловных многоточечных моментных функций, которые ранее были использованы для обработки микрошлифов металлов \cite{bvv} и стеклопластиков \cite{vs}, требуют существенной модификации. Это обусловлено тем, что реализация данных алгоритмов связана с построением вспомогательных координатных сеток, определением принадлежности каждого узла этих сеток одной из фаз материала, требует значительных аппаратных и программных затрат. + +{\itЦелью} дипломного проекта является развитие математических основ решения стохастических краевых задач механики структурно-неоднородных сред, получение и анализ аналитических выражений для определения моментов второго и третьего порядка, оценка характерного размера представительного объема двухфазного композита с учетом характера усредненного многочастичного взаимодействия в системе армирующих элементов и прогнозирование эффективных упругих свойств однонаправленно армированных стеклопластиков. + +Основные результаты проведенных исследований докладывались и обсуждались на 3-й Всероссийской научно-технических конференциях ``Математическое моделирование'' (Самара, 2006). + +Дипломный проект выполнен в соответствии с планом научных исследованиями, проводимыми на кафедре Механика композиционных материалов и конструкций ПГТУ. + +Теоретические разработки нашли отражение в спецкурсах ``Методы исследования микроструктуры и свойств композитов'', ``Синтез и анализ случайных структур композитов'', читаемых в Пермском государственном техническом университете студентам специальности 121000 --- ``Конструирование и производство изделий из композиционных материалов''. + +Автор выражают признательность научному руководителю, доценту кафедры Механика композиционных материалов и конструкций ПГТУ, к.ф.-м.н. А.В.~Зайцеву, а также С.В.~Мельникову, Ю.В.~Соколкину и А.А.~Ташкинову за внимание к работе и обсуждение представленных результатов. + +\pagebreak +\section{Моментные функции случайной структуры\\двухфазных однонаправленно-армированных\\композитов} +\subsection{Использование многопроцессорных систем для построения\\моментных функций} + +Случайная структура однонаправленно армированных волокнистых композитов, которая исследуется экспериментально путем обработки микрошлифов, а также на основе анализа модельных плоских или пространственных структур, полученных при помощи компьютерного синтеза, может быть описана совокупностью условных и безусловных моментных функций \cite{vs}. Пусть $\lambda(\bf{r})$ --- случайная индикаторная функция, которая принимает значение, равное единице, в случае, если точка $\bf{r}$ принадлежит дискретной фазе --- волокну (объемная доля которых равна $\nu_f$), и нулю --- если эта точка принадлежит непрерывной фазе --- матрице. + +Прогнозирование эффективных деформационных свойств и определение статистических характеристик случайных полей напряжений и деформаций в компонентах двухфазных волокнистых и дисперсно-упрочненных композитов связаны с необходимостью решения стохастически нелинейных краевых задач, для построения приближенных решений которых (например, полного корреляционного приближения) требуются описывающие многочастичное взаимодействие в системе армирующих элементов двух- и трехточечные моментные функции структурных модулей упругости второго, третьего, четвертого и пятого порядков. Эти функции определяются центральными моментами соответствующих порядков случайного индикатора $\lambda(\bf{r})$. + +Моментные функции второго порядка $K_\lambda^{(2)}(\bf{r_1,r_2})$ позволяют определить степень взаимодействия и характер упорядоченности между соседними и удаленными друг от друга элементами структуры; третьего порядка $K_\lambda^{(3)}(\bf{r_1,r_2,r_3})$ характеризуют форму, а четвертого порядка $K_\lambda^{(4)}(\bf{r_1,r_2,r_3})$ позволяют установить, как группируются включения \cite{ber}. + +Стохастический характер структуры композитов обусловлен случайностью формы, взаимного расположения и ориентации волокон, разбросом характерных размеров частиц армирующего наполнителя. В настоящее время существует потребность определения «скрытых» параметров порядка стохастических структур (например, детерминированных периодических и квази-детерминированных составляющих), практически не отражающихся на эффективных упругих характеристиках этих материалов, но предопределяющих сценарии развития процесса разрушения. + +Исследование закономерностей случайных структур будем проводить на основе анализа сгенерированных плоских фрагментов (синтез которых связан со случайным размещением непересекающихся гладких дисков на плоскости \cite{vs,zlt,zlt3}), считая, что волокна двухфазного композита имеют круглое поперечное сечение (рис. 1). При моделировании структур композитов будем предполагать, что координаты центров размещаемых внутри синтезируемого фрагмента дисков (поперечных сечений волокон) являются независимыми равномерными случайными величинами, а характерные размеры волокон описываются одномодальными статистическими законами распределения: симметричными (нормальный) и несимметричными (логнормальный). Кроме того, ограничим законы распределения диаметров слева заданным минимальным значением $D_{min}$ (которое для всех генерируемых структур будет равно $D_{min}=/2$) и будем считать неизменным отношение $/L=0.01$ среднего диаметра волокон $D$ к характерному размеру фрагмента $L$. + +\begin{figure}[!h] +\label{struct1} +\includegraphics[width=0.5\textwidth]{ris/str2} +\caption{Фрагменты модельных структур ($k_D=0.60, d/=0.0$) волокнистых композитов с предельной объемной долей волокон, диаметры которых описываются нормальным (а) и логнормальным (б) законами} +\end{figure} + +Эти предположения и ограничения согласуются с результатами построения законов распределения диаметров волокон однонаправленно армированного стеклопластиков на основе эпоксидной \cite{vs} и ненасыщенной полиэфирной смолы горячего отверждения ПН–1 \cite{fkg}, оправданы организацией технологического процесса получения волокнистых наполнителей. Как свидетельствуют представленные в этих работах результаты, проверка по критерию $\chi^2$ Пирсона показала, что нормальный закон может быть принят с вероятностью ошибочного отклонения гипотезы не более 6\%, а логнормальный — не более 2\%. + +Расположение волокон в сечении может быть охарактеризовано распределением длин промежутков между волокнами, измеренных в произвольном направлении. В монографии \cite{vs} приведены результаты, показывающие, что с вероятностью не более 6\% по критерию $\chi^2$ Пирсона можно отклонить гипотезу о нормальном распределении минимальных расстояний между волокнами. Вместе с тем, авторами \cite{fkg} была доказана эквивалентность законов распределения длин промежутков и минимальных расстояний между волокнами. + +Предположение о том, что диаметры волокон двухфазных однонаправленно армированных композитов являются случайными не вносит существенных корректировок в алгоритмы синтеза структур этих материалов. Для достижения объемных наполнений, близких к предельным, генерация структуры может, при необходимости, сопровождать дополнительным взаимным перемещением, вновь и ранее размещаемых волокон \cite{vs,zlt,zlt3}, а также запрещением выхода какой-либо части поперечного сечения армирующего элемента за границы области. + +Однако возможна дополнительная модификация алгоритмов \cite{vs,zlt,zlt3} процедурой предварительной сортировки (по возрастанию значений) последовательности псевдослучайных диаметров волокон, распределенных по заданному статистическому закону \cite{zlt4}. Последующее случайное размещение волокон внутри фрагмента происходит в порядке уменьшения диаметров. Использование предварительной сортировки оправдано необходимостью строгого соблюдения соответствия заданного теоретического и эмпирического (построенного по сгенерированной случайной структуре) статистических законов распределения характерных размеров включений. Можно предположить, что невыполнение этого условия может наблюдаться у материалов, которые содержат как крупные, так и мелкие фракции (поскольку вероятность размещения внутри синтезируемого фрагмента волокон с малыми диаметрами намного больше вероятности расположения волокон большого диаметра). + +Вычисление моментных функций можно проводить при помощи алгоритмов, использованных для экспериментального построения этих функций по микрошлифам структуры металлов [13] и стеклопластиков [1]. Реализация этих алгоритмов связана с построением вспомогательных координатных сеток, определением принадлежности каждого узла этих сеток одной из фаз материала, требует значительных аппаратных и программных затрат даже в случае сведения задачи построения моментных функций к нахождению геометрических вероятностей [1]. Действительно, обозначив $Prob(\xi)$ вероятность события $\xi$ , можно преобразовать первое слагаемое равенства + +\begin{equation} +\label{k2_beg} +K_\lambda^{(2)}(\bf{r_1,r_2})=\left<\lambda^\circ(\bf{r_1})\lambda^\circ(\bf{r_2})\right>=\left<\lambda(\bf{r_1})\lambda(\bf{r_2})\right>-\nu_f^2 +\end{equation} + +следующим образом + +\begin{equation} +\label{ll} +\left<\lambda(\bf{r_1})\lambda(\bf{r_2})\right>=Prob({\bf r_1}\in\Omega_f\land{\bf r_2}\in\Omega_f)=Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)\nu_f, +\end{equation} + +а условную вероятность $Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)$ определим отношением + +\begin{equation} +\label{prob} +Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)\cong\alpha/\beta. +\end{equation} +Здесь $\alpha$ --- число событий ${\bf r}\in\Omega_f\land{\bf r'}\in\Omega_f$ и $\beta$ --- число событий ${\bf r}\in\Omega_f$. + +Моментные функции, построенные этим при помощи реализации этого алгоритма, очень чувствительны к шагу вспомогательной сетки. Однако, уменьшение шага вспомогательной сетки, приводит к значительному росту аппаратных и программных затрат для проведения статистического анализа каждого сгенерированного фрагмента случайной структуры. + +Одним из путей преодоления данной проблемы является использование параллельных вычислений с применением технологии MPI (The Message Passing Interface), которая представляет хорошо стандартизованный механизм построения параллельных алгоритмов в модели обмена сообщениями. Кроме того, в настоящее время разработаны стандартные «привязки» MPI к языкам программирования С/С++ и Fortran 77/90, свободные и коммерческие реализации для большинства многопроцессорных платформ, а также для сетей рабочих станций UNIX и Windows NT. + +Для реализации технологии MPI был использован пакет MPICH, который поддерживает стандарт MPI 1.2 и некоторые элементы стандарта MPI 2.0. + +Для решения задачи был использованы два алгоритма: в первом на нескольких процессорах определяется только принадлежность точки волокну или матрицы методом прямого перебора, а во втором помимо этого каждый процессор вычислял значения моментной функции для своего, заранее определенного направления. Многопроцессорная реализация заключалась в том, что каждый процессор перебирал точки, начиная со своего порядкового номера, с шагом, равным числу процессоров. В результате весь промежуток делился на число участков, равное числу всех процессоров. + +Блок-схема алгоритма 1 приведена на рис. \ref{bs}: + +\begin{figure} +\label{bs} +%\include{ris/bs.tex} +\includegraphics[width=\textwidth]{ris/bs} +\caption{Блок-схема алгоритма построения моментных функций для многопроцессорной системы МВС-1000} +\end{figure} + +В результате исполнения программы с разным числом процессоров была получена следующая зависимость отношения времени счета на одном процессоре к времени счета на n процессорах от числа процессоров n (таблица \ref{time_zatr}). + +\begin{table}[!h] +\footnotesize{ +\caption{Временные затраты на реализацию алгоритма построения моментных функций второго порядка случайной структуры однонаправленно армированного композита} +\label{time_zatr} +\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|} +\hline +&\multicolumn{12}{|c|}{Число процессоров}\\ +\cline{2-13} +&1&2&3&4&5&6&7&8&9&10&11&12\\ +\hline +Алгоритм 1&1.000&0.867&0.800&0.667&0.599&0.599&0.533&0.533&0.467&0.467&0.533&0.599\\ +\hline +Алгоритм 2&1.000&0.857&0.714&0.571&0.429&0.357&0.283&0.214&0.286&0.357&---&---\\ +\hline +\end{tabular} +} +\end{table} + +Вычисления были выполнены на многопроцессорной системе МВС–1000, основу которой составляет масштабируемый массив процессорных узлов. Каждый узел содержит микропроцессор Alpha 21164 с производительностью 2 GFLOPS при тактовой частоте 500 MHz и оперативную память объемом 128 MB с возможностью расширения. Процессорные узлы взаимодействуют через коммуникационные процессоры TMS320C44, имеющие по 4 внешних канала с общей пропускной способностью 80 Мбайт/с. Для управления массивом процессоров и внешними устройствами, а также для доступа к системе извне был использован хост-компьютер на базе процессора Intel с операционной системой Linux. + +Отметим, что для первого алгоритма при счете на 10 процессорах время счета уменьшается более чем в два раза. Увеличение числа процессоров более 10 приводит к увеличению времени счета, что связано с затратами на пересылку данных между процессорами. Для второго алгоритма оптимальное число процессоров равняется восьми и время счета уменьшается почти в пять раз. + +\pagebreak +\subsection{Метод геометрических вероятностей для определения моментных функций третьего порядка} + +Для снижения программно-аппаратных затрат, которые возникают в результате применения вышеописанного алгоритма, появляется потребность разработки способов аналитического построения условных и безусловных моментных функций произвольного порядка. + +Рассмотрим фрагмент случайной структуры двухфазного однонаправленно армированного или дисперсно-упрочненного композита (рис. \ref{struct2}). + +\begin{figure}[!h] +\label{struct2} +\begin{center} +\includegraphics[angle=-90, width=0.6\textwidth]{ris/struct2} +\caption{Геометрический смысл функций $Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)$} +\end{center} +\end{figure} + + +{\bf Определение~1.} Назовем {\it прообразом} фрагмент, представляющий конкретную реализацию случайной структуры. Будем считать, что прообраз условно неподвижен. +{\bf Определение~2.} Фрагмент, геометрически идентичный прообразу, над которым могут быть осуществлены преобразования трансляции и произвольного пространственного разворота как жесткого целого, назовем {\it образом} случайной структуры. + +На рис. \ref{struct2} представлен прообраз $\hat{\Omega}$ случайной структуры однонаправленно армированного волокнистого композита и образ ${\hat{\Omega}}'$, полученный в результате параллельного переноса на расстояние, определяемое вектором трансляции ${\bf\Delta r}$, направляющие косинусы которого относительно неподвижной системы координат, связанной с $\hat{\Omega}$ равны $n_1$ и $n_2$ ($n_1^2 + n_2^2 = 1)$. + +Геометрический смысл условной вероятности $Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)$ в уравнении (\ref{ll}) может быть определен как отношение меры пересечения множеств $\Omega_f$ и $\Omega'_f$ точек, принадлежащих включениям прообраза $\hat{\Omega}$ и образа $\hat{\Omega}'$ соответственно к мере множества $\Omega_f$ ($mes\Omega_f=\nu_f mes\hat{\Omega})$: + + +\begin{equation} +\label{probr1r2} +\begin{array}{ll} +Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)&=\frac{mes(\Omega_f\cap\Omega'_f)}{mes\Omega_f}={}\\ +{}&=\frac{\sum_{i=1}^N\sum_{j=1}^N mes(\Omega_i\cap\Omega'_j)}{\sum_{i=1}^N mes\Omega_f^{(i)}}= +\frac{\sum_{i=1}^N\sum_{j=1}^N mes(\Omega_i\cap\Omega'_j)}{\nu_f mes\hat{\Omega}}. +\end{array} +\end{equation} + +Здесь $\Omega_i\subset\Omega_f$ и $\Omega'_j\subset\Omega'_f$ --- конкретные включения, принадлежащие $\hat{\Omega}$ и $\hat{\Omega}'$, а $N$ --- количество включений. + +Важными характеристиками структуры однонаправленно армированных волокнистых и дисперсно-упрочненных композитов являются диаметры включений $D$ и минимальные расстояния между частицами армирующего наполнителя $d$, которые являются детерминированными или случайными, распределенными по заданным статистическим законам. Очевидно, что для каждого сгенерированного фрагмента $D\in[{D_{min},D_{max}}]$, $d\in[{d_{min},d_{max}}]$. Тогда необходимым и достаточным условием ненулевого пересечения будет являться одновременное удовлетворение неравенств + +\begin{equation} +\label{krit} +R < r_i + r_j , +\quad +r_i < R + r_j , +\quad +r_j < R + r_i , +\quad +R \le D_{min} , +\quad +R = |\bf{\Delta r}|. +\end{equation} + +Слагаемые $mes(\Omega_i\cap\Omega'_j)$ из формулы \ref{probr1r2} находятся по следующим соотношениям: + +\begin{equation} +\label{mesij} +mes(\Omega_i\cap\Omega'_j)=\left\{ +\begin{array}{ll} +0, &d_{ij}\ge r_i+r_j;\\ +\pi r_i^2, &r_j\ge d_{ij}+r_i;\\ +\pi r_j^2, &r_i\ge d_{ij}+r_j;\\ +\kappa, &(d_{ij}\le r_i+r_j)\lor(r_i\le d_{ij}+r_j)\lor(r_j\le d_{ij}+r_i), +\end{array}\right. +\end{equation} + +\begin{equation} +\label{kappa} +\begin{array}{ll} +\kappa&=r_i^2\arccos\left[\frac{1}{2d_{ij}r_i}\left(r_i^2+r_j^2-d_{ij}^2\right)\right]+r_j^2\arccos\left[\frac{1}{2d_{ij}r_j}\left(r_i^2+r_j^2-d_{ij}^2\right)\right]-{}\\ +{}&-2\sqrt{p(p-r_i)(p-r_j)(p-d_{ij})}, +\end{array} +\end{equation} + +\begin{equation} +\label{pdij} +p=\frac{1}{2}(r_i+r_j+d_{ij}), d_ij=\sqrt{(x_i-x'_j-Rn_1)^2+(y_i-y'_j-Rn_2)^2} +\end{equation} + +Здесь $d_{ij}$ --- расстояние между центрами $i$ включения прообраза $\hat{\Omega}$ ($x_i$ и $x_j$ --- координаты центра) и $j$ включения образа $\hat{\Omega}'$ (${x}'_i$ и ${x}'_j$ --- координаты центра), радиусы $\hat{\Omega }$ и $\hat{\Omega}'$ равны $r_i$ и $r_j$ соответственно; $R = |\bf{\Delta r}|$. + + +Если расстояния $R=|\bf{\Delta r}|$ таковы, что выполнимо неравенство + +\begin{equation} +\label{mrasst} +R\le min[D_{min},d_{min}], +\end{equation} + +то будет иметь место пересечение $i$-го включения образа с $i$-м включением прообраза ($\Omega_i\cap\Omega'_i$). Этот случай мы в дальнейшем будем называть приближением "малых расстояний". В приближении "малых расстояний" соотношения (\ref{probr1r2}) значительно упрощаются: + +\begin{equation} +\label{probr1r2_m} +Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)=\frac{\sum_{i=1}^N mes(\Omega_i\cap\Omega'_i)}{\sum_{i=1}^N mes\Omega_f^{(i)}}=\frac{\sum_{i=1}^N mes(\Omega_i\cap\Omega'_i)}{\nu_fmes\Omega}. +\end{equation} + +Меры пересечений $mes(\Omega_i\cap\Omega'_i)$ в приближении "малых расстояний" находятся по следующим формулам: + +\begin{equation} +\label{mes_m} +mes(\Omega_i\cap\Omega'_i)=\left\{ +\begin{array}{ll} +2r_i^2\arccos\left(\frac{R}{2r_i}\right)-\frac{R}{2}\sqrt{r_i^2-\frac{R^2}{4}}, &R<2r_i;\\ +0, &R\ge 2r_i.\\ +\end{array}\right. +\end{equation} + +Если двухфазные композиты содержат включения одинакового радиуса ($r_i=r_j=r$), то из выражения (\ref{mes_m}) будет следовать равенство, полученное ранее авторами \cite{it2}: + +\begin{equation} +mes(\Omega_f\cap\Omega'_f)=\left\{ +\begin{array}{ll} +2r^2\arccos\left(\frac{R}{2r}\right)-2\sqrt{r^2-\frac{R^2}{4}}, &R<2r;\\ +\pi r^2, &R\equiv 0;\\ +0, &R=2r.\\ +\end{array}\right. +\end{equation} + +В приближении "малых расстояний" вид моментной функции $\tilde{K}_\lambda^{(2)}(R)$ значительно упрощается: + +\begin{equation} +\label{k2m} +\tilde{K}_\lambda^{(2)}(R)=\frac{\sum_{i=1}^N\left[mes(\Omega_i\cap\Omega'_i)\right]-\nu_f^2 mes\hat\Omega}{\nu_f(1-\nu_f)mes\hat{\Omega}}. +\end{equation} + +Аналитическое соотношение в виде ряда (\ref{k2m}) для нормированных корреляционных функций случайных структур двухфазных композитов матричного типа в приближении "малых" расстояний позволяет получить точные выражения для производных этих функций при значениях аргумента, равных нулю. + +Определение производных нормированных моментных функций является важным этапом на пути решения проблемы идентификации статистических моделей механики струк-\\турно-неоднородных сред. Аналитическое соотношение в виде ряда (\ref{k2m}) для нормированных корреляционных функций случайных структур двухфазных композитов матричного типа в приближении "малых" расстояний позволяет получить точные выражения для производных этих функций при значениях аргумента, равных нулю. Учитывая в равенстве (\ref{k2m}) меры $mes(\Omega_i\cap\Omega'_i)$, представленные в виде (\ref{mes_m}), получим формулу для производной $\tilde{K}_\lambda^{(2)}(R)$ при $R=0$: + + +\begin{equation} +\label{proizv2} +\left.\frac{d}{dR}\tilde{K}_\lambda^{(2)}(R)\right|_{R=0}=-\frac{3}{2\pi(1-\nu_f)\left}\equiv +-\frac{3\aleph}{4\pi\nu_f(1-\nu_f)mes\hat{\Omega}}. +\end{equation} + +Здесь $R=|\bf{\Delta r}|$. Как видим, для плоских случайных структур искомая производная при любых объемных наполнениях имеет отрицательный знак, определяется отношением меры $\aleph=mes\partial\Omega_f$, связанной с межфазной границей (суммарный периметр) к площади прообраза $mes\hat{\Omega}$ ($mes\Omega_f = v_f mes\hat{\Omega})$. Кроме того, значение производной при корреляционной функции при $R = 0$ не зависит от направляющих косинусов углов ориентации вектора трансляции $\bf{\Delta r}$. + +Для случайных структур армирующие элементы которых имеют детерминированные размеры ($r_i = r)$, выражение (\ref{proizv2}) значительно упрощается: + +\begin{equation} +\label{proizv2rir} +\left.\frac{d}{dR}\tilde{K}_\lambda^{(2)}(R)\right|_{R=0}=-\frac{3}{2\pi(1-\nu_f)r}. +\end{equation} + +Как видим, значение производной в рассматриваемом частном случае определяется только объемным наполнением $v_f $ и характерным размером включений. + +Обобщим полученные результаты с целью получения аналитических выражений для моментных функций третьего порядка. Задача построения моментных функций третьего порядка случайной структуры двухфазных композитов матричного типа также связана с нахождением геометрических вероятностей \ref{l1l2}. + +\begin{equation} +\label{k3} +\begin{array}{ll} +K_\lambda^{(3)}(r_1,r_2,r_3)&\equiv\left<\lambda^\circ(r_1)\lambda^\circ(r_2)\lambda^\circ(r_3)\right>=\left<\lambda(r_1)\lambda(r_2)\lambda(r_3)\right>-{}\\ +{}&-\nu_f\left[\left<\lambda(r_1)\lambda(r_2)\right>+\left<\lambda(r_1)\lambda(r_3)\right>+\left<\lambda(r_2)\lambda(r_3)\right>\right]+2\nu_f^3 +\end{array} +\end{equation} + +Введем множества точек $\Omega_f$ , $\Omega'_f$ и $\Omega''_f$, принадлежащих включениям прообраза $\hat{\Omega}$ и образов $\hat{\Omega}'$ и $\hat{\Omega}''$ соответственно. Образы $\hat{\Omega}'$ и $\hat{\Omega}''$ получены в результате параллельного переноса $\hat{\Omega}$ на расстояния, определяемые векторами трансляции Взаимная ориентация этих векторов относительно неподвижной системы координат, связанной с $\hat{\Omega}$, определяется углом $\Theta$, а взаимная ориентация --- углом $\phi$ (рис. \ref{obraz}). Обратим внимание на то, для статистически изотропного случайного поля структуры аргументами моментной функции третьего порядка будут $|\Delta r_1|$, $|\Delta r_2|$ и $\phi$. + +\begin{figure}[!h] +\begin{center} +\label{obraz} +\includegraphics[width=0.6\textwidth]{ris/struct3} +\caption{Геометрический смысл функций $Prob\left[(\bf{r_1}\in\Omega_f\land\bf{r_2}\in\Omega_f)\mid\bf{r_3}\in\Omega_f\right]$} +\end{center} +\end{figure} + +Тогда геометрическим смыслом условной вероятности $Prob\left[r_1 \in \Omega_f\mid(r_2\in\Omega_f\land r_3\in\Omega_f)\right]$, которая содержится в выражении (\ref{k3}) в виде произведения + +\begin{equation} +\label{ml3} +\begin{array}{ll} +\left<\lambda({\bf r_1})\lambda({\bf r_2})\lambda({\bf r_3})\right>& +=Prob\left({\bf r_1}\in\Omega_f\land {\bf r_2}\in\Omega_f\land {\bf r_3}\in\Omega_f\right)={}\\ +{}&=Prob\left[{\bf r_1} \in\Omega_f\mid({\bf r_2}\in\Omega_f\land {\bf r_3}\in\Omega_f)\right]\times{}\\ +{}&\times Prob\left[{\bf r_2}\in\Omega_f\mid {\bf r_3}\in\Omega_f\right]Prob\left[{\bf r_3}\in\Omega_f\right] +\end{array} +\end{equation} + +является мера $mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)$ множества точек, получаемых при пересечении включений, принадлежащих $\hat{\Omega}$, $\hat{\Omega}'$ и $\hat{\Omega}''$ (рис. \ref{obraz}). Следовательно, используя для $Prob\left[{\bf r_2}\in\Omega_f\mid {\bf r_3}\in\Omega_f\right]$ представление (\ref{prob_r1r2}) и, учитывая, что $Prob\left[{\bf r_3}\in\Omega_f\right]\equiv\nu_f$, из соотношения (\ref{ml3}) получим: + +\begin{equation} +\label{ml3_p} +\left<\lambda({\bf r_1})\lambda({\bf r_2})\lambda({\bf r_3})\right>=\frac{mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)}{mes\Omega''_f}\nu_f. +\end{equation} + +Таким образом, заменяя в выражении (\ref{k3}) группу слагаемых, содержащих общий множитель $\nu_f$, соответствующими представлениями (\ref{ml2}) и принимая во внимание (\ref{ml3_p}), представим моментную функции третьего порядка случайной структуры двухфазного композита запишем следующим образом: + +\begin{equation} +\label{k3_end} +\begin{array}{ll} +K_\lambda^{(3)}(R_1,R_2,\phi)&=\nu_f\left\{\frac{mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)}{mes\Omega''_f}-\right.\\ +{}&\left.-\nu_f\left[\frac{mes(\Omega_f\cap\Omega'_f)}{mes\Omega'_f}+\frac{mes(\Omega_f\cap\Omega''_f)}{mes\Omega''_f}+\frac{mes(\Omega'_f\cap\Omega''_f)}{mes\Omega''_f}\right]\right\}+2\nu_f^3 +\end{array} +\end{equation} + +или + +\begin{equation} +\label{k3_mr} +\begin{array}{ll} +K_\lambda^{(3)}(R_1,R_2,\phi)&=\frac{\nu_f}{mes\Omega_f}\left\{mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)-\nu_f\left[mes(\Omega_f\cap\Omega'_f)+\right.\right.{}\\ +{}&\left.\left.+mes(\Omega_f\cap\Omega''_f)+mes(\Omega'_f\cap\Omega''_f)\right]\right\}+2\nu_f^3={}\\ +{}&\frac{\nu_f}{mes\Omega_f}\sum_{i=1}^N\left\{mes(\Omega_i\cap\Omega'_i\cap\Omega''_i)-\nu_f\left[mes(\Omega_i\cap\Omega'_i)+\right.\right.{}\\ +{}&\left.\left.+mes(\Omega_i\cap\Omega''_i)+mes(\Omega'_i\cap\Omega''_i)\right]\right\}+{}\\ +{}&+\frac{\nu_f}{mes\Omega_f}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}{N}(1-\delta_{ijk})\left\{mes(\Omega_i\cap\Omega'_j\cap\Omega''_k)-\right.{}\\ +{}&\left.-\nu_f\left[mes(\Omega_i\cap\Omega'_j)+mes(\Omega_i\cap\Omega''_k)+mes(\Omega'_j\cap\Omega''_k)\right]\right\}+2\nu_f^3. +\end{array} +\end{equation} + +Здесь $R_1=|{\bf \Delta r_1}|$ и $R_2=|{\bf \Delta r_2}|$; $\delta_{ijk}$ --- коэффициенты, принимающие значения 1 при совпадающих индексах и 0, если хотя бы один из трех индексов отличается от двух других различны (при $i,j=1\dots3$ эти коэффициенты являются обобщенными символами Кронекера). + +Из анализа условий сходимости рядов (\ref{k3_mr}) также может быть получена информация о характере затухания статистических моментов третьего порядка. Существование конечного передела позволит определить асимптоты, вокруг которых происходит осцилляция моментных функций, а анализ знака сумм, входящих в выражения (\ref{k3_mr}) --- определить наличие или отсутствие периодических составляющих в случайных полях структуры. + +Для конкретизации слагаемых, входящих в выражение \ref{k3_mr}, рассмотрим ряд вспомогательных геометрических задач по пересечению трех окружностей. Возможные варианты пересечения включений показаны на рис. \ref{cross}: + +\begin{figure}[!h] +\label{cross} +\includegraphics[width=\textwidth]{ris/circles} +\caption{Возможные варианты пересечения включений прообраза $\Omega$ с включениями образов $\Omega'$ и $\Omega''$} +\end{figure} + + +\begin{enumerate} +\item Все три включения совпадают если одновременно выполняются следующие условия: +$$ +r_i=r_j, r_j=r_k, r_i=r_k, R_{ij}=0.0, R_{ik}=0.0, R_{jk}=0.0; +$$ +В этом случае площадь пересечения трех окружностей равна $\pi\cdot r_i^2.$ + +\item Одно из включений не пересекается с другими если выполняется одно из следующих условий: +$$ +R_{ij}\ge r_i+r_j, R_{ik}\ge r_i+r_k, R_{jk}\ge r_j+r_k. +$$ +Площадь пересечения трех окружностей равна нулю. + +$i$-е включение совпадает с $j$-м включением и лежит внутри $k$-го включения (рис. \ref{cross}, а): +$$ +r_i=r_j, R_{ij}=0.0, r_k \ge R_{jk}+r_j. +$$ +Площадь пересечения трех окружностей равна $\pi\cdot r_i^2.$ + +\item $i$-е включение совпадает с $j$-м включением и $k$-е включение лежит внутри (рис. \ref{cross}, б): +$$ +r_i=r_j, R_{ij}=0.0, r_i\ge R_{jk}+r_k. +$$ +Площадь пересечения трех окружностей: $\pi\cdot r_k^2.$ + +\item $i$-е включение лежит внутри пересечения $j$-го и $k$-го включений (рис. \ref{cross}, в): +$$ +R_{jk}R^{(1)}_{jx_{ik}}, r_j>R^{(2)}_{jx_{ik}} +$$ + +Площадь пресечения равна: +\begin{equation} +\begin{array}{lr} +k=&r_i^2\cdot\arccos\left[\frac{1.0}{2\cdot R_{ij}\cdot r_i}\cdot\left(r_i^2-r_j^2+R_{ij}^2\right)\right]+{}\\ +{}+&r_j^2\cdot\arccos\left[\frac{1.0}{2\cdot R_{ij}\cdot r_j}\cdot\left(r_j^2-r_i^2+R_{ij}^2\right)\right]-2\cdot\sqrt{p\cdot(p-r_i)\cdot(p-r_j)\cdot(p-R_{ij})}, +\end{array} +\end{equation} +$$ +p=\frac{1}{2}\cdot(r_i+r_j+R_{ij}). +$$ + +\item Случай, показанный на рис. \ref{cross}, ж - если выполняются оба условия: + +$R^{(l)}_{jx_{ik}}$ - расстояние от центра $j$ окружности до $l$-той точки пересечения $i$ и $k$ окружностей. +$$ +r_jR^{(2)}_{jx_{ik}} +$$ + +\end{enumerate} + +\begin{figure}[!h] +\label{funct1} +\includegraphics[width=\textwidth]{ris/funct1} +\caption{Частный случай нормированных моментных функций третьего порядка случайной структуры однонаправленно армированного волокнистого композита} +\end{figure} + + +\pagebreak +\subsection{Частные случаи моментных функций третьего порядка} + +Рассмотрим некоторые частные случаи: + +\begin{enumerate} + +\item Найдем выражение для двухточечного момента третьего порядка $K_\lambda^{(3)}({\bf r_i,r_i,r_k})$: + +\begin{equation} +\label{rierj} +K_{\lambda}^{(3)}({\bf r_i,r_i,r_k}) = \left<\lambda({\bf r_i})\lambda({\bf r_i})\lambda({\bf r_k})\right>-\nu_f \left[\left<\lambda({\bf r_i})\lambda({\bf r_i})\right>+2\left<\lambda({\bf r_i})\lambda({\bf r_k})\right>\right]+2\nu_f^3 +\end{equation} + +Среднее от произведения трех индикаторных функций $\left<\lambda({\bf r_i})\lambda({\bf r_i})\lambda({\bf r_k})\right>$ можно представить в виде условных вероятностей следующим образом (\ref{lililkf}): + +\begin{equation} +\label{lililkf} +\begin{array}{rcl} +\left<\lambda({\bf r_i})\lambda({\bf r_i})\lambda({\bf r_k})\right> = +Prob({\bf r_i}\in\Omega_f\land {\bf r_i}\in\Omega_f\land {\bf r_k}\in\Omega_f)={}\\ +{}=Prob\left[{\bf r_i}\in\Omega_f\mid({\bf r_i}\in\Omega_f\land {\bf r_k}\in\Omega_f)\right]\times{}\\ +{}\times Prob\left[{\bf r_i}\in\Omega_f\mid {\bf r_k}\in\Omega_f\right]Prob\left[{\bf r_k}\in\Omega_f\right], +\end{array} +\end{equation} + +или + +\begin{equation} +\label{lililks} +\begin{array}{rcl} +\left<\lambda({\bf r_i})\lambda({\bf r_i})\lambda({\bf r_k})\right> = +Prob({\bf r_i}\in\Omega_f\land {\bf r_k}\in\Omega_f)Prob({\bf r_i}\in\Omega_f\mid {\bf r_k}\in\Omega_f)Prob({\bf r_k}\in\Omega_f)={}\\ +{}=Prob({\bf r_i}\in\Omega_f\mid {\bf r_k}\in\Omega_f)\cdot\nu_f\cdot Prob({\bf r_i}\in\Omega_f\mid {\bf r_k}\in\Omega_f)Prob({\bf r_k}\in\Omega_f). +\end{array} +\end{equation} + +Заменяя условные вероятности отношениями мер пересечения включений получаем: + +\begin{equation} +\label{lililkt} +\left<\lambda({\bf r_i})\lambda({\bf r_i})\lambda({\bf r_k})\right> = +\frac{mes(\Omega_f \cap \Omega''_f)}{mes(\Omega_f)}\cdot\nu_f\cdot\frac{mes(\Omega_f \cap \Omega''_f)}{mes\Omega_f}\cdot\nu_f=\left(\frac{mes(\Omega_f \cap \Omega''_f)}{mes\hat{\Omega}}\right)^2 +\end{equation} + +Представим в виде условных вероятностей слагаемые содержащие произведение двух индикаторных функций: + +\begin{equation} +\label{lili} +\left<\lambda({\bf r_i})\lambda({\bf r_i})\right>=Prob({\bf r_i}\in\Omega_f\land {\bf r_i}\in\Omega_f)=Prob({\bf r_i}\in\Omega_f)=\nu_f +\end{equation} + +и + +\begin{equation} +\label{lilk} +\left<\lambda({\bf r_i})\lambda({\bf r_k})\right>=Prob({\bf r_i}\in\Omega_f\mid {\bf r_k}\in\Omega_f)\cdot\nu_f=\frac{mes(\Omega_f \cap \Omega''_f)}{mes\hat{\Omega}} +\end{equation} + +подставляя выражения (\ref{lililkt})--(\ref{lilk}) в (\ref{rierj}) получаем: + +\begin{equation} +\label{kiikend} +\begin{array}{rcl} +K_{\lambda}^{(3)}({\bf r_i,r_i,r_k})= +\left[\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}\right]^2- +\nu_f\cdot\left[\nu_f+2\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}\right]+2\nu_f^3={}\\ +{}=\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}\left[\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}-2\nu_f\right]-\nu_f^2+2\nu_f^3 +\end{array} +\end{equation} + + +\item Рассмотрим случай, когда один из аргументов моментной функции равен нулю: + +\begin{equation} +\label{rk0} +K_\lambda^{(3)}({\bf r_i,r_j,}\vec{0})=\left<\lambda({\bf r_i})\lambda({\bf r_j})0\right>-\nu_f\left[\left<\lambda({\bf r_i})\lambda({\bf r_j})\right>+\left<\lambda({\bf r_i})0\right>+\left<\lambda({\bf r_j})0\right>\right]-2\nu_f^3=-\nu_f\left[\left<\lambda({\bf r_i})\lambda({\bf r_j})\right>+2\nu_f^2\right]. +\end{equation} + +Представим среднее от произведения индикаторных функций в виде условных вероятностей: + +\begin{equation} +\label{lilj0} +\left<\lambda({\bf r_i})\lambda({\bf r_j})\right> =Prob({\bf r_i}\in\Omega_f\mid {\bf r_j}\in\Omega_f)\nu_f=\frac{mes(\Omega_f\cap\Omega'_f)}{mes\hat{\Omega}}. +\end{equation} + +В результате подстановки (\ref{lilj0}) в (\ref{rk0}) получим: + +\begin{equation} +K_\lambda^{(3)}({\bf r_i,r_j,}\vec{0})=-\nu_f\left(\frac{mes(\Omega_f\cap\Omega'_f)}{mes\hat\Omega}+2\nu_f^2\right). +\end{equation} + +\end{enumerate} + +\pagebreak +\subsection{Выводы по разделу} + +Получены аналитические выражения в виде рядов для моментных функций второго и третьего порядка случайной структуры двухфазных композитов матричного типа. + +Получены точные выражения для производных моментных функций второго и третьего порядка случайной структуры двухфазных композитов, которые определяются отношением мер, связанных с межфазной границей (поверхностью) и частицами армирующего наполнителя. + +Подготовлен пакет прикладных программ для построения моментных функций второго и третьего порядков (для одно- и многопроцессорных систем). + +\pagebreak +\section{Заключение} + +Основные результаты дипломного проекта заключаются в следующем: + +\begin{enumerate} +\item Получены новые аналитические выражения в виде рядов для моментных функций второго и третьего порядка случайной структуры двухфазных композитов матричного типа, позволяющие в явном виде выделить слагаемые, соответствующие приближению ``малых'' расстояний. Определены производные этих функций и проанализировано влияние типа закона распределения диаметров включений на угол наклона корреляционных функций в точке, соответствующей нулевому значению аргумента. + +\item В работе проведен расчет экономической эффективности программного продукта для построения моментных функций третьего порядка случайной структуры двухфазных однонаправленно армированных композитов. + +\item Проведен расчет допустимого уровня шума на рабочем месте инженера-программиста. +центров. +\end{enumerate} diff --git a/bibliography.tex b/bibliography.tex new file mode 100755 index 0000000..efae1a8 --- /dev/null +++ b/bibliography.tex @@ -0,0 +1,19 @@ +\begin{thebibliography}{99} +\bibitem{bvv} Богачев И.Н., Вайнштейн А.А., Волков С.Д. Статистическое металловедение. – М.: Металлургия, 1984. – 176 с. +\bibitem{vs} Волков С.Д., Ставров В.П. Статистическая механика композитных материалов. – Минск: Изд-во БГУ, 1978. – 208 с. +\bibitem{ber} Беран М. Применение статистических теорий для определения тепловых, электрических и магнитных свойств неоднородных материалов // Композиционные материалы. Т. 2. Механика композиционных материалов. – М.: Мир, 1978. – С. 242–286. +\bibitem{dab} Debye P., Anderson H.R., Brumberger H. Scattering by an inhomogeneous sold. II. The correlation function and its application // J. Appl. Phys., 1957, Vol. 28, No 6. – P. 679–683. +\bibitem{it} Иванов Д.С., Ташкинов А.А. Физические поля в компонентах композитов с псевдослучайной структурой // Физическая мезомеханика. – 2001. – Т. 4, No 2. – С.29–36. +\bibitem{zlt} Зайцев А.В., Лукин А.В., Трефилов Н.В. Компьютерный синтез случайной структуры однонаправленно армированных волокнистых композитов // Молодежная наука Прикамья. – 2001. – Вып. 1. – С. 78–87. +\bibitem{zlt2} Зайцев А.В., Лукин А.В., Трефилов Н.В. Закономерности случайных полей структуры двухфазных однонаправленно армированных волокнистых композитов // Математ. моделирование систем и проц. – 2003. – Вып. 11. – С. 29–37. +\bibitem{zlt3} Зайцев А.В., Лукин А.В., Трефилов Н.В. Статистическое описание структуры двухфазных волокнистых композитов // Математ. моделирование систем и проц. – 2002. – Вып. 10. – С. 52–62. +\bibitem{zltt}Зайцев А.В., Лукин А.В., Ташкинов А.А., Трефилов Н.В. Случайные структуры двухфазных композитов: синтез, закономерности, новая оценка характерных размеров представительных объемов // Математ. моделирование систем и проц. – 2004. – Вып. 12. – С. 30–44. +\bibitem{zdp}Зайцев А.В., Дедков Д.В., Покатаев Я.К. Моментные функции, описывающие структуру волокнистых и дисперсно-упрочненных композитов // Вестник ПГТУ. Аэрокосмическая техника. --- 2005. --- Вып. 25. --- С. 3--11. +\bibitem{it2} Иванов Д.С., Ташкинов А.А. Физические поля в компонентах композитов с псевдослучайной структурой // Физическая мезомеханика. – 2001. – Т. 4, No 2. – С.29–36. +\bibitem{zlt4} Зайцев А.В., Лукин А.В., Трефилов Н.В. Компьютерный синтез, закономерности случайных структур и моделирование процессов разрушения волокнистых композитов при продольном сдвиге // Физическая мезомеханика. – 2004. – Т. 7, No 5. – С. 73–79. +\bibitem{zat} Зайцев А.В., Абашев Д.Ш., Трефилов Н.В. Влияние «скрытых» закономерностей случайных структур на микро- и макроразрушение однонаправленно армированных волокнистых композитов // Аэрокосмическая техника и высокие технологии – 2005: Материалы VIII Всероссийск. науч.-техн. конф. – Пермь: ПГТУ, 2005. – С. 61. +\bibitem{zdpt} Зайцев А.В., Дедков Д.В., Покатаев Я.К., Трефилов Н.В. Статистическое описание структуры двухфазных волокнистых и дисперсно-упрочненных композитов // Аэрокосмическая техника и высокие технологии – 2005: Материалы VIII Всероссийск. науч.-техн. конф. – Пермь: ПГТУ, 2005. – С. 62. +\bibitem{fkg} Ван Фо Фы Г.А., Клявин В.В., Гордиенко В.П. Исследование распределения волокон в ориентированных стеклопластиках // Механика полимеров. – 1969. – No 2. – С. 282–287. +\bibitem{sn4088} Микроклимат производственных помещений. СН 4088–86. +\bibitem{ssvn} Охрана труда в вычислительных центрах / Ю.Г. Сибаров, Н.Н. Сколотнев, В.К. Васин, В.Н. Нагинаев. – М: Машиностроение, 1990. – 192 с. +\end{thebibliography} diff --git a/dip_plan.tex b/dip_plan.tex new file mode 100755 index 0000000..bdde18a --- /dev/null +++ b/dip_plan.tex @@ -0,0 +1,53 @@ +\documentclass{report} +\batchmode +\usepackage{latexsym} +\usepackage[T2A]{fontenc} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\righthyphenmin=2 +\oddsidemargin=0pt +\textwidth=15cm +\topmargin=0cm +\textheight=23cm +\begin{document} + +\section{Введение} + +\begin{itemize} +\item Актуальность работы +\item Описание структуры +\end{itemize} + +\section{Моментные функции случайной структуры двухфазных однонаправленно армированных композитов} + +\subsection{Использование многопроцессорных систем для определения моментных функций} + +\begin{itemize} +\item Принцип работы многопроцессорных систем +\item Общее описание алгоритма +\item Блок-схема +\item Сравнительная таблица временных затрат +\end{itemize} + +\subsection{Метод геометрических вероятностей для определения моментных функций третьего порядка} + +\begin{itemize} +\item Описание метода +\item Алгоритм поиска моментных функций третьего порядка +\item (??блок схема??) +\item (??Приближение малых расстояний??) +\end{itemize} + +\subsection{Частные случаи моментных функций третьего порядка} + +\begin{itemize} +\item Формулы для некоторых частных случаев моментов третьего порядка +\end{itemize} + +\section{Выводы} + +\begin{itemize} +\item Выводы по диплому +\end{itemize} + +\end{document} diff --git a/diplom.kilepr b/diplom.kilepr new file mode 100755 index 0000000..44812dc --- /dev/null +++ b/diplom.kilepr @@ -0,0 +1,113 @@ +[General] +def_graphic_ext= +img_extIsRegExp=false +img_extensions=.eps .jpg .jpeg .png .pdf .ps .fig .gif +kileprversion=2 +kileversion=2.1.0 +lastDocument=diplom.tex +masterDocument= +name=diplom +pkg_extIsRegExp=false +pkg_extensions=.cls .sty .bbx .cbx .lbx +src_extIsRegExp=false +src_extensions=.tex .ltx .latex .dtx .ins + +[Tools] +MakeIndex= +QuickBuild= + 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\include{end/econom} +\include{zakl} +\include{bibliography} +%} +\end{document} diff --git a/econom.tex b/econom.tex new file mode 100755 index 0000000..e69de29 diff --git a/in.tex b/in.tex new file mode 100755 index 0000000..5b67830 --- /dev/null +++ b/in.tex @@ -0,0 +1,18 @@ +\pagebreak +\section{Введение} + +Прогнозирование эффективных деформационных свойств и определение статистических характеристик случайных полей напряжений и деформаций в компонентах волокнистых и дисперсно-упрочненных композитов связаны с необходимостью решения стохастически нелинейных краевых задач, для построения приближенных решений которых (например, полного корреляционного приближения) требуются описывающие многочастичное взаимодействие в системе армирующих элементов моментные функции структурных модулей упругости второго, третьего, четвертого и пятого порядков. + +При построении приближенных решений нелинейных стохастических краевых задач используются различные (но очень часто не вполне обоснованные) гипотезы о характере многочастичного взаимодействия в ансамбле частиц армирующего наполнителя (например, предельная локальность) и аппроксимации центральных моментов случайного индикатора. Поэтому, во-первых, существует потребность в идентификации и "отбраковке" соответствующих статистических моделей механики структурно-неоднородных сред. Во-вторых, "традиционные" алгоритмы построения условных и безусловных многоточечных моментных функций, которые ранее были использованы для обработки микрошлифов металлов \cite{bvv} и стеклопластиков \cite{vs}, требуют существенной модификации. Это обусловлено тем, что реализация данных алгоритмов связана с построением вспомогательных координатных сеток, определением принадлежности каждого узла этих сеток одной из фаз материала, требует значительных аппаратных и программных затрат. + +Данная работа является продолжением серии работ по исследованию случайных структур двухфазных однонаправленно армированных композитов, которые проводились авторами \cite{zlt,zlt2,zlt3,zltt}. В этих работах были неполностью рассмотрены моментные функции третьего порядка, которые позволяют более точно описать случайную структуру и свойства композиционных материалов. Соответственно {\it целью} дипломного проекта является развитие математических основ решения стохастических краевых задач механики структурно-неоднородных сред, получение и анализ аналитических выражений для определения моментов третьего порядка, оценка характерного размера представительного объема двухфазного композита с учетом характера усредненного многочастичного взаимодействия в системе армирующих элементов и прогнозирование эффективных упругих свойств однонаправленно армированных стеклопластиков. + +В данной работе были получены аналитические выражения для безусловных трехточечных и условных двухточечных моментов третьего порядка, получены выражения для производных условных двухточечных моментов третьего порядка, а также разработан и протестирован программный пакет для многопроцессорной системы МВС-1000 для построения моментных функций. + +По теме дипломного проекта опубликованы две печатные работы \cite{zdp,zdpt}. Основные результаты проведенных исследований докладывались и обсуждались на 3-й Всероссийской научно-технической конференции "Математическое моделирование и краевые задачи" (Самара, 2006), а также на "Поздеевских чтениях" (Пермь, 2006 г.). + +Дипломный проект выполнен в соответствии с планом научных исследованиями, проводимыми на кафедре Механика композиционных материалов и конструкций ПГТУ. + +Теоретические разработки нашли отражение в спецкурсах "Методы исследования микроструктуры и свойств композитов", "Синтез и анализ случайных структур композитов", читаемых в Пермском государственном техническом университете студентам специальности 121000 --- "Конструирование и производство изделий из композиционных материалов". + +Автор выражают признательность научному руководителю, доценту кафедры Механика композиционных материалов и конструкций ПГТУ, к.ф.-м.н. А.В.~Зайцеву, а также С.В.~Мельникову, Ю.В.~Соколкину и А.А.~Ташкинову за внимание к работе и обсуждение представленных результатов. diff --git a/krit.tex b/krit.tex new file mode 100755 index 0000000..14f26d3 --- /dev/null +++ b/krit.tex @@ -0,0 +1,274 @@ + +\section{Построение моментных функций третьего порядка случайной структуры волокнистых композитов} + +Обобщим полученные результаты с целью получения аналитических выражений для моментных функций третьего порядка. Задача построения моментных функций третьего порядка случайной структуры двухфазных композитов матричного типа также связана с нахождением геометрических вероятностей \ref{l1}. + +\begin{equation} +\label{k3} +\begin{array}{ll} +K_\lambda^{(3)}(r_1,r_2,r_3)&\equiv\left<\lambda^\circ(r_1)\lambda^\circ(r_2)\lambda^\circ(r_3)\right>=\left<\lambda(r_1)\lambda(r_2)\lambda(r_3)\right>-{}\\ +{}&-\nu_f\left[\left<\lambda(r_1)\lambda(r_2)\right>+\left<\lambda(r_1)\lambda(r_3)\right>+\left<\lambda(r_2)\lambda(r_3)\right>\right]+2\nu_f^3 +\end{array} +\end{equation} + +Введем множества точек $\Omega_f$ , $\Omega'_f$ и $\Omega''_f$, принадлежащих включениям прообраза $\widehat{\Omega}$ и образов $\widehat{\Omega}'$ и $\widehat{\Omega}''$ соответственно. Образы $\widehat{\Omega}'$ и $\widehat{\Omega}''$ получены в результате параллельного переноса $\widehat{\Omega}$ на расстояния, определяемые векторами трансляции Взаимная ориентация этих векторов относительно неподвижной системы координат, связанной с $\widehat{\Omega}$, определяется углом $\Theta$, а взаимная ориентация --- углом $\phi$ (рис. \ref{obraz}). Обратим внимание на то, для статистически изотропного случайного поля структуры аргументами моментной функции третьего порядка будут $|\Delta r_1|$, $|\Delta r_2|$ и $\phi$. + +Тогда геометрическим смыслом условной вероятности $Prob\left[r_1 \in \Omega_f\mid(r_2\in\Omega_f\land r_3\in\Omega_f)\right]$, которая содержится в выражении (\ref{k3}) в виде произведения + +\begin{equation} +\label{ml3} +\begin{array}{ll} +\left<\lambda(r_1)\lambda(r_2)\lambda(r_3)\right>& +=Prob\left(r_1\in\Omega_f\land r_2\in\Omega_f\land r_3\in\Omega_f\right)={}\\ +{}&=Prob\left[r_1 \in\Omega_f\mid(r_2\in\Omega_f\land r_3\in\Omega_f)\right]\times{}\\ +{}&\times Prob\left[r_2\in\Omega_f\mid r_3\in\Omega_f\right]Prob\left[r_3\in\Omega_f\right] +\end{array} +\end{equation} + +является мера $mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)$ множества точек, получаемых при пересечении включений, принадлежащих $\widehat{\Omega}$, $\widehat{\Omega}'$ и $\widehat{\Omega}''$ (рис. \ref{obraz}). Следовательно, используя для $Prob\left[r_2\in\Omega_f\mid r_3\in\Omega_f\right]$ представление (\ref{prob_r1r2}) и, учитывая, что $Prob\left[r_3\in\Omega_f\right]\equiv\nu_f$, из соотношения (\ref{ml3}) получим: + +\begin{equation} +\label{ml3_p} +\left<\lambda(r_1)\lambda(r_2)\lambda(r_3)\right>=\frac{mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)}{mes\Omega''_f}\nu_f. +\end{equation} + +Таким образом, заменяя в выражении (\ref{k3}) группу слагаемых, содержащих общий множитель $\nu_f$, соответствующими представлениями (\ref{ml2}) и принимая во внимание (\ref{ml3_p}), представим моментную функции третьего порядка случайной структуры двухфазного композита запишем следующим образом: + +\begin{equation} +\label{k3_end} +\begin{array}{ll} +K_\lambda^{(3)}(R_1,R_2,\phi)&=\nu_f\left\{\frac{mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)}{mes\Omega''_f}-\right.\\ +{}&\left.-\nu_f\left[\frac{mes(\Omega_f\cap\Omega'_f)}{mes\Omega'_f}+\frac{mes(\Omega_f\cap\Omega''_f)}{mes\Omega''_f}+\frac{mes(\Omega'_f\cap\Omega''_f)}{mes\Omega''_f}\right]\right\}+2\nu_f^3 +\end{array} +\end{equation} + +или + +\begin{equation} +\label{k3_mr} +\begin{array}{ll} +K_\lambda^{(3)}(R_1,R_2,\phi)&=\frac{\nu_f}{mes\Omega_f}\left\{mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)-\nu_f\left[mes(\Omega_f\cap\Omega'_f)+\right.\right.{}\\ +{}&\left.\left.+mes(\Omega_f\cap\Omega''_f)+mes(\Omega'_f\cap\Omega''_f)\right]\right\}+2\nu_f^3={}\\ +{}&\frac{\nu_f}{mes\Omega_f}\sum_{i=1}^N\left\{mes(\Omega_i\cap\Omega'_i\cap\Omega''_i)-\nu_f\left[mes(\Omega_i\cap\Omega'_i)+\right.\right.{}\\ +{}&\left.\left.+mes(\Omega_i\cap\Omega''_i)+mes(\Omega'_i\cap\Omega''_i)\right]\right\}+{}\\ +{}&+\frac{\nu_f}{mes\Omega_f}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}{N}(1-\delta_{ijk})\left\{mes(\Omega_i\cap\Omega'_j\cap\Omega''_k)-\right.{}\\ +{}&\left.-\nu_f\left[mes(\Omega_i\cap\Omega'_j)+mes(\Omega_i\cap\Omega''_k)+mes(\Omega'_j\cap\Omega''_k)\right]\right\}+2\nu_f^3. +\end{array} +\end{equation} + +Здесь $R_1=|\Delta r_1|$ и $R_2=|\Delta r_2|$; $\delta_{ijk}$ --- коэффициенты, принимающие значения 1 при совпадающих индексах и 0, если хотя бы один из трех индексов отличается от двух других различны (при $i,j=1\dots3$ эти коэффициенты являются обобщенными символами Кронекера). + +Преимуществом аналитического представления (\ref{k3_mr}) очевидны. В выражении (\ref{k3_mr}) появляется возможность в явном виде выделить слагаемые, соответствующие приближению "малых расстояний" + +\begin{equation} +\label{mr} +\begin{array}{ll} +K_\lambda^{(3)}(R_1,R_2,\phi)&=\frac{\nu_f}{mes\Omega_f}\sum_{i=1}^N\left\{mes(\Omega_i\cap\Omega'_i\cap\Omega''_i)-\nu_f\left[mes(\Omega_i\cap\Omega'_i)+\right.\right.{}\\ +{}&\left.\left.+mes(\Omega_i\cap\Omega''_i)+mes(\Omega'_i\cap\Omega''_i)\right]\right\}+2\nu_f^3 +\end{array} +\end{equation} + +при наложении на аргументы $R_1$ и $R_2$ ограничений + +\begin{equation} +\label{usl} +R_1\le min\left[D_{min},d_{min}\right], R_2\le min\left[D_{min},d_{min}\right] +\end{equation} + +при произвольном выборе угла $\phi$. Эти условия обеспечивают пересечение $i$-го включения образа с $i$-ми включениями прообразов $(\Omega_i\cap\Omega'_i\cap\Omega''_i)$. + +Из анализа условий сходимости рядов (\ref{k3_mr}) также может быть получена информация о характере затухания статистических моментов третьего порядка. Существование конечного передела позволит определить асимптоты, вокруг которых происходит осцилляция моментных функций, а анализ знака сумм, входящих в выражения (\ref{k3_mr}) --- определить наличие или отсутствие периодических стравляющих в случайных полях структуры. + +Для конкреттизации слагаемых, входящих в выражение \ref{k3_mr}, рассмотрим ряд вспомогательных геометрических задач по пересечению трех окружностей. Возможны следующие варианты пересечения окружностей: + +\begin{enumerate} +\item Все три включения совпадают если одновременно выполняются следующие условия: +$$ +r_i=r_j, r_j=r_k, r_i=r_k, R_{ij}=0.0, R_{ik}=0.0, R_{jk}=0.0; +$$ +В этом случае площадь пересечения трех окружностей равна $\pi\cdot r_i^2.$ + +\item Одно из включений не пересекается с другими если выполняется одно из следующих условий: +$$ +R_{ij}\ge r_i+r_j, R_{ik}\ge r_i+r_k, R_{jk}\ge r_j+r_k. +$$ +Площадь пересечения трех окружностей равна нулю. + +\item +\begin{figure}[!h] +\caption{$i$ совпадает с $j$ и лежит внутри $k$} +\label{p3} +\includegraphics{./ris/3} +\end{figure} + +$i$ совпадает с $j$ и лежит внутри $k$ (рис. \ref{p3}): +$$ +r_i=r_j, R_{ij}=0.0, r_k \ge R_{jk}+r_j. +$$ +Площадь пересечения трех окружностей равна $\pi\cdot r_i^2.$ + +\item +\begin{figure}[!h] +\caption{$i$ совпадает с $j$ и $k$ лежит внутри} +\label{p4} +\includegraphics{./ris/4} +\end{figure} + +$i$ совпадает с $j$ и $k$ лежит внутри (рис. \ref{p4}): +$$ +r_i=r_j, R_{ij}=0.0, r_i\ge R_{jk}+r_k. +$$ +Площадь пересечения трех окружностей: $\pi\cdot r_k^2.$ + +\item +\begin{figure}[!h] +\caption{$i$ лежит внутри пересечения $j$ и $k$} +\label{p5} +\includegraphics{./ris/5} +\end{figure} + +$i$ лежит внутри пересечения $j$ и $k$ (рис. \ref{p5}): +$$ +R_{jk}R^{(2)}_{jx_{ik}} +$$ + +\item Случай, показанный на рис. \ref{p9} - если выполняются оба условия: + +$R^{(l)}_{jx_{ik}}$ - расстояние от центра $j$ окружности до $l$-той точки пересечения $i$ и $k$ окружностей. +$$ +r_jR^{(1)}_{jx_{ik}}, r_j>R^{(2)}_{jx_{ik}} +$$ + +Площадь пресечения равна: +\begin{equation} +\begin{array}{lr} +k=&r_i^2\cdot\arccos\left[\frac{1.0}{2\cdot R_{ij}\cdot r_i}\cdot\left(r_i^2-r_j^2+R_{ij}^2\right)\right]+r_j^2\cdot\arccos\left[\frac{1.0}{2\cdot R_{ij}\cdot r_j}\cdot\left(r_j^2-r_i^2+R_{ij}^2\right)\right]-{}\\ +&{}-2\cdot\sqrt{p\cdot(p-r_i)\cdot(p-r_j)\cdot(p-R_{ij})}, +\end{array} +\end{equation} +$$ +p=\frac{1.0}{2.0}\cdot(r_i+r_j+R_{ij}). +$$ +\end{enumerate} + +\begin{figure}[!h] +\caption{$i$ совпадает с $j$ и лежит внутри $k$} +\label{p3} +\includegraphics{./ris/3} +\end{figure} +\begin{figure}[!h] +\caption{$i$ совпадает с $j$ и $k$ лежит внутри} +\label{p4} +\includegraphics{./ris/4} +\end{figure} +\begin{figure}[!h] +\caption{$i$ лежит внутри пересечения $j$ и $k$} +\label{p5} +\includegraphics{./ris/5} +\end{figure} +\begin{figure}[!h] +\caption{$i$ лежит внутри $j$ и $k$, которые не совпадают} +\label{p6} +\includegraphics{./ris/6} +\end{figure} +\begin{figure}[!h] +\caption{$i$ и $j$ лежат внутри $k$ и пересекаются} +\label{p7} +\includegraphics{./ris/7} +\end{figure} +\begin{figure}[!h] +\caption{Взаимное пересечение трех включений} +\label{p8} +\includegraphics{./ris/8} +\end{figure} +\begin{figure}[!h] +\caption{Взаимное пересечение трех включений} +\label{p9} +\includegraphics{./ris/9} +\end{figure} +\begin{figure}[!th] +\caption{Взаимное пересечение трех включений} +\label{p10} +\includegraphics{./ris/10} +\end{figure} diff --git a/mes_3.ps b/mes_3.ps new file mode 100755 index 0000000..deb622e --- /dev/null +++ b/mes_3.ps @@ -0,0 +1,2066 @@ +%!PS-Adobe-3.0 +%%Pages: (atend) +%%BoundingBox: 0 0 582 731 +%%HiResBoundingBox: 0.000000 0.000000 581.900000 730.100000 +%......................................... +%%Creator: GNU Ghostscript 705 (pswrite) +%%CreationDate: 2006/05/09 14:26:47 +%%DocumentData: Clean7Bit +%%LanguageLevel: 2 +%%EndComments +%%BeginProlog +% This copyright applies to everything between here and the %%EndProlog: +% Copyright (C) 2002 artofcode LLC, Benicia, CA. 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+%%PageTrailer +%%Trailer +%%Pages: 1 +%%EOF diff --git a/mes_3.tex b/mes_3.tex new file mode 100755 index 0000000..a6e254c --- /dev/null +++ b/mes_3.tex @@ -0,0 +1,125 @@ + +\documentclass{report} +%\batchmode +\usepackage{latexsym} +\usepackage[T2A]{fontenc} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{mathtext} +\righthyphenmin=2 +\oddsidemargin=0pt +\textwidth=18cm +\topmargin=0cm +\textheight=23cm +\begin{document} + +$S_{12}$ -- площадь пересечения 1 и 2 окружностей; + +$S_{23}$ -- площадь пересечения 2 и 3 окружностей; + +$S_2$ -- площадь 2й окружности. + +Площадь фигуры найдем по формуле \ref{first}: + +\begin{equation} +\label{first} +{\bf mes}\:\Omega_{inters} = S_2 - (S_2-S_{12}) - (S_2-S_{23}) +\end{equation} + +Или, после раскрытия скобок: + +\begin{equation} +\label{second} +{\bf mes}\:\Omega_{inters} = S_{12}+S_{23}-S_2 +\end{equation} + +$S_{12}$ и $S_{23}$ были найдены ранее, для моментов II порядка, $S_2=\pi r_2^2$. В результате подстановки получаем формулу \ref{end}: + +\begin{equation} +\label{end} +\begin{array}{rcl} +{\bf mes}\:\Omega_{inters} = r_2^2\cdot +\left( +\arccos\left[\frac{1}{2R_{12}r_2}\cdot\left(r_2^2-r_1^2+R_{12}^2\right)\right]+ +\arccos\left[\frac{1}{2R_{23}r_2}\cdot\left(r_2^2-r_1^2+R_{23}^2\right)\right]-\pi +\right)+\\ +{}+ +r_1^2\arccos\left[\frac{1}{2R_{12}r_1}\cdot\left(r_1^2-r_2^2+R_{12}^2\right)\right]+ +r_3^2\arccos\left[\frac{1}{2R_{23}r_3}\cdot\left(r_3^2-r_2^2+R_{23}^2\right)\right]-\\ +{}-2\cdot\left( +\sqrt{p_1(p_1-r_1)(p_1-r_2)(p_1-R_{12})}+ +\sqrt{p_2(p_2-r_3)(p_2-r_2)(p_2-R_{23})} +\right) +\end{array} +\end{equation} + +\begin{equation} +\label{p1} +p_1=\frac{1}{2}\cdot\left(r_1+r_2+R_{12}\right) +\end{equation} + +\begin{equation} +\label{p2} +p_2=\frac{1}{2}\cdot\left(r_3+r_2+R_{23}\right) +\end{equation} + +Рассмотрим частные случаи: + +\begin{enumerate} + +\item $r_i=r_j \ne r_k, i \ne j \ne k$: + +\begin{equation} +\label{rierj} +\bf K_{\lambda}^{(3)}(r_i,r_i,r_k) = \left<\lambda(r_i)\lambda(r_i)\lambda(r_k)\right>-\nu_f \left[\left<\lambda(r_i)\lambda(r_i)\right>+2\left<\lambda(r_i)\lambda(r_k)\right>\right]+2\nu_f^3 +\end{equation} + +\begin{equation} +\label{lililkf} +\begin{array}{rcl} +\bf\left<\lambda(r_i)\lambda(r_i)\lambda(r_k)\right> = +Prob(r_i \in \Omega_f \land r_i \in \Omega_f \land r_k \in \Omega_f) = {}\\ +\bf{}=Prob \left[r_i \in \Omega_f \mid (r_i \in \Omega_f \land r_k \in \Omega_f)\right]\times{}\\ +\bf{}\times Prob\left[r_i \in \Omega_f \mid r_k \in \Omega_f\right]Prob\left[r_k \in \Omega_f\right] +\end{array} +\end{equation} + +\begin{equation} +\label{lililks} +\begin{array}{rcl} +\bf\left<\lambda(r_i)\lambda(r_i)\lambda(r_k)\right> = +Prob(r_i \in \Omega_f \land r_k \in \Omega_f)Prob(r_i \in \Omega_f \mid r_k \in \Omega_f)Prob(r_k \in \Omega_f)={}\\ +\bf{}=Prob(r_i \in \Omega_f \mid r_k \in \Omega_f) \cdot\nu_f\cdot Prob(r_i \in \Omega_f \mid r_k \in \Omega_f)Prob(r_k \in \Omega_f) +\end{array} +\end{equation} + +\begin{equation} +\label{lililkt} +\bf\left<\lambda(r_i)\lambda(r_i)\lambda(r_k)\right> = +\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes(\Omega_f)}\cdot\nu_f\cdot\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\Omega_f}\cdot\nu_f=\left(\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}\right)^2 +\end{equation} + +\begin{equation} +\label{lili} +\bf\left<\lambda(r_i)\lambda(r_i)\right>=Prob(r_i \in \Omega_f \land r_i \in \Omega_f)=Prob(r_i \in \Omega_f)=\nu_f +\end{equation} + +\begin{equation} +\label{lilk} +\bf\left<\lambda(r_i)\lambda(r_k)\right>=Prob(r_i \in \Omega_f \mid r_k \in \Omega_f)\cdot\nu_f=\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}} +\end{equation} + +\begin{equation} +\label{kiikend} +\begin{array}{rcl} +\bf K_{\lambda}^{(3)}(r_i,r_i,r_k)= +\left[\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}\right]^2- +\nu_f\cdot\left[\nu_f+2\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}\right]+2\nu_f^3={}\\ +\bf{}=\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}\left[\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}-2\nu_f\right]-\nu_f^2+2\nu_f^3 +\end{array} +\end{equation} + +\end{enumerate} + + +\end{document} diff --git a/ogl.txt b/ogl.txt new file mode 100755 index 0000000..78ca4e9 --- /dev/null +++ b/ogl.txt @@ -0,0 +1,86 @@ +Оглавление + +Введение 2 +Опасные и вредные факторы 2 +Электрический ток 2 +Влияние на человека 3 +Предупреждение воздействия 3 +Защитные меры 4 +Освещение 5 +Влияние на человека 5 +Предупреждение воздействия 6 +Защитные меры 7 +Электростатические поля 7 +Воздействие на человека 7 +Предупреждение воздействия 9 +Защитные меры 9 + +Введение +В данном дипломном проекте проводится исследование термонапряжённого состояния углерод-углеродного материала, это исследование с точки зрения БЖД можно рассмотреть как работу в офисе, при этом используются персональный компьютер, у которого приходится проводить много времени, что может сказаться на здоровье работающих и результатах их труда. В связи с этим необходимо учесть влияние вредных факторов действующих в офисе на здоровье, и эффективность труда людей. +Опасные и вредные факторы +Среди опасных и вредных факторов действующих на человека в офисе можно выделить следующие: +Электрический ток1 +пониженная освещённость +электромагнитное излучение +Электрический ток +Источником воздействия могут служить электрические установки, к которым относится практически все оборудование ЭВМ, представляют для человека большую потенциальную опасность, так как в процессе эксплуатации или проведения профилактических работ человек может коснуться частей, находящихся под напряжением. Специфическая опасность электроустановок: токоведущие про­водники, корпуса стоек ЭВМ и прочего оборудования, оказавшего­ся под напряжением в результате повреждения (пробоя) изоляции, не подают каких-либо сигналов, которые предупрежда­ли бы человека об опасности. Так же источником опасности поражения электрическим током может стать неисправная проводка, электрические осветительные приборы, кондиционеры и прочее оборудование. +Влияние на человека +Проходя через тело человека, электрический ток оказывает на него сложное воздействие, вызывая термическое, электролитиче­ское, механическое и биологическое действие. Термическое действие тока проявляется в ожогах отдельных участков тела, нагреве тканей и биологических сред, что вызывает в них функциональные расстройства. Электролитическое действие тока выражается в разложении органической жидкости, крови и проявляется в изменении их физико-химического состава. Механическое действие тока приводит к разрыву мышечных тканей. Биологическое действие тока заключается в способности тока раздражать и возбуждать живые ткани организма. +Любое из перечисленных воздействий тока может привести к электрической травме, т. е. к повреждению организма, вызванному воздействием электрического тока или электрической дуги   (ГОСТ  12.1.009—76). +Реакция человека на электрический ток возникает лишь при протекании последнего через тело человека. +Предупреждение воздействия +Исключительно важное значение для предотвра­щения электротравматизма имеет правильная организация обслуживания действующих электроустановок ВЦ, проведение ремонтных, монтажных и профилактических работ. При этом под правильной организацией понимается строгое выполнение ряда организационных и технических мероприятий и средств, установленных действующими «Правилами технической эксплуа­тации электроустановок потребителей и правила техники безо­пасности при эксплуатации электроустановок потребителей» (ПТЭ и ПТБ потребителей) и «Правилами устройства электроустано­вок»   (ПУЭ). +Защитные меры +Различают организационные и технические меры защиты от поражения электрическим током. +К организационным относятся: +1.разделение персонала на группы допуска +2.оформление работы нарядом или устным распоряжением +3.надзор во время работы +4.оформление перерыва в работе, переводов на другое рабочее место, окончания работы +К техническим мерам защиты относятся: +5.отключение оборудования на участке, выделенном для производства работ, и принятие мер    против ошибочного    или    самопроизвольного включения +6.ограждение при необходимости рабочих мест и  оставшихся пол напряжением токоведущих частей +7.вывешивание предупредительных    плакатов    и    знаков    безопасности +8.проверка    отсутствия напряжения - наложение заземления +Технические средства защиты: +1.электрическую изоляцию токоведущих частей +2.защитное заземле­ние +3.зануление +4.выравнивание потенциалов +5.защитное отключение +6.электрическое разделение сети +7.малое напряжение +8.двойная изоляция +Освещение + + + +Влияние на человека +Свет оказывает существенное влияние на жизнедеятельность человека и результаты вы­полняемой им работы. Около 90 % всех сведений об окружающем мире человек получает за счет зрения. Существующие условия производства вызывают повышенное напряжение зрительного анализатора человека. На организм человека воздействует как пониженная освещённость, так и слишком яркий свет, так же желательно избежать мерцания света, которое недопустимо для некоторых видов работ. +Понижение освещенности ведет к снижению производительности труда, причем не только ручного, но и умственного, требующего напряжения памяти, логического мышления. Например, снижение освещенности на величину до 50 % от нормативного значения может привести к зрительному утомлению и снижению производительности труда на 3...10 %  с одновременным ростом брака продукции. +Одной из сторон предупреждения зрительного и общего утомления, создания благоприятных условий для безопасной трудовой деятельности человека является организация хорошей освещенности. +Предупреждение воздействия +Освещение должно быть гигиенически рациональным, т.е. обес­печивать достаточную освещенность рабочих поверхностей, постоянство равномерной освещенности во времени, равномерное распределение яркости в окружающем пространстве и отсутствие слепящего действия. +Ориентация светопроемов для помещений с ЭВМ ВДТ должна быть северо-восточной или северной, с КЕО 1,5-1,0%. +В качестве источников искусственного освещения должны использоваться люминесцентные лампы типа ЛБ и компактные люминисцентные лампы (КЛЛ) ДРЛ. Освещенность в горизонтально плоскости должно быть не ниже 300 лк для системы общего освещения и не ниже 750 лк для системы комбинированного освещения; на монтажных столах инженеров-электронщиков по ремонту и отладке блоков —1000 лк; при одновременной работе с документацией и видеотерминалом горизонтальная освещенность —500 лк. Показатель ослепленности для источников общего искусственного освещения в производственном помещении не более 20. Коэффициент пульсации не должен превышать 5 %. +Требования к видеотерминалу +Яркость экрана не менее 100 кд/м. Освещенность поверхности экрана не должна быть более 300 лк. Яркость бликов на экране — не более 40 кд/м2. Размер светящейся точ­ки — не более 0,4 мм для монохромного дисплея и не более 0,56— для цветного. Контраст изображения знака — не менее 0,8. +Режим работы +Продолжительность работы с ВДТ при вводе дан­ных, редактировании программ; чтении информации с экрана не должна превышать 4 ч при 8-часовом рабочем дне. Через каждый час работы — перерыв на 5-10 мин, а через 2 ч — на 15 мин. + +Защитные меры +Защитных мер от пониженного освещения не существует. +От слишком яркого технологического излучения необходимо защищаться с помощью светофильтров, снижающих интенсивность излучения и изменяющих его спектральный состав. +Электростатические поля + +В условиях офиса электростатический заряд может накапливаться на мониторах Персональных ЭВМ, электризующихся покрытиях и стекать на любые массивные, заземлённые или имеющие иной потенциал поверхности. +Воздействие на человека +Воздействие электростатического поля (ЭСП) — статического электричества — на человека связано с протеканием через него слабого тока (несколько микроампер). При этом электротравм никогда не наблюдается. Однако вследствие рефлекторной реакции на электрический ток (резкое отстранение от заряженного тела) возможна механическая травма при ударе о рядом расположенные элементы конструкций, падений с высоты и т. д. +Исследование биологических эффектов показало, что наиболее чувствительны к электростатическому полю ЦНС, сердечно-сосудистая система, анализаторы. Люди, работающие в зоне воздействия ЭСП, жалуются на раздражительность, головную боль, нарушение сна и др. Характерны своеобразные «фобии», обусловленные страхом ожидаемого разряда, склонность к психосоматическим расстройствам с повышенной эмоциональной возбудимостью и быстрой истощаемостью, неустойчивость показателей пульса и артериального давления. + +Предупреждение воздействия +Для предупреждения воздействия необходимо препятствовать накоплению электростатического заряда на поверхностях, с которыми контактирует работник офиса. ПК должен быть заземлён, а пол в помещении должен быть покрыт не накапливающим заряд покрытием. Так же нужно следить за влажностью в помещении, поскольку электростатический заряд накапливается в большей степени в сухих помещениях. + +Защитные меры +Защитой от статических зарядов, накапливающихся на мониторе, может служить токопроводящее покрытие, подключённое к контуру заземления. Накоплению зарядов на предметах находящихся в помещении может препятствовать токопроводящее покрытие пола. + diff --git a/out.tex b/out.tex new file mode 100755 index 0000000..5903f04 --- /dev/null +++ b/out.tex @@ -0,0 +1,13 @@ +\pagebreak +\section{Заключение} + +Основные результаты дипломного проекта заключаются в следующем: + +\begin{enumerate} +\item Получены новые аналитические выражения в виде рядов для безусловных трехточечных и условных двухточечных моментных функций третьего порядка случайной структуры двухфазных однонаправленно армированных композитов матричного типа, позволяющие в явном виде выделить слагаемые, соответствующие приближению "малых" расстояний. Определены производные условных двухточечных моментных функций и проанализировано влияние типа закона распределения диаметров включений на угол наклона корреляционных функций в точке, соответствующей нулевому значению аргумента, а также разработан и протестирован пакет прикладных программ для многопроцессорной системы МВС-1000 для построения моментных функций. + +\item В работе проведен расчет экономической эффективности программного продукта для построения моментных функций третьего порядка случайной структуры двухфазных однонаправленно армированных композитов. + +\item Проведен расчет допустимого уровня шума на рабочем месте инженера-программиста. + +\end{enumerate} diff --git a/plakat/1.odt b/plakat/1.odt new file mode 100755 index 0000000..6f8b60d Binary files /dev/null and b/plakat/1.odt differ diff --git a/plakat/2.odt b/plakat/2.odt new file mode 100755 index 0000000..89c7afc Binary files /dev/null and b/plakat/2.odt differ diff --git a/plakat/3.odt b/plakat/3.odt new file mode 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+1,467 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 34 336 289 807 +%%Title: cross3 +%%CreationDate: Sun Mar 26 20:05:53 2006 +%%Creator: Tgif-4.1.42-QPL written by William Chia-Wei Cheng (bill.cheng@acm.org) +%%ProducedBy: (unknown) +%%Pages: 1 +%%DocumentFonts: (atend) +%%EndComments +%%BeginProlog + +/tgifdict 86 dict def +tgifdict begin + +/tgifellipsedict 6 dict def +tgifellipsedict /mtrx matrix put + +/TGEL % tgifellipse + { tgifellipsedict begin + /yrad exch def + /xrad exch def + /y exch def + /x exch def + /savematrix mtrx currentmatrix def + x y translate + xrad yrad scale + 0 0 1 0 360 arc + savematrix setmatrix + end + } def + +/tgifarcdict 8 dict def +tgifarcdict /mtrx matrix put + +/TGAN % tgifarcn + { tgifarcdict begin + /endangle exch def + /startangle exch def + /yrad exch def + /xrad exch def + /y exch def + /x exch def + /savematrix mtrx currentmatrix def + x y translate + xrad yrad scale + 0 0 1 startangle endangle arc + savematrix setmatrix + end + } def + +/TGAR % tgifarc + { tgifarcdict begin + /endangle exch def + /startangle exch def + /yrad exch def + /xrad exch def + /y exch def + /x exch def + /savematrix mtrx currentmatrix def + x y translate + xrad yrad scale + 0 0 1 startangle endangle arcn + savematrix setmatrix + end + } def + +/tgifpatdict 10 dict def + +/tgifpatbyte + { currentdict /retstr get exch + pat i cellsz mod get put + } def + +/tgifpatproc + { 0 1 widthlim {tgifpatbyte} for retstr + /i i 1 add def + } def + +/TGPF % tgifpatfill + { tgifpatdict begin + /h exch def + /w exch def + /lty exch def + /ltx exch def + /cellsz exch def + /pat exch def + + /widthlim w cellsz div cvi 1 sub def + /retstr widthlim 1 add string def + /i 0 def + + tgiforigctm setmatrix + ltx lty translate + w h true [1 0 0 1 0 0] {tgifpatproc} imagemask + ltx neg lty neg translate + end + } def + +/pat3 <8000000008000000> def +/pat4 <8800000022000000> def +/pat5 <8800220088002200> def +/pat6 <8822882288228822> def +/pat7 def +/pat8 <77dd77dd77dd77dd> def 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+/SU { sub } bd +/PO { pop } bd +/EX { exch } bd +/CO { concat } bd +/CL { clip } bd +/EC { eoclip } bd +/EF { eofill } bd +/IM { image } bd +/IMM { imagemask } bd +/ARY { array } bd +/SG { setgray } bd +/RG { setrgbcolor } bd +/SD { setdash } bd +/W { setlinewidth } bd +/SM { setmiterlimit } bd +/SLC { setlinecap } bd +/SLJ { setlinejoin } bd +/SH { show } bd +/FF { findfont } bd +/MS { makefont setfont } bd +/AR { arcto 4 {pop} repeat } bd +/CURP { currentpoint } bd +/FLAT { flattenpath strokepath clip newpath } bd +/TGSM { tgiforigctm setmatrix } def +/TGRM { savematrix setmatrix } def + +end + +%%EndProlog +%%Page: 1 1 + +%%PageBoundingBox: 34 336 289 807 +tgifdict begin +/tgifsavedpage save def + +1 SM +1 W + +0 SG + +72 0 MU 72 11.695 MU TR +72 128 DI 100.000 MU 100 DI DU NE SC + +GS + +/tgiforigctm matrix currentmatrix def + +% OVAL +0 SG +GS + GS + NP 224 224 160 160 TGEL + S + GR +GR + +% OVAL +0 SG +GS + GS + NP 288 352 160 160 TGEL + S + GR +GR + +% OVAL +0 SG +GS + GS + NP 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+ + 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+яяяяяяуууяяяяяяяяяяяяюююшшшяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяэээяяяюююяяяяяяэээяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяшшшяяяжжжяяяяяяяяяфффщщщяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяьььяяяяяятттяяяъъъцццяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяъъъщщщуууяяяЫЫЫяяя 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яяяяяяэээяяящщщяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяъъъяяяяяяьььюююъъъяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяяьььяяяъъъыыыяяя яяясссююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююяяя яяяхххяяяцццяяящщщююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююююяяянннщщщыыыяяяяяяннн + + 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+%%Trailer +end +%%EOF diff --git a/ris/struct.jpg b/ris/struct.jpg new file mode 100755 index 0000000..d42bb9e Binary files /dev/null and b/ris/struct.jpg differ diff --git a/ris/struct.ps b/ris/struct.ps new file mode 100755 index 0000000..e60f7f5 --- /dev/null +++ b/ris/struct.ps @@ -0,0 +1,9767 @@ +%!PS-Adobe-3.0 +%%Creator: (ImageMagick) +%%Title: (/home/sage/diplom/ris/struct.ps) +%%CreationDate: (Sun Jun 18 18:19:53 2006) +%%BoundingBox: 0 0 201 100 +%%HiResBoundingBox: 0 0 201.43 100 +%%DocumentData: Clean7Bit +%%LanguageLevel: 1 +%%Orientation: Portrait +%%PageOrder: Ascend +%%Pages: 1 +%%EndComments + +%%BeginDefaults +%%EndDefaults + +%%BeginProlog +% +% Display a color image. The image is displayed in color on +% Postscript viewers or printers that support color, otherwise +% it is displayed as grayscale. +% +/DirectClassPacket +{ + % + % Get a DirectClass packet. + % + % Parameters: + % red. + % green. + % blue. + % length: number of pixels minus one of this color (optional). + % + currentfile color_packet readhexstring pop pop + compression 0 eq + { + /number_pixels 3 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add 3 mul def + } ifelse + 0 3 number_pixels 1 sub + { + pixels exch color_packet putinterval + } for + pixels 0 number_pixels getinterval +} bind def + +/DirectClassImage +{ + % + % Display a DirectClass image. + % + systemdict /colorimage known + { + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { DirectClassPacket } false 3 colorimage + } + { + % + % No colorimage operator; convert to grayscale. + % + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { GrayDirectClassPacket } image + } ifelse +} bind def + +/GrayDirectClassPacket +{ + % + % Get a DirectClass packet; convert to grayscale. + % + % Parameters: + % red + % green + % blue + % length: number of pixels minus one of this color (optional). + % + currentfile color_packet readhexstring pop pop + color_packet 0 get 0.299 mul + color_packet 1 get 0.587 mul add + color_packet 2 get 0.114 mul add + cvi + /gray_packet exch def + compression 0 eq + { + /number_pixels 1 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add def + } ifelse + 0 1 number_pixels 1 sub + { + pixels exch gray_packet put + } for + pixels 0 number_pixels getinterval +} bind def + +/GrayPseudoClassPacket +{ + % + % Get a PseudoClass packet; convert to grayscale. + % + % Parameters: + % index: index into the colormap. + % length: number of pixels minus one of this color (optional). + % + currentfile byte readhexstring pop 0 get + /offset exch 3 mul def + /color_packet colormap offset 3 getinterval def + color_packet 0 get 0.299 mul + color_packet 1 get 0.587 mul add + color_packet 2 get 0.114 mul add + cvi + /gray_packet exch def + compression 0 eq + { + /number_pixels 1 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add def + } ifelse + 0 1 number_pixels 1 sub + { + pixels exch gray_packet put + } for + pixels 0 number_pixels getinterval +} bind def + +/PseudoClassPacket +{ + % + % Get a PseudoClass packet. + % + % Parameters: + % index: index into the colormap. + % length: number of pixels minus one of this color (optional). + % + currentfile byte readhexstring pop 0 get + /offset exch 3 mul def + /color_packet colormap offset 3 getinterval def + compression 0 eq + { + /number_pixels 3 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add 3 mul def + } ifelse + 0 3 number_pixels 1 sub + { + pixels exch color_packet putinterval + } for + pixels 0 number_pixels getinterval +} bind def + +/PseudoClassImage +{ + % + % Display a PseudoClass image. + % + % Parameters: + % class: 0-PseudoClass or 1-Grayscale. + % + currentfile buffer readline pop + token pop /class exch def pop + class 0 gt + { + currentfile buffer readline pop + token pop /depth exch def pop + /grays columns 8 add depth sub depth mul 8 idiv string def + columns rows depth + [ + columns 0 0 + rows neg 0 rows + ] + { currentfile grays readhexstring pop } image + } + { + % + % Parameters: + % colors: number of colors in the colormap. + % colormap: red, green, blue color packets. + % + currentfile buffer readline pop + token pop /colors exch def pop + /colors colors 3 mul def + /colormap colors string def + currentfile colormap readhexstring pop pop + systemdict /colorimage known + { + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { PseudoClassPacket } false 3 colorimage + } + { + % + % No colorimage operator; convert to grayscale. + % + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { GrayPseudoClassPacket } image + } ifelse + } ifelse +} bind def + +/DisplayImage +{ + % + % Display a DirectClass or PseudoClass image. + % + % Parameters: + % x & y translation. + % x & y scale. + % label pointsize. + % image label. + % image columns & rows. + % class: 0-DirectClass or 1-PseudoClass. + % compression: 0-none or 1-RunlengthEncoded. + % hex color packets. + % + gsave + /buffer 512 string def + /byte 1 string def + /color_packet 3 string def + /pixels 768 string def + + currentfile buffer readline pop + token pop /x exch def + token pop /y exch def pop + x y translate + currentfile buffer readline pop + token pop /x exch def + token pop /y exch def pop + currentfile buffer readline pop + token pop /pointsize exch def pop + /Times-Roman findfont pointsize scalefont setfont + x y scale + currentfile buffer readline pop + token pop /columns exch def + token pop /rows exch def pop + currentfile buffer readline pop + token pop /class exch def pop + currentfile buffer readline pop + token pop /compression exch def pop + class 0 gt 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Пусть $\lambda(\bf{r})$ --- случайная индикаторная функция, которая принимает значение, равное единице, в случае, если точка $\bf{r}$ принадлежит дискретной фазе --- волокну (объемная доля которых равна $\nu_f$), и нулю --- если эта точка принадлежит непрерывной фазе --- матрице. + +Прогнозирование эффективных деформационных свойств и определение статистических характеристик случайных полей напряжений и деформаций в компонентах двухфазных волокнистых и дисперсно-упрочненных композитов связаны с необходимостью решения стохастически нелинейных краевых задач, для построения приближенных решений которых (например, полного корреляционного приближения) требуются описывающие многочастичное взаимодействие в системе армирующих элементов двух- и трехточечные моментные функции структурных модулей упругости второго, третьего, четвертого и пятого порядков. Эти функции определяются центральными моментами соответствующих порядков случайного индикатора $\lambda(\bf{r})$. + +Моментные функции второго порядка $K_\lambda^{(2)}(\bf{r_1,r_2})$ позволяют определить степень взаимодействия и характер упорядоченности между соседними и удаленными друг от друга элементами структуры; третьего порядка $K_\lambda^{(3)}(\bf{r_1,r_2,r_3})$ характеризуют форму, а четвертого порядка $K_\lambda^{(4)}(\bf{r_1,r_2,r_3})$ позволяют установить, как группируются включения \cite{ber}. + +Стохастический характер структуры композитов обусловлен случайностью формы, взаимного расположения и ориентации волокон, разбросом характерных размеров частиц армирующего наполнителя. В настоящее время существует потребность определения «скрытых» параметров порядка стохастических структур (например, детерминированных периодических и квази-детерминированных составляющих), практически не отражающихся на эффективных упругих характеристиках этих материалов, но предопределяющих сценарии развития процесса разрушения. + +Исследование закономерностей случайных структур будем проводить на основе анализа сгенерированных плоских фрагментов (синтез которых связан со случайным размещением непересекающихся гладких дисков на плоскости \cite{vs,zlt,zlt3}), считая, что волокна двухфазного композита имеют круглое поперечное сечение (рис. 1). При моделировании структур композитов будем предполагать, что координаты центров размещаемых внутри синтезируемого фрагмента дисков (поперечных сечений волокон) являются независимыми равномерными случайными величинами, а характерные размеры волокон описываются одномодальными статистическими законами распределения: симметричными (нормальный) и несимметричными (логнормальный). Кроме того, ограничим законы распределения диаметров слева заданным минимальным значением $D_{min}$ (которое для всех генерируемых структур будет равно $D_{min}=/2$) и будем считать неизменным отношение $/L=0.01$ среднего диаметра волокон $D$ к характерному размеру фрагмента $L$. + +\begin{figure}[!h] +\label{struct1} +\begin{center} +\includegraphics[width=0.5\textwidth]{ris/struct2} +\caption{Фрагмент модельной структуры волокнистого композита, диаметры которых описываются нормальным законом} +\end{center} +\end{figure} + +Эти предположения и ограничения согласуются с результатами построения законов распределения диаметров волокон однонаправленно армированного стеклопластиков на основе эпоксидной \cite{vs} и ненасыщенной полиэфирной смолы горячего отверждения ПН–1 \cite{fkg}, оправданы организацией технологического процесса получения волокнистых наполнителей. Как свидетельствуют представленные в этих работах результаты, проверка по критерию $\chi^2$ Пирсона показала, что нормальный закон может быть принят с вероятностью ошибочного отклонения гипотезы не более 6\%, а логнормальный — не более 2\%. + +Расположение волокон в сечении может быть охарактеризовано распределением длин промежутков между волокнами, измеренных в произвольном направлении. В монографии \cite{vs} приведены результаты, показывающие, что с вероятностью не более 6\% по критерию $\chi^2$ Пирсона можно отклонить гипотезу о нормальном распределении минимальных расстояний между волокнами. Вместе с тем, авторами \cite{fkg} была доказана эквивалентность законов распределения длин промежутков и минимальных расстояний между волокнами. + +Предположение о том, что диаметры волокон двухфазных однонаправленно армированных композитов являются случайными не вносит существенных корректировок в алгоритмы синтеза структур этих материалов. Для достижения объемных наполнений, близких к предельным, генерация структуры может, при необходимости, сопровождать дополнительным взаимным перемещением, вновь и ранее размещаемых волокон \cite{vs,zlt,zlt3}, а также запрещением выхода какой-либо части поперечного сечения армирующего элемента за границы области. + +Однако возможна дополнительная модификация алгоритмов \cite{vs,zlt,zlt3} процедурой предварительной сортировки (по возрастанию значений) последовательности псевдослучайных диаметров волокон, распределенных по заданному статистическому закону \cite{zlt4}. Последующее случайное размещение волокон внутри фрагмента происходит в порядке уменьшения диаметров. Использование предварительной сортировки оправдано необходимостью строгого соблюдения соответствия заданного теоретического и эмпирического (построенного по сгенерированной случайной структуре) статистических законов распределения характерных размеров включений. Можно предположить, что невыполнение этого условия может наблюдаться у материалов, которые содержат как крупные, так и мелкие фракции (поскольку вероятность размещения внутри синтезируемого фрагмента волокон с малыми диаметрами намного больше вероятности расположения волокон большого диаметра). + +Вычисление моментных функций можно проводить при помощи алгоритмов, использованных для экспериментального построения этих функций по микрошлифам структуры металлов [13] и стеклопластиков [1]. Реализация этих алгоритмов связана с построением вспомогательных координатных сеток, определением принадлежности каждого узла этих сеток одной из фаз материала, требует значительных аппаратных и программных затрат даже в случае сведения задачи построения моментных функций к нахождению геометрических вероятностей [1]. Действительно, обозначив $Prob(\xi)$ вероятность события $\xi$ , можно преобразовать первое слагаемое равенства + +\begin{equation} +\label{k2_beg} +K_\lambda^{(2)}(\bf{r_1,r_2})=\left<\lambda^\circ(\bf{r_1})\lambda^\circ(\bf{r_2})\right>=\left<\lambda(\bf{r_1})\lambda(\bf{r_2})\right>-\nu_f^2 +\end{equation} + +\noindent следующим образом: + +\begin{equation} +\label{ll} +\left<\lambda(\bf{r_1})\lambda(\bf{r_2})\right>=Prob({\bf r_1}\in\Omega_f\land{\bf r_2}\in\Omega_f)=Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)\nu_f, +\end{equation} + +\noindent а условную вероятность $Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)$ определим отношением + +\begin{equation} +\label{prob} +Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)\cong\alpha/\beta. +\end{equation} +Здесь $\alpha$ --- число событий ${\bf r}\in\Omega_f\land{\bf r'}\in\Omega_f$ и $\beta$ --- число событий ${\bf r}\in\Omega_f$. + +Моментные функции, построенные этим при помощи реализации этого алгоритма, очень чувствительны к шагу вспомогательной сетки. Однако, уменьшение шага вспомогательной сетки, приводит к значительному росту аппаратных и программных затрат для проведения статистического анализа каждого сгенерированного фрагмента случайной структуры. + +Одним из путей преодоления данной проблемы является использование параллельных вычислений с применением технологии MPI (The Message Passing Interface), которая представляет хорошо стандартизованный механизм построения параллельных алгоритмов в модели обмена сообщениями. Кроме того, в настоящее время разработаны стандартные «привязки» MPI к языкам программирования С/С++ и Fortran 77/90, свободные и коммерческие реализации для большинства многопроцессорных платформ, а также для сетей рабочих станций UNIX и Windows NT. + +Для реализации технологии MPI был использован пакет MPICH, который поддерживает стандарт MPI 1.2 и некоторые элементы стандарта MPI 2.0. + +Для решения задачи был использованы два алгоритма: в первом на нескольких процессорах определяется только принадлежность точки волокну или матрицы методом прямого перебора, а во втором помимо этого каждый процессор вычислял значения моментной функции для своего, заранее определенного направления. Многопроцессорная реализация заключалась в том, что каждый процессор перебирал точки, начиная со своего порядкового номера, с шагом, равным числу процессоров. В результате весь промежуток делился на число участков, равное числу всех процессоров. + +Блок-схема алгоритма 1 приведена на рис. \ref{bs}. + +\begin{figure} +\label{bs} +%\include{ris/bs.tex} +\includegraphics[width=\textwidth]{ris/bs} +\caption{Блок-схема алгоритма построения моментных функций для многопроцессорной системы МВС-1000} +\end{figure} + +В результате исполнения программы с разным числом процессоров была получена следующая зависимость отношения времени счета на одном процессоре к времени счета на n процессорах от числа процессоров n (таблица \ref{time_zatr}). + +\begin{table}[!h] +\footnotesize{ +\caption{Временные затраты на реализацию алгоритма построения моментных функций второго порядка случайной структуры однонаправленно армированного композита} +\label{time_zatr} +\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|} +\hline +&\multicolumn{12}{|c|}{Число процессоров}\\ +\cline{2-13} +&1&2&3&4&5&6&7&8&9&10&11&12\\ +\hline +Алгоритм 1&1.000&0.867&0.800&0.667&0.599&0.599&0.533&0.533&0.467&0.467&0.533&0.599\\ +\hline +Алгоритм 2&1.000&0.857&0.714&0.571&0.429&0.357&0.283&0.214&0.286&0.357&---&---\\ +\hline +\end{tabular} +} +\end{table} + +Вычисления были выполнены на многопроцессорной системе МВС–1000, основу которой составляет масштабируемый массив процессорных узлов. Каждый узел содержит микропроцессор Alpha 21164 с производительностью 2 GFLOPS при тактовой частоте 500 MHz и оперативную память объемом 128 MB с возможностью расширения. Процессорные узлы взаимодействуют через коммуникационные процессоры TMS320C44, имеющие по 4 внешних канала с общей пропускной способностью 80 Мбайт/с. Для управления массивом процессоров и внешними устройствами, а также для доступа к системе извне был использован хост-компьютер на базе процессора Intel с операционной системой Linux. + +Отметим, что для первого алгоритма при счете на 10 процессорах время счета уменьшается более чем в два раза. Увеличение числа процессоров более 10 приводит к увеличению времени счета, что связано с затратами на пересылку данных между процессорами. Для второго алгоритма оптимальное число процессоров равняется восьми и время счета уменьшается почти в пять раз. + +\pagebreak +\subsection{Безусловные трехточечные моментные функции третьего порядка} + +Для снижения программно-аппаратных затрат, которые возникают в результате применения вышеописанного алгоритма, появляется потребность разработки способов аналитического построения условных и безусловных моментных функций произвольного порядка. + +Рассмотрим фрагмент случайной структуры двухфазного однонаправленно армированного (рис. \ref{struct2}). + +\begin{figure}[!h] +\label{struct2} +\begin{center} +\includegraphics[width=0.6\textwidth]{ris/struct3} +\caption{Геометрический смысл функций $Prob\left[(\bf{r_1}\in\Omega_f\land\bf{r_2}\in\Omega_f)\mid\bf{r_3}\in\Omega_f\right]$} +\end{center} +\end{figure} + + +{\bf Определение~1.} Назовем {\it прообразом} фрагмент, представляющий конкретную реализацию случайной структуры. Будем считать, что прообраз условно неподвижен. + +{\bf Определение~2.} Фрагмент, геометрически идентичный прообразу, над которым могут быть осуществлены преобразования трансляции и произвольного пространственного разворота как жесткого целого, назовем {\it образом} случайной структуры. + +Безусловная трехточечная моментная функция третьего порядка может быть представлена следующим образом (\ref{k3}): + +\begin{equation} +\label{k3} +\begin{array}{ll} +K_\lambda^{(3)}(r_1,r_2,r_3)&\equiv\left<\lambda^\circ(r_1)\lambda^\circ(r_2)\lambda^\circ(r_3)\right>=\left<\lambda(r_1)\lambda(r_2)\lambda(r_3)\right>-{}\\ +{}&-\nu_f\left[\left<\lambda(r_1)\lambda(r_2)\right>+\left<\lambda(r_1)\lambda(r_3)\right>+\left<\lambda(r_2)\lambda(r_3)\right>\right]+2\nu_f^3 +\end{array} +\end{equation} + +Введем множества точек $\Omega_f$ , $\Omega'_f$ и $\Omega''_f$, принадлежащих включениям прообраза $\hat{\Omega}$ и образов $\hat{\Omega}'$ и $\hat{\Omega}''$ соответственно. Образы $\hat{\Omega}'$ и $\hat{\Omega}''$ получены в результате параллельного переноса $\hat{\Omega}$ на расстояния, определяемые векторами трансляции ${\bf \Delta r_1}$ и ${\bf |Delta r_2}$. Ориентация этих векторов относительно неподвижной системы координат, связанной с $\hat{\Omega}$, определяется углом $\Theta$, а взаимная ориентация --- углом $\phi$ (рис. \ref{struct2}). Обратим внимание на то,что для статистически изотропного случайного поля структуры аргументами моментной функции третьего порядка будут $|\Delta r_1|$, $|\Delta r_2|$ и $\phi$. + +Тогда геометрическим смыслом условной вероятности $Prob\left[r_1\in\Omega_f\mid(r_2\in\Omega_f\land r_3\in\Omega_f)\right]$, которая содержится в выражении (\ref{k3}) в виде произведения + +\begin{equation} +\label{ml3} +\begin{array}{ll} +\left<\lambda({\bf r_1})\lambda({\bf r_2})\lambda({\bf r_3})\right>& +=Prob\left({\bf r_1}\in\Omega_f\land {\bf r_2}\in\Omega_f\land {\bf r_3}\in\Omega_f\right)={}\\ +{}&=Prob\left[{\bf r_1} \in\Omega_f\mid({\bf r_2}\in\Omega_f\land {\bf r_3}\in\Omega_f)\right]\times{}\\ +{}&\times Prob\left[{\bf r_2}\in\Omega_f\mid {\bf r_3}\in\Omega_f\right]Prob\left[{\bf r_3}\in\Omega_f\right] +\end{array} +\end{equation} + +\noindent является мера $mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)$ множества точек, получаемых при пересечении включений, принадлежащих $\hat{\Omega}$, $\hat{\Omega}'$ и $\hat{\Omega}''$ (рис. \ref{struct2}). Геометрический смысл условной вероятности $Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)$ в уравнении (\ref{ml3}) может быть определен как отношение меры пересечения множеств $\Omega_f$ и $\Omega'_f$ точек, принадлежащих включениям образа $\hat{\Omega}'$ и образа $\hat{\Omega}''$ соответственно к мере множества $\Omega_f'$ ($mes\Omega'_f=\nu_f mes\hat{\Omega}')$: + +\begin{equation} +\label{probr1r2} +\begin{array}{ll} +Prob({\bf r_1}\in\Omega'_f\mid{\bf r_2}\in\Omega'_f)&=\frac{mes(\Omega'_f\cap\Omega''_f)}{mes\Omega'_f}={}\\ +{}&=\frac{\sum_{i=1}^N\sum_{j=1}^N mes(\Omega'_i\cap\Omega''_j)} +{\sum_{i=1}^N mes\Omega_f^{(i)}}= +\frac{\sum_{i=1}^N\sum_{j=1}^N mes(\Omega'_i\cap\Omega''_j)}{\nu_f mes\hat{\Omega}'}. +\end{array} +\end{equation} + +Здесь $\Omega'_i\subset\Omega'_f$ и $\Omega''_j\subset\Omega''_f$ --- конкретные включения, принадлежащие $\hat{\Omega}'$ и $\hat{\Omega}''$, а $N$ --- количество включений. + +Принимая во внимание что + +$$ +\begin{array}{ll} +Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)&=\frac{mes(\Omega_f\cap\Omega'_f)}{mes\Omega_f}={}\\ +{}&=\frac{\sum_{i=1}^N\sum_{j=1}^N mes(\Omega_i\cap\Omega'_j)}{\sum_{i=1}^N mes\Omega_f^{(i)}}= +\frac{\sum_{i=1}^N\sum_{j=1}^N mes(\Omega_i\cap\Omega'_j)}{\nu_f mes\hat{\Omega}}, +\end{array} +$$ + +а также то, что $Prob[r_3\in\Omega_f]=\nu_f$ можно записать следующее выражение: + +\begin{equation} +\label{ml3_p} +\left<\lambda({\bf r_1})\lambda({\bf r_2})\lambda({\bf r_3})\right>=\frac{mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)}{mes\Omega''_f}\nu_f. +\end{equation} + +Для нахождения меры $mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)$, необходимо рассмотреть ряд вспомогательных геометрических задач по пересечению трех кругов различного диаметра. Возможные варианты пересечения включений показаны на рис. \ref{cross}: + +\begin{figure}[!h] +\label{cross} +\includegraphics[width=\textwidth]{ris/circles} +\caption{Возможные варианты пересечения включений прообраза $\Omega$ с включениями образов $\Omega'$ и $\Omega''$} +\end{figure} + +Рассмотрим подробнее эти варианты: + +\begin{enumerate} +\item Все три включения совпадают если одновременно выполняются следующие условия: +$$ +r_i=r_j,\qquad r_j=r_k,\qquad r_i=r_k, R_{ij}=0.0, R_{ik}=0.0, R_{jk}=0.0; +$$ +В этом случае $mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)=\pi\cdot r_i^2.$ + +\item Отсутствие пересечений реализуется при выполнении одного из следующих условий: +$$ +R_{ij}\ge r_i+r_j, R_{ik}\ge r_i+r_k, R_{jk}\ge r_j+r_k. +$$ + +\noindent$mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)=0$. + +$i$-е включение совпадает с $j$-м включением и лежит внутри $k$-го включения (рис. \ref{cross}, а): +$$ +r_i=r_j, R_{ij}=0.0, r_k \ge R_{jk}+r_j. +$$ +Мера $mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)=\pi\cdot r_i^2.$ + +\item $i$-е включение совпадает с $j$-м включением и $k$-е включение лежит внутри (рис. \ref{cross}, б): +$$ +r_i=r_j, R_{ij}=0.0, r_i\ge R_{jk}+r_k. +$$ +Мера $mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)=\pi\cdot r_k^2.$ + +\item $i$-е включение лежит внутри пересечения $j$-го и $k$-го включений (рис. \ref{cross}, в): +$$ +R_{jk}R^{(1)}_{jx_{ik}}, r_j>R^{(2)}_{jx_{ik}} +$$ + +Мера +\begin{equation} +\begin{array}{ll} +mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)&=r_i^2\cdot\arccos\left[\frac{1.0}{2\cdot R_{ij}\cdot r_i}\cdot\left(r_i^2-r_j^2+R_{ij}^2\right)\right]+{}\\ +{}&+r_j^2\cdot\arccos\left[\frac{1.0}{2\cdot R_{ij}\cdot r_j}\cdot\left(r_j^2-r_i^2+R_{ij}^2\right)\right]-{}\\ +{}&-2\cdot\sqrt{p\cdot(p-r_i)\cdot(p-r_j)\cdot(p-R_{ij})}, +\end{array} +\end{equation} +$$ +p=\frac{1}{2}\cdot(r_i+r_j+R_{ij}). +$$ + +\item Случай, показанный на рис. \ref{cross}, ж - если выполняются оба условия: + +$R^{(l)}_{jx_{ik}}$ - расстояние от центра $j$ окружности до $l$-той точки пересечения $i$ и $k$ окружностей. +$$ +r_jR^{(2)}_{jx_{ik}} +$$ + +\end{enumerate} + +Конкретизируем остальные слагаемые, входящие в выражение (\ref{k3})\cite{zlt3}: + +\begin{equation} +\label{ll1} +\left<\lambda(\bf{r_1})\lambda(\bf{r_2})\right>=Prob({\bf r_1}\in\Omega_f\land{\bf r_2}\in\Omega_f)=Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)\nu_f, +\end{equation} + +Условная вероятность $Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)$ определяется отношением + +\begin{equation} +\label{probr1r2} +\begin{array}{ll} +Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)&=\frac{mes(\Omega_f\cap\Omega'_f)}{mes\Omega_f}={}\\ +{}&=\frac{\sum_{i=1}^N\sum_{j=1}^N mes(\Omega_i\cap\Omega'_j)}{\sum_{i=1}^N mes\Omega_f^{(i)}}= +\frac{\sum_{i=1}^N\sum_{j=1}^N mes(\Omega_i\cap\Omega'_j)}{\nu_f mes\hat{\Omega}}. +\end{array} +\end{equation} + +Здесь $\Omega_i\subset\Omega_f$ и $\Omega'_j\subset\Omega'_f$ --- конкретные включения, принадлежащие $\hat{\Omega}$ и $\hat{\Omega}'$, а $N$ --- количество включений. + +Важными характеристиками структуры однонаправленно армированных волокнистых и дисперсно-упрочненных композитов являются диаметры включений $D$ и минимальные расстояния между частицами армирующего наполнителя $d$, которые являются детерминированными или случайными, распределенными по заданным статистическим законам. Очевидно, что для каждого сгенерированного фрагмента $D\in[{D_{min},D_{max}}]$, $d\in[{d_{min},d_{max}}]$. Тогда необходимым и достаточным условием ненулевого пересечения включений будет являться одновременное удовлетворение неравенств + +\begin{equation} +\label{krit} +R < r_i + r_j , +\quad +r_i < R + r_j , +\quad +r_j < R + r_i , +\quad +R \le D_{min} , +\quad +R = |\bf{\Delta r}|, +\end{equation} + +\noindent для меры $mes(\Omega_f\cap\Omega'_f)$. Для мер $mes(\Omega'_f\cap\Omega''_f)$ и $mes(\Omega_f\cap\Omega''_f)$ необходимые и достаточные условия аналогичны. + +Принимая во внимание выражения (\ref{ll1}) и (\ref{probr1r2}) можно записать выражение для безусловного трехточечного момента $K_\lambda^{(3)}(R_1,R_2,\phi)$ следующим образом: + +\begin{equation} +\label{k3_end} +\begin{array}{ll} +K_\lambda^{(3)}(R_1,R_2,\phi)&=\nu_f\left\{\frac{mes(\Omega_f\cap\Omega'_f\cap\Omega''_f)}{mes\Omega''_f}-\right.\\ +{}&\left.-\nu_f\left[\frac{mes(\Omega_f\cap\Omega'_f)}{mes\Omega'_f}+\frac{mes(\Omega_f\cap\Omega''_f)}{mes\Omega''_f}+\frac{mes(\Omega'_f\cap\Omega''_f)}{mes\Omega''_f}\right]\right\}+2\nu_f^3, +\end{array} +\end{equation} + +\noindent или + +\begin{equation} +\label{k3_mrij} +\begin{array}{ll} +K_\lambda^{(3)}(R_1,R_2,\phi)&=\frac{\nu_f}{mes\Omega_f}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}{N}\left\{mes(\Omega_i\cap\Omega'_j\cap\Omega''_k)-\right.{}\\ +{}&\left.-\nu_f\left[mes(\Omega_i\cap\Omega'_j)+mes(\Omega_i\cap\Omega''_k)+mes(\Omega'_j\cap\Omega''_k)\right]\right\}+2\nu_f^3. +\end{array} +\end{equation} + + +Здесь $R_1=|{\bf \Delta r_1}|$ и $R_2=|{\bf \Delta r_2}|$. + +Меры пресечения $mes(\Omega_i\cap\Omega'_j)$ из формулы (\ref{k3_end}) определяются следующим образом: + +\begin{equation} +\label{mesij} +mes(\Omega_i\cap\Omega'_j)=\left\{ +\begin{array}{ll} +0, &d_{ij}\ge r_i+r_j;\\ +\pi r_i^2, &r_j\ge d_{ij}+r_i;\\ +\pi r_j^2, &r_i\ge d_{ij}+r_j;\\ +\kappa_1, &(d_{ij}\le r_i+r_j)\lor(r_i\le d_{ij}+r_j)\lor(r_j\le d_{ij}+r_i), +\end{array}\right. +\end{equation} + +$$ +\begin{array}{ll} +\kappa_1&=r_i^2\arccos\left[\frac{1}{2d_{ij}r_i}\left(r_i^2+r_j^2-d_{ij}^2\right)\right]+r_j^2\arccos\left[\frac{1}{2d_{ij}r_j}\left(r_i^2+r_j^2-d_{ij}^2\right)\right]-{}\\ +{}&-2\sqrt{p_1(p_1-r_i)(p_1-r_j)(p_1-d_{ij})}, +\end{array} +$$ + +$$ +p_1=\frac{1}{2}(r_i+r_j+d_{ij}), d_{ij}=\sqrt{(x_i-x'_j-Rn_1)^2+(y_i-y'_j-Rn_2)^2}, +$$ + +\noindent Здесь $d_{ij}$ --- расстояние между центрами $i$ включения прообраза $\hat{\Omega}$ ($x_i$ и $x_j$ --- координаты центра) и $j$ включения образа $\hat{\Omega}'$ (${x}'_i$ и ${x}'_j$ --- координаты центра), радиусы $\hat{\Omega }$ и $\hat{\Omega}'$ равны $r_i$ и $r_j$ соответственно; $R = |\bf{\Delta r}|$. + +Аналогичные выражения записываются и для двух других мер пересечения. Для $mes(\Omega'_j\cap\Omega''_k)$: + +\begin{equation} +\label{mesjk} +mes(\Omega'_j\cap\Omega'_k)=\left\{ +\begin{array}{ll} +0, &d_{jk}\ge r_j+r_k;\\ +\pi r_j^2, &r_k\ge d_{jk}+r_j;\\ +\pi r_k^2, &r_j\ge d_{jk}+r_k;\\ +\kappa_2, &(d_{jk}\le r_j+r_k)\lor(r_j\le d_{jk}+r_k)\lor(r_k\le d_{jk}+r_j), +\end{array}\right. +\end{equation} + +$$ +\begin{array}{ll} +\kappa_2&=r_j^2\arccos\left[\frac{1}{2d_{jk}r_j}\left(r_j^2+r_k^2-d_{jk}^2\right)\right]+r_k^2\arccos\left[\frac{1}{2d_{jk}r_k}\left(r_j^2+r_k^2-d_{jk}^2\right)\right]-{}\\ +{}&-2\sqrt{p_2(p_2-r_j)(p_2-r_k)(p_2-d_{jk})}, +\end{array} +$$ + +$$ +p_2=\frac{1}{2}(r_j+r_k+d_{jk}), d_{jk}=\sqrt{(x'_j-x''_k-Rn_1)^2+(y'_j-y''_k-Rn_2)^2}, +$$ + +И для $mes(\Omega_f\cap\Omega''_f)$: + +\begin{equation} +\label{mesik} +mes(\Omega_i\cap\Omega''_k)=\left\{ +\begin{array}{ll} +0, &d_{ik}\ge r_i+r_k;\\ +\pi r_i^2, &r_k\ge d_{ik}+r_k;\\ +\pi r_k^2, &r_i\ge d_{ik}+r_k;\\ +\kappa_3, &(d_{ik}\le r_i+r_k)\lor(r_i\le d_{ik}+r_k)\lor(r_k\le d_{ik}+r_i), +\end{array}\right. +\end{equation} + +$$ +\begin{array}{ll} +\kappa_3&=r_i^2\arccos\left[\frac{1}{2d_{ik}r_i}\left(r_i^2+r_k^2-d_{ik}^2\right)\right]+r_k^2\arccos\left[\frac{1}{2d_{ik}r_k}\left(r_i^2+r_k^2-d_{ik}^2\right)\right]-{}\\ +{}&-2\sqrt{p_3(p_3-r_i)(p_3-r_k)(p_3-d_{ik})}, +\end{array} +$$ + +$$ +p_3=\frac{1}{2}(r_i+r_k+d_{ik}), d_{ik}=\sqrt{(x_i-x''_k-Rn_1)^2+(y_i-y''_k-Rn_2)^2}. +$$ + +Из анализа условий сходимости рядов (\ref{k3_end}) также может быть получена информация о характере затухания статистических моментов третьего порядка. Существование конечного передела позволит определить асимптоты, вокруг которых происходит осцилляция моментных функций, а анализ знака сумм, входящих в выражения (\ref{k3_mr}) --- определить наличие или отсутствие периодических составляющих в случайных полях структуры. + +Выражение (\ref{k3_end}) можно также записать в следующем виде: + +\begin{equation} +\label{k3_mr} +\begin{array}{ll} +K_\lambda^{(3)}(R_1,R_2,\phi)&=\frac{\nu_f}{mes\Omega_f}\sum_{i=1}^N\left\{mes(\Omega_i\cap\Omega'_i\cap\Omega''_i)-\nu_f\left[mes(\Omega_i\cap\Omega'_i)+\right.\right.{}\\ +{}&\left.\left.+mes(\Omega_i\cap\Omega''_i)+mes(\Omega'_i\cap\Omega''_i)\right]\right\}+{}\\ +{}&+\frac{\nu_f}{mes\Omega_f}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}{N}(1-\delta_{ijk})\left\{mes(\Omega_i\cap\Omega'_j\cap\Omega''_k)-\right.{}\\ +{}&\left.-\nu_f\left[mes(\Omega_i\cap\Omega'_j)+mes(\Omega_i\cap\Omega''_k)+mes(\Omega'_j\cap\Omega''_k)\right]\right\}+2\nu_f^3. +\end{array} +\end{equation} + +Здесь $\delta_{ijk}$ --- коэффициенты, принимающие значения 1 при совпадающих индексах и 0, если хотя бы один из трех индексов отличается от двух других различны (при $i,j=1\dots3$ эти коэффициенты являются обобщенными символами Кронекера). + +В данном выражении в отдельную группу выделены слагаемые, соответствующие приближению "малых расстояний", которое реализуется для $R=|\bf{\Delta r}|$, удовлетворяющих неравенству + +\begin{equation} +\label{mrasst} +R\le min[D_{min},d_{min}]. +\end{equation} + +В этом случае будет иметь место пересечение $i$-х включений образов с $i$-м включением прообраза ($\Omega_i\cap\Omega'_i\cap\Omega''_i$). В приближении "малых расстояний" соотношения (\ref{probr1r2}) значительно упрощаются: + +\begin{equation} +\label{probr1r2_m} +Prob({\bf r_1}\in\Omega_f\mid{\bf r_2}\in\Omega_f)=\frac{\sum_{i=1}^N mes(\Omega_i\cap\Omega'_i)}{\sum_{i=1}^N mes\Omega_f^{(i)}}=\frac{\sum_{i=1}^N mes(\Omega_i\cap\Omega'_i)}{\nu_fmes\Omega}. +\end{equation} + +Меры пересечений $mes(\Omega_i\cap\Omega'_i)$ в приближении "малых расстояний" находятся по следующим формулам: + +\begin{equation} +\label{mes_m} +mes(\Omega_i\cap\Omega'_i)=\left\{ +\begin{array}{ll} +2r_i^2\arccos\left(\frac{R}{2r_i}\right)-\frac{R}{2}\sqrt{r_i^2-\frac{R^2}{4}}, &R<2r_i;\\ +0, &R\ge 2r_i.\\ +\end{array}\right. +\end{equation} + +Если двухфазные композиты содержат включения одинакового радиуса ($r_i=r_j=r$), то из выражения (\ref{mes_m}) будет следовать равенство, полученное ранее авторами \cite{it2}: + +\begin{equation} +mes(\Omega_f\cap\Omega'_f)=\left\{ +\begin{array}{ll} +2r^2\arccos\left(\frac{R}{2r}\right)-2\sqrt{r^2-\frac{R^2}{4}}, &R<2r;\\ +\pi r^2, &R\equiv 0;\\ +0, &R=2r.\\ +\end{array}\right. +\end{equation} + +В приближении "малых расстояний" вид моментной функции $\tilde{K}_\lambda^{(3)}(R)$ значительно упрощается: + +\begin{equation} +\label{k3m} +\begin{array}{ll} +K_\lambda^{(3)}(R_1,R_2,\phi)&=\frac{\nu_f}{mes\Omega_f}\sum_{i=1}{N}\left\{mes(\Omega_i\cap\Omega'_i\cap\Omega''_i)-\nu_f\left[mes\Omega_i\cap\Omega'_i+\right.\right.{}\\ +{}&\left.\left.+mes(\Omega_i\cap\Omega''_i)+mes(\Omega'_f\cap\Omega''_f)\right]\right\}+2\nu_f^3. +\end{array} +\end{equation} + +Аналитическое соотношение в виде ряда (\ref{k3m}) для нормированных корреляционных функций случайных структур двухфазных композитов матричного типа в приближении "малых" расстояний позволяет получить точные выражения для производных этих функций при значениях аргумента, равных нулю. + +\pagebreak +\subsection{Условные двухточечные моментные функции третьего порядка} + +Условная двухточечная моментная функция третьего порядка может быть получена из безусловной трехточечной, если в ней приравнять друг другу два любых аргумента. Заменим в выражении (\ref{k3}) ${\bf r_2}$ на ${\bf r_1}$, в результате получим следующее выражение (\ref{k3usl}): + +\begin{equation} +\label{k3usl} +K_\lambda^{(3)}({\bf r_1,r_1,r_3})=\nu_f\left[-\nu_f-2\left(<\lambda({\bf r_1})\lambda({\bf r_3})>-\nu_f^2\right)\right]. +\end{equation} + +Можно увидеть, что в данном случае условный двухточечный момент третьего порядка может быть выражен через момент второго порядка. Примеры условных двухточечных моментов приведены на рисунке \ref{m2f}. + +\begin{figure}[!h] +\label{m2f} +\includegraphics[width=\textwidth]{ris/funct2} +\caption{Условные двухточечные моментные функции третьего порядка} +\end{figure} + +Принимая во внимание выражения (\ref{ml3}) и (\ref{probr1r2}) можно записать выражение для условного двухточечного момента следующим образом: + +\begin{equation} +\label{k3u_mes} +K_\lambda^{(3)}({\bf r_1,r_1,r_3})=\nu_f\left[-\nu_f-2\left(\frac{\sum_{i=1}^N\sum_{j=1}^N mes(\Omega_i\cap\Omega''_j)}{mes\hat{\Omega}}-\nu_f^2\right)\right], +\end{equation} + +\noindent или, в приближении малых расстояний: + +\begin{equation} +\label{k3u_mes_mr} +K_\lambda^{(3)}({\bf r_1,r_1,r_3})=\nu_f\left[-\nu_f-2\left(\frac{\sum_{i=1}^N mes(\Omega_i\cap\Omega''_i)}{mes\hat{\Omega}}-\nu_f^2\right)\right]. +\end{equation} + +Дифференцируя выражение (\ref{k3u_mes_mr}) можно получить выражение для производной условной моментной функции третьего порядка в точках, соответствующих нулевым значениям аргумента. Эта производная полностью совпадает с производной момента второго порядка: + +\begin{equation} +\label{proizv2} +\left.\frac{d}{dR}\tilde{K}_\lambda^{(3)}(R)\right|_{R=0}=-\frac{3}{2\pi(1-\nu_f)\left}\equiv +-\frac{3\aleph}{4\pi\nu_f(1-\nu_f)mes\hat{\Omega}}. +\end{equation} + +Здесь $R=|\bf{\Delta r}|$. Как видим, для плоских случайных структур искомая производная при любых объемных наполнениях имеет отрицательный знак, определяется отношением меры $\aleph=mes\partial\Omega_f$, связанной с межфазной границей (суммарный периметр) к площади прообраза $mes\hat{\Omega}$ ($mes\Omega_f = \nu_f mes\hat{\Omega})$. Кроме того, значение производной при корреляционной функции при $R = 0$ не зависит от направляющих косинусов углов ориентации вектора трансляции $\bf{\Delta r}$. + +Для случайных структур армирующие элементы которых имеют детерминированные размеры ($r_i = r)$, выражение (\ref{proizv2}) значительно упрощается: + +\begin{equation} +\label{proizv2rir} +\left.\frac{d}{dR}\tilde{K}_\lambda^{(3)}(R)\right|_{R=0}=-\frac{3}{2\pi(1-\nu_f)r}. +\end{equation} + +Как видим, значение производной в рассматриваемом частном случае определяется только объемным наполнением $\nu_f $ и характерным размером включений. + +\begin{table}[!h] +\label{table1} +\caption{Значение первой производной нормированных корреляционных функций $\frac{d}{dR}\tilde{K}_\lambda^{(3)}(R)$ для случайных структур однонаправленно армированных композитов с круглыми в поперечном сечении волокнами ($\nu_f=0.4$ и $d=0.25$)} +\begin{tabular}{|c|c|c|c|c|c|c|c|c|} +\hline +$k_D$&0,600&0,500&0,400&0,300&0,200&0,100&0,000\\ +\hline +Нормальный&–0,097&–0,104&–0,112&–0,121&–0,128&–0,132&-0,133\\ +\hline +Логнормальный&–0,096&–0,104&–0,114&–0,121&–0,128&–0,132&–0,133\\ +\hline +\end{tabular} +\end{table} + +Обратим внимание на то, что выражение (\ref{proizv2}) справедливо для структурно-неоднородных сред, армированных частицами произвольной формы и, что особенно важно, не зависят от взаимного расположения и связности включений. + +\begin{table}[!h] +\label{table2} +\caption{Значение первой производной нормированных корреляционных функций $\frac{d}{dR}\tilde{K}_\lambda^{(3)}(R)$ для случайных структур однонаправленно армированных волокнистых композитов с предельным для $d/=0$ наполнением} +\begin{tabular}{|c|c|c|c|c|c|c|c|} +\hline +$k_D$&0,600&0,500&0,400&0,300&0,200&0,100&0,000\\ +\hline +Нормальный&–0,352&–0,375&–0,397&–0,420&–0,436&–0,443&-0.499\\ +\hline +Логнормальный&–0,370&–0,392&–0,411&–0,427&–0,437&–0,443&–0,499\\ +\hline +\end{tabular} +\end{table} + +Поскольку знаменатель отношения $/$ входящего в выражение (\ref{proizv2}), чувствительны к изменению радиусов включений, различных плоских случайных структур двухфазных однонаправленно армированных композитов матричного типа оценим виляние типа закона распределения характерных размеров волокон на значения производных нормированных корреляционных функций. Результаты, представленные в табл. \ref{table1} свидетельствуют, что при заданной объемной доле волокон при коэффициентах вариации диаметров $k_D$ от 0,0 до 0,6 значения производных не зависят от типа закона распределения. Вместе с тем, для структур с предельным для заданного $k_D$ объемным наполнением при $k_D>4$ значения производных существенно зависит от типа закона распределения (табл. \ref{table2}). Это прежде всего связано с более неоднородным фракционным составом материала, который предопределяется несимметричным логнормальным законом распределения диаметров волокон. + +\pagebreak +\subsection{Выводы по разделу} + +Получены аналитические выражения в виде рядов для безусловных трехточечных и условных двухточечных моментных функций третьего порядка случайной структуры двухфазных однонаправленно армированных композитов матричного типа. + +Получены точные выражения для производных условных двухточечных моментных функций третьего порядка случайной структуры двухфазных однонаправленно армированных композитов, которые определяются отношением мер, связанных с межфазной границей (поверхностью) и частицами армирующего наполнителя. + +Подготовлен пакет прикладных программ для построения моментных функций второго и третьего порядков (для одно- и многопроцессорных систем). diff --git a/sqr.tex b/sqr.tex new file mode 100755 index 0000000..e0c6d77 --- /dev/null +++ b/sqr.tex @@ -0,0 +1,45 @@ + +\documentclass{report} +%\batchmode +\usepackage{latexsym} +\usepackage[T2A]{fontenc} +\usepackage[cp1251]{inputenc} +\usepackage[russian]{babel} +\usepackage{mathtext} +\righthyphenmin=2 +\oddsidemargin=0pt +\textwidth=18cm +\topmargin=0cm +\textheight=23cm +\begin{document} + +\begin{equation} +\label{first} +\left\{\begin{array}{l}(x-x_1)^2+(y-y_1)^2=R^2\\y=\tg(\alpha)(x-x_c)+y_c\end{array}\right. +\end{equation} + +\begin{equation} +\label{second} +x^2-2\cdot x \cdot x_1 + x_1^2 + \left[\tg(\alpha)x-\tg(\alpha)x_c+y_c-y_1\right]^2=R^2 +\end{equation} + +\begin{equation} +\label{third} +x^2-2\cdot x \cdot x_1 + x_1^2+(\tg(\alpha)x)^2+2(\tg(\alpha)x)(y_c-y_1-\tg(\alpha)x_c)+(y_c-y_1-\tg(\alpha)x_c)^2=R^2 +\end{equation} + +\begin{equation} +\label{fourth} +x^2(1+\tg^2(\alpha))+x\left(2\tg(\alpha)(y_c-y_1-\tg(\alpha)x_c)-2x_1\right)+x_1^2+(y_c-y_1-\tg(\alpha)x_c)^2-R^2=0 +\end{equation} + +\begin{equation} +\label{ABC} +\left\{\begin{array}{l} +a=1+\tg^2(\alpha)\\ +b=2\tg(\alpha)(y_c-y_1-\tg(\alpha)x_c)\\ +c=x_1^2+(y_c-y_1-\tg(\alpha)x_c)-R^2 +\end{array}\right. +\end{equation} + +\end{document} diff --git a/titul.odt b/titul.odt new file mode 100755 index 0000000..5065f13 Binary files /dev/null and b/titul.odt differ diff --git a/titul.sxw b/titul.sxw new file mode 100755 index 0000000..215aa10 Binary files /dev/null and b/titul.sxw differ diff --git a/vved.tex b/vved.tex new file mode 100755 index 0000000..2950c93 --- /dev/null +++ b/vved.tex @@ -0,0 +1,18 @@ +\pagebreak +\section{Введение} + +Прогнозирование эффективных деформационных свойств и определение статистических характеристик случайных полей напряжений и деформаций в компонентах волокнистых и дисперсно-упрочненных композитов связаны с необходимостью решения стохастически нелинейных краевых задач, для построения приближенных решений которых (например, полного корреляционного приближения) требуются описывающие многочастичное взаимодействие в системе армирующих элементов моментные функции структурных модулей упругости второго, третьего, четвертого и пятого порядков. + +При построении приближенных решений нелинейных стохастических краевых задач используются различные (но очень часто не вполне обоснованные) гипотезы о характере многочастичного взаимодействия в ансамбле частиц армирующего наполнителя (например, предельная локальность) и аппроксимации центральных моментов случайного индикатора. Поэтому, во-первых, существует потребность в идентификации и "отбраковке" соответствующих статистических моделей механики структурно-неоднородных сред. Во-вторых, "традиционные" алгоритмы построения условных и безусловных многоточечных моментных функций, которые ранее были использованы для обработки микрошлифов металлов \cite{bvv} и стеклопластиков \cite{vs}, требуют существенной модификации. Это обусловлено тем, что реализация данных алгоритмов связана с построением вспомогательных координатных сеток, определением принадлежности каждого узла этих сеток одной из фаз материала, требует значительных аппаратных и программных затрат. + +Дипломный проект является продолжением серии работ по исследованию случайных структур двухфазных однонаправленно армированных композитов, которые проводились авторами \cite{zlt,zlt2,zlt3,zltt}. В этих работах были неполностью рассмотрены моментные функции третьего порядка, которые позволяют описать особенности геометрии частиц армирующих элементов, составляющих случайную структуру композиционных материалов. Соответственно {\it целью} дипломного проекта является развитие математических основ решения стохастических краевых задач механики структурно-неоднородных сред, получение и анализ аналитических выражений для определения моментов третьего порядка. + +В данной работе были получены аналитические выражения для безусловных трехточечных и условных двухточечных моментов третьего порядка, получены выражения для производных условных двухточечных моментов третьего порядка, а также разработан и протестирован программный пакет для многопроцессорной системы МВС-1000 для построения моментных функций. + +По теме дипломного проекта опубликованы две печатные работы \cite{zdp,zdpt}. Основные результаты проведенных исследований докладывались и обсуждались на 3-й Всероссийской научно-технической конференции "Математическое моделирование и краевые задачи" (Самара, 2006), а также на "Поздеевских чтениях" (Пермь, 2006 г.). + +Дипломный проект выполнен в соответствии с планом научных исследованиями, проводимыми на кафедре Механика композиционных материалов и конструкций ПГТУ. + +Теоретические разработки нашли отражение в спецкурсах "Методы исследования микроструктуры и свойств композитов"; "Синтез и анализ случайных структур композитов", читаемых в Пермском государственном техническом университете студентам специальности 121000 --- "Конструирование и производство изделий из композиционных материалов". + +Автор выражают признательность научному руководителю, доценту кафедры Механика композиционных материалов и конструкций ПГТУ, к.ф.-м.н. А.~В.~Зайцеву, а также С.~В.~Мельникову и Е.~А.~Митюшову за внимание к работе и обсуждение представленных результатов. diff --git a/zakl.tex b/zakl.tex new file mode 100755 index 0000000..5903f04 --- /dev/null +++ b/zakl.tex @@ -0,0 +1,13 @@ +\pagebreak +\section{Заключение} + +Основные результаты дипломного проекта заключаются в следующем: + +\begin{enumerate} +\item Получены новые аналитические выражения в виде рядов для безусловных трехточечных и условных двухточечных моментных функций третьего порядка случайной структуры двухфазных однонаправленно армированных композитов матричного типа, позволяющие в явном виде выделить слагаемые, соответствующие приближению "малых" расстояний. Определены производные условных двухточечных моментных функций и проанализировано влияние типа закона распределения диаметров включений на угол наклона корреляционных функций в точке, соответствующей нулевому значению аргумента, а также разработан и протестирован пакет прикладных программ для многопроцессорной системы МВС-1000 для построения моментных функций. + +\item В работе проведен расчет экономической эффективности программного продукта для построения моментных функций третьего порядка случайной структуры двухфазных однонаправленно армированных композитов. + +\item Проведен расчет допустимого уровня шума на рабочем месте инженера-программиста. + +\end{enumerate}