From 04dac789e1319b9e9d55edd02eddbf2acfca5bda Mon Sep 17 00:00:00 2001 From: "Denis V. Dedkov" Date: Sun, 30 Sep 2012 19:24:00 +0600 Subject: [PATCH] Dissertation was finished --- bibliography.bib | 6 +- c1.tex | 16 +- c2.tex | 94 +- c3.tex | 580 +++- common.tex | 48 +- disser.kilepr | 26 +- end.tex | 25 + fig/concentrators/v2/scheme1/d1d3.png | Bin 0 -> 543579 bytes fig/concentrators/v2/scheme1/d2d4.png | Bin 0 -> 537480 bytes fig/concentrators/v2/scheme2/d1d3.png | Bin 0 -> 543773 bytes fig/concentrators/v2/scheme2/d2d4.png | Bin 0 -> 541671 bytes fig/concentrators/v2/scheme3/d1d3.png | Bin 0 -> 544666 bytes fig/concentrators/v2/scheme3/d2d4.png | Bin 0 -> 533555 bytes fig/geometry/v2/bc.png | Bin 0 -> 196896 bytes fig/geometry/v2/bc.svg | 3779 +++++++++++++++++++++++++ fig/geometry/v2/d1_in_matrix.png | Bin 0 -> 175513 bytes fig/geometry/v2/d1d3.png | Bin 0 -> 349587 bytes fig/geometry/v2/d2d4.png | Bin 0 -> 344502 bytes fig/geometry/v2/regular_slice.png | Bin 0 -> 99343 bytes fig/mesh/v2/fibers.png | Bin 0 -> 157429 bytes fig/mesh/v2/matrix.png | Bin 0 -> 298016 bytes fig/mesh/v2/matrix_zoom.png | Bin 0 -> 89505 bytes fig/vmis_v2_s1.png | Bin 0 -> 149977 bytes fig/vmis_v2_s2.png | Bin 0 -> 154893 bytes fig/vmis_v2_s3.png | Bin 0 -> 132364 bytes intro.tex | 22 +- stress_concentartors.tex | 1 + 27 files changed, 4372 insertions(+), 225 deletions(-) create mode 100644 end.tex create mode 100644 fig/concentrators/v2/scheme1/d1d3.png create mode 100644 fig/concentrators/v2/scheme1/d2d4.png create mode 100644 fig/concentrators/v2/scheme2/d1d3.png create mode 100644 fig/concentrators/v2/scheme2/d2d4.png create mode 100644 fig/concentrators/v2/scheme3/d1d3.png create mode 100644 fig/concentrators/v2/scheme3/d2d4.png create mode 100644 fig/geometry/v2/bc.png create mode 100644 fig/geometry/v2/bc.svg create mode 100644 fig/geometry/v2/d1_in_matrix.png create mode 100644 fig/geometry/v2/d1d3.png create mode 100644 fig/geometry/v2/d2d4.png create mode 100644 fig/geometry/v2/regular_slice.png create mode 100644 fig/mesh/v2/fibers.png create mode 100644 fig/mesh/v2/matrix.png create mode 100644 fig/mesh/v2/matrix_zoom.png create mode 100644 fig/vmis_v2_s1.png create mode 100644 fig/vmis_v2_s2.png create mode 100644 fig/vmis_v2_s3.png diff --git a/bibliography.bib b/bibliography.bib index a3f2881..11445c2 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -1,6 +1,7 @@ @ARTICLE{bib:surovikin, Author = {Суровикин~В.~Ф. and Суровикин~Ю.~В. and Цеханович~М.~С.}, - Title = {Новые направления в технологии получения углерод-углеродных материалов. Применение углерод-углеродных материалов.}, + Title = {Новые направления в технологии получения углерод-углеродных +материалов. Применение углерод-углеродных материалов.}, Journal = {Рос. хим. ж-л. (Ж-л Рос. хим. об-ва им. Д.~И.~Менделеева}, Volume = {4}, Pages = {111--118}, @@ -64,7 +65,8 @@ @ARTICLE{bib:dedkov1, Author = {Дедков~Д.~В. and Зайцев~А.~В. and Ташкинов~А.~А. }, - Title = {Концентрация напряжений в слое тканого композита с закрытыми внутренними технологическими порами}, + Title = {Концентрация напряжений в слое тканого композита с закрытыми +внутренними технологическими порами}, Journal = {Вестник ПНИПУ. Механика}, Volume = {4}, Pages = {29--36}, diff --git a/c1.tex b/c1.tex index 8d5096f..f3ec7c9 100644 --- a/c1.tex +++ b/c1.tex @@ -1,15 +1,7 @@ \chapter{Зависимость деформационных и прочностных свойств тканых композитов с поликристаллической матрицей от наличия локальных концентраторов напряжений} -В главе рассматриваются технологические операции изготовления конструкций из -тканых композиционных материалов с поликристаллической матрицей. Описывается -процесс изготовления волокон, рассматриваются типы тканей и способы их -производства, а так же совмещение тканого каркаса с поликристаллической -матрицей. - -Также описаны методы контроля качества конструкций их тканых материалов с -поликристаллической матрицей и типы дефектов, возникающие во время -технологических процессов, выявляемых с их помощью. +В главе\infirsttext \section{Технологические операции изготовления конструкций из тканых композиционных материалов, приводящие к появлению локальных концентраторов @@ -363,7 +355,7 @@ $6\dots100$~МПа при температуре $550\dots 650^\circ\mathrm{C}$. матрицей} Особенностью тканых композитов с поликристаллической матрицей является то, что -наряду сдефектами, присущими традиционным материалам, такими как трещины, поры, +наряду с дефектами, присущими традиционным материалам, такими как трещины, поры, посторонние включения, могут образовываться дефекты, характерные только для данного вида материала, связанные с особенностями структуры ткани и методом формирования матрицы. Такие дефекты могут быть различными для каждого этапа @@ -371,7 +363,7 @@ $6\dots100$~МПа при температуре $550\dots 650^\circ\mathrm{C}$. Дефекты связанные с отклонениями от расчетных параметров структуры возникают на этапе изготовления ткани. К числу таких дефектов можно отнести отклонения в -напралении армирующих нитей, пропуски нитей в направлении армирования (рис. +направлении армирующих нитей, пропуски нитей в направлении армирования (рис. \ref{fig:c1:no_fiber}). На этапе формирования матрицы могут возникнуть дефекты связанные с отклонением @@ -381,7 +373,7 @@ $6\dots100$~МПа при температуре $550\dots 650^\circ\mathrm{C}$. трещины и внутренние поры (рис. \ref{fig:c1:pore}). Разрывы волокон утка или основы (рис. \ref{fig:c1:break}) могут возникать на -каждом из этапов: на этапе формирования ткани --- вследствии очень тесного +каждом из этапов: на этапе формирования ткани --- вследствие очень тесного размещения нитей, в процессе сшивки слоев ткани при формировании конструкции, на этапе формирования матрицы --- из-за внутренний напряжений, возникающих в материале во время его изготовления. diff --git a/c2.tex b/c2.tex index e0dbd3d..17472d8 100644 --- a/c2.tex +++ b/c2.tex @@ -1,10 +1,13 @@ \chapter{Локальные поля напряжений и деформаций в представительных объемах тканого композита с поликристаллической матрицей} +В главе\insecondtext + \section{Математическая модель упруго-хрупкого поведения тканого композита с поликристаллической матрицей} \subsection{Геометрическая модель слоя тканого композита} +\label{c1:geometry} Будем моделировать слой тканого композита с армирующим каркасом полотняного переплетения образованного волокнами круглого поперечного сечения @@ -36,7 +39,7 @@ NUclear REactor SIMulations), предназначенного для полно рис.~\ref{fig:c2:geometry}, формируется сегмент волокна, из которого, в свою очередь, с помощью операций трансляции и зеркалирования формируется фрагмент ткани (рис.~\ref{fig:c2:regular}~а). Матрица моделируется с помощью операции -вычитания из твердотельного прямоугольного параллилепипеда фрагмента ткани, +вычитания из твердотельного прямоугольного параллелепипеда фрагмента ткани, после чего матрица и фрагмент ткани совмещаются для получения твердотельной модели тканого композита с поликристаллической матрицей (рис.~\ref{fig:c2:regular}~б) \cite{bib:salome:geom, @@ -174,7 +177,7 @@ C_{ijkl}^{m} \left[ 1-\lambda({\bf r}) \right ] \right\} \label{fig:c2:b_cond} \end{figure} -Полости, вызванные наличием локальных дефектов и незаполенные матрицей имеют +Полости, вызванные наличием локальных дефектов и незаполненные матрицей имеют внутреннюю поверхность $\Gamma_8$, на которой отсутствуют ограничения на перемещения, а сама поверхность свободна от напряжений: @@ -217,17 +220,7 @@ C_{ijkl}^{m} \left[ 1-\lambda({\bf r}) \right ] \right\} матрицы слоя модельного тканого композита полотняного переплетения. Конечно-элементная сетка волокон представлена на рис.~\ref{fig:mesh:fibers}. -Степень дискретизации выбиралась таким образом, чтобы чтобы полученные значения -структурных перемещений, деформаций и напряжений в слое тканого композита без -локальных дефектов и с несовершенствами ни качественно, ни количественно не -изменялись при уменьшении характерных размеров конечных элементов. - -Из таблицы~{\ref{tab:convergence}}, в которой показана зависимость максимальных -интенсивностей напяжений от количества конечных элементов, видно, что -расхождение между двумя последними строками не превышает $1\%$. Это говорит о -достаточной степени дискретизации модели. - -\begin{figure}[ht!] +\begin{figure}[!ht] \centering \includegraphics[width=17cm]{mesh/v1/matrix} \caption{Пример дискретизации матрицы} @@ -241,6 +234,16 @@ C_{ijkl}^{m} \left[ 1-\lambda({\bf r}) \right ] \right\} \label{fig:mesh:fibers} \end{figure} +Степень дискретизации выбиралась таким образом, чтобы чтобы полученные значения +структурных перемещений, деформаций и напряжений в слое тканого композита без +локальных дефектов и с несовершенствами ни качественно, ни количественно не +изменялись при уменьшении характерных размеров конечных элементов. + +Из таблицы~{\ref{tab:convergence}}, в которой показана зависимость максимальных +интенсивностей напряжений от количества конечных элементов, видно, что +расхождение между двумя последними строками не превышает $1\%$. Это говорит о +достаточной степени дискретизации модели. + \begin{table}[ht!] \caption{Зависимость максимальных интенсивностей напряжений от количества \newline конечных элементов} @@ -278,21 +281,21 @@ C_{ijkl}^{m} \left[ 1-\lambda({\bf r}) \right ] \right\} & Тетраэдральные элементы & Гексаэдральные элементы \\ \hline \hline - Идеальная периодическая структура & 298 255 & 77 760 \\ + Идеальная периодическая структура & 298~255 & 77~760 \\ \hline - Тунельная пора & 285 664 & 69 984 \\ + Тунельная пора & 285~664 & 69~984 \\ \hline - Туннельная пора с доуплотнением & 266 314 & 69 984 \\ + Туннельная пора с доуплотнением & 266~314 & 69~984 \\ \hline - Разрыв волокна основы & 285 466 & 75 168 \\ + Разрыв волокна основы & 285~466 & 75~168 \\ \hline - Разрыв волокна основы с доуплотнением & 296 499 & 75 168 \\ + Разрыв волокна основы с доуплотнением & 296~499 & 75~168 \\ \hline - Разрыв волокон основы и утка & 279 276 & 72 576 \\ + Разрыв волокон основы и утка & 279~276 & 72~576 \\ \hline - Разрыв волокон основы и утка с доуплотнением & 276 175 & 72 576 \\ + Разрыв волокон основы и утка с доуплотнением & 276~175 & 72~576 \\ \hline - Внутренняя технологическая пора & 287 934 & 77 760 \\ + Внутренняя технологическая пора & 287~934 & 77~760 \\ \hline \end{tabular} \label{tab:discr} @@ -303,7 +306,7 @@ C_{ijkl}^{m} \left[ 1-\lambda({\bf r}) \right ] \right\} Упругие модули поликристаллической матрицы были выбраны следующими: $E_m = 0{,}28$~ГПа и коэффициент Пуассона $\nu_m = 0,40$. -Распределения интесивностей напряжений в слое тканого композита с идеальной +Распределения интенсивностей напряжений в слое тканого композита с идеальной периодической структурой, полученные в ходе решения задачи показаны на рис.~\ref{fig:vmis_v1_s1}. @@ -317,7 +320,7 @@ C_{ijkl}^{m} \left[ 1-\lambda({\bf r}) \right ] \right\} Из рисунка видно, что распределение искомых полей в рассматриваемом случае удовлетворяют условиям симметрии и периодичности геометрической модели и приложенной внешней нагрузке, что говорит о корректно построенной -геометрической модели и корректности полученного численного решенеия. +геометрической модели и корректности полученного численного решения. Максимальных значений интенсивность напряжений достигает в местах наибольшей кривизны волокон. @@ -333,7 +336,7 @@ C_{ijkl}^{m} \left[ 1-\lambda({\bf r}) \right ] \right\} простым и, в то же время, мощным интерпретируемым объектно-ориентированным языком программирования. Он предоставляет структуры данных высокого уровня, имеет изящный синтаксис и использует динамический контроль типов, что делает -его деальным языком для быстрого написания различных приложений, работающих +его идеальным языком для быстрого написания различных приложений, работающих на большинстве распространенных платформ \cite{bib:rossum}. Максимальные значения коэффициентов концентрации напряжений представлены в @@ -391,7 +394,7 @@ C_{ijkl}^{m} \left[ 1-\lambda({\bf r}) \right ] \right\} На рис.~\ref{fig:k_d1d2_s1}~--~\ref{fig:k_d5_s1} показаны распределения коэффициентов концентрации интенсивностей напряжений в слое тканого композита с -искривленными волокнами и поликристалической матрицей при наличии различных +искривленными волокнами и поликристаллической матрицей при наличии различных типов технологических дефектов и с учётом дополнительной пропитки композита материалом матрицы. Расположение областей, в которых интенсивность напряжений достигает максимальных значений в местах, где искривленные волокна основы или @@ -528,19 +531,9 @@ $\sigma_{12}$ и $\sigma_{23}$ и нормальная составляющая На рис.~\ref{fig:k_d1d2_s2}~--~\ref{fig:k_d5_s2} показаны распределения коэффициентов концентрации интенсивностей напряжений в слое тканого композита с -искривленными волокнами и поликристалической матрицей при наличии различных +искривленными волокнами и поликристаллической матрицей при наличии различных типов технологических дефектов и с учётом дополнительной пропитки композита -материалом матрицы под воздействием сдвиговых нагрузок. Вблизи локальных -дефектов интенсивности напряжений превышают соответствующие интенсивности -напряжений определенное для композита идеальной периодической структуры в -$1{,}2$ раза при наличии внутренней технологической поры, в $1{,}3$ раза для -слчая пропуска или разрыва волокна основы и в $1{,}6$ раз для одновременного -разрыва волокон основы и утка. При этом, в случае разрыва волокна основы -или волокон основы и утка, значение коэффициентов концентрации интенсивностей -напряжений ожет быть снижено до $1{,}2$ и $1{,}5$ соответственно, с помощью -дополнительных операций доуплотнения поликристаллической матрицы. - -\pagebreak +материалом матрицы под воздействием сдвиговых нагрузок. \begin{figure}[ht!] \includegraphics[width=17cm]{concentrators/v1/scheme2/d1d2} @@ -576,6 +569,16 @@ $1{,}2$ раза при наличии внутренней технологич \label{fig:k_d5_s2} \end{figure} +Вблизи локальных дефектов интенсивности напряжений превышают соответствующие +интенсивности напряжений определенное для композита идеальной периодической +структуры в $1{,}2$ раза при наличии внутренней технологической поры, в $1{,}3$ +раза для случая пропуска или разрыва волокна основы и в $1{,}6$ раз для +одновременного разрыва волокон основы и утка. При этом, в случае разрыва волокна +основы или волокон основы и утка, значение коэффициентов концентрации +интенсивностей напряжений может быть снижено до $1{,}2$ и $1{,}5$ +соответственно, с помощью дополнительных операций доуплотнения +поликристаллической матрицы. + \subsection{Коэффициенты концентрации напряжений при одноосном растяжении} В случае, если граничные условия \ref{eq:b_cond} в краевой задаче @@ -657,16 +660,14 @@ $1{,}2$ раза при наличии внутренней технологич $\sigma_{13}$ и $\sigma_{23}$ и нормальная составляющая $\sigma_{33}$. Исключение составляет случай, когда в слое тканого композита присутствует внутренняя технологическая пора. В этом случае значение касательной -компоненты тензора напряжений превышает соответсвующее значение в и деальной +компоненты тензора напряжений превышает соответствующее значение в и идеальной периодической структуре в $4{,}59$ раз. Распределение коэффициентов концентрации интенсивностей напряжений в слое -тканого композита с искривленными волокнами и поликристалической матрицей при +тканого композита с искривленными волокнами и поликристаллической матрицей при наличии различных типов технологических дефектов и с учётом дополнительной пропитки композита материалом матрицы под воздействием сдвиговых нагрузок -представлены на рис.~\ref{fig:k_d1d2_s3}~--~\ref{fig:k_d5_s3}. - -\pagebreak +представлены на рис.~\ref{fig:k_d1d2_s3}~--~\ref{fig:k_d5_s3}. \begin{figure}[ht!] \includegraphics[width=17cm]{concentrators/v1/scheme3/d1d2} @@ -706,7 +707,7 @@ $\sigma_{13}$ и $\sigma_{23}$ и нормальная составляющая Из рисунков видно что вблизи локальных дефектов интенсивности напряжений превышают соответствующие интенсивности напряжений определенное для композита идеальной периодической структуры в $1{,}2$ раза при наличии внутренней -технологической поры или разрыва волокна основы, в $1{,}3$ раза для слчая +технологической поры или разрыва волокна основы, в $1{,}3$ раза для случая пропуска или разрыва волокна основы и в $1{,}4$ раз для одновременного разрыва волокон основы и утка. При этом, в случае пропуска или разрыва волокна основы, значение коэффициентов концентрации интенсивностей напряжений может быть @@ -718,15 +719,16 @@ $\sigma_{13}$ и $\sigma_{23}$ и нормальная составляющая \begin{enumerate} \item Построена твердотельная модель фрагмента слоя тканого композита с -искривленными волокнами и поликристалической матрицей с идеальной +искривленными волокнами и поликристаллической матрицей с идеальной периодической структурой и локальными технологическими дефектами, такими как пропуск волокна основы, разрыв волокна основы, разрыв волокон основы и утка и внутренняя технологическая пора. \item Получены численные решения краевых задач на двухосное растяжение в плоскости слоя, чистый сдвиг и одноосное растяжение в направлении утка. \item Вычислены безразмерные коэффициенты концентрации напряжений, вызванные -наличием локальных технологическх дефектов в виде пропуска волокна основы, +наличием локальных технологических дефектов в виде пропуска волокна основы, разрыва волокна основы, разрыва волокон основы и утка, а также внутренней технологической поры. - \item Определены механизмы инициирующие разрушение матрицы. + \item Определены механизмы инициирующие разрушение матрицы в слое тканого +композита с искривленными волокнами. \end{enumerate} diff --git a/c3.tex b/c3.tex index 86d44c1..3759394 100644 --- a/c3.tex +++ b/c3.tex @@ -1,189 +1,469 @@ \chapter{Влияние локальных полей напряжений на прочностные свойства тканых композитов с поикристаллической матрицей с учётом трения между волокнами} +В главе\inthirdtext + \section{Математическая модель упруго-хрупкого поведения слоят тканого композита с поликристаллической матрицей при наличии контакта с трением между волокнами} -В случае, если в модельном материале не исключается возможность контакта -нитей основы и утка, на соответствующих контактных поверхностях -$\Gamma_9$ (положение и геометрия которых считается заданными и неизменными -в процессе нагружения слоя) будем считать справедливыми условия контакта -с кулоновским трением. На $\Gamma_9$ следует задать 2 условия: +\subsection{Геометрическая модель слоя тканого композита с контактом между +волокнами} -\noindent если $\left[\sigma_{n\tau} {\bf (r)} -\right] |_{\Gamma_9^{+}} < \left[ {f | \sigma_{nn} {\bf (r)} |} -\right] |_{\Gamma_9^{-}}$, то +В процессе изготовления тканого композиционного материала с поликристаллической +матрицей не всегда удается исключить соприкосновение нитей основы и утка, +вследствие чего в конструкции могут возникать такие локальные дефекты как +разрывы нитей основы, разрывы нитей основы и утка, а также внутренние +технологические поры. Поэтому необходимо построение моделей, где волокна основы +и утка не всегда окружены гарантированной прослойкой поликристаллической +матрицы. + +Геометрические параметры модели аналогичны указанным в +разделе~\ref{c1:geometry}, за исключением того что расстояние между волокнами в +точках максимальных кривизн равно нулю (рис.~\ref{fig:c3:fibers}), а в матрице, +вблизи максимальных кривизн волокон всегда присутствуют внутренние +технологические поры из-за невозможности заполнить это простаранство материалом +матрицы (рис.~\ref{fig:c3:matrix}). + +\begin{figure}[ht] + \includegraphics[width=17cm]{geometry/v2/regular_slice} + \caption{Фрагмент слоя ткани с контактом между волокнами} + \label{fig:c3:fibers} +\end{figure} + +\begin{figure}[ht!] + \centering + \includegraphics[width=10cm]{geometry/v2/matrix} + \caption{Фрагмент поликристаллической матрицы слоя тканого композита с +внутренними технологическими порами} + \label{fig:c3:matrix} +\end{figure} + +В качестве дефектов, вызывающих концентрации напряжений будем рассматривать +типичные дефекты, возникающие вследствии очень плотного расположения волокон +--- разрыв волокна основы (рис.~\ref{fig:c3:d1d3}~а) и разрывы волокон основы и +утка (рис.~\ref{fig:c3:d2d4}~а). Кроме того рассмотрим случаи когда пора в +матрице, образованная дефектом заполняется материалом матрицы в ходе +дополнительных технологических операций (рис.~\ref{fig:c3:d1d3}~б и +\ref{fig:c3:d2d4}~б). + +\begin{figure} + \includegraphics[width=17cm]{geometry/v2/d1d3} + \caption{Разрыв волокна основы в тканом композите с поликристаллической +матрицей при наличии контакта между волокнами~(а) с дополнительной +пропиткой~(б)} + \label{fig:c3:d1d3} +\end{figure} + +\begin{figure} + \includegraphics[width=17cm]{geometry/v2/d2d4} + \caption{Разрыв волокон основы и утка в тканом композите с поликристаллической +матрицей при наличии контакта между волокнами~(а) с дополнительной +пропиткой~(б)} + \label{fig:c3:d2d4} +\end{figure} + + +\subsection{Постановка краевой задачи теории упругости при наличии контакта с +трением} + +Краевая задача теории упругости для случая когда в материале возникает контакт +с трением между волокнами основы и утка в местах наибольших кривизн +волокон аналогична краевой задаче \ref{eq:Eqvilibrium}~--~\ref{eq:Guck} с +граничными условиями \ref{eq:b_cond}~--~\ref{eq:b_cond_free}, за исключением +того, что соответствующих контактных поверхностях $\Gamma_9$ +(рис.~\ref{fig:c3:bc}) необходимо задать дополнительные граничные условия. + +\begin{figure}[!ht] + \centering + \includegraphics[width=10cm]{geometry/v2/bc} + \caption{Граничные условия краевой задачи теории упругости при наличии контакта +с трением между волокнами} + \label{fig:c3:bc} +\end{figure} + +Положение и геометрия контактных поверхностей считается заданными и неизменными +в процессе нагружения слоя, кроме того будем считать справедливыми условия +контакта с кулоновским трением, тогда на $\Gamma_9$ следует задать 2 условия: + +\noindent если $\left[\sigma_{n\tau} {\bf (r)} \right] |_{\Gamma_9^{+}} < \left[ +{f | \sigma_{nn} {\bf (r)} |} \right] |_{\Gamma_9^{-}}$, то \begin{equation} \left[\sigma_{n\tau} {\bf (r)} n_{n} \right] |_{\Gamma_9^{+}} = \left[\sigma_{n\tau} {\bf (r)} n_{n} \right] |_{\Gamma_9^{-}}, \quad -\left[u_n {\bf (r)}\right]|_{\Gamma_9^{+}} = \left[u_n {\bf -(r)}\right]|_{\Gamma_9^{-}} , +\left[u_n {\bf (r)}\right]|_{\Gamma_9^{+}} = +\left[u_n {\bf (r)}\right]|_{\Gamma_9^{-}} , \label{eq:b_cond_Colomb_1} \end{equation} -\noindent а, если $\left[\sigma_{n\tau} {\bf (r)} \right] -|_{\Gamma_9^{+}} \geq \left[ {f | \sigma_{nn} {\bf (r)} |} \right ] -|_{\Gamma_9^{-}}$, то +\noindent а, если $\left[\sigma_{n\tau} {\bf (r)} \right]|_{\Gamma_9^{+}} \geq +\left[ {f | \sigma_{nn} {\bf (r)} |} \right]|_{\Gamma_9^{-}}$, то \begin{equation} -\left[\sigma_{n\tau} {\bf (r)} \right] |_{\Gamma_9^{+}} \geq \left[ -f | \sigma_{nn} {\bf (r)} | \right ] |_{\Gamma_9^{-}}, \quad -\left[u_n {\bf (r)}\right]|_{\Gamma_9^{+}} = \left[u_n {\bf -(r)}\right]|_{\Gamma_9^{-}} , \label{eq:b_cond_Colomb_2} +\left[\sigma_{n\tau} {\bf (r)} \right] |_{\Gamma_9^{+}} \geq +\left[f|\sigma_{nn} {\bf (r)}| \right] |_{\Gamma_9^{-}},\quad +\left[u_n {\bf (r)}\right]|_{\Gamma_9^{+}} = +\left[u_n {\bf (r)}\right]|_{\Gamma_9^{-}}, +\label{eq:b_cond_Colomb_2} \end{equation} \noindent где $f$ --- статический коэффициент трения, а индексы $n$ и $\tau$ --- определяют направление внешней нормали и касательной к поверхности $\Gamma_9$. -Внутренние поры имеют место в слое композита в случае, если не -исключается соприкосновение волокон. Это герметичные полости, недоступные -для материала матрицы, +В случае если в слое тканого композита с поликристаллической матрицей не +исключено соприкосновение волокон, вблизи мест с максимальной +кривизной волокон остаются герметичные полости, незаполненные материалом +матрицы. На поверхностях этих пор отсутствуют ограничения на перемещения, а +сама поверхность свободна от напряжений. Граничные условия на этих поверхностях +аналогичны граничным условиям \ref{eq:b_cond_free}. -\begin{figure}[!ht] - \centering -% \includegraphics[width=0.83\linewidth]{img/matrix} - \caption{Фрагмент тканого композита с искривленными волокнами} - \label{fig:matrix} +\subsection{Численное решение краевой задачи упругости} + +Для численного решения задачи равнокомпонетного растяжения тканого композита с +искривленными волокнами и поликристаллическкой матрицей в плоскости слоя +необходимо задать свойства материала. Модуль Юнга $E_f += 280$~ГПа и коэффициент Пуассона $\nu_f = 0,20$ волокон зададим +в соответствии с данными работы \cite{bib:tarnapolsky}, а упругие модули +поликристаллической матрицы выберем следующими: $E_m = 0,28$~ГПа и +коэффициент Пуассона $\nu_m = 0,40$. Так как между волокнами присутствует +контакт с трением, необходимо задать статический коэффициент трения +\cite{bib:code-aster:contact}: $f = 0,12$, который соответствует случаю +скольжения волокна по поверхности поликристаллической матрицы. + +Матрицу будем разбивать 14-узловыми тетраэдральными элементами +(рис.~\ref{fig:c3:mesh:matrix}), а волокно --- 20-узловыми гексаэдральными +элементами (рис.~\ref{fig:c3:mesh:fibers}). Степень дискретизации +конечно-элементной сетки будем выбирать таким образом, чтобы дальнейшее +ументшение характерных размеров элементов ни качественно ни количественно не +влияло на значения структурных перемещений, деформаций и напряжений в слое +тканого композита. Параметры сеток, удовлетворяющих этим условиям показаны в +таблице~\ref{tab:c3:discr}. + +\begin{figure}[ht] + \includegraphics[width=17cm]{mesh/v2/matrix} + \caption{Пример дискретизации матрицы} + \label{fig:c3:mesh:matrix} \end{figure} -На рис.~\ref{fig:sigma} показаны распределения интенсивностей напряжений -в искривленных нитях основы и утка при равнокомпонентном двухосном -однородном деформировании слоя модельного тканого композита -идеальной периодической структуры в собственной плоскости. Модуль Юнга $E_f -= 280$~ГПа и коэффициент Пуассона $\nu_f = 0,20$ волокон соответствовали -данным работы \cite{bib:tarnapolsky}. Упругие модули поликристаллической -матрицы ыли выбраны следующими: $E_m = 0,28$~ГПа и коэффициент Пуассона $\nu_m - = 0,40$. Статический коэффициент трения $f = 0,12$ соответствовал -случаю скольжения волокна по поверхности поликристаллической матрицы. Как -видим, распределение искомых полей в рассматриваемом случае -удовлетворяет условиям симметрии и периодичности геометрической модели -и приложенной внешней нагрузке. Это свидетельствует о корректно -построенной модели и корректности полученного численного решения. Кроме -того, обращает на себя внимание концентрация напряжений в местах, -где искривленные нити основы и утка имеют наибольшую кривизну. - -\begin{figure} - \centering -% \includegraphics[width=0.75\linewidth]{img/vmis} - \caption{Поля интенсивности напряжений в нитях основы и утка} - \label{fig:sigma} +\begin{figure}[ht!] + \includegraphics[width=15cm]{mesh/v2/fibers} + \caption{Пример дискретизации волокон} + \label{fig:c3:mesh:fibers} \end{figure} +\begin{table}[ht] + \caption{Параметры конечно-элементной сетки} + \begin{tabular}{|p{8.25cm}||>{\centering}p{3.45cm}|p{3.45cm}<{\centering}|} + \hline + & Тетраэдральные элементы & Гексаэдральные элементы \\ + \hline + \hline + Идеальная периодическая структура & 405~480 & 77~760 \\ + \hline + Разрыв волокна основы & 405~480 & 75~168 \\ + \hline + Разрыв волокна основы с доуплотнением & 355~341 & 75~168 \\ + \hline + Разрыв волокон основы и утка & 405~480 & 72~576 \\ + \hline + Разрыв волокон основы и утка с доуплотнением & 390~862 & 72~576 \\ + \hline + \end{tabular} + \label{tab:c3:discr} +\end{table} -\begin{table} +Решив задачу \ref{eq:Eqvilibrium}~--~\ref{eq:Koshi} с граничными условиями +\ref{eq:b_cond}~--~\ref{eq:b_cond_Colomb_2} методом конечных элементов получим +поля интенсивностей напряжений в искривленных нитях основы и утка слоя +модельного тканого композита идеальной периодической структуры, показанных +на рис.~\ref{fig:c3:vmis_v2_s1}. Как видим, распределение искомых полей в +рассматриваемом случае удовлетворяет условиям симметрии и периодичности +геометрической модели и приложенной внешней нагрузке. Это свидетельствует о +корректно построенной модели и корректности полученного численного решения. +Кроме того, обращает на себя внимание концентрация напряжений в местах, где +искривленные нити основы и утка имеют наибольшую кривизну. + +В табл. \ref{tab:c3:max_k_s1} представлены максимальные безразмерные +коэффициенты $K_{\sigma_{ij}} = \sigma_{ij}({\bf r}) / +\sigma_{ij}^{per}({\bf r})$, определяемые отношением компонент тензора +напряжений в слое модельного тканого композита с локальным дефектом к +соответствующим компонентам в слое материала идеальной периодической +структуры. Обратим внимание на то, что наибольший вклад в коэффициенты +концентрации вносят касательные составляющие тензора напряжений $\sigma_{13}$. +Напряжения для этих компонент, в 10--48 раз превышают соответствующие значения +для модельного материала с идеальной периодической структурой. + +\begin{figure}[t!] + \centering + \includegraphics[width=17cm]{vmis_v2_s1} + \caption{Поля интенсивности напряжений в нитях основы и утка при +равнокомпонентном двухосном растяжении} + \label{fig:c3:vmis_v2_s1} +\end{figure} + +\begin{table}[t!] \centering \caption{Максимальные коэффициенты концентрации напряжений в матрице слоя тканого композита} -\begin{tabular}{p{6cm}||c|c|c|c|c|c} -\hline - & $K_{\sigma_{11}}$ & $K_{\sigma_{22}}$ & $K_{\sigma_{33}}$ & -$K_{\sigma_{12}}$ & $K_{\sigma_{13}}$ & $K_{\sigma_{23}}$ \\ -\hline \hline - Разрыв нити основы & $\frac{1{,}29} {4{,}57}$ & $\frac {1{,}63} {3{,}61}$ & -$\frac {1{,}30} {4{,}37}$ & $\frac {1{,}25} - {6{,}87}$ & $\frac {2{,}31} {10{,}87}$ & $\frac {1{,}44} {3{,}69}$ \\ -\hline - Разрыв нити основы (доуплотнение) & $\frac{1{,}26}{4{,}07}$ & -$\frac{1{,}49}{4{,}69}$ & $\frac{1{,}27}{3{,}75}$ & $\frac{1{,}25}{8{,}72}$ - & $\frac{2{,}20}{16{,}46}$ & $\frac{1{,}32}{7{,}27}$ \\ -\hline\hline - Разрыв нитей основы и утка & $\frac{1{,}50} {4{,}01}$ & $\frac{1{,}92} -{3{,}73}$ & $\frac{1{,}56} {5{,}92}$ & $\frac{1{,}58} {6{,}59}$ - & $\frac{2{,}53} {48{,}08}$ & $\frac{1{,}70} {3{,}70}$ \\ -\hline - Разрыв нитей основы и утка (доуплотнение) & $\frac{1{,}35}{3{,}93}$ & -$\frac{1{,}68}{4{,}38}$ & $\frac{1{,}41}{3{,}57}$ - & $\frac{1{,}41}{8{,}42}$ & $\frac{2{,}21}{16{,}06}$ & $\frac{1{,}50}{3{,}85}$ -\\ -\hline -\end{tabular} -\label{tab:k} + \begin{tabular}{|p{8cm}||c|c|c|c|c|c|} + \hline + & $K_{\sigma_{11}}$ + & $K_{\sigma_{22}}$ + & $K_{\sigma_{33}}$ + & $K_{\sigma_{12}}$ + & $K_{\sigma_{13}}$ + & $K_{\sigma_{23}}$ \\ + \hline + \hline + Разрыв нити основы + & $4{,}57$ & $3{,}61$ & $4{,}37$ & $6{,}87$ & $\bf 10{,}87$ & $3{,}69$ \\ + \hline + Разрыв нити основы (доуплотнение) + & $4{,}07$ & $4{,}69$ & $3{,}75$ & $8{,}72$ & $\bf 16{,}46$ & $7{,}27$ \\ + \hline + \hline + Разрыв нитей основы и утка + & $4{,}01$ & $3{,}73$ & $5{,}92$ & $6{,}59$ & $\bf 48{,}08$ & $3{,}70$ \\ + \hline + Разрыв нитей основы и утка (доуплотнение) + & $3{,}93$ & $4{,}38$ & $3{,}57$ & $8{,}42$ & $\bf 16{,}06$ & $3{,}85$ \\ + \hline + \end{tabular} +\label{tab:c3:max_k_s1} \end{table} -В табл. \ref{tab:k} представлены максимальные безразмерные -коэффициенты $K_{\sigma _{ij} } = {\sigma _{ij} \left( {\rm {\bf r}} -\right)} \mathord{\left/ {\vphantom {{\sigma _{ij} \left( {\rm {\bf r}} -\right)} {\sigma _{ij}^{\mbox{per}} \left( {\rm {\bf r}} \right)}}} -\right. \kern-\nulldelimiterspace} {\sigma_{ij}^{\mbox{per}} \left( {\rm {\bf -r}} \right)}$, определяемые отношением компонент тензора напряжений в -слое модельного тканого композита с локальным дефектом к -соответствующим компонентам в слое материала идеальной периодической -структуры. Значения в числителе были определены в случае, когда каждая -нить армирующего каркаса окружена гарантированным слоем матрицы, а в -знаменателе --- в случае, когда нити основы и утка имеют общую -поверхность контакта с трением, а между участками с наибольшей -кривизной располагается внутренняя пора. Обратим внимание на то, что -наибольший вклад в коэффициенты концентрации вносят касательные -составляющие тензора напряжений $\sigma_{13}$ и $\sigma_{23}$. Кроме -того, коэффициенты концентрации для этих компонент, определенные для -слоя композита, содержащего внутренние поры, в 5--16 раз -превышают соответствующие значения для материала, в котором каждая нить -окружена гарантированным слоем поликристаллической матрицы. - -\begin{figure} - \begin{minipage}[h]{0.47\linewidth} -% \center{\includegraphics[width=1\linewidth]{img/d3_k}} \\ а) - \end{minipage} - \hfill - \begin{minipage}[h]{0.47\linewidth} -% \center{\includegraphics[width=1\linewidth]{img/d3_k_fric}} \\ б) - \end{minipage} - \caption{Распределение коэффициентов концентрации интенсивности напряжений - в поликристаллической матрице тканого композита с локальным разрывом нити -утка} - \label{fig:k_rasp_1} -\end{figure} - -\begin{figure} - \begin{minipage}[h]{0.47\linewidth} -% \center{\includegraphics[width=1\linewidth]{img/d4_k}} \\ а) - \end{minipage} - \hfill - \begin{minipage}[h]{0.47\linewidth} -% \center{\includegraphics[width=1\linewidth]{img/d4_k_fric}} \\ б) - \end{minipage} - \caption{Распределение коэффициентов концентрации интенсивности напряжений - в поликристаллической матрице тканого композита с локальным разрывом нитей - основы и утка} - \label{fig:k_rasp_2} -\end{figure} - -На рис.~\ref{fig:k_rasp_1} и \ref{fig:k_rasp_2} представлены +На рис.~\ref{fig:c3:k_d1d3_s1} и \ref{fig:c3:k_d2d4_s1} представлены распределения коэффициентов концентрации интенсивностей напряжений для -слоя модельного тканого композита с различными локальными дефектами. -Расположение областей, в которых интенсивность напряжений достигает -максимальных значений в местах, где искривленные волокна основы или утка -имеют наибольшую кривизну, строго периодично. Исключение составляют +слоя модельного тканого композита с разрывом волокна основы и разрывом волокон +основы и утка. Расположение областей, в которых интенсивность напряжений +достигает максимальных значений в местах, где искривленные волокна основы или +утка имеют наибольшую кривизну, строго периодично. Исключение составляют области, расположенные вблизи локального разрыва утка или одновременного -разрыва основы и утка, где интенсивность напряжений превышает -соответствующее значение, определенное для композита идеальной -периодической структуры в $1,4$ и $1,6$ раз в случае, если нить -армирующего каркаса окружена гарантированным слоем -матрицы (рис.~\ref{fig:k_rasp_1},~б и \ref{fig:k_rasp_2}~б). Если в слое -тканого композита не исключена возможность контакта с кулоновским -трением искривленных нитей, а также присутствуют локальные поры в -местах наибольших кривизн волокон, то коэффициенты концентрации -для рассматриваемых случаев увеличиваются до $2,5$. +разрыва основы и утка, где интенсивность напряжений превышает соответствующее +значение, определенное для композита идеальной периодической структуры в $2{,}1$ +раза. Стоит заметить, что заполнение поры, образовавшейся вследствие дефекта, +материалом поликристаллической матрицы путем дополнительной пропитки или +осаждения матрицы из газовой фазы приводит к увеличению коэффициентов +концентрации интенсивностей напряжений до $2{,}8$, при разрыве волокна основы +(рис.~\ref{fig:c3:k_d1d3_s1}~б) и $3{,}1$ при разрыве волокон основы и утка +одновременно (рис.~\ref{fig:c3:k_d2d4_s1}~б). -\section{Выводы к третьей главе} +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v2/scheme1/d1d3} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с разрывом волокна основы~(а) и с учётом +доуплотнения~(б)} + \label{fig:c3:k_d1d3_s1} +\end{figure} -На основе построенной модели слоя тканого композита с искривленными волокнами -и поликристаллической матрицей определены коэффициенты концентрации -напряжений, вызванные наличием локальных технологических дефектов в виде -разрыва нити утка, одновременного разрыва нитей основы и утка, наличия -закрытых пор при двухосном равнокомпонентном деформировании, -определены механизмы, инициирующие разрушение матрицы. +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v2/scheme1/d2d4} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с разрывом волокон основы и утка~(а) и с учётом +доуплотнения~(б)} + \label{fig:c3:k_d2d4_s1} +\end{figure} -Полученные результаты позволяют сделать вывод о том, что для -повышения способности тканым композитом сопротивляться внешнему -силовому воздействию необходимо предусмотреть в технологическом -процессе операции, обеспечивающие проникновение связующего в -полости технологических локальных дефектов, а также дополнительную -пропитку связующим, доуплотнение и карбонизацию, доосаждение -поликристаллической матрицы из газовой фазы в случае, если в -результате ультразвукового контроля готового изделия обнаруживаются -закрытые внутренние поры. В противном случае возможно развитие дефектов -и последующее разрушение материала матрицы по механизмам сдвигов. \ No newline at end of file +\subsection{Коэффициенты концентрации напряжений в слое тканого композита с +соприкасащимися волокнами при чистом сдвиге} + +Решим задачу \ref{eq:Eqvilibrium}~--~\ref{eq:Guck} с граничными условиями +\ref{eq:b_cond_ideal}~--~\ref{eq:b_cond:s2}, +соответствующими чистому сдвигу, дополненными граничными условиями +\ref{eq:b_cond_Colomb_1} и \ref{eq:b_cond_Colomb_2}, задающими трения между +волокнами основы и утка тканого композита с поликристаллической матрицей. + +Поля интенсивностей напряжений, полученные в результате решения такой задачи, +показанные на рис.~\ref{fig:c3:vmis_v2_s2}, строго периодичны, что говорит о +корректности полученного решения. + +\begin{figure}[ht] + \includegraphics[width=15cm]{vmis_v2_s2} + \caption{Поля интенсивности напряжений в слое тканого композита с идеальной +периодической структурой при чистом сдвиге и наличии контакта между волокнами +основы и утка} + \label{fig:c3:vmis_v2_s2} +\end{figure} + +Максимальные значения коэффициентов концентрации напряжений представлены в +таблице~\ref{tab:c3:max_k_s2}. Как видно из таблицы, наибольший вклад в +коэффициенты концентрации напряжений вносят касательная составляющая +$\sigma_{13}$ и нормальная составляющая $\sigma_{33}$ тензора напряжений. +Значения этих составляющих в материале с дефектом в $10$~--~$29$ раз превышают +соответствующие значения в материале с идеальной периодической структуре. + +\begin{table}[t!] + \centering + \caption{Максимальные коэффициенты концентрации напряжений в матрице слоя +тканого композита при чистом сдвиге} + \begin{tabular}{|p{7cm}||c|c|c|c|c|c|} + \hline + & $K_{\sigma_{11}}$ + & $K_{\sigma_{22}}$ + & $K_{\sigma_{33}}$ + & $K_{\sigma_{12}}$ + & $K_{\sigma_{13}}$ + & $K_{\sigma_{23}}$ \\ + \hline + \hline + Разрыв нити основы + & $3{,}66$ & $5{,}70$ & $9{,}64$ & $5{,}16$ & $\bf12{,}30$ & $\bf10{,}54$\\ + \hline + Разрыв нити основы (доуплотнение) + & $3{,}36$ & $4{,}87$ & $\bf15{,}83$ & $6{,}53$ &$\bf11{,}59$ & $10{,}97$\\ + \hline + \hline + Разрыв нитей основы и утка + & $3{,}76$ & $7{,}53$ & $\bf29{,}34$ & $5{,}82$ & $\bf28{,}39$ & $8{,}02$\\ + \hline + Разрыв нитей основы и утка (доуплотнение) + & $3{,}33$ & $6{,}95$ & $\bf15{,}03$ & $5{,}85$ & $\bf12{,}71$ & $7{,}31$\\ + \hline + \end{tabular} +\label{tab:c3:max_k_s2} +\end{table} + +На рис.~\ref{fig:c3:k_d1d3_s2} и \ref{fig:c3:k_d2d4_s2} представлены +распределения коэффициентов концентрации интенсивностей напряжений, вызванных +наличием дефекта в виде разрыва волокна основы и разрыва волокон основы и утка в +слое тканого композита с поликристаллической матрицей при чистом сдвиге. + +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v2/scheme2/d1d3} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с разрывом волокна основы~(а) и с учётом +доуплотнения~(б) при чистом сдвиге} + \label{fig:c3:k_d1d3_s2} +\end{figure} + +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v2/scheme2/d2d4} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с разрывом волокон основы и утка~(а) и с учётом +доуплотнения~(б) при чистом сдвиге} + \label{fig:c3:k_d2d4_s2} +\end{figure} + +Распределение коэффициентов концентрации интенсивностей напряжений строго +периодично и достигает максимальных значений в местах, расположенных вблизи +локальных дефектов. При разрыве волокна основы максимальное значение +коэффициентов концентрации интенсивностей напряжений равно $2{,}3$ +(рис.~\ref{fig:c3:k_d1d3_s2}~а), а при одновременном разрыве волокон основы и +утка --- $3{,}0$ (рис.~\ref{fig:c3:k_d2d4_s2}~б). При этом, заполнение поры, +образовавшейся в результате одновременного разрыва волокон основы и утка, +материалом поликристаллической матрицы путем дополнительной пропитки или +осаждения матрицы из газовой фазы позволяет снизить коэффициенты концентрации +интенсивностей напряжений до $2{,}6$ (рис.~\ref{fig:c3:k_d2d4_s2}~б). + +\subsection{Коэффициенты концентрации напряжений в слое тканого композита с +соприкасащимися волокнами при одноосном растяжении} + +Решим задачу \ref{eq:Eqvilibrium}~--~\ref{eq:Guck} с граничными условиями +\ref{eq:b_cond_ideal}, \ref{eq:b_cond_free} и \ref{eq:b_cond:s3}, +соответствующими одноосному растяжению в направлении утка, дополненными +граничными условиями \ref{eq:b_cond_Colomb_1} и \ref{eq:b_cond_Colomb_2}, +задающими трения между волокнами основы и утка тканого композита с +поликристаллической матрицей. + +Поля интенсивностей напряжений, полученные в результате решения такой задачи, +показанные на рис.~\ref{fig:c3:vmis_v2_s3}, строго периодичны, что говорит о +корректности полученного решения. + +В таблице \ref{tab:c3:max_k_s3} показаны макисмальные безразмерные коэффициенты +концентрации напряжений, вызванные наличием разрыва волокна основы и разрывов +волокон основы и утка в слое тканого композита с поликристаллической матрицей +при одноосном растяжении. Максимальный вклад в коэффициенты концентраций вносит +касательная составляющая тензора напряжений $\sigma_{13}$, значения которой в +материале с локальным дефектом превышают соответствующие значения в материале с +идеальной периодической структурой в $11$~--~$16$ раз. + +\begin{figure}[ht!] + \includegraphics[width=15cm]{vmis_v2_s3} + \caption{Поля интенсивности напряжений в слое тканого композита с идеальной +периодической структурой при чистом сдвиге и наличии контакта между волокнами +основы и утка} + \label{fig:c3:vmis_v2_s3} +\end{figure} + +\begin{table}[t!] + \centering + \caption{Максимальные коэффициенты концентрации напряжений в матрице слоя +тканого композита при одноосном растяжении} + \begin{tabular}{|p{7cm}||c|c|c|c|c|c|} + \hline + & $K_{\sigma_{11}}$ + & $K_{\sigma_{22}}$ + & $K_{\sigma_{33}}$ + & $K_{\sigma_{12}}$ + & $K_{\sigma_{13}}$ + & $K_{\sigma_{23}}$ \\ + \hline + \hline + Разрыв нити основы + & $3{,}84$ & $10{,}08$ & $6{,}56$ & $4{,}31$ & $\bf14{,}00$ & $6{,}22$ \\ + \hline + Разрыв нити основы (доуплотнение) + & $3{,}59$ & $4{,}50$ & $6{,}32$ & $3{,}98$ & $\bf11{,}21$ & $7{,}12$ \\ + \hline + \hline + Разрыв нитей основы и утка + & $3{,}82$ & $9{,}27$ & $6{,}96$ & $4{,}50$ & $\bf16{,}02$ &$\bf16{,}89$\\ + \hline + Разрыв нитей основы и утка (доуплотнение) + & $3{,}42$ & $4{,}22$ & $6{,}37$ & $3{,}87$ & $\bf11{,}47$ & $6{,}89$ \\ + \hline + \end{tabular} +\label{tab:c3:max_k_s3} +\end{table} + +Распределение коэффициентов концентрации интенсивностей напряжений, вызванные +наличием разрыва волокна основы и разрывов волокон основы и утка, показаны на +рис.~\ref{fig:c3:k_d1d3_s3} и \ref{fig:c3:k_d2d4_s3}. + +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v2/scheme3/d1d3} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с разрывом волокна основы~(а) и с учётом +доуплотнения~(б) при одноосном растяжении} + \label{fig:c3:k_d1d3_s3} +\end{figure} + +\begin{figure}[ht!] + \includegraphics[width=17cm]{concentrators/v2/scheme3/d2d4} + \caption{Распределение коэффициентов концентрации интенсивностей напряжений в +слое тканого композита с разрывом волокон основы и утка~(а) и с учётом +доуплотнения~(б) при одноосном растяжении} + \label{fig:c3:k_d2d4_s3} +\end{figure} + +Максимальных значений коэффициенты концентрации достигают в местах наибольшей +кривизны волокон. Для материала с локальным разрывом волокна основы значения +коэффициентов концентрации интенсивностей напяжений достигают $2{,}3$, а при +одновременном разрыве волокон основы и утка --- $2{,}5$, причем заполнение +поры, образовавшейся вследствие наличия локального дефекта, материалом +поликристаллической матрицы, путем дополнительной пропитки конструкции или +осаждения матрицы из газовой фазы, приводит к увеличению коэффициентов +концентрации до $2{,}6$ и $3{,}7$ для случаев разрыва волокна основы и +одновременного разрыва волокон основы и утка соответственно. + +\section*{Выводы к третьей главе} +\addcontentsline{toc}{section}{Выводы к третьей главе} + +\begin{enumerate} + \item Построена твердотельная модель фрагмента слоя тканого композита с +искривленными волокнами и поликристалической матрицей с идеальной +периодической структурой и локальными технологическими дефектами, такими как +разрыв волокна основы и одновременный разрыв волокон основы и утка с учетом +контакта с трением между волокнами. + \item Получены численные решения краевых задач на двухосное растяжение в +плоскости слоя, чистый сдвиг и одноосное растяжение в направлении утка с учетом +контакта с трением между волокнами. + \item Вычислены безразмерные коэффициенты концентрации напряжений, вызванные +наличием локальных технологических дефектов в виде разрыва волокна основы и +одновременного разрыва волокон основы и утка. + \item Определены механизмы, инициирующие разрушение матрицы в слое тканого +композита с искривленными волокнами и наличием контакта с трением между +волокнами. +\end{enumerate} \ No newline at end of file diff --git a/common.tex b/common.tex index 3405cf3..13bb7c8 100644 --- a/common.tex +++ b/common.tex @@ -129,8 +129,54 @@ $n_2$-ти тезисах докладов~\citemy{bib:dedkov1}. научного руководителя. } -\mkcommonsect{struct}{Структура и объем диссертации}{% +\mkcommonsect{struct}{Структура и объем диссертации.}{% Диссертационная работа состоит из введения, $n$-х частей, заключения, выводов и списка литературы. Полный объем составляет $n_1$ страниц. Библиография включает $n_2$ наименований. } + +\mkcommonsect{inintro}{Во введении}{ +приведен краткий обзор, отражающий современное +состояние вопросов исследования, сделано заключение об актуальности +темы диссертационной работы. Сформулирована цель данной работы, полученные +в ней новые научные результаты, применение и практическая ценность, +приведена аннотация содержания глав диссертационной работы. +} + +\mkcommonsect{infirst}{В первой главе}{ +рассматриваются технологические операции изготовления конструкций из +тканых композиционных материалов с поликристаллической матрицей. Описывается +процесс изготовления волокон, рассматриваются типы тканей и способы их +производства, а так же совмещение тканого каркаса с поликристаллической +матрицей, методы контроля качества конструкций их тканых материалов с +поликристаллической матрицей и типы дефектов, возникающие во время +технологических процессов, выявляемых с их помощью. +} + +\mkcommonsect{insecond}{Во второй главе}{ +рассматриваются математические модели упруго-хрупкого поведения +слоя тканого композита с искривленными волокнами и поликристаллической матрицей +при двухосном растяжении в плоскости слоя, чистом сдвиге и одноосном растяжении +слоя в направлении утка. Приведены значения безразмерных коэффициентов +концентрации напряжений, вызванных наличием таких локальных дефектов как +пропуск волокна основы, разрыв волокна основы, разрыв волокон основы и утка, а +также наличием внутренней технологической поры. Определены механизмы, +приводящие к разрушению поликристалллической матрицы. +} + +\mkcommonsect{inthird}{В третьей главе}{ +рассматриваются математические модели упруго-хрупкого поведения слоя такного +композита с искривленными волокнами и поликристаллической матрицей при +двухосном растяжении в плоскости слоя, чистом сдиге и одноосном +растяжении в направлении утка, с учетом наличия контакта с трением между +волокнами. Приведены значения безразмерных коэффициентов концентрации +напряжений, вызванных наличием таких локальных дефектов как разрыв волокна +основы и разрыв волокон основы и утка, учитывая внутренние технологические +поры, образовавшиеся в результате недостаточной пропитки ткани материалом +матрицы в местах наибольшей кривизны волокон. Определены механизмы, +приводящие к разрушению поликристалллической матрицы. +} + +\mkcommonsect{inend}{В заключении}{ +изложены основные результаты диссертационной работы в целом. +} \ No newline at end of file diff --git a/disser.kilepr b/disser.kilepr index 69f9ca3..530462e 100644 --- a/disser.kilepr +++ b/disser.kilepr @@ -4,7 +4,7 @@ img_extIsRegExp=false img_extensions=.eps .jpg .jpeg .png .pdf .ps .fig .gif kileprversion=2 kileversion=2.1.0 -lastDocument=c2.tex +lastDocument=c3.tex masterDocument= name=disser pkg_extIsRegExp=false @@ -88,10 +88,10 @@ ReadWrite=true [item:bibliography.bib] archive=true -column=11 +column=0 encoding=UTF-8 highlight=BibTeX -line=127 +line=68 mode=BibTeX open=true order=3 @@ -108,20 +108,20 @@ order=4 [item:c2.tex] archive=true -column=60 +column=35 encoding=UTF-8 highlight=LaTeX -line=730 +line=732 mode=LaTeX open=true order=5 [item:c3.tex] archive=true -column=0 +column=75 encoding=UTF-8 highlight=LaTeX -line=1 +line=455 mode=LaTeX open=true order=6 @@ -167,8 +167,8 @@ open=true order=0 [view-settings,view=0,item:bibliography.bib] -CursorColumn=11 -CursorLine=127 +CursorColumn=0 +CursorLine=68 JumpList= ViMarks= @@ -179,14 +179,14 @@ JumpList= ViMarks= [view-settings,view=0,item:c2.tex] -CursorColumn=60 -CursorLine=730 +CursorColumn=35 +CursorLine=732 JumpList= ViMarks= [view-settings,view=0,item:c3.tex] -CursorColumn=0 -CursorLine=1 +CursorColumn=75 +CursorLine=455 JumpList= ViMarks= diff --git a/end.tex b/end.tex new file mode 100644 index 0000000..4cb11e9 --- /dev/null +++ b/end.tex @@ -0,0 +1,25 @@ +\chapter*{Заключение} +\addcontentsline{toc}{chapter}{Заключение} + +Основные результаты, полученные в диссертации сводятся к следующему: + +\begin{enumerate} + \item Рассмотрены основные технологические процессы изготовления конструкций +из тканых композиционных материалов с поликристаллической матрицей, приводящие +к появлению локальных технологических дефектов, определены типы дефектов, +возникающие на каждой из стадиях этих технологических процессов. Описаны методы +контроля качества, выявляющие локальные дефекты. + \item Построены твердотельные модели слоя тканого композиционного материала с +искривленными волокнами и поликристаллической матрицей с гарантированной +прослойкой матрицы между волокнами, а также с наличием контакта с трением между +волокнами. + \item Определены концентраторы напряжений в слое тканого композита с +искривленными волокнами и поликристаллической матрицей, вызванные наличием +таких локальных дефектов как пропуск волокна основы, разрыв волокна основы, +разрыв волокон основы и утка, наличие внутренней технологической поры при +двухосном растяжении в плоскости слоя, чистом сдвиге, а также одноосном +растяжении в направлении волокон основы. + \item Определены механизмы, инициирующие разрушение матрицы в слое тканого +композита при наличии гарантированной прослойки матрицы между волокнами, а +также при наличии контакта с трением между волокнами. +\end{enumerate} diff --git 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+\inthirdsection +\inthirdtext + +% Содержание заключения +\inendsection +\inendtext \ No newline at end of file diff --git a/stress_concentartors.tex b/stress_concentartors.tex index 0871e5f..b63eebc 100644 --- a/stress_concentartors.tex +++ b/stress_concentartors.tex @@ -37,6 +37,7 @@ \input{c1} \input{c2} \input{c3} +\input{end} \bibliography{bibliography} \bibliographystyle{disser}

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