Моделирование функционально-механического поведения пористого сплава с памятью формы на основе аппроксимации его структуры как балочной конструкции тема диссертации и автореферата по ВАК РФ 01.02.04, кандидат наук Япарова Елизавета Николаевна

• Япарова Елизавета Николаевна
• кандидат наук
• 2020

Оглавление диссертации кандидат наук Япарова Елизавета Николаевна

Введение

1 Пористые сплавы с памятью формы и методы их описания

1.1 Особенности свойств пористых сплавов с памятью формы

1.2 Методы получения пористых сплавов с памятью формы

1.3 Модели функционально-механического поведения СПФ

1.4 Подходы к расчету функционально-механических свойств пористых СПФ

2 Микроструктурное моделирование пористых СПФ с использованием

методов сопротивления материалов

2.1 Пористый СПФ с вертикальной ориентацией поровых каналов

2.2 Пористый СПФ с горизонтальной ориентацией поровых каналов

2.3 Пористый СПФ с неупорядоченной ориентацией поровых каналов

2.4 Пористый СПФ, полученный методом селективного лазерного плавления

2.5 Определение геометрических параметров моделей

3 Верификация модели на образцах различной пористой структуры

3.1 Определение материальных констант микроструктурной модели

3.2 Расчет одноосного сжатия и эффекта памяти формы в пористом образце из СПФ со структурой, приближенной каскадом криволинейных балок

3.3 Моделирование пористого образца из СПФ со структурой, аппроксимированной плоскими прорезными пружинами

3.4 О механизмах неупругого деформирования пористых СПФ

3.5 Расчет изотермической деформации пористого СПФ с неупорядоченной структурой

3.6 Моделирование поведения пористого СПФ с упорядоченной структурой, полученного селективным лазерным плавлением

Заключение

Список литературы

Введение диссертации (часть автореферата) на тему «Моделирование функционально-механического поведения пористого сплава с памятью формы на основе аппроксимации его структуры как балочной конструкции»

Введение Актуальность темы

Последние десятилетия сплавы с эффектом памяти формы (СПФ), как сплошные, так и пористые, привлекают к себе большое внимание в связи с возможностями их применения в самых разных областях: от аэрокосмической промышленности до хирургических инструментов и медицинских имплантатов. Среди СПФ наиболее широко используются сплавы на основе Т1№ благодаря своим высоким физико-механическим характеристикам и эффекту памяти формы. Получаемые из смеси порошков Т1 и N1 или Т1М пористые СПФ обладают хорошими функциональными свойствами при простоте и высокой производительности методов их получения. Эти материалы проявляют эффекты памяти формы и псевдоупругости (сверхэластичности), обладают высокой демпфирующей способностью, а также специфической для высокопористых материалов особенностью - проницаемостью. Таким образом, пористый Т1№ имеет черты как сплавов с памятью формы, широко применяемых в технике и медицине, так и пористых проницаемых металлических материалов. Наиболее важными с практической точки зрения являются изделия из пористого Т1М, применяемые в медицине: челюстно-лицевые и зубные имплантаты, заменители шейных и поясничных отделов позвоночника, заменители суставов, костных пластин, фиксаторы позвоночника при переломе.

Востребованность этого материала и его недостаточная изученность способствовали проведению исследований по разработке моделей пористых СПФ. Для того, чтобы прогнозировать поведение пористых СПФ, необходимо было построить одновременно простую и точную модель описания его функционально -механического поведения.

Практически все имеющиеся в настоящее время работы по моделированию механического поведения пористого СПФ ограничены описанием изотермического деформирования с использованием макроскопических феноменологических

моделей деформации СПФ. Расчет деформации пористого СПФ при изменении температуры представлен лишь в единичных работах. Кроме того, существующие модели лишь косвенно учитывают особенности строения поровых каналов в материале, в то время как они оказывают значительное влияние на деформационное поведение образцов. Поэтому моделирование функционально-механических свойств пористого СПФ с учетом особенностей пористой структуры является актуальной темой научных исследований.

Методика исследований

При создании модели для расчета напряженно-деформированного состояния отдельных структурных элементов, а также для их связи с макровеличинами использовались уравнения и гипотезы сопротивления материалов. Для различных типов структур аппроксимация выполнялась при помощи конструкции из балок различных конфигураций. Расчет неупругой деформации выполняли для наиболее напряженных участков структурных элементов при помощи микроструктурной модели СПФ. В этой модели определяющие уравнения формулируются для микроуровня, а деформация представительного объема макроуровня рассчитывается посредством усреднения микродеформаций. Аппроксимация сложной пористой структуры выполнялась на основании анализа микрофотографий сечений пористых образцов Т1М, полученных методами самораспространяющегося высокотемпературного синтеза (СВС) и селективного лазерного плавления (СЛП). Необходимые значения геометрических параметров модели получены при статистическом анализе размеров межпоровых перегородок (для образцов, полученных СВС), либо могут быть известны при измерении малого представительного объема образца (для образцов, полученных СЛП). Материальные константы для сравнения с экспериментальными результатами взяты из литературных данных. Алгоритм расчета поведения пористых СПФ реализован при помощи программы на языке С++.

Цель работы и задачи

Цель данной работы - построение модели функционально-механического поведения образцов из пористого сплава с памятью формы, учитывающей особенности структуры материала.

Для достижения цели были поставлены следующие задачи:

1. Анализ структуры образцов из сплава на основе Т1М, полученных методами СВС и СЛП, с различным строением поровых каналов и аппроксимация этих структур конструкциями, состоящими из балок.

2. Определение геометрических параметров, характеризующих поровую структуру образца.

3. Получение соотношений, связывающих средние деформации и напряжения с деформациями и напряжениями в структурных элементах для различных типов пористых структур.

4. Верификация модели для расчета функционально-механического поведения образцов из пористого СПФ на различных типах пористых структур.

5. Выявление особенностей функционально-механического поведения пористых СПФ, связанных со спецификой пористой структуры.

Научная новизна

1. Впервые было выполнено моделирование поведения пористого СПФ с различными ориентациями поровых каналов на основе их аппроксимации балочными конструкциями и разработана методика определения их геометрических параметров на основе статистического анализа микрофотографий.

2. Теоретически получены диаграммы деформирования сжатием пористых образцов с различной структурой, находящихся в различных фазовых состояниях.

3. Найдены зависимости обратимой и необратимой деформации от числа циклов термомеханического нагружения для образцов, полученных методом СЛП.

4. Впервые выполнено моделирование изменения деформации в высокопористом СПФ при охлаждении и нагреве под нагрузкой (эффект памяти формы).

Положения, выносимые на защиту

1. Модель функционально-механического поведения пористых СПФ с поровыми каналами, ориентированными параллельно оси образца, основанная на аппроксимации пористой структуры каскадами криволинейных балок.

2. Модель функционально-механического поведения пористых СПФ с поровыми каналами, ориентированными перпендикулярно оси образца, основанная на аппроксимации пористой структуры плоскими прорезными пружинами.

3. Модель функционально-механического поведения пористых СПФ с неупорядоченной ориентацией поровых каналов, основанная на представлении пористой структуры как конструкции из горизонтальных балок, соединенных криволинейными балками по типу плоской прорезной пружины.

4. Модель функционально-механического поведения пористого СПФ, полученного методом СЛП, основанная на введении эффективного напряжения, учитывающего пористость.

5. Результаты компьютерного расчета изменения деформации при сжатии в разных фазовых состояниях, при охлаждении и нагреве под нагрузкой (эффект памяти формы), расчет обратимой и необратимой деформации при термомеханическом циклическом нагружении.

Достоверность полученных результатов

Достоверность полученных результатов достигается благодаря сравнению результатов моделирования с экспериментальными данными, использованию классических методов сопротивления материалов и соотношений апробированной микроструктурной модели СПФ. Благодаря достаточно точному представлению

структуры пористого образца аппроксимирующими конструкциями, расчетные зависимости хорошо согласуются с результатами экспериментов.

Теоретическая и практическая значимость работы

Разработанная модель может быть использована для описания изменения деформации пористого СПФ, полученного различными методами, при изотермическом деформировании, а также при термо- и механоциклировании. При помощи данной модели можно выяснить влияние вкладов различных механизмов деформации на функционально-механическое поведение пористого TiNi. Практическая значимость работы заключается в том, что данная модель может быть использована при оценке свойств изделий из пористого СПФ при произвольных режимах изменения напряжения и температуры.

Апробация работы

Результаты работы были представлены на семинарах кафедры теории упругости СПбГУ и на факультете информатики и математики Кильского университета в г. Киль (Великобритания), в Доме ученых им. М. Горького РАН и на следующих всероссийских и международных конференциях:

1. «Седьмые Поляховские чтения», г. Санкт-Петербург, 2015 г.;

2. «European Symposium on Martensitic Transformations» (ESOMAT-2015), г. Антверпен, Бельгия, 2015 г.;

3. LVII международная конференция «Актуальные проблемы прочности», г. Севастополь, 2016 г.;

4. VIII международная конференция «Микромеханизмы пластичности, разрушения и сопутствующих явлений» (MPFP - 2016), г. Тамбов, 2016 г.;

5. Вторая международная научная конференция «Сплавы с эффектом памяти формы» к 85-летию со дня рождения В. А. Лихачева, г. Санкт-Петербург, 2016 г.;

6. LVIII Международная конференция «Актуальные проблемы прочности», г. Пермь, 2017 г.;

7. VIII Международная школа «Физическое материаловедение» с элементами научной школы для молодежи, г. Тольятти, 2017 г.;

8. «Восьмые Поляховские чтения», г. Санкт-Петербург, 2018 г.;

9. Третья международная научная конференция «Сплавы с эффектом памяти формы», г. Челябинск, 2018 г.;

10. 11th European Symposium on Martensitic Transformations (ESOMAT 2018), г. Мец, Франция, 2018 г.;

11. Международный симпозиум "Перспективные материалы и технологии", Брест, Беларусь, 2019 г.;

12. «Intermetallics 2019», Бад Штаффельштайн, Германия, 2019 г.

Структура и объем работы

Работа состоит из введения, трех глав и заключения, и включает в себя 105 страниц и 46 рисунков. Список литературы содержит 207 библиографических ссылок.

Публикации по теме исследования

а) Публикации в рецензируемых изданиях, входящих в базы Scopus и Web of Science:

1. Volkov A.E., Evard M.E., Iaparova E.N. A beam model of porous shape memory alloy deformation // Materials Today: Proceedings. 2017. V. 4. No. 3. P. 4631-4636.

2. Volkov A.E., Evard M.E., Iaparova E.N. Modeling of functional properties of porous shape memory alloy // MATEC Web of Conferences. 2015. V. 33. P. 02006.

б) Публикации в изданиях, входящих в перечень ВАК:

1. Волков А.Е., Евард М.Е., Япарова Е.Н. О выборе граничных условий при компьютерном моделировании функционально-механического поведения пористых образцов из сплава с памятью формы // Вектор науки Тольяттинского государственного университета. 2017. Т. 42. № 4. С. 26-31.

2. Волков А.Е., Евард М.Е., Япарова Е.Н. Моделирование изотермического сжатия пористых образцов из сплава TiNi с продольной и поперечной ориентацией пор // Деформация и разрушение материалов. 2017. Т. 4. С. 9-14.

3. Волков А.Е., Евард М.Е., Япарова Е.Н. Деформация пористого образца из сплава с памятью формы с поперечной ориентацией пор относительно оси нагружения // Вестник Тамбовского Университета. Серия: естественные и технические науки. 2016. Т. 21. № 3. С. 913-916.

в) Другие публикации:

1. Iaparova E., Volkov A., Evard M., Belyaev F. Simulation of cyclic functional and mechanical behavior of porous NiTi samples obtained by selective laser melting // Intermetallics 2019: Programme and Abstracts. 2019. P. 184-185.

2. Япарова Е.Н., Волков А.Е., Евард М.Е., Беляев Ф.С. Моделирование поведения пористого TiNi с регулярной структурой при циклических термомеханических нагрузках // Перспективные материалы и технологии: сборник материалов международного симпозиума, Брест, 27 - 31 мая 2019 г. Витебск: УО «ВГТУ». 2019. С. 417.

3. Volkov A., Evard M., Iaparova E. Elucidation of the role of the structure of porous TiNi for its mechanical and functional properties // ESOMAT 2018: 11th European Symposium on Martensitic Transformations, August 27-31, 2018. Book Of Abstracts. 2018. P. 21-22.

4. Волков А.Е., Евард М.Е., Япарова Е.Н. Механическая модель пористого образца из сплава с памятью формы с неупорядоченной структурой // Сплавы с эффектом памяти формы. Третья Международная научная конференция (Челябинск, Россия, 16-20 авг. 2018 г.): сб. матер. конф. Челябинск: Изд-во Челяб. гос. ун-та. 2018. С. 75.

5. Волков А.Е., Евард М.Е., Япарова Е.Н. О механизмах неупругого деформирования пористых образцов из сплавов с памятью формы TiNi // Восьмые Поляховские чтения: Тезисы докладов Международной научной конференции по механике, Санкт-Петербург, 30 января - 2 февраля 2018 г. СПб.: Изд-во СПбГУ. 2018. С. 192-193.

6. Evard M., Volkov A., Iaparova E. Microstructural modeling of functional properties of porous shape memory alloy // International ECCOMAS Thematic Conference "Computational modeling of complex materials across the scales (CMCS)" (7-9.11.2017, Paris, France): Abstracts. 2017. P. 156126.

7. Волков А.Е., Евард М.Е., Япарова Е.Н. Применение балочной модели для расчета деформации пористого образца из сплава с памятью формы // Сплавы с эффектом памяти формы. Вторая международная научная конференция. К 85-летию со дня рождения В. А. Лихачева. Санкт-Петербург. 20-23 сентября 2016 г. Тезисы докладов. СПб.: Изд-во ВВМ. 2016. С. 19.

8. Волков А.Е., Евард М.Е., Япарова Е.Н. Деформация пористого образца из сплава с памятью формы с поперечной ориентацией пор относительно оси нагружения // Актуальные проблемы прочности: сборник тезисов LVII международной конференции, 24-27 мая, 2016 г. Севастополь: СевГУ. 2016. С. 203.

9. Волков А.Е., Евард М.Е., Япарова Е.Н. Расчет деформации пористого сплава с памятью формы // Седьмые Поляховские чтения: Тезисы докладов Международной научной конференции по механике, Санкт-Петербург, 2-6 февраля 2015 г. СПб.: Изд-во СПбГУ. 2015. С. 162.

1 Пористые сплавы с памятью формы и методы их описания 1.1 Особенности свойств пористых сплавов с памятью формы

Функционально-механическое поведение пористых СПФ обусловлено свойствами соответствующих непористых сплавов. Известно, что в СПФ при охлаждении происходит рост термоупругих кристаллов мартенсита (прямое мартенситное превращение), а при нагреве - их уменьшение и исчезновение (обратное мартенситное превращение). Важными параметрами фазового превращения являются его характеристические температуры (рис.1):

• М5 - температура начала прямого мартенситного превращения,

• М^ - температура конца прямого мартенситного превращения,

• А3 - температура начала обратного мартенситного превращения,

• А^ - температура конца обратного мартенситного превращения.

При охлаждении до температуры Ы/ исходная кубическая решетка аустенита

Тетрегв^ге

Рисунок 1 - Схема фазового превращения [1].

переходит в некубическую решетку мартенсита. Как только температура вновь превышает значение А^, кристаллическая решетка снова приобретает упорядоченную кубическую структуру аустенита [2 - 4].

Если продеформировать образец из СПФ в охлажденном мартенситном состоянии, это приведет к переориентации кристаллической решетки и

образованию ориентированного мартенсита. Новая форма образца будет сохраняться до тех пор, пока образец не нагреют до температуры начала обратного фазового превращения, при этом в процессе нагрева сплав перейдет в аустенитную фазу, восстанавливая прежнюю форму. Это явление называют однократным эффектом памяти формы или просто эффектом памяти формы (рис. 2, 3а).

Эффект памяти формы можно также инициировать путем деформирования образца в аустенитном состоянии. В этом случае форма образца может быть также восстановлена при охлаждении и нагреве под постоянной нагрузкой (рис. 3б).

Рисунок 2 - Изменение деформации в процессе реализации эффекта памяти формы

[5].

Эффект псевдоупругости (сверхупругости, сверхэластичности) реализуется в изделии из СПФ в аустенитном состоянии-если его нагрузить при высокой температуре, то после снятия нагрузки оно вернет себе исходную форму (рис. 4). Образцы из СПФ могут быть сверхупруго деформированы на 7 - 8% относительной

1

Температура

Температура

(а)

(б)

Рисунок 3 - Эффект памяти формы при охлаждении и нагреве под постоянным напряжением (а), при нагреве после деформирования и разгрузки в мартенситном состоянии (б).

длины, запасая в десятки раз большую энергию, чем обычная пружина.

Первые шаги к исследованию сплавов с памятью формы были сделаны в первой половине XX века, когда А. Оландер в 1932 г. заметил псевдоупругое поведение сплава АиСё [2]. Гренингер и Мурадиан (1938) [3] заметили появление и исчезновение кристаллов мартенсита в сплавах Си7п и С^п при изменении температуры. Само явление эффекта памяти формы, вызванное термоупругим мартенситным превращением, было описано десять лет спустя Курдюмовым и Хандросом [6], а затем в работе Чанга и Рида [7]. Открытый эффект быстро приобрел известность по всему миру, и к настоящему времени найдено более 120

Деформация

Рисунок 4 - Эффект сверхэластичности в СПФ.

подобных сплавов. Это сплавы на основе металлических систем AuCd, CuZnAl, CuAlNi, FeMnS, FeNi, CuAl, CuMn, CoNi, TiNi, NiAl и других.

Наконец, в начале 1960-х годов Бюлер с коллегами из Лаборатории ВМС США (U.S. Naval Ordinance Laboratory) открыли эффект памяти формы в эквиатомном сплаве никеля и титана [8]. Это событие стало прорывом в области материалов с памятью формы и послужило началом многочисленных исследований, посвященных природе необычного поведения СПФ. Оказалось, что сплавы на основе TiNi достаточно технологичны в обработке, устойчивы к коррозии и обладают отличными физико-механическими характеристиками: например, предел прочности TiNi находится в пределах 770 - 1100 МПа, что соответствует аналогичным характеристикам большинства сталей, и имеет демпфирующую способность выше, чем чугун. Поверхность элементов, как и у элементов из многих титановых сплавов, покрыта диоксидом титана, что предопределяет их высокую коррозионную стойкость к воздействию морской воды, большинства кислот и щелочей, а также биологическую совместимость [9].

Благодаря своим уникальным свойствам, сплав нашел широкое применение в различных отраслях промышленности: в кораблестроении [10, 11], аэрокосмической технике [10, 12, 13], в создании мартенситных приводов [14, 15], соединительных муфт [10, 11] и др. Сплавы на основе TiNi уникальны с точки зрения возможностей применения в медицине - из них изготавливают стенты, хирургические инструменты, имплантаты [10, 14, 16 - 21]. Среди кандидатов на замену костной ткани наиболее подходящим является пористый TiNi. Пористый TiNi - материал, в котором поровое пространство имеет важное функциональное значение, поскольку в процессе эксплуатации оно заполняется жидкостями и живыми тканями организма [18]. Наличие открытой пористости в имплантате способствует росту костной ткани и в дальнейшем улучшает фиксацию между поверхностью имплантата и кости. По этим причинам при исследовании пористого TiNi большое внимание уделяется описанию порового пространства. Сама металлическая матрица вступает в комплексное взаимодействие с тканями и жидкостями, включая механическое, электрохимическое, тепловое,

гидродинамическое [18]. Наконец, пористость уменьшает нестыковку между механическими свойствами костной ткани и имплантата, обеспечивая их

Рисунок 5 - Сравнение диаграмм деформирования костных тканей, пористого Т1М и пористого Т [17].

механическую совместимость. Изделия из ТМ с пористостью 30 - 80% могут обладать эффективным модулем Юнга, сравнимым с модулем Юнга кортикальной кости (12 - 17 ГПа) (рис. 5) [17].

Пористый Т1М может послужить материалом для заменителей позвонков в шейном и поясничном отделах позвоночника, костных пластин, для челюстно-лицевых и зубных имплантатов (рис. 6). Предполагается, что эти передовые

Рисунок 6 - Применение пористого Т1М при лечении травмы лобной кости [22].

материалы получат и другие применения, например, для вибрационной и сеймической изоляции в космической технике, в гражданских инженерных сооружениях. В связи с этим большое внимание научного сообщества, вдохновленного перспективами применения этих материалов, направлено на производство и моделирование пористых СПФ.

Для описания пористого СПФ используют характеристики, обычно применяемые для других пористых материалов. Главным понятием при изучении пористых материалов является пористость. Пористостью р называется отношение объема Ур пустот материала к его полному объему V. Ее определяют по формуле

Р = Ур/У. (1.1.1)

На практике для измерения пористости часто применяется метод секущих. Вдоль разных хорд замеряются длины участков, занятых порами, и общая длина хорды. Пористость вычисляется по формуле:

р = 100%, (1.1.2)

где Ьр - суммарная длина всех пор, Ь - длина выбранной хорды. Распределение пористости в материале или изделии можно определить следующими способами:

• методом микрофотографий;

• методом измерения расхода газа при его фильтрации через отдельные участки пористой поверхности;

• разрезкой материала на отдельные элементы с последующим определением пористости каждого их них.

Форма пор сложна и зависит от формы и размеров частиц, метода и условий получения. Поровые каналы имеют по всей длине большое число сужений и расширений и соединяются между собой многочисленными межпоровыми перегородками [23].

Такие параметры структуры пористых СПФ, как пористость, форма, размер и ориентация поровых каналов, полностью зависят от способа и условий получения пористого материала. Характеристики пористых СПФ на основе ^М, полученных разными методами, могут значительно отличаться и будут иметь разные свойства. Понимание влияния способа получения на структуру очень важно для создания материала с заданными свойствами и прогнозирования его поведения.

1.2 Методы получения пористых сплавов с памятью формы

Технологии получения пористых СПФ делятся на методы порошковой металлургии и методы аддитивных технологий [18]. История применений методов порошковой металлургии при производстве пористых СПФ насчитывает более двадцати лет. К этим методам относят простое спекание [24], самораспространяющийся высокотемпературный синтез (СВС) [25, 26], спекание при повышенном давлении с помощью горячего изостатического прессования (ГИП) [27], искровое плазменное спекание (ИПС) [28], инжекционное формование металла (ИФМ) [29] и микроволновое спекание [30].

Спекание (Conventional sintering, CS)

При обычном спекании порошки Ni и Ti смешиваются и затем спекаются при температурах, близких к температуре плавления. Благодаря диффузии частиц Ni и Ti получается сплав TiNi. Максимально возможная пористость при таком процессе - 40% [24, 31, 32]. Поры в образце, полученном спеканием с наполнителем, имеют форму и размер частиц наполнителя, в качестве которого используют NaCl [33], NH4HCO3 [34 - 36], (CO(NH2))2 [37] (рис. 7), NaF [38], Mg [39]. Преимуществами данного метода является его низкая стоимость, хорошая размерная точность,

Рисунок 7 - Микрофотографии пористого Т1М, полученного методом спекания с наполнителем с разными размерами частиц наполнителя, пористость 41,4 - 47,7% [37].

высокая производительность и отсутствие необходимости во вторичной обработке [24, 40]. Однако при использовании этого метода требуется долгое время нагрева [41], имеются трудности, связанные с удалением наполнителя, а производимые образцы ограничены в форме и размере. Кроме того, нельзя точно задавать размер пор и пористость без наполнителя, а в составе продукта имеются вторичные фазы [42, 43].

Самораспространяющийся высокотемпературный синтез (СВС) (Self-Propagating High-Temperature Synthesis, SHS)

Для получения пористого TiNi методом самораспространяющегося высокотемпературного синтеза порошки Ni и Ti спрессовывают в форме и подогревают, а затем поджигают шихту при помощи локального источника возгорания (вольфрамовая катушка, лазерный луч, микроволновая печь и т.д. [25, 44, 45]. Экзотермическая реакция, возникшая в области нагрева, порождает волну горения, послойно проходящую через всю заготовку, формируя пористую структуру образца. Параметры синтеза, такие как температура нагрева шихты, пористость заготовки, скорость нагрева и давление компактирования, сильно влияют на пористость готового продукта, его микроструктуру, функциональные и механические свойства [46 - 53]. Изменение условий синтеза также позволяет в некоторой степени контролировать средний размер и распределение пор [54, 55] (рис. 8). Равномерность температурного профиля в образце при синтезе сильно

Рисунок 8 - Пористый TiNi, полученный методом СВС [60].

влияет на однородность изделия. Реакция в образцах часто происходит не полностью из-за своей быстроты и высоких скоростей нагрева, в результате чего готовые образцы содержат включения, в частности Ti2Ni, Ni3Ti и Ni4Ti3 [42, 56]. Температура нагрева шихты влияет на количество переходной жидкой фазы, присутствующей на фронте горения, излишний нагрев может привести к анизотропии пористой структуры [53]. СВС может инициироваться двумя разными подходами: локальным началом реакции, которая затем распространяется по всему образцу [26, 53, 57], и объемным горением [58] - нагревом всего образца до температуры реакции, что дает реакции начаться во всем образце одновременно [53]. Метод является не энергозатратным, он относительно прост с технологической точки зрения и имеет низкую стоимость, а производимый продукт характеризуется высокой чистотой [25, 26], но неоднородной структурой. Пористые образцы, получаемые методом СВС, отличаются высокой пористостью (30 - 70 %, иногда до 80% [46] и большим размером пор [42, 59, 60].

Горячее изостатическое прессование (ГИП) (Hot Isostatic Pressure, HIP)

Горячее изостатическое прессование представляет собой метод спекания с повышенным давлением. Смесь элементарных частиц порошка инкапсулируется в вакуумированную сварную канистру и одновременно подвергается изостатическому давлению при повышенной температуре. В качестве инертной среды может быть использован аргон, тогда стадия диффузии под высоким давлением приводит к появлению пор, наполненных аргоном. Спекание продукта при пониженном давлении приводит к расширению газа и образованию в конечном продукте околосферических пор (рис. 9).

По своему составу образцы, полученные ГИП, относительно однородные, однако, в нем присутствуют частицы NiTi2 и Ni3Ti [42, 61]. По сравнению с обычным спеканием поры в образцах, полученных безкапсульным ГИП распределены более однородно [62 - 64]. Преимуществами этого метода являются короткое время диффузии, возможность изменения размеров и формы пор [42], низкая температура спекания [65], термодинамически стабильная и

Рисунок 9 - Микрофотографии пористого TiNi, полученного методом ГИП, пористость 39,2% [37].

Список литературы диссертационного исследования кандидат наук Япарова Елизавета Николаевна, 2020 год

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SAINT PETERSBURG STATE UNIVERSITY

Manuscript copyright

Elizaveta N. Iaparova

MODELLING OF FUNCTIONAL AND MECHANICAL BEHAVIOR OF POROUS SHAPE MEMORY ALLOY BASED ON THE APPROXIMATION OF ITS STRUCTURE BY

BEAM CONSTRUCTIONS

Dissertation is submitted for the degree of Candidate of Physics and Mathematics

01.02.04 - Solid Mechanics

Translation from Russian

Supervisor: Professor, Doctor of Science Alexandr E. Volkov

Saint Petersburg 2020

Contents

Introduction.....................................................................................108

1 Porous shape memory alloys and methods for their description...............115

1.1 Features of porous shape memory alloys' properties..........................115

1.2 Methods for producing porous shape memory alloys.........................121

1.3 Models of functional and mechanical behavior of SMA.....................130

1.4 Approaches to the calculation of the functional and mechanical properties of porous SMA.....................................................................136

2 Microstructural modelling of porous SMA using methods of the strength of materials...................................................................................142

2.1 Porous SMA with vertical orientation of pore channels.......................143

2.2 Porous SMA with horizontal orientation of pore channels..................153

2.3 Porous SMA with disordered orientation of pore channels...................158

2.4 Porous SMA obtained by the method of selective laser melting.............162

2.5 Determination of geometric parameters of the models.......................164

3 Verification of the model on samples of various porous structure.................166

3.1 Determination of the material constants of the microstructural model.....166

3.2 Calculation of uniaxial compression and shape memory effect in a porous SMA sample with the structure approximated by a cascade

of curved beams.................................................................... 168

3.3 Modelling of porous SMA sample with the structure approximated

by flat slotted springs...............................................................171

3.4 Concerning the mechanisms of inelastic strain of porous SMA............... 175

3.5 Calculation of the isothermal strain of porous SMA with disordered structure...............................................................................179

3.6 Modelling of the behavior of porous SMA with ordered structure obtained by selective laser melting.........................................................183

Conclusion....................................................................................... 187

References........................................................................................189

Introduction Relevance of the topic

In recent decades, shape memory alloys (SMA), both dense and porous, have attracted a lot of attention due to the possibilities of their use in a wide variety of fields from the aerospace industry to surgical instruments and medical implants. Among SMA, TiNi-based alloys are most widely used due to their high physical and mechanical characteristics and shape memory effect. Porous SMA obtained from a mixture of Ti and Ni or TiNi powders have good functional properties with the simplicity and high productivity of the methods for their preparation. These materials exhibit shape memory effect and pseudoelasticity (superelasticity), have a high damping capacity and a permeability specific to highly porous materials. Thus, porous TiNi has the features of both SMA widely used in technology and medicine, as well as porous permeable metallic materials. Porous TiNi products used in medicine are the most important from the practical point of view - they are used as materials for maxillofacial and dental implants, substitutes for the cervical and lumbar spine, substitutes for joints, bone plates, spinal fixators for fractures.

The demand for this materials and insufficiency of knowledge of its properties contributed to research on the development of models of porous SMA. In order to predict the behavior of porous SMA, it is necessary to create both simple and accurate model to describe its functional and mechanical behavior.

Almost all of currently available works on modelling of the behavior of porous SMA is limited to the description of isothermal deformation using macroscopic phenomenological models of SMA strain. The calculation of the porous SMA strain due to the temperature changes is presented only in a few works. In addition, existing models take into account the structural features of pore channels in the material only indirectly, while they significantly affect the deformation behavior of the samples. Therefore, modelling of the functional and mechanical properties of porous SMA, taking into account the features of the porous structure, is an urgent topic of scientific research.

Methods of research

To create a model for calculation of the stress-strain behavior of individual structural elements, as well as for its relation with macroscale values, equations and hypotheses of the strength of materials were used. For various types of structures, the approximation was performed using beam structures of various configurations. Inelastic strain was calculated for the most stressed sections of the structural elements using the microstructural model of SMA. In this model, the constitutive equations were formulated for the microlevel, and the strain of the representative volume of the macrolevel was calculated by averaging the microstrains. The complex porous structure was approximated based on the analysis of micrographs of cross sections of porous TiNi samples obtained by the methods of self-propagating high-temperature synthesis (SHS) and selective laser melting (SLM). The necessary values of the geometric parameters of the model were obtained by statistical analysis of the measurements of interporous ligaments (for samples obtained by SHS), or could be known by measuring of a small representative volume of the sample (for samples obtained by SLM). Material constants for comparison with the available experimental results were taken from the literature data. The algorithm for calculation the behavior of porous SMA was implemented using a program in C ++.

The aim and objectives of the research

The aim of this work was to create a model of the functional and mechanical behavior of porous SMA samples taking into account the structural features of the material.

To achieve this aim, the following objectives were set:

1. Perform the analysis of the structure of the porous TiNi-based samples, produced by SHS and SLM, with various orientation of pore channels, and approximation of these structures by constructions consisting of beams.

2. Determination of geometric parameters characterizing the porous structure of the samples.

3. Obtaining equations relating the average strain and stress to the strain and stress in structural elements for various types of porous structures.

4. Verification of models for calculating the functional and mechanical behavior of porous SMA samples for various types of porous structures.

5. Detection of the features of the functional and mechanical behavior of porous SMA related to the specifics of the porous structure.

Scientific novelty

1. For the first time, the behavior of porous SMA with different orientations of pore channels was modelled on the base of their approximation by beam structures, and the principle of determining their geometric parameters was developed using statistical analysis of micrographs.

2. Theoretically obtained stress-strain diagrams under compression for porous samples with different types of structure in different phase states.

3. The dependences of recoverable and irrecoverable strain on the number of thermomechanical loading cycles for the porous samples fabricated by SLM were found.

4. For the first time, a simulation of the strain variation in a highly porous SMA during cooling and heating under load (shape memory effect) was performed.

Provisions to be defended

1. The model of the functional and mechanical behavior of porous SMA with pore channels oriented parallel to the specimen's axis, based on the approximation of the porous structure by cascades of curved beams.

2. The model of the functional and mechanical behavior of porous SMA with pore channels oriented perpendicular to the sample axis, based on the approximation of the porous structure by flat slotted springs.

3. The model of the functional and mechanical behavior of porous SMA with disordered orientation of pore channels, based on the approximation of the porous structure by the construction of horizontally oriented beams connected to each other by curved beams, as a flat slotted spring.

4. The model of the functional and mechanical behavior of porous SMA obtained by SLM, based on the introduction of an effective stress considering porosity.

5. The results of calculations of porous samples' strain under compression in various phase states and during cooling and heating under stress (shape memory effect), the calculation of recoverable and irrecoverable strain under thermomechanical cyclic loading.

Reliability of the results

The reliability of the obtained results is achieved due to their correspondence to the available experimental data, using classical methods of the strength of materials and time-tested microstructural model of SMA. Due to a fairly accurate representation of the structure of the sample by approximating structures, the calculated dependences are in good agreement with experimental results.

Theoretical and practical significance of the work

The developed model can be used to describe the strain variation of porous SMA obtained by various methods during isothermal deformation, as well as during thermomechanical cycling. Using this model, one can determine the influence of the contributions of various deformation mechanisms on the functional and mechanical behavior of porous TiNi. The practical significance of the work lies in the fact that this model can be used to assess the properties of porous SMA products under arbitrary modes of stress and temperature variations.

Approbation of work

The results of the work were presented at the seminars of the Department of the Theory of Elasticity of SPbU and School of Computing and Mathematics in Keele University (Keele, UK), in the House of scientists of the RAS, and at all-Russian and international conferences:

1. "The 7th Polyakhov's Reading", Saint Petersburg, Russia, 2015;

2. "10th European Symposium on Martensitic Transformations" (ESOMAT-2015), Antwerp, Belgium, 2015;

3. LVII International Conference "Actual problems of strength", Sevastopol, Russia, 2016;

4. VIII International Conference "Micromechanisms of Plasticity, Fracture and Accompanied Phenomena" (MPFP-2016), Tambov, Russia, 2016;

5. Second International Conference "Shape Memory Alloys II" (SMA II), Saint Petersburg, Russia, 2016;

6. LVIII International Conference "Actual problems of strength", Perm, Russia,

2017;

7. VIII International School "Physical Materials Science" with elements of a scientific school for youth, Tolyatti, Russia, 2017;

8. "The 8th Polyakhov's Reading", Saint Petersburg, Russia, 2018;

9. Third International Conference "Shape Memory Alloys III" (SMA III), Chelyabinsk, Russia, 2016;

10. "11th European Symposium on Martensitic Transformations" (ESOMAT 2018), Metz, France, 2018;

11. International Symposium "Perspective Materials and Technologies", Brest, Belarus, 2019;

12. «Intermetallics 2019», Bad-Staffelstein, Germany, 2019.

Structure and scope of work

The work consists of the introduction, three chapters and conclusion, and contains 102 pages and 46 drawings. The list of references contains 207 items.

Publications on the research topic

a) Publications in journals indexed in Scopus and Web of Science:

1. Volkov A.E., Evard M.E., Iaparova E.N. A beam model of porous shape memory alloy deformation // Materials Today: Proceedings. 2017. V. 4. No. 3. P. 4631-4636.

2. Volkov A.E., Evard M.E., Iaparova E.N. Modeling of functional properties of porous shape memory alloy // MATEC Web of Conferences. 2015. V. 33. P. 02006.

b) Publications in journals included in the List of VAK:

1. Volkov A.E., Evard M.E., Iaparova E.N. Concerning the selection of boundary conditions during the computer modelling of functional-mechanical behavior of porous shape memory alloy samples // Vector of Science TSU. 2017. V. 4. No. 42. P. 26-31.

2. Volkov A.E., Evard M.E., Iaparova E.N. Modeling of isothermal compression of porous TiNi samples with longitudinal and transversal orientation of porous channels// Deformation and Fracture of Materials. 2017. V. 4. P. 9-14.

3. Volkov A.E., Evard M.E., Iaparova E.N. Deformation of porous shape memory alloy sample with pores transversally oriented relative to the load direction // Tambov University Reports. 2016. V. 21. No. 3. P. 913-916.

c) Other publications:

1. Iaparova E., Volkov A., Evard M., Belyaev F. Simulation of cyclic functional and mechanical behavior of porous NiTi samples obtained by selective laser melting // Intermetallics 2019: Programme and Abstracts. 2019. P. 184-185.

2. Iaparova E.N., Volkov A.E., Evard M.E., Belyaev F.S. Modelling of the behavior of porous TiNi with regular structure under cyclic thermomechanical loads // Perspective Materials and Technologies: Proceedings of the International Symposium (Brest, May 27-31, 2019). Vitebsk: EE "VSTU" 2019. P. 417.

3. Volkov A., Evard M., Iaparova E. Elucidation of the role of the structure of porous TiNi for its mechanical and functional properties // ESOMAT 2018: 11th European Symposium on Martensitic Transformations (August 27-31, 2018). Book Of Abstracts. 2018. P. 21-22.

4. Volkov A.E., Evard M.E., Iaparova E.N. Mechanical model of porous shape memory alloy sample with disordered structure // Shape Memory Alloys. Third International Conference (Chelyabinsk, Russia, August 16-20, 2018): Conference Proceedings. Chelyabinsk: CSU Publishing office. 2018. P. 75.

5. Volkov A.E., Evard M.E., Iaparova E.N. Concerning mechanisms of inelastic deformation of TiNi shape memory alloy porous samples // The Eighth Polyakhov's

Reading: Book of abstracts of the International Scientific Conference on Mechanics, Saint Petersburg (January 30 -February 2, 2018). St. Petersburg: Publishing house of SPbU. 2018. P. 192-193.

6. Evard M., Volkov A., Iaparova E. Microstructural modeling of functional properties of porous shape memory alloy // International ECCOMAS Thematic Conference "Computational modeling of complex materials across the scales (CMCS)" (Paris, France, September, 7-9, 2017): Abstracts. 2017. P. 156126.

7. Volkov A.E., Evard M.E., Iaparova E.N. The use of the beam model for calculating of porous shape memory alloy sample // Shape Memory Alloys. Second International Conference (Saint Petersburg, Russia, September 20-23, 2016). Book of Abstracts. St. Petersburg: Publishing house VVM. 2016. P. 19.

8. Volkov A.E., Evard M.E., Iaparova E.N. Deformation of porous shape memory alloy sample with transversal to the load direction pores' orientation // Actual Problems of Strength: Book of Abstracts of the LVII International Conference (May 24-27, 2016). Sevastopol: SevSU. 2016. P. 203.

9. Volkov A.E., Evard M.E., Iaparova E.N. Calculation of the strain of a porous shape memory alloy // The Seventh Polyakhov's Reading: Book of Abstracts of the International Scientific Conference on Mechanics, Saint Petersburg (February 2-6, 2015). St. Petersburg: Publishing house of SPbU. 2015. P. 162.

1 Porous shape memory alloys and methods for their description 1.1 Features of porous shape memory alloys' properties

The functional and mechanical behavior of porous SMA is conditioned by the properties of the corresponding non-porous alloys. It is known that upon cooling, the growth of thermoelastic martensite crystals occurs (forward martensitic transformation), and upon heating, they decrease and disappear (reverse martensitic transformation). Characteristic temperatures of transformation are important parameters of the phase transformation (Fig. 1):

• MS - start temperature of the forward martensitic transformation,

• Mf - finish temperature of the forward martensitic transformation,

• As - start temperature of the reverse martensitic transformation,

• Af - finish temperature of the reverse martensitic transformation.

TempereKire

Figure 3 - Phase transformation scheme [1].

On cooling to a temperature Mf, the initial cubic austenite lattice transforms into a non-cubic martensite lattice. As soon as the temperature exceeds Af again, the crystal lattice acquires the ordered cubic structure of austenite [2 - 4].

If the SMA sample is subjected to a deformation in martensite state, this leads to a reorientation of the crystal lattice and the formation of oriented martensite. The new shape of the sample persists until the sample is heated to the reverse martensite transformation

start temperature, and during the heating process, the material is transforming into the austenitic phase, restoring the previous shape. This phenomenon is called the one-way shape memory effect or shape memory effect (Fig. 2, 3a).

The shape memory effect can also be initiated by deforming the sample in the austenitic state. In this case, the sample shape can be restored by cooling and heating under a constant load (Fig. 3b).

Figure 4 - Stress-strain-temperature data exhibiting the shape memory effect [5].

The effect of pseudo-elasticity (superelasticity) is realized in SMA samples in the austenitic state: if it is loaded at high temperature, then it returns to its original shape after unloading (Fig. 4). SMA samples can be superelastically deformed by 7-8%, storing tens of times more energy than an ordinary spring.

The first steps in the study of SMA were made in the first half of the 20th century, when A. Olander in 1932 noticed the pseudoelastic behavior of the AuCd alloy [2]. Greninger and Mooradian (1938) [3] observed the formation and extinction of martensite crystals in CuZn and CuSn alloys caused by a change in temperature.

Temperatuifc Temperature

(a) (b)

Figure 3 - Shape memory effect occurs (a) during cooling and heating at a constant stress, (b) during heating after deformation and unloading in the martensitic state.

Strain

Figure 4 - Superelasticity effect in SMA.

The phenomenon of the shape memory effect caused by the thermoelastic martensitic transformation was described ten years later by Kurdjumov and Khandros [6], and then in the work of Chang and Read [7]. The discovered effect quickly gained fame around the world, and to date, more than 120 similar alloys have been found. These are alloys based on metal systems AuCd, CuZnAl, CuAlNi, FeMnS, FeNi, CuAl, CuMn, CoNi, TiNi, NiAl and others.

Finally, in the early 1960s, Buehler and his colleagues from the U.S. Naval Ordinance Laboratory discovered the shape memory effect in an equiatomic alloy of

nickel and titanium [8]. This event was a breakthrough in the field of materials with shape memory effect and gave rise to the numerous studies on the nature of the unusual behavior of SMA. It turned out that TiNi-based alloys are quite technologically advanced in processing, resistant to corrosion and have excellent physical and mechanical characteristics: for example, the tensile strength of TiNi is in the range of 770 - 1100 MPa, which corresponds to the similar characteristics of most steels, and has a damping capacity higher than cast iron. The surface of the elements, like that of elements from many titanium alloys, is coated with titanium dioxide, which determines their high corrosion resistance to sea water, most acids and alkalis, as well as biological compatibility [9].

Due to its unique properties, the alloy has found wide application in various industries: in shipbuilding [10, 11], aerospace engineering [10, 12, 13], in the producing of martensitic actuators [14, 15], couplings [10, 11] and others. The TiNi-based alloys are unique in terms of medical applications — stents, surgical instruments, and implants are made of them [10, 14, 16-21]. Among candidates for bone tissue replacement, porous TiNi is the most suitable. Porous TiNi is a material in which the pore space has important functional significance, since during work it is filled with body fluids and living tissues [18]. The presence of an open porosity in the implant promotes bone growth and further improves fixation between the surface of the implant and the bone. For these reasons, much attention in the study of porous TiNi is paid to the description of the pore space. The metal matrix itself enters into a complex interaction with tissues and fluids, including mechanical, electrochemical, thermal, and hydrodynamic [18]. Finally, the pores reduce the inconsistency between the mechanical properties of the bone tissue and the implant, ensuring their mechanical compatibility. Products made of TiNi with a porosity of 30 -80% may have an effective Young's modulus comparable to the Young's modulus of the cortical bone (12 - 17 GPa) (Fig. 5) [17].

Porous TiNi can be used as a material for vertebral substitutes in the cervical and lumbar spine, bone plates, for maxillofacial and dental implants (Fig. 6). It is assumed that these advanced materials will receive other applications, for example, for vibration and seismic isolation in space technology, in civil engineering structures. Thereby, much

attention of the scientific community, inspired by the prospects of using these materials, is directed to the production and modelling of porous SMA.

Strain [%]

Figure 5 - Comparison of stress-strain curves of bone tissues, porous TiNi and porous Ti [17].

Figure 6 - The use of porous TiNi in the treatment of frontal bone injury [22].

Characteristics commonly used for other porous materials are used to describe porous SMA. The main concept in the study of porous materials is porosity. Porosity p is the ratio of the volume Vp of the voids of the material to its total volume V. It is determined by the formula

P = vp/v.

(1.11)

In practice, the secant method is often used to measure the porosity. Along different chords, the lengths of the areas occupied by the pores and the total length of the chord are measured. Porosity is calculated by the formula:

p = ^*100%, (1.1.2)

where Lp is the total length of all pores, L is the length of the selected chord. The pore distribution in a material or product can be determined in the following ways:

• method of microphotographs;

• the method of measuring gas flow during its filtration through separate sections of the porous surface;

• cutting material into separate elements with subsequent determination of the porosity of each of them.

The shapes of pores are complex and depend on the shape and size of the particles, the methods and preparation conditions. Pore channels have a large number of constrictions and extensions along the entire length and are interconnected by numerous interporous ligaments [23].

Such parameters of the structure of porous SMA as the porosity, shape, size and orientation of pore channels completely depend on the methods and conditions of obtaining the porous material. The characteristics of porous TiNi-based SMA obtained by different methods can vary significantly and, accordingly, these samples have different properties. Understanding the influence of the production method on the structure is very important for creating a material with desired properties and predicting its behavior.

1.2 Methods for producing porous shape memory

The technologies for producing porous SMA are divided into methods of powder metallurgy and additive manufacturing methods [18]. The history of the application of powder metallurgy methods in the production of porous SMA has more than twenty years. These methods include conventional sintering [24], self-propagating high-temperature synthesis (SHS) [25, 26], sintering at elevated pressure using hot isostatic pressing (HIP) [27], spark plasma sintering (SPS) [28], injection molding metal (IFM) [29] and microwave sintering [30].

Conventional sintering (CS)

In conventional sintering, Ni and Ti powders are mixed and then sintered at temperatures close to the melting temperature. Due to the diffusion of Ni and Ti particles, a TiNi alloy is obtained. The maximum possible porosity in this process is 40% [24, 31, 32]. The pores in the sample obtained by sintering with a space-holder have the shape and size of the space-holder particles, which are NaCl [33], NH4HCO3 [34 - 36], (CO (NH2))2 [37] (Fig. 7), NaF [38], Mg [39]. The advantages of this method are its low cost, good dimensional accuracy, high productivity and lack of need for secondary processing [24,

Figure 7 - Microphotographs of porous TiNi samples obtained by conventional sintering with different sizes of space-holder particles, porosity - 41.4 - 47.7% [37].

40]. However, using this method requires a long heating time [41]. There are difficulties associated with the removal of the space-holder particles, and the produced samples are limited in shape and size. In addition, it is impossible to accurately set the pore size and porosity without space-holder, and the final product contains precipitates [42, 43].

Self-Propagating High-Temperature Synthesis (SHS)

To obtain porous TiNi by the method of self-propagating high-temperature synthesis (SHS), Ni and Ti powders are pressed into a mold and heated, and then the mixture is burnt using a local ignition source (tungsten coil, laser beam, microwave oven, etc. [25, 44, 45]. The exothermic reaction occurred in the heating region generates a combustion wave, passing layer-by-layer through the green specimen and forms the porous structure of the sample. Synthesis parameters, such as the preheating temperature, the porosity of the green specimen, the heating rate and pressure of compacting, strongly influence the porosity of the final product, its microstructure and functional and mechanical properties [46 - 53]. Changing the synthesis conditions also allows to control the average pore size and distribution [54, 55] (Fig 8). The uniformity of the temperature profile in the sample during synthesis strongly affects the homogeneity of the product. The reaction in the samples often does not occur completely due to its rapidity and high heating rates; as a result, the final samples contain precipitates, in particular, Ti2Ni, Ni3Ti,

Figure 8 - Porous TiNi obtained by SHS [60].

and Ni4Ti3 [42, 56]. The preheating temperature affects the amount of the transitional liquid phase present at the combustion front; excessive heating can lead to anisotropy of the porous structure [53]. The SHS process can be initiated by two different ways: the local onset of the reaction, which then spreads throughout the sample [26, 53, 57], and volume combustion [58] - by heating the entire sample to the reaction temperature, which allows the reaction to begin in the entire sample simultaneously [53]. The method is not energy-consuming, it is relatively simple from a technological point of view and has a low cost, and the produced product is characterized by high purity [25, 26], but with a heterogeneous structure. The porous samples obtained by the SHS method are characterized by high porosity (30-70%, sometimes up to 80% [46] and a large pore size [42, 59, 60].

Hot Isostatic Pressure (HIP)

Hot isostatic pressing is a high pressure sintering method. A mixture of elementary powder particles is encapsulated in a vacuum welded canister and at the same time subjected to isostatic pressure at elevated temperature. Argon can be used as an inert medium, and then the stage of diffusion under high pressure leads to the appearance of pores filled with argon. Sintering of the product under reduced pressure leads to gas expansion and the formation of near-spherical pores in the final product (Fig. 9).

The composition of the samples obtained by HIP is relatively homogeneous, however, NiTi2 and Ni3Ti particles can be observed in it [42, 61]. Compared to the

Figure 9 - Microphotographs of porous TiNi obtained by HIP, porosity - 39.2% [37].

conventional sintering, pores in the samples obtained by capsule-free HIP are more uniformly distributed [62-64]. The advantages of this method are the short diffusion time, the ability to vary the size and shape of the pores [42], the low sintering temperature [65], the thermodynamically stable and controlled reaction, the ability to obtain samples of large size and or complex shape, high efficiency of utilization of materials. However, the equipment has a high cost and low productivity [40].

Spark Plasma Sintering (SPS)

In spark plasma sintering, pre-alloyed powders of nickel and titanium are placed in a graphite capsule and pressed. A large pulsed current is supplied to the pressed mass, transferring a large amount of energy to it, initiating sintering of the powders [28, 66]. SPS occurs at a low temperature and for a short time, this avoids any undesirable reaction products [67]. The method is simple to operate and has high energy efficiency, temperature and pressure changes can be controlled with great accuracy, and sintering itself occurs evenly [28, 68] (Fig. 10). The disadvantages of SPS are the high cost of a pulsed current generator [67] and the possibility of obtaining samples with only a simple form [69].

Figure 10 - Microphotographs of porous TiNi obtained by SPS [66], porosity - 18%, 39%, 42%.

Metal Injection Molding (MIM)

The MIM technology (Fig. 11) consists of four stages: production of raw materials, injection molding, binder removal and sintering [17, 70, 71]. In the manufacture of raw

Figure 11 - Porous TiNi obtained by MIM [17], porosity - 51%.

materials, the powder is mixed with a binder of a certain composition and percentage. Then the raw material is placed in the mold, its appearance affects the pressure and temperature distribution during further sintering. Removing the binder is carried out in a salt bath, as well as in vacuum at elevated temperature to evaporate the binder from the product. The MIM process is used for the mass production of small parts of complex shape at an adequate cost [29, 40, 72]. The disadvantages of the method are the high sintering temperature, a large number of impurity phases [65] and expensive equipment [40].

Microwave Sintering (MS)

A mixture of Ni and Ti powders is pressed at low temperature, then placed in the microwave machine. The green specimen absorbs electromagnetic energy, which turns into heat, and the mixture is heated to sintering temperature [30]. Compared to conventional sintering, the microwave sintering method consumes less energy, has a higher heating rate [73 - 75] and, as a result, shorter sintering time, improved diffusion of elements and improved physical and mechanical properties of sintered materials.

The method allows obtaining a porous structure without the use of any pore-forming substances. The pore size and porosity can be controlled by temperature and sintering time. The porosity of the final product producing by this method is 22 - 62% (Fig. 12).

Figure 12 - Microphotographs of porous TiNi obtained by MIM with different size of particles of space-holder [75], porosity of the samples - (a) 22%, (b) 41%, (c) 50%, (d) 62%.

Additive manufacturing

The general term additive manufacturing (AM) describes the processes where details are constructed by adding sequential layers of material, so there is no excessive consumption of raw materials in this production. Each layer is printed in accordance with precisely defined geometry defined by a three-dimensional computer model (3D CAD). The advantages of AM are the ability to create parts with complex geometry. These are economically and energy efficient, environmentally friendly production processes.

Selective Laser Sintering (SLS)

To create a product by the SLS method, a powder layer is applied to the surface of the substrate and evenly distributed using a rotating roller. The laser beam focused by the optical system scans the resulting surface according to the 3D model and forms layer by layer by sintering. Thus, areas corresponding to the current section of the sample are printed. Then the movable bottom of the chamber is lowered to a distance equal to the thickness of the powder layer, proceeding to the next section of the product [76]. For the

synthesis of porous TiNi, the combined SLS-SHS processes are also used [77 - 79]. The combination of processes allows to obtain an implant with a better chemical composition with better biocompatibility. Products made from TiNi require subsequent machining or polishing [80, 81]. In the SLS process, preheating is also necessary to reduce powder requirements, as well as to prevent thermal expansion deformation and shrinkage. Finally, in order to avoid deformation and oxidation of the finished product, a controlled cooling cycle is necessary [76].

Selective Laser Melting (SLM)

In selective laser melting, the working surface is covered with a thin layer of powder, and the laser beam, in accordance with the 3D model, melts the powder. After the end of the first layer, the working surface is lowered by a distance equal to the thickness of the layer, and the next layer is built. The process goes on until the entire product is completely built [82 - 84]. The use of long-wavelength lasers that are better tuned to the absorption capacity of metal powders is one of the characteristic features of the SLM method. Almost all SLM machines today have more affordable fiber lasers, they are compact and energy efficient. Another key advantage of SLM is a different laser scanning scheme, the use of lenses that minimize beam distortion during scanning, and a low oxygen content in an inert atmosphere [76]. The advantages of SLM are also the absence of a clear binder of the molten phase (for example, it is possible to produce individual parts of the sample from polymers, metals, and ceramics) [76, 80, 85], the absence of the need for long and costly post-processing in the furnace, infiltration or post-sintering, and the efficiency of use materials [86, 87]. However, an increase in the scanning speed leads to geometric discrepancies of the obtained material with the original model, although it improves its mechanical properties [20]. Therefore, scanning is performed at a low speed, the surface of the products has low quality and large residual distortions [88], and high energy costs are required for the operation of the laser [80, 85]. The porous samples obtained by the SLM method are shown in Fig. 13.

Figure 13 - Porous TiNi, obtained by SLM [89], porosity (from left to right) - 32 %, 45 % and 58 %.

Laser Engineered Net Shaping (LENS)

In a LENS system, a molten bath is formed under the influence of a fiber laser beam, into which metal powder enters. When the building of the layer is completed, the head moves in the vertical direction [76, 90]. However, the surface of the product requires both polishing and finishing. Powder particles cannot be removed from closed pores, but they are not in direct contact with bone tissue [91]. Laser processing prevents the appearance of extraneous intermetallic compounds such as Ti2Ni, Ni4Ti3 and Ni3Ti, but partial melting of the powders near the pores can reduce the solidification rate due to small temperature gradients and poor thermal conductivity, which leads to a larger grain size.

Electron Beam Melting (EBM)

In electron beam melting, a product is made by layer-by-layer melting of metal powders using an electron beam. The device uses a thermionic emission gun, in which a tungsten filament is used to create an electron beam [87, 91]. The manufacture of the details in a vacuum chamber ensures the absence of impurities from the atmosphere in the details [82, 86, 87]. Compared to a laser beam, an electron beam has a higher energy density, which leads to a reduction in the part creation time and lower production costs [76, 80, 85]. Unlike other methods of AM, which are based on laser sintering, EBM is able to produce completely dense products without pores and inclusions [91]. The

strength and modulus of elasticity can be selectively optimized for greater bone compatibility in the samples produced by EBM. EBM technology is only suitable for completely dense metal products, has low dimensional accuracy and poor surface finish. The disadvantage is the high cost of environment for maintaining a vacuum [80].

The variety of methods for producing porous SMA makes it possible to create products from the materials with different structures and properties. In this regard, it is necessary to have methods for predicting the behavior of these materials depending on their internal structure, and it is important to take into account the deformation mechanisms that occur in SMA. The existing constitutive relations are often used in the modelling of the behavior of porous SMA. They were developed earlier to describe the functional and mechanical behavior of dense SMA.

1.3 Models of functional and mechanical behavior of SMA

The models describing the features of SMA behavior use various potentials or molecular dynamics methods [92 - 94]. The earliest microstructural models are considered to be the work of F. Falk (1980, 1983) [95, 96]. The Landau-Devonshire free energy was used due to the similarity of the electromagnetic curves of ferromagnetic materials with the deformation behavior of SMA under uniaxial load [95]. This was followed by the works of G. Barsh and J. Krumhansl (1988) [97], J.M. Ball and R.D. James (1989) [98], R. Abeyaratne and J.K. Knowles (1993) [99] which were based on the Ginzburg-Landau theory.

Micro-macro models for describing the behavior of materials at the micro and meso levels are based on micromechanics, and scaling up the constitutive relations for the macro level [100 - 105]. The development of these models requires the use of appropriate state variables (usually temperature and stress or strain) and internal variables. Volume fraction of the martensite and transformation strain are usually taken as internal variables. The two main micro-macro modelling approaches are micromechanical and microplane or microspherical modelling.

In the micromechanical models, the behavior of the SMA polycrystal at the macro level is calculated by strain averaging over grains. In this case, methods of large-scale transition, such as the Mori-Tanaki scheme [106 - 108] and self-consistent method [109 - 121] are used. The transition between scale levels can be carried out numerically, for example, by the finite element method, to take into account the complex interaction between neighboring grains [112 - 116]. Early micromechanical models of SMA were proposed by L. Delay et al. (1987) [117], F. Fischer and K. Tanaka (1992) [118] and B. Raniecki (1992) [119]. Micromechanical models considering plastic strain were proposed by Q. Sun (1991) [120], A. Sadjadpour and K. Bhattacharya (2007) [104], X. Wang (2008) [121] and C. Yu (2012) [122]. E. Patoor (1988, 1994) [109, 123] used crystallographic data to describe the martensitic transformation of an SMA polycrystal. The model described pseudoelastic behavior, but could not describe the self-accommodation of martensite. Z. Lu and G. Weng (1997) [124] developed a similar

model with one variant of martensite, while B. Goo and C. Lexcellent, (1997) [125] proposed a multivariate formulation considering both chemical and surface energies. Alternative expressions for the interaction energy in SMA were obtained by M. Huang and L. Brinson (1998) [126], and N. Siredey (1999) [107]. Other micromechanical models were developed by K. Gall and H. Sehitoglu (1999) [111], T. Lim and D. McDowell (2002) [114], X. Wang (2006, 2008) [121, 127], C. Yu et al. (2012) [122], V.I. Levitas (2013) [128] and Y. Zu (2014) [129].

The microplane theory was developed in 1938 by G. Taylor (1938) [130], it described the multiaxial macroscopic behavior of SMA as a superposition of uniaxial responses inside different planes of different orientations, called "microplanes", under the assumption of static [131] or kinematic constraints [132]. With this approach, the macroscopic stress tensor at a material point was projected onto each microplane passing through this point, and one-dimensional constitutive relations were set at the micro level. Then the homogenization procedure was used to generalize the one-dimensional case to the three-dimensional one. This modelling method is thermodynamically consistent and is capable to simulate disproportionate loading paths, anisotropic behavior, and tensioncompression asymmetries in SMA [133]. Using this approach, R. Ostwald, (2014) [134] calculated pseudoelastic and pseudoplastic behavior, as well as strain curves. This work was finalized [135] to simulate the behavior of SMA under cyclic loading. In conclusion, micro-macro models allow carrying out accurate calculations based on physical hypotheses; however, it is necessary to set many state variables and material constants to use them, which makes the modelling time-consuming.

Macromodels (phenomenological models) describe the behavior of a SMA polycrystal based on phenomenological assumptions, simplified micro-macro thermodynamics, or direct fitting of experimental data. K. Tanaka and S. Nagaki (1982) [136] established the first phenomenological model using internal variables to describe thermoelastic phase transformation in SMA under uniaxial loading.

To circumvent the need for explicit restrictions on the volume fractions of different phases, C. Liang and C. Rogers (1990) [137] proposed a 3D model using the evolutionary cosine function for the volume fraction. L. Brinson et al. (1993) [138, 139] expanded this

model, dividing the volume fraction into two parts: one included temperature, the other included stress.

The model based on the application of approaches of statistical physics to calculate the strain recovery during phase transformation was developed by J.G. Boyd and D.C. Lagoudas (1996) [140]. This phenomenological model was the first model that took into account the reorientation of martensite due to the introduction of the inelastic strain tensor. D.C. Lagudas and P. Enchev (2000) [141] developed a three-dimensional, rate-independent thermomechanical constitutive model for SMA that took into account the accumulation of plastic strain upon inducing the martensitic phase. P. Popov, D.C. Lagoudas, (2007) [142] presented a three-dimensional constitutive model of polycrystalline SMA, which took into account the transition from austenite to twinned martensite and the twinning of self-accommodated martensite.

A.C. Souza et al. (1998) [143] proposed a 3D phenomenological model for calculation of the behavior of a CuZnAlMn polycrystal by writing down the Helmholtz free energy in terms of total strain and temperature. This model was improved by F. Auricchio and L. Petrini (2002) [144], who proposed an algorithm for calculating the shape memory effect and pseudoelasticity under uneven loading, and then V. Evangelista (2009) [145] developed 3D and 1D sequential models, where the role of various material parameters was emphasized. Subsequently, temperature energy [146] was introduced into the model [143]. S. Leclercq and C. Lexcellent (1996) [147], generalized the model proposed by Ranieki [118] to create the first macromodel that took into account the reorientation of martensite. The authors examined the differentiation of martensitic crystals into oriented and self-accommodating, and obtained the constitutive relations based on the thermodynamics equations.

The most famous approaches to the modelling of SMA behavior developed in Russia and former USSR are the models of A.A. Movchan and Likhachev-Malinin. In the model of A.A. Movchan [148, 149] an internal variable was introduced to determine the volume fraction of martensite, for which temperature dependences were constructed and phase deformation was calculated. A specific feature of the model is the ability to take into account the influence of strain in the oriented transformation.