Probabilistic Fatigue Crack Initiation and Propagation Fields Using the Strain Energy Density

Recently, Huffman developed a strain energy density based on Walker-like stress life and fatigue crack growth behavior. In this paper, the Huffman model based on local strain energy density is used to predict the fatigue crack initiation and propagation for the P355NL1 pressure vessel steel. This mo...

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Дата:	2018
Автори:	Correia, J.A.F.O., Huffman, P.J., De Jesus, A.M.P., Lesiuk, G., Castro, J.M., Calcada, R.A.B., Berto, F.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут проблем міцності ім. Г.С. Писаренко НАН України 2018
Назва видання:	Проблемы прочности
Теми:	Научно-технический раздел
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/173940
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Probabilistic Fatigue Crack Initiation and Propagation Fields Using the Strain Energy Density / J.A.F.O. Correia, P.J. Huffman, A.M.P. De Jesus, G. Lesiuk, J.M. Castro, R.A.B. Calcada, F. Berto // Проблемы прочности. — 2018. — № 4. — С. 128-145. — Бібліогр.: 32 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1739402025-02-09T14:21:11Z Probabilistic Fatigue Crack Initiation and Propagation Fields Using the Strain Energy Density Построение вероятностных полей зарождения и роста усталостных трещин по критерию плотности энергии деформации Correia, J.A.F.O. Huffman, P.J. De Jesus, A.M.P. Lesiuk, G. Castro, J.M. Calcada, R.A.B. Berto, F. Научно-технический раздел Recently, Huffman developed a strain energy density based on Walker-like stress life and fatigue crack growth behavior. In this paper, the Huffman model based on local strain energy density is used to predict the fatigue crack initiation and propagation for the P355NL1 pressure vessel steel. This model is combined with the generalized probabilistic fatigue model proposed by Correia aiming the generation of probabilistic fatigue crack initiation and propagation fields. In this study, the local stress and strains at the crack tip were obtained combining linear-elastic and elastoplastic analyses. The probabilistic fatigue crack growth rates fields for several stress R-ratios are estimated considering strain, SWT and equivalent stress amplitude damage parameters. A comparison between the experimental fatigue crack growth (FCG) data and the generated probabilistic FCG fields is made with very satisfactory correlations being found. В настоящее время Хаффман разработал концепцию плотности энергии деформации, основанную на циклической долговечности и развитии усталостной трещины по типу Волкера. Модель Хаффмана, базирующаяся на локальной плотности деформации, используется для прогнозирования инициирования и распространения усталостной трещины в стали P355NL1, применяемой в сосудах высокого давления. Данная модель сочетается с обобщенной вероятностной моделью усталости, ранее предложенной одним из соавторов, суть которой состоит в генерации вероятностных полей инициирования и распространения усталостных трещин. Получены локальные напряжения и деформации в вершине трещины, объединяющие их линейно-упругие и упругопластические составляющие. Для нескольких значений коэффициента асимметрии цикла напряжений R оценены вероятностные поля скоростей роста усталостных трещин с учетом параметров повреждения, параметра Смита–Ватсона–Топпера (SWT) и эквивалентных параметров амплитуды напряжения. Сравнение экспериментальных данных по приращению усталостной трещины с прогнозируемыми вероятностными полями ее роста показывает их тесную корреляцию. На сьогодні Хаффман розробив концепцію густини енергії деформації на основі циклічної довговічності і поширення тріщини від утомленості по типу Волкера. Модель Хаффмана, що базується на локальній густині енергії деформації, використовується для прогнозування ініціювання і поширення тріщини від утомленості в сталі P355NL1, яка використовується для виготовлення посудин високого тиску. Дана модель поєднується з узагальненою імовірнісною моделлю утоми, раніше запропонованою одним із співавторів, суть якої полягає в генерації імовірнісних полів ініціювання і поширення тріщин від утомленості. Отримано локальні напруження і деформації у вершині тріщини, що об’єднують їх лінійно-пружні і пружнопластичні складові. Для декількох значень коефіцієнта асиметрії циклу напружень R оцінено імовірнісні поля швидкостей росту тріщин від утомленості з урахуванням параметрів пошкодження, параметра Сміта–Ватсона–Топпера (SWT) й еквівалентних параметрів амплітуди напруження. Порівняння експериментальних даних щодо приросту тріщини від утомленості з прогнозованими імовірнісними полями її росту свідчить про їх тісну кореляцію. The authors of this work would like to express their gratitude to the SciTech-Science and Technology for Competitive and Sustainable Industries, R&D project NORTE-01-0145-FEDER-000022 co-financed by Programme Operational Regional do Norte (“NORTE2020”) through Fundo Europeu de Desenvolvimento Regional (FEDER) and the Portuguese Science Foundation (FCT) through the post-doctoral grant SFRH/BPD/107825/2015 the for their collaboration, financial and technical support during this research works. 2018 Article Probabilistic Fatigue Crack Initiation and Propagation Fields Using the Strain Energy Density / J.A.F.O. Correia, P.J. Huffman, A.M.P. De Jesus, G. Lesiuk, J.M. Castro, R.A.B. Calcada, F. Berto // Проблемы прочности. — 2018. — № 4. — С. 128-145. — Бібліогр.: 32 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173940 539.4 en Проблемы прочности application/pdf Інститут проблем міцності ім. Г.С. Писаренко НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Научно-технический раздел Научно-технический раздел
spellingShingle	Научно-технический раздел Научно-технический раздел Correia, J.A.F.O. Huffman, P.J. De Jesus, A.M.P. Lesiuk, G. Castro, J.M. Calcada, R.A.B. Berto, F. Probabilistic Fatigue Crack Initiation and Propagation Fields Using the Strain Energy Density Проблемы прочности
description	Recently, Huffman developed a strain energy density based on Walker-like stress life and fatigue crack growth behavior. In this paper, the Huffman model based on local strain energy density is used to predict the fatigue crack initiation and propagation for the P355NL1 pressure vessel steel. This model is combined with the generalized probabilistic fatigue model proposed by Correia aiming the generation of probabilistic fatigue crack initiation and propagation fields. In this study, the local stress and strains at the crack tip were obtained combining linear-elastic and elastoplastic analyses. The probabilistic fatigue crack growth rates fields for several stress R-ratios are estimated considering strain, SWT and equivalent stress amplitude damage parameters. A comparison between the experimental fatigue crack growth (FCG) data and the generated probabilistic FCG fields is made with very satisfactory correlations being found.
format	Article
author	Correia, J.A.F.O. Huffman, P.J. De Jesus, A.M.P. Lesiuk, G. Castro, J.M. Calcada, R.A.B. Berto, F.
author_facet	Correia, J.A.F.O. Huffman, P.J. De Jesus, A.M.P. Lesiuk, G. Castro, J.M. Calcada, R.A.B. Berto, F.
author_sort	Correia, J.A.F.O.
title	Probabilistic Fatigue Crack Initiation and Propagation Fields Using the Strain Energy Density
title_short	Probabilistic Fatigue Crack Initiation and Propagation Fields Using the Strain Energy Density
title_full	Probabilistic Fatigue Crack Initiation and Propagation Fields Using the Strain Energy Density
title_fullStr	Probabilistic Fatigue Crack Initiation and Propagation Fields Using the Strain Energy Density
title_full_unstemmed	Probabilistic Fatigue Crack Initiation and Propagation Fields Using the Strain Energy Density
title_sort	probabilistic fatigue crack initiation and propagation fields using the strain energy density
publisher	Інститут проблем міцності ім. Г.С. Писаренко НАН України
publishDate	2018
topic_facet	Научно-технический раздел
url	https://nasplib.isofts.kiev.ua/handle/123456789/173940
citation_txt	Probabilistic Fatigue Crack Initiation and Propagation Fields Using the Strain Energy Density / J.A.F.O. Correia, P.J. Huffman, A.M.P. De Jesus, G. Lesiuk, J.M. Castro, R.A.B. Calcada, F. Berto // Проблемы прочности. — 2018. — № 4. — С. 128-145. — Бібліогр.: 32 назв. — англ.
series	Проблемы прочности
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fulltext	UDC 539.4 Probabilistic Fatigue Crack Initiation and Propagation Fields Using the Strain Energy Density J. A. F. O. Correia, a,1 P. J. Huffman, b A. M. P. De Jesus, a G. Lesiuk, c J. M. Castro, a R. A. B. Calcada, a and F. Berto d a Faculty of Engineering, University of Porto, Porto, Portugal b John Deere, One John Deere Place, Moline, IL, USA c Laboratory of Materials Science and Engineering, University of Cantabria, Santander, Cantabria, Spain d Department of Industrial and Mechanical Design, Norwegian University of Science and Technology, Trondheim, Norway 1 jacorreia@inegi.up.pt ÓÄÊ 539.4 Ïîñòðîåíèå âåðîÿòíîñòíûõ ïîëåé çàðîæäåíèÿ è ðîñòà óñòàëîñòíûõ òðåùèí ïî êðèòåðèþ ïëîòíîñòè ýíåðãèè äåôîðìàöèè Æ. À. Ô. Î. Êîððåéÿ à , Ï. Äæ. Õàôôìàí á , À. Ì. Ï. Äå Õåñóñ à , Ã. Ëåñþê â , Æ. Ì. Êàñòðî à , Ð. À. Á. Êàëüñàäà à , Ô. Áåðòî ã à Ìàøèíîñòðîèòåëüíûé ôàêóëüòåò, Óíèâåðñèòåò Ïîðòó, Ïîðòó, Ïîðòóãàëèÿ á Øòàá-êâàðòèðà Äæîí Äèð, Ìîëèí, Èëëèíîéñ, ÑØÀ â Ëàáîðàòîðèÿ ìàòåðèàëîâåäåíèÿ è ìàøèíîñòðîåíèÿ, Êàíòàáðèéñêèé óíèâåðñèòåò, Ñàíòàíäåð, Êàíòàáðèÿ, Èñïàíèÿ ã Ôàêóëüòåò ïðîìûøëåííîãî è èíæåíåðíîãî ïðîåêòèðîâàíèÿ, Íîðâåæñêèé óíèâåðñèòåò åñòåñò- âåííûõ è òåõíè÷åñêèõ íàóê, Òðîíõåéì, Íîðâåãèÿ Ðîñò óñòàëîñòíîé òðåùèíû äîñòàòî÷íî õîðîøî èçó÷åí è îïèñàí ñ ïîìîùüþ íåñêîëüêèõ ìîäåëåé, íàèáîëåå èçâåñòíûìè èç êîòîðûõ ÿâëÿþòñÿ óðàâíåíèÿ Ïàðèñà. Õîòÿ ïðîöåññû èíèöèèðîâàíèÿ è ðàñïðîñòðàíåíèÿ óñòàëîñòíûõ òðåùèí èçó÷àþòñÿ îòäåëüíî, ðÿä èññëåäî- âàòåëåé ïîäðîáíî ðàññìàòðèâàþò âçàèìîñâÿçü ìåæäó ýòèìè äâóìÿ óñòàëîñòíûìè ïðîöåñ- ñàìè. Â ýòîì íàïðàâëåíèè ïðåäëîæåíî íåñêîëüêî ìîäåëåé ðîñòà óñòàëîñòíûõ òðåùèí, îñíîâàííûõ íà ëîêàëüíûõ ïîäõîäàõ, â ÷àñòíîñòè ìîäåëü åäèíîãî ðîñòà (UniGrow). Äëÿ íåêîòîðîãî ýëåìåíòàðíîãî îáúåìà ìàòåðèàëà ïðîöåññ ðîñòà óñòàëîñòíîé òðåùèíû ìîæíî ïðåäñòàâèòü â âèäå ïîñëåäîâàòåëüíîñòè ïîâòîðíîãî çàðîæäåíèÿ (èíèöèèðîâàíèÿ) òðåùèíû. Â íàñòîÿùåå âðåìÿ Õàôôìàí ðàçðàáîòàë êîíöåïöèþ ïëîòíîñòè ýíåðãèè äåôîðìàöèè, îñíî- âàííóþ íà öèêëè÷åñêîé äîëãîâå÷íîñòè è ðàçâèòèè óñòàëîñòíîé òðåùèíû ïî òèïó Âîëêåðà. Ìîäåëü Õàôôìàíà, áàçèðóþùàÿñÿ íà ëîêàëüíîé ïëîòíîñòè äåôîðìàöèè, èñïîëüçóåòñÿ äëÿ ïðîãíîçèðîâàíèÿ èíèöèèðîâàíèÿ è ðàñïðîñòðàíåíèÿ óñòàëîñòíîé òðåùèíû â ñòàëè P355NL1, ïðèìåíÿåìîé â ñîñóäàõ âûñîêîãî äàâëåíèÿ. Äàííàÿ ìîäåëü ñî÷åòàåòñÿ ñ îáîáùåííîé âåðîÿò- íîñòíîé ìîäåëüþ óñòàëîñòè, ðàíåå ïðåäëîæåííîé îäíèì èç ñîàâòîðîâ, ñóòü êîòîðîé ñîñòî- èò â ãåíåðàöèè âåðîÿòíîñòíûõ ïîëåé èíèöèèðîâàíèÿ è ðàñïðîñòðàíåíèÿ óñòàëîñòíûõ òðå- ùèí. Ïîëó÷åíû ëîêàëüíûå íàïðÿæåíèÿ è äåôîðìàöèè â âåðøèíå òðåùèíû, îáúåäèíÿþùèå èõ ëèíåéíî-óïðóãèå è óïðóãîïëàñòè÷åñêèå ñîñòàâëÿþùèå. Äëÿ íåñêîëüêèõ çíà÷åíèé êîýôôèöè- åíòà àñèììåòðèè öèêëà íàïðÿæåíèé R îöåíåíû âåðîÿòíîñòíûå ïîëÿ ñêîðîñòåé ðîñòà óñòà- © J. A. F. O. CORREIA, P. J. HUFFMAN, A. M. P. DE JESUS, G. LESIUK, J. M. CASTRO, R. A. B. CALCADA, F. BERTO, 2018 128 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 ëîñòíûõ òðåùèí ñ ó÷åòîì ïàðàìåòðîâ ïîâðåæäåíèÿ, ïàðàìåòðà Ñìèòà–Âàòñîíà–Òîïïåðà (SWT) è ýêâèâàëåíòíûõ ïàðàìåòðîâ àìïëèòóäû íàïðÿæåíèÿ. Ñðàâíåíèå ýêñïåðèìåíòàëüíûõ äàííûõ ïî ïðèðàùåíèþ óñòàëîñòíîé òðåùèíû ñ ïðîãíîçèðóåìûìè âåðîÿòíîñòíûìè ïîëÿìè åå ðîñòà ïîêàçûâàåò èõ òåñíóþ êîððåëÿöèþ. Êëþ÷åâûå ñëîâà: óñòàëîñòü, èíèöèèðîâàíèå òðåùèíû, ðîñò òðåùèíû, ëîêàëüíàÿ ýíåðãèÿ äåôîðìàöèè, îáîáùåííàÿ âåðîÿòíîñòíàÿ ìîäåëü, ðåçóëüòàòû èñïûòàíèé íà âûíîñëèâîñòü. Introduction. Fatigue crack growth (FCG) process is associated with the formation of new crack faces. Formation of new crack surfaces need the activation energy. During fatigue process, this energy can be delivered to the process zone due to work of external loading. The dissipated damage energy is responsible for “jump like” fatigue crack growth. This concept is reflected in several formulas and ideas of fatigue crack growth [1–8], where the fatigue crack growth process is treated as a discrete process where the local fatigue crack growth is associated with a specified “unit” of fatigue crack growth (see Fig. 1): da dN d N f � * . (1) Generally, the fatigue crack growth can be postulated as a process involving the following assumptions [2, 4, 5, 7, 8]: (i) the material is composed of simple particles of a finite dimension, �, which represents the elementary material block size, below which material cannot be regarded as a continuum (Fig. 1); (ii) ahead crack tip exist two plastic zones; static and cyclic (Fig. 1); (iii) the fatigue crack tip is supposed to be equivalent to a notch with radius, �; (iv) the fatigue crack growth process is considered as representing successive crack increments (after N f cycles) due to crack re-initiations over the distance, d* , �. The mentioned assumptions are already reflected in several fatigue crack propagation models based on low-cycle fatigue (LCF) data [9–14]. Another relevant aspect in these local approaches is the calculation of local stresses and strains at the crack tip. The most commonly used local approaches are based on Coffin–Manson & Morrow relationships [15–17] with or without the influence of the mean stress effects. Glinka [2], Peeker and Niemi [18], Noroozi et al. [7, 8], Hurley and Evans [19], among others, used in their studies the strain fatigue damage parameter. More Fig. 1. Discrete nature of fatigue fracture process. ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 129 Probabilistic Fatigue Crack Initiation and Propagation Fields ... recently, Correia et al. [20], Hafezi et al. [21], and De Jesus and Correia [22] have used the Smith–Watson–Topper (SWT) local damage parameter [23]. The equivalent stress amplitude damage parameter, � ar , has been used for the fatigue crack growth modelling based on local approaches by Correia and Huffman [12, 13]. In this paper, probabilistic fatigue crack initiation and propagation fields are presented based on strain energy approach proposed by Huffman [24] and considering the generalized probabilistic fatigue model suggested by Correia et al. [25]. The Huffman fatigue crack growth model based on strain energy density is analyzed considering several fatigue damage parameter such as strain, Smith–Watson–Topper (SWT), and equivalent stress amplitude. An analysis of results from the application of the fatigue crack growth model proposed by Huffman using the different damage parameters and experimental data is done. The generalized probabilistic fatigue model is used aiming at generating probabilistic FCG fields, for the fatigue damage parameters under consideration. In this analysis, experimental fatigue data of the P355NL1 steel from the low-cycle fatigue and fatigue crack growth tests are used [26–28]. 1. Huffman Fatigue Crack Initiation and Growth Model. The fatigue crack growth relations are established with the linear elastic fracture mechanics (LEFM) to evaluate the stress distributions in a material. The variables such as the geometry and loading are considered to obtain the fatigue crack growth rates. Glinka [2] was pioneer to relate the local stress and strain at the crack tip, by relating the strain energy distribution with the LEFM approach, and the fatigue failure of the material ahead the crack tip. This so called local approach is used to generate the fatigue crack growth for several crack lengths and calculating the stress or strain lives at the crack tip. These stresses and strains are used to obtain the number of cycles to failure, N f , leading to the crack growth extension, �a. Huffman [24] proposed a fatigue damage parameter, D, based on strain energy density concepts and calculated it from cyclic stress-strain properties as given by U U U U D N N e d c p d c f� � � � � � � � � � � � � � � � * , 2 2 (2) where Ue is the elastic strain energy density, U p * is the complementary plastic strain energy density, �c is the critical dislocation density, and Ud is the strain energy density that can be estimated by U E bd � � � � � � � � 2 1 2 ( ) \| \| , � � (3) where � is the Poisson’s ratio and � b is the Burger’s vector. The value of \| \| � b is equal to 2 52 1010. m for iron, steel or similar metals as can be found in [24]. Finding the strain energy density by integrating the Ramberg–Osgood stress-strain relationship, results the damage equation in terms of materials parameters: 2 1 1 2 2 2 1 U n n K d c a � � � � � � � � � � � � � � � � � � � � � � � ( )( )max n n f D N � � � 1 2 , (4) where � max is the maximum stress and � a is the stress amplitude. This equation can be used to generate the stress–life, strain–life, and fatigue crack growth curves from cyclic stress-strain properties. 130 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 J. A. F. O. Correia, P. J. Huffman, A. M. P. De Jesus, et al. The strain-life or Morrow parameters can be equated based on Eq. (3) resulting: � � � � � � � � � � � � � � � � � � � � � � f d c n n nE U K n n E 2 1 12 2 1 1 2 / ( )/� � � � � � � � n n1 3 , (5) b n n � � � � �1 3 , (6) � � � � � � � � � � � � � � � � � � � � � f d c n E K U E n n 2 1 3 2 2 1 1 3( ) , (7) c n � � � � 1 1 3 , (8) where �� f is the fatigue strength coefficient, E is the elastic modulus, �K and �n are the cyclic Ramberg–Osgood parameters [29], b is the fatigue strength exponent, �� f is the fatigue ductility coefficient, c is the fatigue ductility exponent, and the �c parameter can be determined by fitting the resultant strain–life curve to low-cycle strain–life data by solving the Eq. (4): �c f d nN U EN n n K � � � � � � � � � � � � � � � � � � � �2 2 1 1 2 1( ) ( ( / ) � �max )( ) .2 1 a n n � � � � � � � (9) Acording to [24], this parameter has been found between 1 10 3 1015 16 � � �c m�2. The Huffman fatigue crack growth rates using the �a calibrator can be given by �a U n n K d c a2 2 2 1 1 1 � � � � � � � � � � � � � � � � � � � � � � ( )( )max � � � � � � � � � n n f a N da dN � 0 . (10) In Eq. (10), the stresses, � a and � max , are the local stresses near the crack tip that can be related to the load, geometry and fatigue crack growth parameters as proposed by Noroozi et al. [7, 8]. The fatigue crack growth rate considering the driving force is given by the following expression: da dN C� ( ) ,�� (11) where � is the fatigue crack growth rate exponent, �� is the fatigue crack growth driving force, and C is the fatigue crack growth rate coefficient that is given by C a U n n K E d c n n n � � � � � � � � � � � � � � � � � � �� 2 2 2 1 4 2 1� ( ) ( ) � � � � � � � � � �� 1 1 1 3 11 2 1 2 n n n x� . (12) The fatigue crack growth driving force can be calculated as � �� K Kp p max ( ) ,1 (13) ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 131 Probabilistic Fatigue Crack Initiation and Propagation Fields ... K K Kapplied rmax max, ,� � (14) � �K K Kapplied r� � , (15) p n n � � � � 2 1 3 , (16) �� 2 6 1 2n n b c , (17) where �K is the total stress intensity factor range, �K applied is the applied stress intensity factor range, K appliedmax, is the maximum applied stress intensity factor, and K r is the residual stress intensity factor. The adjustment of the Huffman fatigue crack growth model to the experimental results is made using the values of �a and x as calibrators. Several authors such as Correia et al. [20, 26, 27], Hafezi et al. [21], and De Jesus and Correia [22] proposed the use of the finite element method to obtain the stresses and strains fields based on linear-elastic and elastoplastic analyses. These numerical analyses to calculate the stresses and strains ahead of the crack tip can be used for fitting the fatigue crack growth model proposed by Huffman [24] to the experimental FCG data. 2. Probabilistic Fatigue Damage Fields. The probabilistic fatigue damage fields can be obtained using the generalized probabilistic model for several fatigue damage parameters proposed by Correia et al. [25]. Originally, the probabilistic model was proposed by Castillo and Fernández-Canteli [30] for stress- and strain-based fatigue damage parameters using the Weibull or Gumbel distributions. These probabilistic distributions satisfy the statistical and physical requirements of the fatigue data. Correia et al. [25] proposed the generalization of the fatigue probabilistic model for several damage parameters, considering the similarity between the power relations used for the deterministic damage representation and the hyperbolic probabilistic fields proposed by Castillo and Fernández-Canteli [30]: �� q N f( ), (18) � � � � � �( ) ,2 0N f (19) where � is a fatigue damage parameter, �0 is the fatigue (endurance) limit, � and � are material constants, q is the decreasing function of total fatigue life, and finally, 2N f or N f represent the reversals to failure and the number of cycles to failure, respectively. The deterministic power-law model is shown in Fig. 2. Several fatigue damage parameters based on stress, strain or energy criteria, describing the same schematic representation of Fig. 2, can be used in Eqs. (18) and (19). Thus, fatigue damage parameters such as strain amplitude, �a , walker-like strain, �w , Smith–Watson–Topper, SWT, as well as the equivalent stress amplitude, � ar , among others, can be described by the generalized fatigue damage parameter, �: � �� a , (20) � � � � � � � � � � � � � � � w a R w 2 1 1 , (21) 132 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 J. A. F. O. Correia, P. J. Huffman, A. M. P. De Jesus, et al. � � �� SWT amax , (22) � � � � � � � � � ar a w w( ) ( ) ,max 1 (23) where the Walker exponent is defined as �w n n � � � � � 1 1 3 . (24) According to [31], the main advantage of the �w parameter is to describe the material mean stress fatigue behavior sensitivity, where for the SWT-life model, this parameter is equal to 0.5. In this way, the generalized probabilistic model to describe the fatigue failure criterion for several damage parameters was proposed by Correia et al. [25], based on Castillo and Fernández-Canteli probabilistic model [30], is given by P F N N N f ar f � � � � �� ! ( ; ) exp log( ) log( ) * * � � � " # $ 1 0 0 % ! & ' ! ( ! , (25) log( ) log( ) ,N Nf 0 0� � ") (26) where P is the probability of failure, N 0 and �0 are values for normalizing, and $, #, and " are the dimensionless Weibull parameters. The normalizing parameters N 0 and �0 are the threshold value for life and the corresponding value for the equivalent stress, respectively. The maximum likelihood method is used to estimate the Weibull parameters, where the details can be found in [30]. 3. Application of the Strain Energy Density Based Crack Propagation Model to Generate Probabilistic Fatigue Crack Initiation and Propagation Data. 3.1. Procedure to generate Probabilistic Fatigue Crack Initiation and Propagation Fields. The procedure proposed to generate probabilistic fatigue crack initiation and propagation fields may be summarized in the following steps: Fig. 2. Schematic representation of the deterministic power-law fatigue failure criterion [25]. ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 133 Probabilistic Fatigue Crack Initiation and Propagation Fields ... 3.1.1. Probabilistic Fatigue Crack Initiation Fields: (i) estimation of the Weibull parameters for the generalized probabilistic fatigue model based on strain, stress or energy damage parameters (Section 2), using experimental strain-life data from smooth specimens; (ii) application of the Huffman fatigue crack initiation model to estimate the Morrow constants (Section 1); (iii) computation of the probabilistic Huffman fatigue crack initiation models for several fatigue damage parameters. 3.1.2. Probabilistic Fatigue Crack Propagation Fields: (i) the first step considered in probabilistic fatigue crack initiation fields (Section 2); (ii) application of the Huffman fatigue crack propagation model based on strain energy density with probabilistic fatigue damage models (Section 1); (iii) Computation of the P da dN K R� � �� fields. 3.2. Probabilistic Fatigue Crack Initiation Data. The parameters of the cyclic elastoplastic stress–strain curves and elastic properties of the P355NL1 steel were collected in references [26–28]. These parameters were obtained based on experimental low-cycle fatigue tests of smooth specimens performed under strain controlled conditions carried out according the ASTM E606 standard. In Table 1 the cyclic stress-strain properties for the P355NL1 steel are shown. The Morrow constants obtained using the Coffin–Manson & Morrow relation [15–17] are presented in Table 2. In this table the Morrow constants estimates according to the assumption in the Huffman model [24] presented in Section 1 are also shown. This model allows evaluating the fatigue crack initiation phase based on strain energy. Equations (5)– (8) were used to estimate the Morrow constants and Eq. (9) used to obtain the critical dislocation density. A good agreement for the Morrow strain-life constants between the Huffman model and the Coffin–Manson & Morrow relation is verified (see Table 2). The critical dislocation density, �c , estimated and presented in Table 2, is similar when compared with other materials of identical mechanical properties (see [24]). In Fig. 3, the stress–life, strain–life, and SWT–life curves are presented, being compared the experimental results with the Huffman fatigue crack initiation model predictions. In these figures, a good agreement between the experimental results and the fatigue crack initiation model predictions by Huffman [24] can be observed. T a b l e 1 Cyclic Stress-Strain Properties for P355NL1 Steel [26–28] Strain ratio R � E, GPa � �K , MPa �n 0 205.20 0.275 913.6 0.1459 �1 1022.3 0.1682 “�1” � “0” 948.35 0.1533 T a b l e 2 Morrow Constants for P355NL1 Steel Calculated as per Eqs. (5)–(8) Strain ratio R � Model �� f , MPa b �� f c �c , m/m3 “�1” � “0” Morrow equation 1005.5 �0.1033 0.3678 �0.5475 – Huffman model 959.0 �0.1050 1.08 �0.6850 7 0 1015. 134 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 J. A. F. O. Correia, P. J. Huffman, A. M. P. De Jesus, et al. a b Fig. 3. Stress–life (a), strain–life (b), and SWT–life (c) curves of P355NL1 steel. c ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 135 Probabilistic Fatigue Crack Initiation and Propagation Fields ... Figure 4 shows, respectively, the probabilistic fields for several fatigue damage parameters as strain amplitude, SWT, and equivalent stress amplitude, using the generalized probabilistic model [Eqs. (25) and (26)] presented in Section 2 and proposed by Correia et al. [25]. The Walker exponent for the P355NL1 steel is equal to 0.79 and was used to generate the probabilistic field (Fig. 4c). These probabilistic fields were used to generate the probabilistic Huffman fatigue crack growth fields. 3.3. Probabilistic Fatigue Crack Propagation Data. The fatigue crack propagation data of the P355NL1 steel collected in [26–28] were used to evaluate the fatigue crack growth model based on strain energy proposed by Huffman [24]. Fatigue crack growth (FCG) rates for several stress R-ratios under constant amplitude loading conditions were obtained using the ASTM E647 standard and considering the Paris law [32]. The CT specimens of P355NL1 steel were built with a width W � 40 mm and a thickness B� 4.5 mm. For the P355NL1 steel, the FCG tests were carried out for stress R-ratios, R � � 0, 0.5, and 0.7. All FCG tests were carried out for stress R-ratios, R � � 0, 0.5, and 0.7, in air at room temperature under a sinusoidal waveform at a maximum frequency of 20 Hz. In this research, the results from the linear-elastic and elastoplastic finite element analyses ahead of the crack tip of the CT geometry are required and were generated by De Jesus and Correia [20, 22, 27]. In those analyses, the cyclic elastoplastic curve obtained experimentally was considered by De Jesus and Correia [20, 22, 27]. The residual stress intensity factor results, K r , are presented in Fig. 5 against the applied stress intensity factor range, �K applied , for CT specimen built in P355NL1 steel, which were calculated by De Jesus and Correia [20, 22, 27]. The numerical model of the CT specimen was highly refined at the crack tip region in order to estimate the crack increment, �a. De Jesus and Correia [20, 22, 26, 27] in their investigations estimated the �a parameter as equal to 4 5 10 3. � m. This value was used in this research. The fatigue crack growth model based on strain energy proposed by Huffman [24] with the generalized probabilistic model for several fatigue damage parameters was applied to CT specimen geometry made of P355NL1 steel. The fatigue crack growth rate constants for several fatigue damage parameters, such as, strain amplitude, SWT, and Walker stress amplitude, were obtained from Eqs. (10) and (11)–(17) are shown in Table 3. The probabilistic Huffman fatigue crack growth rates based on strain, SWT, and Walker stress amplitude fatigue damage parameters for the P355NL1 pressure vessel steel taking into account several stress R-ratios, are shown in Figs. 6–8. Figure 9 shows that the models appear to be more sensitive to stress ratio effects compared to the measured fatigue crack growth rates. It can be seen that models which demonstrate high levels of stress ratio sensitivity show particularly poor correlation at high stress R-ratios, R � , for materials which seem insensitive to stress R-ratio effects. When compared to the stress-life or strain-life, however, these models don’t demonstrate to be so inaccurate. Although the P355NL1 steel was tested at 2 strain R-ratios, R � , the stress R-ratios, R � , were not drastically different. Because of this, differences in the stress-strain behavior of the material at different stress R-ratios may not have been observed even if they exist. If the material demonstrates stress ratio-sensitive stress-strain behavior, it could make a difference in how the residual stress would be calculated at a crack tip. A difference in the behavior of the residual stress would influence the residual stress intensity factor, K r , and therefore would influence the calculated stress ratio dependent fatigue crack growth rates. This would appear as a lack of stress-ratio effect in a material, when it could in actuality be the same stress ratio effect as far as what the material itself goes through at the crack tip, but it would be disguised by a change in cyclic stress-strain behavior influencing the residual stress profile differently at different stress-ratios. 136 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 J. A. F. O. Correia, P. J. Huffman, A. M. P. De Jesus, et al. a b c Fig. 4. Probabilistic �a fN� (a), SWT N f� (b), and �ar fN� (c) fields of P355NL1 steel. ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 137 Probabilistic Fatigue Crack Initiation and Propagation Fields ... It is interesting to note that the strain amplitude probability field appears to be less sensitive to stress ratio. This could be evidence in support of using strain-life based calculations preferentially over stress-life based calculations under certain circumstances, though much further investigation would be necessary to determine that. These results are also evidence that care should be taken when selecting a mean stress effect, as overly sensitive models can lead to over-designed structures or under-designed structures depending on the expected load cases. T a b l e 3 Fatigue Crack Growth Rate Constants from Eqs. (10)–(17) for P355NL1 Steel Fatigue damage parameter Stress ratio R � �a, m x, m C , MPa m� � �/ 2 1 � p Strain amplitude �a 0 4 5 10 3. � 0 7 10 5. � 8 488 10 11. � 2.606 0.210 0.5 7 927 10 11. � 2.627 0.210 0.7 7 004 10 11. � 2.665 0.210 SWT 0 4 5 10 3. � 3 0 10 5. � 2 872 10 10. � 2.571 0.210 0.5 2 774 10 10. � 2.582 0.210 0.7 2 598 10 10. � 2.602 0.210 Walker stress amplitude �ar 0 4 5 10 3. � 2 5 10 5. � 1609 10 11. � 2.611 0.210 0.5 1495 10 11. � 2.631 0.210 0.7 1288 10 11. � 2.685 0.210 Fig. 5. Residual stress intensity factor Kr as a function of the �Kapplied for the CT geometry made in P355NL1 steel. 138 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 J. A. F. O. Correia, P. J. Huffman, A. M. P. De Jesus, et al. Fig. 6. Probabilistic prediction of the fatigue crack growth for P355NL1 steel based on Huffman FCG model and P Na� �� field for R � � 0 (a), 0.5 (b), and 0.7 (c). a b c ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 139 Probabilistic Fatigue Crack Initiation and Propagation Fields ... a b c Fig. 7. Probabilistic prediction of the fatigue crack growth for P355NL1 steel based on Huffman FCG model and P SWT N� � field for R � � 0 (a), 0.5 (b), and 0.7 (c). 140 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 J. A. F. O. Correia, P. J. Huffman, A. M. P. De Jesus, et al. a b c Fig. 8. Probabilistic prediction of the fatigue crack growth for P355NL1 steel based on Huffman FCG model and P Nar� �� field for R � � 0 (a), 0.5 (b), and 0.7 (c). ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 141 Probabilistic Fatigue Crack Initiation and Propagation Fields ... a b c Fig. 9. Fatigue crack growth rates for the probability of failure equal to 50% based on Huffman model and P Na� �� (a), P SWT N� � (b), and P Nar� �� (c) fields for several stress R-ratios. 142 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 4 J. A. F. O. Correia, P. J. Huffman, A. M. P. De Jesus, et al. C o n c l u s i o n s 1. The strain energy based fatigue crack initiation model proposed by Huffman can be used to estimate the Morrow constants giving good results when compared with experimental results. 2. The fatigue crack propagation model based on strain energy density proposed by Huffman can be used with results from the linear-elastic and elastoplastic numerical analyses and a good agreement is observed. 3. The Huffman model uses the calibration parameters, �a and x, which are required for the fit of the Huffman FCG model to the experimental FCG data. 4. The combination of the Huffman fatigue crack growth model with the generalized probabilistic model allowed to generate the probabilistic fields for the fatigue crack propagation rates. 5. The probabilistic crack propagation fields generated using several fatigue damage parameters such as strain, SWT, and equivalent stress amplitude, allowed to conclude that the strain damage parameter is the most suitable for the P355NL1 steel, taking into account the comparison that is made with the experimental crack propagation data, showing that this material is not sensitive to the mean stress effects. Acknowledgements. The authors of this work would like to express their gratitude to the SciTech-Science and Technology for Competitive and Sustainable Industries, R&D project NORTE-01-0145-FEDER-000022 co-financed by Programme Operational Regional do Norte (“NORTE2020”) through Fundo Europeu de Desenvolvimento Regional (FEDER) and the Portuguese Science Foundation (FCT) through the post-doctoral grant SFRH/BPD/107825/2015 the for their collaboration, financial and technical support during this research works. Ð å ç þ ì å Ð³ñò òð³ùèíè â³ä óòîìëåíîñò³ äîñòàòíüî äîáðå âèâ÷åíî é îïèñàíî çà äîïîìîãîþ äåê³ëüêîõ ìîäåëåé, íàéá³ëüø â³äîìèìè ñåðåä ÿêèõ º ð³âíÿííÿ Ïàð³ñà. Õî÷à ïðîöåñè ³í³ö³þâàííÿ ³ ïîøèðåííÿ òð³ùèí â³ä óòîìëåíîñò³ âèâ÷àþòüñÿ îêðåìî, äåÿê³ äîñë³ä- íèêè äåòàëüíî ðîçãëÿäàþòü âçàºìîçâ’ÿçîê ì³æ öèìè äâîìà óòîìíèìè ïðîöåñàìè. Ó öüîìó íàïðÿìêó çàïðîïîíîâàíî äåê³ëüêà ìîäåëåé ðîñòó òð³ùèí â³ä óòîìëåíîñò³, ùî áàçóþòüñÿ íà ëîêàëüíèõ ï³äõîäàõ, çîêðåìà ìîäåëü ºäèíîãî ðîñòó (UniGrow). Äëÿ äåÿêîãî åëåìåíòàðíîãî îá’ºìó ìàòåð³àëó ïðîöåñ ðîñòó òð³ùèíè â³ä óòîìëåíîñò³ ìîæíà ïðåäñòàâèòè ó âèãëÿä³ ïîñë³äîâíîñò³ ïîâòîðíîãî çàðîäæåííÿ (³í³ö³þâàííÿ) òð³ùèíè. Íà ñüîãîäí³ Õàôôìàí ðîçðîáèâ êîíöåïö³þ ãóñòèíè åíåðã³¿ äåôîðìàö³¿ íà îñíîâ³ öèêë³÷íî¿ äîâãîâ³÷íîñò³ ³ ïîøèðåííÿ òð³ùèíè â³ä óòîìëåíîñò³ ïî òèïó Âîë- êåðà. Ìîäåëü Õàôôìàíà, ùî áàçóºòüñÿ íà ëîêàëüí³é ãóñòèí³ åíåðã³¿ äåôîðìàö³¿, âèêîðèñòîâóºòüñÿ äëÿ ïðîãíîçóâàííÿ ³í³ö³þâàííÿ ³ ïîøèðåííÿ òð³ùèíè â³ä óòîìëå- íîñò³ â ñòàë³ P355NL1, ÿêà âèêîðèñòîâóºòüñÿ äëÿ âèãîòîâëåííÿ ïîñóäèí âèñîêîãî òèñêó. Äàíà ìîäåëü ïîºäíóºòüñÿ ç óçàãàëüíåíîþ ³ìîâ³ðí³ñíîþ ìîäåëëþ óòîìè, ðàí³- øå çàïðîïîíîâàíîþ îäíèì ³ç ñï³âàâòîð³â, ñóòü ÿêî¿ ïîëÿãàº â ãåíåðàö³¿ ³ìîâ³ðí³ñíèõ ïîë³â ³í³ö³þâàííÿ ³ ïîøèðåííÿ òð³ùèí â³ä óòîìëåíîñò³. Îòðèìàíî ëîêàëüí³ íàïðó- æåííÿ ³ äåôîðìàö³¿ ó âåðøèí³ òð³ùèíè, ùî îá’ºäíóþòü ¿õ ë³í³éíî-ïðóæí³ ³ ïðóæíî- ïëàñòè÷í³ ñêëàäîâ³. Äëÿ äåê³ëüêîõ çíà÷åíü êîåô³ö³ºíòà àñèìåòð³¿ öèêëó íàïðóæåíü R îö³íåíî ³ìîâ³ðí³ñí³ ïîëÿ øâèäêîñòåé ðîñòó òð³ùèí â³ä óòîìëåíîñò³ ç óðàõóâàííÿì ïàðàìåòð³â ïîøêîäæåííÿ, ïàðàìåòðà Ñì³òà–Âàòñîíà–Òîïïåðà (SWT) é åêâ³âàëåíòíèõ ïàðàìåòð³â àìïë³òóäè íàïðóæåííÿ. Ïîð³âíÿííÿ åêñïåðèìåíòàëüíèõ äàíèõ ùîäî ïðè- ðîñòó òð³ùèíè â³ä óòîìëåíîñò³ ç ïðîãíîçîâàíèìè ³ìîâ³ðí³ñíèìè ïîëÿìè ¿¿ ðîñòó ñâ³ä÷èòü ïðî ¿õ ò³ñíó êîðåëÿö³þ. ISSN 0556-171X. 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Probabilistic Fatigue Crack Initiation and Propagation Fields Using the Strain Energy Density

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