Effect of the Loading Rate on the Brittle Fracture of Terfenol-D Specimens in Magnetic Field: Strain Energy Density Approach

Effect of the Loading Rate on the Brittle Fracture of Terfenol-D Specimens in Magnetic Field: Strain Energy Density Approach

The aim of the present study is to characterize the fracture behaviour of the giant magnetostrictive alloy commercially named Terfenol-D, both experimentally and numerically. Three-point bending tests have been carried out on single-edge precracked specimens and fracture loads have been measured at...

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Datum:2016
Hauptverfasser: Colussi, M., Berto, F., Mori, K., Narita, F.
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Veröffentlicht: Інститут проблем міцності ім. Г.С. Писаренко НАН України 2016
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spelling irk-123456789-1735572020-12-13T01:26:07Z Effect of the Loading Rate on the Brittle Fracture of Terfenol-D Specimens in Magnetic Field: Strain Energy Density Approach Colussi, M. Berto, F. Mori, K. Narita, F. Научно-технический раздел The aim of the present study is to characterize the fracture behaviour of the giant magnetostrictive alloy commercially named Terfenol-D, both experimentally and numerically. Three-point bending tests have been carried out on single-edge precracked specimens and fracture loads have been measured at different loading rates,in the presence and absence of a magnetic field. In the recent years it has been shown that Strain Energy Density (SED), averaged in a finite control volume, can successfully predict brittle failures of cracked, U- and V-notched specimens made out of several materials. By performing coupled-field finite element analyses, the effect of the magnetic field and of the loading rate on Terfenol-D failures have been analyzed and the capability of SED criterion to capture these effects has been discussed.A relationship between the size of the SED’s control volume and the loading rate has been proposed and failures have then been predicted in terms of averaged SED. Экспериментально и путем расчета оценены характерные особенности разрушения инновационного магнитострикционного сплава Терфенол-Д. Плоские образцы из данного материала с исходной краевой трещиной испытывали на статический трехточечный изгиб с различными скоростями нагружения при наличии и отсутствии магнитного поля. Исследования показывают, что для описания характеристик разрушения образцов с трещиной либо надрезом V- или U-подобной формы достаточно перспективным является такой параметр, как плотность энергии деформации, осредненный по некоторому контрольному объему. С использованием конечноэлементного подхода проанализировано влияние магнитного поля и скорости нагружения на механизм разрушения сплава Терфенол-Д, а также возможность их учета с помощью критерия плотности энергии деформации. Предложена зависимость между размером контрольного объема осреднения и скоростью нагружения, а также выполнен прогноз разрушения исследуемого сплава в рамках указанного подхода. Експериментально і шляхом розрахунку оцінено характерні особливості руйнування іноваційного магнітострикційного сплаву Терфенол-Д. Плоскі зразки з даного матеріалу з вихідною крайовою тріщиною випробовували на статичний триточковий згин із різними швидкостями навантаження за наявності та відсутності магнітного поля. Дослідження показують, що для опису характеристик руйнування зразків із тріщиною або надрізом V- чи U-подібної форми досить перспективним є такий параметр, як щільність енергії деформації, осереднений по деякому контрольному об'єму. Із використанням скінченноелементного підходу проаналізовано вплив магнітного поля і швидкості навантаження на механізм руйнування сплаву Терфенол-Д та можливість їх урахування за допомогою критерію щільності енергії деформації. Запропоновано залежність між розміром контрольного об'єму осереднення і швидкістю навантаження та виконано прогноз руйнування досліджуваного сплаву в рамках указаного підходу. 2016 Article Effect of the Loading Rate on the Brittle Fracture of Terfenol-D Specimens in Magnetic Field: Strain Energy Density Approach / M. Colussi, F. Berto, K. Mori, F. Narita // Проблемы прочности. — 2016. — № 6. — С. 73-83. — Бібліогр.: 21 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/173557 539.4 en Проблемы прочности 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
Colussi, M.
Berto, F.
Mori, K.
Narita, F.
Effect of the Loading Rate on the Brittle Fracture of Terfenol-D Specimens in Magnetic Field: Strain Energy Density Approach
Проблемы прочности
description The aim of the present study is to characterize the fracture behaviour of the giant magnetostrictive alloy commercially named Terfenol-D, both experimentally and numerically. Three-point bending tests have been carried out on single-edge precracked specimens and fracture loads have been measured at different loading rates,in the presence and absence of a magnetic field. In the recent years it has been shown that Strain Energy Density (SED), averaged in a finite control volume, can successfully predict brittle failures of cracked, U- and V-notched specimens made out of several materials. By performing coupled-field finite element analyses, the effect of the magnetic field and of the loading rate on Terfenol-D failures have been analyzed and the capability of SED criterion to capture these effects has been discussed.A relationship between the size of the SED’s control volume and the loading rate has been proposed and failures have then been predicted in terms of averaged SED.
format Article
author Colussi, M.
Berto, F.
Mori, K.
Narita, F.
author_facet Colussi, M.
Berto, F.
Mori, K.
Narita, F.
author_sort Colussi, M.
title Effect of the Loading Rate on the Brittle Fracture of Terfenol-D Specimens in Magnetic Field: Strain Energy Density Approach
title_short Effect of the Loading Rate on the Brittle Fracture of Terfenol-D Specimens in Magnetic Field: Strain Energy Density Approach
title_full Effect of the Loading Rate on the Brittle Fracture of Terfenol-D Specimens in Magnetic Field: Strain Energy Density Approach
title_fullStr Effect of the Loading Rate on the Brittle Fracture of Terfenol-D Specimens in Magnetic Field: Strain Energy Density Approach
title_full_unstemmed Effect of the Loading Rate on the Brittle Fracture of Terfenol-D Specimens in Magnetic Field: Strain Energy Density Approach
title_sort effect of the loading rate on the brittle fracture of terfenol-d specimens in magnetic field: strain energy density approach
publisher Інститут проблем міцності ім. Г.С. Писаренко НАН України
publishDate 2016
topic_facet Научно-технический раздел
url http://dspace.nbuv.gov.ua/handle/123456789/173557
citation_txt Effect of the Loading Rate on the Brittle Fracture of Terfenol-D Specimens in Magnetic Field: Strain Energy Density Approach / M. Colussi, F. Berto, K. Mori, F. Narita // Проблемы прочности. — 2016. — № 6. — С. 73-83. — Бібліогр.: 21 назв. — англ.
series Проблемы прочности
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fulltext UDC 539.4 Effect of the Loading Rate on the Brittle Fracture of Terfenol-D Specimens in Magnetic Field: Strain Energy Density Approach M. Colussi, a F. Berto, b,1 K. Mori, c and F. Narita d a Department of Management and Engineering, University of Padua, Vicenza, Italy b Department of Engineering Design and Materials, Norwegian University of Science and Technology (NTNU), Trondheim, Norway c Department of Mechanical Engineering, Ibaraki University, Hitachi, Japan d Department of Material Processing, Tohoku University, Sendai, Japan 1 berto@gest.unipd.it ÓÄÊ 539.4 Âëèÿíèå ñêîðîñòè íàãðóæåíèÿ íà õðóïêîå ðàçðóøåíèå îáðàçöîâ èç Òåðôåíîë-Ä ïîä âîçäåéñòâèåì ìàãíèòíîãî ïîëÿ íà îñíîâå ïîäõîäà ïëîòíîñòè ýíåðãèè äåôîðìàöèè Ì. Êîëóññè à , Ô. Áåðòî á , Ê. Ìîðè â , Ô. Íàðèòà ã à Ïàäóàíñêèé óíèâåðñèòåò, Âè÷åíöà, Èòàëèÿ á Íîðâåæñêèé óíèâåðñèòåò åñòåñòâåííûõ è òåõíè÷åñêèõ íàóê, Òðîíõåéì, Íîðâåãèÿ â Èáàðàêè óíèâåðñèòåò, Õèòà÷è, ßïîíèÿ ã Òîõîêó óíèâåðñèòåò, Ñåíäàé, ßïîíèÿ Ýêñïåðèìåíòàëüíî è ïóòåì ðàñ÷åòà îöåíåíû õàðàêòåðíûå îñîáåííîñòè ðàçðóøåíèÿ èííîâà- öèîííîãî ìàãíèòîñòðèêöèîííîãî ñïëàâà Òåðôåíîë-Ä. Ïëîñêèå îáðàçöû èç äàííîãî ìàòåðèàëà ñ èñõîäíîé êðàåâîé òðåùèíîé èñïûòûâàëè íà ñòàòè÷åñêèé òðåõòî÷å÷íûé èçãèá ñ ðàçëè÷- íûìè ñêîðîñòÿìè íàãðóæåíèÿ ïðè íàëè÷èè è îòñóòñòâèè ìàãíèòíîãî ïîëÿ. Èññëåäîâàíèÿ ïîêàçûâàþò, ÷òî äëÿ îïèñàíèÿ õàðàêòåðèñòèê ðàçðóøåíèÿ îáðàçöîâ ñ òðåùèíîé ëèáî íàäðå- çîì V- èëè U-ïîäîáíîé ôîðìû äîñòàòî÷íî ïåðñïåêòèâíûì ÿâëÿåòñÿ òàêîé ïàðàìåòð, êàê ïëîòíîñòü ýíåðãèè äåôîðìàöèè, îñðåäíåííûé ïî íåêîòîðîìó êîíòðîëüíîìó îáúåìó. Ñ èñïîëü- çîâàíèåì êîíå÷íîýëåìåíòíîãî ïîäõîäà ïðîàíàëèçèðîâàíî âëèÿíèå ìàãíèòíîãî ïîëÿ è ñêîðîñòè íàãðóæåíèÿ íà ìåõàíèçì ðàçðóøåíèÿ ñïëàâà Òåðôåíîë-Ä, à òàêæå âîçìîæíîñòü èõ ó÷åòà ñ ïîìîùüþ êðèòåðèÿ ïëîòíîñòè ýíåðãèè äåôîðìàöèè. Ïðåäëîæåíà çàâèñèìîñòü ìåæäó ðàçìå- ðîì êîíòðîëüíîãî îáúåìà îñðåäíåíèÿ è ñêîðîñòüþ íàãðóæåíèÿ, à òàêæå âûïîëíåí ïðîãíîç ðàçðóøåíèÿ èññëåäóåìîãî ñïëàâà â ðàìêàõ óêàçàííîãî ïîäõîäà. Êëþ÷åâûå ñëîâà: ïëîòíîñòü ýíåðãèè äåôîðìàöèè, ñêîðîñòü âûñâîáîæäåíèÿ ýíåðãèè äåôîðìàöèè, “èíòåëëåêòóàëüíûå” ìàòåðèàëû, ñâåðõìàãíèòîñòðèêöèîííûå ìàòåðèàëû, òðåùèíîñòîéêîñòü, ìàãíèòíîå ïîëå. N o t a t i o n a – crack depth for cracked specimens B – magnetic induction vector Bi – ith component of the magnetic induction © M. COLUSSI, F. BERTO, K. MORI, F. NARITA, 2016 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 73 dkij – magnetoelastic constants E – the Young modulus G – strain energy release rate Gc – critical strain energy release rate h – thickness of the specimen H – intensity vector of the magnetic field Hi – ith component of the intensity vector of the magnetic field J – J-integral value l – length of the specimen K I – Mode I stress intensity factor K cI – material fracture toughness n – exit path normal Rc – radius of the control volume sijkl H – elastic compliance T – surface tension vector u – displacement vector ui – ith component of the displacement vector w – width of the specimen W – strain energy density W – averaged strain energy density Wc – critical strain energy Wm – magnetic enthalpy � – strain tensor � ij – ijth component of the strain tensor � ij T – magnetic permittivity � – Poisson’s ratio � – the Cauchy stress tensor � ij – ijth component of the stress tensor � t – tensile strength � – magnetic potential � – area of the control volume Introduction. Magnetostriction is the change in shape of materials under the influence of an external magnetic field. This effect was firstly described in the 19th century by the famous physicist James Joule. The so-called giant magnetostrictive materials exhibit a large deformation in response to external magnetic fields and a substantial magnetization changes in response to applied forces. Terfenol-D, which is a rare earth element alloy of iron, terbium, and dysprosium, is a widespread commercial material for magnetostrictive applications. This material has attracted much interest over the years, particularly because it provides one of the largest known magnetostrictive elongation and a high energy density storing capacity at room temperature, which advantages ensure a great potential in many M. Colussi, F. Berto, K. Mori, and F. Narita 74 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 applications [1]. A large field of use is in high power ultrasonic transducers. Such material is also increasingly employed in automotive industry, avionics and robotics, where magnetostriction-based actuators and sensors are commonly used [2] because of their robustness and responsiveness. A Terfenol-D-based sensor has been recently proposed for stress monitoring of steel cables in suspended bridges [3] and this material is also expected to be used in energy harvesting devices [4–6]. In spite of all these promising applications, it is well known that Terfenol-D is a brittle alloy and, as a consequence, Terfenol-D-based devices are susceptible to in-service fracture [7]. Therefore, defects caused by manufacturing and cracking can have a strong influence on the material performance. Moreover, quite a few works can be found in literature dealing with this topic, probably because Terfenol-D is a very expensive alloy, not easy to produce, both for the high reactivity of the raw materials and for the impurities contained. It is therefore of interest to gain insight into the defect sensitivity of giant magnetostrictive materials, in particular, Terfenol-D alloy. In recent years, the strain energy density (SED) based criterion has been proposed to predict brittle failures, which occur without any plastic deformation [8]. The criterion implies that a brittle fracture occurs when the strain energy density averaged in a circular control volume, which includes a crack or notch tip, reaches the critical value, which is dependent on the material. Thereafter many researchers worked on this criterion and proved that it can successfully predict brittle and high cycle fatigue failures of cracked, U- and V-notched specimens made of several materials, including metals and ceramics [9–11]. The effect of a magnetic field on fracture behavior of Terfenol-D has recently been studied, both experimentally and numerically, by means of the energy release rate [12], and it has been shown that the material fracture resistance, under Mode I loading condition, is greater in absence of the magnetic field and decreases with the increase of the latter. It has also been proved that the resistance decrease may be related to the increase of the energy release rate with increasing magnetic fields. The extension of the SED criterion to the assessment of brittle behavior of giant magnetostrictive materials, under the same loading condition, has recently been shown in [13]. In the present work, experimental data sets on fracture behavior of Terfenol-D specimens under three-point bending have been extended and fracture loads were measured in presence and absence of the magnetic field and at different loading rates. By performing coupled-field numerical analyses the effect of the magnetic field and of the loading rate on Terfenol-D brittle failure have been discussed. The capability of the SED criterion to capture these effects has then been analyzed and, for this purpose, a relationship between the radius of the control volume and the loading rate has also been proposed. 1. Analysis. 1.1. Basic Equations of the Material. The basic equations for magnetostrictive materials are here outlined. Considering a Cartesian coordinate system, x x x1 2 3, , , the equilibrium equations are given by � ji j, ,� 0 � ijk k jH , ,� 0 Bi i, ,� 0 (1) where � ij , Hi , and Bi are the components of the stress tensor, the intensity vector of the magnetic field, and the magnetic induction vector, respectively, whereas � ijk is the Levi-Civita symbol. A comma followed by an index denotes partial differentiation with respect to the spatial coordinate xi and the Einstein summation convention for repeated tensor indices is applied. The constitutive laws are given as � �ij ijkl H kl kij ks d H� , B d Hi ikl kl ik T k� � � , (2) Effect of the Loading Rate on the Brittle Fracture ... ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 75 where � ij are the components of the strain tensor and sijkl H , dikl , and � ik T are the magnetic field elastic compliance, the magnetoelastic constants and the magnetic permittivity, respectively. Valid symmetry conditions are s s s sijkl H jikl H ijlk H klij H� � � , d dkij kji� , � �ij T ji T� . (3) The relation between the strain tensor and the displacement vector ui is � ij j i i ju u� 1 2 ( )., , (4) The magnetic field intensity, named � the potential, is written as Hi i��, . (5) For Terfenol-D, the constitutive relations can be written as � � � � � � 11 22 33 23 31 12 11 12 1 2 2 2 � � � � � � � � � � � � � � � � � s s sH H 3 12 11 13 13 13 33 44 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 H H H H H H H H s s s s s s s s 4 66 11 22 33 23 31 12 0 0 0 0 0 0 H Hs � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 0 0 0 0 0 0 0 0 0 0 0 0 0 31 31 33 15 15 d d d d d � � � � � � � � � � � � � � � � � � � � � H H H 1 2 3 , (6) B B B d d d d d 1 2 3 15 15 31 31 33 0 0 0 0 0 0 0 0 0 0 0 0 0 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 11 22 33 23 31 12 11 0 0 0 T 11 33 1 2 3 0 0 0 T T H H H� � � � � � � � � � � � � � � � � � � , (7) where � � � � � � � � � � � � 23 32 31 13 12 21 23 32 31 13 12 21 � � � � � � � , , , , , , (8) s s s s s s s sH H H H H H H H 11 1111 2222 12 1122 13 1133 2233� � � � �, , , s s s s s s s sH H H H H H H 33 3333 44 2323 3131 66 1212 14 4 4 2� � � � �, , ( 1 12 H Hs� � ), (9) d d d15 131 2232 2� � , d d d31 311 332� � , d d33 333� . (10) The relationship between magnetostriction and magnetic field intensity is nonlinear. The nonlinearity arises from the motion of magnetic domain walls [14]. To take into account this behavior, the constants d15, d31, and d33 are written as a function of second-order magnetoelastic constants: 76 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 M. Colussi, F. Berto, K. Mori, and F. Narita d d m 15 15� , d d m Hm z31 31 31� , d d m Hm z33 33 33� , (11) where d m 15 , d m 31 , and d m 33 are the piezomagnetic constants, whereas m31 and m33 are the second-order magnetoelastic constants. 1.2. Averaged Strain Energy Density Approach. According to Lazzarin and Zambardi [8], the brittle failure of a component occurs when the total strain energy, W , averaged in a specific control volume located at a notch or crack tip, reaches the critical value Wc . In agreement with the Beltrami criterion [15], where � t is the ultimate tensile strength under elastic stress field conditions and E is the Young modulus of the material, the critical value of the total strain energy can be determined by the relation: W E c t� � 2 2 . (12) The control volume takes different shapes based on the considered kind of notch. If the notch is represented by a crack, its opening angle is equal to zero and the control volume is circular and centered on the crack tip. This being the case, the radius Rc of the control volume can be evaluated by means of the following expression [16]: R K c c� � � � �� � � �� ( )( ) , 1 5 8 4 2 � � � � I t (13) where K cI is the material fracture toughness, � is Poisson’s ratio, and � t is the ultimate tensile strength of an unnotched specimen. If the material fracture toughness is unknown, an empirical approach can be a good alternative for determining Rc , as described in the following sections. 1.3. Finite Element Model. The strain energy density averaged in the control volume, W , can be computed directly by means of a finite element analysis. Analyses were performed by means of ANSYS R14.5 finite element software, both in plane strain and plane stress conditions depending on the specimen width. For this purpose, solid models were used to determine the most appropriate condition. The basic equations for magnetostrictive materials are mathematically equivalent to those of the piezoelectric materials [17], so four nodes PLANE13 and eight nodes SOLID5 coupled-field solid elements from ANSYS library were used, respectively for plane and solid models, and the magnetic field has been introduced by a voltage difference. Figure 1 shows the schematic representation of the boundary conditions and the adopted reference system. The coordinate axes x x� 1 and z x� 3 are chosen such that the y x� 2 axis coincides with the thickness direction and such that the easy axis of magnetization is the z-direction. Because of this symmetry, only a half of the model was used. ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 77 Fig. 1. Schematic representation of the model and boundary conditions. Effect of the Loading Rate on the Brittle Fracture ... The mesh adopted to compute W had the same grade of refinement adopted in a previous work by the present authors [13], in which models with 6400 elements were used to evaluate the energy release rate by means of the J-integral for the same geometry. 2. Experimental Procedure. Among giant magnetostrictive materials, the commercially named Terfenol-D (Tb0.3Dy0.7Fe1.9) alloy supplied by Etrema Products, Inc. (USA) was used in all tests. The material properties are listed in Table 1. Tests were performed with the aim to measure the fracture load, Pc , of single-edge precracked specimens, subjected to three-point bending, in presence and in absence of the magnetic field and at various loading rates. The testing apparatus is reported in Fig. 2. Specimens were 5 mm thick, 3 mm wide, and 15 mm long. Before testing, all specimens were weakened on one side by a 0.5 mm deep crack, which was introduced using a tungsten cutter. The tested specimen is shown in Fig. 3. The load P has been applied at the midpoint of the specimens, which were simply supported with span of 13 mm, by means of a 250 N load cell (resolution: 0.01 N). The load was applied for different loading rates: 0.05, 0.5, and 3.0 N/s. A uniform magnetic field, with magnetic induction B0, has been applied in the longitudinal direction through an electromagnet. As devices in which Terfenol-D is employed commonly work in magnetic induction range varying from 0.02 to 0.05 T, the representative value of 0.03 T has been adopted in all tests. It noteworthy that, insofar as alloying elements in Terfenol-D are terbium and dysprosium, which are very expensive rare earth elements, the number of tested specimens was limited: from two to three for each condition. 78 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 T a b l e 1 Terfenol-D Material Properties Material Elastic compliance (10 12� m2/N) Piezo-magnetic constants (10 9� mA�1) Magnetic permeability (10 6� H/m) Density (kg/m3) sH 11 sH 33 sH 44 sH 12 sH 13 dm 31 dm 33 dm 15 �11 T �33 T � Terfenol-D 17.9 17.9 26.3 �5.88 �5.88 �5.3 11 28 6.29 6.29 9250 Fig. 2. Schematic representation of the experimental setup. M. Colussi, F. Berto, K. Mori, and F. Narita By means of an experimental procedure it has also been possible to evaluate the nonlinear trend of Terfenol-D magnetostriction versus magnetic field intensity. Let consider a Cartesian coordinate system, x y z, , , whose origin is located at the top center of an uncracked specimen. Since the specimen dimension in the z-direction (longitudinal direction) exceeds the other two dimensions by over three times, the longitudinal magnetostriction is prevailing, and it can be assumed that m31 is equal to zero [18]. Varying the intensity of the magnetic field applied in the z-direction, the trend of magnetostriction has been measured by a strain gauge located at x y z� � � 0. By comparison between the measured strain � zz and the numerically obtained one, it has been found that the proper value for the second-order magnetoelastic constant m33 is 4 82 10 12. � m2/A2. This value has been used in the analyses to compute the SED. Figure 4 shows the trend of strain versus the applied magnetic field intensity. The full dots represent the experimental data, whereas the solid line represents the numerical trend after the second-order magnetoelastic constant has been incorporated. 3. Results and Discussion. Fracture loads, Pc , in presence and absence of the magnetic field have been experimentally measured at each loading rate. Data, in terms of the fracture load, are summarized in Table 2. Bold numbers represent the average values for each condition, whereas numbers in brackets indicate the relative standard deviations. The average fracture loads are depicted in Fig. 5. The error bars indicate the maximum and minimum values of Pc . In the presence of the magnetic field, the average fracture loads at the load rates of 0.05, 0.50, and 3.0 N/s drop by 7, 9, and 14%, respectively. It has also been found that Terfenol-D exhibits the fracture load reduction as the loading rate decreases. A similar behavior has been observed for other materials, such as TiAl alloys [19] and piezoelectric ceramics [20, 21]. ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 79 Fig. 3. Specimen geometry and testing condition. Fig. 4. Trend of strain versus magnetic field intensity in Terfenol-D. Effect of the Loading Rate on the Brittle Fracture ... Since Terfenol-D material properties are loading rate-dependent, here it is assumed that the critical radius Rc , which depends on the material, varies with the loading rate. By plotting the averaged SED related to the mean values of critical loads in Table 2, in presence or absence of the magnetic field, as a function of the control volume radius, it is possible to determine different intersections for each loading rate. The intersections have been found at 0.050, 0.056, and 0.1 mm, respectively, for the loading rates of 0.05, 0.5, and 3.0 N/s. This means that at the critical load level the material is characterized by the strain energy density averaged in the control volume having the size variable with the loading rate, which is independent of the ratio between the applied load and the magnetic field intensity. A good fit of Rc versus loading rate to a linear model has been found by adopting a simple linear regression model. The following relationship (evaluated for the loading rates from 0.05 to 3.0 N/s) is proposed R dP dt c � 00195 005. . . (14) The critical radius of 0.07 mm, obtained from (13) and proposed in [13] with no account of the loading rate, falls into the proposed variation range. 80 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 T a b l e 2 Measured Fracture Loads as a Function of the Loading Rate and the Magnetic Field Intensity dP dt, N/s Pc , N B0 0� B0 � 0.03 T 0.05 58.3 65.8 74.7 66.3 (5.81) 59.2 61.9 64.6 61.9 (1.91) 0.5 66.6 68.5 67.5 (0.78) 60.7 61.6 61.1 (0.37) 3.0 71.0 79.2 – 75.1 (3.35) 74.2 59.3 60.0 64.5 (5.95) Fig. 5. Mean fracture load as a function of the loading rate and the magnetic field intensity B0. M. Colussi, F. Berto, K. Mori, and F. Narita Figure 6 shows a summary of the experimental data in terms of the square root of the ratio between the averaged strain energy density, W , and the critical value of strain energy, Wc . This parameter has been chosen because of its proportionality to the fracture load. The averaged strain energy density, W , has been computed in control volumes having radius given by (14), whereas a critical strain energy equal to 0.02 MJ/m3 is assumed. This critical value is obtained from Eq. (12), assuming the Young modulus equal to 30 GPa, Poisson’s ratio equal to 0.25, and tensile strength equal to 34 MPa, which are the medium characteristics provided by the material supplier (Etrema Products, Inc.). Here, the Young modulus is assumed to be independent from the applied magnetic field intensity. This assumption seems to be reasonable for the variation range of the applied magnetic field intensity. In Fig. 4, the experimental data from [12] have also been summarized. Data refer to fracture loads measured under three-point bending, with and without the 0.03 T magnetic field, at the following loading rate: 0.2 and 3.0 N/s. Specimens were 3 mm thick, 5 mm wide, and 15 mm long. The crack depth was 0.5 mm. Due to the different geometry (ratio between width, w, and thickness, h, equal to 5/3 instead of 3/5) the plane strain condition instead of the plane stress condition turned out to be more appropriate for their finite element modeling. It has been found that nearly all experimental data fit in a narrow scatter band, whose limits are drawn here from 0.80 to 1.20 (4 data over 35 being outside of this range). A few data ouside this band fall, however, into the safety region of the plot. It can be concluded that the data scatter is quite limited, and the averaged SED criterion appears to be suitable for the fracture strength assessment of cracked specimens of Terfenol-D alloy under Mode I condition, in the presence or absence of the magnetic field and with variable loading rate. In the authors’ opinion this is a promising result and the SED criterion seems suitable for a reliable assessment of Terfenol-D brittle failure. The proposed relationship between the size of the control volume and the loading rate also permits to take into account the loading rate by means of static finite element analyses. Conclusions. Defect sensitivity of Terfenol-D rare earth element alloy was studied both numerically and experimentally. Under three-point bending conditions it has been found that Terfenol-D fracture resistance is greater in the absence of the magnetic field. This behavior is justified by the increase in the strain energy density around the crack tip ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 6 81 Fig. 6. Summary of results on Terfenol-D specimens with various loading rates in the presence and absence of the magnetic field B0. Effect of the Loading Rate on the Brittle Fracture ... when the magnetic field is acting. It has also been proven that Terfenol-D fails at decreasing fracture loads as the loading rate decreases. The averaged SED criterion captures this behavior and is able to predict Terfenol-D failures, provided a linear relationship between the size of the control volume and the loading rate is assumed. Ð å ç þ ì å Åêñïåðèìåíòàëüíî ³ øëÿõîì ðîçðàõóíêó îö³íåíî õàðàêòåðí³ îñîáëèâîñò³ ðóéíóâàííÿ ³íîâàö³éíîãî ìàãí³òîñòðèêö³éíîãî ñïëàâó Òåðôåíîë-Ä. Ïëîñê³ çðàçêè ç äàíîãî ìàòå- ð³àëó ç âèõ³äíîþ êðàéîâîþ òð³ùèíîþ âèïðîáîâóâàëè íà ñòàòè÷íèé òðèòî÷êîâèé çãèí ³ç ð³çíèìè øâèäêîñòÿìè íàâàíòàæåííÿ çà íàÿâíîñò³ òà â³äñóòíîñò³ ìàãí³òíîãî ïîëÿ. Äîñë³äæåííÿ ïîêàçóþòü, ùî äëÿ îïèñó õàðàêòåðèñòèê ðóéíóâàííÿ çðàçê³â ³ç òð³ùè- íîþ àáî íàäð³çîì V- ÷è U-ïîä³áíî¿ ôîðìè äîñèòü ïåðñïåêòèâíèì º òàêèé ïàðàìåòð, ÿê ù³ëüí³ñòü åíåð㳿 äåôîðìàö³¿, îñåðåäíåíèé ïî äåÿêîìó êîíòðîëüíîìó îá’ºìó. ²ç âèêîðèñòàííÿì ñê³í÷åííîåëåìåíòíîãî ï³äõîäó ïðîàíàë³çîâàíî âïëèâ ìàãí³òíîãî ïîëÿ ³ øâèäêîñò³ íàâàíòàæåííÿ íà ìåõàí³çì ðóéíóâàííÿ ñïëàâó Òåðôåíîë-Ä òà ìîæëèâ³ñòü ¿õ óðàõóâàííÿ çà äîïîìîãîþ êðèòåð³þ ù³ëüíîñò³ åíåð㳿 äåôîðìàö³¿. Çàïðîïîíîâàíî çàëåæí³ñòü ì³æ ðîçì³ðîì êîíòðîëüíîãî îá’ºìó îñåðåäíåííÿ ³ øâèäê³ñòþ íàâàíòàæåííÿ òà âèêîíàíî ïðîãíîç ðóéíóâàííÿ äîñë³äæóâàíîãî ñïëàâó â ðàìêàõ óêàçàíîãî ï³äõîäó. 1. G. Engdahl, Handbook of Giant Magnetostrictive Materials, Academic Press, New York (1999). 2. F. Calkins, A. B. Flatau, and M. J. Dapino, “Overview of magnetostrictive sensor technology,” J. Intel. Mat. Syst. Str., 18, 1057–1066 (2007). 3. R. Zhang, Y. Duan, S. Wing Or, and Y. Zhao, “Smart elasto-magneto-electric (EME) sensors for stress monitoring of steel cables: design theory and experimental validation,” Sensors, 14, 13644–13660 (2014). 4. X. Zhao and D. G. Lord, “Application of the Villari effect to electric power harvesting,” J. Appl. Phys., 99, 08M703–08M703-3 (2006). 5. P. Li, Y. Wen, P. Liu, et al., “A magnetoelectric energy harvester and management circuit for wireless sensor network,” Sensor. Actuat. A – Phys., 157, 100–106 (2010). 6. K. Mori, T. Horibe, S. Ishikawa, et al., “Characteristics of vibration energy harvesting using giant magnetostrictive cantilevers with resonant tuning,” Smart Mater. Struct., 24, 125032 (2015). 7. D. T. Peterson, J. D. Verhoeven, O. D. McMasters, and W. A. Spitzig, “Strength of Terfenol-D,” J. Appl. Phys., 65, 3712–3713 (1989). 8. P. Lazzarin and R. Zambardi, “A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches,” Int. J. Fracture, 112, 275–298 (2001). 9. F. Berto, A. Campagnolo, and P. Gallo, “Brittle failure of graphite weakened by V-notches: a review of some recent results under different loading modes,” Strength Mater., 47, No. 3, 488–506 (2015). 10. F. Berto and P. Lazzarin, “Recent developments in brittle and quasi-brittle failure assessment of engineering materials by means of local approaches,” Mat. Sci. Eng. R, 75, 1–48 (2014). 11. F. Berto and P. 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Sato, “Dynamic fatigue behavior of cracked piezoelectric ceramics in three-point bending under AC electric fields,” J. Eur. Ceram. Soc., 32, 3759–3766 (2012). Received 29. 06. 2016 ISSN 0556-171X. 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<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> /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.) >> /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ << /AsReaderSpreads false /CropImagesToFrames true /ErrorControl /WarnAndContinue /FlattenerIgnoreSpreadOverrides false /IncludeGuidesGrids false /IncludeNonPrinting false /IncludeSlug false /Namespace [ (Adobe) (InDesign) (4.0) ] /OmitPlacedBitmaps false /OmitPlacedEPS false /OmitPlacedPDF false /SimulateOverprint /Legacy >> << /AddBleedMarks false /AddColorBars false /AddCropMarks false /AddPageInfo false /AddRegMarks false /ConvertColors /ConvertToCMYK /DestinationProfileName () /DestinationProfileSelector /DocumentCMYK /Downsample16BitImages true /FlattenerPreset << /PresetSelector /MediumResolution >> /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ] >> setdistillerparams << /HWResolution [2400 2400] /PageSize [612.000 792.000] >> setpagedevice

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