Measured Hardness by an Indenter Having Ellipsoidal Shape

Measured Hardness by an Indenter Having Ellipsoidal Shape

Если при исследовании твердости в качестве индентора использовалось твердое тело эллипсоидальной формы, выражение для статической твердости рассматривалось как функция от глубины и радиусов отпечатка индентора. При этом применялась общая формула, связывающая статическую твердость с отношением нормал...

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Date:2016
Main Authors: Boudilmi, A., Loucif, K.
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Language:English
Published: Інститут проблем міцності ім. Г.С. Писаренко НАН України 2016
Series:Проблемы прочности
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Научно-технический раздел
Online Access:http://dspace.nbuv.gov.ua/handle/123456789/173479
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Cite this:Measured Hardness by an Indenter Having Ellipsoidal Shape / A. Boudilmi, K. Loucif // Проблемы прочности. — 2016. — № 3. — С. 99-106. — Бібліогр.: 12 назв. — англ.

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spelling irk-123456789-1734792020-12-06T01:26:25Z Measured Hardness by an Indenter Having Ellipsoidal Shape Boudilmi, A. Loucif, K. Научно-технический раздел Если при исследовании твердости в качестве индентора использовалось твердое тело эллипсоидальной формы, выражение для статической твердости рассматривалось как функция от глубины и радиусов отпечатка индентора. При этом применялась общая формула, связывающая статическую твердость с отношением нормальной силы, прилагаемой к индентору, к реальной площади его отпечатка. При получении конечной формулы для расчета твердости использовались геометрический и математический подходы. Якщо при дослідженні твердості за індентор слугувало тверде тіло еліпсоїдної форми, вираз для статичної твердості розглядався як функція від глибини і радіусів відбитка індентора. При цьому використовувалась загальна формула, що зв язувала статичну твердість із відношенням нормальної сили, прикладеної до індентора, до реальної площі його відбитка. При отриманні кінцевої формули для розрахунку твердості використовувались геометричний і механічний підходи. 2016 Article Measured Hardness by an Indenter Having Ellipsoidal Shape / A. Boudilmi, K. Loucif // Проблемы прочности. — 2016. — № 3. — С. 99-106. — Бібліогр.: 12 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/173479 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
Boudilmi, A.
Loucif, K.
Measured Hardness by an Indenter Having Ellipsoidal Shape
Проблемы прочности
description Если при исследовании твердости в качестве индентора использовалось твердое тело эллипсоидальной формы, выражение для статической твердости рассматривалось как функция от глубины и радиусов отпечатка индентора. При этом применялась общая формула, связывающая статическую твердость с отношением нормальной силы, прилагаемой к индентору, к реальной площади его отпечатка. При получении конечной формулы для расчета твердости использовались геометрический и математический подходы.
format Article
author Boudilmi, A.
Loucif, K.
author_facet Boudilmi, A.
Loucif, K.
author_sort Boudilmi, A.
title Measured Hardness by an Indenter Having Ellipsoidal Shape
title_short Measured Hardness by an Indenter Having Ellipsoidal Shape
title_full Measured Hardness by an Indenter Having Ellipsoidal Shape
title_fullStr Measured Hardness by an Indenter Having Ellipsoidal Shape
title_full_unstemmed Measured Hardness by an Indenter Having Ellipsoidal Shape
title_sort measured hardness by an indenter having ellipsoidal shape
publisher Інститут проблем міцності ім. Г.С. Писаренко НАН України
publishDate 2016
topic_facet Научно-технический раздел
url http://dspace.nbuv.gov.ua/handle/123456789/173479
citation_txt Measured Hardness by an Indenter Having Ellipsoidal Shape / A. Boudilmi, K. Loucif // Проблемы прочности. — 2016. — № 3. — С. 99-106. — Бібліогр.: 12 назв. — англ.
series Проблемы прочности
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fulltext UDC 539.4 Measured Hardness by an Indenter Having Ellipsoidal Shape A. Boudilmi a,b,1 and K. Loucif b a Département de Génie Mécanique, Faculté de Technologie, Université de M’sila, M’sila, Algérie b Laboratoire des Matériaux Nonmétalliques, Institut d’Optique et Mécanique de Précision, Université Ferhat Abbas de Sétif 1, Sétif, Algérie 1 boudissa2@yahoo.fr ÓÄÊ 539.4 Èçìåðåíèå òâåðäîñòè ñ ïîìîùüþ èíäåíòîðà ýëëèïñîèäàëüíîé ôîðìû À. Áóäèëìè à , Ê. Ëîócèô á à Òåõíîëîãè÷åñêèé ôàêóëüòåò, Óíèâåðñèòåò ã. Ìñèëà, Àëæèð á Ëàáîðàòîðèÿ íåìåòàëëè÷åñêèõ ìàòåðèàëîâ, Óíèâåðñèòåò èì. Ôåðõàòà Àááàñà, Ñåòèô, Àëæèð Åñëè ïðè èññëåäîâàíèè òâåðäîñòè â êà÷åñòâå èíäåíòîðà èñïîëüçîâàëîñü òâåðäîå òåëî ýëëèï- ñîèäàëüíîé ôîðìû, âûðàæåíèå äëÿ ñòàòè÷åñêîé òâåðäîñòè ðàññìàòðèâàëîñü êàê ôóíêöèÿ îò ãëóáèíû è ðàäèóñîâ îòïå÷àòêà èíäåíòîðà. Ïðè ýòîì ïðèìåíÿëàñü îáùàÿ ôîðìóëà, ñâÿçû- âàþùàÿ ñòàòè÷åñêóþ òâåðäîñòü ñ îòíîøåíèåì íîðìàëüíîé ñèëû, ïðèëàãàåìîé ê èíäåíòîðó, ê ðåàëüíîé ïëîùàäè åãî îòïå÷àòêà. Ïðè ïîëó÷åíèè êîíå÷íîé ôîðìóëû äëÿ ðàñ÷åòà òâåðäîñòè èñïîëüçîâàëèñü ãåîìåòðè÷åñêèé è ìàòåìàòè÷åñêèé ïîäõîäû. Êëþ÷åâûå ñëîâà: èçìåðåíèå òâåðäîñòè, ìåõàíè÷åñêèå èñïûòàíèÿ, îáúåìíàÿ äåôîð- ìàöèÿ, ãåîìåòðè÷åñêàÿ ìîäåëü. N o t a t i o n A B C, , – semi-axes of ellipsoid d – diameter of the area of the project imprint of a revolution ellipsoid indenter He – static hardness of the ellipsoidal indenter Hec – measured hardness by a body of revolution, in the case of a circular imprint Her – measured hardness by a body of revolution, in the case of a real imprint F – applied load S – imprint surface a b� �, – semi-axes of the projected elliptic surface h – depth of the imprint r – radius of the projected surface of the imprint Introduction. The different dynamic and static hardness tests play a major role to define the hardness of massive and covered materials. They consist to penetrate a harder indenter in a less hard body [1]. © A. BOUDILMI, K. LOUCIF, 2016 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 99 The hardness of material is a very important property in the fields of materials industry. It is considered as a significant mechanical characteristic in a world of technologies. It is defined as the mechanical resistance of material under test, which opposes the penetration of another harder material [2–10]. The hardness of covered and massive materials is an old problem, which had already been a subject of many theoretical and experimental studies. There are a lot of geometrical and mathematical models, to determine the hardness of different materials. The static hardness He is expressed by the ratio of the applied load F to the imprint surface S [2– 10]. Its mathematical expression is given by the following expression: H F S e � . In the hardness tests, the geometry of the indenter (pyramid, cone, sphere, etc.) is a very significant factor because of the geometrical form of the resulting imprint and the phenomena occurring during and after the tests (cracking, deformation, etc.). In the present theoretical study, an indenter of an ellipsoidal geometrical shape is used to measure the static hardness of materials, where it has been calculated according to semi-axes A, B, and C of the indenter, semi-axes of the projected imprint a� , b� , and r, applied load F , and depth h of the imprint. Furthermore, the hardness has been theoretically calculated by using geometrical approaches for the real imprint cap. The paper is organized as follows: First, general theoretical concepts related to the indenter theory are highlighted. Secondly, the area of the imprint is calculated as a function of its semi-axes and its depth. Finally, mathematical and geometrical assumptions have been made to reduce and give a useful expression of the hardness expression. 1. Mathematical Concepts. 1.1. Ellipse. An ellipse is formed by cutting a three-dimensional cone with a slanted plane. It has radius that changes in between A along the x axis and B along the y axis [11]. The standard equation of an ellipse (Fig. 1) with a center at the origin of a Cartesian coordinates system and aligned with the axes is x A y B 2 2 2 2 1� � . (1) Thus, the area of the ellipse (S ) is S AB� � . (2) 1.2. Ellipsoid. The standard characteristic equation of an ellipsoid centered at the origin of a Cartesian coordinates system and aligned with the axes is [11]: x A y B z C � � � � � � � � � � � � � � � 2 2 2 1. (3) For an ellipsoid of revolution of semi-axes (R R C, , ) respectively, along axes Ox, Oy, and Oz (Fig. 2), its characteristic equation is written as the following: x R y R z C � � � � � � � � � � � � � � � 2 2 2 1. (4) A. Boudilmi and K. Loucif 100 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 We put x y r2 2 2� � . (5) By introducing Eq. (5) into Eq. (4), we get r R z C � � � � � � � � � � 2 2 1. (6) 1.3. Surface of a Body of Revolution. The side area of a body generated by a revolution of a curve of a characteristic equation z f x� ( ), around an axis Ox and ranging between the points a and b (Fig. 3) is expressed by [11]: S zdl a b � � 2� , (7) where dl is the differential of the curve arc, it is given by the formula: dl dz dx� �( ) ( ) .2 2 (8) By introducing Eq. (8) in Eq. (7), we get S z dz dx dx a b � � � � � �� 2 1 2 � with dz dx df x dx � ( ) . (9) Then, Eq. (7) may be written as S f x df x dx dx a b � � � � � �� 2 1 2 � ( ) ( ) . (10) 2. Hardness Measured by an Ellipsoidal Indenter. 2.1. Principle of Penetration. In the case of hardness tests by using an ellipsoidal indenter, we used an indenter of an ellipsoidal shape of semi-axes A, B, and C , under the action of a known constant force applied perpendicular to the indenter and under defined conditions. We measure the dimensions of the imprint (transverse and depth) and deduct the hardness. Measured Hardness by an Indenter Having Ellipsoidal Shape ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 101 Fig. 1. The ellipse. Fig. 2. The ellipsoid. 2.2. Static Hardness. The static hardness He is expressed by the ratio of the load applied F to the projected surface S of the imprint [12], as following: H F S e � . (11) For an elliptical indenter, the mathematical expression of the area of the projected imprint of the semi-axes; a� and b� is given by the following relation: S a b� � �� . (12) Introducing Eq. (12) in Eq. (11), the static hardness becomes: H F a b e � � �� . (13) For an indenter having a geometrical form of an ellipsoidal (Figs. 4 and 5), the semi-axes a� and b� of the projected surface of the imprint can be written according of the penetration h and the semi-axes of the indenter A, B, and C as following: a A h C h C � � � � � � � 2 2 , (14) b B h C h C � � � � � � � 2 2 . (15) Then the expression of hardness becomes H C F AB hC h e � � 2 22� ( ( ) ) . (16) 102 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 A. Boudilmi and K. Loucif Fig. 3. A body of revolution. 2.3. The Hardness when the Indenter is a Revolution Ellipsoid. 2.3.1. A Circular Imprint. For an ellipsoidal indenter of circular section (C R B� � ), the projected surface of the imprint becomes a circular disk of a radius r d� 2 (Fig. 6). We can write the expression of the radius according of the semi-axes of the body of revolution A and R, and the imprint depth h, as follows: d r R h A h A2 2 2 � � � � � � � . (17) Then, the hardness takes following form: H F r F d A F R hA h ec � � � �� � �2 2 2 2 2 4 2( ( ) ) . (18) 2.3.2. A True Imprint. For an ellipsoidal indenter (body of revolution) of a circular section (C B R� � ), the imprint is a cap of circular basis of a diameter d and a depth h (Fig. 4). Accordingly, the surface of this imprint gets the formula: S z dz dx dx A h A � � � � � � � � 2 1 2 � . (19) ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 103 Measured Hardness by an Indenter Having Ellipsoidal Shape Fig. 4. Penetration of ellipsoid. Fig. 5. Area of project imprint of ellipsoid indenter. Fig. 6. Area of the project imprint of a revolution ellipsoid indenter. From the characteristic equation (1), we deduce: z R x A 2 2 2 2 1� � � � � � � �. (20) The derivative of equation (20) leads to following expression: zz R x A �� 2 2 . (21) By introducing Eqs. (20) and (21) in Eq. (19), we get S z dz dx dx z z dz dx dx A h A A h A � � � � � � � � � � � � � � � �2 1 2 2 2 2 � � � � � � � � � � � � �2 1 1 2 2 2 2 �R x A R A dx A h A . Let t x A � � with �2 2 2 2 � �A R A . Then, the area of the imprint becomes S CA t dt A h A � � � �2 1 2� � � � ( ) . By integration, we get S RA t t t A h A � � � � � � � � � � [ arcsin ] .1 2 Thus, S RA RA A h A A h A � � � �� � � � � �� � � � � � � � ��� � � � �[ arcsin1 12 2 � �� � � � � � � � � � � � �arcsin � A h A � � � � � � �� � � � �� � � � � � �AR AR R A h A h A 1 12 2 arcsin ( ) � � �� � � � �� � � � � � � ( ) arcsin . A h A h A A h A (22) For a modest applied load, the imprint is very small (A h�� ) so A h A � � 1 S Rh� � � � � �� ��� � � � 1 2 arcsin . (23) 104 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 A. Boudilmi and K. Loucif Then, the hardness of a penetration (h) gets following expression: H F Rh er � � � � � � � �( arcsin ) . 1 2 (24) From Eq. (17), the penetration expression h becomes a function of the both indenter dimensions (A R, ) and the radius r of the projected imprint, h AR A R r R � � �2 2 (accepted), and h AR A R r R A� � � � 2 2 (rejected). Thus, h A r R � � � � � � � � � � � � �1 1 2 . (25) For a small imprint (cases: microhardness or nanohardness), the ratio r R � � � � is negligible [12]: 1 1 1 2 2 2 � � � � � � � � � � �� � � � � � r R r R . (26) By introducing Eq. (26) in Eq. (25), we get h A r R � � � � � 2 2 . (27) Then, the expression of the imprint surface becomes S Ar R � � � � � �� ��� � � � 2 2 2 1 arcsin . (28) Then, the hardness expression will be given by the formula H R F Ar G F r er � � � � 2 12 2 2 � � � � �( arcsin ) (29) with the constant G R A � � � 2 1 2 � � � � �( arcsin ) . (30) C o n c l u s i o n s 1. In the present paper, the most important result of using an indenter of ellipsoidal geometric form to measure the material hardness is the derived mathematical expressions of the imprint in different geometric shapes. ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 105 Measured Hardness by an Indenter Having Ellipsoidal Shape 2. In the determination of hardness expression, when the indenter has a geometrical form of an ellipsoidal, the considered surface of the resulting imprint has been taken as an elliptical section (projection of the real print) in the first case. The related mathematical was simple, but in the second case, where the surface of the imprint is considered as an ellipsoidal cap, the hardness expression is more complex. Thus, a geometrical approach has been made, to ease the mathematical expression of the resulting imprint and hardness. 3. As a main conclusion, the indenter of ellipsoidal shape presents theoretical and experimental interests, which will lead to widen the field of applications of various methods of the hardness tests. 4. In this theoretical study , the hardness expression of a spherical indenter can be deduced. Also, it can be seen the differences between an ellipsoid indenter and spherical indenter. Ð å ç þ ì å ßêùî ïðè äîñë³äæåíí³ òâåðäîñò³ çà ³íäåíòîð ñëóãóâàëî òâåðäå ò³ëî åë³ïñî¿äíî¿ ôîðìè, âèðàç äëÿ ñòàòè÷íî¿ òâåðäîñò³ ðîçãëÿäàâñÿ ÿê ôóíêö³ÿ â³ä ãëèáèíè ³ ðàä³óñ³â â³äáèòêà ³íäåíòîðà. Ïðè öüîìó âèêîðèñòîâóâàëàñü çàãàëüíà ôîðìóëà, ùî çâ’ÿçóâàëà ñòàòè÷íó òâåðä³ñòü ³ç â³äíîøåííÿì íîðìàëüíî¿ ñèëè, ïðèêëàäåíî¿ äî ³íäåíòîðà, äî ðåàëüíî¿ ïëîù³ éîãî â³äáèòêà. Ïðè îòðèìàíí³ ê³íöåâî¿ ôîðìóëè äëÿ ðîçðàõóíêó òâåðäîñò³ âèêîðèñòîâóâàëèñü ãåîìåòðè÷íèé ³ ìåõàí³÷íèé ï³äõîäè. 1. D. Tabor, The Hardness of Metals, Clarendon Press, Oxford (1951). 2. B. Jönsson and S. Hogmark, “Hardness measurements on thin films,” Thin Solid Films, 114, 257–269 (1984). 3. P. Morisset et P. Salmon, Chromage Dur et Décoratif, CETIM, Senlis (Oise) (1988). 4. J. P. Mercier, G. Zambelli, et W. Kurz, Introduction à la Science des Matériaux, PPUR, Lausanne (1999). 5. C. Barlier et L. Girardin, Matériaux et Usinage, Paris (1999). 6. E. S. Puchi-Cabrera, “A new model for the computation of the composite hardness of coated systems,” Surf. Coat. Technol., 160, 177–186 (2002). 7. A. M. Korsunsky, M. R. McGurk, S. J. Bull, and T. F. Page, “On the hardness of coated systems,” Surf. Coat. Technol., 99, 171–183 (1998). 8. J. V. Fernandes, A. C. Trindade, L. F. Menezes, and A. Cavaleiro, “A model for coated surface hardness,” Surf. Coat. Technol., 131, 457–461 (2000). 9. D. Chicot, L. Gil, K. Silva, et al., “Thin film hardness determination using indentation loading curve modelling,” Thin Solid Films, 518, 5565–5571(2010). 10. J. Qin, Y. Huang, K. C. Hwang, et al., “The effect of indenter angle on the microindentation hardness,” Acta Mater., 55, 6127–6132 (2007). 11. N. Piskounov, Calcul Diférentiel et Intégral, Office des Publication Universitaires, Algeria (1991). 12. L. Chambadal, Formulaire de Mathématique, BORDAS, Paris (1985). Received 06. 10. 2014 106 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 A. Boudilmi and K. 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