EXTRACT THE PLASTIC PROPERTIES OF METALS US- ING REVERSE ANALYSIS OF NANOINDENTATION TEST

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1 47 3 Vol.47 No Ê ACTA METALLURGICA SINICA Mar. 211 pp ±Á Æ ½ Å³Æ ¹ 1 Î 1 ÏÍ 1 1 Ì 2 Ë 1 1 ¾ Þº, ¾ ¾ ³» Í Þº, ¾ 324 Æ ± Ó Ó ÆÏÞØ, Â ¼ ± È Á ÅÛ ÖÝÛ, Ó Ó Ï ¼ ±. º Ì Ï, Á ÅÛ ÖÝÛ Ï È, Ç È ÏÐ Ç ¾, Ó Ó ÏÁ ; º ±Ï, ¼ ÏÞØÓ Ó ÏÁ ; ¾ Æ Ó Ó Ï ±, ÊÒÓ Ó Ï. Ü, ÞØ ÏÚ. ÏÐ, Û, Á, Á, ± Ê TG115.5 µ A Ç ( EXTRACT THE PLASTIC PROPERTIES OF METALS US- ING REVERSE ANALYSIS OF NANOINDENTATION TEST MA Yong 1, YAO Xiaohong 1, TIAN Linhai 1, ZHANG Xiangyu 1, SHU Xuefeng 2, TANG Bin 1 1 Institute of Surface Engineering, Taiyuan University of Technology, Taiyuan Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 324 Correspondent: TANG Bin, professor, Tel: ( , tangbin645@sina.com Supported by High Technology Research and Development Program of China (No.27AA3Z521, National Natural Science Foundation of China (No and Shanxi Province Science and Technology Key Project (No Manuscript received , in revised form ABSTRACT Using traditional methods to evaluate mechanical properties of bulk materials is not applicable for metal surface studying and metals with very small volume. Nanoindentation testing at very low load is a new successful technique for study of mechanical properties on small scales or near surfaces. However, so far there is not a robust approach to determine plastic properties of metal materials using nanoindentation test. The aim of this paper is to present a method for determining the plastic properties, e.g. the true plastic stress true plastic strain relation of metals combining nanonindentation test and finite element simulation. This methodology contains three main parts. Firstly, considering the special case of metals without strain hardening, the representative stress σ r is determined by varying assumed representative stress over a wide range until the reverse and forward loading curves are consistent. Then, also by comparing the reverse and forward loading curves, the representative strain ε r is obtained, but with different values of strain hardening exponent n, which are in the range of.6. Secondly, a series of simulations are performed for 124 combinations of each parameter (E, σ y, n, ν expressing the elastic plastic behaviors of the universal engineering metals. From the computational results, a dimensionless function u is constructed, and then the strain hardening exponent is determined. At last, substituting the strain hardening exponent n into the power law constitution, the yield stress σ y of metals is acquired. The examination of 5 kinds of metals from the forward analysis metal materials indicates that the dimensionless function u has generality and the strain hardening exponent has stability and uniqueness. The accuracy of this method is also examined by comparing the elasto plastic properties of practical metal AISI 34 steel obtained from nanoindentation test and finite element simulation with the tensile test results. In order to make * ±Ë Ä ßÖÄÆ 27AA3Z521, ±ËÞÕÅ Ó ¼ß ÅÄ Ü Ê : , Ê : Ç Ð : º, Ã, 198 Ë, DOI: /SP.J

322 Ñ 47 the reverse analysis results get higher precision, in the practical application of this technique, the test error of nanoindentation should be maximally reduced.

2 322 Ñ 47 the reverse analysis results get higher precision, in the practical application of this technique, the test error of nanoindentation should be maximally reduced. KEY WORDS forward analysis, reverse analysis, representative stress, representative strain, strain hardening exponent ¹ÛÔ µ«æ Ü ÁÔ Æ Ø, ÌÙ Ð µ µðç. Ø Å, Â «Ü ½ ÍÔ Á Ô µ ÐÇ. ½Â «ÜÅ ÍÔ µ H ÆÐ E [1,2], Ä Ë ÐÂ Ô µ ÐÐÇ²² ß. µ ÑÌ, Ô ÐÐÇ²² Æ Ö Â Ðµ Â Ðµ (σ r, ε r Å [3 13]. É Í À¾ Tabor [14], Dao [3] Ð Þ²Ñ, ÈÔ µ ÆÐ E ½ (σ r, ε r, ÍËÍ É» Ä, µü ² Ô µ ÐÐ Ç²² 6 ĐÌ ²., Antunes [6] Ýµ É, ² µæð, ¼ µ (σ r, ε r µ µ ÐµÂ «ÜÖ µ ² ¼. É, Ô µ µ ²½, Antunes [6] ² É µ, ²Ã Ì ² ÔÐ, Ä ÑÀ ² ²³, Ã ÏØ. µæ ÐÐÇ, ½ ÙÐÂ «Ü Þ É ( Ñ Í» Ä Å ³ ¹ È Ð. «Æ ½ Ñ ³ Â Å ³, Ð ÞÜ, ÐÔÔ µµ ²½ ßÙ. Ñ ½ Ô µ ÆÐ E 55 6 GPa, σ y.1 1 GPa, µ ² n.6, Poisson ν.3. Ý AISI 34, ßÙ Û. 1 Â Å»Ë ÔÐ¾, ÕÁ É½ÞÁ É Û [15], ½ Ý ANSYS v.1.. ÞÕÁ, Ç. Berkovich «Ó Ù 7.3 «ÐÙ«, Ú Berkovich «Ä ². Ã Ú ½ÕÁ 8 Ô Ã, Ð«Ø Ã Þ², «½ ² ÕÁÔÐ¾ Ú, 484 Ã ½ Ð Æ Í x ß Ñ½ y ßÑ, Ð¾Ô ÍÐ¾. Í ½ ½Ä 2 ÐßÆ, «Æ ½Ä Í ßÆ. ² ÍÐ Ñ É² ½ Í Ç Û, Ò Ý ² 4 [16]. É Ô µ ÐÐÇ ½ Ú, Von Mises ÚÂ, Ù 2 ¼. Ð Ð µ, Æ Ð ËÅµ µ (σ ε Ù : Eε (σ σ y σ = n (1 Rε n = σ y (1 + E σ y ε p (σ > σ y Æ, R ², n µ ², σ y 1 ÔÀÓ ½ ÙÏ Fig.1 Two dimensional finite element mesh 2 Å Û Ê Fig.2 The power law elasto plastic true stress true strain curve (E elastic modulus, R strengthen coefficient, σ r representative stress, ε r representative strain, n strain hardening exponent, σ y initial yield stress, ε y corresponding yield strain

3 3 ¾ ¹ :»À ÚÎÚ Đ Ò Ò ³ÎµßßÅ 323, ε p Ðµ, Ðµ 2 ε y (=σ y /E. 2 º Ã 2.1 À²Ä À ¾ ½ É, Antunes [6] ² µ H ½ ÆÐ E [1] r Â Ðµ σ r Æ H =.231 ( Er σ r (2 Lee [7] ½ É, Í ÆÐ Â Ðµ Â Ðµ Æ ( ε r = exp /σ r (3 É Ñ ½ É Ô µ, Æ ÐÐ Ç²² : E=26.3 GPa, σ y =4.2 GPa, n=.38. Ñ Ö, ¾ Oliver Pharr Ù [1] Í Ô H=11.7 GPa. Ü ½ 2 Æ Ô µ Â Ðµ ½ Â Ðµ, Ö µü É ÞĐ. ÅÜ ÉĐ : Ç Ú Þ É, Ö Ü Ñ ³ Î, Ð Ç Þ È Î Þ É, Ù Ñ, ËÜ Ñ ¼Í» Ä ËÈ. Â Ðµ ½Â Ðµ, ² Ç, Â Ðµ É ½ n= É ( 3. ÝµÜ ÉĐ, È σ r = 7.53 GPa, ÉÍ Ñ ³ Í² ¹ È, Ù 4 ¼. Ñ Ü «Ü F max (forward=13.77 mn, F max (reverse= mn, È ÐÏº.14%. Ô²Â Ðµ, ½ < n.6( 3, Ã µ«ß ÜÆ Â Ðµ, ¼ Â Ðµ Ô. ÝµÜ ÉÐ, È ε r =.42, É» Ä Í µ ², Í² Ñ ³, Ù 5 ¼. Ü «Ü F max (reverse= mn, Ñ ³ ÐÏºÕ.8%. ÔÂ Ðµ Ïº Í, ½Ü ÉĐ Â Ðµ µ Û. 6 5 Ö Ð Ð., ÆÐ Ë²µ ², ÃÆÐ ¹Ûµ ² Ã ÓÃ. 2.2 À» É ¼ ¾ ½Ü Ô²Â Ðµ ½Â Ðµ, Íµ ² n, Ö ½ÂÜÆ (1 Ã Í µô µ σ y, ÓÔ Ô Ð«ß. ÆÔÂ Ðµ ½Â Ðµ ßÙ, µü ÉĐ «Ü Ñ ¼ÍÖ È, ÔÔ µ µ ², Ä ¼Đ. ² Ê, Dao [3], «Æ ½ Ñ Á² µ ² Ì ² Forward Reverse ÏÐ Û»ÌÏÌ Ï º Ã Fig.4 Loading curves from forward and reverse finite element simulations Forward Reverse (n=.15 Reverse (n=.25 Reverse (n=.35 Reverse (n=.45 Reverse (n= ± n Ï Fig.3 Scheme of stress strain curves controlled by n (ε e elastic strain ± ÏÐ Û»ÌÏ º Ã Fig.5 Load displacement curves from forward and reverse analysis with different values of n

4 324 Ñ 47 P u = P u (h, h max,, σ r, n = h 2 u ( hmax h, σ r, n (4 Æ, P u Ö», h «, h max «. È P u =, ÐÅÖ, ² h = h r (h r ³ «, ÆØ h r = ( σr u, n h max (5 ²ÉÆ (5 Æ, Ð, ½ 124 ĐÆ ÐÐÇ²² (E, σ y, n, ν Þ Ñ, Ö ÌÐ Ñ ³, ½ ÔÂ Ðµ ßÙ Đ²²Ðµ Â Ðµ. 7 Ð 124 Đ Ð Ç µ É Ðµ hr h max σr. Ç h r /h max,»²ô µ Óµ. ¼Å «ÌÒ µä Ï, «ÌÒ µä Ï, É Â «Ü Ü 2 Ð Æ. ½ MATLAB Ð É² Þ É, ÍÌ ² u ÆÙ : u( σ r, n = h r h max = (.25n n 2 + [.168n.48 ln( σ ] 3 r + (.253n 3 + [.14359n n Forward Reverse (n=.15 Reverse (n=.25 Reverse (n=.35 Reverse (n=.45 Reverse (n=.55 ln( σ r ] ± Õ ÏÐ Û ²Ï Fig.6 Amplificatory unloading segments from forward and reverse analysis with different values of n (.611n n.65421ln( σ r + (.58211n n.6729 (6 Æ (6 µ½ Æ¼ Ô µ, Æ (1 Ü Ç ÉÔ µ µ ² n=.384, σ y =4.18 GPa, Ñ Ðµ ÐÏº 1.5% ½.48%. Ö, Ïº ¾µ ² Ç ÏºÀ, Óµ ² Ç Ï º Â Ðµ ½µ Û ÁÆ (6 ½Ð. ¾ Æ, Â Ðµ ½µ Ïº ¼Ò, ², µ ² Ç ÏºÖ ¾É ¼ÍÌ ² u ½Ð. ² u ½Ð Á¼ µ ², Ñ Ô µý Ä¹ ± 124 Đ 5 ĐÔ µ ÞÜ, Ç ³ÁÏºÙ 1 ¼., Ü ³ Ñ Ô µ ÐÐÇ²² Ïº Ò, 3% Ä, ² Ì ² u ½Ð, ¼ µ ² ÉÐ. ß, ßÙ ÅÉµ½ : ¾Â «ÜÙ ÍÔ µ ÆÐ ½, Ö µ Đ ÉÔÂ Ðµ ½Â Ðµ, ½Ì ²½Æ (1 Íµ ²½. Î µ È ²Ð «½Ì ² À Ïº, ² ÏØ». 8 Ù ¹. h r /h max r / n= n=.1 n=.2 n=.3 n=.4 n=.5 n=.6 7 Ë Å ± u ÏÓ Fig.7 Determination of dimensionless function u 1 5 Å Æ±± ¾Ó Û ²ÏÎ¹ Table 1 Error distribution of reverse analysis results of 5 kinds of metals with known elasto plastic properties Matel Elesto plastic parameter Plastic parameter obtained from reverse analysis E, GPa σ y, GPa n σ r, GPa ε σ y, GPa Error of σ y, % n Error of n, %

3 ¾ ¹ :»À ÚÎÚ Đ Ò Ò ³ÎµßßÅ 325 3 È ²µ½ÅÉÔ ßÙ ÛÔÐ, ÌÐ AISI 34 Þ² Đ½Â «Ü. Đ ½ ASTM Ú Đ, Đµ ½ 1 3 s 1, ¼Í AISI 34 Æ ÐÐÇ²²Ù 2 ¼. Â «Ü ½ Nano Indenter G2 Ú Â «Ç, «2 µm Ù Ç µ, ÁÜ ³ 2.

5 3 ¾ ¹ :»À ÚÎÚ Đ Ò Ò ³ÎµßßÅ È ²µ½ÅÉÔ ßÙ ÛÔÐ, ÌÐ AISI 34 Þ² Đ½Â «Ü. Đ ½ ASTM Ú Đ, Đµ ½ 1 3 s 1, ¼Í AISI 34 Æ ÐÐÇ²²Ù 2 ¼. Â «Ü ½ Nano Indenter G2 Ú Â «Ç, «2 µm Ù Ç µ, ÁÜ ³ 2. 9 ½ 1» ² ½Ü ÉĐ Ô AISI 34 Â Ðµ ½Â Ðµ µ. Ü Ñ «Ü ÐÏº Ð.7% ½.1%. µð 2 ² Í, ½ É 8 ¼Á ÅÛ À Û Ó Ó Ï Æ±± Fig.8 Nanoindentation test and finite element simulation reverse analysis for the prediction of the plastic constitution 2 ¼ 2 ÞØÌÊÏ AISI 34 ÏÅ Æ±± Table 2 Elasto plastic property of AISI 34 steel obtained by two different methods Experimental Reverse Parameter Tensile test Nanoindentation and Error, % reverse analysis E, GPa H, GPa 3.4 σ r, GPa.725 ε r.235 σ y, GPa n Á ÅÛ Û»ÌÏ AISI 34 Ì Ï º Ã Fig.9 Loading curves of AISI 34 by nanoindentation test and reverse analysis

6 326 Ñ Experimental Reverse (n= Reverse (n=.8 Reverse (n=.18 Reverse (n=.28 Reverse (n=.38 Reverse (n=.48 Reverse (n= ± AISI 34 È»ÌÏ º Ã ¼»Ì Fig.1 Load displacement curves from reverse analysis and experimental results for AISI 34 with different values of n ÐÂ «Ü ÞÜ, ÛÔ ÔÔ µ Æ ÐÐÇ²². Æ, Ü µ Ïº ¾ Ü ÉĐ ÁÌ ² u ÍËÆ, ², ½Â «Ü Ü ÅÉÔ µ ÐÐÇ ²², Â «Ü Û «Ü ¼ ÍÔ µ ÐÐÇ²² Û. 4 ½Â «Ü ² ÉÁ, ² ÐÔÔ µ ½µ ² Ü Ù. µ Đ ³Ð, ² ßÙ Û. ÌÐÌÙ ½ Þ Ô µ, ßÙ Ñ Å½Î. Ö Õ : µâ «Ü ÍÔ µ ÆÐ ÆÐ Á ; ½ ÞÜ Đ É, Ô Ô Â Ðµ ½Â Ðµ ; ½Ì ² Ô µ µ ²; ½ «ß Ô µ. «[1] Oliver W C, Pharr G M. J Mater Res, 1992; 7: 1564 [2] Pharr G M. Mater Sci Eng, 1998; A253: 151 [3] Dao M, Chollacoop N, Van Vliet K J, Venkatesh T A, Sure S. Acta Mater, 21; 49: 3899 [4] Bucaille J L, Stauss S, Felder E, Michler J. Acta Mater, 23; 51: 1663 [5] Chollacoop N, Dao M, Suresh S. Acta Mater, 23; 51: 3713 [6] Antunes J M, Fernandes J V, Menezes L F, Chaparro B M. Acta Mater, 27; 55: 69 [7] Lee J, Lee C, Kim B. Mater Des, 29; 3: 3395 [8] Cao Y P, Lu J. Acta Mater, 24; 52: 423 [9] Lee J H, Kim T, Lee H. Int J Solids Struct, 21; 47: 647 [1] Chollacoop N, Ramamurty U. Scr Mater, 25; 53: 247 [11] Cao Y P, Qian X Q, Huber N. Mater Sci Eng, 27; A : 1 [12] Kim J Y, Lee K W, Lee J S, Kwon D. Surf Coat Technol, 26; 21: 4278 [13] Branch N A, Subhash G, Arakere N K, Klecka M A. Acta Mater, 21; 58: 6487 [14] Tabor D. The Hardness of Metals. London: Oxford University Press, 1951: 12 [15] Choi Y, Lee H S, Kwon D. J Mater Res, 24; 19: 337 [16] Soare S, Bull S J, O Neil A G, Wright N, Horsfall A, Santos J M M dos. Surf Coat Technol, 24; : 497

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