Study on nonlinear deformation of binary interstitial alloy AB with BCC structure under pressure

1. Introduction When the object is deformed, its properties change, especially its mechanical properties. There are two types of deformation: elastic deformation and nonlinear deformation. When the load is too large, the object will be deformed nonlinearly and the material itself will have a residual deformation. Nonlinear deformation causes many desired and unwanted effects on the mechanical properties of the material. So this is a problem that material researchers need to solve and it pays the attention of many scientists and technologists [1, 2]. There are various theoretical methods in studying the deformation of metals and alloys including the statistical moment method (SMM) [3, 4]. The SMM has been successfully applied to study the elastic and non-linear deformation of metals and subtitution alloys, the elastic deformation of interstitial alloys [3-12]. However, the application of this method to study nonlinear deformation of interstitial alloys is an open question. In this paper, we build the theory of nonlinear deformation for interstitial AB with bodycentered cubic (BCC) structure under pressure by the statistical moment method (SMM) [4, 8, 9, 12].

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57 HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0029 Natural Sciences 2018, Volume 63, Issue 6, pp. 57-65 This paper is available online at STUDY ON NONLINEAR DEFORMATION OF BINARY INTERSTITIAL ALLOY AB WITH BCC STRUCTURE UNDER PRESSURE Nguyen Quang Hoc 1 , Nguyen Thi Hoa 2 , Nguyen Duc Hien 3 and Dang Quoc Thang 1 1 Faculty of Physics, Hanoi National University of Education 2 University of Transport and Communications, 3 Mac Dinh Chi High School, Gia Lai province Abstract: The analytic expressions of the free energy, the mean nearest neighbor distance between two atoms and the nonlinear deformation quantities such as the density of deformation energy, the maximum real stress and the limit of elastic deformation for interstitial alloy AB with BCC structure under pressure are derived from the statistical moment method. The nonlinear deformations of main metal A is special case of nonlinear deformation for interstitial alloy AB. Keywords: Interstitial alloy, nonlinear deformation, density of deformation energy, maximum real stress, limit of elastic deformation, statistical moment method. 1. Introduction When the object is deformed, its properties change, especially its mechanical properties. There are two types of deformation: elastic deformation and nonlinear deformation. When the load is too large, the object will be deformed nonlinearly and the material itself will have a residual deformation. Nonlinear deformation causes many desired and unwanted effects on the mechanical properties of the material. So this is a problem that material researchers need to solve and it pays the attention of many scientists and technologists [1, 2]. There are various theoretical methods in studying the deformation of metals and alloys including the statistical moment method (SMM) [3, 4]. The SMM has been successfully applied to study the elastic and non-linear deformation of metals and subtitution alloys, the elastic deformation of interstitial alloys [3-12]. However, the application of this method to study nonlinear deformation of interstitial alloys is an open question. In this paper, we build the theory of nonlinear deformation for interstitial AB with body- centered cubic (BCC) structure under pressure by the statistical moment method (SMM) [4, 8, 9, 12]. 2. Content In interstitial alloy AB with BCC structure, the cohesive energy of the atom B (in face centers of cubic unit cell) with the atoms A (in body center and peaks of cubic unit cell) in the approximation of three coordination spheres with the center B and the radii 1 1 1, 2, 5r r r is determined by [4, 8, 9, 12]. Received April 2, 2018. Revised June 27, 2018. Accepted July 6, 2018. Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn Nguyen Quang Hoc, Nguyen Thi Hoa, Nguyen Duc Hien and Dang Quoc Thang 58        0 1 1 1 1 1 1 2 4 2 8 5 2 2 i B AB AB AB n iAB i r r ru r                 1 1 12 2 4 5 ,AB AB ABr r r    (2.1) where AB is the interaction potential between the atom A and the atom B, in is the number of atoms on the ith coordination sphere with the radius ( 1,2,3),ir i  11 1 01 0 ( )B B Ar r r y T   is the nearest neighbor distance between the interstitial atom B and the metallic atom A at temperature T, 01Br is the nearest neighbor distance between the interstitial atom C and the metallic atom A at 0 K and is determined from the minimum condition of the cohesive energy 0Bu , 10 ( )Ay T is the displacement of the atom A1(the atom A stays in the body center of cubic unit cell) from equilibrium position at temperature T. The alloy’s parameters for the atom B in the approximation of three coordination spheres have the form [4, 8, 9, 12]       2 2 (2) (1) (1) 1 1 1 1 1 1 2 2 16 2 5 , 5 5 AB i i eq B AB AB ABk u r r r r r                    4 4 (4) (2) 1 1 12 1 (1) (4) (3) 1 1 13 1 1 1 48 1 1 ( ) 2 24 8 2 1 4 5 2 ( 2) ( 5), 16 150 125 AB i i eq B AB AB AB AB AB u r r r r r r r r                        4 2 2 (3) (2) (1) 2 1 1 12 3 1 1 1 1 1 (3) 1 (2) (1) (4) (3) 1 1 1 12 3 1 1 6 48 1 1 5 2 ( ) ( ) ( ) 2 4 4 8 2 2 5 5 5 ( ) 8 1 1 2 3 ( ) ( ) ( ) ( ) 8 8 25 25 AB i i i eq B AB AB AB AB AB AB AB AB u u r r r r r r r r r r r r r r r                                 (2) (1) 1 12 3 1 1 5 5 5 2 3 ( ) ( ), 25 25 AB ABr r r r     1 24 ,B B B    (2.2) where   ,.,,,4,3,2,1(/)()( zyxmrr miiAB mm AB and iu  is the displacement of the ith atom in the direction . The cohesive energy of the atom A1 (which contains the interstitial atom B on the first coordination sphere) with the atoms in crystalline lattice and the corresponding alloy’s parameters in the approximation of three coordination spheres with the center A1 is determined by [4, 8, 9, 12]   1 10 0 1 ,A A AB Au u r     1 1 1 1 1 1 2 2 1 (2) (1) 1 1 1 2 5 , 2i A AB A A A i Aeq AB A AB A r r k k u r k r r                    Study on nonlinear deformation of binary interstitial alloy AB with BCC structure under pressure 59 1 1 1 1 1 1 1 1 4 1 1 14 2 3 1 1 (4) (2) (1) 1 1 1 1 48 1 1 8 8 1 ( ) ( ) ( ), 24i A AB A A A i A Aeq AB A AB A AB A r r u r r r r r                         1 1 11 1 1 1 4 22 2 1 1 (3) (2) (1) 2 2 1 1 12 3 1 1 1 6 48 1 3 3 ( ) ( ) ( ). 2 4 4i AB A i i eq A A A AB A AB A AB A A A A r r u u r r r r r r                               1 1 11 2 4 ,A A A    (2.3) where 11 1A B r r is the nearest neighbor distance between the atom A1 and atoms in crystalline lattice. The cohesive energy of the atom A2 (which contains the interstitial atom B on the first coordination sphere) with the atoms in crystalline lattice and the corresponding alloy’s parameters in the approximation of three coordination spheres with the center A2 is determined by [4, 8, 9, 12]   2 20 0 1 ,A A AB Au u r      2 2 2 2 1 2 2 2 (2) (1) 1 1 1 1 2 , 4 2 i A AB A A i eq A AB A AB A A r r k k u k r r r                   2 2 2 2 1 2 2 2 2 2 4 1 1 14 (4) (3) 1 1 1 (2) (1) 1 12 3 1 1 1 48 1 1 ( ) ( ) 24 4 1 1 ( ) ( ), 8 8 i A AB A A A i eq AB A AB A A r r AB A AB A A A u r r r r r r r                             2 2 1 2 4 (4) 2 2 12 2 2 6 48 1 ( ) 8i A AB A A AB A i i eq A r r r u u                      2 2 2 2 2 2 2 1 1 (3) (2) (1) 1 1 13 1 1 3 4 8 3 ( ) ( ) ( ), 8A A AB A AB A AB A Ar r r r r r        2 2 21 2 4 ,A A A    (2.4) where 2 2 2 1 01 0 01( ),A A B Ar r y T r  is the nearest neighbor distance between the atom A2 and atoms in crystalline lattice at 0K and is determined from the minimum condition of the cohesive energy 20 0 , ( )A Bu y T is the displacement of the atom C at temperature T. In Eqs. (2.3) and (2.4), 0 1 2, , ,A A A Au k   are the coressponding quantities in clean metal A in the approximation of two coordination sphere [4, 8, 9, 12]. In the action of rather large external force F, the alloy transfers to the process of nonlinear deformation. When the interstititial alloy AB is deformed, the nearest neighbour distance  1 1 2, , , F Xr X A A A B at temperature T has the form Nguyen Quang Hoc, Nguyen Thi Hoa, Nguyen Duc Hien and Dang Quoc Thang 60    1 1 01 01 1 011 2 , F X X X X X Xr r r r r r           (2.5) where ( E     is the stress and E is the Young modulus),  1 1 ,X Xr r P T is the nearest neighbour distance in alloy before deformation. When the alloy is deformed, the mean nearest neighbour distance 01 F Xr at 0 K has the form  01 01 1 F X Xr r   (2.6) The equation of state for interstitial alloy AB with BCC structure at temperature T and pressure P is written in the form 0 1 1 1 1 1 . 6 2 u k Pv r xcthx r k r            (2.7) At 0 K and pressure P, this equation has the form 0 0 1 1 1 . 4 u k Pv r r k r           (2.8) If knowing the interaction potential i0 , the equation (2.8) permits us to determine the nearest neighbour distance   1 1 2,0 , , , Xr P X B A A A at pressure P and temperature 0K. After finding  1 ,0Xr P , we determine     21 1,0 ,0 1 2FX Xr P r P     (2.9) and then determine the parameters 1 2( ,0), ( ,0), ( ,0), ( ,0) F F F F X X X Xk P P P P   at pressure P and 0K for each case of X when alloy is deformed. Then, the displacement 0 ( , ) F Xy P T of atom from the equilibrium position at temperature T and pressure P is calculated by 2 0 3 2 ( ,0) ( , ) ( , ) 3 ( ,0) , F F FX X XF X P y P T A P T k P    5 2 1 12 2 2 3 2 3 4 2 3 2 3 4 4 , 1 2 , , , , 2 13 47 23 1 25 121 50 16 1 , - 3 6 6 2 3 6 3 3 2 43 93 169 83 22 1 3 2 3 3 4 3 F X i F F F F F F F F FX X X X iX X X X XF i X F F F F F F F F F X X X X X X X X X F F F F F X X X X X X Y A a a k m x a k a Y Y Y a Y Y Y Y a Y Y Y Y Y                                      5 , F 2 3 4 5 6 5 103 749 363 733 148 53 1 - , 3 6 3 3 3 6 2 F F F F F F F X X X X X X Xa Y Y Y Y Y Y             2 3 4 5 6 7 6 , 561 1489 927 733 145 31 1 65 coth . 2 3 2 3 2 3 2 F F F F F F F F F F F X X X X X X X X X X Xa Y Y Y Y Y Y Y Y x x         (2.10) When alloy is deformed, the nearest neighbour distance  1 , F Xr P T is determined by Study on nonlinear deformation of binary interstitial alloy AB with BCC structure under pressure 61 11 1 1 1 ( , ) ( ,0) ( , ), ( , ) ( ,0) ( , ),F F F F F FB B A A A Ar P T r P y P T r P T r P y P T    1 2 21 1 1 1 ( , ) ( , ), ( , ) ( ,0) ( , ).F F F F FA B A A Br P T r P T r P T r P y P T   (2.11) When alloy is deformed, the mean nearest neighbour distance 1 ( , ) F Ar P T has the form    1 1( , ) ,0 , , F F F A Ar P T r P y P T     21 1 1 1 1( ,0) 1 ( ,0) ( ,0) 1 2 , ( ,0) 3 ( ,0),F F F F FA B A B A A Br P c r P c r P r P r P            1 2 ( , ) 1 7 ( , ) ( , ) 2 ( , ) 4 ( , ),F F F F FB A B A B A B Ay P T c y P T c y P T c y P T c y P T     (2.12) where 1 ( , ) F Ar P T is the mean nearest neighbor distance between atoms A in the deformed interstitial alloy AB at pressure P and temperature T, 1 ( ,0) F Ar P is the mean nearest neighbor distance between atoms A in the deformed interstitial alloy AB at pressure P and temperature 0K, 1 ( ,0) F Ar P is the nearest neighbor distance between atoms A in the deformed clean metal A at pressure P and temperature 0K, 1 ( ,0) F Ar P is the nearest neighbor distance between atoms A in the zone containing the interstitial atom B when the alloy AB is deformed at pressure P and temperature 0K and Bc is the concentration of interstitial atoms B. The free energy of alloy AB with BCC structure before deformation with and the condition B Ac c has the form   1 2 1 7 2 4 .AB B A B B B A B A cc c c c TS          2 2 1 0 0 22 2 3 1 3 2 X X X X X X X X Y U N Y k                         3 2 2 2 1 1 24 2 4 1 2 2 1 1 , 3 2 2 X X X X X X X X X Y Y Y Y k                           2 0 3 ln(1 ) , coth , Xx X X X X XN x e Y x x        (2.13) where X is the free energy là an atom X in clean metal A or interstitial alloy AB before deformation and Sc is the configuration entropy of interstitial alloy AB before deformation. The free energy of alloy AB with BCC structure after deformation with and the condition B Ac c has the form   1 2 1 7 2 4 .F F F F FAB B A B B B A B A cc c c c TS          2 2 1 0 0 22 2 3 1 3 2 F F F F F F F X X X X X X XF X Y U N Y k                          3 2 2 2 1 1 24 2 4 1 2 2 1 1 , 3 2 2 F F F F F F F FX X X X X X X XF X Y Y Y Y k                             Nguyen Quang Hoc, Nguyen Thi Hoa, Nguyen Duc Hien and Dang Quoc Thang 62 2 0 3 ln(1 ) , coth , F XxF F F F F X X X X XN x e Y x x         (2.14) where ψFX is the free energy là an atom X in clean metal A or interstitial alloy AB after deformation and Sc is the configuration entropy of interstitial alloy AB after deformation.However, this entropy is constant. When the process of nonlinear deformation in alloy happens, the relation between the stress and the strain is decribed by . 1 AB F lAB oAB F       (2.15) Here, oAB and AB are constant depending on every interstitial alloy. We can find the strain F corresponding to the maximum value of the real stress through the density of deformation energy. In order to determine the stress – strain dependence according to the above formula, it is necessary to determine two constants oAB và AB for every intestitial alloy. Therefore, we can calculate the density of deformation energy of interstititial alloy AB in the form   1 1 ( ) 1 7 F F F AB AB AB AB A A AB BF F F AB AB AB AB AB AB f c V V N v v N v v                          1 1 2 22 4 . F FF A A A AA A B B BF F F AB AB AB AB AB AB c c c v v v v v v                              (2.16) Since normally  is very small ( << 1), we can expand the expression of the Helmholtz free energy  1 2 , , , F X X A A A B  in terms of the strain  in the form of series and approximately, 2 2 2 1 ( ) . 2 F F F X X X X T T                        (2.17) Therefore,   2 2 2 1 ( ) 1 7 2 F F A A A A AB B F F F AB AB AB ABT T f c N v v v v                                1 1 1 1 2 2 2 2 2 2 2 2 2 F F B B B B B F F F AB AB AB ABT T F F A A A AB F F F AB AB AB AB T T c N v v v v c N v v v v                                                              2 2 2 2 2 2 2 4 . 2 F F A A A AB F F F AB AB AB AB T T c N v v v v                                (2.18) Applying the following formulas Study on nonlinear deformation of binary interstitial alloy AB with BCC structure under pressure 63 1 1 , F F F X X X F X r r         , 2 2 2 2 1 1 1 2 2 2 1 1 1 . F F F F F F F X X X X X X X F F F X X X r r r r r r                                 (2.19) From that,   2 2 22 1 1 1 2 2 1 1 1 1 1 1 1 1 ( ) 1 7 2 1 1 F F F F F F A A A A A A AB B A F F F F F F AB AB AB A AB A AT T T F F B B B B F F F AB AB AB B T r r r f c N v v v r v r r c r N v v v r                                                                                 1 1 1 1 1 1 1 1 2 2 22 1 1 2 2 1 1 2 22 1 1 2 1 1 1 2 2 1 1 2 F F F F B B B B F F F AB B BT T F F F F F A A A A AB A F F F F F AB AB AB A AB A A T T r r v r r r rc N v v v r v r r                                                                               1 1 2 1 2 F A F T r                   2 2 2 2 2 2 2 2 2 2 2 2 22 1 1 1 2 2 1 1 1 4 1 1 2 F F F F F F A A A A A AB A F F F F F F AB AB AB A AB A A T T T r r rc N v v v r v r r                                                            (2.20) From that we have 1 01 01 2 1 012 2 (1 ) 2 , 2 . F FX X X F X X r r r r r              (2.21) Therefore,     22 2 01 01 012 1 1 1 22 01 1 21 1 1 ( ) 1 7 2 2 2 21 1 2 F F F F F FA A A A AB B A A AF F F F F F AB AB AB A AB A AT T T F F BB B B F F F F AB AB AB B ABT r f c r r N v v v r v r r rc N v v v r v                                                                       1 1 1 1 1 1 1 1 1 1 2 01 012 1 1 22 201 01 012 1 1 1 2 2 22 1 1 2 2 2 F F F FB B B BF F B BT T F F F F A A A AF FB A A AF F F F F F AB AB AB A AB A A T T T r r r r rc r r N v v v r v r r                                                                             2 2 2 2 2 2 2 2 2 2 22 201 01 012 1 1 1 24 1 1 2 2 . 2 F F F F A A A AF FB A A AF F F F F F AB AB AB A AB A A T T T rc r r N v v v r v r r                                               (2.22) Here, the derivatives of the Helmholtz free energy in respect to the neighbour distance have the form 0 1 1 1 1 1 3 coth , 2 2 FF F F FXX X X XF F F F X X X XT u k N x x r r k r                  Nguyen Quang Hoc, Nguyen Thi Hoa, Nguyen Duc Hien and Dang Quoc Thang 64 2 22 2 0 2 2 2 1 1 1 1 31 1 . 2 4 2 FF F F F XX X X X F F F F F F X X X X X XT u k k N r r k r k r                              . (2.23) Similar to metal when the deformation rate is constant, the density of deformation energy of alloy has the form fAB() = CAB.AB., (2.24) where CAB is the proportional factor. The function fAB() gets its maximum at the strain F AB . This means that max max( ) F F AB AB AB AB AB ABf f C    . (2.25) Then, we can find the maximum stress ABmax and the real stress 1ABmax max max , AB AB F AB AB f C    max max 1 max 1 (1 ) AB AB AB F F F AB AB AB AB f C          . (2.26) From the maximum condition of stress 0 F AB ABl          we determine the strain F AB corresponding to the maximum value of the real stress as follows 1 F AB AB AB      (2.27) Therefore,   max 0 . 1 ABF AB lAB AB F AB        (2.28) The proportional factor CAB is determined from the experimental condition of the stress 0,2AB in alloy in the form 0,2 0,2 0,2 ( ) . AB AB AB f C     . (2.29) In interstitial alloy AB, if the concentration cB of interstitial atoms is equal to zero, we obtain the expression of the density of deformation energy for main metal A. After having the value of the strain F AB corresponding to the maximum value of the density of deformation energy, we can find the expression to describe the relation between the stress and the strain in the process of nonlinear deformation of interstitial alloy AB. 3. Conclusion The analytic expressions of the free energy, the mean nearest neighbor and the nonlinear deformation quantities such as the density of deformation energy