- Email: [email protected]
Finite element analysis of peening process with plasticity deforming shot
Journal of
Materials Processing Technology ELSEVIER
J. Mater. Process. Technol. 45 (1994) 607-612
Finite Element Analysis of Peening Process with Plastically Deforming Shot K e n - i c h i r o Mori, Kozo Osakada and Naoki Matsuoka Department of Mechanical Engineering, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan To simulate not only plastic deformation of a workpiece but also that of a shot in peening processes, the effect of the interaction between the shot and workpiece in the collision is included in the dynamic viscoplastic finite element method. In the formulation, the equilibrium equations of nodal forces and the boundary condition for displacement are simultaneously satisfied under a sliding condition at the interface between the workpiece and shot. Axi-symmetric plastic deformation in peening of a circular workpiece with a single shot is computed. The calculated shapes of the workpiece and the shot are compared with the experimental ones for the workpiece and the shot made from plasticine whose flow stress is controlled by the temperature. It is shown that almost no plastic deformation of the shot occurs under actual shot-peening conditions for steel workpieces when the flow stress ratio of the shot to the workpiece is larger than two.
1. INTRODUCTION In shot-peening, the surface of the metal workpiece is pelted with hard and round shots at a high velocity and is plastically deformed to cause the residual stress after the rebound of the shots. For high-strength steel products, the use of shot-peening recently tends to increase, e.g. gears, springs and shafts. Since the increase in the strength of the workpiece brings about plastic deformation and fracture of the shots, it is desirable in industry to predict the deformation behaviour of the shots as well as that of the workpiece. In shot-peening processes, the interaction between the shot and the workpiece has a great influence on the deformation behaviour. During the collision of the workpiece with the shot, the plastically deforming region gradually expands, and thus non-steady-state deformation occurs. The finite element method is suitable for simulating n o n - s t e a d y sate deformation in forming processes because the calculation is performed incrementally. Moreover, the effects of the inertia, the workhardening characteristic, the strain-rate
sensitive and the friction on the deformation behaviour can be included in the formulation. Al-Obaid [1] has used the dynamic elasticplastic finite element method to analyse a residual stress distribution in shot-peening. The authors [2] have simulated plastic deformation of the workpiece by the dynamic viscoplastic finite element method, and have calculated the residual stress distribution from the elastic finite element method by using the stress distribution obtained from the viscoplastic analysis. In these simulations, however, deformation of the shot has been neglected. In the present study, peening processes with a plastically deforming shot are simulated by a dynamic viscoplastic finite element method. The interaction of plastic deformation between the shot and the workpiece in the collision is introduced into the formulation. The effects of working conditions on plastic deformation of the shot in peening are examined.
2. METHOD OF SIMULATION
0924-0136/94/S07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0924-0136(94)00248-7
608
2.1 Dynamic viscoplastic finite element method The present viscoplastic finite element method is formulated on the basis of the plasticity theory for a material with slight compressibility [3,4]. Due to the slight compressibility, the stress components {cr} can be calculated directly from the strain-rate components {~}; {cr} = [DP]{~}, (1) where [D p] is the matrix correlating the stress with the strain-rate for the compressible viscoplastic deformation. To deal with the inertial effect due to the collision of the shot with the workpiece, the finite deformation theory of plasticity [4,5] is employed. In the finite deformation formulation, the equilibrium equations are satisfied at the end of each deformation step. Therefore, the acceleration, the change in the velocity, can be taken into consideration [2]. The differential equations of equilibrium for high speed deformation problems are given by Ooij vn-vi0
axj - P--ST- = o,
(2)
where P is the density, At is the time interval and vi0, v n are the velocities at the beginning and the end of each deformation step, respectively. In equation (2), the acceleration is approximated by using the difference between the velocities at the beginning and the end of each deformation step. For the finite element formulation of the equilibrium equations (2), the Galerkin method is used as follows;
Oxj{N}
- p - - ~ t ] a v = 0,
(3)
where {N} is the vector of the shape function of the element. The left hand side of equation (3) is partially integrated and equation (1) is substituted; [ f v ([B]T[DP][B]+{N}T{N}A-~t dV]{vl} =
l / { N } f { N } - - ~ - t d ~ { v o } + { F },
(4)
where {v0}, {vl} are the nodal velocity vectors at the beginning and the end of each deformation step, respectively and {F} is the nodal force vector. In these simultaneous equations, {vl} are variables and {vo} arc constants obtained at the previous deformation step.
2.2 Interaction between shot and workpiece For axi-symmetric deformation, the interaction of high speed plastic deformation between the shot and workpiece in the collision is taken into consideration in the viscoplastic finite element formulation. Consider that node B slides on the interface between the shot and the workpiece as shown in Fig. 1. To locate node B on the line AC at the end of each deformation step; the boundary condition for displacement is given by (rck-rA0(Zm-ZA1)-(Zc~-zA0(rm-r~)= 0 (5) rA~ = rA0 + At(Var0+VAr~)/2 Equation (5) is a nonlinear function in terms of the nodal velocities at the end of each deformation step. Beginning of each deformation step
%,%
/ "-,,,,,,
,Jr " ........
.Z?" ,
/ '..
(re0, zoo)
Z°o)C\ ............. ,
,,
\ '"),
,
,
End of each deformation step Fig.1 Sliding of node on interface between shot and workpiece.
609
The Coulomb friction is employed as a frictional law in the interface. The tangential nodal force FBt is provided by the normal nodal force FB, and the coefficient of friction ~t; Fat = gFsn, (6) where Fan is an independent variable and F~ is a dependent variable determined from equation (6). The nodal forces at node B are decomposed by using the shape function of the element, and are equilibrated with the nodal forces at nodes A and C; FAr = -(1-~B)FB]2 F~, : -(1-~B)F,J2 Fcr= -(I+~B)Fs]2 (7) Fcz = -(I+~B)FBz/2, where Fa~ and FA~ are the nodal forces in the radial and axial directions at node A, and ~ is the normalized co-ordinate having a value between - 1 and 1 at point B in the end of each deformation step. FBr and FBz are expressed in a term of FB~ by the use of equation (6) and the co-ordinate transformarion. Equation (7) is substituted into the vector {F} in the second term on the righthand side of equation (4), and FAr, FAz, Fc~ and Fc~ are eliminated. In this method, the nodal forces are equilibrated at nodes A and C and not at node B. When nodal points come newly in contact with the interface, the same treatment is carried out. The number of the independent variables, the normal nodal forces, in the interface is equal to the number of contact nodal points. The equilibrium equations (4) and the displacement boundary conditions (5) are simultaneously solved to deal with the effect of interaction between the shot and workpiece. Since the nodal velocities and the normal nodal forces in the interface are independent variables in the simultaneous equations, the number of the equations is equal to that of the total independent variables. In addition, the obtained matrix becomes unsymmetric because of equation (5) and the normal nodal forces at the interface.
3. PEENING WITH SINGLE SHOT 3.1 Comparison between calculated and experimental results for plasticine Axi-symmetric plastic deformation in peening of a circular workpiece with a single shot is computed under the conditions shown in Table 1. The shot collides perpendicularly with the workpiece. The calculated shapes of the workpieces and the shots are compared with the experimental ones for the workpieces and the shots made from plasticine. Since the deformation behaviour is greatly affected by the flow stress ratio of the shot to the workpiece, the flow stress ratio is controlled Table 1 Working conditions used in simulation of peening for plasticine workpiece and shot. Flow stress / kPa
289~° 14E°°7(15"C)
(plasticine)
198~°16g°1°(25"C) 130~°22~° °4(35"C)
Temperature of shot
15, 25, 35
T~ / "C Temperature of workpiece / "C
25
Impact speed v,i / ms -1
7.5
Diameter of shot d/mm
50
Diameter of workpiece / mm
50
Height of workpiece h
62.5
/ mm Coefficient of friction
0.20
Density / gmm -3
1.95x10 -3
610
3.2 Under
in the plasticine experiment by changing the temperature T s of the shot and fixing that of the workpiece. The element mesh in axi-symmetric peening for the shot temperature T,=15"C is illustrated in Fig. 2. Both of the workpiece and the shot are divided into quadrilateral ring elements. Not only the workpiece but also the shot undergo plastic deformation in the collision. The relationship between the amount of plastic deformation and the flow stress ratio Fs/Fw is given in Fig. 3, where F~ and Fw are the F - v a l u e s of the shot.and workpiece in the flow stress curve ~=F~"~ ~, respectively. The flow stress ratio is assumed to be F{Fw, although the n - and m - v a l u e s vary with the temperature. The amount of plastic deformation for the shot decreases as the flow stress ratio increases, whereas that of the workpiece increases. The amount of plastic deformation for the shot is larger than that for the workpiece when the flow stress ratio is 1. The calculated results agree well with the experimental ones for plasticine.
actual working speed
Since the impact speed in actua] shotpeening processes is between 50 and 100ms-~,
0.12
.'t i(--) \
0.08
Shot
<1_0.06
_c \ , - - 0.04 <~
Calculated \ ---=- :-S::--- - ~__ Experimental " ~ i ~
Work~
0.02 0.00
0.5
1,0 Fs / Fw
::--:
:
IIl-r---~,i~:~ , I i I
,'h~"~i,'iiii' ,~ IIIIilllilli I i I I IItllllfillll I I I I
..ii.ii i"
:
;
Iltltt1111111 ',IIIIItUlIIIItll l,lllllllilll
1
1.5
Fig. 3 Relationship between amount of plastic deformation and flow stress ratio.
r
it;
i !
~l,i']~'l
l
,~i-~:fi'i i ~1i~l!5'111111 tlltlflllllU I I I I
i1111tlIIIItl llllltllflll /IIt111,11
; l
I
l
~
IW Uh'ltllllll I I 1~~ f l t l l l l l
I t
IHII11LI L tHllllll 1 tlOllILIIII I
('a) Before deformation (b) During deformation (c) After deformation Fig. 2 Element mesh in axi-symmetric peening of plasticine for T<=t5"C.
611
The distributions of the calculated equivalent strain in the workpiece and shot are shown in Fig. 5. For the shot of Fs/Fw= 1.2, small plastic deformation appears near the interface, whereas only the workpiece is deforming plastically in the case of F]Fw=2. The fatigue strength is improved by not only the compressive residual stress but also the increase in hardness predicted from the equivalent strain. The distributions of the calculated residual radial and hoop stress components in the workpiece and shot for Fs/Fw=2.0 are shown
the simulation is carried out for the steel workpiece in a higher speed to examine the effect of the flow stress ratio on the deformation behaviour (see Table 2). The relationship between the amount of plastic deformation and the flow stress ratio for the impact speed vsi=100ms -I is shown in Fig. 4. It is found that almost no plastic deformation of the shot occurs when the flow stress ratio is larger than 2. Table 2 Working conditions used in simulation of peening for steel workpiece and shot. Flow stress / GPa(workpiece)
[email protected]
Impact speed vsi / ms -1
100
Diameter of shot d / mm
2.0
Diameter of workpiece / m m
2.0
Height of workpiece h / m m
2.5
Coefficient of friction
0.20
Density / gmm -3
7.79x10 -3
"o 0.06
Workpiec~:~.__.
\
0.04
.ff \
..c
0.02 -
0.0(~.0
1.2
1.4
1.6
0. I
.
2
~
2.0
Fig. 4 Relationship between amount of plastic deformation and flow stress ratio for impact speed v,~=lOOms-1.
O.l~
0
1.8
Fs/Fw
0.7 0.
0. k
-
v
~
--------'t0.1 (a) F,/F~,=I.2 (b) F,/Fw=2.0 Fig. 5 Distributions of calculated equivalent strain in workpiece and shot.
612
in Fig. 6. The residual stress components are obtained from the elastic finite element simulation by using the stress components at the time of the maximum inertial force in the viscoplastic simulation [2]. By releasing the stress components for plastic deformation, the compressive residual stress components remain near the interface. 4. CONCLUSIONS
To simulate shot-peening processes of high-strength steels, not only plastic deformation of the workpiece but also that of the shot were dealt with in the dynamic visco-plastic finite element approach. The interaction of high speed plastic deformation I
!
between the shot and the workpiece in the collision was included in this approach~ The sliding between the shot and the workpiece in the interface is allowed for under the Coulomb friction. The forces at the nodal point in contact with the interface were decomposed, and were equilibrated with those at the neighboring nodal points The equilibrium equations of nodal forces were solved simultaneously with the displacement boundary conditions at the interface. The effect of the flow stress ratio of the shot to the workpiece on the deformation behaviour of the shot was examined from the finite element simulation. REFERENCES 1.
~
0
i0
0
~
0
GPa
4
.0~~
(a) Radial stress (b) Hoop stress Fig. 6 Distributions of calculated residual stress components in workpiece and shot FJFw=I.2.
5.
Y,F. AI-Obaid, "Three-dimensional dynamic finite element analysis for shot-peening mechanics". Comput. Struct., 36 (1990) 681. K. Mori and K. Osakada, "Application of dynamic viscoplastic finite element method to shot-peening process", Trans. of NAMRI/SME; 22 (1994). K. Osakada, J. Nakano, and K. Mori, "Finite element method for rigid-plastic analysis of metal forming -- E~)rmula--tion for finite deformation", Int. J Mech. Sci., 24 (1982) 459. K. Mori, K. Osakada and T. Oda, "Simulation of plane-strain rolling by the rigid-plastic finite element method", Int. J. Mech. Sci., 24 (1982) 5~9. K. Mori and K. Osakada, "Application of finite deformation theory in rigid-plastic finite element simulation", Proc. 3rd Int. Conf. T e c h Plasticity, Kyoto, (1990) 877.
Materials Processing Technology ELSEVIER
J. Mater. Process. Technol. 45 (1994) 607-612
Finite Element Analysis of Peening Process with Plastically Deforming Shot K e n - i c h i r o Mori, Kozo Osakada and Naoki Matsuoka Department of Mechanical Engineering, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan To simulate not only plastic deformation of a workpiece but also that of a shot in peening processes, the effect of the interaction between the shot and workpiece in the collision is included in the dynamic viscoplastic finite element method. In the formulation, the equilibrium equations of nodal forces and the boundary condition for displacement are simultaneously satisfied under a sliding condition at the interface between the workpiece and shot. Axi-symmetric plastic deformation in peening of a circular workpiece with a single shot is computed. The calculated shapes of the workpiece and the shot are compared with the experimental ones for the workpiece and the shot made from plasticine whose flow stress is controlled by the temperature. It is shown that almost no plastic deformation of the shot occurs under actual shot-peening conditions for steel workpieces when the flow stress ratio of the shot to the workpiece is larger than two.
1. INTRODUCTION In shot-peening, the surface of the metal workpiece is pelted with hard and round shots at a high velocity and is plastically deformed to cause the residual stress after the rebound of the shots. For high-strength steel products, the use of shot-peening recently tends to increase, e.g. gears, springs and shafts. Since the increase in the strength of the workpiece brings about plastic deformation and fracture of the shots, it is desirable in industry to predict the deformation behaviour of the shots as well as that of the workpiece. In shot-peening processes, the interaction between the shot and the workpiece has a great influence on the deformation behaviour. During the collision of the workpiece with the shot, the plastically deforming region gradually expands, and thus non-steady-state deformation occurs. The finite element method is suitable for simulating n o n - s t e a d y sate deformation in forming processes because the calculation is performed incrementally. Moreover, the effects of the inertia, the workhardening characteristic, the strain-rate
sensitive and the friction on the deformation behaviour can be included in the formulation. Al-Obaid [1] has used the dynamic elasticplastic finite element method to analyse a residual stress distribution in shot-peening. The authors [2] have simulated plastic deformation of the workpiece by the dynamic viscoplastic finite element method, and have calculated the residual stress distribution from the elastic finite element method by using the stress distribution obtained from the viscoplastic analysis. In these simulations, however, deformation of the shot has been neglected. In the present study, peening processes with a plastically deforming shot are simulated by a dynamic viscoplastic finite element method. The interaction of plastic deformation between the shot and the workpiece in the collision is introduced into the formulation. The effects of working conditions on plastic deformation of the shot in peening are examined.
2. METHOD OF SIMULATION
0924-0136/94/S07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0924-0136(94)00248-7
608
2.1 Dynamic viscoplastic finite element method The present viscoplastic finite element method is formulated on the basis of the plasticity theory for a material with slight compressibility [3,4]. Due to the slight compressibility, the stress components {cr} can be calculated directly from the strain-rate components {~}; {cr} = [DP]{~}, (1) where [D p] is the matrix correlating the stress with the strain-rate for the compressible viscoplastic deformation. To deal with the inertial effect due to the collision of the shot with the workpiece, the finite deformation theory of plasticity [4,5] is employed. In the finite deformation formulation, the equilibrium equations are satisfied at the end of each deformation step. Therefore, the acceleration, the change in the velocity, can be taken into consideration [2]. The differential equations of equilibrium for high speed deformation problems are given by Ooij vn-vi0
axj - P--ST- = o,
(2)
where P is the density, At is the time interval and vi0, v n are the velocities at the beginning and the end of each deformation step, respectively. In equation (2), the acceleration is approximated by using the difference between the velocities at the beginning and the end of each deformation step. For the finite element formulation of the equilibrium equations (2), the Galerkin method is used as follows;
Oxj{N}
- p - - ~ t ] a v = 0,
(3)
where {N} is the vector of the shape function of the element. The left hand side of equation (3) is partially integrated and equation (1) is substituted; [ f v ([B]T[DP][B]+{N}T{N}A-~t dV]{vl} =
l / { N } f { N } - - ~ - t d ~ { v o } + { F },
(4)
where {v0}, {vl} are the nodal velocity vectors at the beginning and the end of each deformation step, respectively and {F} is the nodal force vector. In these simultaneous equations, {vl} are variables and {vo} arc constants obtained at the previous deformation step.
2.2 Interaction between shot and workpiece For axi-symmetric deformation, the interaction of high speed plastic deformation between the shot and workpiece in the collision is taken into consideration in the viscoplastic finite element formulation. Consider that node B slides on the interface between the shot and the workpiece as shown in Fig. 1. To locate node B on the line AC at the end of each deformation step; the boundary condition for displacement is given by (rck-rA0(Zm-ZA1)-(Zc~-zA0(rm-r~)= 0 (5) rA~ = rA0 + At(Var0+VAr~)/2 Equation (5) is a nonlinear function in terms of the nodal velocities at the end of each deformation step. Beginning of each deformation step
%,%
/ "-,,,,,,
,Jr " ........
.Z?" ,
/ '..
(re0, zoo)
Z°o)C\ ............. ,
,,
\ '"),
,
,
End of each deformation step Fig.1 Sliding of node on interface between shot and workpiece.
609
The Coulomb friction is employed as a frictional law in the interface. The tangential nodal force FBt is provided by the normal nodal force FB, and the coefficient of friction ~t; Fat = gFsn, (6) where Fan is an independent variable and F~ is a dependent variable determined from equation (6). The nodal forces at node B are decomposed by using the shape function of the element, and are equilibrated with the nodal forces at nodes A and C; FAr = -(1-~B)FB]2 F~, : -(1-~B)F,J2 Fcr= -(I+~B)Fs]2 (7) Fcz = -(I+~B)FBz/2, where Fa~ and FA~ are the nodal forces in the radial and axial directions at node A, and ~ is the normalized co-ordinate having a value between - 1 and 1 at point B in the end of each deformation step. FBr and FBz are expressed in a term of FB~ by the use of equation (6) and the co-ordinate transformarion. Equation (7) is substituted into the vector {F} in the second term on the righthand side of equation (4), and FAr, FAz, Fc~ and Fc~ are eliminated. In this method, the nodal forces are equilibrated at nodes A and C and not at node B. When nodal points come newly in contact with the interface, the same treatment is carried out. The number of the independent variables, the normal nodal forces, in the interface is equal to the number of contact nodal points. The equilibrium equations (4) and the displacement boundary conditions (5) are simultaneously solved to deal with the effect of interaction between the shot and workpiece. Since the nodal velocities and the normal nodal forces in the interface are independent variables in the simultaneous equations, the number of the equations is equal to that of the total independent variables. In addition, the obtained matrix becomes unsymmetric because of equation (5) and the normal nodal forces at the interface.
3. PEENING WITH SINGLE SHOT 3.1 Comparison between calculated and experimental results for plasticine Axi-symmetric plastic deformation in peening of a circular workpiece with a single shot is computed under the conditions shown in Table 1. The shot collides perpendicularly with the workpiece. The calculated shapes of the workpieces and the shots are compared with the experimental ones for the workpieces and the shots made from plasticine. Since the deformation behaviour is greatly affected by the flow stress ratio of the shot to the workpiece, the flow stress ratio is controlled Table 1 Working conditions used in simulation of peening for plasticine workpiece and shot. Flow stress / kPa
289~° 14E°°7(15"C)
(plasticine)
198~°16g°1°(25"C) 130~°22~° °4(35"C)
Temperature of shot
15, 25, 35
T~ / "C Temperature of workpiece / "C
25
Impact speed v,i / ms -1
7.5
Diameter of shot d/mm
50
Diameter of workpiece / mm
50
Height of workpiece h
62.5
/ mm Coefficient of friction
0.20
Density / gmm -3
1.95x10 -3
610
3.2 Under
in the plasticine experiment by changing the temperature T s of the shot and fixing that of the workpiece. The element mesh in axi-symmetric peening for the shot temperature T,=15"C is illustrated in Fig. 2. Both of the workpiece and the shot are divided into quadrilateral ring elements. Not only the workpiece but also the shot undergo plastic deformation in the collision. The relationship between the amount of plastic deformation and the flow stress ratio Fs/Fw is given in Fig. 3, where F~ and Fw are the F - v a l u e s of the shot.and workpiece in the flow stress curve ~=F~"~ ~, respectively. The flow stress ratio is assumed to be F{Fw, although the n - and m - v a l u e s vary with the temperature. The amount of plastic deformation for the shot decreases as the flow stress ratio increases, whereas that of the workpiece increases. The amount of plastic deformation for the shot is larger than that for the workpiece when the flow stress ratio is 1. The calculated results agree well with the experimental ones for plasticine.
actual working speed
Since the impact speed in actua] shotpeening processes is between 50 and 100ms-~,
0.12
.'t i(--) \
0.08
Shot
<1_0.06
_c \ , - - 0.04 <~
Calculated \ ---=- :-S::--- - ~__ Experimental " ~ i ~
Work~
0.02 0.00
0.5
1,0 Fs / Fw
::--:
:
IIl-r---~,i~:~ , I i I
,'h~"~i,'iiii' ,~ IIIIilllilli I i I I IItllllfillll I I I I
..ii.ii i"
:
;
Iltltt1111111 ',IIIIItUlIIIItll l,lllllllilll
1
1.5
Fig. 3 Relationship between amount of plastic deformation and flow stress ratio.
r
it;
i !
~l,i']~'l
l
,~i-~:fi'i i ~1i~l!5'111111 tlltlflllllU I I I I
i1111tlIIIItl llllltllflll /IIt111,11
; l
I
l
~
IW Uh'ltllllll I I 1~~ f l t l l l l l
I t
IHII11LI L tHllllll 1 tlOllILIIII I
('a) Before deformation (b) During deformation (c) After deformation Fig. 2 Element mesh in axi-symmetric peening of plasticine for T<=t5"C.
611
The distributions of the calculated equivalent strain in the workpiece and shot are shown in Fig. 5. For the shot of Fs/Fw= 1.2, small plastic deformation appears near the interface, whereas only the workpiece is deforming plastically in the case of F]Fw=2. The fatigue strength is improved by not only the compressive residual stress but also the increase in hardness predicted from the equivalent strain. The distributions of the calculated residual radial and hoop stress components in the workpiece and shot for Fs/Fw=2.0 are shown
the simulation is carried out for the steel workpiece in a higher speed to examine the effect of the flow stress ratio on the deformation behaviour (see Table 2). The relationship between the amount of plastic deformation and the flow stress ratio for the impact speed vsi=100ms -I is shown in Fig. 4. It is found that almost no plastic deformation of the shot occurs when the flow stress ratio is larger than 2. Table 2 Working conditions used in simulation of peening for steel workpiece and shot. Flow stress / GPa(workpiece)
[email protected]
Impact speed vsi / ms -1
100
Diameter of shot d / mm
2.0
Diameter of workpiece / m m
2.0
Height of workpiece h / m m
2.5
Coefficient of friction
0.20
Density / gmm -3
7.79x10 -3
"o 0.06
Workpiec~:~.__.
\
0.04
.ff \
..c
0.02 -
0.0(~.0
1.2
1.4
1.6
0. I
.
2
~
2.0
Fig. 4 Relationship between amount of plastic deformation and flow stress ratio for impact speed v,~=lOOms-1.
O.l~
0
1.8
Fs/Fw
0.7 0.
0. k
-
v
~
--------'t0.1 (a) F,/F~,=I.2 (b) F,/Fw=2.0 Fig. 5 Distributions of calculated equivalent strain in workpiece and shot.
612
in Fig. 6. The residual stress components are obtained from the elastic finite element simulation by using the stress components at the time of the maximum inertial force in the viscoplastic simulation [2]. By releasing the stress components for plastic deformation, the compressive residual stress components remain near the interface. 4. CONCLUSIONS
To simulate shot-peening processes of high-strength steels, not only plastic deformation of the workpiece but also that of the shot were dealt with in the dynamic visco-plastic finite element approach. The interaction of high speed plastic deformation I
!
between the shot and the workpiece in the collision was included in this approach~ The sliding between the shot and the workpiece in the interface is allowed for under the Coulomb friction. The forces at the nodal point in contact with the interface were decomposed, and were equilibrated with those at the neighboring nodal points The equilibrium equations of nodal forces were solved simultaneously with the displacement boundary conditions at the interface. The effect of the flow stress ratio of the shot to the workpiece on the deformation behaviour of the shot was examined from the finite element simulation. REFERENCES 1.
~
0
i0
0
~
0
GPa
4
.0~~
(a) Radial stress (b) Hoop stress Fig. 6 Distributions of calculated residual stress components in workpiece and shot FJFw=I.2.
5.
Y,F. AI-Obaid, "Three-dimensional dynamic finite element analysis for shot-peening mechanics". Comput. Struct., 36 (1990) 681. K. Mori and K. Osakada, "Application of dynamic viscoplastic finite element method to shot-peening process", Trans. of NAMRI/SME; 22 (1994). K. Osakada, J. Nakano, and K. Mori, "Finite element method for rigid-plastic analysis of metal forming -- E~)rmula--tion for finite deformation", Int. J Mech. Sci., 24 (1982) 459. K. Mori, K. Osakada and T. Oda, "Simulation of plane-strain rolling by the rigid-plastic finite element method", Int. J. Mech. Sci., 24 (1982) 5~9. K. Mori and K. Osakada, "Application of finite deformation theory in rigid-plastic finite element simulation", Proc. 3rd Int. Conf. T e c h Plasticity, Kyoto, (1990) 877.