- Email: [email protected]
Finite element solution of Prandtl's flat punch problem
Finite Elements in Analysis and Design 6 (1989) 173-186
173
Elsevier
FINITE ELEMENT SOLUTION OF PRANDTL'S FLAT PUNCH PROBLEM T.-M. TAN, S. LI and P.C. CHOU Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, U.S.A. Received January, 1989 Revised August, 1989 Abstract. In this paper, Prandtl's fiat punch problem is analyzed using an elasto-plastic finite element program. Detailed studies are made on the development of both plastic zones and velocity fields for punches of different surface roughness. The finite element solution, obtained by assuming the semi-infinite body to be elastic-perfectly-plastic, is compared with the slip-line solutions of both Prandtl and Hill. It is found that, although the friction condition on the punch surface has some effect on the development of plastic zone and velocity field, the finite element results in general agree very well with Prandtl's slip-line solution in all cases studied. On the other hand, no conclusive evidence supporting Hill's solution could be found for either the smooth or the rough punch cases.
Introduction
The problem of the indentation of a semi-infinite body by a rigid flat punch in the form of an infinite strip was first investigated by Prandtl in 1920 [17]. By considering the semi-infinite body to be rigid-perfectly-plastic and the contact surface between the punch and body to be perfectly lubricated, Prandtl proposed a solution to the problem in which the stress field can be defined by the slip-lines shown in Fig. l(a). Assuming that the pressure between the punch and the semi-infinite body is uniformly distributed, one can readily show that the limiting load F, which causes incipient plastic flow, is uniquely given by
F= 2ap = 4ak(1 + x~r),
(1)
where p is the uniform pressure exerted by the punch, 2a is the width of the punch, and k is
~
FigureI a
,
~
a lines
PranotrlSkO-bne ~kdion
~
F a lines
~,/~V
G,~._ AI ",,\
v~2V
V"
. L3lines
IA yG
,,. " , /
0168-874X/89/$3.50 © 1989, Elsevier Science Publishers B.V.
Fig. 1. (a) Prandtl's slip-line solution. (b) Hill's slip-line solution.
174
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
the yield stress of the semi-infinite body in pure shear. The velocity field associated with this state of stress can be obtained by Geiringer equations [6], as shown in Fig. l(a). It is noted that the triangular region AA'O under the punch behaves like a "dead metal", moving downward with the same velocity as that of the punch. In addition, region AOB (A'OB') moves outward along circular arc OB while region ABC (A'B'C') moves in the direction of BC (B'C'), both with a velocity 0.707 times that of the punch. Hill in 1949 [7] suggested that the velocity field in Prandtl's solution is indeterminate, and proposed a different slip-line solution as depicted in Fig. l(b). It can be shown that the state of stress in each of regions AO'D, ADE and AEG of Hill's solution is identical to that in regions AA' O, AOB and ABC, respectively, of Prandtl's solution. The limiting load F for the onset of plastic flow, therefore, is still given by (1). However, for the same punch velocity, Hill's solution indicates that region AEG would move in direction EG with a velocity twice that of region ABC of Prandtrs solution. In addition, Hill's solution does not require a "dead metal" zone under the punch. Subsequent investigation of the flat punch problem by several authors [10,11,15,16] suggested that the validity of Prandtl's or Hill's solution may depend on whether the punch surface is rough or smooth. Furthermore, due to the nature of the slip-line method, an infinite number of velocity and slip-line fields could be constructed between these two limiting solutions. It should be pointed out that the slip-line method only allows one to estimate the load required to create a state of unconstrained plastic flow, but does not. give the details of the actual deformation process of the material. In addition, the fact that the slip-line solution requires both the velocity and pressure conditions at the punch surface be specified implies that it is not a well-defined boundary value problem. Therefore, to obtain the "true" solution of a flat punch problem, it is necessary to solve the complete elasto-plastic problem, using the Prandtl-Reuss relations, together with properly specified boundary conditions at the punch surface. The solution of this more realistic formulation is rather intractable before advanced numerical techniques, such as the finite element method, became available. The finite element method was first developed in the 1950s for structural analysis, and has been extended to solve elasto-plastic problems during the 1960s [1,13,14,19]. Lee and Kobayashi [13] formulated a finite element procedure for the elasto-plastic analysis of plane-strain and axisymmetric flat punch indentation problems. The development of the plastic zones, the load-displacement relationships and stress and strain distributions with various friction conditions on the punch surface were obtained. However, the velocity field, which is the major difference between the Prandtl's and Hill's solutions, was not reported in their work. Furthermore, the thickness of the region that was modelled and analyzed in their work was too thin to be considered as semi-infinite. In this paper, a two-dimensional finite element Lagrangian program DEFEL [5] is employed to study the flat punch problem. The semi-infinite body is idealized by an area of sufficiently large size to rule out the boundary effect due to finite dimension. The effect of friction condition on the punch surface is examined by analyzing results that simulate smooth and rough punches, respectively. The development of plastic zone and velocity field, and the contour of effective strain at different stages of loading are obtained and compared with the slip-line solutions. It is found that, although the friction condition on the punch surface has some effect on the development of plastic zone and velocity field, the finite element results in general agree quite well with Prandtl's slip-line solution for both smooth and rough punches. Description of DEI~JLcomputer program
DEFEL is a two-dimensional finite element computer program originally developed for dynamic analysis of impact and explosive-metal interaction problems. The formulation of
T.-M. Tan et aL / Finiteelementsolutionof Prandtl'sflat punchproblem
175
DEFEL is based on the hydro-code approach similar to that used in EmC code [9]. In recent years several new features have been incorporated in DEFEL, including various material models for elasto-plastic analysis, dynamic relaxation procedure for quasi-static or static analysis, and slide line routine for modelling frictional contact/slide boundary conditions [4,12,18]. The basic element used in DEFEL code is the triangular constant strain element. Following standard finite element procedures [2,20], the equations of motion for the finite element model can be written as
M i I ' = I - P,
(2)
where M is the mass matrix, q is the nodal displacement vector, f is the external force vector, and P is the equivalent nodal force vector calculated from the element stresses. The superposed dot denotes differentiation with respect to time. In general, the vector P is a function of the current nodal displacements and velocities for non-linear problems, and can be expressed in terms of the current element stress vector o by
e = E f BTO dV, ~v~
(3)
where B is the strain-displacement matrix. The advantage of this formulation is that ~the process of assembling the global stiffness matrix can be bypassed if an explicit integration scheme and a lumped mass matrix are used. In that event, the system of equations (2) are uncoupled into
m,i], = f~ - P,,
(4)
in which the subscript i corresponds to the ith component of the nodal displacement vector q. An explicit integration scheme with central difference quadrature is used in the DEFEL code to integrate the differential equations in time. For this purpose, equation (4) is rewritten in the following form m//], = ,f~ _ ,- atp~,
(5)
where At is the time increment, and the left superscript t refers to time instant t. The velocity and displacement of the ith node are then given by
'+0.Satoi=t-O.SAtoi + At t~i=t-O.SAtoi + At[ tfi--t-atpi ]]
(6)
t+atqi = 'qi + At '+°sat0i,
(7)
and
respectively. Combining (5), (6) and (7), the displacement of node i at time instant t + At can be obtained as
t+Atqi = (At)2 [tfi -t--Atpi] -- 2 tqi- At t--atqi.
(8)
mi Due to the conditionally stable nature of the explicit integration scheme, the time increment, At, may need to be readjusted at each time step to ensure numerical stability. In DEFV.Lcode, the following stability criterion is used [12] 0.7
H.../i ~ + 2#
(9)
In equation (9), Hmin is the minimum height of the element, p is the density, and A and ~ are the Lame constants of the material.
176
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
Once the nodal displacements and velocities are solved, the program then proceeds to perform the elasto-plastic stress/strain calculation. The spherical and deviatoric strain rates are computed from the nodal velocities, and the stress increments and total stress components are then computed. Proportional unloading technique is used to pull the over-estimated stress components back to the current yield surface, and various work hardening and softening models [12] can then be used to establish a new yield surface. This process is repeated at each time increment. For static or quasi-static analyses, the program uses the dynamic relaxation method [4], which artificially adds a damping term into the equations of motion, to quickly damp out the effect of inertial terms. With a damping term added, (4) becomes (10)
mii], + CCli= fi - Pi,
where c is the damping constant. Since the amount of damping for a given value of c depends on m and At, a non-dimensionalized constant to, known as relaxation parameter, is used instead in the DEFEL program to specify damping. The relationship between to and c is given in the following equation:
(
to = 2 / 1 + 2~m, ] '
(11)
where At is the average of the two consecutive time increments before and after time t. By using the relaxation parameter, the velocity equation of the ith node becomes
--
m,
1'
(12)
and the displacement is then computed using (7). The value of to may vary from 1 to 2, corresponding to the value of the damping constant c varying from 2 m J A t to zero. The choice of the value for to depends on the nature of the problem being solved. For all the cases reported in this paper, to = 1.995 has been used.
Modelling and analysis of the flat punch problem Finite element model
Figure 2 shows schematically the geometry of the model used in this study. For a plane-strain finite element analysis, the semi-infinite body is assumed to be infinitely long in the z-direction and only its cross section in the x - y plane is modelled. The cross section is idealized as a rectangular area of width 2W and height h, both of which must be much larger than the punch width, 2a, to minimize the boundary effect on the solutions due to finite dimension. In this study, both W / a and h / a ratios are chosen to be 8.8, as has been suggested by several authors [3,8]. Due to symmetry, only one half of this rectangular area is modelled and analyzed. The finite element mesh, as shown in Fig. 3, consists of 3,600 triangular elements and 1,861 nodes. The mesh grid is finer near the punch, and becomes progressively coarser approaching to the boundaries. Under the punch there are six nodes (from node 1 to node 6), called punch nodes, on which the prescribed punch forces or punch displacements are to be applied. In addition to the symmetrical conditions applied along the edge x = 0, nodes along the edge x = W are constrained against horizontal displacements while those along the edge y -- 0 are constrained against vertical displacements.
177
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
II
Punch
J
2W Semi-infinite body
' W
Fig. 2. Indentation of a semi-infinite body by a flat punch.
X
~,
Fig. 3. Finite elementmesh.
Material properties
Material properties similar to those of commercial aluminum 1100-0 are used for the semi-infinite body. These properties are: Young's modulus E: Poisson's ratio v: Yield stress Y:
68.95 GPa (10 × 106 psi) 0.33 89.6 MPa (13,000 psi)
Beyond yield point the material is assumed to be perfectly plastic (i.e., non-hardening). In a preliminary study a material with Young's modulus 10 times as large was used to examine the effect of initial deformation. No appreciable difference in solution has been found. Friction and loading conditions
As mentioned before it was suggested by many authors who investigated the flat punch problem that the validity of Prandtl's or Hill's solution may depend on the friction condition on the punch surface. To verify this point, two sets of boundary conditions, one simulating a smooth punch and the other simulating a rough punch, have been applied to the punch nodes in this study. In the smooth punch case these nodes are allowed to move freely in the horizontal direction while in the rough punch case they are constrained against any horizontal movement. When a rigid punch is pressed against the surface of an elasto-plastic body the deformation of the surface in general does not follow the shape of the punch face, nor will the pressure exerted on the surface of the body be uniformly distributed. The slip-line solutions, obtained by assuming the semi-infinite body to be rigid-perfectly-plastic (i.e., no deformation could occur before the yield stress is reached) and the punch pressure to be uniformly distributed, therefore, are not the solutions of a realistic problem. In this study, the effect of different loading conditions on the development of the velocity field and plastic zone has also been examined by using two loading procedures. In the first procedure nodal forces equivalent to a uniformly
178
T.-M. Tan et a L / Finite element solution of Prandtl's flat punch problem V 1.2 Vmax1.0
PRESCRIBEDUNIFORM PUNCHVELOCI ~ TY C
0.8
IDct 0.6
0.20'4 t.
time
ment.
. . . . .
""~"
~ODE6
0.0 u.O
V 1.2 Vmax1.0
~
0.2
~
0.4
ASE:
NODES1TO5
0,6
0.8
1,0 tt~'" 1.2
N~UNCV HELOC~ PRESCRIBEDUNIFORM
0.8 0.6
t~
time
Fig. 4. (a) Prescribed punch pessure. (b) Prescribed
punch velocity and displacement.
0.2 0.0
u.O
\
0.2
it""
7.~11~
0.4
UN,FOR~RESSURECASE: NODES1TO5
0.6
0.8
1.0 t/tv 1.2
Fig. 5. (a) Comparison of smooth punch velocity. (b) Comparison of rough punch velocity.
applied punch pressure are specified at the punch nodes, while in the second procedure a uniform velocity in the negative z-direction is prescribed at each punch node. For both smooth and rough punches, the pressure-prescribed loading cases are solved first. Forces applied at punch nodes initially increase linearly in time, and become constant after the total punch pressure exceeds a critical value, Per, at t = tp, as shown in Fig. 4(a). For the smooth punch, a critical pressure that yields a limiting load approximately 3% higher than the value predicted by the slip-line theory, is used to ensure that certain amount of plastic flow would occur. In the case of rough punch, it is found that a much higher limiting load is needed to induce enough plastic deformation, as has been reported in the work of Lee and Kobayashi [13]. In this study, a value approximately 25% higher than that of slip-line solution is found to be appropriate, and is used as the critical pressure. In the velocity-prescribed cases, the variation of punch velocity in time is assumed to follow a sine curve, as shown schematically in Fig. 4(b) together with the corresponding displacement function. The prescribed punch velocity is determined in such a way that its maximum velocity, occurring at t = t o, would roughly coincide with the average maximum punch velocity obtained in the pressure prescribed case. For each of the smooth and rough punches, a comparison between the prescribed punch velocity and the velocities of punch nodes computed in the pressure-prescribed case, normalized with respect to the maximum prescribed velocity, Vmax, is shown in Figs. 5(a) and 5(b), respectively. It is noted that the velocity of the outermost punch node (node 6) in the pressure-prescribed case generally is much smaller than those of other punch nodes (node 1 to node 5), and is therefore not used in computing the average punch velocity. Results and discussions Rough punch
The main differences between Prandtl's and Hill's solutions, as discussed in the earlier section, are in their velocity fields and whether a "dead metal" zone exists under the punch. It
T.-M. Tan et aL / Finite element solution of Prandtl's flat punch problem
......
l
::°
:!:!:!::::.:'.'
179
::::::::::::::::::::::::::::::::::::::
~:,i~iiii?i?i?i?i i i ?i : ~X~t~":':':-:-:" :" :" :" :" ~,,,,:.:.:-:-:.:.:
~.'-.:-.:.:.:.:.:.
• :. : . •
:. :. :
,,,,,:,-.:.:.-.:. :. -. - . . . : .
!:
',i,1,'.',2,'.i,2.2.2,2.2•:,:•2.:.:,;,;.'." ~ , = , , ° o o ,
. . . . . . . . .
• °,oO,Oo,o°oOoO . o . OoOoO .oOoO .oO • o . . . . . . . . . . . .
t - o.~8 ~,
, , , ¢ ~
-
_
,
t - 0.56t,
,
~,~-::!:i:i:i:i:~:!:!:!:
iiii!i!iiiiii!!i!!! !i! !!i ii
~~:-:'~:::::::::::::
: : : : : i '
::::::::::::::::::::::::
:-::
ii!iiiiii' iiiiiiii
: : :
!: =
",'::;',,'-::'::::::, ;: : : • : : : : : . : : :
I..
::::::::::::::::::::::::::::::" :: : : : : : : :. : : " • ,-,,:,,,,,,,;,,,,,......
Illllt~ltttttl~tti~t~tb~tt~ttt~lttttttt l l l l t i l l ~ t ~ l t l t b i l
t b t .
0 0 0 , ' o ~ 0 ~ 0 0 , = , ' 0 ' 0 ~ 0 = = ~ ' ' ' ' ' ~ ° °
, . 0 , . 0 , , . • ~ o 0 , 0 0 o . , , • ~ ,
.
. .
. .
. .
. .
.
t
. .
. .
. .
.
•
. .
~ ~ ~
.
" . • .
.
i : i : :: :
•
. .
• ".'.'.'.'.'•'.'.'.'.'.'.'.'.'.'.'.".
.
.
~
=
.
•
o
•
.
.
.
.
•.
.
I
.
.
.
.
.
•
*
•
,
•
I
=
•
°
•
~ l l ~ L~ • • . • - o % -
i~o=°e.'~'~•o'•°.°.°o°o°o°o'°°o°o°°°o ° •" , o , o , o O • - o - o - o - Oo-oO - . o . ° OoOo-o - • • o • ° . . . . . . . . . . . . . . . . . . , . o o,~o• Oo-oO.O OoOoOo. - . ° o ° o O . O o . • •
.
•
•
. .
O.OoO.OoOoOoO • • o • o ° • • • •
°
•
•
. .
". t - 0.97t~
t ,- 1.53t v
Fig. 6. Development of velocity field for a rough punch with prescribed uniform punch velocity.
is therefore of great interest to monitor the development of velocity field during the loading process. Figures 6(a)-(d) show the vectorial plots of nodal velocity at different times for rough punch under the condition of prescribed uniform punch velocity. At an earlier loading stage (e.g. t = 0.28 to), most material points are moving downward, only those near the punch comer are moving slightly outward. As the punch being pressed further into the body (e.g. t = 0.56 t o and t = 0.97 to), material points near the punch comer start turning upward. Finally at t = 1.53 t o, the material points in the triangular region just outside of punch comer are seen to move in the direction approximately 45 * to the free surface of the semi-infinite body, while those in the triangular region under the punch are continuing to move downward• A centered fan area between these two triangular regions can be clearly observed. When overlapping the Prandtl's slip-line field on this velocity plot, as shown in Fig. 6(d), one observes that the matching between these two solutions is remarkable well. It is noted that the direction of nodal velocity changes abruptly along the "slipline" between the triangular area under the punch and the centered fan area, indicating that the tangential component of velocity is discontinuous• Figure 7 shows the contour of nodal velocity at t = 1.53 t o for rough punch with prescribed punch velocity. Two triangular areas of constant velocity, one under the punch and the other outside of the punch comer, can be clearly seen. This indicates again that Prandtl's solution
180
T.-M. Tan et a L / Finite element solution of Prandtl's flat punch problem
RESULTANT VELOCITY
t = 1.53t v
I.D. 1 2 3 4 5 6 7
VALUE (VN~n¢,) 0.00 0.17 0.35 0.52 0.69 0.87 1.04
Fig. 7. Contour of nodal velocity for a rough punch with prescribed uniform punch velocity.
,$
t - 0.14t v
t - 0.28t v
t -, 1 . 2 5 t v
t,= 1.53t v
Fig. 8. Development of plastic zone for a rough punch with prescribed uniform punch velocity.
181
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
J
:.~.:.:':+~'?:-:-2-?'-'"" Z+':
• ••
....
t = 1.53~
....
EFF. PLASTIC STRAIN I.D. VALUE 1 0.0000 2 0.0181 3 0.0362 4 0.0543 5 0.0724 6 0.0905 7 0.1086
Fig. 9. Contour of effective plastic strain for a rough punch with prescribed uniform punch velocity.
.'.'.'.'.'.'.'.'.'.'..
t=
1.67t v
Fig. 10. Typical velocity field for a rough punch with prescribed uniform punch pressure.
seems to yield better prediction for rough punch. The magnitude of average nodal velocity in the outside triangular area, however, is only about 0.4 times of the average nodal velocity under the punch, somewhat smaller than the value of 0.707 predicted by Prandtl's solution. On the other hand, Hill's solution which predicts that the velocity in the outside triangular area to be greater than (i.e., 1.414 times) the velocity of punch, has never been observed in this case. Figures 8(a)-(d) show the development of plastic zone in the semi-infinite body at different times for the velocity-prescribed case. The plastic zone initially occurs near the comer of the punch, then gradually develops toward the center along a 45 o line. As the loading continues to increase, the plastic zone further develops, eventually reaches the stage shown in Fig. 8(d) at t = 1.53 t o. A triangular areas of "elastic zone" can be seen to exist under the punch. The
1111[1 TTTTTT
RESULTANT VELOCITY I.D
VALUEo.o0(VNmr=) 0.18
0.36 0.54 0.72 0.89
1.07
Fig. 11. Contour of nodal velocity for a rough punch with prescribed uniform punch pressure,
t=
1.67t v
EFF. PLASTIC STRAIN I.D. VALUE 1 0.0000 2 0.0102 3 0.0204 4 0.0306 5 0.0409 6 0.0511 7 0.0613
Fig. 12. Contour of effective strain for a rough punch with prescribed uniform punch pressure.
182
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
contour of effective plastic strain at the same time, as shown in Fig. 9, also suggests that the triangular area under the punch does behaves like a "dead-metal" for rough punch with prescribed velocity. In case of rough punch with prescribed uniform punch pressure, the results are very similar to those of prescribed punch velocity. For comparison, typical results in terms of a vectorial plot of nodal velocity and contours of velocity and effective plastic strain are shown in Figs. 10-12. It can be seen that the vectorial plot of nodal velocity in this case also resembles Prandtl's slip-line field. Contours of nodal velocity and effective plastic strain suggest that a triangular "dead-metal" zone, although not as obvious, also seems to exist under the punch.
Smooth punch For smooth punch where punch nodes are free to move horizontally, the vectorial plots of nodal velocity at different times for velocity prescribed case are shown in Figs. 13(a)-(d). The material points in the area under the punch, although having noticeably larger horizontal components in their velocities comparing with the rough punch cases, are still mostly moving
illlilC,~,~,~,.:~.:.,...
I!!
l ll~lllb:,:,:::::::::.:....... ,UlI!,,,,,,,,,',',',".'.':':. I q ~ | t ,bbbbb+:':":':':':':':'" | t t t ~ l t • • .°o . . - . . . . . . . . o - . lllll
t I t t ~t
leftist
i|stst~
tt
•~.• ...
*'° . . . . . . . .
. . . . . . . . ...... .%%-
,11d~t%~%•.%%%%'.'.'.'.'.'...
.%..%%%%N°.°.-°....%'.'.'.'.'. .%%..%..%°°°.%..%'°'.'.'.'.'.'."
•.
d.%%.°°.%.°... °.°o..... - • •. *. .o°......°°....°..'.'''" . • • .°.. °.. '." • " • ' "."
:'Z';':';+:':':':';':':':':'Z':':':"
;" :- :" :- :- : "
, , , o , . , , , = , , , , , , o , . ,
, . . , , , . , °
....
,.,
.
.
.
.
.
.
.
.
' ' ° ' . ' . ' ' ° ' ' ' ' . ° ' . ' ' . ' . ' ° . ' . ' . ' . " .
t -
.
.
.
.
.
.
.
.
" ° ".
..................
0.27t v
.
. ".
t -
"
.
0.53 tv
//IJls . . . . . . . . . . . . .
i i ::i i ::: ,,
i
i
I=!!!:!:!!!!!!!!!!!!i:i:!!!:!!:!!!i!!!!!:!: !:!!: Z.Z'Z'Z'Z-3Z'Z'?2"Z'""
t - 0.80 t,
"
".'.
t = 0.93t v
Fig. 1.% Development of velocity field for a smooth punch with prescribed uniform punch pressure.
183
T.-M. Tan et a L / Finite element solution of Prandtl's flat punch problem
t - 0.27t,,
t = 0.13t v
i :ii iiiiii!iiiiiiiiiii!iii!i !
t = 0.93t,,
t - 0.80 t,
Fig. 14. Development of plastic zone for a smooth punch with prescribed uniform punch velocity.
i
RESULTANT VELOCITY I.D. 1 2 3 4 5 t = 0.93t v
EFF. PLASTIC STRAIN I.D. VALUE
VALUE (VN~¢',) 0.00 0.18 0.35 0.53 0.70
6
0.88
7
1.05
Fig. 15. Contour of nodal velocity for a smooth punch with prescribed uniform punch velocity.
t = 0.93t v
1 2 3 4 5 6 7
0.0000 0.0210 0.0419 0.0529 0.0838 0.1048 0.1257
Fig. 16. Contour of effective strain for a smooth punch with prescribed uniform punch velocity.
184
T.-M. Tan et aL / Finite element solution of Prandtl's flat punch problem
i!i!!i!i!ili!ilili!i! ! ! i! i! i! i! i i
Io.,...~°-°-°o-°*o°°o°°-°'.°.°
"°
-.
° . - o
° . - .
~ ~ ° ~ • , . . . ° . ° * . - . - . O ° ° o ' ° * ° o ° . - . ° . * . * . * . * . - . - . - . - . .
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
. .
.
.
. . .
.
. . .
.
-.-.-.-.-.-.-...
. .
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . .
. .
.
. .
.
.
.
. .
.
.
. . . .
.
.....
. . . .
. .
,
.
•
,
•
~
'
•
.
•
.
*
,
*
o
. .
.
°
,
° .
.
°
.
.
*
,
°
*
°
•
•
•
,
•
•
•
°
.
.
,
°
Fig. 17. Typical velocity field for a smooth punch with prescribed uniform punch pressure.
t = 0.93 t~ .
downward throughout the entire loading process. At t = 0.98 t o (Fig. 13(d)), where the material points in the area outside of punch c o m e r have turned to the 45 o direction, the magnitude of average velocity in this area is again less than one half of the punch velocity, and the velocity field is also quite close to Prandtl's slip-line field. Yielding of material also occurs first at the c o m e r of the punch, then gradually propagates toward the center along a 45 o line, as shown in Figs. 14(a)-(d). Unlike the rough punch case where a small area under the punch remains elastic, the plastic zone in this case quickly expands and covers the entire area under the punch. Contours of nodal velocity and effective plastic strain, as shown in Figs. 15 and 16, respectively, also indicate that the existence of a " d e a d metal" zone, while clearly observed under a rough punch, is not as obvious in this case. Figure 17 shows the vectorial plot of nodal velocity for smooth punch with prescribed uniform punch pressure at t = 1.63 t o. It is interesting to see that, even though the punch nodes
l l t t l _ T T V T T T
.i
t = 0 . 9 3 tv
RESULTANT VELOCITY I.D. VALUE (VN~.~) 1 0.00 2 0.17 3 0.35 4 0.52 5 0.70 6 0.87 7 1.05
Fig. 18. Contour of nodal velocity for a smooth punch with prescribed uniform punch pressure,
••
t = 0.93~
EFF. I.D. 1 2 3 4 5 6 7
PLASTIC STRAIN VALUE 0.0000 0.0089 0.0178 0.0267 0.0356 0.0445 0.0533
Fig. 19. Contour of effective plastic strain for a smooth punch with prescribed uniform punch pressure.
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
185
in this case are allowed to move freely in the horizontal direction, the material points under the punch are moving almost straightly downward, and there appears to have a 45 ° line of discontinuity in velocity. Contours of velocity and effective plastic strain, shown in Figs. 18 and 19, respectively, also suggest that a triangular "dead metal" zone may exist under the punch.
Conclusion The indentation of a flat punch into an elastic-perfectly-plastic semi-infinity body has been studied using an elasto-plastic finite element program. The development of plastic zone, velocity field and effective plastic strain in the semi-infinite body are monitored during the entire loading process. Two types of boundary condition have been used at punch nodes to separately simulate smooth and rough punch surfaces. In addition, two loading procedures, one with prescribed uniform punch pressure and the other with prescribed uniform punch velocity, have been applied to study the effect of different loading conditions. It is found that for all the cases studied here, the finite element results, in terms of velocity field, existence of triangular dead metal zone under the punch, and magnitude of velocity at free surface, resemble Prandtl's slip-line solution quite well. This is particularly so for the rough punch cases. On the other hand, no conclusive evidence that supports Hill's slip-line solution could be found for either smooth or rough punch. Yielding of the material always occurs first near the corner of the punch, then gradually extends toward the center along a 45 o line. The limiting load predicted by both Prandtl's and Hill's slip-line solutions is found to be quite accurate for the smooth punch case. For the rough punch case, a much higher limiting load is necessary to induce plastic deformation in the semi-infinite body.
Acknowledgement This study was partially supported by a grant from the Ben Franklin Technology Center (formerly Advanced Technology Center) of Southeastern Pennsylvania and Dyna East Corporation through the Ben Franklin Partnership Program. The first author would like to thank Professor Alan Lau for his helpful comments.
References [1] AKYUZ, F.A. and J.E. MERWlH, "Solution of nonlinear problems of elastoplasticity by finite element method", AIAA, J. 6 (10), pp. 1825-1831, 1968. [2] BATHE, K.-J., Finite Element Procedures in Engineering Analysis, Prentice Hall, Engiewood Cliffs, N J, 1982. [3] BISHOP, J.F.W., " O n the complete solution to problems of deformation of a plastic-rigid material", J. Mech. Phys. Solids 2, pp. 43-53, 1953. [4] CHOU, P.C. and L. Wu,"A dynamic relaxation finite element method for metal forming processes", Int. J. Mech. Sci. 28 (4), pp. 231-250, 1986. [5] FLIS, W.J., S. MILLER and W.J. CLARK, DEFEL: A finite element hydrodynamic computer code, Dyna East Corporation Technical Report DE-TR-84-05, 1986. [6] GEIRINGER, H., "Complete solutions to the plane plasticity problem", Proc. 3rd Int. Congr. Appl. Mech. 2, pp. 185-190, 1930. [7] HILL, R., "The plastic yielding of notched bars under tension", Q. J. Mech. Appl. Math. 2 (1), pp. 40-52, 1949. [8] HILL, R., The Mathematical Theory of Plasticity, Oxford, 1950. [9] JOHNSON, G.R., "Analysis of elasto-plastic impact involving severe distortion", J. Appl. Mech. 98, pp. 439-444, 1976. [10] JOHNSOH, W. and P.B. MELLOR, Engineering Plasticity, Ellis Horwood, Chichester, UK, 1983.
186
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
[11] KACHANOV,L.M., Fundamentals of the Theory of Plasticity, Mir Publishers, Moscow, 1974. [12] LAU, A.C.W., R. SHIVPURI,and P.C. CHOU, "An explicit time integration elastic-plastic finite element algorithm for analysis of high speed rolling", Int. J. Mech. Sci., 31 (7), pp. 483-497, 1989. [13] LEE, C.H. and S. KOBAYASm,"Elastoplastic analysis of plane-strain and axisymmetric flat punch indentation by the finite element method", Int. J. Mech. ScL 12 (4), pp. 349-370, 1970. [14] MARCAL, P.V. and I.P. lONG, "Elastic-plastic analysis of two-dimensional stress system by the finite element method", Int. J. Mech. Sci. 9, pp. 143-155, 1967. [15] MENDELSON,I., Plasticity: Theory and Application, Robert E. Krieger Publishing Company, Inc., 1983. [16] PRAGER,W. and P.G. HODOE, Theory of Perfectly Plastic Solids, Dover Publications, New York, 1968. [17] PRANDTL,L., "Concerning the hardness of plastic bodies", Nachr. Ges. Wiss. GSttingen, p. 74-85, 1920. [18] SmvPum, R., P.C. CHou and C.W. LAU, "Finite element investigation of curhng in nonsymmetric rolling of fiat stock", Int. J. Mech. Sci. 30 (9), pp. 625-635, 1988. [19] YAMADA,Y., N. YOSHIMURAand T. SAKURAI,"Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by the finite element method", Int. J. Mech. Sci. 10, pp. 343-354, 1968. [20] ZIENKmWICZ,O.C., The Finite Element Method, 3rd ed., McGraw-Hill, New York, 1977.
173
Elsevier
FINITE ELEMENT SOLUTION OF PRANDTL'S FLAT PUNCH PROBLEM T.-M. TAN, S. LI and P.C. CHOU Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, U.S.A. Received January, 1989 Revised August, 1989 Abstract. In this paper, Prandtl's fiat punch problem is analyzed using an elasto-plastic finite element program. Detailed studies are made on the development of both plastic zones and velocity fields for punches of different surface roughness. The finite element solution, obtained by assuming the semi-infinite body to be elastic-perfectly-plastic, is compared with the slip-line solutions of both Prandtl and Hill. It is found that, although the friction condition on the punch surface has some effect on the development of plastic zone and velocity field, the finite element results in general agree very well with Prandtl's slip-line solution in all cases studied. On the other hand, no conclusive evidence supporting Hill's solution could be found for either the smooth or the rough punch cases.
Introduction
The problem of the indentation of a semi-infinite body by a rigid flat punch in the form of an infinite strip was first investigated by Prandtl in 1920 [17]. By considering the semi-infinite body to be rigid-perfectly-plastic and the contact surface between the punch and body to be perfectly lubricated, Prandtl proposed a solution to the problem in which the stress field can be defined by the slip-lines shown in Fig. l(a). Assuming that the pressure between the punch and the semi-infinite body is uniformly distributed, one can readily show that the limiting load F, which causes incipient plastic flow, is uniquely given by
F= 2ap = 4ak(1 + x~r),
(1)
where p is the uniform pressure exerted by the punch, 2a is the width of the punch, and k is
~
FigureI a
,
~
a lines
PranotrlSkO-bne ~kdion
~
F a lines
~,/~V
G,~._ AI ",,\
v~2V
V"
. L3lines
IA yG
,,. " , /
0168-874X/89/$3.50 © 1989, Elsevier Science Publishers B.V.
Fig. 1. (a) Prandtl's slip-line solution. (b) Hill's slip-line solution.
174
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
the yield stress of the semi-infinite body in pure shear. The velocity field associated with this state of stress can be obtained by Geiringer equations [6], as shown in Fig. l(a). It is noted that the triangular region AA'O under the punch behaves like a "dead metal", moving downward with the same velocity as that of the punch. In addition, region AOB (A'OB') moves outward along circular arc OB while region ABC (A'B'C') moves in the direction of BC (B'C'), both with a velocity 0.707 times that of the punch. Hill in 1949 [7] suggested that the velocity field in Prandtl's solution is indeterminate, and proposed a different slip-line solution as depicted in Fig. l(b). It can be shown that the state of stress in each of regions AO'D, ADE and AEG of Hill's solution is identical to that in regions AA' O, AOB and ABC, respectively, of Prandtl's solution. The limiting load F for the onset of plastic flow, therefore, is still given by (1). However, for the same punch velocity, Hill's solution indicates that region AEG would move in direction EG with a velocity twice that of region ABC of Prandtrs solution. In addition, Hill's solution does not require a "dead metal" zone under the punch. Subsequent investigation of the flat punch problem by several authors [10,11,15,16] suggested that the validity of Prandtl's or Hill's solution may depend on whether the punch surface is rough or smooth. Furthermore, due to the nature of the slip-line method, an infinite number of velocity and slip-line fields could be constructed between these two limiting solutions. It should be pointed out that the slip-line method only allows one to estimate the load required to create a state of unconstrained plastic flow, but does not. give the details of the actual deformation process of the material. In addition, the fact that the slip-line solution requires both the velocity and pressure conditions at the punch surface be specified implies that it is not a well-defined boundary value problem. Therefore, to obtain the "true" solution of a flat punch problem, it is necessary to solve the complete elasto-plastic problem, using the Prandtl-Reuss relations, together with properly specified boundary conditions at the punch surface. The solution of this more realistic formulation is rather intractable before advanced numerical techniques, such as the finite element method, became available. The finite element method was first developed in the 1950s for structural analysis, and has been extended to solve elasto-plastic problems during the 1960s [1,13,14,19]. Lee and Kobayashi [13] formulated a finite element procedure for the elasto-plastic analysis of plane-strain and axisymmetric flat punch indentation problems. The development of the plastic zones, the load-displacement relationships and stress and strain distributions with various friction conditions on the punch surface were obtained. However, the velocity field, which is the major difference between the Prandtl's and Hill's solutions, was not reported in their work. Furthermore, the thickness of the region that was modelled and analyzed in their work was too thin to be considered as semi-infinite. In this paper, a two-dimensional finite element Lagrangian program DEFEL [5] is employed to study the flat punch problem. The semi-infinite body is idealized by an area of sufficiently large size to rule out the boundary effect due to finite dimension. The effect of friction condition on the punch surface is examined by analyzing results that simulate smooth and rough punches, respectively. The development of plastic zone and velocity field, and the contour of effective strain at different stages of loading are obtained and compared with the slip-line solutions. It is found that, although the friction condition on the punch surface has some effect on the development of plastic zone and velocity field, the finite element results in general agree quite well with Prandtl's slip-line solution for both smooth and rough punches. Description of DEI~JLcomputer program
DEFEL is a two-dimensional finite element computer program originally developed for dynamic analysis of impact and explosive-metal interaction problems. The formulation of
T.-M. Tan et aL / Finiteelementsolutionof Prandtl'sflat punchproblem
175
DEFEL is based on the hydro-code approach similar to that used in EmC code [9]. In recent years several new features have been incorporated in DEFEL, including various material models for elasto-plastic analysis, dynamic relaxation procedure for quasi-static or static analysis, and slide line routine for modelling frictional contact/slide boundary conditions [4,12,18]. The basic element used in DEFEL code is the triangular constant strain element. Following standard finite element procedures [2,20], the equations of motion for the finite element model can be written as
M i I ' = I - P,
(2)
where M is the mass matrix, q is the nodal displacement vector, f is the external force vector, and P is the equivalent nodal force vector calculated from the element stresses. The superposed dot denotes differentiation with respect to time. In general, the vector P is a function of the current nodal displacements and velocities for non-linear problems, and can be expressed in terms of the current element stress vector o by
e = E f BTO dV, ~v~
(3)
where B is the strain-displacement matrix. The advantage of this formulation is that ~the process of assembling the global stiffness matrix can be bypassed if an explicit integration scheme and a lumped mass matrix are used. In that event, the system of equations (2) are uncoupled into
m,i], = f~ - P,,
(4)
in which the subscript i corresponds to the ith component of the nodal displacement vector q. An explicit integration scheme with central difference quadrature is used in the DEFEL code to integrate the differential equations in time. For this purpose, equation (4) is rewritten in the following form m//], = ,f~ _ ,- atp~,
(5)
where At is the time increment, and the left superscript t refers to time instant t. The velocity and displacement of the ith node are then given by
'+0.Satoi=t-O.SAtoi + At t~i=t-O.SAtoi + At[ tfi--t-atpi ]]
(6)
t+atqi = 'qi + At '+°sat0i,
(7)
and
respectively. Combining (5), (6) and (7), the displacement of node i at time instant t + At can be obtained as
t+Atqi = (At)2 [tfi -t--Atpi] -- 2 tqi- At t--atqi.
(8)
mi Due to the conditionally stable nature of the explicit integration scheme, the time increment, At, may need to be readjusted at each time step to ensure numerical stability. In DEFV.Lcode, the following stability criterion is used [12] 0.7
H.../i ~ + 2#
(9)
In equation (9), Hmin is the minimum height of the element, p is the density, and A and ~ are the Lame constants of the material.
176
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
Once the nodal displacements and velocities are solved, the program then proceeds to perform the elasto-plastic stress/strain calculation. The spherical and deviatoric strain rates are computed from the nodal velocities, and the stress increments and total stress components are then computed. Proportional unloading technique is used to pull the over-estimated stress components back to the current yield surface, and various work hardening and softening models [12] can then be used to establish a new yield surface. This process is repeated at each time increment. For static or quasi-static analyses, the program uses the dynamic relaxation method [4], which artificially adds a damping term into the equations of motion, to quickly damp out the effect of inertial terms. With a damping term added, (4) becomes (10)
mii], + CCli= fi - Pi,
where c is the damping constant. Since the amount of damping for a given value of c depends on m and At, a non-dimensionalized constant to, known as relaxation parameter, is used instead in the DEFEL program to specify damping. The relationship between to and c is given in the following equation:
(
to = 2 / 1 + 2~m, ] '
(11)
where At is the average of the two consecutive time increments before and after time t. By using the relaxation parameter, the velocity equation of the ith node becomes
--
m,
1'
(12)
and the displacement is then computed using (7). The value of to may vary from 1 to 2, corresponding to the value of the damping constant c varying from 2 m J A t to zero. The choice of the value for to depends on the nature of the problem being solved. For all the cases reported in this paper, to = 1.995 has been used.
Modelling and analysis of the flat punch problem Finite element model
Figure 2 shows schematically the geometry of the model used in this study. For a plane-strain finite element analysis, the semi-infinite body is assumed to be infinitely long in the z-direction and only its cross section in the x - y plane is modelled. The cross section is idealized as a rectangular area of width 2W and height h, both of which must be much larger than the punch width, 2a, to minimize the boundary effect on the solutions due to finite dimension. In this study, both W / a and h / a ratios are chosen to be 8.8, as has been suggested by several authors [3,8]. Due to symmetry, only one half of this rectangular area is modelled and analyzed. The finite element mesh, as shown in Fig. 3, consists of 3,600 triangular elements and 1,861 nodes. The mesh grid is finer near the punch, and becomes progressively coarser approaching to the boundaries. Under the punch there are six nodes (from node 1 to node 6), called punch nodes, on which the prescribed punch forces or punch displacements are to be applied. In addition to the symmetrical conditions applied along the edge x = 0, nodes along the edge x = W are constrained against horizontal displacements while those along the edge y -- 0 are constrained against vertical displacements.
177
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
II
Punch
J
2W Semi-infinite body
' W
Fig. 2. Indentation of a semi-infinite body by a flat punch.
X
~,
Fig. 3. Finite elementmesh.
Material properties
Material properties similar to those of commercial aluminum 1100-0 are used for the semi-infinite body. These properties are: Young's modulus E: Poisson's ratio v: Yield stress Y:
68.95 GPa (10 × 106 psi) 0.33 89.6 MPa (13,000 psi)
Beyond yield point the material is assumed to be perfectly plastic (i.e., non-hardening). In a preliminary study a material with Young's modulus 10 times as large was used to examine the effect of initial deformation. No appreciable difference in solution has been found. Friction and loading conditions
As mentioned before it was suggested by many authors who investigated the flat punch problem that the validity of Prandtl's or Hill's solution may depend on the friction condition on the punch surface. To verify this point, two sets of boundary conditions, one simulating a smooth punch and the other simulating a rough punch, have been applied to the punch nodes in this study. In the smooth punch case these nodes are allowed to move freely in the horizontal direction while in the rough punch case they are constrained against any horizontal movement. When a rigid punch is pressed against the surface of an elasto-plastic body the deformation of the surface in general does not follow the shape of the punch face, nor will the pressure exerted on the surface of the body be uniformly distributed. The slip-line solutions, obtained by assuming the semi-infinite body to be rigid-perfectly-plastic (i.e., no deformation could occur before the yield stress is reached) and the punch pressure to be uniformly distributed, therefore, are not the solutions of a realistic problem. In this study, the effect of different loading conditions on the development of the velocity field and plastic zone has also been examined by using two loading procedures. In the first procedure nodal forces equivalent to a uniformly
178
T.-M. Tan et a L / Finite element solution of Prandtl's flat punch problem V 1.2 Vmax1.0
PRESCRIBEDUNIFORM PUNCHVELOCI ~ TY C
0.8
IDct 0.6
0.20'4 t.
time
ment.
. . . . .
""~"
~ODE6
0.0 u.O
V 1.2 Vmax1.0
~
0.2
~
0.4
ASE:
NODES1TO5
0,6
0.8
1,0 tt~'" 1.2
N~UNCV HELOC~ PRESCRIBEDUNIFORM
0.8 0.6
t~
time
Fig. 4. (a) Prescribed punch pessure. (b) Prescribed
punch velocity and displacement.
0.2 0.0
u.O
\
0.2
it""
7.~11~
0.4
UN,FOR~RESSURECASE: NODES1TO5
0.6
0.8
1.0 t/tv 1.2
Fig. 5. (a) Comparison of smooth punch velocity. (b) Comparison of rough punch velocity.
applied punch pressure are specified at the punch nodes, while in the second procedure a uniform velocity in the negative z-direction is prescribed at each punch node. For both smooth and rough punches, the pressure-prescribed loading cases are solved first. Forces applied at punch nodes initially increase linearly in time, and become constant after the total punch pressure exceeds a critical value, Per, at t = tp, as shown in Fig. 4(a). For the smooth punch, a critical pressure that yields a limiting load approximately 3% higher than the value predicted by the slip-line theory, is used to ensure that certain amount of plastic flow would occur. In the case of rough punch, it is found that a much higher limiting load is needed to induce enough plastic deformation, as has been reported in the work of Lee and Kobayashi [13]. In this study, a value approximately 25% higher than that of slip-line solution is found to be appropriate, and is used as the critical pressure. In the velocity-prescribed cases, the variation of punch velocity in time is assumed to follow a sine curve, as shown schematically in Fig. 4(b) together with the corresponding displacement function. The prescribed punch velocity is determined in such a way that its maximum velocity, occurring at t = t o, would roughly coincide with the average maximum punch velocity obtained in the pressure prescribed case. For each of the smooth and rough punches, a comparison between the prescribed punch velocity and the velocities of punch nodes computed in the pressure-prescribed case, normalized with respect to the maximum prescribed velocity, Vmax, is shown in Figs. 5(a) and 5(b), respectively. It is noted that the velocity of the outermost punch node (node 6) in the pressure-prescribed case generally is much smaller than those of other punch nodes (node 1 to node 5), and is therefore not used in computing the average punch velocity. Results and discussions Rough punch
The main differences between Prandtl's and Hill's solutions, as discussed in the earlier section, are in their velocity fields and whether a "dead metal" zone exists under the punch. It
T.-M. Tan et aL / Finite element solution of Prandtl's flat punch problem
......
l
::°
:!:!:!::::.:'.'
179
::::::::::::::::::::::::::::::::::::::
~:,i~iiii?i?i?i?i i i ?i : ~X~t~":':':-:-:" :" :" :" :" ~,,,,:.:.:-:-:.:.:
~.'-.:-.:.:.:.:.:.
• :. : . •
:. :. :
,,,,,:,-.:.:.-.:. :. -. - . . . : .
!:
',i,1,'.',2,'.i,2.2.2,2.2•:,:•2.:.:,;,;.'." ~ , = , , ° o o ,
. . . . . . . . .
• °,oO,Oo,o°oOoO . o . OoOoO .oOoO .oO • o . . . . . . . . . . . .
t - o.~8 ~,
, , , ¢ ~
-
_
,
t - 0.56t,
,
~,~-::!:i:i:i:i:~:!:!:!:
iiii!i!iiiiii!!i!!! !i! !!i ii
~~:-:'~:::::::::::::
: : : : : i '
::::::::::::::::::::::::
:-::
ii!iiiiii' iiiiiiii
: : :
!: =
",'::;',,'-::'::::::, ;: : : • : : : : : . : : :
I..
::::::::::::::::::::::::::::::" :: : : : : : : :. : : " • ,-,,:,,,,,,,;,,,,,......
Illllt~ltttttl~tti~t~tb~tt~ttt~lttttttt l l l l t i l l ~ t ~ l t l t b i l
t b t .
0 0 0 , ' o ~ 0 ~ 0 0 , = , ' 0 ' 0 ~ 0 = = ~ ' ' ' ' ' ~ ° °
, . 0 , . 0 , , . • ~ o 0 , 0 0 o . , , • ~ ,
.
. .
. .
. .
. .
.
t
. .
. .
. .
.
•
. .
~ ~ ~
.
" . • .
.
i : i : :: :
•
. .
• ".'.'.'.'.'•'.'.'.'.'.'.'.'.'.'.'.".
.
.
~
=
.
•
o
•
.
.
.
.
•.
.
I
.
.
.
.
.
•
*
•
,
•
I
=
•
°
•
~ l l ~ L~ • • . • - o % -
i~o=°e.'~'~•o'•°.°.°o°o°o°o'°°o°o°°°o ° •" , o , o , o O • - o - o - o - Oo-oO - . o . ° OoOo-o - • • o • ° . . . . . . . . . . . . . . . . . . , . o o,~o• Oo-oO.O OoOoOo. - . ° o ° o O . O o . • •
.
•
•
. .
O.OoO.OoOoOoO • • o • o ° • • • •
°
•
•
. .
". t - 0.97t~
t ,- 1.53t v
Fig. 6. Development of velocity field for a rough punch with prescribed uniform punch velocity.
is therefore of great interest to monitor the development of velocity field during the loading process. Figures 6(a)-(d) show the vectorial plots of nodal velocity at different times for rough punch under the condition of prescribed uniform punch velocity. At an earlier loading stage (e.g. t = 0.28 to), most material points are moving downward, only those near the punch comer are moving slightly outward. As the punch being pressed further into the body (e.g. t = 0.56 t o and t = 0.97 to), material points near the punch comer start turning upward. Finally at t = 1.53 t o, the material points in the triangular region just outside of punch comer are seen to move in the direction approximately 45 * to the free surface of the semi-infinite body, while those in the triangular region under the punch are continuing to move downward• A centered fan area between these two triangular regions can be clearly observed. When overlapping the Prandtl's slip-line field on this velocity plot, as shown in Fig. 6(d), one observes that the matching between these two solutions is remarkable well. It is noted that the direction of nodal velocity changes abruptly along the "slipline" between the triangular area under the punch and the centered fan area, indicating that the tangential component of velocity is discontinuous• Figure 7 shows the contour of nodal velocity at t = 1.53 t o for rough punch with prescribed punch velocity. Two triangular areas of constant velocity, one under the punch and the other outside of the punch comer, can be clearly seen. This indicates again that Prandtl's solution
180
T.-M. Tan et a L / Finite element solution of Prandtl's flat punch problem
RESULTANT VELOCITY
t = 1.53t v
I.D. 1 2 3 4 5 6 7
VALUE (VN~n¢,) 0.00 0.17 0.35 0.52 0.69 0.87 1.04
Fig. 7. Contour of nodal velocity for a rough punch with prescribed uniform punch velocity.
,$
t - 0.14t v
t - 0.28t v
t -, 1 . 2 5 t v
t,= 1.53t v
Fig. 8. Development of plastic zone for a rough punch with prescribed uniform punch velocity.
181
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
J
:.~.:.:':+~'?:-:-2-?'-'"" Z+':
• ••
....
t = 1.53~
....
EFF. PLASTIC STRAIN I.D. VALUE 1 0.0000 2 0.0181 3 0.0362 4 0.0543 5 0.0724 6 0.0905 7 0.1086
Fig. 9. Contour of effective plastic strain for a rough punch with prescribed uniform punch velocity.
.'.'.'.'.'.'.'.'.'.'..
t=
1.67t v
Fig. 10. Typical velocity field for a rough punch with prescribed uniform punch pressure.
seems to yield better prediction for rough punch. The magnitude of average nodal velocity in the outside triangular area, however, is only about 0.4 times of the average nodal velocity under the punch, somewhat smaller than the value of 0.707 predicted by Prandtl's solution. On the other hand, Hill's solution which predicts that the velocity in the outside triangular area to be greater than (i.e., 1.414 times) the velocity of punch, has never been observed in this case. Figures 8(a)-(d) show the development of plastic zone in the semi-infinite body at different times for the velocity-prescribed case. The plastic zone initially occurs near the comer of the punch, then gradually develops toward the center along a 45 o line. As the loading continues to increase, the plastic zone further develops, eventually reaches the stage shown in Fig. 8(d) at t = 1.53 t o. A triangular areas of "elastic zone" can be seen to exist under the punch. The
1111[1 TTTTTT
RESULTANT VELOCITY I.D
VALUEo.o0(VNmr=) 0.18
0.36 0.54 0.72 0.89
1.07
Fig. 11. Contour of nodal velocity for a rough punch with prescribed uniform punch pressure,
t=
1.67t v
EFF. PLASTIC STRAIN I.D. VALUE 1 0.0000 2 0.0102 3 0.0204 4 0.0306 5 0.0409 6 0.0511 7 0.0613
Fig. 12. Contour of effective strain for a rough punch with prescribed uniform punch pressure.
182
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
contour of effective plastic strain at the same time, as shown in Fig. 9, also suggests that the triangular area under the punch does behaves like a "dead-metal" for rough punch with prescribed velocity. In case of rough punch with prescribed uniform punch pressure, the results are very similar to those of prescribed punch velocity. For comparison, typical results in terms of a vectorial plot of nodal velocity and contours of velocity and effective plastic strain are shown in Figs. 10-12. It can be seen that the vectorial plot of nodal velocity in this case also resembles Prandtl's slip-line field. Contours of nodal velocity and effective plastic strain suggest that a triangular "dead-metal" zone, although not as obvious, also seems to exist under the punch.
Smooth punch For smooth punch where punch nodes are free to move horizontally, the vectorial plots of nodal velocity at different times for velocity prescribed case are shown in Figs. 13(a)-(d). The material points in the area under the punch, although having noticeably larger horizontal components in their velocities comparing with the rough punch cases, are still mostly moving
illlilC,~,~,~,.:~.:.,...
I!!
l ll~lllb:,:,:::::::::.:....... ,UlI!,,,,,,,,,',',',".'.':':. I q ~ | t ,bbbbb+:':":':':':':':'" | t t t ~ l t • • .°o . . - . . . . . . . . o - . lllll
t I t t ~t
leftist
i|stst~
tt
•~.• ...
*'° . . . . . . . .
. . . . . . . . ...... .%%-
,11d~t%~%•.%%%%'.'.'.'.'.'...
.%..%%%%N°.°.-°....%'.'.'.'.'. .%%..%..%°°°.%..%'°'.'.'.'.'.'."
•.
d.%%.°°.%.°... °.°o..... - • •. *. .o°......°°....°..'.'''" . • • .°.. °.. '." • " • ' "."
:'Z';':';+:':':':';':':':':'Z':':':"
;" :- :" :- :- : "
, , , o , . , , , = , , , , , , o , . ,
, . . , , , . , °
....
,.,
.
.
.
.
.
.
.
.
' ' ° ' . ' . ' ' ° ' ' ' ' . ° ' . ' ' . ' . ' ° . ' . ' . ' . " .
t -
.
.
.
.
.
.
.
.
" ° ".
..................
0.27t v
.
. ".
t -
"
.
0.53 tv
//IJls . . . . . . . . . . . . .
i i ::i i ::: ,,
i
i
I=!!!:!:!!!!!!!!!!!!i:i:!!!:!!:!!!i!!!!!:!: !:!!: Z.Z'Z'Z'Z-3Z'Z'?2"Z'""
t - 0.80 t,
"
".'.
t = 0.93t v
Fig. 1.% Development of velocity field for a smooth punch with prescribed uniform punch pressure.
183
T.-M. Tan et a L / Finite element solution of Prandtl's flat punch problem
t - 0.27t,,
t = 0.13t v
i :ii iiiiii!iiiiiiiiiii!iii!i !
t = 0.93t,,
t - 0.80 t,
Fig. 14. Development of plastic zone for a smooth punch with prescribed uniform punch velocity.
i
RESULTANT VELOCITY I.D. 1 2 3 4 5 t = 0.93t v
EFF. PLASTIC STRAIN I.D. VALUE
VALUE (VN~¢',) 0.00 0.18 0.35 0.53 0.70
6
0.88
7
1.05
Fig. 15. Contour of nodal velocity for a smooth punch with prescribed uniform punch velocity.
t = 0.93t v
1 2 3 4 5 6 7
0.0000 0.0210 0.0419 0.0529 0.0838 0.1048 0.1257
Fig. 16. Contour of effective strain for a smooth punch with prescribed uniform punch velocity.
184
T.-M. Tan et aL / Finite element solution of Prandtl's flat punch problem
i!i!!i!i!ili!ilili!i! ! ! i! i! i! i! i i
Io.,...~°-°-°o-°*o°°o°°-°'.°.°
"°
-.
° . - o
° . - .
~ ~ ° ~ • , . . . ° . ° * . - . - . O ° ° o ' ° * ° o ° . - . ° . * . * . * . * . - . - . - . - . .
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
. .
.
.
. . .
.
. . .
.
-.-.-.-.-.-.-...
. .
.
. .
.
.
. .
.
.
. .
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . .
. .
.
. .
.
.
.
. .
.
.
. . . .
.
.....
. . . .
. .
,
.
•
,
•
~
'
•
.
•
.
*
,
*
o
. .
.
°
,
° .
.
°
.
.
*
,
°
*
°
•
•
•
,
•
•
•
°
.
.
,
°
Fig. 17. Typical velocity field for a smooth punch with prescribed uniform punch pressure.
t = 0.93 t~ .
downward throughout the entire loading process. At t = 0.98 t o (Fig. 13(d)), where the material points in the area outside of punch c o m e r have turned to the 45 o direction, the magnitude of average velocity in this area is again less than one half of the punch velocity, and the velocity field is also quite close to Prandtl's slip-line field. Yielding of material also occurs first at the c o m e r of the punch, then gradually propagates toward the center along a 45 o line, as shown in Figs. 14(a)-(d). Unlike the rough punch case where a small area under the punch remains elastic, the plastic zone in this case quickly expands and covers the entire area under the punch. Contours of nodal velocity and effective plastic strain, as shown in Figs. 15 and 16, respectively, also indicate that the existence of a " d e a d metal" zone, while clearly observed under a rough punch, is not as obvious in this case. Figure 17 shows the vectorial plot of nodal velocity for smooth punch with prescribed uniform punch pressure at t = 1.63 t o. It is interesting to see that, even though the punch nodes
l l t t l _ T T V T T T
.i
t = 0 . 9 3 tv
RESULTANT VELOCITY I.D. VALUE (VN~.~) 1 0.00 2 0.17 3 0.35 4 0.52 5 0.70 6 0.87 7 1.05
Fig. 18. Contour of nodal velocity for a smooth punch with prescribed uniform punch pressure,
••
t = 0.93~
EFF. I.D. 1 2 3 4 5 6 7
PLASTIC STRAIN VALUE 0.0000 0.0089 0.0178 0.0267 0.0356 0.0445 0.0533
Fig. 19. Contour of effective plastic strain for a smooth punch with prescribed uniform punch pressure.
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
185
in this case are allowed to move freely in the horizontal direction, the material points under the punch are moving almost straightly downward, and there appears to have a 45 ° line of discontinuity in velocity. Contours of velocity and effective plastic strain, shown in Figs. 18 and 19, respectively, also suggest that a triangular "dead metal" zone may exist under the punch.
Conclusion The indentation of a flat punch into an elastic-perfectly-plastic semi-infinity body has been studied using an elasto-plastic finite element program. The development of plastic zone, velocity field and effective plastic strain in the semi-infinite body are monitored during the entire loading process. Two types of boundary condition have been used at punch nodes to separately simulate smooth and rough punch surfaces. In addition, two loading procedures, one with prescribed uniform punch pressure and the other with prescribed uniform punch velocity, have been applied to study the effect of different loading conditions. It is found that for all the cases studied here, the finite element results, in terms of velocity field, existence of triangular dead metal zone under the punch, and magnitude of velocity at free surface, resemble Prandtl's slip-line solution quite well. This is particularly so for the rough punch cases. On the other hand, no conclusive evidence that supports Hill's slip-line solution could be found for either smooth or rough punch. Yielding of the material always occurs first near the corner of the punch, then gradually extends toward the center along a 45 o line. The limiting load predicted by both Prandtl's and Hill's slip-line solutions is found to be quite accurate for the smooth punch case. For the rough punch case, a much higher limiting load is necessary to induce plastic deformation in the semi-infinite body.
Acknowledgement This study was partially supported by a grant from the Ben Franklin Technology Center (formerly Advanced Technology Center) of Southeastern Pennsylvania and Dyna East Corporation through the Ben Franklin Partnership Program. The first author would like to thank Professor Alan Lau for his helpful comments.
References [1] AKYUZ, F.A. and J.E. MERWlH, "Solution of nonlinear problems of elastoplasticity by finite element method", AIAA, J. 6 (10), pp. 1825-1831, 1968. [2] BATHE, K.-J., Finite Element Procedures in Engineering Analysis, Prentice Hall, Engiewood Cliffs, N J, 1982. [3] BISHOP, J.F.W., " O n the complete solution to problems of deformation of a plastic-rigid material", J. Mech. Phys. Solids 2, pp. 43-53, 1953. [4] CHOU, P.C. and L. Wu,"A dynamic relaxation finite element method for metal forming processes", Int. J. Mech. Sci. 28 (4), pp. 231-250, 1986. [5] FLIS, W.J., S. MILLER and W.J. CLARK, DEFEL: A finite element hydrodynamic computer code, Dyna East Corporation Technical Report DE-TR-84-05, 1986. [6] GEIRINGER, H., "Complete solutions to the plane plasticity problem", Proc. 3rd Int. Congr. Appl. Mech. 2, pp. 185-190, 1930. [7] HILL, R., "The plastic yielding of notched bars under tension", Q. J. Mech. Appl. Math. 2 (1), pp. 40-52, 1949. [8] HILL, R., The Mathematical Theory of Plasticity, Oxford, 1950. [9] JOHNSON, G.R., "Analysis of elasto-plastic impact involving severe distortion", J. Appl. Mech. 98, pp. 439-444, 1976. [10] JOHNSOH, W. and P.B. MELLOR, Engineering Plasticity, Ellis Horwood, Chichester, UK, 1983.
186
T.-M. Tan et al. / Finite element solution of Prandtl's flat punch problem
[11] KACHANOV,L.M., Fundamentals of the Theory of Plasticity, Mir Publishers, Moscow, 1974. [12] LAU, A.C.W., R. SHIVPURI,and P.C. CHOU, "An explicit time integration elastic-plastic finite element algorithm for analysis of high speed rolling", Int. J. Mech. Sci., 31 (7), pp. 483-497, 1989. [13] LEE, C.H. and S. KOBAYASm,"Elastoplastic analysis of plane-strain and axisymmetric flat punch indentation by the finite element method", Int. J. Mech. ScL 12 (4), pp. 349-370, 1970. [14] MARCAL, P.V. and I.P. lONG, "Elastic-plastic analysis of two-dimensional stress system by the finite element method", Int. J. Mech. Sci. 9, pp. 143-155, 1967. [15] MENDELSON,I., Plasticity: Theory and Application, Robert E. Krieger Publishing Company, Inc., 1983. [16] PRAGER,W. and P.G. HODOE, Theory of Perfectly Plastic Solids, Dover Publications, New York, 1968. [17] PRANDTL,L., "Concerning the hardness of plastic bodies", Nachr. Ges. Wiss. GSttingen, p. 74-85, 1920. [18] SmvPum, R., P.C. CHou and C.W. LAU, "Finite element investigation of curhng in nonsymmetric rolling of fiat stock", Int. J. Mech. Sci. 30 (9), pp. 625-635, 1988. [19] YAMADA,Y., N. YOSHIMURAand T. SAKURAI,"Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by the finite element method", Int. J. Mech. Sci. 10, pp. 343-354, 1968. [20] ZIENKmWICZ,O.C., The Finite Element Method, 3rd ed., McGraw-Hill, New York, 1977.