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Repeated impact and rebound of an elastic-plastic bar from a rigid surface
Compufns & Stn~ctuns, Vol.7. pp.391-397.Pcr@mon Press 1977.
Printed in Great Britain
REPEATED IMPACT AND REBOUND OF AN ELASTIC-PLASTIC BAR FROM A RIGID SURFACE? HYMANGARNETt and HARRYARMEN Research Department, Grumman Aerospace Corporation, Bethpage, NY 11714,U.S.A. (Received 15November 1975) Abstract-The repeated impact, or “bouncing,” of an elastic-plastic rod on a rigid surface, is treated by means of finite element analysis. A computationally convenient substructuring technique is utilized in the analytical treatment to account for the changing kinematic constraints. The time dependence is accounted for by means of a variable time step, direct integration procedure that has been successfully applied to a number of elasto-plastic dynamic response problems. This time integration procedure is at present being employed in a nonlinear crashworthiness study. It has been proven capable of providing an accurate prediction of a variety of multidimensional dynamic elastic-plastic responses. Additionally, the procedure has demonstrated its abiity to account for an elastic-plastic dynamic response that includes plastic reflections from boundaries and interaction of plastic wave fronts. The merging of the substructuring technique and numerical time integration procedure has been accomplished. The cases discussed include an unloaded bar traveling at uniform velocity before impact with a rigid barrier, and a bar traveling toward the rigid surface and subject to forces that are applied and removed periodically. Both cases are motivated by questions concerning the elastic-plastic dynamic response of a crashed vehicle. The latter case, which suggests the term “bouncing problem,” has, in addition, applications to problems involving pneumatically driven tools. The response is described, starting with the instant of contact at the impacting end, is followed while elastic/plastic wave propagation extends to the free end, and continues to be treated as elastic/plastic reflections occur and return to tbe impacting end. Finally, the behavior of the rod as it leaves and possibly returns to the wall is discussed, and the dwell time as a function of elastic and elastic-plastic properties of the material is determined.
1.INTRODUCTION
of nodal displacements, [m] is the mass matrix, [k] the elastic stiffness matrix and {F(tfi the vector defining the applied load history. The terms [k] and {AeP}are the initial strain stiffness matrix and incremental plastic strain vector, respectively; their product defines the plastic load vector. In some treatments, the vector of plastic strains, {a~~}),on the right hand side are based on the results of the previous step. In other treatments, eqn (1) is reformulated by relating {AeP}to the “current” value of {AU}, and obtaining a tangent modulus matrix of stiffness influence coefficients that premultiplies {AU}on the left hand side. Results presented herein have been obtained by a procedure equivalent to the tangent modulus formulation. This is demonstrated subsequently in the discussion of the time integration procedure. The elements of the matrices cited above have been exhibited in many earlier papers and textbooks dealii with finite element analysis.
An elastic-plastic unstrained bar traveling at constant velocity impinges on a rigid surface (Fig. 1). Depending on
the impacting velocity, an elastic or a plastic wave is generated in the bar. Compressive in nature, this wave is
propagated to the free end of the bar, where it is reflected as a tensile wave. The reflected tensile wave has the effect of canceling out the compressive wave generated by the impact. If the impacting velocity generates an elastic wave, the bar leaves the wall when the reflected wave front returns to the impacting end. Therefore, in the elastic case, the “dwell time”-the time during which the bar remains in contact with the wall-is equal to twice the time required for an elastic wave to travel the length of the bar. When the impact is plastic, the dwell time can be anticipated to exceed two transit times. Results presented substantiate this expectation, and also indicate that for an elastic-plastic impact the dwell time is a function of the impacting velocity, and the nonlinear hardening properties of the material.
(b) Time dependent partitioning
kinematic
constraints
and matrix
Referring to Fig. 1, we note that the current problem involves a time dependent kinematic constraint. At initial
2. ANALYsls
(a) Equations of motion The incremental equations of motion governing wave propagation in a one dimensional elasto-plastic bar are discussed in [l] and 121and are given in matrix form as A) ACTUAL IMPACT
ROD IN RIGID
BODY
MOTION
PRIOR
TO
in which {AU}is a column vector denoting the increments (1)
tpresented at the Second National Symposium on Computerized StructuraI Analysis and Design at the School of Engineering and Applied Science, George Washington University, Washington, D.C., 29-31 March 1976. SResearch Scientist. OHead,Applied Mechanics Branch.
(2)
(3)
cl000000 +lPi--
(n-1)
(n)
--clakI
El
FINITE
ELEMENT
IDEALIZATION
OF ROD
ha = L)
Fig. 1. Elastic-plastic bar, at constant velocity, uO,prior to impact with rigid surface.
391
392
H. GARNET and H. ARMEN
time, t = fO,the unloaded bar is moving with constant velocity, uO.The right end of the bar, node R, is at a distance, do, from the rigid wall. If we choose as a reference frame a fixed coordinate system, x, coinciding with the bar at to, and displacements of the nodes are designated as u, then the right end of the bar is in contact with the wall when uR = d,. Therefore the time dependent kinematic conditions at impact and during subsequent contact are uR = d,,
tin = 0.
(2)
The dot over the symbol u denotes the time derivative. Therefore, the symbol & designates particle velocity. No conditions are prescribed at node R when it is not in contact with the wall. The contact condition is determined by monitoring the value of uR. In the cases discussed in this paper, where the bar impinges on a rigid wall, the governing equations, in the form given by eqn (1) are probably just as convenient as any other form. But when the bar impinges against a deformable object, matrix partitioning renders the problem more tractable. With the goal of extending the present study to accommodate impact between deformable bodies, eqn (1) is recast in partitioned form as
If the terms on the right hand side are known, then eqns (Sa) and (5b) become easily evaluated algebraic equations for the determination of the increments of acceleration {Ati,},{A&},or more compactly the incremental acceleration vector {Aii} (7) The values of the increments of displacements {Au,}, {Au*}are obtained by means of the direct time integration scheme-discussed in a subsequent section. These increments of displacement are used in the straindisplacement relations to compute increments of total strain. Having the values of total strain, the plasticity relations are used to determine the plastic strain components and thereby construct the plastic load vector. In this manner the terms on the right hand side of eqns (5) are determined. It is noted that eqn (5b) is evaluated first for {A&},which is then used on the right hand side of eqn (5a). (c) Plasticity relations
For simplicity, a bilinear stress-strain law has been assumed (Fig. 2). Because of the nature of the problem the E, = ELASTIC E: = LINEAR
MODULUS HARDENING
MODULUS
(3) where on the right hand side the product [i]{Aq} appearing in eqn (1) has been rewritten, in partitioned form, as the plastic load vector {AP}.The subscripts 1 and 2 refer to nodes free of constraint, and nodes subject to constraint, respectively. In this case Au, consists of a single quantity, namely, the increment of displacement of node R, i.e. Au,. If we perform the indicated multiplications on the submatrices, we are led to the following two systems of matrix equations [m,,lbW
+ [m,,l&Q + [k,,l~AuJ + [k&M = WJ
[m&W+
[WWzI
+ [k&W
+ WJ
(da)
+ [kzlIW = {AF2}t {APz}.
(4b)
Equation (4a) is used to eliminate the {Aii,} term from eqn (4b), leading to the following set of equations [n,J{Aii,X+, = {AF,]i+l + IAPJi+, - [k,,lW,li - [m,,lWiJ,+, - [k,&%
(54
[&]{A&}i+, = {A&},+,+{APz}i+,- [M,lW,Ii+, + ([Msl[kd - [kz,IWJ~
+ ([MsI[M - [k,,lHWi
Ob)
in which
[h&l = [m221- [m2J[mJ’[~~~l [&I = [mzll[mIJ’.
(6)
The subscripts i t 1 and i, included here, correspond to the ith and (i + l)th time steps in the incremental process.
Fig. 2. Bilinear stress-strain relation and hardening rule.
analysis is required to predict unloading from a plastic state and subsequent reloading into the plastic range loading states. Toward this end, a hardening rule, based on Prager’s kinematic hardening rule[3], has been employed. This rule is illustrated schematically in Fig. 2. Note that after impact, and subsequent loading and unloading, the values of the tensile and compressive yield stresses and strains change. Usually, it is the elastic stress range that is kept constant. In the present investigation, the elastic-strain range was kept constant, and, as indicated in Fig. 2, the hardening rule was enforced on the basis of shifting the tensile and compressive yield strains. This approach, while producing results no different from those obtained by keeping the elastic stress range, seems more natural, since strains are a directly observed experimental quantity. (d) Time integration procedure Direct integration methods are usually employed to handle the time dependence in eqn (1). Terms involving derivatives are replaced by various approximating difference expressions, and the resulting recurrence relations are employed in a step forward manner. The more popular of these schemes have been Houbolt’s, the NewmarkBeta, the central difference and Wilson’s method. These
393
Impact and rebound of an elastic-plastic bar
procedures, all employing a constant time step, have been surveyed in a number of publications, including the present authors [4], and by Nickell[S]. A serious drawback is common to all constant time step methods. The acceptability of these procedures depends heavily on the proper choice of the time increment. In physical terms, this means that the dynamic analysis may have to be repeated several times before acceptable results are achieved. For a modest size problem this procedure can be prohibitively costly. This drawback can be eliminated by employing a variable time step method. In the present paper the authors employ a previously developed version of the modified Adams metbod[6]. The procedure has been employed earlier by the present authors, to account for one and two dimensional elasto-plastic wave propagation, including those cases involving waves reflected from boundaries[l, 2,7]. The method employs Taylor series expansions to obtain predictor and corrector expressions, truncated to difference form, for the solution to a system of first order nonlinear differential equations. The corresponding system of second order equations of motion (eqn l), is employed to construct the corrector expression. The procedure is briefly outlined here and described in more detail in [6]. If the unpartitioned system of equations (eqn l), is of order n, a column vector of order 2n is defined as
used to determine the corrector solution
Note that because of the operations carried out in eqns (7) and (13), {j},,, is known. Therefore, all quantities on the right are known and the corrector expression may be evaluated. Comparison of {y}r+,, {y}:, (displacements and velocities) determines whether or not the results are acceptable. Depending on the error tolerance selected, one of the following situations may exist: (1) the result is acceptable, and the time integration may continue to the next time interval ti+*;(2) the result is unacceptable and the time interval, At, is halved; and (3) the agreement exceeds accuracy requirements, the result is acceptable and forward integration proceeds at double the interval, At, i.e. 2At. Because the plastic strains, which contribute to the plastic load vector, eqn (l), or eqn (5), are computed from displacement based on the current time step (i.e. the predictor solution), the solution procedure is equivalent to the tangent modulus method. (e) Stress at impact
The means for computing the stress generated in the rod at the instant of impact will now be derived. In Refs. [ 1 and 21 an expression was developed for the momentumimpulse balance of an element of the rod. Because of its simplicity the derivation will be repeated here. Figure 3
{Yl=[f]
{j}=(j). Predictor and corrector expressions are employed as follows. A predicted value is generated from the following expression
Fig. 3. Element of rod includinginertia forces.
{Y]r+l={Y}i +At{jJi +$(IiL
-UJi-l).
(10)
The subscripts correspond to the ith time step in the incremental process. Note that at time ti, all quantities on the right hand side are known. Also {Y};+l= [;),+,.
(11)
depicts a free body diagram of an element of the rod, including inertia forces. The rod is of constant cross section A, and is assumed to possess mass density p. The symbols u and u denote stress level and particle velocity level, respectively. The condition of momentum impulse balance for the element may be obtained from D’Lambert’s principle and written as (p dx) du = (da)(dt).
Next, eqns (5) are employed to obtain the increments of acceleration, {Ati},,,. Toward this end, the upper block of the predictor solution, eqn (lo), is used to determine the vector {Au}~appearing in eqns (5) {Au}~= {u}:+,-{u}:.
(12)
The increments of acceleration and velocity are then summed to determined the total acceleration of the system. Before using the corrector expression, the following vector is generated
where the upper half of {j} is obtained from the lower matrix block in eqn (ll), and the lower half is obtained from the solution to eqn (5). The following expression is
(15)
If c is the speed of wave propagation during the time interval, dt, then dx = c dt and eqn (15) becomes do =$.
(16)
Equation (16), which in essence is an expression of momentum-impulse balance between change in particle velocity and stress level, will be employed to determine the stress intensity at impact. If the unloaded rod is traveling at constant velocity, v,,, prior to impact, then its valocity immediately after impact, ul, is obtained by using eqn (16), and enforcing the kinematic constraint u = 0 at impact. Then, integrating eqn (16), we obtain (17)
394
H.
GARNETand
where the symbols c, and c, denote wave speeds based on the elastic and plastic portion of the bilinear stress-strain curve, respectively, i.e.
H.
ARMEN
3,
Go
3,
I? V
R A) ROD PRIOR
G = V%%lp), c,, = V\/(E,/p).
3,
+ooooo~ TO IMPACT
(18)
The integration on the right consists of two terms to account for the eventuality that the intensity of a, may exceed o,, the yield stress. After integrating eqn (17), dividing both sides of the resulting equation by the elastic modulus, E,, and using eqn (18), the following nondimensional expression is obtained for the impact stress
VR Bl ROD AT
7
0
IMPACT
Go
3,3,
$
0
REFLECTED COMPRESSIVE WAVE FRONT
3, 0
010 4
C) CONTINUED
0
v=ov=o
CONTACT
WITH
WALL
in which e, is the yield strain +-d e,
REFLECTED
COMPRESSIVE
WAVE
FRONT
I
ES,
4
Et
IV
In the elastic case, EJE, = 1. and eqn (19) becomes
0
0
=“o
0) ROD COMPLETELY IN CONTACT WITH
0
0
v=o AT RESl WALL REFLECTED TENSILE WAVE FRONT k
Qooooo;
That this is the proper limiting value can be verified by recalling that solutions to the case of one dimensional elastic wave propagation are of the form u = f(x 2 CJ) where f is an arbitrary function. By computing v = du/dt, and (T= au/ax, relations are obtained that result in eqn (21). It follows from eqn (19) that the impact will be elastic when 3< e,. C,
(22)
Because the response is nonlinear, and path dependent, the derivation of analytical expressions similar to eqn (19) by means of the method of eqn (16) becomes increasingly complex as time progresses. Therefore no further attempts to present such expressions to account for the many possible interactions that occur will be made. Their primary purpose in this paper, as well as in [I], and [2], has been to validate a numerical procedure designed to cope with a multi-dimensional complex nonlinear response. 3.
DISCUSSIONOF RESULTS
The mechanics of the elastic case are discussed first. Figure 4 depicts several stages in the elastic impact and rebound process. Figure 4(a) shows the rod in rigid body motion before impact. All nodes are moving with constant velocity, va. In Fig. 4(b), the rod is shown at the instant of impact; node R, at the wall, has zero velocity. All other nodes are still moving to the right with velocity, v,,.At this instant an elastic compressive wave is generated at the impacting end. This wave propagates along the rod, to the left, with wave speed c,, and particle velocity, vO.In a compressive wave, the particle velocity is in the same direction as the propagation of the wave. Figure 4c represents conditions when the wave has not yet reached the free end. The particle velocity associated with the compressive wave has served to bring to a standstill those nodes passed by the wave front, i.e. that part of the rod to the right of the wave front is at rest (when the wave front
*v,/v=o
--Vcl E) CONTINUED
CONTACT
WITH
v-o’ WALL
~00000Q
-vo F) ROD
REBOUNDS
tV, FROM
R WALL
Fig.4. Mechanicsof initialelasticimpact. reaches the free end all particles are at rest). The one dimensional compressional elastic wave will be reflected from the free end as a tensile wave of equal numerical magnitude. This tensile wave will propagate to the right. However, the particle velocity associated with a tensile wave is in a direction opposite to the motion of the wave front. The reflected tensile wave will accomplish two things. First, it will cancel the compressive impacting wave, returning the bar to a stress free state. Additionally, as it progresses to the right, it will impart a particle velocity, of magnitude v,,, to each node-in a direction opposite to the original motion. Figure 4(d) depicts this situation. When the reflected wave reaches the wall, the bar will be stress-free and all particles of the bar will be moving to the left with velocity v,,.The bar then leaves the wall, in a direction opposite to its incident motion, with velocity of magnitude vO(Fig. 4e). Some typical stress response results are shown for an elastic case (Fig. 5). In this and all subsequent figures involving stress response, the ordinates represent the nondimensional stress values u/E,, while abscissas are nondimensional positions, x/L, where L is the length of the bar and x is measured from the left end of the bar. A nondimensional time parameter t is employed to denote the position of the elastic wave front resulting from the impact. A value of t = 1 corresponds to the time required for an elastic wave to undergo a single transit of the bar. Figure 5(a) shows the stress response in the bar before the compressive impacting wave has reached the free end.
395
Impact and rebound of an elastic-plastic bar b
* COMPUTED VALUES THEORETICAL VALUES
-
x ^Cl0 d r
THEORETICAL
IMPACT
STRESS\
.
w
9
.
US
E 1=6
POSITION OF ELASTIC COMPRESSIVE WAVE FRONT
.
f
p O!&O 0.125
B
20
0.000
0.125
0.250
0.375
0.500 X/L
0.625
0.750
0.875
0.260
0.375
0.600
0.626
0.750 0.875
A. COMPRESSIVE WAVE FRONT HAS NOT YET REACHED FREE END IT = 0.6231
l
x
THEORETICAL
5r
COMPUTED IMPACT
VALUES
STRESS
3
. COMPUTED VALUES THEORETICAL VALUES
b c
i$ 0~000’0.125 ;/ ;.
IMPACT
STRESS
$4-
0.250 .~. 0.375 ;b!iy:._.; 0.500
0.625
ki 3-
0.750
0.875 1.000
k < 5
2-
g
l-
INTERACTIVE
STRESS
u* 0.125
:
..
I 0.250
. --
_~
. *
LEVEL
.** +
z EO, 5 0.000 z
i
LOCATION OF REFLECTED TENSILE WAVE FRONT
Lu
w THEORETICAL
1.000
X/L A. COMPRESSIVE WAVE FRONT HAS NOT YE7 REACHED FREE END (T-0.623)
1.000
* 9. .
LOCATION OF COMPRESSIVE PLASTIC WAVE FRONT I 0.375
I 0.500
I 0.625
I 0.750
I I 0.875 ,.OdO
X/L
8. REFLECTIONS FROM FREE END HAVE OCCURRED (i = 1.780)
“0 X/L 8. REFLECTIONS (T = 1.420) Fii.
FROM FREE END HAVE OCCURRED
5.
Elastic impact stress response (a) Compressive wave front has not yet reached the free end; (b)Reflectionsfromthe end have occurred.
Solid lines denote theoretical values. Plotted points are those obtained by means of the computer program. Note that the sharp wave front is approximated by a ramp. As previously discussed in [ 1,2,71 the steepness of the ramp may be increased by refining the finite element grid. The presence of the ramp approximation results from the numerical process and cannot be eliminated entirely. Figure S(b) shows the response after the compressive wave has reflected as a tensile wave. Theoretically, the region to the left of the tensile wave front is stress free; numerically, values of an order of magnitude lower than the impacting stress are obtained. Plastic impact results when the magnitude of the incident velocity exceeds the absolute value of the product of the yield strain and the elastic wave speed, i.e. when uO/c,> e,. The ensuing plastic stress level, al, is given by eqn (19). Figure 6(a) depicts a typical stress response to a plastic impact. Shown are conditions prior to the arrival of the compressive wave front at the free end. During plastic impact two wave fronts are generated. One travels at yield stress intensity, and elastic wave speed, c, (eqn I@, and the other travels at the slower plastic wave speed, c,, (eqn 18). This phenomenon was discussed in an early paper by Donnell@], and also more recently by the present authors [21.
x ^y.5 c‘ % k4 t;
YIELD
STRESS
i3
O.ooO 0.125
0.250
0.375 0.500 0.625 0.750 0.875 X/L C. BAR HAS JUST LEFT WALL (i = 2.102)
1.000
Fig. 6. Plastic impact stress response (a) Compressive wave front has not yet reached the free end; (b) Reflections from the end have occurred;(c) Bar has just left wall.
The mechanics of the rod subsequent to plastic impact differ in several respects from those of elastic impact, already discussed. In both cases, the particles to the left of the elastic wave front are still moving to the right with velocity uO.However, for the plastic case, in the region of the rod passed by the elastic wave front, and not yet reached by the elastic wave front, the particles have been slowed down. In the corresponding elastic case, the particle velocity of the generated elastic wave is equal in magnitude to the incident velocity. Therefore, the arrival of the elastic wave front brings a particle to rest. During plastic impact, the magnitude of the particle velocity, associated with the elastic wave front, II,, is less than the incident velocity uO.This can be seen by referring to eqns
3%
H.
GARNET
and H. ARMEN
(19) and (22), i.e. u, < uO. The diminished particle velocities have the magnitude uO- u,, and in the case shown, they are still moving to the right. Finally, we note that the particle velocities associated with stress levels equal to u, must have magnitude uO.This follows from the manner in which the value u, was obtained in eqn (19) i.e. stopping the bar at the rigid wall, that has the effect of producing a particle velocity equal in magnitude and opposite in direction to the incident velocity. The region traversed by the plastic wave front has therefore been brought to rest. Several observations are now made concerning the reason for the increase in dwell time after plastic impact. Since plastic deformation has taken place, only the elastic portion of the total strain is recoverable. The bar will not leave the wall until this has occurred. And because the value of the recoverable elastic strain, during plastic impact, exceeds the corresponding values of strain during the elastic case, a greater time must elapse during this recovery. Therefore a greater dwell time results. The action following plastic impact is complex. Figure 6a depicts an early stage of the process in which the elastic and plastic wave fronts of the impact are clearly shown. Further stages involve the passage of the compressive elastic wave front (at yield stress) to the free end, where it is reflected as a tensile wave with yield stress magnitude. During the process, the plastic wave front, traveling at a speed lower than the elastic wave front, progresses toward the free end. Interaction takes place with the returning tensile wave and the interactive stress level associated with this state is shown in Fig. 6(b). The reflected tensile wave eventually reaches those portions of the bar that have previously been brought to rest by the progressing plastic wave front. Several reflections may be necessary to bring about the required change in particle velocity necessary for motion sufficient to release the bar from the wall. Figure 6(c) depicts the bar shortly after it has left the wall. Note that although the bar is still partially stressed, all particle motions are to the left. The magnitudes of these particle velocities are indicated on the figure. AS expected, the bar is leaving the wall with a velocity of magnitude lower than that prescribed at impact. The dwell time for a given impact velocity appears to be a function of the hardening properties of the material. In Fig. 7, dwell time is plotted against impact velocity for
two different materials. One, for which E,/E,= 0.1, and a second for which E,/E, = 0.3. It is seen that longer dwell times occur for the material having the ratio EJE, = 0.I (more plastic). Note that if the impact velocity is high enough, large particle velocities and displacements can be induced, so that the bar does not leave the wall. This expectation of this phenomenon is substantiated by the results shown in Fig. 8. This case deals with an elastic-perfectly plastic material, in which infinite strains may be induced. The dwell time is seen to rise rapidly with increasing values of u,Jc,. The computations led to a description of nodes passing through each other for the values shown. (Of course, the theory is not applicable then.) The time history sequence for a repeated impact problem is shown in Fig. 9. Here a rectangular pulse.
THEORY FAILS (LARGE DEFORMATIONS, NODES CROSS)
Fig. 8. Dwell time versus impact velocity for elastic-ideally plastic case.
To
o’ 1
2
3
TR
TP
To
TI
TS
4
Fig. 7. Dwell time versus impact velocity for two ratios of IL/E,.
Fig. 9. Time history sequence during repeated impact.
397
Impactand reboundof an elastic-plasticbar whose magnitude is equal to the yield stress of the bar, is applied at some time, T,, after the bar has rebounded from the wall after its initial impact. The pulse remains on the free end of the rod for some arbitrarily prescribed period of time, (To - Tp). If the pulse is of sufficient magnitude and is applied for a sufficient period of time, the rod will reverse its direction and be propelled back to the rigid surface at a velocity different from its initial impact value, uO.The mechanics of this sequence of events, including’its subsequent rebound off the wall, is schematically depicted in Fig. 10.
APPLIED
IMPULSE,
F(t) = YIELDSTRESS . ... .. . ... ....
..
*
..**** : .** I
I
5
,125
I
,250
,375
I
,500
1.
I
I
.625
,750
,875
1.0
X/L L
-
C
“L
Vi
-VR
R
A) IMPULSE APPLIED AT FREE END AFTER ROD HAS REBOUNDED FROM VL+
-L
A.
f T=Tp
QOOOOOOOOOOO~
F(t)-
RESULTS
AT TIME
T= (Tp + To)/2
COMPRESSIVE
WALL
”
Vi
INITIAL
I
IMPACT
STRESS\
T,,
F(t)-
B) PARTICLE TIME
MOTION
OF ROD DURING
INTERMEDIATE
Tp
TQ
OF ROD AFTER
MOTION
Cl PARTICLE REMOVED
IMPULSE
TENSILE 8. D) ROD AT IMPACT
-
DWELLTIME
RESULTS
AT TIME
T = (TI + TJ2
BEGINS
Fig. 11. Plastic stress responsefor repeatedimpact(a) Resultsat
time T = (T, + T,)/2; (b)Resultsat time T = (T, + T,)/2.
_Q+[;;:;
CONTACT
WITH
WALL
QOOOOOOOOOOO~ CVi
-“L F) ROD REBOUNDS
FROM
1 T>T, +“R
WALL
Fig. 10. Mechanicsof repeatedimpact. Plastic impact stress response curves corresponding to the situations associated with Figs. 10(b) and (e), are depicted in Figs. 11(a) and (b), respectively. The nondimensional initial impact velocity Q/C, corresponds to a value of Q/C, = 1.913~4and a linear hardening value of E,/E, = 0.3 was chosen for the material. The time history sequence of this case, corresponding to the time values schematically shown in Fig. 9, are as follows: T,/T, = 1.05; T,/T, = 1.92; T,lTR = 2.04 and T,/T, = 2.43. In addition, the applied pulse has a value equal to the yield stress of the material. Note that the rod in Fig. 10(b) is in a state of compression throughout its entire length, as can be anticipated. The rod in Fig. 10(e) is, however, in a state of combined tension and compression. The impact stress at the wall (x/L = 1.0) is less than that corresponding to initial impact since it has restruck the wall with a lower average velocity than the initial value of D,,. 4.
The
CONCLUDING RJMARKS
“bouncing” of an elasto-plastic rod from a rigid
wall has been examined. The feasibility of employing finite element-numerical direct time step integration procedures to investigate this problem has been established. The next step in this problem area is the study of the case of a surface with deformable properties. In that case an additional parameter of interest would be depth of penetration and its relationship to the other variables dominating the problem. RFZXRENCES
1. H. Garnet and H. Armen, A variable time step method for determining plastic stress reflections from boundaries. AIAA J. 13(4),532-534(1975). 2. H. Garnet and H. Armen, One dimensional elasto-plastic wave interaction and boundary reflections. Cornput. Struct. S(5). ,. 327-334(1975). 3. W. Prager, A new method of analyzing stress and strains in work-hardening plastic solids. J. ADD/. Mech. 23. 493 11956). 4. H. Garnet and-H. Armen, Eva&ion of numerical time integration methods as applied to elastic-plastic dynamic problems involving wave propagation. Grumman Research Department Report RE-475 (Mar. 1974). 5. R. E. Nickel], Direct integration methods in structural dynamics, 1. Engng Mech. h’u. EM2 Proc. Am. Sot. Civil Engrs. 303-317(1973). 6. G. R. Stoodley and D. J. Ball, Mathematical background of two numerical integration techniques for ordinary differential equations. Grumman Research Department Memorandum RM-192(1961).
Printed in Great Britain
REPEATED IMPACT AND REBOUND OF AN ELASTIC-PLASTIC BAR FROM A RIGID SURFACE? HYMANGARNETt and HARRYARMEN Research Department, Grumman Aerospace Corporation, Bethpage, NY 11714,U.S.A. (Received 15November 1975) Abstract-The repeated impact, or “bouncing,” of an elastic-plastic rod on a rigid surface, is treated by means of finite element analysis. A computationally convenient substructuring technique is utilized in the analytical treatment to account for the changing kinematic constraints. The time dependence is accounted for by means of a variable time step, direct integration procedure that has been successfully applied to a number of elasto-plastic dynamic response problems. This time integration procedure is at present being employed in a nonlinear crashworthiness study. It has been proven capable of providing an accurate prediction of a variety of multidimensional dynamic elastic-plastic responses. Additionally, the procedure has demonstrated its abiity to account for an elastic-plastic dynamic response that includes plastic reflections from boundaries and interaction of plastic wave fronts. The merging of the substructuring technique and numerical time integration procedure has been accomplished. The cases discussed include an unloaded bar traveling at uniform velocity before impact with a rigid barrier, and a bar traveling toward the rigid surface and subject to forces that are applied and removed periodically. Both cases are motivated by questions concerning the elastic-plastic dynamic response of a crashed vehicle. The latter case, which suggests the term “bouncing problem,” has, in addition, applications to problems involving pneumatically driven tools. The response is described, starting with the instant of contact at the impacting end, is followed while elastic/plastic wave propagation extends to the free end, and continues to be treated as elastic/plastic reflections occur and return to tbe impacting end. Finally, the behavior of the rod as it leaves and possibly returns to the wall is discussed, and the dwell time as a function of elastic and elastic-plastic properties of the material is determined.
1.INTRODUCTION
of nodal displacements, [m] is the mass matrix, [k] the elastic stiffness matrix and {F(tfi the vector defining the applied load history. The terms [k] and {AeP}are the initial strain stiffness matrix and incremental plastic strain vector, respectively; their product defines the plastic load vector. In some treatments, the vector of plastic strains, {a~~}),on the right hand side are based on the results of the previous step. In other treatments, eqn (1) is reformulated by relating {AeP}to the “current” value of {AU}, and obtaining a tangent modulus matrix of stiffness influence coefficients that premultiplies {AU}on the left hand side. Results presented herein have been obtained by a procedure equivalent to the tangent modulus formulation. This is demonstrated subsequently in the discussion of the time integration procedure. The elements of the matrices cited above have been exhibited in many earlier papers and textbooks dealii with finite element analysis.
An elastic-plastic unstrained bar traveling at constant velocity impinges on a rigid surface (Fig. 1). Depending on
the impacting velocity, an elastic or a plastic wave is generated in the bar. Compressive in nature, this wave is
propagated to the free end of the bar, where it is reflected as a tensile wave. The reflected tensile wave has the effect of canceling out the compressive wave generated by the impact. If the impacting velocity generates an elastic wave, the bar leaves the wall when the reflected wave front returns to the impacting end. Therefore, in the elastic case, the “dwell time”-the time during which the bar remains in contact with the wall-is equal to twice the time required for an elastic wave to travel the length of the bar. When the impact is plastic, the dwell time can be anticipated to exceed two transit times. Results presented substantiate this expectation, and also indicate that for an elastic-plastic impact the dwell time is a function of the impacting velocity, and the nonlinear hardening properties of the material.
(b) Time dependent partitioning
kinematic
constraints
and matrix
Referring to Fig. 1, we note that the current problem involves a time dependent kinematic constraint. At initial
2. ANALYsls
(a) Equations of motion The incremental equations of motion governing wave propagation in a one dimensional elasto-plastic bar are discussed in [l] and 121and are given in matrix form as A) ACTUAL IMPACT
ROD IN RIGID
BODY
MOTION
PRIOR
TO
in which {AU}is a column vector denoting the increments (1)
tpresented at the Second National Symposium on Computerized StructuraI Analysis and Design at the School of Engineering and Applied Science, George Washington University, Washington, D.C., 29-31 March 1976. SResearch Scientist. OHead,Applied Mechanics Branch.
(2)
(3)
cl000000 +lPi--
(n-1)
(n)
--clakI
El
FINITE
ELEMENT
IDEALIZATION
OF ROD
ha = L)
Fig. 1. Elastic-plastic bar, at constant velocity, uO,prior to impact with rigid surface.
391
392
H. GARNET and H. ARMEN
time, t = fO,the unloaded bar is moving with constant velocity, uO.The right end of the bar, node R, is at a distance, do, from the rigid wall. If we choose as a reference frame a fixed coordinate system, x, coinciding with the bar at to, and displacements of the nodes are designated as u, then the right end of the bar is in contact with the wall when uR = d,. Therefore the time dependent kinematic conditions at impact and during subsequent contact are uR = d,,
tin = 0.
(2)
The dot over the symbol u denotes the time derivative. Therefore, the symbol & designates particle velocity. No conditions are prescribed at node R when it is not in contact with the wall. The contact condition is determined by monitoring the value of uR. In the cases discussed in this paper, where the bar impinges on a rigid wall, the governing equations, in the form given by eqn (1) are probably just as convenient as any other form. But when the bar impinges against a deformable object, matrix partitioning renders the problem more tractable. With the goal of extending the present study to accommodate impact between deformable bodies, eqn (1) is recast in partitioned form as
If the terms on the right hand side are known, then eqns (Sa) and (5b) become easily evaluated algebraic equations for the determination of the increments of acceleration {Ati,},{A&},or more compactly the incremental acceleration vector {Aii} (7) The values of the increments of displacements {Au,}, {Au*}are obtained by means of the direct time integration scheme-discussed in a subsequent section. These increments of displacement are used in the straindisplacement relations to compute increments of total strain. Having the values of total strain, the plasticity relations are used to determine the plastic strain components and thereby construct the plastic load vector. In this manner the terms on the right hand side of eqns (5) are determined. It is noted that eqn (5b) is evaluated first for {A&},which is then used on the right hand side of eqn (5a). (c) Plasticity relations
For simplicity, a bilinear stress-strain law has been assumed (Fig. 2). Because of the nature of the problem the E, = ELASTIC E: = LINEAR
MODULUS HARDENING
MODULUS
(3) where on the right hand side the product [i]{Aq} appearing in eqn (1) has been rewritten, in partitioned form, as the plastic load vector {AP}.The subscripts 1 and 2 refer to nodes free of constraint, and nodes subject to constraint, respectively. In this case Au, consists of a single quantity, namely, the increment of displacement of node R, i.e. Au,. If we perform the indicated multiplications on the submatrices, we are led to the following two systems of matrix equations [m,,lbW
+ [m,,l&Q + [k,,l~AuJ + [k&M = WJ
[m&W+
[WWzI
+ [k&W
+ WJ
(da)
+ [kzlIW = {AF2}t {APz}.
(4b)
Equation (4a) is used to eliminate the {Aii,} term from eqn (4b), leading to the following set of equations [n,J{Aii,X+, = {AF,]i+l + IAPJi+, - [k,,lW,li - [m,,lWiJ,+, - [k,&%
(54
[&]{A&}i+, = {A&},+,+{APz}i+,- [M,lW,Ii+, + ([Msl[kd - [kz,IWJ~
+ ([MsI[M - [k,,lHWi
Ob)
in which
[h&l = [m221- [m2J[mJ’[~~~l [&I = [mzll[mIJ’.
(6)
The subscripts i t 1 and i, included here, correspond to the ith and (i + l)th time steps in the incremental process.
Fig. 2. Bilinear stress-strain relation and hardening rule.
analysis is required to predict unloading from a plastic state and subsequent reloading into the plastic range loading states. Toward this end, a hardening rule, based on Prager’s kinematic hardening rule[3], has been employed. This rule is illustrated schematically in Fig. 2. Note that after impact, and subsequent loading and unloading, the values of the tensile and compressive yield stresses and strains change. Usually, it is the elastic stress range that is kept constant. In the present investigation, the elastic-strain range was kept constant, and, as indicated in Fig. 2, the hardening rule was enforced on the basis of shifting the tensile and compressive yield strains. This approach, while producing results no different from those obtained by keeping the elastic stress range, seems more natural, since strains are a directly observed experimental quantity. (d) Time integration procedure Direct integration methods are usually employed to handle the time dependence in eqn (1). Terms involving derivatives are replaced by various approximating difference expressions, and the resulting recurrence relations are employed in a step forward manner. The more popular of these schemes have been Houbolt’s, the NewmarkBeta, the central difference and Wilson’s method. These
393
Impact and rebound of an elastic-plastic bar
procedures, all employing a constant time step, have been surveyed in a number of publications, including the present authors [4], and by Nickell[S]. A serious drawback is common to all constant time step methods. The acceptability of these procedures depends heavily on the proper choice of the time increment. In physical terms, this means that the dynamic analysis may have to be repeated several times before acceptable results are achieved. For a modest size problem this procedure can be prohibitively costly. This drawback can be eliminated by employing a variable time step method. In the present paper the authors employ a previously developed version of the modified Adams metbod[6]. The procedure has been employed earlier by the present authors, to account for one and two dimensional elasto-plastic wave propagation, including those cases involving waves reflected from boundaries[l, 2,7]. The method employs Taylor series expansions to obtain predictor and corrector expressions, truncated to difference form, for the solution to a system of first order nonlinear differential equations. The corresponding system of second order equations of motion (eqn l), is employed to construct the corrector expression. The procedure is briefly outlined here and described in more detail in [6]. If the unpartitioned system of equations (eqn l), is of order n, a column vector of order 2n is defined as
used to determine the corrector solution
Note that because of the operations carried out in eqns (7) and (13), {j},,, is known. Therefore, all quantities on the right are known and the corrector expression may be evaluated. Comparison of {y}r+,, {y}:, (displacements and velocities) determines whether or not the results are acceptable. Depending on the error tolerance selected, one of the following situations may exist: (1) the result is acceptable, and the time integration may continue to the next time interval ti+*;(2) the result is unacceptable and the time interval, At, is halved; and (3) the agreement exceeds accuracy requirements, the result is acceptable and forward integration proceeds at double the interval, At, i.e. 2At. Because the plastic strains, which contribute to the plastic load vector, eqn (l), or eqn (5), are computed from displacement based on the current time step (i.e. the predictor solution), the solution procedure is equivalent to the tangent modulus method. (e) Stress at impact
The means for computing the stress generated in the rod at the instant of impact will now be derived. In Refs. [ 1 and 21 an expression was developed for the momentumimpulse balance of an element of the rod. Because of its simplicity the derivation will be repeated here. Figure 3
{Yl=[f]
{j}=(j). Predictor and corrector expressions are employed as follows. A predicted value is generated from the following expression
Fig. 3. Element of rod includinginertia forces.
{Y]r+l={Y}i +At{jJi +$(IiL
-UJi-l).
(10)
The subscripts correspond to the ith time step in the incremental process. Note that at time ti, all quantities on the right hand side are known. Also {Y};+l= [;),+,.
(11)
depicts a free body diagram of an element of the rod, including inertia forces. The rod is of constant cross section A, and is assumed to possess mass density p. The symbols u and u denote stress level and particle velocity level, respectively. The condition of momentum impulse balance for the element may be obtained from D’Lambert’s principle and written as (p dx) du = (da)(dt).
Next, eqns (5) are employed to obtain the increments of acceleration, {Ati},,,. Toward this end, the upper block of the predictor solution, eqn (lo), is used to determine the vector {Au}~appearing in eqns (5) {Au}~= {u}:+,-{u}:.
(12)
The increments of acceleration and velocity are then summed to determined the total acceleration of the system. Before using the corrector expression, the following vector is generated
where the upper half of {j} is obtained from the lower matrix block in eqn (ll), and the lower half is obtained from the solution to eqn (5). The following expression is
(15)
If c is the speed of wave propagation during the time interval, dt, then dx = c dt and eqn (15) becomes do =$.
(16)
Equation (16), which in essence is an expression of momentum-impulse balance between change in particle velocity and stress level, will be employed to determine the stress intensity at impact. If the unloaded rod is traveling at constant velocity, v,,, prior to impact, then its valocity immediately after impact, ul, is obtained by using eqn (16), and enforcing the kinematic constraint u = 0 at impact. Then, integrating eqn (16), we obtain (17)
394
H.
GARNETand
where the symbols c, and c, denote wave speeds based on the elastic and plastic portion of the bilinear stress-strain curve, respectively, i.e.
H.
ARMEN
3,
Go
3,
I? V
R A) ROD PRIOR
G = V%%lp), c,, = V\/(E,/p).
3,
+ooooo~ TO IMPACT
(18)
The integration on the right consists of two terms to account for the eventuality that the intensity of a, may exceed o,, the yield stress. After integrating eqn (17), dividing both sides of the resulting equation by the elastic modulus, E,, and using eqn (18), the following nondimensional expression is obtained for the impact stress
VR Bl ROD AT
7
0
IMPACT
Go
3,3,
$
0
REFLECTED COMPRESSIVE WAVE FRONT
3, 0
010 4
C) CONTINUED
0
v=ov=o
CONTACT
WITH
WALL
in which e, is the yield strain +-d e,
REFLECTED
COMPRESSIVE
WAVE
FRONT
I
ES,
4
Et
IV
In the elastic case, EJE, = 1. and eqn (19) becomes
0
0
=“o
0) ROD COMPLETELY IN CONTACT WITH
0
0
v=o AT RESl WALL REFLECTED TENSILE WAVE FRONT k
Qooooo;
That this is the proper limiting value can be verified by recalling that solutions to the case of one dimensional elastic wave propagation are of the form u = f(x 2 CJ) where f is an arbitrary function. By computing v = du/dt, and (T= au/ax, relations are obtained that result in eqn (21). It follows from eqn (19) that the impact will be elastic when 3< e,. C,
(22)
Because the response is nonlinear, and path dependent, the derivation of analytical expressions similar to eqn (19) by means of the method of eqn (16) becomes increasingly complex as time progresses. Therefore no further attempts to present such expressions to account for the many possible interactions that occur will be made. Their primary purpose in this paper, as well as in [I], and [2], has been to validate a numerical procedure designed to cope with a multi-dimensional complex nonlinear response. 3.
DISCUSSIONOF RESULTS
The mechanics of the elastic case are discussed first. Figure 4 depicts several stages in the elastic impact and rebound process. Figure 4(a) shows the rod in rigid body motion before impact. All nodes are moving with constant velocity, va. In Fig. 4(b), the rod is shown at the instant of impact; node R, at the wall, has zero velocity. All other nodes are still moving to the right with velocity, v,,.At this instant an elastic compressive wave is generated at the impacting end. This wave propagates along the rod, to the left, with wave speed c,, and particle velocity, vO.In a compressive wave, the particle velocity is in the same direction as the propagation of the wave. Figure 4c represents conditions when the wave has not yet reached the free end. The particle velocity associated with the compressive wave has served to bring to a standstill those nodes passed by the wave front, i.e. that part of the rod to the right of the wave front is at rest (when the wave front
*v,/v=o
--Vcl E) CONTINUED
CONTACT
WITH
v-o’ WALL
~00000Q
-vo F) ROD
REBOUNDS
tV, FROM
R WALL
Fig.4. Mechanicsof initialelasticimpact. reaches the free end all particles are at rest). The one dimensional compressional elastic wave will be reflected from the free end as a tensile wave of equal numerical magnitude. This tensile wave will propagate to the right. However, the particle velocity associated with a tensile wave is in a direction opposite to the motion of the wave front. The reflected tensile wave will accomplish two things. First, it will cancel the compressive impacting wave, returning the bar to a stress free state. Additionally, as it progresses to the right, it will impart a particle velocity, of magnitude v,,, to each node-in a direction opposite to the original motion. Figure 4(d) depicts this situation. When the reflected wave reaches the wall, the bar will be stress-free and all particles of the bar will be moving to the left with velocity v,,.The bar then leaves the wall, in a direction opposite to its incident motion, with velocity of magnitude vO(Fig. 4e). Some typical stress response results are shown for an elastic case (Fig. 5). In this and all subsequent figures involving stress response, the ordinates represent the nondimensional stress values u/E,, while abscissas are nondimensional positions, x/L, where L is the length of the bar and x is measured from the left end of the bar. A nondimensional time parameter t is employed to denote the position of the elastic wave front resulting from the impact. A value of t = 1 corresponds to the time required for an elastic wave to undergo a single transit of the bar. Figure 5(a) shows the stress response in the bar before the compressive impacting wave has reached the free end.
395
Impact and rebound of an elastic-plastic bar b
* COMPUTED VALUES THEORETICAL VALUES
-
x ^Cl0 d r
THEORETICAL
IMPACT
STRESS\
.
w
9
.
US
E 1=6
POSITION OF ELASTIC COMPRESSIVE WAVE FRONT
.
f
p O!&O 0.125
B
20
0.000
0.125
0.250
0.375
0.500 X/L
0.625
0.750
0.875
0.260
0.375
0.600
0.626
0.750 0.875
A. COMPRESSIVE WAVE FRONT HAS NOT YET REACHED FREE END IT = 0.6231
l
x
THEORETICAL
5r
COMPUTED IMPACT
VALUES
STRESS
3
. COMPUTED VALUES THEORETICAL VALUES
b c
i$ 0~000’0.125 ;/ ;.
IMPACT
STRESS
$4-
0.250 .~. 0.375 ;b!iy:._.; 0.500
0.625
ki 3-
0.750
0.875 1.000
k < 5
2-
g
l-
INTERACTIVE
STRESS
u* 0.125
:
..
I 0.250
. --
_~
. *
LEVEL
.** +
z EO, 5 0.000 z
i
LOCATION OF REFLECTED TENSILE WAVE FRONT
Lu
w THEORETICAL
1.000
X/L A. COMPRESSIVE WAVE FRONT HAS NOT YE7 REACHED FREE END (T-0.623)
1.000
* 9. .
LOCATION OF COMPRESSIVE PLASTIC WAVE FRONT I 0.375
I 0.500
I 0.625
I 0.750
I I 0.875 ,.OdO
X/L
8. REFLECTIONS FROM FREE END HAVE OCCURRED (i = 1.780)
“0 X/L 8. REFLECTIONS (T = 1.420) Fii.
FROM FREE END HAVE OCCURRED
5.
Elastic impact stress response (a) Compressive wave front has not yet reached the free end; (b)Reflectionsfromthe end have occurred.
Solid lines denote theoretical values. Plotted points are those obtained by means of the computer program. Note that the sharp wave front is approximated by a ramp. As previously discussed in [ 1,2,71 the steepness of the ramp may be increased by refining the finite element grid. The presence of the ramp approximation results from the numerical process and cannot be eliminated entirely. Figure S(b) shows the response after the compressive wave has reflected as a tensile wave. Theoretically, the region to the left of the tensile wave front is stress free; numerically, values of an order of magnitude lower than the impacting stress are obtained. Plastic impact results when the magnitude of the incident velocity exceeds the absolute value of the product of the yield strain and the elastic wave speed, i.e. when uO/c,> e,. The ensuing plastic stress level, al, is given by eqn (19). Figure 6(a) depicts a typical stress response to a plastic impact. Shown are conditions prior to the arrival of the compressive wave front at the free end. During plastic impact two wave fronts are generated. One travels at yield stress intensity, and elastic wave speed, c, (eqn I@, and the other travels at the slower plastic wave speed, c,, (eqn 18). This phenomenon was discussed in an early paper by Donnell@], and also more recently by the present authors [21.
x ^y.5 c‘ % k4 t;
YIELD
STRESS
i3
O.ooO 0.125
0.250
0.375 0.500 0.625 0.750 0.875 X/L C. BAR HAS JUST LEFT WALL (i = 2.102)
1.000
Fig. 6. Plastic impact stress response (a) Compressive wave front has not yet reached the free end; (b) Reflections from the end have occurred;(c) Bar has just left wall.
The mechanics of the rod subsequent to plastic impact differ in several respects from those of elastic impact, already discussed. In both cases, the particles to the left of the elastic wave front are still moving to the right with velocity uO.However, for the plastic case, in the region of the rod passed by the elastic wave front, and not yet reached by the elastic wave front, the particles have been slowed down. In the corresponding elastic case, the particle velocity of the generated elastic wave is equal in magnitude to the incident velocity. Therefore, the arrival of the elastic wave front brings a particle to rest. During plastic impact, the magnitude of the particle velocity, associated with the elastic wave front, II,, is less than the incident velocity uO.This can be seen by referring to eqns
3%
H.
GARNET
and H. ARMEN
(19) and (22), i.e. u, < uO. The diminished particle velocities have the magnitude uO- u,, and in the case shown, they are still moving to the right. Finally, we note that the particle velocities associated with stress levels equal to u, must have magnitude uO.This follows from the manner in which the value u, was obtained in eqn (19) i.e. stopping the bar at the rigid wall, that has the effect of producing a particle velocity equal in magnitude and opposite in direction to the incident velocity. The region traversed by the plastic wave front has therefore been brought to rest. Several observations are now made concerning the reason for the increase in dwell time after plastic impact. Since plastic deformation has taken place, only the elastic portion of the total strain is recoverable. The bar will not leave the wall until this has occurred. And because the value of the recoverable elastic strain, during plastic impact, exceeds the corresponding values of strain during the elastic case, a greater time must elapse during this recovery. Therefore a greater dwell time results. The action following plastic impact is complex. Figure 6a depicts an early stage of the process in which the elastic and plastic wave fronts of the impact are clearly shown. Further stages involve the passage of the compressive elastic wave front (at yield stress) to the free end, where it is reflected as a tensile wave with yield stress magnitude. During the process, the plastic wave front, traveling at a speed lower than the elastic wave front, progresses toward the free end. Interaction takes place with the returning tensile wave and the interactive stress level associated with this state is shown in Fig. 6(b). The reflected tensile wave eventually reaches those portions of the bar that have previously been brought to rest by the progressing plastic wave front. Several reflections may be necessary to bring about the required change in particle velocity necessary for motion sufficient to release the bar from the wall. Figure 6(c) depicts the bar shortly after it has left the wall. Note that although the bar is still partially stressed, all particle motions are to the left. The magnitudes of these particle velocities are indicated on the figure. AS expected, the bar is leaving the wall with a velocity of magnitude lower than that prescribed at impact. The dwell time for a given impact velocity appears to be a function of the hardening properties of the material. In Fig. 7, dwell time is plotted against impact velocity for
two different materials. One, for which E,/E,= 0.1, and a second for which E,/E, = 0.3. It is seen that longer dwell times occur for the material having the ratio EJE, = 0.I (more plastic). Note that if the impact velocity is high enough, large particle velocities and displacements can be induced, so that the bar does not leave the wall. This expectation of this phenomenon is substantiated by the results shown in Fig. 8. This case deals with an elastic-perfectly plastic material, in which infinite strains may be induced. The dwell time is seen to rise rapidly with increasing values of u,Jc,. The computations led to a description of nodes passing through each other for the values shown. (Of course, the theory is not applicable then.) The time history sequence for a repeated impact problem is shown in Fig. 9. Here a rectangular pulse.
THEORY FAILS (LARGE DEFORMATIONS, NODES CROSS)
Fig. 8. Dwell time versus impact velocity for elastic-ideally plastic case.
To
o’ 1
2
3
TR
TP
To
TI
TS
4
Fig. 7. Dwell time versus impact velocity for two ratios of IL/E,.
Fig. 9. Time history sequence during repeated impact.
397
Impactand reboundof an elastic-plasticbar whose magnitude is equal to the yield stress of the bar, is applied at some time, T,, after the bar has rebounded from the wall after its initial impact. The pulse remains on the free end of the rod for some arbitrarily prescribed period of time, (To - Tp). If the pulse is of sufficient magnitude and is applied for a sufficient period of time, the rod will reverse its direction and be propelled back to the rigid surface at a velocity different from its initial impact value, uO.The mechanics of this sequence of events, including’its subsequent rebound off the wall, is schematically depicted in Fig. 10.
APPLIED
IMPULSE,
F(t) = YIELDSTRESS . ... .. . ... ....
..
*
..**** : .** I
I
5
,125
I
,250
,375
I
,500
1.
I
I
.625
,750
,875
1.0
X/L L
-
C
“L
Vi
-VR
R
A) IMPULSE APPLIED AT FREE END AFTER ROD HAS REBOUNDED FROM VL+
-L
A.
f T=Tp
QOOOOOOOOOOO~
F(t)-
RESULTS
AT TIME
T= (Tp + To)/2
COMPRESSIVE
WALL
”
Vi
INITIAL
I
IMPACT
STRESS\
T,,
F(t)-
B) PARTICLE TIME
MOTION
OF ROD DURING
INTERMEDIATE
Tp
TQ
OF ROD AFTER
MOTION
Cl PARTICLE REMOVED
IMPULSE
TENSILE 8. D) ROD AT IMPACT
-
DWELLTIME
RESULTS
AT TIME
T = (TI + TJ2
BEGINS
Fig. 11. Plastic stress responsefor repeatedimpact(a) Resultsat
time T = (T, + T,)/2; (b)Resultsat time T = (T, + T,)/2.
_Q+[;;:;
CONTACT
WITH
WALL
QOOOOOOOOOOO~ CVi
-“L F) ROD REBOUNDS
FROM
1 T>T, +“R
WALL
Fig. 10. Mechanicsof repeatedimpact. Plastic impact stress response curves corresponding to the situations associated with Figs. 10(b) and (e), are depicted in Figs. 11(a) and (b), respectively. The nondimensional initial impact velocity Q/C, corresponds to a value of Q/C, = 1.913~4and a linear hardening value of E,/E, = 0.3 was chosen for the material. The time history sequence of this case, corresponding to the time values schematically shown in Fig. 9, are as follows: T,/T, = 1.05; T,/T, = 1.92; T,lTR = 2.04 and T,/T, = 2.43. In addition, the applied pulse has a value equal to the yield stress of the material. Note that the rod in Fig. 10(b) is in a state of compression throughout its entire length, as can be anticipated. The rod in Fig. 10(e) is, however, in a state of combined tension and compression. The impact stress at the wall (x/L = 1.0) is less than that corresponding to initial impact since it has restruck the wall with a lower average velocity than the initial value of D,,. 4.
The
CONCLUDING RJMARKS
“bouncing” of an elasto-plastic rod from a rigid
wall has been examined. The feasibility of employing finite element-numerical direct time step integration procedures to investigate this problem has been established. The next step in this problem area is the study of the case of a surface with deformable properties. In that case an additional parameter of interest would be depth of penetration and its relationship to the other variables dominating the problem. RFZXRENCES
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