HAL Id: jpa-00226662
https://hal.archives-ouvertes.fr/jpa-00226662
Submitted on 1 Jan 1987
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
IMPACT ANALYSIS OF LATTICES WITH CLEAN SURFACES
J. Tasi
To cite this version:
J. Tasi. IMPACT ANALYSIS OF LATTICES WITH CLEAN SURFACES. Journal de Physique Colloques, 1987, 48 (C4), pp.C4-315-C4-323. �10.1051/jphyscol:1987423�. �jpa-00226662�
JOURNAL DE PHYSIQUE
Colloque ~ 4 , suppldment au n a g , Tome 48, septembre 1987
IMPACT ANALYSIS OF LATTICES WITH CLEAN SURFACES
J. TASI
Department of MechanicaZ Engineering, State University of New York at Stony Brook, Stony Brook, New York 11794, U.S.A.
Rksum6.- L'ktude porte sur l'effet d'impact normal sur deux rkseaux cristallins identiques. On suppose que le r6seau cristallin qui produit l'impact avance ?une vitesse initiale uniforme, frappant un i
reseau cristallin qui au dkbut est sans mouvement. En l'absence de toutes impuretks de surface, des forces de traction de longue portke produisent entre les deux surfaces un effet d'adhe'sion des deux rkseaux cristallins, et il est montrk qu'il attire le rkseau cristallin, qui au dkbut est sans mouvement, vers le rkseau cristallin qui frappe avec des vitesses de particule de surface bien plus grandes que la vitesse de l'impact. Avec la venue de la rkgion compressive de l'impact, le rkseau cristallin frappk invertit sa direction et oscille autour de la velocitk attendue selon la thkorie classique du mouvement continu. Dans les deux rkgions-temps on obtient des solutions analytiques d'asymptote de forme simple pour dkcrire le mouvement de surface. Nous illustrons des solutions particulikres pour l'interaction Morse entre les surfaces d'impact.
Abstract, A study is made of noxmal impact of two identical lattices. The incoming lattice is assumed to move with uniform initial speed, impacting an initially motionless lattice. In the absence of any surface impurities, long range tensile forces between the surfaces result in adhesion of the two lattices, and it is shown that they draw the initially motionless lattice toward the im~acting lattice. with surface particle velocities of much greater magnitude than impact speed. With the ^subse&ent &ent of comvressive region of imuact, the imvacted lattice reverses its motion and oscillates about the veloci6 predicted-by classiial continuum theory. In both time regions, asymptotic analytical solutions of a simple form are obtained to describe surface motion. Specific solutions are illustrated for Morse interaction between impact surfaces.
1. Introduction
The problem of normal impact response of two identical perfect lattices has been studied in Refs. [I]-[5]. As first outlined by Tsai [2], calculations in impact studies have assumed an incoming lattice moves with uniform initial speed, with no interaction allowed between the two lattices until they are within an equilibrium distance between lattice planes. Then, compressive interaction occurs between the two impact surfaces and elastic waves propagate into the interiors of the two lattices.
The restriction of allowing only compressive interaction omits long range tensile forces that exist between lattices having clean surfaces. In the absence of any surface impurities, tensile forces between the two surfaces will result in adhesion of the two lattices and have a marked effect on the nature of the dynamic response. In this paper, a lattice moving with uniform initial speed is assumed to impact an initially motionless lattice having identical physical properties. It is shown that before the compressive region of impact occurs, tensile forces draw the initially motionless surface toward the impacting lattice, achieving particle velocities noticeably greater than impact velocity. With the advent of compressive interaction, the surface of the impacted lattice reverses its motion and oscillates about the velocity predicted by classical continuum theory.
Although the response to a general interaction force at the surface is more complicated than for only compressive impact response, it is shown that asymptotic analytical solutions of a simple form can be obtained for the general time dependent impact response at absolute zero degrees. Specific solutions are illustrated for Morse interaction between impact surfaces, and comparisons are made between numerical integrations and the asymptotic approximations.
2. Governing Equations for Clean Surfaces
Zwanzig [6] has shown that the problem of collision of a gas atom with a one-dimensional lattice at absolute zero degrees is governed by a differential-integra1 equation for the relative displacement between an incident atom and the first atom in the lattice chain. This has allowed some analytical determinations of conditions under which an adsorbtion of a gas atom would occur. A
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987423
JOURNAL DE PHYSIQUE
similar equation has been shown [4] to govern the time -dependent seperation distance between surfaces during impact of one lattice on an identical lattice that is initially at rest. The physical configuration is illustrated in Figure 1.
Fig.1. Normal impact of two identical one-dimensional lattices
Dimensionless variables are defined as in [4]. Displacement un of the nth atom, and distances, are considered dimensionless with respect to the original equilibrium spacing between atoms, and dimensionless time t is defined so that a unit of time is the amount required for a disturbance to propagate from one atom to its neighbor at Newtonian sound speed The
dimensionle'ss seperation distance r1,0 between atoms at the impact surfaces is then related to the compressive strain u(t) between those atoms by u = -(r-l,O-l) . The equation governing u(t) is [4]
t
0
where J2 is a Bessel function of the first kind of order 2 , and F(u) is an arbitrary interaction force between surface atoms of the two lattices.
In the derivation of Eq.(l), the strains within each lattice were linearized in order to analytically relate the time-dependent propagation of elastic waves into the interior of the lattices to F(u) , yielding the convolution integral shown. This was shown in [4] to affect the time phase of surface response, but not amplitudes. The only restriction on F(u) is that whatever its long range nature in the tensile region, its magnitude must diminish to an insignificant value when the separation distance between the two lattices is large. This enables one to define a cut-off distance rc at which impact starts, giving initial conditions for (1) as
~ ( 0 ) = -(rc-1) , du dt\O) = vi , (2) where vi is the impact velocity (relative to the Newtonian sound speed) of the incoming lattice.
Once Eq.(l) is integrated subjected to (2), the surface velocity response at n = 0 is
determined. The dimensionless particle velocity at any nth point in the impacted lattice is denoted by 1 dun
v (t) =--
n v H d t ' " ' O ( 3 )
in which vH = vi/2 is the particle velocity that would arise from a classical continuum analysis of the compressive impact problem: One can calculate the surface velocity vo of the impacted lattice from
3. General Solutions
The Laplace transform pair for u(t) is defined by
00
L(u) = j u(t)exp(-pt)dt , Re p > 0 , 0
c - i -
Application of the Laplace transform to (I), using (2), gives
c-i- or
0
Equation (8) is an implicit relation between duldt and F(u) , and, it or (I), can be used numerically to find u(t) . In order to obtain analytical approximations, we shall look for solutions of (7) in two distinct time regions where the character of the impact solutions differ. Region (i), 0 5 t I to , is defined to be the time period for exclusively tensile interaction. This region ends when the incoming lattice arrives at exactly an equilibrium spacing from the impacted lattice, at which point u = 0 and t to. Region (ii), defined by t >to , consists mostly but not exclusively of compressive interaction. In this second time region, the initiation of compressive resistance between the lattices causes an oscillatory behavior in the neighborhood of equilibrium spacing. Both compressive and tensile interactions alternately arise in this region, with the two lattices adhering to each other.
Region B) Analvsis : 0 -< t < t 0 '
For a long-time solution of (7), we let p 0. Then,
which is appropriate for a nonoscillatory response, with the result
du dF
- dt - vi + 2F[u(t)] - + 0(d2~/dt2) .
With
Eq. (10) gives an asymptotic differential equation governing u(t) as
Integrating (1 1) by separation of variables, we have
U
-(rc-l)
which can be explicitly integrated once an interaction potential is specified for F(u) . The time to required for the incoming lattice to first arrive at an equilibrium distance (u = 0) from the impacted lattice is then calculated from
o
C4-318 JOURNAL DE PHYSIQUE
The omission of o($) in Eq.(9), resulting in the omission of d2Fldt2 in Eq.(lO),
becomes inappropriate for the evaluation of duldt as u approaches zero. When the spacing between the two lattices approaches the equilibrium spacing, then t - tO, F(u) 4 -u and dF/du - -1 ,
and Eq. (11) gives du/dt approaches infinity. Thus, to given by (13) also defines a "breakdown"
time at which a solution ignoring the oscillatory contributions of higher order rates of change of F with respect to time become inadequate. If one were to consider the next term in the expansion of (9), we would have
with (10) replaced by
- vi + 2F[u(t)] - + $ d2Fldt2 + 0(d4~/dt4) . dt
Using
d 2 ~ dF(u) d2u d2F du
-- --
dt2 - du dt2 + z(a?)z ' (16)
we can replace Q . ( l l ) by a second-order nonlinear differential equation governing u(t), given by
The coefficient of the highest-order derivative in (17) is zero at dF/du = 0 and the equation becomes singular. The inclusion of the second-order derivative in (17) makes it difficult to obtain a solution that would be more accurate than (12), but it does enable us to calculate, explicitly, the
response when the tensile force F(t) reaches its maximum value. At the maximum, we denote t = t , u = u m with dF/du = 0, and tm < to. Then, Eq. (17) may be solved directly for duldt. That solution, combined with (4), gives the surface velocity of the impacted lattice at tm to be
1 - 1 - 2(F(u) + vi/2) d 2 ~ / d u 2 112
vo(tm) = 1 - 2 i ' vid2F/du2 I (18)
This is easily evaluated once a potential is specified for F(u). For example, the force given by the Morse potential is
F(u) = - 1 [exp (AMu) - exp (2AMu)] ,
AM (19)
with the physical constant AM available for many metals (7). This is illustrated in Fig.2 for Copper.
Fig. 2. - Morse interaction force between copper atoms, AM = 2.45.
The maximum F(u) occurs at
um = - (ln2)/AM, (20) giving
vo(tm) = 1 - =[(20 M i 1 + 8 ~ ~ v ~ ) l / ~ - 4 I . (21)
For weak shocks AMvi << 1, and Eq. (21) illustrates a large negative surface velocity of the impacted lattice. Although the anival t i e s of the maximum force and largest negative velocity differ, the difference is not great. Therefore, the explicit value for vo(tm) that can be obtained from (18) is a close approximation of the greatest magnitude of negative surface velocity that can be achieved during the tensile part of impact response of 1attices.The omission of higher-order derivatives of F[u(t)] in the asymptotic expansion of (8) is not acceptable in the oscillatory region of behavior that occurs when t > 6. Therefore, a different analytical approach is required in Region (ii).
Repion (ii) Analvsis : t > to
For t > to , compression of the impact surfaces takes place, thereby generating lattice oscillations in the neighborhood of an equilibrium distance between the two lattices. Since motion now occurs in the neighborhood of an equilibrium spacing, a linearized force interaction between the two lattices is a reasonable fust approximation. This is consistent with having used a linearized force relation for the interior atoms of the two lattices during the derivation of Eq. (1) in Reference 4. As a result, we let
i FT(u) t 2 to
F(u)= - u t > f 0 (22)
in Eq. (9), where FT (u) refers to the tensile interaction force that occurred in time Region (i).
Equation ( 9) then becomes
0 to
Introduce the change of variables
t' = t - t 0 ' (244
y ' = y - t
0 ' (24b)
where t' is a time variable that starts when Region (ii) starts. Using (24) in (23), we have +'
with
-- J1 [ 2 ( t ' - ~ '11
du vi + Y(t,) - 2
dt' - (,,-,,) u(y')dy' ,
0
Since t(u) is known from Eq. (12) for 0 < t I to , its inverse gives u(t), with F,[u(y)l and Y(t') then known once a potential is specified for ~ ( u ) .
Equation (25) is an inhomogenwus linear fust order differential-integral equation governing u and can be solved by applying a Laplace transform with respect to t'. Let
00
L'(u) = J u(t')exp(-pt')dt' , (27)
C4-320 JOURNAL DE PHYSIQUE
By definition of to , we have u(t' = 0) = 0. Application of the Laplace transform to (25), with the use of Theorem I1 in Ref. [8], gives
v.
L, (g,) = 1 pL'[YO')I
2 ( l t 4 ; y n ' 2 ( 1 +gy" -
The terms in (29) are inverted by the identities
where 6(t') is the Dirac 6 -function. The inverse transform solution of (29) is then explicitly available, with the final result for vO(t') in Region (ii) given by
t'
0
The unit term in (31) corresponds to the classical unit jump in velocity vn . n 2 0, that
would propagate throughout a solid in a continuum approximation. The second term in (31) provides the correction for a lattice model when the impact interaction between the two solids is solely compressive (i.e., a half-harmonic oscillator approximation) and was obtained in Reference 4. When an attractive force is allowed to exist prior to the compression region in time, the surface response is generalized to Equation (31). The two perfect semi-infinite lattices have then combined to form an infinite lattice, with the solutions in Regions (i) and (ii) governing the dynamics of adhesion of the two Lattices.
4. Example for Morse Potential
The general impact solutions of Section 3 for the two time regions are illustrated for a Morse potential between surface particles . For F(u) provided by (19), the integral in (12) for Region (i) gives the relation between t and u as
with
112
[(I + 2AMvi) -1 + 2A M v.] 1
C1 =
1 + 2AMVi)li2 +1 + 2 ~ ' ~ ~ ~ 1
For any chosen value of u , the time required to achieve the response is given by (32) and combines with
from Eq.(l I), to complete the lowest order asymptotic approximation. The time of duration to for Region (i) is given as a function of A d i and A d c by setting u = 0 in (32). Similarly, tm is found by substitution of (20) into (32), giving the time of arrival of the maximum force.
6 5 4 3 2 1 0 0
3 -I-'
-0 1- -1
-I >I -2
11 -3
> 0 -4
-5 -6 -7 -8 -9 -10 -11
0 25 50 75 100 125 150 175 200
TlME , t
Fig. 3. - Asymptotic solutions for vo , with AM = 2.45.
v. = 0.02 , rc = 4.0
I
6 5 4 3 2 1
0 0
3 4 -0 10 -1
-I P -2
11 -3
> 0 -4
-5 -6 -7 -8 -9 -10
-11 0 25 50 75 100 125 150 175 200
TlME , t
Fig. 4. - Numerical solution of Eq. (1) for vo , with AM = 2.45,
V. = 0.02 , rc = 4.0
JOURNAL DE PHYSIQUE
0.10 0.08 0.06 0.04
A A 0.02 LI
0.00 -0.02
-0.06 4
-0.08
0 25 50 75 100 125 150 175 200
TlME , t
Fig. 5. - Asymptotic solutions for F(t) with AM = 2.45 ,
v. = 0.02 , r = 4.0
I C
-0.08 ' I I I I 1
0 25 50 75 100 125 150 175 200
TlME , t
Fig. 6. - Numerical solution of Eq. (1) for F(t) , with AM = 2.45, v; = 0.02 , rc = 4.0
Although the asymptotic relation between u and dddt given by (1 1) breaks down at u = 0, the relation between t and u , and hence between t and F , is well behaved. With u(t) known, Eqs. (8) and (4) provide an alternate means of calculating vo(t) for Region (i) by
t
0
The integral given by (35) was numerically evaluated using Simpson's rule with equidistantly tabulated values of F[u(y)]. Numerical values of u(t) at equidistant values of time between 0 and to were found from (32) using Newton-Raphson iteration. The initial approximation for u(t) in the iteration was chosen as u at the previous time value, starting from u(t = 0) = -(rc -1). With a time increment chosen as 0.05, the Newton-Raphson method required'only one iteration at each new time; except as time approaches to , in which case at most two iterations are required to determine u at each new increased value of time.
Solutions for vo(t) found from (35) are illustrated in Fig. 3 by the solid curve line and continued into Region (ii) by using Eq. (31). The dashed line in Fig. 3 represents vo(t) from (34), with its breakdown clearly seen. In addition, the solution of Eq. (1) was found numehcally and is illustrated in Fig. 4, in order to provide a comparison with the asymptotic shock results for surface velocity. The corresponding results for F(t) in the two time regions are given in Fig. (5) for
asymptotic response and Fig. 6 from numerical integration of Eq.(l). The overall asymptotic profiles show a good correspondence with solutions of Eq.(l), with the exception of the limiting oscillations as time grows greatly beyond to. This difference is due to the linearization of the Morse force in the compression region and has animportant long time effect. The limiting solution for vo(t) as t approaches infinity is 2 for the complete Morse force and is a unit value when the interaction force is linearized in the compression region of response .
Specific comparisons in the vicinity of breakdown time are given in Table I for A,, = 2.45 ,
A,.
where,in addition to the terms previously defined, solutions at the time of minimum negative velocity are denoted by tmv. The value of v(tm) given by Eq. (21) is found to be -9.53, and provides a good lower bound for the amplitude of the large negative velocity.
Asvmutotic Eq.(l)
Table I. Comparison of asymptotic solutions and numerically exact results of Eq.(l) v. = 0.02 , r = 4.0 , A = 2.45
c M
[I] D. H. Tsai and C.W Beckett, in Behavior of Dense Media Under High Dynamic Pressures (Gordon and Breach, 1968), p. 99.
[2] D. H. Tsai, in Accurate Characterization of the High-Pressure Environment, edited by E. C. Lloyd (Natl. Bur. Stand.Spec. Pub. 326, 1971), p. 105
[3] D. H. Tsai and R. A. Mac Donald, High Temp. High Pressures 8,403 (1976).
[4] J. Tasi, J. Appl. Phys. 51, 5804 (1980).
[5] B. L. Holian, H. Flaschka, and D. W. McLaughlin, Phys. Rev. A 24, 2595 (1981).
[6] R. W. Zwanzig, J. Chem. Phys. 32, 1173 (1960).
[7] L. A. Girifalco and V. G. Weizer, Phys. Rev. 114, 687 (1959).
[8] H. S. Carslaw and J. C. Jaeger, Operational Methods in Applied Mathematics (Dover, 1963).
