AFWL-TR-68-46 AFWL-TR- 68-46 EXPONENTIALLY DECAYING PRESSURE PULSE MOVING WITH SUPERSEISMIC VELOCITY ON THE SURFACE OF A HALF SPACE OF VON MISES ELASTO-PLASTIC MATERIAL Hans H. Bleich Alva Matthews Paul Weidlinger, Consulting Engineer _________ New York, New York 10017 : Contract F29601-67-C-0091 TECHNICAL REPORT NO. AFWL-TR-68-46 August 1968 A!R FORCE WEAPONS LABORATORY Research and Technology Division Air Force Systems Command Kirtland Air Force Base New Mexico This document is subject to special export controls and each transmittal to foreign goverp n, or foreign nationals may be made only with prior approval of AFWL i , /Artland AFB, Nil, 87117. - _ _ _ _ _ __ _ _ _ _9
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EXPONENTIALLY DECAYING PRESSURE PULSE ...1 Step Pressure moving over Half-Space 2 2 Decaying Pressure moving over Half-Space 2 3 Configuration for Step Pressure p0 > PL 5 4 Parameters
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AFWL-TR-68-46 AFWL-TR-68-46
EXPONENTIALLY DECAYING PRESSURE PULSE
MOVING WITH SUPERSEISMIC VELOCITY
ON THE SURFACE OF A HALF SPACE OF
VON MISES ELASTO-PLASTIC MATERIAL
Hans H. Bleich Alva Matthews
Paul Weidlinger, Consulting Engineer
_________ New York, New York 10017 :
Contract F29601-67-C-0091
TECHNICAL REPORT NO. AFWL-TR-68-46
August 1968
A!R FORCE WEAPONS LABORATORY
Research and Technology DivisionAir Force Systems Command
Kirtland Air Force BaseNew Mexico
This document is subject to special export controls and each transmittalto foreign goverp n, or foreign nationals may be made only with priorapproval of AFWL i , /Artland AFB, Nil, 87117.
- _ _ _ _ _ __ _ _ _ _9
AFWL-TR-68-46
EXPONENTIALLY DECAYING PRESSURE PULSE
MOVING WITH SUPERSEISMIC VELOCITY
ON THE SURFACE OF A HALF SPACE
OF VON MISES ELASTO-PLASTIC MATERIAL
Hans H. Bleich Alva Matthews
Paul Weidlinger, Consulting Engineer
New York, New York 10017
Contract F29601-67-C-0091
TECHNICAL REPORT NO. AFWL-TR-68-46
This document is subject to special export controlsand each transmittal to foreign governments orforeign nationals may be made only with priorapproval of AFWL (WLDC), Kirtland AFB, NMex 87117.Distribution is limited because of the technology
discussed in the report.
I4." mm~
AFWL-TR-68-46
FOREWORD
This report was prepared by Paul Weidlinger, Consulting Engineer, New York,N. Y. under Contract F29601-67-C-0091. The research was performed underProgram Element 6.16.46.01.41, Project 5710, Subtask RSS2144, and was fundedby the Defense Atomic Support Agency (DASA).
Inclusive dates of research were October 1967 to May 1968. The report wassubmitted 18 June 1968 by the AFWL Project Officer, Dr. Henry F. Cooper, Jr.(WLDC).
This report has been reviewed and is approved.
DR. HENRY F. COOPER, JR.Project Officer
ROBERT E. CRAWFO "Y/GEJf CDARB3 Y% JR -Lt Col US A Colonel USAFChief, Civil Engineering Branch Chief, Development Division
I11B
Is
I __ ____ ____ ____ _ i
ABSTRACT
(Distribution Limitation Statement 2)
An approximate solution is given for the effect of anexponentially decaying pressure pulse traveling with super-seismic velocity on the surface of a half-space. The half-
space is an elastic-plastic material of the von Mises type.The effect of a step wave for this geometry and medium wastreated previously. For that case, the peak pressures donot decrease with increase in depth, while such a decrease
i s obtained for a decaying surface load. The prime purposeof this investigation is to determine the magnitude of this
attenuation. The approximate solutions obtained are valid
for a limited distance behind the wave front, and are tabu-
lated for different sets of parameters pertaining to the
material and velocity. The tabulated results show that thepeak pressures in the case of the decaying surface load do
decrease with depth, but that the decrease is less than onemight intuitively expect. On the other hand the attenuation
is in general larger than that encountered in the similar
problem of an elasto-plastic material of the Coulomb type.
iii
AFWL-AR-6 8-46
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iv
ICONTENTS
Section Page
I Introduction ........ ................ 1
II Formulation of the Approximate Analysis . . 15
III Solution of the Differential Equations byTaylor Series ..... ............... .. 24
IV Numerical Results and Discussion ...... ... 31
1. Discussion of Typical Results ..... . 32
2. Range of Depth for which Results Apply. 72
V Conclusions ..... ................ 74
APPENDIX I Special Case, m = 0 ... ............ . 75
APPENDIX II Elastic Solutions ... ............. . 77
DISTRIBUTION ......... ....................... 79
i[
V
LIST OF ILLUSTRATIONS
Figure Page
1 Step Pressure moving over Half-Space 2
2 Decaying Pressure moving over Half-Space 2
3 Configuration for Step Pressure p0 > PL 5
4 Parameters for which Tables are given
in Section IV 7
5 Range I Solutions, pE < Po < PL 8
6 15ange II Solutions, pE < Po < PL 8
7 Configuration of Elastic Solutions 9
8 Configuration for Approximate Solution
(V > 1/8) 12
9 Yield Violation at the Surface for a
Step Wave 34
10 through 18 Principal stresses a 1 /k at 4 for
different values of v and V/cp 36
vi
I ___________ - --- - -.
LIST OF TABLES
Ia Exact Stresses for po/k = 4.41 35
Ib Approximate Stresses for p0 /k = 4.41 35
II through XXVIII Results for selected values of v,
V/cp , Po/k 37
vii
4 " ' -- m _
LIST OF SYMBOLS
ai , b i , ci Coefficients of Taylor series expansion.
Cp , cS , c Velocity of propagation of elastic P-waves,
S-waves, and inelastic shock fronts,
resDectively.
f i 'Arbitrary functions of x or y.
Shear modulus.
Jl ' J2 Invariants, Eqs. (51) and (2).
k Material parameter related to yield stress
in shear.
M = V, = I
n Ratio defined by Eq. (11).NI ,..., NI0 Punctions defined by Eqs. (82).
p(x - Vt) Surface pressure.
Po Initial value of the surface pressure.
t Time.
u, v, , ' Particle displacements and velocities in
x and y directions, respectively.
UN 'UT Particle velocities normal and tangential,
respectively to fronts of discontinuity.
V Velocity of surface pressure.
x, V Cartesian coordinates.
Aa, A6, AT Increments of C, , t, etc., at a front.
viii
Strain rates.
=x -Vt Coordinate.
Angle between direction of principal stress
a and x-axis.
11 Decay constant of surface pressure, Eq. (46).
VPoisson's ratio.
p Mass density of medium.
G i ) T Normel and shear stresses, respectively.
Principal stresses.
TN ' 1N ' YT Stresses, respectively in the plane, and
perpendicular to the plane of a shock front.
Position angle of element, Fig. 8.
P ' S'Position angle of elastic P- and S- and
inelastic shock fronts, respectively.
4, j, j, r Potential functions.
ixill
AFWL-TR-68-46
This page intentionally left blank.
i1
t
x
SECTION I
INTRODUCTION
In a preceding analysis, Refs. [I] and [2] , the effect
of a step Dressure, Fig. 1, rrogressing with a superseismic
velocity V on the surface of a half-space has been studied
for an elastic-plastic material subject to the von Mises yield
condition
2J2 =0 (i)
where k is the yield stress in shear and J2 is the invariant
12
1 (2)2 = sijsij
The present report considers the more general problem
of a decaying pressure pul p(x - Vt), Fig. 2, moving
with a superseismic, constant velocity V. The problem under
study here is more realictic than the one treated in Refs. []
and [2], but it is also considerably more complex. Because
of the complexity only an approximate solution will be derived,
which is valid for a limited distance from the moving front.
* [] Bleich, H.H. and Matthews, A.T., "Step Load Moving
with Superseismic Velocity on the Surface of an
Elastic-Plastic Half-Space", Int. J. Solids andStructures, 3, 819-852, 1967. (Also Office of Naval
[21 Matthews, A.T. and Bleich, H.H., "Stresses in anElastic-Plastic Half-Space Due to a SupereeismicStep Load", TBL, Ballistics Research Laboratory,Aberdeen, Md., Tech. Rpt. 4, Co~itract DA-30--069-AMC-8(R),March 1966.
P~ p0H(t-x)
y
FIG. i
Step Pressure moving over Half-Space
VELOCITYV
-.-- X
y
Decaying'FG Pressure moving over Half-Space
The approximation employed is somewhat similar to the one
utilized and discussed in Ref. [3] for a material subject
to a different yield condition.
Because the velocity V of the load is constant, the
resulting stress and velocity fields remain unchanged if con-
sidered in a coordinate system moving with velocity V. In
other words, Refs. [1], (2] and [31 and the present paper
consider steady-state solutions, which may be used as approxi-
mations to actual situations when the velocity V varies
gradually.
It was found in Refs. (1] and (2] that the response to
the step pressure changes in character depending on the range
in which the values of Poisson ratio V, of the velocity ratio
V/cp , and of the intensity of the surface step load p are
situated. As introduction to the problem with a decay of
pressure, some aspects of the results of the analysis for the
steD pressure are restated in the following.
Defining the location of a field point by the polar
coordinates c arid r, Fig. 1, where the origin of the coordi-
nate system lies at and moves with the front of the surface
load, it was found that the stresses and velocities axe
functions of the angle only. For a step pressure there is
[3] Bleich, H.H. and Matthews, A.T., "ExponentiallyDecaying Pressure Pulse Moving with SuperseismicVelocity on the Surface of a Half-Space of GranularMaterial", Tech. Rpt. AFWL-TR-67-21, rontractAF29(601)-7082, July 1967.
3
therefore no dependence of stresses or velocities on the
radius r, so Lhat there is no attenuation with depth. It was
found, further, that for sufficiently large values of the
surface load, p> PL the solution has the character of
Fig. 3, where the value of PL depends on the parameters V
and V/c .
The configuration shown in Fig. 3 shows two discon-
tinuities, a P-front and an S-front, moving with the veloci-
ties of elastic P- and S-waves
2 2(l-) G (3)
P 1 - 2v p
2 G (4)S =(
respectively. The locations of the fronts are defined by
the related angles
7 -l 2(l-v)G 15= - sin-i (i-2v)p
7S - sin- 1 [ (6)
The changes in stress and velocity at these fronts are
entirely elastic. There are four "neutral" regions of
uniform stress (and uniform velocity), i.e., regions where
the yield condition, Eq. (15, is just satisfied, but no yield
occurs at the particular instant. There are two plastic
regions, i to , and from 43 to 14 where plastic defor-
mations occur at the particular instant,and where the stresses
4
NETRL EGO
TIREIt
STRESSLESS REGIONAHEAD OF P -FRONT
F * NEUTRAL REGIONS OFUNIFORM STRESSSATISFYING EQ.1
CONFIGURATION FOR STEP PRESSURES p >- L
FIG.3
(and velocities) vary as functions of the location €. One
of the plastic regions is always located between the P- and
S-fronts, and one between the S-front and the surface.
If the surface load lies below the limit pL , but above
another value PE < PL ' so that
PE < Po < PL (7)
only one of the two plastic regions occurs, which one occurs
depends on the parameters v and V/cp . If these parameters
lie in Range I or II of Fig. 4, the configurations of Fig. 5
or 6 apply, respectively. In these cases two of the neutral
regions shown in Fig. 3 coalesce and become a region of
uniform stress below yield as noted in Figs. 5 and 6. The
damarkation line between Ranges I and II in Fig. 4 is given
by the relation
2
V/c = (-)i) (8)P (l-v)(l-3v)
If, finally, po S PE ' the solution becomes entirely elastic,
Fig. 7.
One detail obtained in Ref. [1] for the step pressure
is crucial for the present purpose. If P is not only larger,
but very much larger than PT , it was found that one of the
two plastic regions in Fig. 3 is of much greater importance
than the other. This means that the energy dissipated and
the changes in stress and velocity in the "important" region
6
v/c p
V/ce__ I-v1- ,fv )(-03 v)
4.0
3.0 -RAN( E I RANGE X
* 2.04
10-
0 0.1 0.2 03 0.4 0.5
*PARAMETERS FOR WHICH TABLES ARE GIVENIN SECTION IV.
FIG. 4
V
S -FRONT
RANGE I SOLUTIONS, P E PO P L
FIG. 5
vI
P- FRONT
RANGE]I SOLUTIONS, PO'Q~
FIG. 6
i1
. p (X-Vt) v
P -FRONT
CONFIGURATION OF ELASTIC SOLUTIONS
FIG. 7
are very much larger than in the other plastic region. In
the "important" region the bulk of energy dissipation and of
the changes in stress and velocity occur for values of very
close to a critical angle for which the basic differential
equations arc capable of a singular, discontinuous solution.
The critica1 angle is connected with the fact, that in the
von Mises material considered, plastic shock fronts may exist,
having a velocity
-2 2(l+V) G (9)c - 3(1-2\) P
As discussed in Ref. [1], at such shock fronts the state of
stress ahead and behind the front must be at yield. but - in
addition - the major principal stress must be at a right angle
to the front and the other two principal stresses must be
equal. It is shown in Ref. [1] that at such a plastic shock
front all three principal stresses change by the same amount
Aa, while the component of the velocity normal to the front
changes as required by conservation of momentum.
In the present steady-state problem the angle may be
obtained from the value for c,
sin-1 2(+)G3(10)!V
The location of this front depends on the value of Poisson's
ratio V. If V > 1/8 the critical angle 4 lies between p
and S
__10
While in the actual problem the above stated conditions
for a discontinuity are not exactly satisfied and no shock
front occurs, the conditions ar,, nearly satisfied when
PO >> PL" As a result, very rapid changes were found in
Ref. [11 in an extremely narrow range of the angle 4 neai
when p exceeds about 5. (See the tables in Ref. [2].)0
The fact that for large values of p the plastic effects
are nearly a "shock front" at 4 = 4, suggests that approximate
solutions could be obtained by compressing (approximately)
all plastic effects into a discontinuity in normal pressure
and velocity at 4, and treating the material elsewhere (ap-
proximately) as entirely elastic, even if the yield condition
is slightly violated. For the case of a step pressure p0 > pL'
this approximate solution for V > 1/8 would have the con-
figuration shown in Fig. 8. In addition to the elastic dis-
continuities at 4S and 4p , there is an assumed discontinuity
at , consisting of a change of all three principal stresses
by an amount Au and an appropriate change of normal velocity.
In the region between these fronts the stresses and velocities
are uniform, but the yield condition in one or two of the
regions will be violated. Such violation is considered ac-
ceptable provided it is not too severe.
In view of the fact that correct solutions for the step
pressure are available, there is no need to consider this
case further, but the equivalent approximations ca.i be made,
and will be made below for the case of a decaying surface load.
1U,e -i
p X-Vt)
-,- V
S - FRONT
PLASTICSHOCK P- FRONT
CONFIGURATION FOR APPROXIMATE SOLUTION(V N--/8)
FIG. 8
12
Assuming the discontinuities stated above, at p I 4s and 4,
it will be assumed that the elastic differential equations
apply in each of the three wedge shaped regions shown In
Fig. 8. Solving the boundary value problem Dosed by the
prescribed surface load and by momentum or continuity con-
siderations, approximate solutions are obtained. After
fin-'ing the solutions the degree of violation of the yield
condition can be checked by computing the ratio,
n = 1"(II)
k
starting near the leading edge of the load. If the solution
in this location and within some distance r from the wave
front is acceptable, the hyperbolic nature of the problem
permits one to accept the solution in the range r < r, even
if serious violations of the yield condition occur elsewhere
for r > r. A further restraint on the range of validity, r,
is the fact that the compressive stress at the shock front
S= 4 must inherently 4ncrease and the solution can only be
accepted within a radius r where this requirement holds.
In the numerical solution obtained later the ratios n
indicating compliance or violation of the yield condition are
stated for each point where stresses are given. For engineering
purposes values up to n = 1.2 were considerel, but a reader may
elect not to use the full range of the solution given if smaller
values of n appear appropriate. It is found that the range of
applicability shrinks as V/c p becomes smaller and anproaches
unity.
~13
The formulation of the problem given below applies only
for values V > 1/8. Therefore the tables given contain only
values V/c > 1.5 and v > 1/8.
14I
1.
SECTION II
FORMULATION OF iHE APPROXIMATE ANALYSIS
Based on the reasoning outlined in Section I, approximate
solutions for the case of the decaying shock wave, Fig. 2,
will be formulated. Only the case V > 1/8 is considered so
that S> 4' In accordance with the discussion in the Intro-
duction, it is assumed (approximately) that plastic deformations
occur in any element only at the time of passing of the plastic
shock front indicated in Fig. 8, while stress changes tnere-
after are elastic. The applicable elastic relations between
stress rates and strain rates for the present case of plain
strain are
TV i V (12)x ;x E x y z 1-
whreae -0wil ,I h
0z V&-C + ](14)z E x y z
xy-2Ly + x ] = (15)
xy 2 0y ax 2Gwhere ax ' cry ' az f T are the stresses, while u, v, are the
displacements in the x and y directions, respectively. In
addition, the equation of motion
x +T (16)
15
I'
T+ Y p (17)ax ay t
must be sacisfied. Equations (12, 13, 15) may be integrated
with respect to time, requiring the addition of arbitrary
functions fi(x,Y) of x and y,
i2
_u i-V2-i-E [a x -- Ca y + fz(x,y) (18)
2
av 1-V V a + Oy] + f 2 (x,y) (19)ay : E [- 1--- x y2
[Tyy + 2v =-- T + f(xy) (20)
while Eq. (14) after integration may be written
a= v(G + a ) + f (x,y) (21)z x y 0
The last equation is the only relation containing a , and
simply defines this quantity. It is noted that the arbitrary
functions f1 , f 2 and f,. are not entirely "arbitrary", but
are related by a compatibility relation, as may be seen by
eliminating u and v from Eqs. (18), (19) and (20).
The solution to the differential equations (16-20) may
be expressed in terms of potentials 4 and T which satisfy the
wave equations
+ (22)
T + T 1 , (23)xx yy 2 tt
cS
16
Subscripts on potentials indicate derivatives. In terms of
these potentials and of two arbitrary functions g and of
x and y, the displacements become
u = P - T + gl(x,y) (24)x y
v = 4) + 'P + g 2(x,y) (25)y x 2
while the velocities 6 and ' are
u =4) -' (26)x y
= D + T (27)y x
The stresses are
ox = 2G [ 1 ) + 4+ -V 2 (28)1-2v xx l-2v yy xy
V 1-Vo= 2G [I-- 4 + 4) +' 1 (29)y 1-2v xx i-2V yy xy
1y
T = 2G [4) + I (' (xy 2 xx Yy
with a given by Eq. (21).z
Substitution of Eqs. (24-25) and (28-30) into Eqs.
(18-20) furnishes relations between the three "arbitrary"
functions f, f 2 and f3 , and the two functions g and 92
i.e.,1g g g2 )
f = 3 = ( + (31)ax- 2 D ' 3 2 a1
These three relations and the Eqs. (28-30) satisfy com-
patibility indentically, leaving the two functions gl(x,y)
and g 2 (xy) as arbitrary functions in lieu of the related
functions f1 , f2 and f3 .
It is now noted that the velocities 'x and ', and the
stresses a , a and T in terms of the potentials D and Tx y
have exactly the same form as in an elastic Dlane problem
and these quantities will therefore have the same values as
in an entirely elastic problem with the same boundary con-
ditions. The plastic deformations at the shock front lead,
however, to different expressions for u, v and a because the
functions g, , g2 and fo now occur. As will be seen later,
these three arbitrary functions follow from conditions at the
P-front and at the plastic front.
Due to the fact that the steady-state problem is con-
sidered, the surface pressure is solely a function of
- = x - Vt (32)
and the solutions are only functions of t and y. The wave
equations for the potentials then become
(Dy (M 2 _ 1) 4 (3
yy 2.l
yy (Ms 1)
where
V M V (34)C S c
18
Noting that the open functions f and g which0 and
are not functions of t, can in the steady-state not depend
on x either, the expressions for displacements and stresses
become
U P -T + g 1 (y) (35)
v 4y + T + g (y) (36)y ~ 2(
cr = G[2(I + -_) 4 - 2T (37)
a= G 2(M 2 2) P + 2T] (38)Oy G( S -
T = G[24 - (M - 2) T (39)
az = v(ax + ay) + f 0 (y) (40)
To formulate the boundary conditions, expressions will
be required for the normal and tangential velocities n N T
and the normal, tangential and shear stresses YN ' OT ) TN
for planes inclined at any angle 4 (see Fig. 1)
U- - =-cos D - cos ' T - sin Ty (41)
UiT
- cos sins Tsi (42)
_N _ 2 1-2v 2U [M (1 1-2v sin 4) - 2 cos 214 -
- 2 sin 24) y 4- 2 cos 2 Ty + (Ms - 2) sin 2T (43)
19
aT 2 1-2v 2 +G [MS + 2 cos 2] (P +
+ 2 sin 24 (P - 2 cos 2- - (M 2) sin 2 Y ' (44)
T N 2TN = (M2
- 2) sin 24) * + 2 cos 24 4 y +
+ 2 sin 24 'y - (M2 - 2) cos 24 T (45)
To state the boundary conditions of the oroblem pronerly,
and to be able to make statements on the character of the
solutions, the differential equations for (P and T must be
discussed. While both differential equations are hyperbolic,
the characteristics for D are steeper than the boundary
defining the plastic front, 4p < $. The solution for P in
the region of interest, $ < 4 < 7r, Fig. 8, is therefore
defined by statements on the boundaries 4 = T and 4 = 7r.
D will be a continuous function if the prescribed conditions
at the boundaries are continuous. This being the case, it
will be possible to use for 'D an expansion in a Taylor series
in 4 and y for a region near the origin 4 y = 0.
On the other hand, the characteristic directions for the
differential equation on T are flatter than the boundary,
S > *. The differential equation does therefore not apply
at A .S.'and thc solutions A o ' > lS , and T for
< < S are not fully defined by prescriptions on =
and 4 =. The two functions T and T must, however, lead to
20
values of stress and velocities which satisfy momentum con-
siderations at .1 = swhich furnish additional conditions
for their determination. While at least some of the deriva-
tives of T and T at ) = 4swill not be continuous, each of
these potentials in its region will be continuous, and each
of the potentials T and T can again be expanded in a Taylor
series.
The functions P, T and T are therefore subject to the
conditions listed below. On the surface the normal and
tangential stresses are prescribed. Thus, for an exponentially
decaying surface load, the stresses are
at y = 0, < 0
a Po p= - e (11 > 0) (46)
0 (47)k
At the plastic front 4 = conseivation of momentum requires
= PC A6 N (48)k k N
k =0 (49)
-= (5o)
N = T z 3 1 (Jl
21
iU
where J is the invariant
1J = + +o (51),Ii x y z
and where the symbol A indicates the jump in stress, or
velocity at a front of discontinuity and where c and $ are
given by Eqs. (9) and (10), respectively. Moreover, energy
considerations require that the signs of A and 3f the total
normal stress O at $ are equal.
Due to the fact that different expressions T and T for
the solution of the second of Eqs. (33) will be used, it is
necessary to consider conditions at the 3ocation S
Momentum considerations require that discontinuities in the
shear stress TN and the tangential component of the velocity
UT are proportional
at =
ATN =P cS 6 (52)
while all the normal and tangential stresses and the normal
velocity at 4 = must be continuous
at @ =
AoN = 0 ( 3)AON 0
AMT = 0 (54)
AO 0 (55)z
A6= 0 (56)
N
22
In the assumed approximate solution, the plastic shock
propagates into a region which has been previously stressed
by the passage of the P-front. In this region, 4 F < < '
the stresses are uniform and just at the yield limit.
Their values are:
for < <
__ - /T (i-v) 2 t2pk / I-2V [1 - 2 sin 2 cot (57)
S i(-v) cos 24 (58)k 1-2v S
Txy= 2/5 (1-v) .n2
k 1-2\ sin cot p.f9
az -
k 2 sin 24p (60)
The above equations complete the available conditions
for the determination of t.,e potentials P, T and P. After
the potentials are obtained the arbitrary function f (y)
appearing in Eq. (40) can be obtained from Eq. (50). The
arbitrary functions gl(y) and g 2 (y) in Eqs. (35) and (36)
are defined by the fact that displacements at the plastic
front may be computed from the velocities in the region
between p and $, using the relations:
for p < <
u - ( + y cot p) (61)
v =- ( + y cot P) (62)
23
SECTION III
SOLUTION OF THE DIFFERENTIAL EQUATIONS BY TAYLOR SERIES
The differential equations for the potential, to be
solved are
$yy M2 1 ) P = 0 (63)
yy-(M 2 1) T 0yy
(64)
Tyy - (MS - 1) T 0
The boundary conditions, Eqs. (46-47) on the surface give
at y = O, E < 0
2 Po (65(N -2) + 2T - e (65)
S G
2y - (M2 2) ' =0 (66)
While the conditions, Eqs. (48-50) at the plastic front give
three equations
at y = tan
G {(M 2 - ) + cot + + =
2 1 I ilV cos - (67)
24
( 12
k (M - 2) sin 2 4 + 2 cos 2$ (Dy + 2 sin 2- y -
- 2- 2) cos 2 = 0 (68)
* S
k,2 - 2) cos 2$ ¢ - 2 sin 2$ t y + 2 cos 2" iy- +
+ (M - 2) sin 2 T -2 cos 2($ - 4p) (69)
Finally, the five conditions, Eqs. (52-56) at the S-front
lead to only one condition on T and
at y = tan
(Y - T) + tan - = 0 (70)
The surface load is expanded in a series in E
Coe 1 in m(
0 PO I . (71)m=0
aoplicable for E < 0, while the potentials D, ' and P are
expanded in double series
ai,
Y b i y (72)
i,j C
ihe differential equations and the conditions, Eqs. (46-56)
are homogeneous, i.e., they contain consistently only second
derivatives of (D, ' and P with respect to and y.
25
Substitution of Eqs. (72) into Eqs. (63), (64) and (66-70)
results therefore in linear equations for the values ai,
b , c i , which couple only those terms with indices
m = i+X-2. The potentials can therefore be conveniently
written as a series of polynomials. Each set of polynomials
of class m can be determined independently from the poly-
nomials for a different value of m.
G (i) m) m (n) i m+2-ik = i= ( -- a. ( y (73)
m=0 i=O
- Om+2
-- = X T(m) T(m) = b(m) i ym+2-i (74)m=0 i=0 1
m+2 (i) i m+2-i
k I T (m) %m)= I c y (75)in=0 1.-
The case m - 0 corresponds to the step load and, while
quite simple, is treated differently, see Appendix I. For
each value m > 1 there are (3m + 9) unknown coefficients
(M) I'(M) (M)a m . i n c i Substituting the second derivatives of
Eqs. (71-75) into the conditions on the boundaries of the
regions, Eqs. (65-70), yieldssix equations for each value
case the value 6 stays at 60.000 for p < < S and at
90.000 for 4S < < W.
Looking at results for other grid points it is seen
that at point 10 in Table XIV the plastic shock has reached
a small tensile value, indicating that the requirement of
compressive shocks is violated from there on.
For the higher loads p /k = 6.0 and 10.0, Tables XV, XVI,
the shock remains compressive in the entire range of the tables.
Table XV for p0 /k = 6.0 is limited only by the condition
> -1.7 due to the truncation of the expansions. For the
highest load po/k = 10.0, the validity is limited to PJ > -1.0
because the yield condition in the surface, beyond point 14
would be violated excessively.
For other values of V and V/c the situation is generally
similar. Violation of the yield condition for low values of
PO/k occurs first on the S-front, while for higher values of
PO/k the violation appears first on the surface.
2. Range of depth for which results apply
The previous discussions considered the validity of results
with respect to a nondimensional depth py. It is appropriate
to consider values of 4 of physical interest to find out for
what actual depth y results have been obtained.
72
Let t = 0.2 sec, V = 4000 ft/sec, then Vt 800 ft.0 0
For the typical values v = 0.25 and V/c = 2.0 discussed above,
information on peak pressures for values of the surface load
Po = 4.Ok and 6.Ok are contained in Tables XIV and XV up to
about My = 0.6 or for a depth of 480 feet. A time history
can be plotted using interpolation for the depths My = 0.5
or 400 feet. For the same values of v and V/cp , but
po/k = 10.0, the limit of applicability is My = 0.3, or about
240 feet.
In general, for larger values of V/c and for larger
values of po/k the depth for which the results apply decreases.
*0
I 7
73
SECTION V
CONCLUSIONS
An analytical approach has been presented to describe
the effect of a decaying pressure moving with superseismic
velocity V > 1.5 c on the surface of a half-space of an
elastic-plastic material subject to the von Mises yield con-
dition. Numerical results have been obtained for a wide
range of surface pressure and material properties. Due to
simplifications in the analysis the results obtained are
approximate and restricted to target points of limited depth,
as discussed in Section IV.
Information on the stress field for exponential decay
of the applied load is contained in Tables II - XXVIII and
in the accompanying figures. Comparing the effects of step
loads given in Ref. [1] with the new results for decaying
loads, it is found that in the latter case the peak and sub-
sequent pressures decrease with depth, which is not the case
for a step load. However, the decrease found at any depth y
is smaller than thu decrease of the applied pressure at the
point on the surface directly above the target point, i.e.,
at a distance equal to (-y/tg c) behind the shock front. This
is graphically shown in Figs. 10 to 18. While the attcnuation
is therefore less important than one might expect, it is still
greater than found in a similar analysis, Ref. [3], for an
elastic-plastic material of the Coulomb type.
74
APPENDIX I - Special Case, m = 0
Although an approach analogous to that used in the text
for m > 0 may be used for m = 0, it is considerabiv simpler
to derive expressions for a step pressure using the simple
configuration shown in Fig. 8, consisLing of three shock
fronts only.
For < <
° 1 AO
A = -(86)k
K2 a 3 V A P (87)
where
DAoe - /Tu-v) (88)
k l-2v
For < < S
o P -1 Ac P A-- + (89)k k +k
02 3 A P V + L-j (90)
k k k '1-v k
where
=O -P + 2(l-v) sin 2p, sin 24-= - - + !-- [ cos 2 + (k k 2 1-2v S cos 2%S
75
For S < < w
l P
k k (92)
c 2 2-2 i - V t a n 2 k S s i n 2 p-3 cos e + + in2 cos + + (93)
03 V 3 %__ ) + A'
k 2 - k (94)
7
APPENDIX II - Elastic Solutions
The exrressions for the stresses in a purely elastic
*material subjected to a step pressure may be found in Ref. [1].
The equivalent expressions for a decaying surface pressure are:
For (P < ( < S
A = P (g-. ,)
k F cos 2(S e (95)
VA.. E 2 2 (96[1 - 2 sin cot e (96)
k k S P
Cr P
z Au P I V i-2P=i - - [ t i_ -) s i n 2 p e ( 9 7 )
kU [2 sin2 cot (p] e (98)
A2 P PO (l-V) cos 2(PS (99)k k 2 2(cos 2S + (1-2\) cos - p) - (1-V)]
= y cos 1P -- sin e (100)
For (S <(P < 1T
- po1Jk -0
(101)k k
77
cx AGP i2¢ ¢ p (li p A S (-,S
- - 2 sin O cot ]p le k sin 2 S e (102)
AG p (-2) (-p
k k 2(1-v) sin 2 p e (103)
T -AciP 2¢ i( - p) tv s r -
= u [2 sin 2 cot *p1e + cos 2¢8 e (104)
ATS 1-2V sin 2 0P AMP
k 2(-v) cos 2 S k- (105)
y Cos € S
S sin s (106)
78
UNCLASSIFIEDSpeurity Classific ,tion
DOCUMENT CONTROL DATA. R & D,Seuruitv rlessiteration of title body of nb~tra, i and ,idesxng dninotatrun nznot be entered when the overall report fi classifled)
I ORIGINATING ACTIVITY (Corporate author) |2a. REPORT SECURITY CLASSIFICATION
Paul Weidlinger, Consulting Engineer IUNCLASSIFIEDNew York, New York 10017
2b. GROUP
3 REPORT TITLE
EXPONENTIALLY DECAYING PRESSURE PULSE MOVING WITH SUPERSEISMICVELOCITY ON THE SURFACE OF A HAL?-SPACE OF VON MISES ELASTO-PLASTIC MATERIAL
4 DESCRIPTIVE NO1 ES (Tfype of report and Inclusive dates)
October 1967 to May 19685 AUTHORIS) (First name, middle initial, last name)
Bleich, Hans H.Matthews, Alva
5 REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFSAugust 1968 98 3
la. CONTRACT OR GRANT NO ga. ORIGINATOR'S REPORT NUMBER(S)
F29601-67-C-0091b. PROJECT NO 5710 AFWL-TR-68-46
c. Sub task No RSS2144 9b. OTHER REPORT NOtS) (Any other numbers that may be assIgnedthis report)
d.
0, DSTRIOUTON STATEMENT This document is subject to special export controls and each
transmittal to foreign governments or foreign nationals may be made only with priorapproval of AFWL (WLDC), Kirtland AFB, NMex 87117. Distribution is limitedbecause of the technology discussed in the report.
11 SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
An approximate solution is given for the effect of an exponentially decayingpressure pulse traveling with superseismic velocity on the surface of a half-space. The half-space is an elastic-plastic material of the von Mises type.
Z The effect of a step wave for this geometry and medium was treated previously.J For that case, the peak pressures do not decrease with increase in depth,
while such a decrease is obtained for a decaying surface load. The primep~upose of this investigation is to determine the magnitude of this attenuation.The approximate solutions obtained are valid for a limited distance behind thewave front, and are tabulated for different sets of parameters pertaining tothe material and velocity. The tabulated results show that the peak pressuresin the case of the decaying surface load do decrease with depth, but that thedecrease is less that one might intuitively expect. On the other hand theattenuation is in general larger than that encountered in the similar problemof an elasto-plastic material of the Coulomb type.