Plane elastic-plastic waves

PLANE ELASTIC-PLASTIC WAVES (0 PLOSKOI UPRUQO-PLASTICHESKOI Phfhf Vol. 29. No. 3, 1965,

VOLNE)

pp. 509515

A.M. SKOBEEV (Moscow)

(Rccaivcd

The problem equations

of oblique

proposed

the elastic

case

plicit

correspond

Problems either

of soil

of material

A study

and transverse

the solution

dynamics stress

that

satisfies

the

is made of motions

waves.

of the problem

of the cubic

result

of the problem,

from which

0 (El is found. has been

which

If the shear

can be written

[l]

in principle,

to describe

in

modulus

and

down in es-

model

are no solutions of the present adopted

1. Let We will

can clearly

problem

an entirely different

has

The experimental law has not been

tested

for testing

solved

Grigorian’s

by Antsiferov

for soil,

they obtained.

an elastic-plastic

lie in the boundary

7

=

and ffz*

halfspace plane

t/z @YU -

(J**),

of the stress

out in [6] led

test.

the equa-

be pointed

[?I. However,

of the present

along

there

The results

It should

system

is directed

II, and v are the velocities

at least

at all, since

and an associated

paper

K = p. dp / dp

are

of coordinates.

into the interior.

the x- and y-axes,

tensor,

to

proposed

the regrettable

carried

model.

and the results

and the x-axis

are components

one,

and Rakhmatulin

u (p),

it leads

However.

to be put to experimental

model

notation:

test

Hooke’s

which

(Jry, ‘Jyy

enables However,

is

symmetry

because

p (E).

different

use the following

CXX, z =

model motion.

the relation

T. The model

and the function

be used

in the

a stricter

condition available

heen

This soil

from those

us consider

The y- and z-axes

equations

the medium. tested.

tensor

the symmetry

complications stress

p ia a

of the stress

condition problem

on the shear

elastic-plastic

thoroughly

that generalizes

paper

significant

to describe

an arbitrary

of these

out that a similar significantly

involves

that the relation

can be obtained

By considering

of the present

form of the plasticity

of this

condition).

from the plasticity

This

relation

as in [2 - 51, that the pressure

one can derive

can be used

fact is that it has not been to a concrete

the assumption This

and that the secondinvariant

not only on E but also

by Grigorian

making

6 is given.

In the formulation

adopted.

the fact that u depends

8 alone

(the plasticity

formulation

solved

or by assuming,

dilatation

of the pressure

condition

are usually

u and the strain

as an experimental

a function

fJ =

composed

is investigated.

to longitudinal

are constant,

the nniaxial

function

they

[l]

28, 1964)

form.

between

tions

on a halfspace

by Grigorian

dp/dp

the quantity

impact

November

and

600

A.M. Skobssv

The following

i.e.,

on the

problem

boundary

y or I, the required We will

will

be considered

u (8) and 7(t) quantitiee

assume

are step

that the quantity

the eonva&ive

terms

to the solution

of the system

functions.

are o (x, t), 7(x,

in the equation

Assuming

that nothing

depends

on

t), and y (x, t).

1 - pe/p

is small

of Grlgorfan

in comparison

can be neglected,

with unity.

Then,

and the problem

leads

of equations

(1.1)

&G’+P_-+$-_-_-__ au with

the above-formulated

been

obtained

P

boundary

and initial

u ? %

at in (1.1) has

ap .LKap

and the continuity

at

(1.2)

equation 6$+&o

(1.3)

F (p) and h were determined

The quantities

in [I].

/p

If the shear cases tions

aqnation

the generalized

in system

Hooke’s

law,

F (p) is an experimentally The pressure

construct

conditions.

Let

in the theory (1.1)

determined

in Fig. some us write

the characteristics

occurs

plastically,

A = 0, and system

is the equation plasticity

to depend

then A > 0

(1.1) then reduces

; in the other

to the usual

equa-

of elasticity.

and the third equation

p is assumed

is shown

2. We will By definition,

aP

T‘he third equation

5 = p.

FIG. 1

p (p) relation

2F (P)

conditions.

J

The first

F’(P)

ax

F(P)

from the relation

f

initial

av

z

F (PI 8%

of motion,

the second

is the plasticity

equation

condition,

is

where

function.

on p and the direction

of the process.

A typical

1. solutions

of system

out the equations of a system

(1.1)

without

attempting

for the characteristics

of first-order

differential

to satisfy

of system eqaations

the

(1.1). are

Plane

curves

x - f (t) having

the following

the syatem are given on such uniquely

defined.

w given

elortic-pltutic

Denoting

properties

a curve,

: if all the unknown

the derivatives

differention

601

wavea

along

of these

the cwve

function9

functiona

f by d/t%,

will

appearing

ln

not be

for an arbitrary

fanction

on f we have (2.1)

System rivatives

(1.1)

contains

of the type (2.1). derivatives Thus, which

the six unknown8

are to be determined Since

the resulting

can be determined the condition

u, v, u, 7,

by the ten equation8 system

uniquely

p, and

of system

is linear

f(t)

partial

de-

and by ten equation0

in the partial

when the determinant

for the characteristics

y; the twenty

(1.1)

derivatives,

all the

is nonzero.

ia the vanishing

of this

determinant,

haa the form

dx 0 0 0 0 0 0 0

dt 0 0 0 0 dx dt. 0 0 0 0 dx 0 0 0 0 0 0 0 0 0 0 0 0 piJ 0 0 --1 0 0 po 0 0 0 as@ 0 a19 0 0110 0 a,‘0 0 0 all1 0 a8l1 0 KOOOOOOOOl

a+P

a19=Gmr,

all0 =

GOfP

G

Expanding zeros)

_;

this

1

determinant it equal

0

0 dt 0 0 0

0 0 dx 0 0

0

0 1 0

0 0 0

0 0 dt 0 0’

0 00 0 0 0 dx 0

0 0 0 dt 0’

0

0

0

0

0 0 0

al00

0

i 0

a1011 0 00

0

1 +

(which

2F@)

11

p)

in not difficult

we obtain

al0

becaase

the eqnation

F’ 10=--I 'Y

al0

-WY-@)

to zero,

(2.1$

(~1

)

,f@

0

UllJ10 0

-F'W 2F@)

9--

F (~1

=911 =

,

0

--I+

G C

(~1

0

-i

0 0 1

a10

aglO =

Z,

-d+ 1 F

and setting

0

as 9=G&,

F (~1 alll =

0 0 00000 dt 0 0 dx 0 0 0 0 0 0

of the large number

for the characteristics

of of the

system

ad-{[I +k($-$-y]K

+4qG+

$G}a2+

+ ‘G + GK Ls’+ h_(~2_ $)‘“I = 0 PO(g)* = a*, It should

be noted [4] that s’,<

I-&

= 3,

1 and k < 1. When y 10,

p& equation

(2.2)

=k (2.2) redncea

to

602

A.M. Skobcav

@

(a”, s) z .’ --[(I + ks) K + 1/S (4 - 2) Gl a2 + GKs (s + k) = 0 Equation

(2.2) has been

doer, not make the problem over,

we will restrict

which

a is constant

istics

are straight In view

motion

obtained

without

K, G, and k. In the following

quantities

ourselves along

because

of the fact

solutions

to the case

equation

Since

concerning

in (1.1)

where y I 0 and will

the characteristics.

that the boundary

a’ depends

conditions

to be self-modelling.

of system

As a consequence

have

Under

of the self-modelling,

this

is nonlinear.

only consider on s alone,

the form of step assumption

all quantities

only on a, because

to depend

only on s.

From the self-modelling

we have

More-

motions

functions,

it is possible

the to const-

will

depend

all quantities

only on x/t appearing

the firat equation

(1.1) can be written au

making

use

of the third

equation

02 ap _ --__---

in (1.1)

in the form

da

(1.1).

(2.5)

we have,

respectively,

3P

at

K

= (I/$.

(2.4)

a2%=z Hence,

for

the character-

a -= at Therefore

the

assumption

(1.1).

It is clear from (2.3) that o depends can be assumed

the second

whatsoever

that G < (1 + k) K; this

assume

lines.

can be assumed

ruct some

linear,

any assumption

we will

(2.3)

(2.6)

at

(2.7) or (2.8) Integration

of (2.8) yields

VP and according

to (2.2)

= k-(po) exp (\

$

(E)

b”

we have

_

;1”+ dE) kc)

(2.9)

K

(2.10) The remaining methods.

quantities

Replacing

tions

of system

value

problems.

(1.1).

appearing

in the system

o in (2.Y) by the roots which

can be used

of equation

can be obtained

by elementary

(2.3),

a family

for the construction

we obtain of solutions

of solu-

of the boundary

elastic-plastic

Plane

Equation ones

(2.3) has

correspond

the solution

obtained

a longitudinal verse

four roots : two are positive

to waves

propagating

wave;

the larger

that obtained

and two are negative.

in the positive

by substituting

603

wavea

direction

positive

by substituting

along

root (2.3)

the smaller

The positive

the x-axis.

into (2.9)

one will

Further,

will

be called

be called

a

trans-

wave. 3. Now, let us examine

Let the larger

of a’ by assuming Solving

(2.3).

by 4

We will

and the smaller

that a* are functions

(2.3).

(1 +

?a2 =

equations

root be denoted

assnme

that (2.3)

is an eqi;ailon

one by a :. We will

obtain

in aa.

an estimate

of s.

we obtain

ks) K +

(3.1)

1/Q (4 -

s2) C h

{{(l

f

ks) K f

‘i,

(4 -

s2) cl2

-

4GKs

(s f

k).l” (3.1)

&3

=

(1 +

1/3 (4 -

ks) K +

s3) G f

{[(I

+

lis )K -

‘i, (4 -

s2) Cla +

4GKh)“’

for s < 1 h E (1 +

ks) ‘/, (4 -

3)

-

s (s +

k) L

1!3 (i -

s’) (4 +

ks) >

0

(3.la)

Therefore

a?< This

follows

(I f

=

have

Further,

a very elementary

the possible

character

(3.2a)

V(a - B)“+S2

a + B*

max(a, B)

z2 >

divide

(3.2)

from the fact that for the relation

xl,2

we always

ap2 > (1 + ks) K

k-4K,

values

analysis

shows

of the parameters

that

dal* / ds > 0

into three

cases

(3.2b) and

a,* <

in accordance

G.

with

One can the

of dd/ds.

(1). If

kK > then

daz2 I ds >

0

and

a22 > K -+ a/&

3 (If 4sk) for all values

G

(3.3)

of a.

(2). If G>kK>;G (3.4) then

a22 >

K +

‘13G and the derivative

dai/ds

changes

sign

in the interval

[0, 11.

(3). If ‘,,C

>

kK

(3.5)

604

A.M. Skobccv

then the derivative

&,a/

4. Now we will cribes will

the motion clarify

aystem

when

(1.1).

ds <

consider

K +

4/3C

1 >

o>

of system

The plasticity

condition

can be employed.

writcl (2.6)

when

of applicability

shear.

condition

we will

a,2 <

the limit

with plastic this

0,

Making

s*, where s* = LICK / 4~;.

(1.1).

System

for shear

use

will

(1.1) des-

be A > 0. We

of the third equation

in

in the form

[a2-

t

(1

-j-;)K]g = +gIzg

With the aid of (2.4) transform

and the first

the second

equation

(4*1)

equation

in (1.1)

in (l.l),

we

into

h=f($__l)$_ (4.2) =-

1/j

a% 1/l

account

where

[a% -

(C -a*)

2

has been

(s + k) K] au -

taken

tiX

s2 of (4.1).

FIG. 2 By using (4.2)

can be transformed

Thus,

if the material

=-+[a2-

is compressed,

elastically cases

(K +

then during

K + 4/3G

a2 > K f a

to show

that

(4.3)

+G)]$

loading

or unloading

the condition

respectively,

a2 <

With

it is not difficult

into the form

h

h > 0 yields,

(2.3)

wave

transverse

during

unloading

for all values

(I), (2) and (3)(s < s+) the shearing

during

anloading;

during

loading.

in case These

(au /

412G

G) the

(a 12 <

(au / 62 < 0)

shearing

differ

occurs

plastically

from the results

0)

(4.5)

plastically

of the parameters. occurs

(3)(s > s+) the shearing

results

ar >

(4.4)

during

loading

With longitudinal during

loading

during

unloading

is elastic

in [4] (p. 84) where

waves

and in

and elastically and plastic

an error has been

made. 5. Now we will ing transverse tion of (1.1). the

construct

We will

(p, p)-diagram we have

2 is the region wave.

condition

the following

ourselves

boundary

that a uniform to the case

(3.5) and the unloading

picture

of propagation

In region

of the original

and assuming

set y 3 0 and limit

ment,

the transverse

waves

satisfied

(z, I) -plane region

the solution

andlongitudinal

1 we have

: regions

value

problem

motion

is also

where branch

the loading satisfies

1, 3, and 5 are regions

of the longitudinal ~7= uu and 7=

wave,

7,, and in region

branch

in

(3.3).

In the

of uniform

move-

and region

and T- T& ; in region 3 we will set ~7= o,, and T= TV. Introducing [F (p)] Y - X, the third equation in (1.1) uniquely determines the relation

D = ol,

by combina solu-

4 that of

5 we have the notation between

u, 7 and z, 7.

Plane

Henceforth

the state

of stress

will

elastic-plastic

be described

605

waves

by the quantities

z, 7, where

we have

set

zi = [F (pi)]? If we consider see

that at first

the time variation

it is constant.

Then,

of the state

wave

to formulas

(2.9)

x0, T,, to x1, v-~. Then, I3

changed

I \

are two families

of these

to different family

of a curve

the curves

the curve fixed x on a transverse five

must

(b) in region also

lie on curve

(zl, 7,) must This

pass

means

compression

possible

cases

tom

l-4 wave,

zu7a

Hence

The

waves waves

3 has

by fixing

family been

the curve

and curve

l-3

to point

za. Let

through

by 1, and

of possible

on a longitudinal

passages compress-

in: (a) region

(1.2.3), l-3

etc.

When the solution

and, in addition, (z,,

of the problem

the curve

of the second

exists,

point

family

7J.

that

we obtain

and

and draw

passing

denoted

(zt, TJ lying

family

(aa = o:) ; the

is made

and second

first

(a’ = a:)

(za, ~a) in the (2, T) -plane

represents

correspond

curves.

a family

in Fig.

as para-

FIG. 5

through

z2 =

from

it is

in the Cz, 7) -plane.

transverse

FIG. 4

(1.3.4),

until

to zt, 7r and remains

to longitudinal

of the first

it. The point

The

changes

with aa = ai

can be regarded

There

us take the point

FIG. 3

and (2.10) of curves

choice z

(2.9)

representations

the second

etc.

of x, we

the state

constant

wave

metric

corresponds

ion wave,

passes,

value

and (2.10)

it remains

by the transverse

Formulas

with

a fixed

constant.

/

E!C-

with

as the longitudinal

according

t

of stress

z. exp Q2 (so, s2),

the equation

for sp

z1 =

z2

expml (s2,SJ

(zl, ~~1

issuing

from

606

rl.M. Skobccv

[Q>,(sr, SJ + CD2(so, sJ1 == z1

z. exp When sr = a,, the left-hand the

left-hand

curve.

side

When So + 1, the

such

that

the

Figures

4 and

considered.

other

boundary

side

5 show The

point

aide

the

than

cases

for system

side.

this is an s1

the matter.

x of z and 7on

solution

provided

above there

settles

analogously.

Their

aide,

7,) lies

Therefore

This

station

be treated (1.1).

and (zI,

to infinity.

at a fixed can

the right-hand

l-4

to the right-hand

dependence

problems

is smaller

on the curve

of (5.11 tends

is equal

remaining

value

of (5.1)

some

left-hand

left-hand

case

aide

determines

(5.1)

time

for the

It is possible

to treat in a similar

is constructed

way. Apart to treat

from the continuous

discontinuous

stability

criterion,

small

disturbances

Hence

it follows

wave

the

that

of (1.11

corresponding

velocity

ahead

if da/dp

exists

solutions

solutions

of a shock

of the wave

a compressive

wave

front shock

considered

to shock

and

must

than

exists

3, it is possible

In accordance

be greater

less

wave

in section

waves.

those

with

than

the velocities

behind

the wave

the of

front.

if do/dp > 0 and a rarefaction

shock

< 0.

We have da _=_-- 1 daa ds 2a ds dp dP From verse that

(2.7)

and

wave.

(3.2)

Now,

it will

from (5.2)

In the physically condition

(3.51,

it is clear

that

a shock

wave

wave

and

ds/dp

can

exist

only

< 0 on a transin case

(2) and

wave.

important

and the

In conclusion

ds/dp > 0 an a longitudinal

we have

be a rarefaction

(5.2)

case

unloading

the author

where

part

wishes

the

part

(3.31,

a shock

satisfies

to thank

of the (p, @-diagram

loading

N.V.

wave

Zvolinskii

cannot

for valuable

satisfies

exist. advice

and

suggestions.

BIBLIOGRAPHY 1.

Crigorian, %A., of soil

2.

dynamics1

Zvolinskii,

N.V.,

0 osnovnykh PMM,

Ob izluchenii

(Radiation of an elastic No. 1, 1960. 3.

4.

Zvollnskii, of a plastic Grigorian.

N.V., wave S.S.,

(One-dimensional

Vol.

wave

predstavleniiakh

dinamiki

gruntov

oprugoi

volny

in the

Rykov, G.V., Otrazhenie plastichaskoi from an obstacle) PMhf, Vol. 27, No.

volny 1, 1963.

quasi-static

from a spherical

pri sfaricheskom explosion

Chemous’ko,

(Basic

concepts

24, No. 6, 1960.

F.L.,

Odnomemye

motions

of soil)

vzryve ground)

PMM, Vol.

ot pregrady

kvazistaticheskie PMM, Vol.

v grunte

25, No.

dvizheniia 1, 1961.

24,

(Reflection

grunta

Plane

5.

Grigorian, (The

6.

S.S., Chernous’ko,

piston

problem

F.L.,

for the soil

Zadacha

equation)

waves

o porshna

Antsiferov, vozmnshchenii tional

in the dynamics VS.

Rakhmatulin,

of soft

disturbances

soils)

in a nonlinear

srede elastic

dlia

uravnaaii

dinamiki

gruntov

(The

Rykov, G.V., Nekotorye gruntov (Some experimental

Dokl. AN SSSR, Vol. 133, No. 6, 1960.

Kh. A., Rasprostranenie

v nelineino-uprugoi

607

PMM, Vol. 25, No. 5, 1961.

Alekseenko, V.D., Grigorian, S.S., Novgorodov, A.F., eksperimental’nye issladovaniia po dinamike miagkikh investigations

7.

elastic-plastic

szhimaiushche

propagation

medium)

sdvigainahchikh

of compressional

PUM, Vol. 28, No.3, Translated

-

dilata-

1964.

by D.B. McV.