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The penetration of thin rigid body in initially elastic media
002~7225/82/010101-11$03.00/0 Pergamon Press Ltd
In1 J Engng Sci Vol. 20, No. 1, pp. 101411, 1982 Prmted in Great Bntain.
THE ‘PENETRATION OF THIN RIGID BODY IN INITIALLY ELASTIC MEDIA K. A. RAHMATULIN Department of fiaculty of Mechanics and Mathematics, Moscow University, U.S.S.R. and A. G. BAGDOEV Institute of Mechanics of the Academy of Sciences of Armenian SSR, U.S.S.R.
1. INTRODUCTION THEPENETRATION of thin
bodies in half-space occupied by compressible fluid was investigated by several authors[l-31. The statement of problem of penetration into soil was done by Rahmatulin[4], who was also developed the model of soil mechanics, named by “plastic gas”. In[4] the solution of the problem of penetration for mentioned model of medium using the plane sections approximation is done. In the present paper the problem of penetration of the rigid body of revolution into initially elastic medium, occupying half-space, is considered. The solution of purely elastic problem [5,6] shows that near the body behaviour of medium is essentially unelastic. Thus, one must introduce the surface S (Fig. I), separating the region of elastic behaviour of material from the region of fracture T which takes place near the body. The method, developed in the present paper consists with consideration of following disturbed regions. (1) Region of elasticity outside the fracture front. (2) Region of fracture of material behind the front. Then one can believe that the fracture front corresponds to the small values of radial coordinate r and one can use the asymptotic formula of solution of elasticity equation in the form of point sources. The matching with solution of nonlinear equations (taking place in fracture region) on the fracture front allows to obtain the distribution of stress tensor components on the body. The method, developed here is quite general, and can be applied to several models of fracture of medium.
X Fig. 1. The principal scheme of penetration of slender body. 101
K. A.
MkTULIN and A. G.
e by the equation
of i
fractwe frc?m s can be
is cowstantand
ig.
where f(k) ise and T surfaces Mises [IO, 111
raecan a%?%ume
isplacement vector,and
in
The penetration of thin rigid body in initially elastic media
103
one can obtain u=rtF(+r:).
(24
Where F(x) is the arbitrary function, satisfying the condition on the body F(O) = 0. For the thin bodies \exx]4 I+] and then for the intensity of the strain rate tensor one can obtain 4 = 2(nJr). In the elastic region out of the surface S one can seek the solution by the sources methodPI and for small r one can write the equation of S in the following form r = Q(X,t)&,, where &, is constant, determined below. It may be supposed, that &,% 1, but for a small rk Q& is still small. Therefore, one can write down the asymptotic formula for small r in the region r 3 r&,
(2.5) One can suppose that the formula (2.4) holds throughout the surface S in the main order of consideration. Then one can match the solution (2.4), written for large values of (r/rk) with the solution (2.5) to obtain
F(? - r”,)= - d(P - rf)
(2.6)
f(x,t) = $
(2.7)
The stress tensor components in elastic region
Then from (2.5) one can obtain
The condition of conservation of mass and momentum on S taking into account the smallness of z+are reduced to the continuity of U = u, and the stress component a, The first condition is satisfied by the solutions (2.5) and (2.4), (2.61, the latter being written for r% rk in the form U = (r:/2r). In plastic region one can obtain instead of (2.1) the relations
The equation of medium motion near the body in the main order
2+--_UFT-
r
ffee _
0
’
o;r--o&3=-27,
after integration yields a;, = 27, In r f p(x, t).
(2.10)
where Q is the arbitrary function. The continuity of cr, for r = rk(roaccording to (2.7), (2.10) yields 27, In r&OS p(x, t) = -w. k
(2.11)
on the body
acc~r~iug~y to (2.H2)
one can supposethat
E =
co
en one can obtain using unitid Cafa
The penetration of thin rigid body in initially elastic media
10s
The last formula yields for r % rk
and v, = (au/at) is equal to (2.3). For Y> r&o (2.5), (2.16) yield relation
On surface r = rk[O u, is continuous and there is the jump of v,., i.e. the fraCtUre SUrfaCe is compression front. It should be noted that the continuity of o, on surface r = r&o gives cp(x,t) = -2(&/r;&) and due to the (2.10), (2.16) takes place solution (2.17) which slightly differs from (2.12), (2.13). On the other hand if one uses the internal energy expressions [12] on both sides of surface I= rk&I I, = lo(T,) t 2&$
1 = !I)(T*)t $ t $,
10(T) = CT.
Where T is the temperature, es is the constant fracture energy, (42~) is the internal energy of plastic flow at initial instant, e.g. on fracture front, one can obtain from energy equation using the relation (2.15)
(The plastic energy rate Tseimust be integrated on t from the value, corresponding to r = rk[O which yields the zero contribution in expression of internal energy on surface r = rkt0). Then one can believe that es + c(T2- T,) is known constant and obtain instead of (2.16) the following relations f
=iark, 2
pa,-
50, 2
1
a=
1 +$
(2.18)
t cT2 - CT,)
Again for small values of dissipation energy with respect to (7:/p) one can obtain from (2.16) the solution (2.7). The solution (2.12) takes place with to given by (2.16). All obtained solutions have the main term in stress distribution near the body
and are slightly distinguished one from another. Therefore, in present paper only the solution (2.12) will be used.
3. THE MAXIMUM
DEPTH OF PENETRATION
Let us consider the problem of penetration of cone with semiangle p, then rk = pcf - x), and supposing that on the body the following condition holds
ax,= k 1af.r
(3.1)
ritingdown the low Qf the
ody
motion m$“(k)= -
P
afteri~tegratiQ~~ one
e condition f’(t) = 0 gives the
where
can obtain
maxi
b = I$ (Fig. 2). one can obtain formaximal dept
enetration phemmena
fromferm~s powder
the composite
with steel fibres was used. T
with mass of 7.9 g and initiai speed of 715 misec. al loading e413p = 4.8 x values T, = 48 kg/cm*, taken fro The main part of t
conical
Fig. 2. The penetration
and 0.3) formula
ofc0r.ecyiinder body.
is appiie
ateriaE made
The penetration of thin rigid body in initially elastic media
107
calculations of a strength characteristics rS and of shear modulus I_Lfor composite material was made accordingly to [13]. The comparison of theoretical curve (Fig. 3) with experimental data shows, that the model of medium fracture along the sliding lines adopted in present paper is more suitable for considered materials than the model of generation of meridional cracks[5,12]. The eqn (3.5) can be written
(3.6) where
For k, = 0 (3.6) yields
&=i+*.
(3.7)
The eqn (3.3) yields 4 = $‘/DP/(P t k&y]. The last equation takes place for 4 < 1, and for 4 > 1 the eqns (3.6), (3.7) hold. For x = x0, x = (p + + k,)/3P there is the continuous transition from (3.3) to (3.6), (3.7) and C$= 1, (d4ldx) = (1/3x0). One can also investigate the dependence of according to (3.3), (3.6) with respect to [ as curve I on Fig. 4 shows. The minimal value of tends to will be for the values f,,,,x--Gl given by (3.3). For [+O fmax k:: and of - Pcf = fmax) infinity for fixed small values of p. The eqns (3.6), (3.7) are obtained in assumption that along the cylindrical part of the body there is the friction between the body and the medium, the friction coefficient be kl. Probably more realistic case consists with assumption that along the cylindrical part kl = 0 and then one can obtain
for
f > [, and (3.6), (3.7) must be replaced by (3.8)
The corresponding curve II for (P/k,) = (22115)is drawn in Fig. 4.
2.5
1.5 0
IO 20 30 40 Fig. 3. The volume concentration of fibres VF%. 1. The curve obtained experimentalIy; 2. The curve obtained by (3.3); 3. The curve for the cone-cylinder body (K = 0 on cylinder); 4. The curve for the cone-cylinder body (Kf 0 on cylinder). (LIIESVol. 20, No 1-H
108
where /3, rekitions
Y
are
constants,b = j3b”~ Then
e
where
The ~or~e§~~n~i~gcurves III an IV are drawn in Fi to (3.6), (3.8) for Y= 1, i.e. for coni formulas (5,3) and (5.4), respectively. friction on cylindrical part of the b
The penetration of thin rigid body in initially elastic media
109
4. THE PENETRATION INTO SOIL TAKING INTO ACCOUNT VIBRATIONAL DISPLACEMENTS
One can also solve the considered problem of penetration for viscoplastic and for brittle materials 151,as well as, for a soil. For the latter case one can use instead’ of the (2.1) the Coulon condition[4,9], l-k (+Bg = mu”
+ &,
70= 27,.
(4.1)
For generality of consideration let us investigate the problem of vibrational penetration of body when there are initial displacement components uj”, u$‘)due to the action of vibrational force. For example, one can suppose that force is applied at the 0 point (Fig. I), and we have for determination of the problem of Lamb x = 0, cr, = -p(r)Re
eimct+Q, o;, = 0
The solution is written in[14], and for small r one can obtain, that u?= - nc, where K and (a~$~)/&) are given functions of X, t. Then we can modify all the relations of Sections 1 and 2, but for brevity we shall write all equations only for a general case of present section. The equation of motion yields instead of (2.9) (4.2) where 6, is the value of o;, on the body (r = rk). Near the body one can neglect vibrational displacements u’p’, uf) and write the l-dimensional solution (2.4). On hand one can believe, that in the elastic region u,,, = u(O)+ u$‘! where the index 1 is the displacements components connected with penetra& and (2.5) yields uy’ = [f(x, can use the energy relation of Section 2 which for small values of E gives v,, = vl,
the small the other related to t)/r]. One
(4.3) The solution ahead and behind the front r = rkto will be, respectively
f u,=-rxtr,u,=rt
( -I+&-x)d(r’-ri).
The second expression for r % rk pass to former expression. The condition of continuity of gW on the fracture front yields
~(1’: k)
(4.5)
to is determined from the condition (4.1), posed on the surface r = rkro which, taking into account that (T$!)= cr$j, g$) = - u$‘j yields
(4.6) In the absence of vibrational displacements U,(‘)= u!“’= 0 and from (4.4), (4.5) one can obtain (4.7)
5. THE ~~~~T~AT~~~
INTO VISCDPLASTIC
MEDIUM
can
also consider the case ns instead of (2.1) can be fa
e r = r& front. The stale
ensity
of medium
material,
w
is t
e dynamical
visc,~siQ
h are constants.
If the condition on surface r = r& depends only from stress ~Qm~~~e~t~[nz] one can put racture front surface is not sin-&r to ne can see, that in e body surface. One can show, t
e resistance force ist~i~~tion on the body can be written
ectsare essential only for f conicai body mctiorn yie
The
last equation is of
[l] S. S. GRIGQRYAN,SomeProblems of ~~~~y~~rn~c~of §lder Bodies. Thesis dissert. MGLJ. Moscow (19%). [2] A.J.§AGQMONYAN, The Penetration. MGU. Moscow (1977). [3]A. 6. BAGDOEV, The Space Unsteady Problems of Continuous Media Motion with Shock -Waves. Yetevan (1961~.
The penetrationof thin rigid body in initially electric media
111
[4] K. A. RAHMATULIN, A. J. SAGOMONYAN and N. A. ALEKSEEV, The Problems of Soil Dynamics. MGU, Moscow (1964). [5] A. G. BAGDOEV, Iso. ANArm. SSR, Mechanica 300) (1977). [6] A. G. BAGDOEV, A. N. MARTIROSYAN and G. A.-SARKISYAN, Zsv. ANSSSR, M.T.T., No. 3. Moscow (1978). [7] M. E. BACKMAN and W. GOLDSMITH, ht. J. of Engng Sci. 16(l) (1978). [8] L. I. SLEPYAN, Physio-technical Problems of Explotation of useful Minerals. Nauka, Novosibirsk (1978). [9] A. JU. ISHLINSKI, P.M.M. 8(3) (1944). [IO] N. V. ZVOLINSKI. G S. POLIAPOLSKI and L. M. FLITMAN, Isv. AN SSSR, Physica Zemli, No. 1 (1973). [I I] V. 0LSH.X Z. MRLiS and P. PESHINA, The Modem State of Plasticity,. Mir, Moscow (1964). [I?] S. S. GRIGORYAN. P.If.AI. 31(4) I 1967). [ 131 A. G. BOGDOEV and B. C. MINASYAN, Isu. AN Arm.SSR Tech. Seri 32(3) (1979). [l-l] \‘. NWACKI. The Elusticirg. Mir. Moscow (1975). [I] \t’. A. ALLEN. E B. MAYFIELD and U. L. MORRISON, J. Appl. Phys. 28(3) (1957). 1161A. A. ILLUSHIN. The Continuum Mechanics. MGU, Moscow (1978).
(Received 23 August 1980)
In1 J Engng Sci Vol. 20, No. 1, pp. 101411, 1982 Prmted in Great Bntain.
THE ‘PENETRATION OF THIN RIGID BODY IN INITIALLY ELASTIC MEDIA K. A. RAHMATULIN Department of fiaculty of Mechanics and Mathematics, Moscow University, U.S.S.R. and A. G. BAGDOEV Institute of Mechanics of the Academy of Sciences of Armenian SSR, U.S.S.R.
1. INTRODUCTION THEPENETRATION of thin
bodies in half-space occupied by compressible fluid was investigated by several authors[l-31. The statement of problem of penetration into soil was done by Rahmatulin[4], who was also developed the model of soil mechanics, named by “plastic gas”. In[4] the solution of the problem of penetration for mentioned model of medium using the plane sections approximation is done. In the present paper the problem of penetration of the rigid body of revolution into initially elastic medium, occupying half-space, is considered. The solution of purely elastic problem [5,6] shows that near the body behaviour of medium is essentially unelastic. Thus, one must introduce the surface S (Fig. I), separating the region of elastic behaviour of material from the region of fracture T which takes place near the body. The method, developed in the present paper consists with consideration of following disturbed regions. (1) Region of elasticity outside the fracture front. (2) Region of fracture of material behind the front. Then one can believe that the fracture front corresponds to the small values of radial coordinate r and one can use the asymptotic formula of solution of elasticity equation in the form of point sources. The matching with solution of nonlinear equations (taking place in fracture region) on the fracture front allows to obtain the distribution of stress tensor components on the body. The method, developed here is quite general, and can be applied to several models of fracture of medium.
X Fig. 1. The principal scheme of penetration of slender body. 101
K. A.
MkTULIN and A. G.
e by the equation
of i
fractwe frc?m s can be
is cowstantand
ig.
where f(k) ise and T surfaces Mises [IO, 111
raecan a%?%ume
isplacement vector,and
in
The penetration of thin rigid body in initially elastic media
103
one can obtain u=rtF(+r:).
(24
Where F(x) is the arbitrary function, satisfying the condition on the body F(O) = 0. For the thin bodies \exx]4 I+] and then for the intensity of the strain rate tensor one can obtain 4 = 2(nJr). In the elastic region out of the surface S one can seek the solution by the sources methodPI and for small r one can write the equation of S in the following form r = Q(X,t)&,, where &, is constant, determined below. It may be supposed, that &,% 1, but for a small rk Q& is still small. Therefore, one can write down the asymptotic formula for small r in the region r 3 r&,
(2.5) One can suppose that the formula (2.4) holds throughout the surface S in the main order of consideration. Then one can match the solution (2.4), written for large values of (r/rk) with the solution (2.5) to obtain
F(? - r”,)= - d(P - rf)
(2.6)
f(x,t) = $
(2.7)
The stress tensor components in elastic region
Then from (2.5) one can obtain
The condition of conservation of mass and momentum on S taking into account the smallness of z+are reduced to the continuity of U = u, and the stress component a, The first condition is satisfied by the solutions (2.5) and (2.4), (2.61, the latter being written for r% rk in the form U = (r:/2r). In plastic region one can obtain instead of (2.1) the relations
The equation of medium motion near the body in the main order
2+--_UFT-
r
ffee _
0
’
o;r--o&3=-27,
after integration yields a;, = 27, In r f p(x, t).
(2.10)
where Q is the arbitrary function. The continuity of cr, for r = rk(roaccording to (2.7), (2.10) yields 27, In r&OS p(x, t) = -w. k
(2.11)
on the body
acc~r~iug~y to (2.H2)
one can supposethat
E =
co
en one can obtain using unitid Cafa
The penetration of thin rigid body in initially elastic media
10s
The last formula yields for r % rk
and v, = (au/at) is equal to (2.3). For Y> r&o (2.5), (2.16) yield relation
On surface r = rk[O u, is continuous and there is the jump of v,., i.e. the fraCtUre SUrfaCe is compression front. It should be noted that the continuity of o, on surface r = r&o gives cp(x,t) = -2(&/r;&) and due to the (2.10), (2.16) takes place solution (2.17) which slightly differs from (2.12), (2.13). On the other hand if one uses the internal energy expressions [12] on both sides of surface I= rk&I I, = lo(T,) t 2&$
1 = !I)(T*)t $ t $,
10(T) = CT.
Where T is the temperature, es is the constant fracture energy, (42~) is the internal energy of plastic flow at initial instant, e.g. on fracture front, one can obtain from energy equation using the relation (2.15)
(The plastic energy rate Tseimust be integrated on t from the value, corresponding to r = rk[O which yields the zero contribution in expression of internal energy on surface r = rkt0). Then one can believe that es + c(T2- T,) is known constant and obtain instead of (2.16) the following relations f
=iark, 2
pa,-
50, 2
1
a=
1 +$
(2.18)
t cT2 - CT,)
Again for small values of dissipation energy with respect to (7:/p) one can obtain from (2.16) the solution (2.7). The solution (2.12) takes place with to given by (2.16). All obtained solutions have the main term in stress distribution near the body
and are slightly distinguished one from another. Therefore, in present paper only the solution (2.12) will be used.
3. THE MAXIMUM
DEPTH OF PENETRATION
Let us consider the problem of penetration of cone with semiangle p, then rk = pcf - x), and supposing that on the body the following condition holds
ax,= k 1af.r
(3.1)
ritingdown the low Qf the
ody
motion m$“(k)= -
P
afteri~tegratiQ~~ one
e condition f’(t) = 0 gives the
where
can obtain
maxi
b = I$ (Fig. 2). one can obtain formaximal dept
enetration phemmena
fromferm~s powder
the composite
with steel fibres was used. T
with mass of 7.9 g and initiai speed of 715 misec. al loading e413p = 4.8 x values T, = 48 kg/cm*, taken fro The main part of t
conical
Fig. 2. The penetration
and 0.3) formula
ofc0r.ecyiinder body.
is appiie
ateriaE made
The penetration of thin rigid body in initially elastic media
107
calculations of a strength characteristics rS and of shear modulus I_Lfor composite material was made accordingly to [13]. The comparison of theoretical curve (Fig. 3) with experimental data shows, that the model of medium fracture along the sliding lines adopted in present paper is more suitable for considered materials than the model of generation of meridional cracks[5,12]. The eqn (3.5) can be written
(3.6) where
For k, = 0 (3.6) yields
&=i+*.
(3.7)
The eqn (3.3) yields 4 = $‘/DP/(P t k&y]. The last equation takes place for 4 < 1, and for 4 > 1 the eqns (3.6), (3.7) hold. For x = x0, x = (p + + k,)/3P there is the continuous transition from (3.3) to (3.6), (3.7) and C$= 1, (d4ldx) = (1/3x0). One can also investigate the dependence of according to (3.3), (3.6) with respect to [ as curve I on Fig. 4 shows. The minimal value of tends to will be for the values f,,,,x--Gl given by (3.3). For [+O fmax k:: and of - Pcf = fmax) infinity for fixed small values of p. The eqns (3.6), (3.7) are obtained in assumption that along the cylindrical part of the body there is the friction between the body and the medium, the friction coefficient be kl. Probably more realistic case consists with assumption that along the cylindrical part kl = 0 and then one can obtain
for
f > [, and (3.6), (3.7) must be replaced by (3.8)
The corresponding curve II for (P/k,) = (22115)is drawn in Fig. 4.
2.5
1.5 0
IO 20 30 40 Fig. 3. The volume concentration of fibres VF%. 1. The curve obtained experimentalIy; 2. The curve obtained by (3.3); 3. The curve for the cone-cylinder body (K = 0 on cylinder); 4. The curve for the cone-cylinder body (Kf 0 on cylinder). (LIIESVol. 20, No 1-H
108
where /3, rekitions
Y
are
constants,b = j3b”~ Then
e
where
The ~or~e§~~n~i~gcurves III an IV are drawn in Fi to (3.6), (3.8) for Y= 1, i.e. for coni formulas (5,3) and (5.4), respectively. friction on cylindrical part of the b
The penetration of thin rigid body in initially elastic media
109
4. THE PENETRATION INTO SOIL TAKING INTO ACCOUNT VIBRATIONAL DISPLACEMENTS
One can also solve the considered problem of penetration for viscoplastic and for brittle materials 151,as well as, for a soil. For the latter case one can use instead’ of the (2.1) the Coulon condition[4,9], l-k (+Bg = mu”
+ &,
70= 27,.
(4.1)
For generality of consideration let us investigate the problem of vibrational penetration of body when there are initial displacement components uj”, u$‘)due to the action of vibrational force. For example, one can suppose that force is applied at the 0 point (Fig. I), and we have for determination of the problem of Lamb x = 0, cr, = -p(r)Re
eimct+Q, o;, = 0
The solution is written in[14], and for small r one can obtain, that u?= - nc, where K and (a~$~)/&) are given functions of X, t. Then we can modify all the relations of Sections 1 and 2, but for brevity we shall write all equations only for a general case of present section. The equation of motion yields instead of (2.9) (4.2) where 6, is the value of o;, on the body (r = rk). Near the body one can neglect vibrational displacements u’p’, uf) and write the l-dimensional solution (2.4). On hand one can believe, that in the elastic region u,,, = u(O)+ u$‘! where the index 1 is the displacements components connected with penetra& and (2.5) yields uy’ = [f(x, can use the energy relation of Section 2 which for small values of E gives v,, = vl,
the small the other related to t)/r]. One
(4.3) The solution ahead and behind the front r = rkto will be, respectively
f u,=-rxtr,u,=rt
( -I+&-x)d(r’-ri).
The second expression for r % rk pass to former expression. The condition of continuity of gW on the fracture front yields
~(1’: k)
(4.5)
to is determined from the condition (4.1), posed on the surface r = rkro which, taking into account that (T$!)= cr$j, g$) = - u$‘j yields
(4.6) In the absence of vibrational displacements U,(‘)= u!“’= 0 and from (4.4), (4.5) one can obtain (4.7)
5. THE ~~~~T~AT~~~
INTO VISCDPLASTIC
MEDIUM
can
also consider the case ns instead of (2.1) can be fa
e r = r& front. The stale
ensity
of medium
material,
w
is t
e dynamical
visc,~siQ
h are constants.
If the condition on surface r = r& depends only from stress ~Qm~~~e~t~[nz] one can put racture front surface is not sin-&r to ne can see, that in e body surface. One can show, t
e resistance force ist~i~~tion on the body can be written
ectsare essential only for f conicai body mctiorn yie
The
last equation is of
[l] S. S. GRIGQRYAN,SomeProblems of ~~~~y~~rn~c~of §lder Bodies. Thesis dissert. MGLJ. Moscow (19%). [2] A.J.§AGQMONYAN, The Penetration. MGU. Moscow (1977). [3]A. 6. BAGDOEV, The Space Unsteady Problems of Continuous Media Motion with Shock -Waves. Yetevan (1961~.
The penetrationof thin rigid body in initially electric media
111
[4] K. A. RAHMATULIN, A. J. SAGOMONYAN and N. A. ALEKSEEV, The Problems of Soil Dynamics. MGU, Moscow (1964). [5] A. G. BAGDOEV, Iso. ANArm. SSR, Mechanica 300) (1977). [6] A. G. BAGDOEV, A. N. MARTIROSYAN and G. A.-SARKISYAN, Zsv. ANSSSR, M.T.T., No. 3. Moscow (1978). [7] M. E. BACKMAN and W. GOLDSMITH, ht. J. of Engng Sci. 16(l) (1978). [8] L. I. SLEPYAN, Physio-technical Problems of Explotation of useful Minerals. Nauka, Novosibirsk (1978). [9] A. JU. ISHLINSKI, P.M.M. 8(3) (1944). [IO] N. V. ZVOLINSKI. G S. POLIAPOLSKI and L. M. FLITMAN, Isv. AN SSSR, Physica Zemli, No. 1 (1973). [I I] V. 0LSH.X Z. MRLiS and P. PESHINA, The Modem State of Plasticity,. Mir, Moscow (1964). [I?] S. S. GRIGORYAN. P.If.AI. 31(4) I 1967). [ 131 A. G. BOGDOEV and B. C. MINASYAN, Isu. AN Arm.SSR Tech. Seri 32(3) (1979). [l-l] \‘. NWACKI. The Elusticirg. Mir. Moscow (1975). [I] \t’. A. ALLEN. E B. MAYFIELD and U. L. MORRISON, J. Appl. Phys. 28(3) (1957). 1161A. A. ILLUSHIN. The Continuum Mechanics. MGU, Moscow (1978).
(Received 23 August 1980)