The principle of virtual work and the third order wave on continua with second order constitutive equations
The generalized form of the principle of virtual work is obtained, when the virtual work is considered as a time integral of virtual power. The corresponding this form Euler – Lagrange equation includes the divergence of the Lie derivative of stress. So, the equation of motion on the stress rate fi...
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| Cite this: | The principle of virtual work and the third order wave on continua with second order constitutive equations / G. Beda // Прикладная механика. — 2010. — Т. 46, № 10. — С. 136-143. — Бібліогр.: 16 назв. — англ. |
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| citation_txt | The principle of virtual work and the third order wave on continua with second order constitutive equations / G. Beda // Прикладная механика. — 2010. — Т. 46, № 10. — С. 136-143. — Бібліогр.: 16 назв. — англ. |
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| description | The generalized form of the principle of virtual work is obtained, when the virtual work is considered as a time integral of virtual power. The corresponding this form
Euler – Lagrange equation includes the divergence of the Lie derivative of stress. So, the equation of motion on the stress rate field is one of the results of this paper. When bying studied the third order wave, a generalization of the acoustic tensor is obtained. The generalized acoustic tensor seems the most important result of these paper. This one can also be found by investigating the acceleration wave.
Отримано узагальнений принцип віртуальних зміщень, коли віртуальні зміщення
розглядаються як інтеграл по часу від віртуальної енергії. Рівняння Ейлера-Лагранжа дають рівняння для дивергенції похідної Лі по напруженнях. Рівняння руху в термінах поля швидкості напружень є одним з нових результатів цієї статті. При вивченні хвилі третього порядку отримано узагальнення акустичного тензора, яке можна вважати найбільшим досягненням у проведеному дослідженні. Цей результат може бути отриманий також при дослідженні хвилі прискорення.
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2010 П РИКЛАДНАЯ МЕХАНИКА Том 46, № 10
136 ISSN0032–8243. Прикл. механика, 2010, 46, № 10
G . B É D A
THE PRINCIPLE OF VIRTUAL WORK AND THE THIRD ORDER WAVE
FOR CONTINUA WITH SECOND ORDER CONSTITUTIVE EQUATIONS
Budapest University of Technology and Economics
Műegyetem rkp. 3-5. H-1111, Budapest, Hungary; e-mail: beda@mm.bme.hu
Abstract: The generalized form of the principle of virtual work is obtained, when the
virtual work is considered as a time integral of virtual power. The corresponding this form
Euler – Lagrange equation includes the divergence of the Lie derivative of stress. So, the
equation of motion on the stress rate field is one of the results of this paper. When bying
studied the third order wave, a generalization of the acoustic tensor is obtained. The general-
ized acoustic tensor seems the most important result of these paper. This one can also be
found by investigating the acceleration wave.
Key words: motion equation, Lie derivative of stress, generalized principle of virtual
work, compatibility conditions, the third order wave, generalized acoustic tensor.
Introduction.
The investigation of the third order wave necessitates the knowledge of the dynamical
compatibility equation. This equation rises from the first equation of motion in case of the
acceleration wave. Now it needs the time derivative of the first equation of motion. The ma-
terial time derivative isn't simple in the current configuration. Using the principle of virtual
power, namely the principle of virtual work, the derivative will be obvious and indisputable.
We assume that the integral of the virtual power with respect to time is the virtual work.
Hence, from the principle of virtual work the time derivative of the first equation of motion
can be obtained and then the dynamical compatibility equation can be calculated. The time
derivative of the first equation of motion will be called the equation of motion on the stress
rate field. Many authors have dealt with this question when the body was in equilibrium [8,
9, 10]. The third order wave can be investigated by using the compatibility equations (dy-
namic, kinematic and constitutive). When the constitutive equation is a system of first order
nonlinear partial differential equations the investigation of wave propagation is more con-
venient by use of the third order wave.
1. The principle of virtual work.
In continuum mechanics, the principle of virtual power is writing as follows
; ,
p
kl k k
k l k k
V V A
t v dV q v dV p v dA
∗ ∗ ∗
= +∫ ∫ ∫ % (1)
where ;, , and kl k
k k l
t v v q
∗ ∗ denote the Cauchy stress, the virtual velocity, the virtual velocity
gradient and the difference between the body force and the force of inertia in domain V ,
respectively, and
k
p~ is the surface force on boundary surface
p
A (
v p
A A A= + , the ve-
locity
k
v
~
is known on A
v
).
137
The stress tensor in the satisfies the second Cauchy equation of motion, that is,
t t
kl lk
= .
Assume as a starting point that the integral of the power for a given period t t1 2,
means the work during this period. Thus, the integrated with respect to time t equation (1)
gives
2 2 2
1 1 1
;
p
t t t
kl k k
k l k k
t V t V t A
t v dVdt q v dVdt p v dAdt
∗ ∗ ∗
= +∫ ∫ ∫ ∫ ∫ ∫ % . (2)
As it can be seen, the virtual deformation rate v
kl
∗
on the left hand side of the equation
has been replaced with virtual velocity gradient v
k l; .∗
This replacement leaves the product
t v
kl
kl
∗
unchanged since t t
kl lk
= . The material time derivative of the deformation gradient
is
p
K
k
p
k
K
xvx ,;, =& .
Then
K
p
k
K
k
p
Xxv ,,;
&= . (3)
With displacement vector u used r R u= + and the derivatives with respect to time
and X
K
are written in indexed form as
, ; , :and
k k k k q k
K q K K
v u x u x u= = ≡& & & &
respectively, the formula (3) becomes
k
p
K
p
k
K
k
p
uXuv ;,:;
&& == . (4)
With the volume integral on the left hand side of (2) transformed to the initial configu-
ration, the integrals with respect to time and over volume V
o
can be interchanged
2 2
1 1
: ,
o o
t t
l k K k
k K l o k o
V t V t
t u X J dt dV J q u dt dV
∗ ∗
= +∫ ∫ ∫ ∫& &
2
1
, ;
o
p
t
kl K o
k l K
tA
J t u X dt dA
∗
+ ∫ ∫ & .
o
dV
dV
J = (5)
Consider now the integrals with respect to time, one after the other:
( ) ( )
2 2
1 1
. .
, : , : , : .
t t
l K k l K k l K k
k l K k l K k l K
t t
J t X u dt J t X u J t X u dt
∗ ∗ ∗ = − ∫ ∫&
The first integral can be calculated from time t1 to t2 on the right hand side
( ) ( )
2 2
2
1
1 1
, : ; ; , :; .
t t
t
p p ll K k l k q p K k
k l K k l q p Kk k k l
t
t t
J t X u dt Jt u J t v t t v X u dt
∗ ∗ ∗
= − + −∫ ∫ && (6)1
After similar transformations, the first integral with respect to time on the right hand
side of (5) is as follows:
138
( ) ( )
2 2
2
1
1 1
; .
t t
t
k k k s k
k k s k
t
t t
J q u dt Jq u J q v q u dt
∗ ∗ ∗
= − +∫ ∫& & (6)2
Here again the virtual displacement are not vanishing u
k
∗
≠ 0 at time 1 2 and t t in
(6)2 .
Finally, after transformation of the second integral on the right hand side of (5)
( ) ( )
2 2
2
1
1 1
, , ; ,; .
t t
t
kl K kl K kp s kp kl p K
l k l k s p kl
t
t t
J t X u dt Jt X u J t v t t v X u dt
∗ ∗ ∗
= − + −∫ ∫ && 3(6)
With (6)1 2 3, , substituted into (5) and after proper rearrangement, the equation of virtual
work is [1]
( ) ( ) ( )
2
2 2
1
1
1
; ; ; ;
p
t
t t
kp kq p kp s lk k k kl
q s k p l k l k
t
t V V A
t
t t v t v u dV dt t q u dV p t n u dA
∗ ∗ ∗ − + = − + + − +
∫ ∫ ∫ ∫& %
( ) ( )
2 2
1 1
; ; ; .
p
t t
k k s kp kp s kl p
s k s p kl
t V t A
q q v u dV dt t t v t v n u dAdt
∗ ∗
+ + + + −∫ ∫ ∫ ∫ &&
Therefore
t q
k l
l
k; + = 0 is the first Cauchy equation of motion,
k kp
p
p t n=% is the dynamic
boundary condition on A
p
.
The principle of virtual work is as follows
( )
2
0
; ; ;
t
kp kq p kp s
q s k p
t V
t t v t v u dV dt
∗
− + =∫ ∫ &
( ) ( )
1 1
0 0
; ;; .
p
t t
k k s kp kh p kp s
s k s p kh
t V t A
q q v u dV dt t t v t v n u dAdt
∗ ∗
= + + − +∫ ∫ ∫ ∫ && (7)
Equation (7) refers to continua and its any part. Otherwise, on the basis of what has
been said above, the equation given below is obtained after suitable mathematical transfor-
mation
( ) ( ); ; ;
; ;
ij hj i ij h i i h i
V h h h
j j
L t t v t v q q v vρ+ + + + =& && (8)
supposing that the Cauchy equations of motion and the boundary condition are satisfied.
Here ( )
ij
V
tL denotes the Lie derivative of Cauchy′s stress tensor, that is,
( ) h
jih
h
ihjijij
V
vtvtttL ;; −−= & ;
(9)
k
q is the body force density; ρ is the mass density and satisfies the continuity equation.
Equation (8) is the equation of motion on the stress rate (or Lie derivative of stress)
field, [8, 9, 10, 12].
2. The third order wave.
When the basic quantities
kl
klk
atv ,, and the first derivatives of them are continuous,
but the second derivatives have a jump by crossing surface ( ) 0, =tx
k
ϕ , we speak about
139
the third order waves [2]. Let us denote the jump of some quantity ; by
k
p
v ; .k
p
v When the
velocity gradient is v
p
k
;
, in case of the wave of order three ; ;0, but 0.k k
p pq
v v= ≠ Thus
in (9) ( ) 0
kp
V
L t = , but ( )
;
0kp
V
q
L t ≠ and so on.
Now the dynamic condition of the third order wave is
( )( ) ; ;
;
kl pq k kj l k
V pq lj
l
L t t v t v vρ+ + = && . (10)
Let the kinematic equation [4, 5] be
( )( ) ( ) ( ); ; ; ; ;
k k k k l
V V ij ij kj i ik j lj kj i lk i i
L L a a a v a v a a v a v v= + + + + + +
�
&& &
( ); ; ; .k k l
il ik l kl i i
a a v a v v+ + +&
When the Lie derivative of the velocity field is L
V
, expression v£
V
L
t
∂
≡ +
∂
in (10) is a
generalization of the velocity [3]. The Euler strain tensor is a
ij
. As it is well known
( )V ij ij
L a v= , for the strain rate , thus the kinematic compatibility condition of the third or-
der wave is
j
k
ik
i
k
kjijij
vavaav ;; ++= &&&& (11)
namely
( )( ) ( ) ; ;
l l
V V ij V ij ij lj i il j
L L a L v v v v v v≡ = + +& .
It can easily be shown that ( ); ;
k k
i i
v v
•
≠& , but ( ); ;
k k
i i
v v
•
=& and this property is the
same for the second derivatives of all other functions.
Let the constitutive equation be
( ) ( ) ( ) ( ) ( )( )
2 2
; ; ;;;
, , , , , , , , , , 0
ij ij ij ij
V V kh V pq V pq pq rs V pq pq r pq pq
rk
f L t L t t L a L a a L v v t a v
α
= , (12)
.6,5,4,3,2,1=a
where ( )( )
2
..
V V V
L L L≡ . The equations contain the second order derivatives with respect to
space and time, hence they are called the second order constitutive equations. The constitu-
tive compatibility conditions can be obtained from equation (12) by calculation after and
before the wave front.
( ) ( ) ( ) ( )( ) ( )( ) 0,....,,...,,....,
2
;;
22
=−++
pq
ij
Vpq
r
pqV
r
pqV
ij
V
ij
V
vtLfvaLaLtLtLf
αα .
(13)
Notation means the jump across the wavefront, for example
( )
∂
∂
+
∂
∂
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
=
q
iqj
p
pji
p
p
p
pijij
V
x
t
x
t
x
v
tx
v
t
tL
ϕ
λ
ϕ
λ
ϕϕϕϕ
γ
~~~
2
2
140
or
;pq rs pq r s
a
x x
ϕ ϕ
α
∂ ∂
=
∂ ∂
% and so on.
The system (13) is the system of first order partial differential equations. Using the
characteristic equation of (13), the constitutive compatibility condition is obtained in form
ˆ
ˆ
ˆ0,..., 1,2,3,4
i
i
f
i
α
ϕ
ϕ
∂
= =
∂
, (14)
where ˆ ˆ 4
if
i i t xx
ϕ ϕ ϕ
ϕ
∂ ∂ ∂
= =
∂ ∂∂
.
In the following we use Cartesian coordinates. When the jumps in the second deriva-
tives of the stress, strain tensors and the velocity on the surface ( ) 0, =tx
k
ϕ are denoted by
, , ,
ij k
ij
γ α λ the unit normal vector of the wavefront is introduced
k
k
pq
p q
x
n
g
x x
∂ϕ
∂
∂ϕ ∂ϕ
∂ ∂
≡
and the wave propagation velocity is denoted
k
k
C c v n= − ,
the equations (10), (11) and (14) lead to the dynamic
kl k
l
n Cγ ρ λ= − ; (15)
kinematic
( ) ( )
1
2 2
2
k
ij i kj kj j ik ik
n a g n a g
C
α λ = − + − (16)
and the constitutive compatibility equations [2 , 4]
( ) ( )
2 2
4 4
ij k kl pq pqr
ij ij k ij k l pq r r s
S C S n C S n n E C E n C E n n
α α α α α α
γ α− + + − + +
+ 4 4 4
1
.
2
r pq pq pq k ij
pq r rpq rpq ij k r
S T E A W G C S n T
α α α α
λ
− − −
+
1
0.
2
pqs pqk
rpqs k rpqs
E A W n G
α α
+ + =
(17)
Here the notations are used:
( )
4 2
;
ij
ij
V
f
S
L t
α
α
∂
=
∂
( )
;
;
k
ij
ij
V
k
f
S
L t
α
α
∂
=
∂
;
ijkl ij
kl
f
S
t
α
α
∂
=
∂
;
( )
4 2
;pq
V pq
f
E
L a
α
α
∂
=
∂
( )
;
;pqr
V pq
r
f
E
L a
α
α
∂
=
∂
pqrs
pqrs
f
E
a
α
α
∂
=
∂
;
141
( )
pqV
pq
vL
f
∂
∂
=
α
α
4 W ;
kpq
pqk
v
f
W
;∂
∂
=
α
α
;
pjiqqjpipijq
nnannaA += ;
pjiqqjpipijq
nngnngG += ;
j
i
p
s
sq
j
i
q
s
ps
ij
pq
ngntngntT += ,
where
pi
g and
p
i
g are the metric tensors and
2
V
L is the second order Lie derivative.
Substituting (15) and (16) into (17) we can write the wave propagation equation for the
stress amplitude
β
γγγ ~jiij
=
{ } 022 234
=+++−
β
αβαβαβαβαβ
γρρ HCDCBCSCS , (18)
where the following notations were used:
( )
;4ij
SS
αβαβ
=
( )
;
k
k
ij
nSS
αβαβ
=
( ) ( )4 4 42 2 ;kh pq pq pq
k h pqp ij q p ij q
B S n n E G S T W G
αβ αβ α α β αβ β
ρ= + − +
( )
( )
( )
,2
qijpk
pqk
ij
pss
k
k
psqijpr
pqr
GnWTnSGnED
βα
β
αβααβ
−+−=
( ) ( )
( )
qijpqijpsr
pqrs
GAnnEH
ββααβ
−= 2 .
The interpretation of index β is: ( )
( )
( )
( )
1
... ;
...
2 ... .
i
ij
i j
if i j
if i j
β
+ +
=
=
≠
The determinant of the matrix in bracket }{ in (18) is zero, because
β
γ is not zero
{} .0det =
This is the equation of the propagation of the third order wave, being the 24-th order al-
gebraical equation for the propagation velocity C . Matrix {} can be considered as the
characteristic matrix (6x6) of a generalization of the acoustic tensor.
The matrix of acoustic tensor can be obtained from (18). Let us denote the coefficients
of C in the form of 6x6 matrices by .HD,B,,SS, The the acoustic matrix
HDBSS ++++ CCCC
234
(19)
gives the wave propagation, if it is multiplied by γγγγ and set equal to zero
( ) 0HDBSS =++++ γγγγCCCC
234
.
The most general acoustic tensor can be obtained from (19), when the coefficients of
CCCC ,,, 234
and C
o
have been denoted in form of 6x6 matrices .HD,B,,SS, By
introducting the inverse
1
S
−
and unit matrices I , the wave propagation equation [6, 7] is
142
=
∗∗∗
∗∗
∗
∗∗∗
∗∗
∗
−−−− γ
γ
γ
γ
γ
γ
γ
γ
SSBSDSHS
I000
0I00
00I0
1111
C
.
Now the generalized acoustic matrix can be written [6] in the following form
-1 -1 -1 -1
-IC I 0 0
0 -IC I 0
0 0 -IC I
S H S D S B S S - IC
. (20)
The elements of the acoustic matrix are 6x6 matrices and the final matrix is 24x24.
3. Two special cases.
i). Let H and D be identical to zero; then equation (18) is
{ } 0γ22 212
=+ρ−ρ BSS CCC .
We can designate such bodies as the quasi-viscoelastic bodies.
ii). Let us write the one dimensional form of equation (18)
( ) ( )
4 3 2
2 2 2
tt tx xx tt tt t
v
f C f C f f f f C
σ σ σ ε σ
ρ ρ σ − + + − + +
( )2 0
tx tx x xx
v
f f f C f
ε σ ε
σ γ+ − + − + =
, (21)
where notations are as usual: stress σ , strain ε , velocity of strain v and subscripts denote
the partial derivatives.
The algebraic equation (21) has at least two positive and negative real roots. The coeffi-
cients of that equation satisfy this condition. Similar conditions can also be supposed to the
equation (18).
These conditions for the material coefficients enable us to approximate the constitutive
equations, when the suitable experiments are performed.
Р Е З ЮМ Е . Отримано узагальнений принцип віртуальних зміщень, коли віртуальні зміщення
розглядаються як інтеграл по часу від віртуальної енергії. Рівняння Ейлера-Лагранжа дають рівняння
для дивергенції похідної Лі по напруженнях. Рівняння руху в термінах поля швидкості напружень є
одним з нових результатів цієї статті. При вивченні хвилі третього порядку отримано узагальнення
акустичного тензора, яке можна вважати найбільшим досягненням у проведеному дослідженні. Цей
результат може бути отриманий також при дослідженні хвилі прискорення.
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P. 123 – 131.
*From the Editorial Board: The article corresponds completely to submitted manuscript.
Поступила 17.06.2009 Утверждена в печать 15.06.2010
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| id | nasplib_isofts_kiev_ua-123456789-95446 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0032-8243 |
| language | English |
| last_indexed | 2025-12-07T13:37:19Z |
| publishDate | 2010 |
| publisher | Інститут механіки ім. С.П. Тимошенка НАН України |
| record_format | dspace |
| spelling | Beda, G. 2016-02-26T18:18:21Z 2016-02-26T18:18:21Z 2010 The principle of virtual work and the third order wave on continua with second order constitutive equations / G. Beda // Прикладная механика. — 2010. — Т. 46, № 10. — С. 136-143. — Бібліогр.: 16 назв. — англ. 0032-8243 https://nasplib.isofts.kiev.ua/handle/123456789/95446 The generalized form of the principle of virtual work is obtained, when the virtual work is considered as a time integral of virtual power. The corresponding this form Euler – Lagrange equation includes the divergence of the Lie derivative of stress. So, the equation of motion on the stress rate field is one of the results of this paper. When bying studied the third order wave, a generalization of the acoustic tensor is obtained. The generalized acoustic tensor seems the most important result of these paper. This one can also be found by investigating the acceleration wave. Отримано узагальнений принцип віртуальних зміщень, коли віртуальні зміщення розглядаються як інтеграл по часу від віртуальної енергії. Рівняння Ейлера-Лагранжа дають рівняння для дивергенції похідної Лі по напруженнях. Рівняння руху в термінах поля швидкості напружень є одним з нових результатів цієї статті. При вивченні хвилі третього порядку отримано узагальнення акустичного тензора, яке можна вважати найбільшим досягненням у проведеному дослідженні. Цей результат може бути отриманий також при дослідженні хвилі прискорення. en Інститут механіки ім. С.П. Тимошенка НАН України Прикладная механика The principle of virtual work and the third order wave on continua with second order constitutive equations Принцип виртуальных перемещений и волны третьего порядка в средах с определяющими уравнениями второго порядка Article published earlier |
| spellingShingle | The principle of virtual work and the third order wave on continua with second order constitutive equations Beda, G. |
| title | The principle of virtual work and the third order wave on continua with second order constitutive equations |
| title_alt | Принцип виртуальных перемещений и волны третьего порядка в средах с определяющими уравнениями второго порядка |
| title_full | The principle of virtual work and the third order wave on continua with second order constitutive equations |
| title_fullStr | The principle of virtual work and the third order wave on continua with second order constitutive equations |
| title_full_unstemmed | The principle of virtual work and the third order wave on continua with second order constitutive equations |
| title_short | The principle of virtual work and the third order wave on continua with second order constitutive equations |
| title_sort | principle of virtual work and the third order wave on continua with second order constitutive equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/95446 |
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