Kounadis A.N., Gdoutos E.E. (Eds.) Recent Advances in Mechanics

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Variational Approach to Static and Dynamic Elasticity Problems 139

Let us consider a functional from such families which has the dimension of

energy

33 3

0, :

ϕϕ

Φ= Ω≥ =

∫

ηξ

(27)

and formulate the following variational problem: to find unknown functions

and σ minimizing the functional

(, ) minΦ→

(28)

under constraints (3), (4),(10)–(12).

It is possible to transform the integral in Eq. (27) to the form

00 0

111

:, :, :

222

WW W

WdWdWd

Φ= + −

=Ω=Ω=Ω

ΩΩΩ

∫∫∫

σ ε σε σε

(29)

where

W and W

are the strain and stress energies, respectively, and W is the

work of external forces. Note that

W and W

are positively defined functionals,

and W is a bilinear functional independent explicitly of elastic material properties.

Integrating by parts the work of external forces W and using equilibrium equa-

tion (3) give

()

2Wdd

=⋅⋅+⋅Ω

∫∫

nu fuσ

(30)

Firstly, let us consider the case when the stress (Eq. (10)) and displacement (Eq.

(11)) boundary conditions are only given. The mixed boundary conditions (12)

(elastic foundation) are absent. Then, after substituting Eq. (30) into Eq. (29), the

functional

Φ will have the form

312

uiiii

W d qud W qud

γγ

Φ= − ⋅ Ω− + − ≥

∑∑

∫∫ ∫

(31)

Taking into account the expressions for the potential energy in Eq. (13) and com-

plementary energy in Eq. (14) one can present inequality (31) as follows

(, ) () () 0

Φ=Π+Π≥uuσσ

(32)

Thus, the following theorem formulated in [18, 21] is valid for this case: for any

equilibrium stress fields σ under boundary conditions (10) and any kinematically

admissible displacement fields

u under boundary conditions (11) the sum of the

potential and complementary energies is nonnegative.

140 G.V. Kostin and V.V. Saurin

Consequently, this constrained minimization problem is equivalent to the inde-

pendent minimization of the potential energy over displacements

u under bound-

ary condition (11) and the complementary energy over equilibrium stresses σ un-

der boundary condition (10)

min ( , ) min ( ) min ( ) 0

Φ=Π+Π=

σσ

(33)

It is worth to emphasize that for such boundary value problems the approximate

displacement fields are found using the minimum principle for potential energy

(13) while based on the minimum principle for complementary energy (14) the

stress fields are obtained. Moreover, the method of integrodifferential relations

enables one to construct effective bilateral estimates for the total elastic energy

stored ( , ) ( ) ( )

WWW

∗∗ ∗ ∗

==uuσσ as a function of the actual displacements

∗

and stresses

∗

σ . In particular, if 0

u = (

∈ ) or

=∅ (

1, 2, 3i =

) then for ar-

bitrary admissible displacements

u and stresses σ the following estimates hold

() ( , ) ()

∗

−Π ≤ ≤ Πuuσσ

(34)

On the other hand, if

q = (

∈ ) or

=∅ (

1, 2, 3i =

) then the inequalities

are valid

() ( , ) ()

∗

−Π ≤ ≤ Πuuσσ

(35)

The positivity of the second variation

Φ with respect to u and σ follows

from the corresponding properties of the second variations

Π≥ and

Π≥ . The equality

Φ= is valid if and only if all the components of the

strain tensor

()

uε and stress tensor

σ are equal to zero [22].

Let us consider the case when the mixed conditions (12) (elastic foundation)

are given on some part of the boundary

(

≠∅). One can show that the de-

composition of the variational problem (28) on two independent problems in dis-

placement and stresses, respectively, is not possible. Indeed, after integrating by

parts the work of external forces W has the form

33 3

122

11 1

123

iii

ii ii ii

ii i

Wd qud qud qud

γγγ

== =

=⋅Ω+ + +

∑∑∑

∫∫ ∫ ∫

(36)

The last term in relations (36) depends explicitly on both the displacements

u and

stresses σ .

Taking into account that the variations

u and

σ on

(

1, 2, 3i =

) are re-

lated by the equality 0

ij j i i

δσ κ δ

+= the second variation

Φ can be pre-

sented as follows

() () ( ) 0, 0

uiii

WW ud

δδδκδγκ

Φ= + + ≥ >

∑

∫

u σ

(37)

Variational Approach to Static and Dynamic Elasticity Problems 141

The equality

Φ= is valid if and only if all components of the strain tensor

()

uε , stress tensor

σ as well as the components

of the displacement vec-

tor on

(

1, 2, 3i =

) are equal to zero.

3.2 MIDR for Dynamical Problems

To introduce uncertainties into the dynamical linear elasticity problem a combined

integral relation instead of stress-strain and velocity-momentum relations (1)–(2)

was proposed in [12] and the following integrodifferential formulation of initial-

boundary value problem (1)–(6) was given: to find such functions

∗

u ,

∗

σ , and

∗

that satisfy integral relation

[]

(,,) 0, : : ;

ddt

ϕϕρ

++ +

Φ= Ω = = ⋅+

∫∫

up Cσηη

ξξ

(38)

−−

=− =−up Cη

εσ

(39)

under equilibrium, kinematical, boundary, and initial conditions (3)–(6). Here the

auxiliary velocity vector η and strain tensor

are introduced.

Note that the integrand

in Eq. (38) has the dimension of energy density and

is nonnegative. Hence, the corresponding integral

Φ is nonnegative for arbitrary

functions

u , σ , and p (0

Φ≥ ). The proposed integrodifferential problem (3)–

(6) and (38) can be reduced to a variational one: to find unknown functions

∗

u ,

∗

σ , and

∗

p minimizing the functional

(, , ) min (,,)0

∗∗∗

Φ=Φ=

up up

σσ

(40)

subject to constraints (3)–(6).

It was shown in previous Subsection for static linear elasticity that the method

of integrodifferential relations give one the possibility to formulate various varia-

tional problems. Analogously to the static case, let us introduce into consideration

a parametric family of quadratic functionals

0, : : , , 0,

ddt

ϕϕαρβαβα

Φ= Ω = = ⋅ + + = ≥

∫∫

Cηη ξ ξ

(41)

which stationary conditions are equivalent to relations (1) and (2). Here

and

are real constants. At

1/2

integral (41) is equivalent to the nonnegative

functional

Φ defined in Eq. (38). If

, 0

≠ the minimization problems

appropriate to corresponding functional [ ]

Φ is formulated in much the same

way as it was made for the functional

Φ . The general variational formulation

can be proposed for all nonzero values of

and

≠ ,

≠

): to find un-

known functions

∗

u ,

∗

σ ,

∗

p providing the stationary value for the functional Φ

142 G.V. Kostin and V.V. Saurin

δδδ

Φ+ Φ+ Φ=

upσ

(42)

and satisfying constraints (3)–(6).

After integrating the first variations by parts and taking into account Eqs. (3)–(6)

the first variations of Φ have the following form

[]

(:)

2: , 2 .

ddt d

ddt

ddt ddt

δαβδαδ

βδγ

δβδ δαδ

•

ΩΩ

⎡⎤

Φ=− + ∇⋅ ⋅ Ω + ⋅ Ω

⎣⎦

+⋅⋅

Φ=− Ω Φ=− ⋅ Ω

∫∫ ∫

∫∫

∫∫ ∫∫

Cu u

nC u

ηξ η

ξσ η

(43)

It is seen from Eq. (43) that the fist variation of the functional Φ is equal to zero

for any admissible variations

u ,

if the following equations equivalent

to Eqs. (1) are valid

00.=η

(44)

The stress tensor σ and momentum density vector p as well as their variation

and

are related each other through the equilibrium equation (3)

δδ

•

=∇⋅p σ

(45)

Introduce the auxiliary tensor

so that

dt=∇⋅ =

∫

+σ

(46)

Then equation (45) is fulfilled automatically and the stationary conditions for the

functional Φ can be rewritten in the forms

()

[]

()

[]

().

f f

ddt ddt

ddt ddt d

δδ

αρ β δ β δ γ

αδ

αβδ α δγβδ

•

Ω Ω

Φ+ Φ=

=− + ∇⋅ ⋅ Ω + ⋅ ⋅

+⋅ Ω

=+ Ω−⋅⋅− Ω

=∇ ∇

∫∫ ∫∫

∫

∫∫ ∫∫ ∫

Cu nCu

ηξ ξ

εξχ ηχ ξχ

εηη+

(47)

Variational Approach to Static and Dynamic Elasticity Problems 143

From Eqs (47) it is following the system of Euler’s equations with the conditions

on the boundary

and at the initial and terminal instants 0,

tt=

0, 0,

:0, 0 :;

:0.

tt tt x x

γγ

αρ β α β ρ

••••

== ∈ ∈

+∇⋅ = + = ⇒ =∇⋅

===⋅ =

ηε

ηξη ξ

(48)

Taking into account the type of boundary and initial conditions(5), (6) and system

(48) it can be proven that if the unique solution

∗

u ,

∗

σ ,

∗

p exists then the station-

ary conditions (47) for the functional Φ are equivalent to relations (1), (2) and to-

gether with constraints (3)–(6) constitute the full system of dynamical equations

for the linear theory of elasticity. Moreover the nonnegative functional Φ for

reaches its absolute minimum on this solution and can serve as integral

quality criteria for any admissible approximations of the displacements, stresses,

and momentum densities u , σ , p . At the same time the integrand

can be used

to estimate the local quality of approximations.

4 Relations of the Dynamical Variational Principles

Consider a functional of the family defined in Eq. (41) at

1/2

=− =

(

1/2

−

Φ=Φ ) in more detail. It is possible to transform the integral

−

Φ to the

form

[]

(, ,) 0, : : ;

2, :

:: , : :

f f

ddt

ddt ddt

ϕϕρ

ρρ

−− −

•

−

•• − −

ΩΩ

Φ= Ω = = ⋅−

⎡⎤

Φ=Θ+Θ −Θ Θ= ⋅ − Ω

⎣⎦

⎡⎤ ⎡ ⎤

Θ= ⋅ − Ω Θ = ⋅ − Ω

⎣⎦ ⎣ ⎦

∫∫

∫∫ ∫∫

up C

uu C pp C

σηηξξ

σε

εε σ σ

(49)

where

Θ ,

Θ , Θ are space-time integrals. Note that the functional

Θ depends

only on the vector function u , whereas

Θ is independent of the displacements.

The bilinear functional Θ does not include explicitly elastic and inertial material

properties of the body, and after integrating by parts and using relations (3) it can

be presented as

[]

dddtddt

ΩΩ

Θ= ⋅ Ω− ⋅ Ω − ⋅

∫∫∫∫∫

pu fu qu

(50)

Let us analyze the case of the boundary conditions when the components

q of

the loading vector q are given on the parts

()k

of the boundary

= ,

144 G.V. Kostin and V.V. Saurin

= in Eq. (5)) whereas on the parts

()k

γγ

⊂ the displacement components are

defined ( 1

= , 0

= ). These patches

()k

of the boundary

satisfy the

following conditions:

() ()kk

γγγ

=∪ ,

() ()kk

γγ

=∅∩ (

1, 2, 3k =

). In this case

mixed boundary conditions like an elastic foundation are absent. Then, after sub-

stituting Eq. (50) into Eq. (49), the functional

−

Φ can be rewritten as follows

[]

()

uiip

ii u

ddt vud dt

qvd dt d

σσ

−

Φ=Θ+ ⋅ Ω + +Θ

+−⋅Ω

∑

∫∫ ∫ ∫

∑

∫∫ ∫

(51)

If at both initial time 0

t = and finale time

tt= either the displacement or mo-

mentum density vector is given, the functional

−

Φ decomposes into two independ-

ent parts

()

uu ii u

pp iiu p

ddt vud dt

qvd dt

σσ

−

Φ=Η+Η

Η=Θ+ ⋅ Ω + +Ξ

Η=Θ+ +Ξ

∑

∫∫ ∫ ∫

∑

∫∫

(52)

Here the functional

Η depends only on the displacement vector u ,

Η is a

function of the stress tensor σ and momentum density vector p ,

Ξ and

Ξ are

volume integrals dependent only on u and p , respectively. In Table 1 the inte-

grals

Ξ and

Ξ are presented for several kinds of initial and terminal conditions.

The first four cases correspond to space-time boundary value problems, and the

last row describes periodic body motions.

The stationary conditions for the functional

−

Φ under equilibrium relation and

space-time boundary conditions discussed above can be written according to rela-

tion (46) as

δδ δ

−

Φ= Η+ Η =

u χ

(53)

Consequently, this constrained variational problem is equivalent to two independ-

ent problems:

Problem 1. To find a displacement vector

∗

u providing the stationary value for

the functional

Variational Approach to Static and Dynamic Elasticity Problems 145

Η=

(54)

and strictly satisfying boundary conditions on

()k

(1,2,3k = ) as well as initial

and/or terminal conditions on the displacement fields shown in Table 1.

Problem 2. To find a stress tensor

∗

σ and momentum density vector

∗

p guaranty-

ing the stationarity for the functional

Η=

(55)

and obeying relations (46), stress conditions on

()k

(1,2,3k = ), and the initial

and/or terminal momentum conditions from Table 1.

Table 1. Volume integrals Ξ

and Ξ

for different initial and terminal conditions



Condition I

Condition II



(0, ) ( )

uu

(,) ()

tx x







⎡



⎢

⎣

⎤



⎦

∫

(0, ) ( )

xpp

(,) ()

tx xpp







⎡



⎢

⎣

⎤

 

⎦

∫

(0, ) ( )

uu

(,) ()

tx xpp





⎡⎤



⎣⎦

∫





⎡⎤

 

⎣⎦

∫

(0, ) ( )

pp

(,) ()

tx xuu





⎡⎤

 

⎣⎦

∫





⎡⎤



⎣⎦

∫

(0, ) ( , )

uu

(0, ) ( , )

pp



The displacement fields

∗

u obtained from Problem 1 let one find correspond-

ing stress tensor : ( )

∗

Cuε and momentum density vector ( )

∗•

u . On the other

hand the stress fields

∗

σ obtained from Problem 2 can be related to the following

strain fields

−∗

C σ . In this case it is difficult enough to identify corresponding

displacements u since one has to solve the overdetermined system of differential

equations (

•

=up,

∇+∇ =uu ε2 ).

It is important to emphasize that if displacements are given at the initial and ter-

minal instants (the first row in Table 1) then Problem 1 is equivalent to Hamilton’s

principle (8) (

Η=Η). If the initial and terminal momentum fields are fixed (the

second row in Table 1) then Problem 2 coincides with complementary principle (9)

146 G.V. Kostin and V.V. Saurin

(

Η=Η). Both classical principles are also valid in the case of periodic condi-

tions (the last row in Table 1).

In contrast to variational principles (8) and (9) which are only formulated for

space-time boundary value problems, the functional

−

Φ can be applied to the ini-

tial-boundary value problem (1)–(6) as well. The initial conditions (6) do not al-

low us to separate this functional, because the last term in relation (51) depends

explicitly on both displacements u and momentum density p . In this case the

variational problem must be solved simultaneously with respect to unknown dis-

placement, stress, and momentum functions.

As it will be seen in Section 6, the value of the nonnegative functional

Φ can

serve to estimate the integral quality of approximate solutions of problem (3)–(6),

(42) for 1/ 2

= ,

1/2

=−

. But even if the decomposition of the variational for-

mulation for

−

Φ onto Problems 1 and 2 is possible, nevertheless, both problems

have to be solved to estimate explicitly the quality of the numerical results.

5 Dynamical Variational Principle in Displacements and

Stresses

As it was shown in the previous sections the variational formulations for the ini-

tial-boundary value problem of controlled motions of an elastic body can be pro-

posed with the displacement vector u , stress tensor σ , and momentum density

vector p to be unknown functions. After introducing into consideration the auxil-

iary tensor

nine independent functions (components of tensor

and vector u )

remain in the system. In numerical approaches to 3D dynamical problems this fact

leads to sufficiently large dimension of system parameters. To decrease the num-

ber of unknown functions in the variational formulation and raise the effectiveness

of numerical computation a special member of functional family (41), namely, the

nonnegative functional Φ at 0

= , 1/ 2

= can be proposed

000

(, ) , : :

ddt

ϕϕ

Φ≡ Φ = Ω =

∫∫

uCσ

ξξ

(56)

To formulate a constrained minimization problem fitting for

Φ the velocity-

momentum vector relation 0=η in Eq. (1) must be considered as an additional

differential constraint. After that the variational problem is to find unknown func-

tions

∗

u and

∗

σ minimizing the following functional

Φ under constraints

010

(, )min (,)0,

() () , ; , 1,2,3,

(0, ) ( ), (0, ) ( ).

kkkkk

xu xq v x k

xx x x

αβ γ

∗∗

••

•−

Φ=Φ=

=∇⋅ +

+=∈=⋅=

uuu p

σσ

(57)

Variational Approach to Static and Dynamic Elasticity Problems 147

In the same way as in Section 3 denote the actual and arbitrary admissible dis-

placements and stresses by

∗

u ,

∗

σ and u , σ , respectively, and specify that

∗

=+uu u,

∗

=+σσ σ then

0000

(, )

δδδ

Φ = Φ+ Φ+ Φ

σ . The second

variation quadratic with respect to

u ,

is nonnegative (

Φ≥ ). The fist

variation of the functional

Φ is equal to zero for any admissible variations

u ,

if 0=

. The displacement vector u and stress tensor σ as well as their

variations

u and

are related through the equilibrium equation shown in Eq.

(57) and

δδ

••

=∇⋅u σ

(58)

Introduce the auxiliary tensor

so that

dtdt

••

=∇⋅ + =

∫∫

(59)

and write down the stationary conditions for

()()( )()

()()

:: ,

::.

ddt

δρδ

δδγ

δδ

••

Φ=

⎡⎤

Φ= Δ− Ω

⎣⎦

+⋅⋅∇⋅∇⋅⋅⋅ +⎡⎤

⎣⎦

⎡⎤

⎣⎦

⎡⎤

Δ= ∇∇⋅ +∇∇⋅

⎣⎦

∫∫

∫

nC C n

χξ

ξχ

ξχ− ξχ

ξχ−ξχ

ξξ

(60)

The system of Euler’s equations with the corresponding conditions on the bound-

ary

and at initial and terminal instants 0,

tt= can be obtained from Eq. (60). It

is possible to show (as in Section 3) that the stationary conditions for the func-

tional

Φ are equivalent to relations 0=

and together with constraint 0=η and

Eqs. (3)–(6) constitute the full system of dynamical equations for linear elasticity.

6 Numerical Algorithm and Error Analysis

Let us describe one of the possible algorithms approximating the solution

∗

u ,

∗

σ ,

∗

p of the variational problem defined by Eq. (42). Constrain ourselves to the case

of zero volume force 0=f . At the beginning the positive integers

N , N

, and

N are chosen and the approximations u



σ , and



of the solution are defined

in the finite dimensional form

148 G.V. Kostin and V.V. Saurin

() () ()

111

(, ), (, ), (, ).

kkk

tx tx tx

ψψψ

===

∑∑∑

uu pp





σσ

(61)

Here

{

}

(, ), 1,2,

tx k

= … is some complete countable system of linearly inde-

pendent functions and

()k

u ,

()k

σ ,

()k

p are unknown real coefficients presented in

the vector and tensor forms. The basis functions

should be chosen so that the

approximations u



σ , and



were able to satisfy exactly the equilibrium equa-

tion in Eq. (3) as well as boundary and initial conditions (5), (6). The admissible

set of the boundary vector v and initial functions

u and

p is defined. It is fol-

lows from Eq. (61) that the vectors v ,

u ,

p must have the structure

() 0(0) 0(0)

11 1

,(0,),(0,),

kk k

ψψ ψ

∈

== =

∑∑ ∑

vv uu pp

where

(0 )k

u , and

(0 )k

p are fixed coefficients;

()k

v are either given constants on

γγ

defined on the boundary patch

In the next step the conditions (3)–(6) are fulfilled with respect to unknown co-

efficients

()k

u ,

()k

σ , and

()k

p and the admissible approximations u



σ , and



obtained are substituted into the functional Φ . Since the functional Φ is quad-

ratic with respect to the parameters

()k

u ,

()k

σ ,

()k

p , and

()k

v , the stationary prob-

lem (42) is reduced to the linear system of the algebraic equations versus unknown

coefficients

()k

u ,

()k

σ , and

()k

p . To estimate the quality of the numerical solution

(, )

txu



, ( , )tx



σ , and ( , )txp



the following criterion is proposed [19]

/, (,,);

,() (), ::

(, ) : : , , :

,::,:.

err err

Wdt W t d

WdWW d

ψψρ

ϕρ ρ

γρ

+++

••

•− −

••••

Δ=Φ Ψ< Φ =Φ

⎡⎤

Ψ= = Ω = ⋅ +

⎣⎦

⎡⎤

=⋅

⎣⎦

⎡⎤

=⋅ + = ⋅ Ω =

⎣⎦

∫∫

uuuC

CupC

qu u C q C









 



 

 











 

εε

ηη+ξ ξ η= − ξ=ε− σ

εσ

(62)

Here

is a small positive constant, 0

Φ≥



is the value of functional

Φ on the

numerical solution, 0Ψ> is the time integral of the total mechanical energy

W ,

and

is the volume density of this energy. The ratio Δ of these two values can

serve as a relative integral error of the approximate solution u



σ , and



whereas function



shows the distribution of local error. The time derivative