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

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Extreme Instability Phenomena in Autonomous Weakly Damped Systems 129

constant amplitude, regardless of the initial conditions. Namely, such a dy-

namic bifurcation behaves in a way similar to that of a

generic Hopf bifurca-

tion. This new type of dynamic bifurcation was also verified via a nonlinear

dynamic analysis.

Nonconservative (asymmetric) systems

• The asymptotic stability can be conveniently established by using the rather

forgotten but very simple and efficient criterion of Liénard and Chipart.

• The geometric locus of the double branching point (η

) corresponding to

various values of k is established via the relation

η versus λ

. The locus is in-

dependent of the mass m whose effect on the dynamic stability is considerable.

The intersection between the curve (102) or (104) and the curve

corresponds

to a

coupled flutter-divergence instability bifurcation.

• The region of flutter (dynamic) instability may disappear for suitable values

of stiffness parameters.

• Contrary to existing, widely accepted results, for a positive definite damp-

ing matrix (of Rayleigh viscous type) the system (under certain combinations

of mass and stiffness distributions) may exhibit local asymptotic instability

even for a load smaller than the 1

buckling (critical) load.

• Double purely imaginary eigenvαlues (leading to a divergent motion) may

occur in case of an indefinite (infinitesimal) damping matrix, while in case of

positive semi-definite damping matrix is excluded.

References

1. Ziegler, H.: Die Stabilitätskriterien der Elastomechanik. Ing. Arch. 20, 49–56 (1952)

2. Nemat-Nasser, S., Herrmann, G.: Some general considerations concerning the destabi-

lizing effect in nonconservative systems. ZAMP 17, 305–313 (1966)

3. Crandall, S.H.: The role of damping in vibration theory. J. Sound Vib. 11, 3–18 (1970)

4. Kounadis, A.N.: Hamiltonian weakly damped autonomous systems exhibiting periodic

attractors. ZAMP 57, 324–350 (2006)

5. Huseyin, K.: Multiple Parameter Stability Theory and its Applications. Oxford Eng.

Sciences, vol. 18. Clarendon Press, Oxford (1986)

6. Huseyin, K.: Vibrations and stability of Damped Mechanical Systems. Sijthoff &

Noordhoff Intern. Publishing, Alphein (1978)

7. Zajac, E.E.: The Kelvin-Tait-Chetaev theorem and extensions. J. Aeronaut. Sci. 11,

46–49 (1964)

8. Zajac, E.E.: Comments on stability of damped mechanical systems, and a further ex-

tension. AIAA J. 3, 1749–1750 (1965)

9. Sygulski, R.: Dynamic stability of pneumatic structures in wind: theory and experi-

ment. J. Fluids Struct. 10, 945–963 (1996)

10. Laneville, A., Mazouzi, A.: Wind-induced ovalling oscillations of cylindrical shells:

critical onset velocity and mode prediction. J. Fluids Struct. 10, 691–704 (1996)

11. Kounadis, A.N.: A geometric approach for establishing dynamic buckling loads of

autonomous potential two-DOF systems. J. Appl. Mech., ASME 66, 55–61 (1999)

130 A.N. Kounadis

12. Kounadis, A.N.: Non-potential dissipative systems exhibiting periodic attractors in re-

gion of divergence. Chaos, Solitons Fractals 8(4), 583–612 (1997)

13. Kounadis, A.N.: Flutter instability and other singularity phenomena in symmetric sys-

tems via combination of mass distribution and weak damping. Int. J. of Non-Linear

Mechanics 42(1), 24–35 (2007)

14. Kounadis, A.N.: The effect of infinitesimal damping on non conservative divergence

instability systems. J. of Mechanics and Materials and Structures 4, 1415–1428 (2009)

15. Gantmacher, F.R.: The Theory of Matrices. Chelsea, New York (1959)

16. Gantmacher, F.R.: Lectures in Analytical Mechanics, Mir, Moscow, Russia (1970)

17. Kounadis, A.N.: On The Failure Of Static Stability Analysis Of Nonconservative Sys-

tems In Regions Of Divergence Instability. Int. J. Solids & Structures 31(15), 2099–

2120 (1994)

18. Sophianopoulos, D.S., Michaltsos, G.T., Kounadis, A.N.: The Effect of Infinitesimal

Damping on the Dynamic Instability Mechanism of Conservative Systems. Mathe-

matical Problems in Engineering, Special Issue on Uncertainties in Nonlinear Struc-

tural Dynamics, vol. 2008, Artic. ID 471080, 25 Pages (2008)

19. Pipes, L.A., Harvill, L.R.: Applied Mathematics for Engineers and Physicists. Interna-

tional Student Edition, 3rd edn. McGraw-Hill, Kogakusha (1970)

20. Bellman, R.: Introduction to Matrix Analysis, 2nd edn. McGraw-Hill, New York

(1970)

21. Fuller, A.T.: Conditions for a matrix to have only characteristic roots with negative

real parts. Journal of Mathematical Analysis and Applications 23, 71–98 (1968)

22. Kounadis, A.N.: Stability of elastically restrained Timoshenko cantilevers with at-

tached masses subjected to follower loads. J. of Applied Mechanics, ASME 44, 731–

736 (1977)

23. Kounadis, A.N., Katsikadelis, G.T.: On the discontinuity of the flutter load for various

types of cantilevers. Int. J. of Solid and Structures 16, 375–383 (1980)

24. Bahder, T.B.: Mathematica for Scientists and Engineers. Addison-Wesley, Reading

(1995)

Variational Approach to Static and Dynamic

Elasticity Problems

Georgy V. Kostin and Vasily V. Saurin

Institute for Problems in Mechanics of the Russian Academy of Sciences,

pr. Vernadskogo 101-1, 119526 Moscow, Russia

kostin@ipmnet.ru, saurin@ipmnet.ru

Abstract. The integrodifferential approach incorporated in variational technique

for static and dynamic problems of the linear theory of elasticity is considered. A

families of statical and dynamical variational principles, in which displacement,

stress, and momentum fields are varied, is proposed. It is shown that the Hamilton

principle and its complementary principle for the dynamical problems of linear

elasticity follow out the variational formulation proposed. A regular numerical al-

gorithm of constrained minimization for the initial-boundary value problem is

worked out. The algorithm allows us to estimate explicitly the local and integral

quality of numerical solutions obtained. As an example, a problem of lateral con-

trolled motions of a 3D rectilinear elastic prism with a rectangular cross section is

investigated.

1 Introduction

Variational principles in mechanics involving the theory of elasticity have been

very thoroughly developed and intensively studied by scientists. Among these

formulations the minimum principles for potential and complementary energy, the

Hamilton principle can be mentioned (see [22], for example). Many different ap-

proaches to deduce variational principles for mechanical problems are presented in

the literature. It can be noted the recent publication concerning the semi-inverse

method suggested by He [7]. The variational formulation of the finite element me-

thod is broadly used in scientific and engineering applications. The mathematical

origin of the method can be traced to a paper by Courant [5]. Other numerical ap-

proaches, e.g., Petrov-Galerkin method [3, 4] or the least squares method [20], are

being actively developed for solid mechanics. In all these methods it is supposed

that some of the elasticity relations (equilibrium equations, boundary conditions in

terms of stresses and etc.) are generalized and the exact solution is approximated

by a finite set of trial functions.

Elastic properties of structure elements can essentially affect the dynamical be-

havior of the whole system. Some parts of mechanical structures with distributed

132 G.V. Kostin and V.V. Saurin

parameters are modeled as elastic bodies with given stiffness and inertia characteris-

tics. A significant number of numerical methods has been developed for modeling

the behavior of dynamical systems described by initial-boundary value problems.

One of the most widespread approaches to solving problems of such kind is the

method of separation of variables [6]. In [1] a regular perturbation method (a small

parameter method) for investigating the dynamics of weakly non-uniform thin rods

with arbitrary distributed loads and different boundary conditions was proposed.

Based on the classical Rayleigh-Ritz approach, a numerical analytical method of fast

convergence was developed in [2] to find precise values of unknown functions for

arbitrary distributed stiffness and inertia characteristics of elastic systems. In model-

ing the elastic systems the methods of finite-dimensional approximations, for exam-

ple the decomposition method and the regularization method, were developed to

reduce an initial-boundary value problem for partial differential equations to a

system of ordinary differential equations [8, 9]. The direct discretization methods in

optimal control problems are also well known (see, e.g. [10]).

The aim of this contribution is to develop the method of integrodifferential

relations (MIDR) for static and dynamic linear elasticity problems based on the in-

tegral formulation of the constitutive equations and to apply this approach to

analysis of 3D elastic body behavior. The basic ideas of MIDR were proposed and

discussed by Kostin and Saurin [11-19, 21].

In the next section, the statements of static and dynamic linear elasticity prob-

lems and its conventional variational formulations are discussed. In the third sec-

tion a parametric family of quadratic functionals is considered and their stationary

conditions equivalent to the constitutive relation are obtained. The relations of

proposed and conventional variational principles are shown in Section 4. In

Section 5 a new variational formulation in displacements and stresses for the ini-

tial boundary value problem of linear elasticity is given. In Section 6 a numerical

algorithm used the piecewise polynomial approximations of unknown functions

(displacements, stresses, and momentums) is developed [19] and the effective in-

tegral and local bilateral estimates of solution quality are obtained relying on the

extremal properties of the finite dimensional variational problem. Sections 7 and 8

are devoted to numerical modeling, optimization, and analysis of controlled

motions of a 3D elastic beam. Concluding remarks are given in the last section.

2 Statement of the Problems

2.1 Dynamical Formulation

Consider an elastic body occupying a bounded domain

Ω with an external piece-

wise smooth boundary

. Taking into account the assumption of the linear theory

of elasticity about smallness of elastic deformations and relative velocities the mo-

tion of the body can be described by the following system of partial differential

equations [16]:

:,= Cσε

(1)

Variational Approach to Static and Dynamic Elasticity Problems 133

•

=pu

(2)

0,∇⋅ − + =pf



(3)

()

=∇+∇uuε

(4)

Here

σ and ε are the stress and strain tensors, p and u are the momentum density

and displacement vectors, f is the vector of volume forces, C is the elastic

modulus tensor, and

is the volume density of the body. The dots above the

symbols denote partial time derivatives, and

123

(/ ,/ ,/ )xxx∇=∂∂ ∂∂ ∂∂ is the

gradient operator in the Cratesian coordinate space

{

}

123

,,xxxx= . The dots and

colons between vectors and tensors point out to their scalar and double scalar

products, respectively. The components

ijkl

C of the tensor C are characterized by

the following symmetry property:

ijkl ijlk klij

CCC==. The superscript T denotes the

transposition operator.

Let us constrain ourselves to the case of the linear boundary conditions ex-

pressed componentwise in the form:

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

kkkkk

xu xq v x k

αβ γ

+=∈=⋅=qnσ

(5)

where

q is the loading vector, n is the unit vector pointing in the direction of the

outward normal to the boundary

u and

q are the components of the vector

functions

u and q,

and

are given coordinate functions defining the type of

boundary conditions. In particular, if 1

= and 0

= on some part of the

boundary

then, according to Eq.(5), the displacement

u is set by the boundary

vector function

v via its component

v . To the contrary, if 0

= and 1

then the external load component

q are defined through

v on

. Conditions (5)

include also various combinations of linear elastic supports if the pair

simultaneously nonzero on a certain part of the boundary.

It is supposed that at the initial instant 0

t = the distribution of momentum den-

sity

p and displacements u are given as sufficiently smooth functions of the coor-

dinates

(0, ) ( ), (0, ) ( ), .xx xxx==∈Ωuupp

(6)

Note that initial conditions (6) and boundary conditions (5) should be consistent

[14]. The components of boundary vector

v are either given functions of the time

t and coordinates x or unknown control inputs defined in some functional space

(, ), \ ; , .

tx x V x

γγ γ

=∈∈∈vv v

(7)

134 G.V. Kostin and V.V. Saurin

Let us formulate two classical variational principles for dynamical boundary value

problems of linear elasticity (see [22]). If the functions of strain energy density

()A u

and kinetic energy density

()K u

exist, the displacements u as a coordinate

vector function are given at the initial ( 0

t = ) and final (

tt= ) instants, and

boundary conditions (5) do not change under displacement variations then Hamil-

ton’s principle follows the principal of virtual work:

[] [ ]

()

() 0

0, , ;

,::;

, , 1,2,3; (0,) , ( ,) .

kk u f

dt Kd A d u v d

uv x k x tx

δδ

ΩΩ

Η= Τ−Π = Τ= Ω Π= − ⋅ Ω−

=⋅ =

=∈ = = =

∑

∫∫∫∫

uu C

uuu u

εε (8)

In this problem the displacement field u must rigorously satisfy the kinematic re-

lation (4) and, according to Eq. (5), the displacement conditions on

()k

γγ

⊂ ,

where 1

= , 0

= . The equilibrium relation (3) and stress boundary condi-

tions on

() ()

γγγ

= , where 0

= , 1

= , appear in the stationary conditions

of the Hamiltonian Η in Eq. (8).

If the functions of stress energy density ( )

A σ and kinetic energy density

()

K p exist for the same type of boundary conditions on

() ()kk

γγ γ

= ∪ , the mo-

mentum vector p is given in the initial ( 0t = ) and final (

tt= ) instants, and

boundary conditions (5) do not change under stress and momentum variations then

the stationary principle for the complementary energy follows the principal of vir-

tual complementary work:

[]

()

() 0

0, , ;

,::;

, , 1,2,3; (0,) , ( ,) .

ccc cccc ii

kk f

dt K d A d q v d

qv x k x tx

δδ

ΩΩ

−

Η= Τ−Π = Τ= Ω Π= Ω−

=⋅ =

=∈ = = =

∑

∫∫∫∫

pp C

ppp p

σσ (9)

According to this principle the fields of stress σ and momentum p must rigor-

ously satisfy the equilibrium equation in Eq. (3) and boundary conditions on

()k

The kinematic equation in (4) together with boundary conditions on

()k

are im-

plicitly equivalent to the stationary conditions of the functional

Η in Eq. (9).

2.2 Formulation for the Static Problem

The static stress-strain state of the body is governed by the differential equations

of linear elasticity (1) - (4), supposing that the momentum dencity vector p is

Variational Approach to Static and Dynamic Elasticity Problems 135

equal to zero. Let us restrict ourselves to the case when the boundary conditions

can be presented in the form

iijji

qnqx

≡= ∈

(10)

uu x

=∈

(11)

0, 0,

iii i

qu x

κκ

+= > ∈

(12)

Here

q and

u are given components of the vector functions of external loads

and boundary displacements

u ;

are given moduli of elastic foundations. The

boundary parts

satisfy the following conditions:

123

iii

γγγγ

=∪∪ ;

γγ

=∅∩ ,

jk≠

(

,, 1,2,3ijk=

Different variational approaches are broadly used for solving the linear elastic-

ity problems. In the case when the function of strain energy density

,introduced

in (8),exists and boundary conditions (10) do not change under displacement vari-

ations, the minimum principle for potential energy follows the principal of virtual

work [22]:

() min

Ad d qud

Π= Ω− ⋅ Ω− →

ΩΩ

∑

∫∫ ∫

ufu

(13)

Note that the displacement fields u must rigorously satisfy kinematic relations

(4), Hooke’s law relation (1), and boundary conditions (11). Equilibrium equation

(3) and boundary conditions (10) are implicitly contained in the functional Π in

Eq. (13) as its stationary conditions.

When the function of stress energy density

(see (9)) exists and boundary

conditions (11) do not change under stress variations, then the minimum principle

for complementary energy follows the principal of virtual complementary work

() min

cc ii

Ad qud

Π= Ω− →

∑

∫∫

(14)

According to this principle the stress fields σ must rigorously satisfy equilibrium

equation in (3), Hooke’s law relation (1), and boundary conditions (10). The ki-

nematic relations (4) and boundary conditions (11) are Euler’s equations for the

functional

Π in Eq. (14).

It is considered [22] that both variational principles are equivalent each other

under the hypotheses of the linear theory of elasticity.

Note that the approximate displacement fields u obtained from the minimum

principle for potential energy let one find corresponding stress fields

()uσ taking

into account the linear kinematic relations in (4) and stress-strain relation in (1):

:()≡ Cuσε

(15)

136 G.V. Kostin and V.V. Saurin

On the other hand by using Eq. (1) the stress fields σ obtained from the minimum

principle for complementary energy can be related to the following strain fields

()εσ

-1

:≡ Cεσ

(16)

It is difficult enough to identify corresponding displacements u since one has to

solve the overdetermined system of differential equations

()

2=∇+∇uuε . In

the general case the stresses

σ and σ found by these approaches may not be

equal to each other (

≠σσ).

3 The Method of Integrodifferential Relations

Relations (1)–(6) describe the deformed state of an elastic body at any internal

point

x and any instant t . In addition, it is assumed that the stresses and dis-

placements at the internal points of the body should tend to boundary stresses and

displacements, i.e. conditions (5) are satisfied. It is implied that the components of

the elastic modulus tensor C and mass density

, defined in the interior of the

domain Ω , continuously pass through the boundary

. Analogously, the dis-

placement and momentum density vectors tend continuously to their initial values

given by relations (6). On the other hand, it is necessary to take into account that

boundary and initial conditions (5), (6) are generated by specific physical and

geometrical factors. For example, some part of the boundary could be an interface

between two or more media (elastic or inelastic). In this case, any boundary point

belongs simultaneously to the body under consideration and the bodies which

generate these boundary conditions. So such points on the boundary belong simul-

taneously to material parts with different mechanical properties. In other words,

the elastic modulus tensor C and mass density function

on such surfaces,

strictly speaking, are not defined.

3.1 MIDR for Static Problems

To introduce this uncertainty into the static linear elasticity problem we propose

the integral formulation of the stress-strain relation [11, 15]:

11 1

0, : ,d

ϕϕ

Φ= Ω= = ≡ −

∫

ηησσ

(17)

where

()= uσσ is the tensor function defined in Eq. (15). Taking into account

Eq. (17) the boundary value problems of the linear theory of elasticity can be re-

formulated: to find the kinematically admissible displacements

u and equilibrium

stresses σ that satisfy relations (3), (4), (17) and boundary conditions (10)–(12).

Variational Approach to Static and Dynamic Elasticity Problems 137

The integrand

in Eq. (17) is nonnegative for arbitrary functions u and σ

(

≥ ). Hence, the corresponding integral

Φ is also nonnegative. Under the cir-

cumstances, the integrodifferential problem (3), (4),(10)–(12), (17) can be reduced

to a variational problem: to find unknown functions

u and σ minimizing the

functional

(, ) minΦ→

(18)

under constraints (3), (4),(10)–(12) on the stress-strain state of the elastic body.

Let us denote the actual and arbitrary admissible displacements and stresses by

∗

u ,

∗

σ and u , σ , respectively, and specify that

∗

=+uu u,

∗

=+σσ σ. Then

1111

(, )

δδδ

Φ = Φ+ Φ+ Φ

σ , where

are the first variations of the

functional

Φ with respect to u , σ and

Φ is its second variation.

Integrating the first variations by parts and taking into account Eqs. (3),

(4),(10)–(12), (17) result in the following relations

2: 2:

δδδ

δδ

Φ=− ∇⋅ ⋅ Ω+ ⋅ ⋅

Φ= Ω

∫∫

∫

Cu nCu

ηη

ησ

(19)

It is seen from Eqs. (19) that the fist variations

and

are equal to zero

for any

u and

σ if the following equations are valid

0, : 0, [ : ] 0

=∇⋅ = ⋅ =CnCηηη

(20)

Hence, Hooke’s local relation (1) is the stationary condition of the functional

and together with constraints (3), (4),(10)–(12) constitutes the complete system of

equations for the linear theory of elasticity.

The second variation

(,) 0d

δϕδδ

Φ= Ω≥

∫

u σ

which is quadratic with respect to the displacement and stress variations

u and

σ , is nonnegative because the integrand

(,)0

ϕδ δ

≥u σ .

It is possible to derive other variational formulations for the linear elasticity

problems using properties of the functional

Φ . If

0Φ= then the corresponding

integral equalities for the components of the stress tensors σ and

σ hold

0, , 1, 2, 3

dkl

Ω= =

∫

(21)

where

are the components of the stress tensor

138 G.V. Kostin and V.V. Saurin

Using Eqs. (15), (16), (21) one can shown that the integral equalities for the

components of the strain tensors ε and

ε is also valid

()

0, , 1, 2, 3

kl kl

dkl

εε

−Ω= =

∫

(22)

Thus, let us introduce into consideration the following nonnegative quadratic

functional

22 2

0, : ,d

ϕϕ

Φ= Ω≥ = ≡−

∫

ξξ ξ

εε

(23)

which reaches its absolute minimum (zero value) on the actual displacements

∗

and stresses

∗

σ . The first variations the functional in Eq. (23) have the form

2::

δδδ

δδ

−

Φ=− ∇⋅⋅ Ω+ ⋅⋅

Φ= Ω

∫∫

∫

unu

ξξ

ξσ

(24)

The stationary condition for the functional

Φ (the same as for

Φ ) is Hooke’s

law rewritten in the form

, and its second variation is also nonnegative.

Taking into account Eqs. (15) and (16), the functional

Φ expressed in

Eq. (23) in terms of the strain tensor

can be rewritten in terms of the compo-

nents of the stress tensor η . In particular, for the 2D case (plane stressed state) of

homogeneous isotropic material the integral

Φ has the form

222

21111222212

22 2

(1 ) 4 2(1 )

(1 ) (1 )

μμ μ

ηηηη η

μμ

⎡⎤

Φ= − + + Ω

⎢⎥

⎣⎦

∫

(25)

Here

is Young’s modulus,

is Poisson’s ratio. The nonnegativity of the func-

tional in Eq. (25) follows the fact that the factor of the mixed term

4/(1 ) 2

μμ

+< because

11/2

−< <

Generalizing the approach, one can introduce into consideration a parametric

family of nonnegative quadratic functionals

()

22 222

11 11 22 22 12

20,||1ca bda

ηηηηη

Φ= − + + Ω≥ <

∫

(26)

for which the stationary conditions are equivalent to Hook’s relation (1). Here a ,

и c are some real constants. Analogous functional families can be constructed

for an arbitrary anisotropic 3D case.