CH.11. VARIATIONAL PRINCIPLES Continuum Mechanics Course (MMC)

Overview

Introduction

Functionals Gâteaux Derivative Extreme of a Functional

Variational Principle Variational Form of a Continuum Mechanics Problem

Virtual Work Principle Virtual Work Principle Interpretation of the VWP VWP in Engineering Notation

Minimum Potential Energy Principle Hypothesis Potential Energy Variational Principle

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Lecture 1

Lecture 2

Lecture 3

Lecture 4

Lecture 5

Lecture 6

Lecture 7

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Ch.11. Variational Principles

11.1. Introduction

Computational Mechanics

In computational mechanics problems are solved by cooperation of mechanics, computers and numerical methods.

This provides an additional approach to problem-solving, besides the theoretical and experimental sciences.

Includes disciplines such as solid mechanics, fluid dynamics, thermodynamics, electromagnetics, and solid mechanics.

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8

Ch.11. Variational Principles

11.2. Functionals

Definition of Functional

Consider a function space : The elements of are functions of an arbitrary tensor order, defined in a subset .

A functional is a mapping of the function space onto the set of the real numbers , : . It is a function that takes an element of the function space as

its input argument and returns a scalar.

X

( ) 3: : m= Ω ⊂ →u xX R R

X ( )u x

3Ω⊂ R

( ) : →uF X R( )uF

RX

( )u x X

X( )u x

R( )

b

a

u x dx∫

( )b

a

u x dx′∫

[ ], ( ), ( )b

a

f x u x u x dx′∫( )uF

( ) [ ]: ,u x a b → R

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Definition of Gâteaux Derivative

Consider : a function space the functional a perturbation parameter a perturbation direction

The function is the perturbed function of in the direction.

3: ( ) : m= Ω⊂ →u xX R R( ) : →uF X R

ε∈R( )∈x Xη

( ) ( )+ ε ∈u x x Xη ( )u x( )xη

Ω0

Ω

t=0

P P’

t

( )u x

( )ε xη( ) ( )+ εu x xη

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Definition of Gâteaux Derivative

The Gâteaux derivative of the functional in the direction is:

( )uF η

( ) ( )( )0

; : dd ε=

δ = + εε

u uF Fη η

Ω0

Ω

t=0

P P’

t

( )u x

( )ε xη( ) ( )+ εu x xη

P’ ( )F u

REMARK The perturbation direction is often denoted as . Do not confuse with the differential . is not necessarily small !!!

not= δuη

( )δu x ( )du x( )δu x

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Example

Find the Gâteaux derivative of the functional

Solution :

( ) ( ) ( ): d dΩ ∂Ω

= ϕ Ω+ φ Γ∫ ∫u u uF

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )0 0

0 0

0

; d d dd dd d d

d dd d

d d

Ω ∂Ωε= ε=

Ω ∂Ωε= ε=

ε=

δ δ = + εδ = ϕ + εδ Ω + φ + εδ Γ =ε ε ε

∂ϕ + εδ + εδ ∂φ + εδ + εδ= ⋅ Ω + ⋅ Γ ∂ ε ∂ ε

∫ ∫

∫ ∫

u u u u u u u u

u u u u u u u uu u

F F

( ) ( ) ( ); d dΩ ∂Ω

∂ϕ ∂φδ δ = ⋅ δ Ω + ⋅ δ Γ

∂ ∂∫ ∫u uu u u u

u uF

= δu = δu

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Ω0

Ω

t=0

P P’

t

( )u x

( )εδu x( ) ( )+ εδu x u x

Gâteaux Derivative with boundary conditions

Consider the function space :

By definition, when performing the Gâteaux derivative on , . Then,

The direction perturbation must satisfy:

V

( ) ( ) ( ) ( ) m *: : ;u∈Γ

= Ω→ =x

u x u x u x u xV R

V( )+ εδ ∈u u V

( ) ( )*

u∈Γ+ εδ =

xu u u x *

u u∈Γ ∈Γ+ ε δ =

x xu u u 0

u∈Γε δ =

xu

u∈Γδ =

xu 0

*= u

u σΓ Γ =∅

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Consider the family of functionals

The Gâteaux derivative of this family of functionals can be written as,

Gâteaux Derivative in terms of Functionals

( ) ( , ( ), ( ))

( , ( ), ( ))

d

dσ

φ

ϕΩ

Γ

= Ω

+ Γ

∫∫

u x u x u x

x u x u x

∇

∇

F

( ); ( , ( ), ( )) ( , ( ), ( ))d dσ

δ δ δ δΩ Γ

= ⋅ Ω + ⋅ Γ∫ ∫u u x u x u x u x u x u x u∇ ∇F E T

REMARK The example showed that for , the

Gâteaux derivative is .

( ) ( ) ( ) d dΩ ∂Ω

∂φ ∂ϕδ = ⋅ δ Ω + ⋅δ Γ

∂ ∂∫ ∫u uu u u

u uF

( ) ( ) ( ): d dΩ ∂Ω

= φ Ω+ ϕ Γ∫ ∫u u uF

u σΓ Γ =∅

u

δδ

∈Γ

∀=

x

uu 0

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A function has a local minimum (maximum) at

Necessary condition:

The same condition is necessary for the function to have extrema (maximum, minimum or saddle point) at .

This concept can be can be extended to functionals.

Extrema of a Function

( ) :f x →R R 0x

Local minimum

( )0

0( ) 0

not

x x

df x f xdx =

′= =

0x

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A functional has a minimum at

Necessary condition for the functional to have extrema at :

This can be re-written in integral form:

Extreme of a Functional. Variational principle

( ) : →uF V R ( )∈u x V

( )u x

( ); 0 |u∈Γ

δ δ = ∀ =x

u u u u 0F δ δ

( ); ( ) ( ) 0d dσ

δ δ δ δΩ Γ

= ⋅ Ω + ⋅ Γ =∫ ∫u u u u u uF E T

Variational Principle u

δδ

∈Γ

∀=

x

uu 0

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17

Ch.11. Variational Principles

11.3.Variational Principle

Variational Principle:

Fundamental Theorem of Variational Calculus: The expression is satisfied if and only if

Variational Principle

( ); 0d dΩ Γ

= ⋅ Ω + ⋅ Γ =∫ ∫u u u uF E Tσ

δ δ δ δ REMARK Note that is arbitrary.

uδ

( , ( ), ( )) 0 σ= ∀ ∈Γx u x u x x∇T

( , ( ), ( )) 0= ∀ ∈Ωx u x u x x∇E Euler-Lagrange equations

Natural boundary conditions

( , ( ), ( )) ( , ( ), ( )) 0d dσ

δ δΩ Γ

⋅ Ω + ⋅ Γ =∫ ∫x u x u x u x u x u x u∇ ∇E T

u

δδ

∈Γ

∀=

x

uu 0

u

δδ

∈Γ

∀=

x

uu 0

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Example

Find the Euler-Lagrange equations and the natural and forced boundary conditions of the functional

( ) ( ) ( ) ( ) [ ] ( ) ( ), , : , ;b

x aa

u x u x u x dx u x a b u x u a pφ=

′= → = = ∫F Rwith

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Example - Solution

Find the Euler-Lagrange equations and the natural and forced boundary conditions of the functional

Solution :

First, the Gâteaux derivative must be obtained. The function is perturbed:

This is replaced in the functional:

( ) ( ) ( ) ( ) ( ), ,b

x aa

u x u x u x dx u x u a pφ=

′= = = ∫F with

( )u x( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )| 0

not

a

u x u x xx u x a

u x u x x→ + ∀ ≡ = =′ ′ ′→ +

ε ηη δ η η

ε η

( ) ( ) ( ), ,b

a

u x u x u x dxεη φ εη εη′ ′+ = + + ∫F

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Example - Solution

The Gâteaux derivative will be

Then, the expression obtained must be manipulated so that it resembles the Variational Principle :

Integrating by parts the second term in the expression obtained:

The Gâteaux derivative is re-written as:

( ) ( )0

;b

a

ddu u dx

u uεεφ φδ η ε η η η

=

∂ ∂ ′= + = + ′∂ ∂ ∫F F

( ); 0d dσ

δ δ δ δΩ Γ

= ⋅ Ω + ⋅ Γ =∫ ∫u u u uF E T

( ) ( )b

b b b

b aa a aa b a

d ddx dx dxu u dx u u u dx uφ φ φ φ φ φη η η η η η∂ ∂ ∂ ∂ ∂ ∂′ = − = − −′ ′ ′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂∫ ∫ ∫

( ) ( ) ( ) ( )

( (

, , ;

; ) ; ) [ ( )]

b

a

b

babu

u x u x u x dx u a p

du u u udx uu dx u uδ

δ δ

φ

φ φ φη δ δ δ≡

=

′= =

∂ ∂ ∂= − +

′ ′∂ ∂ ∂

∫

∫

0aη =

( ) ( ) ( )

( ) ( )

, ,b

a

x a

u x u x u x dx

u x u a p

φ

=

′=

= =

∫F

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Example - Solution

Therefore, the Variational Principle takes the form

If this is compared to , one obtains:

( ; ) 0b

bab

du u u dx uu dx u u

δφ φ φδ δ δ=

∂ ∂ ∂ − + = ′ ′∂ ∂ ∂ ∫

( ); 0d dσ

δ δ δ δΩ Γ

= ⋅ Ω + ⋅ Γ =∫ ∫u u u uF E T

0a

uuδ

δ∀

=

( ) ( ), , 0 ,dx u u x a bu dx uφ φ∂ ∂ ′ ≡ − = ∀ ∈ ′∂ ∂

E Euler-Lagrange Equations

Natural (Newmann) boundary conditions

Essential (Dirichlet) boundary conditions

( ) ( )x a

u x u a p=≡ =

( ), , 0x b

x u uuφ

=

∂′ ≡ =′∂

T

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Consider a continuum mechanics problem with local or strong governing equations given by, Euler-Lagrange equations

with boundary conditions: Natural or Newmann

Forced (essential) or Dirichlet

Variational Form of a Continuum Mechanics Problem

( , ( ), ( )) 0 V= ∀ ∈x u x u x xE ∇

*( , ( ), ( )) ( ) ( ) σ≡ ⋅ − = ∀ ∈Γx u x u x u n t x 0 x∇ ∇T

( ) ( ) u∗= ∀ ∈Γu x u x x REMARK

The Euler-Lagrange equations are generally a set of PDEs.

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The variational form of the continuum mechanics problem consists in finding a field where

fulfilling:

Variational Form of a Continuum Mechanics Problem

( )∈u x X

( ) ( ) ( )

3

30

: : on

( ) : ( ) on

mu

mu

V

Vδ δ

∗= ⊂ → = Γ

= ⊂ → = Γ

u x u x u x

u x u x 0

V R R

V R R

0( , ( ), ( )) ( ) ( , ( ), ( )) ( ) 0 ( )V

dV dσ

δ δ δΓ

⋅ + ⋅ Γ = ∀ ∈∫ ∫x u x u x u x x u x u x u x u x∇ ∇E T V

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Variational Form of a Continuum Mechanics Problem

REMARK 1 The local or strong governing equations of the continuum mechanics are the Euler-Lagrange equation and natural boundary conditions.

REMARK 2 The fundamental theorem of variational calculus guarantees that the solution given by the variational principle and the one given by the local governing equations is the same solution.

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Ch.11. Variational Principles

11.4. Virtual Work Principle

Continuum mechanics problem for a body: Cauchy equation

Boundary conditions

Governing Equations

( ) ( ) ( )2

0 0 2

,, ,

u xx b x

tt t V

tρ ρ

∂+ =

∂in∇ ⋅σ

( ) ( ) ( )( ( ),t)

, t , t , t σ∗⋅ = Γ

u

x n x t x

∇

σ

ε

on

( ) ( ), , ut t∗= Γu x u x on

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( ( ( , )))u x tε ∇

The variational principle consists in finding a displacement field , where

such that the variational principle holds,

where Note: is the space of admissible displacements. is the space of admissible virtual displacements (test functions). The (perturbations of the displacements ) are termed virtual

displacements.

( )2

02; [ ( )] ( ) 0V

dV dt

δ δ ρ δ δ δ∗

Γ

∂= + − ⋅ + − ⋅ ⋅ Γ = ∀ ∈

∂∫ ∫uu u b u t n u uW V∇ ⋅σ σ

σ = T

Variational Principle

( ) ( ) ( ) 3: , : , , onmut V t t∗= ⊂ → = Γu x u x u xV R R

( ) ( ) 30 : : onm

uVδ δ= ⊂ → = Γu x u x 0V R R

δu

= E

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The first term in the variational principle

Considering that

and (applying the divergence theorem): Then, the Virtual Work Principle reads:

Virtual Work Principle (VWP)

( ) ( ) sδ δ δ⋅ = ⋅ −u u u∇ ⋅σ ∇ ⋅ σ σ : ∇

( ) s( dV V

dV dVδ δ δΓ

⋅ ⋅ = ⋅ Γ − ∫ ∫ ∫u n u u∇ ⋅σ ⋅ σ) σ : ∇σ

( ) ( ) * s0; 0

V V

dV d dVδ δ ρ δ δ δ δΓ

= − ⋅ + ⋅ Γ − = ∀ ∈∫ ∫ ∫u u b a u t u u uσ : ∇σ

W V

( ) ( )2

02; [ ( )] 0V

dV dt

δ δ ρ δ δ δ∗

Γ

∂= + − ⋅ + − ⋅ ⋅ Γ = ∀ ∈

∂∫ ∫uu u b u t n u u∇ ⋅σ σW V

σ

= E= T

= a

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Virtual Work Principle (VWP)

REMARK 1 The Cauchy equation and the equilibrium of tractions at the boundary are, respectively, the Euler-Lagrange equations and natural boundary conditions associated to the Virtual Work Principle.

REMARK 2 The Virtual Work Principle can be viewed as the variational principle associated to a functional , being the necessary condition to find a minimum of this functional.

( )uW

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The VWP can be interpreted as:

Interpretation of the VWP

( ) ( ) ( )* s

V V

; 0

*

dV d dVδ δ ρ δ δ δ

δΓ

= − ⋅ + ⋅ Γ − =∫ ∫ ∫u u b a u t u u

b

pseudo - virtualbody forces strains

σ : ∇

εσ

W

Work by the pseudo-body forces and the contact forces.

External virtual work

Work by the virtual strain.

Internal virtual work

intδWextδW

( ) 0ext int;δ δ δ δ δ= − = ∀ ∈u u 0 uW W W V

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Engineering notation uses vectors instead of tensors:

The Virtual Work Principle becomes

VWP in Voigt’s Notation

x x x

y y y

notz z z6 6

xy xy xy

xz xz xz

yz yz yz

; ;2

22

σ δε δεσ δε δε

σ δε δεδ δ

τ δγ δε

τ δγ δετ δγ δε

∈ = ∈ = =

σ σ εR Rε :δ δ δ= ⋅ = ⋅σ ε σ ε ε σ

( ) *00b a u t u u

V V

dV dV dσ

δ δ ρ δ δ δΓ

= ⋅ − − ⋅ + ⋅ Γ = ∀ ∈

∫ ∫ ∫W Vε σ

Total virtual work.

Internal virtual work, . intδW

External virtual work, .

ex tδW

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33

Ch.11. Variational Principles

11.5. Minimum Potential Energy Principle

An explicit expression of the functional in the VWP can only be obtained under the following hypothesis: 1. Linear elastic material. The elastic potential is:

2. Conservative volume forces. The potential for the quasi-static

case under gravitational forces and constant density is:

3. Conservative surface forces. The potential is:

Then a functional, total potential energy, can be defined as

Hypothesis

W

( )a 0=

( ) )GG ∗ ∗∂ (= − ⋅ = −

∂uu t u t

u

( ) )ρ ρ∂φ(φ = − ⋅ = −

∂uu b u b

u

ˆ1 (ˆ( : : :2

uu ∂ )) = = =

∂σC Cε

ε ε ε εε

( ) ( ) ( )ˆ(V V

u dV dV G dSφΓ

= ) + +∫ ∫ ∫u u uεUσ

Elastic energy

Potential energy of the body forces

Potential energy of the surface forces ( ))s u= ε(∇

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The variational form consists in finding a displacement field , such that for any the following

condition holds,

This is equivalent to the VWP previously defined.

Potential Energy Variational Principle

( , )t ∈u x V δ δ = Γu0u u in

( ) ( ) *0; d

V V

dV dVσ

δ δ δ ρ δ δ δΓ

= − − ⋅ − ⋅ Γ ∀ ∈∫ ∫ ∫u u : b a u t u uσ εU V

( );δ δ δ≡ u uW U

( ) ( ) ( ) ( );

ˆ 0S

V V

Gu dV dV dσ

δ δ δ δ δΓ

=∂φ ∂∂

∇ + ⋅ + ⋅ Γ =∂ ∂ ∂∫ ∫ ∫u u

u u: u u u

u uεU

= σ δ= εbρ= −

*= −t

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The VWP is obtained as the variational principle associated with this functional , the potential energy.

The potential energy is This function has an extremum (which can be proven to be a minimum) for

the solution of the linear elastic problem.

The solution provided by the VWP can be viewed in this case as the solution which minimizes the total potential energy functional.

Minimization of the Potential Energy

U

0( ; ) 0δ δ δ= ∀ ∈u u uU V

( ) *1( ) ( ) ( ) ( ) 2

u u u b a u u t uV V

dV dV dσ

ρΓ

= − − ⋅ − ⋅ Γ∫ ∫ ∫U Cε : :

deriving from a potential

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