CH.11. ARIATIONAL PRINCIPLES Continuum Mechanics Course (MMC)
Overview Introduction Functionals Gâteaux Derivative Extreme of a Functional ariational Principle ariational Form of a Continuum Mechanics Problem irtual Work Principle irtual Work Principle Interpretation of the WP WP in Engineering Notation Minimum Potential Energy Principle Hypothesis Potential Energy ariational Principle Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 2
11.1. Introduction Ch.11. ariational Principles 3
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. 6
11.2. Functionals Ch.11. ariational Principles 8
Definition of Functional Consider a function space X : { ( ) } 3 m X: = u x : Ω R R a b F( u) The elements of X are functions u( x) [, ( ), ( )] a b of an arbitrary tensor order, defined in X 3 u ( x) dx a subset Ω R. u ( x ) a u( x) :[ ab, ] R A functional F( u) is a mapping of the function space X onto the set of the real numbers, R : F( u) : X R. It is a function that takes an element u( x) of the function space X as its input argument and returns a scalar. R b u ( x ) dx f x u x u x dx 9
Definition of Gâteaux Derivative Consider : a function space the functional a perturbation parameter 3 m { ux } X: = ( ): Ω R R ( u) : F X R ε R a perturbation direction η( x) X The function ux ( ) +ε ( x) X is the perturbed function of u x in the η x direction. ( ) η ( ) t= t Ω ( ) u x Ω P P ε η u x +εη x ( ) ( ) ( x) 1
Definition of Gâteaux Derivative The Gâteaux derivative of the functional in the η direction is: d δ F = +ε ε ( ) ( u; η) : F( u η) d ε= F ( u) t= F ( u) t P Ω ( ) u x Ω P P ε η u x +εη x ( ) ( ) ( x) REMARK not The perturbation direction is often denoted as η = δu. Do not confuse δu( x ) with the differential du( x). δu( x) is not necessarily small!!! 11
Example Find the Gâteaux derivative of the functional F Solution : ( u ) ( u ) ( u ) : = ϕ d Ω+ φ d Γ Ω Ω Ω t= ( ) u x Ω P P εδu x u x +εδu x ( ) ( ) t ( ) d d d δf( u; δ u) = F( u+ εδ u) = ϕ ( + εδ ) dω + φ ( + εδ ) dγ = dε dε u u dε u u ε= Ω ε= Ω ε= ( u u) d( u u) ( u u) d( u u) ϕ + εδ + εδ φ + εδ + εδ = dω + dγ u dε dε Ω u = δu = δu ε= Ω ε= 12 ϕ( u) φ( u) δf( u; δ u) = δu dω+ δ dγ u u u Ω Ω
Gâteaux Derivative with boundary conditions Consider the function space : { u( x) u( x) R m u( x) u * ( x) } x : = : Ω ; = Γ u Γ Γ = u By definition, when performing the Gâteaux derivative on, ( +εδ ) u u Then, * ( +εδ ) = ( ) x Γ u. * u u u x u +ε δ u = u εδ = x Γ u = u * x Γ u u x Γ u 13 The direction perturbation must satisfy: δ u = x Γ u
Gâteaux Derivative in terms of Functionals Consider the family of functionals F ( u) φ( xux, ( ), ux ( )) Ω = dω + ϕ( xux, ( ), ux ( )) dγ Γ The Gâteaux derivative of this family of functionals can be written as, Γ Γ = u ( ) δ F u ; δu = E( xux, ( ), ux ( )) δu d Ω+ T( xux, ( ), ux ( )) δu d Γ Ω REMARK The example showed that for F Γ u : = φ u d Ω+ ϕ u d Γ, the φ( u) ϕ( u) δ F( u) = δu dω+ δ dγ u u u Gâteaux derivative is. Ω ( ) ( ) ( ) Ω Ω Ω δ u δu = x Γ u 14
Extrema of a Function A function f( x): has a local minimum (maximum) at R R x Necessary condition: df ( x) dx = x x not ( x ) = f = Local minimum The same condition is necessary for the function to have extrema (maximum, minimum or saddle point) at. x This concept can be can be extended to functionals. 15
Extreme of a Functional. ariational principle A functional ( u) : F R has a minimum at u( x) Necessary condition for the functional to have extrema at u x : ( ) δf u; δ u = δu δu = x Γ u ( ) This can be re-written in integral form: ( ) δ F u ; δu = E( u ) δu d Ω+ T( u ) δu d Γ= Ω ariational Principle Γ δ u δu = x Γ u 16
11.3.ariational Principle Ch.11. ariational Principles 17
ariational Principle ariational Principle: δf( u ; δ u ) = E δ u d Ω+ d Ω T δ u Γ= δ u REMARK Γ δu = Note that δu x Γu is arbitrary. Fundamental Theorem of ariational Calculus: The expression E( xux, ( ), ux ( )) δudω+ (, ( ), ( )) δ dγ = Ω T xux ux u δ u Γ δu = x Γu is satisfied if and only if E( xux, ( ), ux ( )) = x Ω T( xux, ( ), ux ( )) = x Γ Euler-Lagrange equations Natural boundary conditions 18
Example F Find the Euler-Lagrange equations and the natural and forced boundary conditions of the functional b ( ) φ, ( ), ( ) with ( ) :[, ] R ; ( ) ( ) u = x u x u x dx u x a b u x = u a = p a x= a 19
Example - Solution Find the Euler-Lagrange equations and the natural and forced boundary conditions of the functional F b ( ) φ, ( ), ( ) with ( ) ( ) u = x u x u x dx u x = u a = p a Solution : First, the Gâteaux derivative must be obtained. The function u( x) is perturbed: u( x) u( x) + εη( x) not ( ) ( ) ( ) a ( ) ( ) ( ) η x δu x η a = η = u x u x + εη x This is replaced in the functional: F b ( ), ( ), ( ) a x= a u + εη = φ x u x + εη u x + εη dx 2
Example - Solution F b ( ) φ, ( ), ( ) u = x u x u x dx a ( ) ( ) u x = u a = p x= a The Gâteaux derivative will be φ φ u = u + = dx dε + u u ( ; ) d F( ) ε = δf η εη η η Then, the expression obtained must be manipulated so that it resembles the ariational Principle δ F u ; δu = E δu d Ω+ T δu d Γ= : ( ) Ω Integrating by parts the second term in the expression obtained: b a Γ 21 b b b b φ φ d φ φ φ d φ η dx = η ( ) η dx = ηb ηa ( ) ηdx u u dx u u u dx u a a a a b a The Gâteaux derivative is re-written as: b ( ) φ, ( ), ( ) ; ( ) u = x u x u x dx u a = p a η a = b φ d φ φ δ ( u; ( η ) = δ u; δu) = [ ( )] δudx + δu a u dx u u b δ u b
Example - Solution Therefore, the ariational Principle takes the form b φ d φ φ δ ( u; δu) = δu dx δub a + = u dx u u b δu δu a = If this is compared to ( ) δ F u ; δu = E δu d Ω+ T δu d Γ= Ω Γ, one obtains: E φ d φ u dx u ( xuu,, ) = x ( ab, ) T φ,, = u ( xuu) x= b ux ( ) ua ( ) = p x= a Euler-Lagrange Equations Natural (Newmann) boundary conditions Essential (Dirichlet) boundary conditions 22
ariational Form of a Continuum Mechanics Problem Consider a continuum mechanics problem with local or strong governing equations given by, Euler-Lagrange equations E( xux, ( ), ux ( )) = x with boundary conditions: Natural or Newmann T * ( xux, ( ), ux ( )) ( u) n t( x) = x Γ Forced (essential) or Dirichlet = Γ ( ) ( ) u u x u x x REMARK The Euler-Lagrange equations are generally a set of PDEs. 23
ariational Form of a Continuum Mechanics Problem The variational form of the continuum mechanics problem consists in finding a field where fulfilling: u( x) X 3 m { u( x) u( x) u ( x) u} : = : R R = on Γ 3 m { δux ( ): δux ( ) on u} = R R = Γ E( xux, ( ), ux ( )) δux ( ) d + T( xux, ( ), ux ( )) δux ( ) dγ= δ ux ( ) Γ 24
ariational 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. 25
11.4. irtual Work Principle Ch.11. ariational Principles 26
Governing Equations Continuum mechanics problem for a body: Cauchy equation 2 u( x, t) ( x, t) + ρb( x, t) = ρ in 2 t ( ε uxt ( (, ))) Boundary conditions (, t) = (, t) Γu ( x ) n( x ) t ( x ) u x u x on,t,t =,t on Γ ( ε ( u),t) 27
ariational Principle The variational principle consists in finding a displacement field, where such that the variational principle holds, 2 u δw( u; δu) = [ + ρ( b )] δud + ( ) δ dγ= δ 2 t t n u u where Note: { ( ) 3 m u x t u( x t) u ( x t) u} : =, : R R, =, on Γ = E is the space of admissible displacements. is the space of admissible virtual displacements (test functions). The (perturbations of the displacements ) δu are termed virtual displacements. Γ = T { ( ) 3 m δu x δu( x) u} : : on = R R = Γ 28
irtual Work Principle (WP) The first term in the variational principle = a 2 u δw( u; δu) = [ + ρ( b )] δud + 2 ( ) δ dγ= δ t t n u u Considering that s ( ) = ( ) : δu δu δu and (applying the divergence theorem): s ( ) d ( ) d : δu = n δu Γ δu d Then, the irtual Work Principle reads: * s ( ) ( ) : Γ δw u; δu = ρ b a δu d + t δu dγ δu d = δu = E Γ Γ = T 29
irtual Work Principle (WP) 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 irtual Work Principle. REMARK 2 The irtual Work Principle can be viewed as the variational principle associated to a functional W ( u), being the necessary condition to find a minimum of this functional. 3
Interpretation of the WP The WP can be interpreted as: * s ( u u) ( b a) u t u : ( u) δw ; δ = ρ δ d + δ dγ δ d = Γ b* δε pseudo - body forces Work by the pseudo-body forces and the contact forces. External virtual work δ W ext virtual strains Work by the virtual strain. Internal virtual work int δ W ext int ( u; u) = = u δw δ δw δw δ 31
WP in oigt s Notation Total virtual work. 32 Engineering notation uses vectors instead of tensors: x δε x δε x δε δε y y y not δε δε τ xy δγ xy 2δε xy τ δγ 2δε 6 z 6 z z { } R ;{ } = { δε } R ; { δε} = = :δε= { } { δε} = { δε} { } xz xz xz τ yz δγ yz 2δε yz The irtual Work Principle becomes * δw = { δε} { } d ρ( b a) δu d + t δu dγ = δu Γ Internal virtual int work,. δ W External virtual ex t work,. δ W
11.5. Minimum Potential Energy Principle Ch.11. ariational Principles 33
Hypothesis 34 An explicit expression of the functional in the WP can only be obtained under the following hypothesis: 1. Linear elastic material. The elastic potential is: 1 uˆ ( ε) uˆ( ε)= ε: C: ε = = C: ε 2 ε 2. Conservative volume forces. The potential for the quasi-static case ( a= ) under gravitational φ( u) forces and constant density is: φ ( ) = ρ = ρ 3. Conservative surface forces. The potential is: Then a functional, total potential energy, can be defined as s = ε( ( u )) U W ( u) = ˆ( ε) + φ ( u) + ( u) u d d G ds Elastic energy Potential energy of the body forces u bu b u Γ G G( u) u = t u = t u ( ) Potential energy of the surface forces
Potential Energy ariational Principle The variational form consists in finding a displacement field ux (,) t, such that for any δ u δ u = in Γ u the following condition holds, ( u) G ( u) uˆ S φ δu( u; δu) = : ( δu) d + δ d + δ dγ= u u ε u u Γ = δε = = ρb = t * * ( ) ε d ( ) d u; u = : b a u t u dγ u δu δ δ ρ δ δ δ This is equivalent to the WP previously defined. ( ; ) δw δu u δu Γ 35
Minimization of the Potential Energy The WP is obtained as the variational principle associated with this functional U, the potential energy. deriving from a The potential energy is potential 1 * U( u) = ε ( u) : C: ( u) d ρ ( b au ( )) ud t udγ 2 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 WP can be viewed in this case as the solution which minimizes the total potential energy functional. δu( u; δu) = δu Γ 36