Second gradient theory P . Seppecher (IMATH Toulon) Sperlonga , - PowerPoint PPT Presentation

Second gradient theory P . Seppecher (IMATH Toulon) Sperlonga , September 2010 P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 1 / 41

Duality in mechanics 1 Second gradient theory 2 A Cauchy-like construction of the theory 3 Second gradient material 4 5 A mechanical error to avoid First example : capillary fluid 6 Second example : the beam in flexion 7 Third example : homogenized network of beams 8 Fourth example : pantographic beam 9 10 Closure of elasticity functionals 11 Conclusion P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 1 / 41

Duality in mechanics (point masses) Dynamics of a point mass is driven by The balance of forces: the external mechanical actions on the mass can be described by a vector F suc that m γ = F or by the principle of virtual powers: the external mechanical actions on the mass can be described by a linear form P such that V ∈ R 3 , m γ · � ∀ � V = P ( � V ) - As well known any linear form on R 3 can be identify to a scalar product : P ( � V ) has the form P ( � V ) = F · � V and the two principles are equivalent. - Generalization to finite number of particles is straightforward. P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 2 / 41

Duality in mechanics (rigid solids) The displacement of a rigid solid is an isometry. The only possible velocity fields � V have to satisfy the equiprojectivity property ∀ ( x , y ) ∈ ( R 3 ) 2 , ( � V ( x ) − � V ( y )) · ( x − y ) = 0 This makes a dimension 6 vector space. Indeed (Ω , W ) �→ ( V : x �→ W +Ω · x ) is an isomorphism with the set SKEW × R 3 . Its dual has a similar structure P ( � V ) = M · Ω+ R · W ( M , R ) is a torque-resultant representation of mechanical actions Generalization to finite number of rigid solids is straightforward. Let us show that the concept of forces is here both unsufficient and superfluous: P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 3 / 41

Superfluous Two opposite forces have no physical meaning inside the theory no power is expanded in any possible motion is 0 in the dual of the space of rigid motions P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 4 / 41

Unsufficient A wheel on sand P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 5 / 41

Unsufficient A wheel on sand in rigid mechanics The applied torque at the contact point is not a force. It corresponds to some expanded power It is a non trivial element of the dual of rigid motions P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 5 / 41

Duality Conclusion The PVP is equivalent to the momentum balance in simple situations It is more precise for systems with “sophisticated” kinematics In continuum mechanics : velocity fields belong to a space of smooth functions. Elements of the dual are distributions . P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 6 / 41

Second gradient theory There are two way for constructing the theory: postulating a form for the internal virtual power and deducing boundary 1 actions postulating a form for boundary interactions and stating a representation 2 theorem for internal stresses Let us start by the first (and easier) method. P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 7 / 41

Second gradient theory We assume the following form for internal virtual power b ij ∂ j V i + ∑ Z D ∑ a i V i + ∑ P int ( V ) = − c ijk ∂ j ∂ k V i � i i , j i , j , k P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 8 / 41

Second gradient theory We assume the following form for internal virtual power b ij ∂ j V i + ∑ Z D ∑ a i V i + ∑ P int ( V ) = − c ijk ∂ j ∂ k V i � i i , j i , j , k b ij ∂ j V i + ∑ Z D ∑ P int ( V ) = − � c ijk ∂ j ∂ k V i i , j i , j , k P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 8 / 41

Second gradient theory We assume the following form for internal virtual power b ij ∂ j V i + ∑ Z D ∑ a i V i + ∑ P int ( V ) = − c ijk ∂ j ∂ k V i � i i , j i , j , k b ij ∂ j V i + ∑ Z D ∑ P int ( V ) = − � c ijk ∂ j ∂ k V i i , j i , j , k Z P int ( V ) = − b ij ∂ j V i + c ijk ∂ j ∂ k V i � D P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 8 / 41

Second gradient theory We assume the following form for internal virtual power b ij ∂ j V i + ∑ Z D ∑ a i V i + ∑ P int ( V ) = − c ijk ∂ j ∂ k V i � i i , j i , j , k b ij ∂ j V i + ∑ Z D ∑ P int ( V ) = − � c ijk ∂ j ∂ k V i i , j i , j , k Z P int ( V ) = − b ij ∂ j V i + c ijk ∂ j ∂ k V i � D Z P int ( V ) = − � b ij V i , j + c ijk V i , jk D P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 8 / 41

Second gradient theory We assume the following form for internal virtual power b ij ∂ j V i + ∑ Z D ∑ a i V i + ∑ P int ( V ) = − c ijk ∂ j ∂ k V i � i i , j i , j , k b ij ∂ j V i + ∑ Z D ∑ P int ( V ) = − � c ijk ∂ j ∂ k V i i , j i , j , k Z P int ( V ) = − b ij ∂ j V i + c ijk ∂ j ∂ k V i � D Z P int ( V ) = − � b ij V i , j + c ijk V i , jk D and apply the principle of virtual power Z P int ( V )+ � P ext ( V ) D ργ i V i = � ∀ V , to explicit external actions P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 8 / 41

Second gradient theory Let us integrate by parts the last term in Z Z Z P ext ( V ) = P int ( V ) = � D ργ i V i − � D ργ i V i + b ij V i , j + c ijk V i , jk D P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 9 / 41

Second gradient theory Let us integrate by parts the last term in Z Z Z P ext ( V ) = P int ( V ) = � D ργ i V i − � D ργ i V i + b ij V i , j + c ijk V i , jk D Z D ρ ∑ Z Z P ext ( V ) = γ i V i + � b ij V i , j − c ijk , k V i , j + c ijk n k V i , j ∂ D D i P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 9 / 41

Second gradient theory Let us integrate by parts the last term in Z Z Z P ext ( V ) = P int ( V ) = � D ργ i V i − � D ργ i V i + b ij V i , j + c ijk V i , jk D Z D ρ ∑ Z Z P ext ( V ) = γ i V i + � b ij V i , j − c ijk , k V i , j + c ijk n k V i , j ∂ D D i Setting σ ij = b ij − c ijk , k , P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 9 / 41

Second gradient theory Let us integrate by parts the last term in Z Z Z P ext ( V ) = P int ( V ) = � D ργ i V i − � D ργ i V i + b ij V i , j + c ijk V i , jk D Z D ρ ∑ Z Z P ext ( V ) = γ i V i + � b ij V i , j − c ijk , k V i , j + c ijk n k V i , j ∂ D D i Setting σ ij = b ij − c ijk , k , Z Z Z P ext ( V ) = D ργ i V i + D σ ij V i , j + � c ijk n k V i , j ∂ D P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 9 / 41

Second gradient theory Z Z Z P ext ( V ) = � D ργ i V i + D σ ij V i , j + c ijk n k V i , j ∂ D P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 10 / 41

Second gradient theory Z Z Z P ext ( V ) = � D ργ i V i + D σ ij V i , j + c ijk n k V i , j ∂ D Let us integrate by parts again Z Z P ext ( V ) = D ( ργ i − σ ij , j ) V i + ∂ D σ ij n j V i + c ijk n k V i , j � P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 10 / 41

Second gradient theory Z Z Z P ext ( V ) = � D ργ i V i + D σ ij V i , j + c ijk n k V i , j ∂ D Let us integrate by parts again Z Z P ext ( V ) = D ( ργ i − σ ij , j ) V i + ∂ D σ ij n j V i + c ijk n k V i , j � Setting f ext = ργ − div ( σ ) , we get Z Z P ext ( V ) = f ext � V i + ∂ D σ ij n j V i + c ijk n k V i , j i D P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 10 / 41

Second gradient theory Z Z P ext ( V ) = f ext ∂ D σ ij n j V i + c ijk n k V i , j � V i + i D Now, let us integrate by parts the last term on the boundary. We need to separate normal and tangent derivatives: V i , j = V n i , j + V t where V n V t i , j = V i ,ℓ n ℓ n j , i , j = V i ,ℓ P ℓ j , P ℓ j = δ ℓ j − n ℓ n j i , j , P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 11 / 41

Second gradient theory Z Z P ext ( V ) = f ext ∂ D σ ij n j V i + c ijk n k V i , j � V i + i D Now, let us integrate by parts the last term on the boundary. We need to separate normal and tangent derivatives: V i , j = V n i , j + V t where V n V t i , j = V i ,ℓ n ℓ n j , i , j = V i ,ℓ P ℓ j , P ℓ j = δ ℓ j − n ℓ n j i , j , Then Z Z Z Z c ijk n k V t c ijk n k ν j V i c ijk n k V i ,ℓ P ℓ q P qj = − ∂ D ( c ijk n k P qj ) ,ℓ P ℓ q V i + i , j = ∂ D ∂ D ∂∂ D P . Seppecher (IMATH Toulon) () Second gradient theory Sperlonga , September 2010 11 / 41