A 3D approximation of the nonlinear * Korns ineg. * The tot. el. - PowerPoint PPT Presentation

A 3D approximation of the nonlinear elasticity system for rods Contents I. Nonlin. el. rod * Dec. rod-def. A 3D approximation of the nonlinear * Korn’s ineg. * The tot. el. eneg. elasticity system for rods * Estimates II. Explicit decomposition * Prel. alg. lemma * New decomp. theorem Georges GRISO. III Simp. G.S.V.’s tensor IV. App. total Laboratoire Jacques-Louis Lions, Paris VI, France energy * Just. of the 3D approx. Rouen, 25-26 Octobre, 2011

A 3D approximation of the nonlinear Contents elasticity system for rods Contents I. Nonlin. el. rod I. Nonlinear elastic thin rod * Dec. rod-def. * Korn’s ineg. 1. Decomposition of the rod deformations * The tot. el. eneg. * Estimates II. Explicit 2. Korn’s inequalities decomposition * Prel. alg. lemma 3. Elastic thin rod : the total elastic energy. Estimates * New decomp. theorem III Simp. G.S.V.’s II. Explicit decomposition of the rod-deformations tensor IV. App. total 1. A preliminary algebraic lemma energy * Just. of the 3D approx. 2. New decomposition for the rod deformations III. Simplification in the Green St Venant’s tensor IV. An approximation of the total energy

A 3D approximation of the nonlinear I. Nonlinear elastic thin rod elasticity system for rods Notations Contents I. Nonlin. el. rod • M 3 the 3 × 3 real matrices, � * Dec. rod-def. * Korn’s ineg. • ||| A ||| = Tr ( A T A ) the Frobenius norm of the matrix A , * The tot. el. eneg. * Estimates • I 3 the unit 3 × 3 matrix, II. Explicit decomposition • I d the identity map of R 3 , * Prel. alg. lemma • ω an open bounded set in R 2 with lipschitz boundary, O ∈ ω , * New decomp. theorem III Simp. G.S.V.’s • ω δ = δω tensor IV. App. total • Ω δ = ω δ × ] 0 , L [ : the rod, energy * Just. of the 3D • the rod is clamped in one extremity approx. Γ 0 ,δ = ω δ × { 0 } , • the set of admissible deformations � � v ∈ H 1 (Ω δ ; R 3 ) | v = I d H δ = on Γ 0 ,δ .

A 3D approximation of the nonlinear Decomposition of elasticity system for rods rod-deformations Contents Theorem 1. I. Nonlin. el. rod * Dec. rod-def. Let v ∈ H 1 (Ω δ ; R 3 ) be a deformation. There exist V ∈ H 1 ( 0 , L ; R 3 ) , * Korn’s ineg. R ∈ H 1 ( 0 , L ; SO ( 3 )) and v ∈ H 1 (Ω δ ; R 3 ) such that * The tot. el. eneg. * Estimates � � II. Explicit decomposition v ( x ) = V ( x 3 ) + R ( x 3 ) x 1 e 1 + x 2 e 2 + v ( x ) for a.e. x ∈ Ω δ * Prel. alg. lemma * New decomp. theorem with the estimates (we set d ( ∇ v ( x )) = dist ( ∇ v ( x ) , SO ( 3 )) ) III Simp. G.S.V.’s δ 2 � � tensor � � � d R � � � ∇ v − R � L 2 ( 0 , L ; R 3 × 3 ) + L 2 (Ω δ ; R 3 × 3 ) ≤ C || d ( ∇ v ) || L 2 (Ω δ ) . IV. App. total � energy dx 3 * Just. of the 3D approx. Corollary. 1 δ || v || L 2 (Ω δ ; R 3 ) + ||∇ v || L 2 (Ω δ ; R 3 × 3 ) ≤ C 1 || d ( ∇ v ) || L 2 (Ω δ ) , � � L 2 ( 0 , L ; R 3 ) ≤ C || d ( ∇ v ) || L 2 (Ω δ ) � d V � � − Re 3 . � dx 3 δ [1] D. Blanchard, G. Griso. Decomposition of deformations of thin rods. Application to nonlinear elasticity, Ana. Appl. 7 (1) (2009).

A 3D approximation of the nonlinear Korn’s inequalities elasticity system for rods If v ∈ H δ then Contents V ( 0 ) = 0 , R ( 0 ) = I 3 and v ( x ) = 0 x ∈ Γ 0 ,δ . I. Nonlin. el. rod * Dec. rod-def. * Korn’s ineg. Hence, from Theorem 1 we obtain * The tot. el. eneg. * Estimates II. Explicit || R − I 3 || L 2 ( 0 , L ; R 3 × 3 ) ≤ CL || d ( ∇ v ) || L 2 (Ω δ ) decomposition . * Prel. alg. lemma δ 2 * New decomp. theorem III Simp. G.S.V.’s Besides, we also have tensor √ IV. App. total || R − I 3 || L 2 ( 0 , L ; R 3 × 3 ) ≤ 2 3 L . energy * Just. of the 3D approx. We obtain the following nonlinear Korn’s inequalities : || v − I d || L 2 (Ω δ ; R 3 ) + ||∇ v − I 3 || L 2 (Ω δ ; R 3 × 3 ) ≤ C δ || d ( ∇ v ) || L 2 (Ω δ ) , � � || v − I d || L 2 (Ω δ ; R 3 ) + ||∇ v − I 3 || L 2 (Ω δ ; R 3 × 3 ) ≤ C δ + || d ( ∇ v ) || L 2 (Ω δ ) . The constants do not depend on δ .

A 3D approximation of the nonlinear The total elastic energy elasticity system for rods → R + ∪ { + ∞} The local elastic energy � W : M 3 − Contents I. Nonlin. el. rod W ( F ) = Q ( F T F − I 3 ) if det ( F ) > 0 , � � * Dec. rod-def. W ( F ) = + ∞ if det ( F ) ≤ 0 . * Korn’s ineg. * The tot. el. eneg. * Estimates where Q is a positive-definite quadratic form. In the case of a St II. Explicit Venant-Kirchhoff’s material the quadratic form is given by decomposition * Prel. alg. lemma * New decomp. � � 2 + µ � E 2 � Q ( E ) = λ theorem tr ( E ) 4 tr , III Simp. G.S.V.’s 8 tensor IV. App. total The total elastic energy J κ,δ ( v ) is ( v ∈ H δ ) energy * Just. of the 3D approx. � � � J κ,δ ( v ) = W ( ∇ v ) − f κ,δ · ( v − I d ) . Ω δ Ω δ where f κ,δ ( x ) = δ 2 κ − 2 f ( x 3 ) if 1 ≤ κ ≤ 2 , for a.e. x ∈ Ω δ . f κ,δ ( x ) = δ κ f ( x 3 ) if κ ≥ 2

A 3D approximation of the nonlinear Estimates elasticity system for rods Lemma 2. Contents I. Nonlin. el. rod Let v ∈ H δ be a deformation satisfying J κ,δ ( v ) ≤ 0 . We have * Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. || d ( ∇ v ) || L 2 (Ω δ ) + ||∇ v T ∇ v − I 3 || L 2 (Ω δ ; R 3 × 3 ) ≤ C δ κ || f || L 2 ( 0 , L ; R 3 ) . * Estimates II. Explicit decomposition The constant is independent of δ . * Prel. alg. lemma * New decomp. As a consequence, there exists a nonpositive constant c which theorem III Simp. G.S.V.’s does not depend on δ such that for any v ∈ H δ we have tensor IV. App. total c δ 2 κ ≤ J κ,δ ( v ) . energy * Just. of the 3D approx. We set m κ,δ = inf v ∈ H δ J κ,δ ( v ) . Then we get c ≤ m κ,δ δ 2 κ ≤ 0 .

A 3D Rescaling Ω δ approximation of the nonlinear elasticity For every measurable function w on Ω δ we define Π δ ( w ) by system for rods Π δ ( w )( X 1 , X 2 , x 3 ) = w ( δ X 1 , δ X 2 , x 3 ) Contents for a.e. ( X 1 , X 2 , x 3 ) ∈ Ω = ω × ] 0 , L [ . I. Nonlin. el. rod * Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. If w ∈ L 2 (Ω δ ) then we have || Π δ ( w ) || L 2 (Ω) = 1 δ || w || L 2 (Ω δ ) , * Estimates II. Explicit 1 || Π δ ( w ) || L 1 (Ω) = δ 2 || w || L 1 (Ω δ ) . decomposition * Prel. alg. lemma * New decomp. Some remarks theorem III Simp. G.S.V.’s • Consider v ∈ H δ such that || d ( ∇ v ) || L 2 (Ω δ ) ≤ C δ κ . tensor IV. App. total energy Theorem 1 gives a field R ∈ H 1 ( 0 , L ; SO ( 3 )) such that * Just. of the 3D approx. ||∇ v − R || L 2 (Ω δ ; R 3 × 3 ) ≤ C || d ( ∇ v ) || L 2 (Ω δ ) ≤ C δ κ , || R − I 3 || L 2 ( 0 , L ; R 3 × 3 ) ≤ C || d ( ∇ v ) || L 2 (Ω δ ) ≤ C δ κ − 2 . δ 2 � � The set A δ = y ∈ Ω | ||| Π δ ( ∇ v )( y ) − R ( y 3 ) ||| ≥ 1 y ∈ Ω has a measure less than C δ κ − 1 .

A 3D approximation of the nonlinear elasticity system for rods Contents • If v satisfies J κ,δ ( v ) ≤ 0. We have I. Nonlin. el. rod * Dec. rod-def. || d ( ∇ v ) || L 2 (Ω δ ) + ||∇ v T ∇ v − I 3 || L 2 (Ω δ ; R 3 × 3 ) ≤ C δ κ . * Korn’s ineg. * The tot. el. eneg. * Estimates We have the identity II. Explicit decomposition * Prel. alg. lemma ( ∇ v ) T ∇ v − I 3 = ( ∇ v − R ) T R + R T ( ∇ v − R )+( ∇ v − R ) T ( ∇ v − R ) . * New decomp. theorem III Simp. G.S.V.’s tensor Then we deduce that IV. App. total � � energy ( ∇ v − R ) T ( ∇ v − R ) || L 1 (Ω; R 3 × 3 ) ≤ C δ 2 ( κ − 1 ) , || Π δ * Just. of the 3D approx. � � ( ∇ v − R ) T R + R T ( ∇ v − R ) || L 1 (Ω; R 3 × 3 ) ≤ C δ κ − 1 . || Π δ

A 3D approximation of the nonlinear II. An explicit decomposition elasticity system for rods of the rod-deformations Contents I. Nonlin. el. rod Theorem 3. * Dec. rod-def. * Korn’s ineg. Any displacement u ∈ H 1 (Ω δ ; R 3 ) is decomposed as * The tot. el. eneg. * Estimates � � II. Explicit u ( x ) = U ( x 3 ) + R ( x 3 ) ∧ x 1 e 1 + x 2 e 2 + u ( x ) for a.e. x ∈ Ω δ . decomposition * Prel. alg. lemma * New decomp. where U ∈ H 1 ( 0 , L ; R 3 ) , R ∈ H 1 ( 0 , L ; R 3 ) , u ∈ H 1 (Ω δ ; R 3 ) and theorem III Simp. G.S.V.’s verifies the conditions tensor � � IV. App. total energy u ( x 1 , x 2 , x 3 ) dx 1 dx 2 = 0 , x α u 3 ( x 1 , x 2 , x 3 ) dx 1 dx 2 = 0 , * Just. of the 3D approx. ω δ ω δ � � � x 1 u 2 ( x 1 , x 2 , x 3 ) − x 2 u 1 ( x 1 , x 2 , x 3 ) dx 1 dx 2 = 0 ω δ for a.e. x 3 ∈ ] 0 , L [ . [2] G. Griso. Decomposition of displacements of thin structures. J. Math. Pures Appl. 89 (2008), 199-233.