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

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

A 3D approximation of the nonlinear elasticity system for rods

Georges GRISO.

Laboratoire Jacques-Louis Lions, Paris VI, France

Rouen, 25-26 Octobre, 2011

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Contents

• I. Nonlinear elastic thin rod
• 1. Decomposition of the rod deformations
• 2. Korn’s inequalities
• 3. Elastic thin rod : the total elastic energy. Estimates
• II. Explicit decomposition of the rod-deformations
• 1. A preliminary algebraic lemma
• 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

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

• I. Nonlinear elastic thin rod

Notations

• M3 the 3 × 3 real matrices,
• |||A||| =
• Tr(ATA) the Frobenius norm of the matrix A,
• I3 the unit 3 × 3 matrix,
• Id the identity map of R3,
• ω an open bounded set in R2 with lipschitz boundary, O ∈ ω,
• ωδ = δω
• Ωδ = ωδ×]0, L[ : the rod,
• the rod is clamped in one extremity

Γ0,δ = ωδ × {0},

• the set of admissible deformations

Hδ =

• v ∈ H1(Ωδ; R3) | v = Id
• n

Γ0,δ

• .

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Decomposition of rod-deformations

Theorem 1. Let v ∈ H1(Ωδ; R3) be a deformation. There exist V ∈ H1(0, L; R3), R ∈ H1(0, L; SO(3)) and v ∈ H1(Ωδ; R3) such that v(x) = V(x3) + R(x3)

• x1e1 + x2e2
• + v(x)

for a.e. x ∈ Ωδ with the estimates (we set d(∇v(x)) = dist(∇v(x), SO(3))) δ2

• dR

dx3

• L2(0,L;R3×3) +
• ∇v − R
• L2(Ωδ;R3×3) ≤ C||d(∇v)||L2(Ωδ).

Corollary. 1 δ ||v||L2(Ωδ;R3) + ||∇v||L2(Ωδ;R3×3) ≤ C1||d(∇v)||L2(Ωδ),

• dV

dx3 − Re3

• L2(0,L;R3) ≤ C ||d(∇v)||L2(Ωδ)

δ . [1] D. Blanchard, G. Griso. Decomposition of deformations of thin

• rods. Application to nonlinear elasticity, Ana. Appl. 7 (1) (2009).

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Korn’s inequalities

If v ∈ Hδ then V(0) = 0, R(0) = I3 and v(x) = 0 x ∈ Γ0,δ. Hence, from Theorem 1 we obtain ||R − I3||L2(0,L;R3×3) ≤ CL||d(∇v)||L2(Ωδ) δ2 . Besides, we also have ||R − I3||L2(0,L;R3×3) ≤ 2 √ 3L. We obtain the following nonlinear Korn’s inequalities : ||v − Id||L2(Ωδ;R3) + ||∇v − I3||L2(Ωδ;R3×3) ≤ C δ ||d(∇v)||L2(Ωδ), ||v − Id||L2(Ωδ;R3) + ||∇v − I3||L2(Ωδ;R3×3) ≤ C

• δ + ||d(∇v)||L2(Ωδ)
• .

The constants do not depend on δ.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

The total elastic energy

The local elastic energy W : M3 − → R+ ∪ {+∞}

• W(F) = Q(FTF − I3) if det(F) > 0,
• W(F) = +∞ if det(F) ≤ 0.

where Q is a positive-definite quadratic form. In the case of a St Venant-Kirchhoff’s material the quadratic form is given by Q(E) = λ 8

• tr(E)

2 + µ 4tr

• E2

, The total elastic energy Jκ,δ(v) is (v ∈ Hδ) Jκ,δ(v) =

• Ωδ
• W(∇v) −
• Ωδ

fκ,δ · (v − Id). where fκ,δ(x) = δ2κ−2f(x3) if 1 ≤ κ ≤ 2, fκ,δ(x) = δκf(x3) if κ ≥ 2 for a.e. x ∈ Ωδ.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Estimates

Lemma 2. Let v ∈ Hδ be a deformation satisfying Jκ,δ(v) ≤ 0. We have ||d(∇v)||L2(Ωδ) + ||∇vT∇v − I3||L2(Ωδ;R3×3) ≤ Cδκ||f||L2(0,L;R3). The constant is independent of δ. As a consequence, there exists a nonpositive constant c which does not depend on δ such that for any v ∈ Hδ we have cδ2κ ≤ Jκ,δ(v). We set mκ,δ = inf

v∈Hδ Jκ,δ(v).

Then we get c ≤ mκ,δ δ2κ ≤ 0.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Rescaling Ωδ For every measurable function w on Ωδ we define Πδ(w) by Πδ(w)(X1, X2, x3) = w(δX1, δX2, x3) for a.e. (X1, X2, x3) ∈ Ω = ω×]0, L[. If w ∈ L2(Ωδ) then we have ||Πδ(w)||L2(Ω) = 1

δ||w||L2(Ωδ),

||Πδ(w)||L1(Ω) =

1 δ2 ||w||L1(Ωδ).

Some remarks

• Consider v ∈ Hδ such that ||d(∇v)||L2(Ωδ) ≤ Cδκ.

Theorem 1 gives a field R ∈ H1(0, L; SO(3)) such that ||∇v − R||L2(Ωδ;R3×3) ≤ C||d(∇v)||L2(Ωδ) ≤ Cδκ, ||R − I3||L2(0,L;R3×3) ≤ C ||d(∇v)||L2(Ωδ) δ2 ≤ Cδκ−2. The set Aδ =

• y ∈ Ω | |||Πδ(∇v)(y) − R(y3)||| ≥ 1 y ∈ Ω
• has a measure less than Cδκ−1.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

• If v satisfies Jκ,δ(v) ≤ 0. We have

||d(∇v)||L2(Ωδ) + ||∇vT∇v − I3||L2(Ωδ;R3×3) ≤ Cδκ. We have the identity (∇v)T∇v−I3 = (∇v−R)TR+RT(∇v−R)+(∇v−R)T(∇v−R). Then we deduce that ||Πδ

• (∇v − R)T(∇v − R)
• ||L1(Ω;R3×3) ≤ Cδ2(κ−1),

||Πδ

• (∇v − R)TR + RT(∇v − R)
• ||L1(Ω;R3×3) ≤ Cδκ−1.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

• II. An explicit decomposition
• f the rod-deformations

Theorem 3. Any displacement u ∈ H1(Ωδ; R3) is decomposed as u(x) = U(x3) + R(x3) ∧

• x1e1 + x2e2
• + u(x)

for a.e. x ∈ Ωδ. where U ∈ H1(0, L; R3), R ∈ H1(0, L; R3), u ∈ H1(Ωδ; R3) and verifies the conditions

• ωδ

u(x1, x2, x3)dx1dx2 = 0,

• ωδ

xαu3(x1, x2, x3)dx1dx2 = 0,

• ωδ
• x1u2(x1, x2, x3) − x2u1(x1, x2, x3)
• dx1dx2 = 0

for a.e. x3 ∈]0, L[. [2] G. Griso. Decomposition of displacements of thin structures. J.

• Math. Pures Appl. 89 (2008), 199-233.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Moreover, there exists C2 independent of δ and L such that δ2

• dR

dx3

• L2(0,L;R3)+||∇u−AR||L2(Ωδ;R3×3) ≤ C2

δ ||(∇u)T+∇u||L2(Ωδ;R3×3). The matrix AR(x3) is defined by ∀x ∈ R3, AR(x3)x = R(x3) ∧ x, for a. e. x3 ∈]0, L[. Corollary. 1 δ ||u||L2(Ωδ;R3) + ||∇u||L2(Ωδ;R3×3) ≤ C2||(∇u)T + ∇u||L2(Ωδ;R3×3),

• dU

dx3 − R ∧ e3

• L2(0,L;R3) ≤ C2

δ ||(∇u)T + ∇u||L2(Ωδ;R3×3).

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

A preliminary algebraic lemma

We would like to obtain the same type of decomposition as for a rod-deformation v ∈ H1(Ωδ; R3), namely v(x) = V(x3) + R(x3)

• x1e1 + x2e2
• + v(x)
• for a.e. x ∈ Ωδ

where V ∈ H1(0, L; R3), R ∈ H1(0, L; SO(3)), v ∈ H1(Ωδ; R3) and verifies the conditions

• ωδ

v(x1, x2, x3)dx1dx2 = 0,

• ωδ

xαv3(x1, x2, x3)dx1dx2 = 0,

• ωδ
• x1v2(x1, x2, x3) − x2v1(x1, x2, x3)
• dx1dx2 = 0

for a.e. x3 ∈]0, L[.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Hence, the field R has to satisfy m1(v)(x3) · R(x3)e2 = m2(v)(x3) · R(x3)e1, m1(v)(x3) · R(x3)e3 = m2(v)(x3) · R(x3)e3 = 0, for a.e. x3 ∈]0, L[ where mα(v)(x3) = 1 δ4

• ωδ

xαv(x1, x2, x3)dx1dx2.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Lemma 4. Let m1 and m2 be two linearly independent vectors in R3. The matrix R ∈ SO(3) defined by Re1 =

• ||m2||2 + ||m1 ∧ m2||
• m1 − (m1 · m2)m2

||m1 ∧ m2||

• ||m1||2 + 2||m1 ∧ m2|| + ||m2||2 ,

Re2 = −(m1 · m2)m1 +

• ||m1||2 + ||m1 ∧ m2||
• m2

||m1 ∧ m2||

• ||m1||2 + 2||m1 ∧ m2|| + ||m2||2 ,

Re3 = m1 ∧ m2 ||m1 ∧ m2|| satisfies the following conditions : m1 · Re2 = m2 · Re1 and m1 · Re3 = m2 · Re3 = 0. Moreover, there exists C such that for any matrices R

′ ∈ SO(3)

we have |||R − R

′||| ≤ C max(I1, I2)

I1I2

• ||m1 − I1R

′e1|| + ||m2 − I2R ′e2||

• .

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

New decomposition theorem

Theorem 5. There exists C∗ which does not depend on δ and L such that any deformations v ∈ H1(Ωδ; R3) satisfying ||d(∇v)||L2(Ωδ) ≤ C∗δ3/2 is decomposed as v(x) = V(x3) + Rv(x3)

• x1e1 + x2e2 + v(x)
• for a.e. x ∈ Ωδ.

where V ∈ H1(0, L; R3), Rv ∈ H1(0, L; SO(3)), v ∈ H1(Ωδ; R3) verifies the relations

• ωδ

v(x1, x2, x3)dx1dx2 = 0,

• ωδ

xαv3(x1, x2, x3)dx1dx2 = 0,

• ωδ
• x1v2(x1, x2, x3) − x2v1(x1, x2, x3)
• dx1dx2 = 0

for a.e. x3 ∈]0, L[.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Moreover, there exists a constant C3 which does not depend on δ and L such that 1 δ ||v||L2(Ωδ;R3) + ||∇v||L2(Ωδ;R3×3) ≤ C3||d(∇v)||L2(Ωδ),

• dRv

dx3

• L2(0,L;R3×3) ≤ C3

δ2 ||d(∇v)||L2(Ωδ),

• dV

dx3 − Rve3

• L2(0,L;R3) ≤ C3

δ ||d(∇v)||L2(Ωδ),

• ∇v − Rv
• L2(Ωδ;R3×3) ≤ C3||d(∇v)||L2(Ωδ).

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Main idea of the proof. From Theorem 1, we have v(x) = V(x3) + R

′(x3)

• x1e1 + x2e2
• + v

′(x),

x ∈ Ωδ. Moreover, we have the estimates 1 δ ||v

′||L2(Ωδ;R3) + ||∇v ′||L2(Ωδ;R3×3) ≤ C1||d(∇v)||L2(Ωδ)

δ

• dR

′

dx3

• L2(0,L;R3×3) +
• dV

dx3 − R

′e3

• L2(0,L;R3) ≤ C

δ ||d(∇v)||L2(Ωδ)

• ∇v − R

′

• L2(Ωδ;R3×3) ≤ C||d(∇v)||L2(Ωδ).

Then mα(v) = IαR

′eα + mα(v ′)

Iα =

• ω

X 2

αdX1dX2

and ||mα(v

′)||L∞(0,L;R3) ≤ 2C1

√Iα δ3/2 ||d(∇v)||L2(Ωδ).

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

We set M(v

′)(x3) =

• ωδ

||v

′(x1, x2, x3)||2dx1dx2

for a.e. x3 ∈]0, L[. We have M(v

′) ∈ H1(0, L) and

||M(v

′)||L∞(0,L) ≤ 5||v ′||L2(Ωδ;R3)||v ′||H1(Ωδ;R3),

• dM(v

′)

dx3

• L2(0,L) ≤ 5||v

′||1/2

L2(Ωδ;R3)||v

′||3/2

H1(Ωδ;R3).

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

• III. The simplified Green St

Venant’s tensor

We have the identity (∇v)T∇v−I3 = (∇v−Rv)TRv+RT

v (∇v−Rv)+(∇v−Rv)T(∇v−Rv).

The simplified Green-St Venant’s strain tensor GVs(v) is defined by GVs(v) = (∇v − Rv)TRv + RT

v (∇v − Rv).

Theorem 6. There exists a constant C∗∗ ≤ C∗ independ of δ and L such that, if v ∈ H1(Ωδ; R3) satisfies ||d(∇v)||L2(Ωδ) ≤ C∗∗δ3/2 then

• dRv

dx3

• L2(0,L;R3×3) ≤ C

δ2 ||GVs(v)||L2(Ωδ;R3×3),

• ∇v − Rv
• L2(Ωδ;R3×3) ≤ C||GVs(v)||L2(Ωδ;R3×3).

The constant C does not depend on δ and L.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Main idea of the proof. Let Av be in L2(0, L; R3×3) defined by dRv dx3 = RvAv. We introduce the displacement u ∈ H1(Ωδ; R3) u(x) = U(x3) + R(x3) ∧ (x1e1 + x2e2) + v(x) for a.e. x ∈ Ωδ where ∀− → x ∈ R3 dR dx3 ∧ − → x = Av− → x , dU dx3 − R ∧ e3 =

• Rv

T dV dx3 − Rve3

• A straightforward calculation gives

(∇u)T + ∇u = 2GVs(v) −   Avv · e1 Avv · e2 Avv · e1 Avv · e2 2Avv · e3   . We have ||Avv||L2(Ωδ;R3) ≤ ||Av||L2(Ωδ;R3×3)||M(v)||1/2

L∞(0,L)

≤ √ 5C3 δ3/2 ||(∇u)T + ∇u||2

L2(Ωδ;R3).

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Corollary. Let v ∈ H1(Ωδ; R3) be a deformation satisfying the condition in Theorem 4. We have c||d(∇v)||L2(Ωδ) ≤ ||GVs(v)||L2(Ωδ;R3×3) ≤ C||d(∇v)||L2(Ωδ). The constants c and C do not depend on δ and L.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

First approximation of the total energy

The approximate total elastic energy J(S)

κ,δ is defined by ( v ∈ Hδ)

J(S)

κ,δ(v) =

• Ωδ

Q

• GVs(v)
• −
• Ωδ

fκ,δ · (v − Id

• =
• Ωδ

Q

• (∇v)TRv + RT

v ∇v − 2I3

• −
• Ωδ

fκ,δ · (v − Id

• if ||d(∇v)||L2(Ωδ) < C∗∗δ3/2 and +∞ otherwise.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Theorem 7. For δ > 0 and κ > 3/2 fixed, there exists vκ,δ ∈ Hδ such that m(S)

κ,δ = J(S) κ,δ(vκ,δ) = min v∈Hδ J(S) κ,δ(v).

We have ||GVs(vκ,δ)||L2(Ωδ;R3×3) ≤ Cδ2κ. The constants do not depend on δ.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Justification of the 3D approximations

Theorem 8. For κ > 3/2, we have mκ = lim

δ→0

mκ,δ δ2κ = lim

δ→0

m(S)

κ,δ

δ2κ . [1] D. Blanchard, G. Griso. Decomposition of deformations of thin

• rods. Application to nonlinear elasticity, Ana. Appl. 7 (1) (2009).

[3] D. Blanchard, G. Griso. Asymptotic behavior of structures made of straight rods (to appear in Journal of elasticity.)

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

The linearized approximation.

The linearized approximate total energy J(L)

κ,δ is

J(L)

κ,δ(u) =

• Ωδ

Q

• (∇u)T + ∇u
• −
• Ωδ

fκ,δ · u where Id + u ∈ Hδ. For δ > 0 and κ ≥ 1 fixed, there exists uκ,δ ∈ Hδ − Id such that m(L)

κ,δ = J(L) κ,δ(uκ,δ) =

min

u∈Hδ−Id J(L) κ,δ(u)

For κ > 3, we have mκ = lim

δ→0

mκ,δ δ2κ = lim

δ→0

m(L)

κ,δ

δ2κ .

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Complement

We have ∀(B, C) ∈ S3 × M3 Q(B + CTC) ≤

• Q(B) + c1|||C|||22

where Q(B) ≤ c2

1|||B|||2.

We denote Q(2) : S3 × M3 − → R the functional defined by Q(2)(B, C) =

• Q(B) + c1|||C|||22.

This functional is convex. We define the approximate local energy

• W (2) : M3 × SO(3) −

→ R+ ∪ {+∞} by

• W (2)(A, R) =
• Q(2)

ATR + RTA − 2I3, A − R

• if

det(A) ≥ 0 + ∞ if det(A) < 0.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

The second approximate total elastic energy J(2)

κ,δ is

J(2)

κ,δ(v) =

• Ωδ
• W (2)(∇v, Rv) −
• Ωδ

fκ,δ · (v − Id

• where v ∈ Wδ = Hδ ∩ W 1,4(Ωδ; R3).

Theorem 9. For δ > 0 and κ > 3/2 fixed, there exists vκ,δ ∈ Wδ such that m(2)

κ,δ = J(2) κ,δ(vκ,δ) = min v∈Wδ J(2) κ,δ(v).

We have ||GVs(vκ,δ)||L2(Ωδ;R3×3) + ||(∇vκ,δ)T∇vκ,δ − I3||L2(Ωδ;R3×3) ≤ Cδ2κ The constants do not depend on δ.

A 3D approximation

• f the nonlinear

elasticity system for rods Contents

• I. Nonlin. el. rod

* Dec. rod-def. * Korn’s ineg. * The tot. el. eneg. * Estimates

• II. Explicit

decomposition

* Prel. alg. lemma * New decomp. theorem

III Simp. G.S.V.’s tensor

• IV. App. total

energy

* Just. of the 3D approx.

Theorem 10. For κ > 3/2, we have mκ = lim

δ→0

mκ,δ δ2κ = lim

δ→0

m(2)

κ,δ

δ2κ .