On Cavitation in Elastodynamics Jan Giesselmann IANS, University of - - PowerPoint PPT Presentation

Institute of Applied Analysis and Numerical Simulations

On Cavitation in Elastodynamics

Jan Giesselmann

IANS, University of Stuttgart joint work with A. Tzavaras (University of Crete) This work was upported by the ACMAC project - European Union FP7 14th International Conference on Hyperbolic Problems

June 25–29 2012

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Institute of Applied Analysis and Numerical Simulations

Outline

Introduction 1D: Concept for discontinuous solutions to second order equations 1D: Energy of solutions 3D: Solution concept and energy Summary & Prospects

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Institute of Applied Analysis and Numerical Simulations

Introduction

Study fracture and cavitation in elastical solids. Once a bar breaks, or a hole forms, coherence is lost and the hypothesis of continuum implicit in the elastodynamics equations is no longer valid. Transition region from a range of loading where the model is valid to a range where it loses validity. Explore this transition at the onset of fracture. Irregular solutions to the compressible, non-linear elastodynamics equations, where a hole forms in the interior. i.e. a discontinuity of the displacement field at one point. Study the energy of the irregular solution compared to trivial solutions (which also exist).

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Institute of Applied Analysis and Numerical Simulations

Introduction

Equations of elastodynamics: Search displacements y : B1♣0q ✂ r0, Tq Ý Ñ Rd such that det♣∇yq → 0. satisfying the wave equation ytt ✁ div♣τ♣∇yqq ✏ 0 (WAVE) with stress response W : Rd✂d

• Ý

Ñ R; τ ✏ ❇W ❇F : Rd✂d

• Ý

Ñ Rd✂d

• .

The wave equation is equivalent to the following system of first order conservation laws with u ✏ ∇y, v ✏ yt: ut ✁ ∇v ✏ 0 vt ✁ div♣τ♣uqq ✏ 0. (CONS)

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Institute of Applied Analysis and Numerical Simulations

Energy and admissibility

Smooth solutions of (WAVE) satisfy the energy equality d dt ✂1 2♣ytq2 W ♣∇yq ✡ ✁ div ♣τ♣∇yqytq ✏ 0. For weak solutions (having discontinuities in ∇y, yt) the classical admissibility criterion is the energy inequality d dt ✂1 2♣ytq2 W ♣∇yq ✡ ✁ div ♣τ♣∇yqytq ↕ 0, which has to be satisfied in a weak sense.

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Institute of Applied Analysis and Numerical Simulations

Solutions with cavitation

We study solutions of the form y♣x, tq ✏ tϕ ✁

⑤x⑤ t

✠

x ⑤x⑤

(ANSATZ) for some ϕ : r0, ✽q Ñ r0, ✽q, with ϕ♣0q → 0, ϕ✶♣sq → 0 ❅ s → 0. ✏ ⑤ ⑤ ♣ q ✏ ♣ q ✏ ❅ P ♣ q ♣ q ✏ ❅ P ❇ ♣ q ↕ ➔ → ♣ q ✏

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Institute of Applied Analysis and Numerical Simulations

Solutions with cavitation

We study solutions of the form y♣x, tq ✏ tϕ ✁

⑤x⑤ t

✠

x ⑤x⑤

(ANSATZ) for some ϕ : r0, ✽q Ñ r0, ✽q, with ϕ♣0q → 0, ϕ✶♣sq → 0 ❅ s → 0. For solutions having the form (ANSATZ) u, v only depend on s :✏ ⑤x⑤

t .

♣ q ✏ ♣ q ✏ ❅ P ♣ q ♣ q ✏ ❅ P ❇ ♣ q ↕ ➔ → ♣ q ✏

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Institute of Applied Analysis and Numerical Simulations

Solutions with cavitation

We study solutions of the form y♣x, tq ✏ tϕ ✁

⑤x⑤ t

✠

x ⑤x⑤

(ANSATZ) for some ϕ : r0, ✽q Ñ r0, ✽q, with ϕ♣0q → 0, ϕ✶♣sq → 0 ❅ s → 0. For solutions having the form (ANSATZ) u, v only depend on s :✏ ⑤x⑤

t .

In 1D: these solutions describe discontinuous shear or fracture, in 3D: such solutions describe opening holes/cavities. ♣ q ✏ ♣ q ✏ ❅ P ♣ q ♣ q ✏ ❅ P ❇ ♣ q ↕ ➔ → ♣ q ✏

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Institute of Applied Analysis and Numerical Simulations

Solutions with cavitation

We study solutions of the form y♣x, tq ✏ tϕ ✁

⑤x⑤ t

✠

x ⑤x⑤

(ANSATZ) for some ϕ : r0, ✽q Ñ r0, ✽q, with ϕ♣0q → 0, ϕ✶♣sq → 0 ❅ s → 0. For solutions having the form (ANSATZ) u, v only depend on s :✏ ⑤x⑤

t .

In 1D: these solutions describe discontinuous shear or fracture, in 3D: such solutions describe opening holes/cavities. Displacement initial boundary value problem y♣x, 0q ✏ λx, yt♣x, 0q ✏ 0 ❅x P B1♣0q, y♣x, tq ✏ λx ❅x P ❇B1♣0q and 0 ↕ t ➔ T for some λ → 0. ♣ q ✏

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Institute of Applied Analysis and Numerical Simulations

Solutions with cavitation

We study solutions of the form y♣x, tq ✏ tϕ ✁

⑤x⑤ t

✠

x ⑤x⑤

(ANSATZ) for some ϕ : r0, ✽q Ñ r0, ✽q, with ϕ♣0q → 0, ϕ✶♣sq → 0 ❅ s → 0. For solutions having the form (ANSATZ) u, v only depend on s :✏ ⑤x⑤

t .

In 1D: these solutions describe discontinuous shear or fracture, in 3D: such solutions describe opening holes/cavities. Displacement initial boundary value problem y♣x, 0q ✏ λx, yt♣x, 0q ✏ 0 ❅x P B1♣0q, y♣x, tq ✏ λx ❅x P ❇B1♣0q and 0 ↕ t ➔ T for some λ → 0. Note: There is the trivial solution y♣x, tq ✏ λx to this IBVP.

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Institute of Applied Analysis and Numerical Simulations

Situation in 1D

Ansatz y♣x, tq ✏ tY ✁x t ✠ with Y ♣✁ξq ✏ ✁Y ♣ξq, Y ✶♣ξq → 0 ❅ ξ → 0, lim

ξÑ0,ξ→0 Y ♣ξq → 0.

Then (WAVE) amounts to ξ2Y ✷ ✏ ♣τ♣Y ✶qq✶

W u

τ u

We impose the conditions W ✷ → 0 and W ✸ ➔ 0.

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Institute of Applied Analysis and Numerical Simulations

Rankine Hugoniot conditions and admissibility

yt♣x, tq ✏ Y ♣ξq ✁ ξY ✶♣ξq ✏: V ♣ξq, yx♣x, tq ✏ Y ✶♣ξq ✏: U♣ξq where ξ ✏ x t We can write down the equations for U, V ξU ✶ V ✶ ✏ 0 ξV ✶ ♣τ♣Uqq✶ ✏ 0 ♣✝q For a shock with speed σ the Rankine–Hugoniot conditions read ✁σrUs ✏ rV s and ✁ σrV s ✏ rτ♣Uqs ù ñ σ ✏ ❞ rτ♣Uqs rUs . The eigenvalues of ♣✝q are ✟ ❛ τ ✶♣Uq and the Lax condition becomes Ul → Ur for 1 ✁ shocks and Ul ➔ Ur for 2 ✁ shocks.

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Institute of Applied Analysis and Numerical Simulations

One class of possible solutions

Thus, we investigate Y ♣ξq :✏ ✩ ✫ ✪ Y ♣0q αξ : 0 ➔ ξ ➔ σ ✁Y ♣0q αξ : ✁σ ➔ ξ ➔ 0 λξ : ⑤ξ⑤ → σ (1D-ANSATZ)

t x x = σt x = −σt −tY (0) + αx tY (0) + αx λx λx

Y ξ Y (0) α λ α λ

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Institute of Applied Analysis and Numerical Simulations

One class of possible solutions

U ✏ 2Y ♣0qδξ✏0 αχt⑤ξ⑤➔σ✉ λχt⑤ξ⑤→σ✉ V ✏ Y ♣0qχt0➔ξ➔σ✉ ✁ Y ♣0qχt✁σ➔ξ➔0✉

α α λ λ U ξ −σ σ

−Y (0) Y (0) V ξ −σ σ 10 / 23

Institute of Applied Analysis and Numerical Simulations

One class of possible solutions

U ✏ 2Y ♣0qδξ✏0 αχt⑤ξ⑤➔σ✉ λχt⑤ξ⑤→σ✉ V ✏ Y ♣0qχt0➔ξ➔σ✉ ✁ Y ♣0qχt✁σ➔ξ➔0✉

α α λ λ U ξ −σ σ

−Y (0) Y (0) V ξ −σ σ

What is the meaning of τ♣Uq in this case?

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Institute of Applied Analysis and Numerical Simulations

slic–solutions

We propose the following notion of solution Definition We call y P C♣r0, Tq, Lp♣B1♣0qq a singular limiting induced from continuum (slic)–solution provided for all ψ P C ✽

0 ♣Rq such that

supp♣ψq ⑨ r✁1, 1s, ➺ 1

✁1

ψ♣xqdx ✏ 1, and ψ♣xq ✏ ψ♣✁xq the following holds lim

nÑ✽ryn tt ✁ ♣τ♣yn x qqxs ✏ 0

in D✶ where yn♣x, tq ✏ y ✝x nψ♣n☎q.

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Institute of Applied Analysis and Numerical Simulations

δ-shocks

This problem is conected to the notion of δ-shocks for hyperbolic conservation laws. There is a vast literature on this field, see the work of Danilov, Shelkovic. In contrast, our notion of solution is based on the underlying structure of the second order problem. For the p-system in fluid mechanics we have limUÑ✽ p♣Uq ✏ 0 such that ♣p♣δx✏0qq♣0q ✏ 0 seems reasonable. See chapter 9.7 in Dafermos’ book.

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Institute of Applied Analysis and Numerical Simulations

Are there slic–solutions?

Lemma Let y P H 1♣r0, Ts, L2♣B1♣0qqq ❳ L2♣r0, Ts, H 1♣B1♣0qqq with essinf yx → 0 satisfy ➺ T ➺

B1♣0qq

ytϕt ✁ τ♣yxqϕx dxdt ✏ 0 ❅ϕ P C 1

0 ♣♣0, Tq ✂ B1♣0qq,

then y is a slic–solution. Thus, slic–solutions generalize standard weak solutions. Lemma A function y given by (1D-ANSATZ) with Y ♣0q ασ ✏ λσ is a slic–solution if and only if Y ♣0q ✏ τ♣λq ✁ τ♣αq σ and lim

uÑ✽

τ♣uq u ✏ 0.

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Institute of Applied Analysis and Numerical Simulations

Energy of slic–solutions

We define the energy at time t of a slic–solution y to be E♣y, tq :✏ lim

nÑ✽

➺

B1♣0q

1 2♣yn

t ♣x, tqq2 W ♣yn x ♣x, tqq dx.

Because of l’Hospitals Theorem it holds lim

uÑ✽

W ♣uq u ✏ lim

uÑ✽ τ♣uq.

Lemma In case limuÑ✽ τ♣uq ✏ ✽ a slic–solution y given by (1D-ANSATZ) satisfies E♣y, tq ✏ ✽ for all t → 0 and E♣y, 0q ➔ ✽.

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Institute of Applied Analysis and Numerical Simulations

Solutions with finite energy

Lemma In case τ✽ ✏ limuÑ✽ τ♣uq ✏ limuÑ✽ τ♣uq ➔ ✽ slic–solutions given by (1D-ANSATZ) satisfy E♣y, tq ✏ 2W ♣λq ❧♦♦♠♦♦♥

initial energy

✁2tσW ♣λq tσY ♣0q2 2tσW ♣αq 2tY ♣0qτ♣αq ❧♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♥

energy dissipation at the shock

2Y ♣0qt♣τ✽ ✁ τ♣αqq. ❧♦♦♦♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♦♦♦♥

energy of the cavity

Sketch of the proof: Decompose the integral over right half of the wave fan into 4 parts

1 n

σt − 1

n

σt + 1

n

For 1

n ➔ ⑤x⑤ ➔ σt ✁ 1 n

⑤yn

t ⑤ ✏ Y ♣0q,

W ♣yn

x q ✏ W ♣αq.

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Institute of Applied Analysis and Numerical Simulations

Solutions with finite energy

Sketch of the proof, continued: For σt ✁ 1

n ➔ ⑤x⑤ ➔ σt 1 n

⑤yn

t ⑤ ↕ Y ♣0q,

⑤W ♣yn

x q⑤ ↕ max uPrα,λs ⑤W ♣uq⑤.

For σt 1

n ➔ ⑤x⑤

⑤yn

t ⑤ ✏ 0,

W ♣yn

x q ✏ W ♣λq.

For ⑤x⑤ ➔ 1

n

⑤yn

t ⑤ ↕ Y ♣0q,

W ♣yn

x ♣xqq ✏ W

✁ α 2tY ♣0qnψ ✁ x n ✠✠ lim

nÑ✽

➺ 1

n

✁ 1

n

1 2Y ♣0q2 W ✁ α 2tY ♣0qnψ ✁ x n ✠✠ dx ✏ lim

nÑ✽

➺ 1

✁1

1 n W ♣α 2tY ♣0qnψ ♣˜ xqq d˜ x ✏ 2tY ♣0qτ✽.

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Institute of Applied Analysis and Numerical Simulations

For (1D-ANSATZ) solutions the energy increases

Lemma In case τ✽ ✏ limuÑ✽

W♣uq u

✏ limuÑ✽ τ♣uq ➔ ✽ all slic–solutions given by (1D-ANSATZ) satisfy d dt E♣y, tq → 0 and ➺

B1♣0q

♣τ♣yn

x qyn t qx dx ✏ 0.

for n sufficiently large. Sketch of the proof: As yn is smooth ➺

B1♣0q

♣τ♣yn

x qyn t qx dx ✏ τ♣yn x ♣1, tqqyn t ♣1, tq ✁ τ♣yn x ♣✁1, tqqyn t ♣✁1, tq

and yn

t ♣1, tq ✏ yn t ♣✁1, tq ✏ 0.

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Institute of Applied Analysis and Numerical Simulations

For (1D-ANSATZ) the energy increases

Recall: E♣y, tq ✏ ♣2 ✁ 2tσqW ♣λq tσY ♣0q2 2tσW ♣αq 2Y ♣0qtτ✽. d dt E♣y, tq ✏ ✁2σW ♣λq σY ♣0q2 2σW ♣αq 2Y ♣0qτ✽ ù ñ d dt E♣y, tq ✏ σY ♣0q2 ❧♦♦♠♦♦♥

→0

✁2 σ♣λ ✁ αq ❧♦♦♦♠♦♦♦♥

✏Y♣0q

W ♣λq ✁ W ♣αq λ ✁ α ❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥

✏W ✶♣¯ uq✏τ♣¯ uq

2Y ♣0qτ✽ ù ñ d dt E♣y, tq → ✁2Y ♣0qτ♣¯ uq 2Y ♣0qτ✽ → 0 as τ ✶ ✏ W ✷ → 0.

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Institute of Applied Analysis and Numerical Simulations

Energies in d ✏ 3 dimensions

We consider ytt ✁ div♣τ♣∇yqq ✏ 0; τ ✏ ❇W ❇F . We assume that W : R3✂3

• Ý

Ñ R is frame indifferent, isotropic and polyconvex, i.e. W ♣Fq ✏ Φ♣λ1, λ2, λ3q where λ1, λ2, λ3 are the eigenvalues of ❄ FF T and Φ : tx P R3 : xi → 0 ❅ i✉ Ý Ñ R is symmetric. We will consider energy densities of the form W ♣Fq ✏ 1 2

• λ2

1 λ2 2 λ2 3

✟ h♣λ1λ2λ3q where h : R Ý Ñ R satisfies h✷ → 0, h✸ ➔ 0.

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Institute of Applied Analysis and Numerical Simulations

slic–solutions

Definition We call y P C♣r0, Tq, Lp♣B1♣0qdq a slic–solution provided for all ψPC ✽

0 ♣R, r0, ✽qq s.t. supp♣ψq ⑨ r✁1, 1s,

➺ 1

✁1

ψ♣xqdx ✏ 1, ψ♣xq ✏ ψ♣✁xq the following holds lim

nÑ✽ryn tt ✁ div♣τ♣∇ynqqs ✏ 0

in D✶ where yn♣x, tq ✏ tϕn ✁

⑤x⑤ t

✠

x ⑤x⑤

and ϕn ✏ ➺ T ➺ ✽ nψ ✂s ✁ ˜ s n ✡ ϕ♣˜ sq d˜ s ✁ ➺ T ➺ 0

✁✽

nψ ✂s ✁ ˜ s n ✡ ϕ♣✁˜ sq d˜ s. Observe that with our regularization ∇yn and yn

t only depend on s.

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Institute of Applied Analysis and Numerical Simulations

Are the existing weak solutions slic–solutions?

Pericak-Spector and Spector ’88 constructed weak solutions of (WAVE) with cavity. In their work the cavity itself does not contribute to the energy. It just makes shocks possible, which in turn dissipate energy. Lemma The solutions constructed by Pericak-Spector and Spector are slic–solutions provided lim

vÑ✽

h✶♣vq

3

❄v ✏ 0. not slic–solutions in case lim inf

vÑ✽

h✶♣vq

3

❄v → 0. There are intermediate cases here, where the answer is not clear.

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Institute of Applied Analysis and Numerical Simulations

Energy of the existing solutions

Lemma With our definition of energy the solutions constructed by Pericak-Spector and Spector satisfy E♣y, tq ✏ ✩ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✫ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✪ ✽ for limvÑ✽

h♣vq v

✏ ✽ 3 2λ2 h♣λ3q ❧♦♦♦♦♦♦♠♦♦♦♦♦♦♥

initial energy

• t3σd4π

3 J ❧♦♦♦♦♠♦♦♦♦♥

energy dissipated at the shock

t34π 3 ϕ♣0q3L ❧♦♦♦♦♦♦♠♦♦♦♦♦♦♥

energy of the cavity

for L :✏ limvÑ✽

h♣vq v

➔ ✽ with J ✏ 1 2♣ϕ✶✁q2 h♣ϕ✶✁λ2q ✁ 1 2λ2 ✁ h♣λ3q 1 2 ✁ ϕ✶✁ h✶♣ϕ✶✁λ2qλ2 λ h✶♣λ3qλ2✠ ♣λ ✁ ϕ✶✁q where ϕ✶✁ ✏ limsÑσ,s➔σ ϕ✶♣sq. In particular: For the known weak solutions the energy is increasing.

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Summary and Prospects

Summary: New solution concept for solutions including δ-shocks. Notion of energy for these solutions. We constructed solutions containing δ-shocks. For these solutions the energy increases in time. Similar situation in 3d. Prospects: Can we use slic–solutions to describe vacuum in gas-dynamics? Are there slic–solutions with cavity, which dissipate energy?

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Institute of Applied Analysis and Numerical Simulations

Summary and Prospects

Summary: New solution concept for solutions including δ-shocks. Notion of energy for these solutions. We constructed solutions containing δ-shocks. For these solutions the energy increases in time. Similar situation in 3d. Prospects: Can we use slic–solutions to describe vacuum in gas-dynamics? Are there slic–solutions with cavity, which dissipate energy? Thank you for your attention!

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