ON A MODEL IN PORO-ELASTICITY Rainer Picard Technische Universitt - - PowerPoint PPT Presentation

A Model in Poro-Elasticity.

ON A MODEL IN PORO-ELASTICITY

Rainer Picard

Technische Universität Dresden rainer.picard@tu-dresden.de Abstract. A modification of the material law associated with the well-known Biot system as suggested by M. A. Murad and J. H. Cushman 1996 for some types of clay and first investigated by R. E. Showalter in 2000 is re-considered in the light of a new approach to a comprehensive class of evolutionary problems and extended to anisotropic, inhomogeneous media. The results presented are based on joint work with D. McGhee, Glasgow.

The Second Najman Conference

• n

Spectral Problems for Operators and Matrices May 10-17, 2009, Dubrovnik, Croatia

A Model in Poro-Elasticity.

Outline

Introduction The Time Derivative Evolutionary Dynamics and Material Laws

General Linear Material Laws Solution Theory A Modified Biot System

Summary Literature

A Model in Poro-Elasticity. Introduction

Introduction

We shall investigate a particular model for poro-elastic media in a more general framework. The model under consideration describes the consolidation of soil as water is draining out of it. We begin by noticing, that on closer inspection of initial boundary value problems of mathematical physics, in particular those describing wave propagation phenomena one is inclined to describe their general form as ❇0V AU ✏ f on R→0, V ♣0q ✏ Φ, where A is skew-selfadjoint in a suitable Hilbert space setting. We shall indeed prefer to consider this problem on the whole real time-line and to by-pass the full construction of associated Sobolev lattices we shall assume – without loss of generality – that Φ ✏ 0. This turns our problem into ❇0V AU ✏ f on R. (1)

A Model in Poro-Elasticity. Introduction

As typical examples we may consider: Elastic Waves: ̺0❇2

0u ✁ DivT ✏ f , T ✏ C Grad u,

where u denotes the displacement field, T the stress tensor, ̺0 mass density, C the elasticity tensor, which is assumed to be modeled as bounded, self-adjoint, strictly positive definite mappings in a Hilbert space Hsym of L2 ♣Ωq-valued, self-adjoint 3 ✂ 3-matrices. This can be transformed into ❇0 ♣̺0vq ✁ DivT ✏ f ❇0

• C ✁1T

✟ ✁ Grad v ✏ 0 where v :✏ ❇0u denotes the velocity field of the displacement field u.

A Model in Poro-Elasticity. Introduction

Electro-Magnetic Waves: ❇0

• D ❇✁1

0 σE

✟ ✁ curl H J ✏ 0, ❇0B curl E ✏ 0. Here E denotes the electric and H the magnetic field, whereas D is known as the displacement current density and B as the magnetic induction. Material relation: D ✏ εE, B ✏ µH.

A Model in Poro-Elasticity. Introduction

Following the lead of these examples, the abstract evolutionary problem ❇0V AU ✏ f on R. is now completed by an additional rule frequently referred to as a “material law”, which for simplicity we assume to be time-translation invariant and more precisely of the form V ✏ M

• ❇✁1

✟ U, (2) where z ÞÑ M ♣zq is bounded-operator-valued and analytic in an

• pen ball BC ♣r, rq with some positive radius r centered at r.

A Model in Poro-Elasticity. The Time Derivative

The Time Derivative

It is well-known that 1

i ❇0 can be established as a selfadjoint

• perator in the space L2 ♣Rq of equivalence classes of

square-integrable complex-valued functions on R. The space ˚ C✽ ♣Rq of smooth complex-valued functions with compact support is densely embedded in the domain. Indeed, this case is

• ccasionally used as a simple example for an explicit spectral

representation, which here is provided by the Fourier transform F given as the unitary extension of ˚ C✽ ♣Rq ❸ L2 ♣Rq Ñ L2 ♣Rq ϕ ÞÑ ♣ ϕ with ♣ ϕ ♣xq ✏ 1 ❵ 2π ➺

R

exp ♣✁i x tq ϕ ♣tq dt, x P R.

As a spectral representation the Fourier transform makes 1

i ❇0

unitarily equivalent to the multiplication by the argument operator m given by ♣mϕq ♣xq ✏ xϕ ♣xq for x P R and ϕ P ˚ C✽ ♣Rq:

✛ ✚ ✘ ✙

1 i ❇0 ✏ F✝m F.

We introduce an exponential weight function t ÞÑ exp ♣✁ν tq, ν P R, and consider the weighted L2-space Hν,0 generated by completion of ˚ C✽ ♣Rq with respect to the inner product ① ☎ ⑤ ☎ ②ν,0 ♣ϕ, ψq ÞÑ ➺

R

ϕ ♣tq✝ ψ ♣tq exp ♣✁2νtq dt. The associated norm will be denoted by ⑤ ☎ ⑤ν,0. The multiplication

• perator

˚ C✽ ♣Rq ❸ Hν,0 Ñ ˚ C✽ ♣Rq ❸ H0,0 ✏ L2 ♣Rq ϕ ÞÑ exp ♣✁νmq ϕ with ♣exp ♣✁νmq ϕq ♣xq ✏ exp ♣✁νxq ϕ ♣xq , x P R, clearly has a unitary extension, which we shall denote by exp ♣✁νmq, where the m serves as a reminder for ’multiplication’.

Its inverse will be denoted by exp ♣νmq. Thus, the operator 1 i ❇ν :✏ exp ♣νmq 1 i ❇0 exp ♣✁νmq defines a unitarily equivalent operator 1

i ❇ν, which is now selfadjoint

in Hν,0. We shall use again the notation ❇0 for the normal operator ❇ν ν, which is justified since indeed ♣❇ν νq ϕ ✏ ❇0ϕ for ϕ P ˚ C✽ ♣Rq. Obviously we have that the spectrum of ❇ν is purely imaginary. In fact, the spectrum σ ♣❇νq is also purely continuous spectrum:

✗ ✖ ✔ ✕

σ ♣❇νq ✏ σc ♣❇νq ✏ i R.

Thus, in particular for ν P R③ t0✉ we have the bounded invertibility

• f ❇0 ✏ ❇ν ν.

With Lν :✏ F exp ♣✁νmq

✛ ✚ ✘ ✙

1 i ❇ν ✏ L✝

νm Lν.

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws General Linear Material Laws

Evolutionary Dynamics and Material Laws

General Linear Material Laws

We shall now consider the initially stated evolutionary problem in precise terms. For this we need to extend the operators ❇0, A to the tensor product spaces Hν,0 ❜ H by interpreting A as 1Hν,0 ❜ A with 1Hν,0 : Hν,0 Ñ Hν,0 as the identity operator in Hν,0 and the time derivative ❇0 as ❇0 ❜ 1H, where 1H : H Ñ H denotes the identity operator in H. In this sense, our aim is to be able to find U P Hν,0 ❜ H such that for a given f P Hν,0 ❜ H we have

• ❇0M
• ❇✁1

✟ A ✟ U ✏ f . (3) The operator M

• ❇✁1

✟ will be referred to as the material operator.

Here ♣M ♣zqqzPBC♣r,rq is a uniformly bounded, holomorphic family of linear operators in H with r ↕

1 2ν . It is

✤ ✣ ✜ ✢

M

• ❇✁1

✟ :✏ L✝

ν M

✂ 1 i m ν ✡ Lν. To warrant a solution theory we require an additional constraint on such causal materials: There should be a constant c P R→0 such that

✛ ✚ ✘ ✙

Re

• z✁1M ♣zq

✟ ➙ c → 0. (posdef) for all z P BC ♣r, rq.

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws Solution Theory

Solution Theory Theorem 1:

Let ♣M ♣zqqzPBC♣r,rq be a holomorphic family of uniformly bounded linear operators on H, ν ➙

1 2r , satisfying

• ur definiteness condition (posdef) and A skew-selfadjoint in

H, then we have for every f P Hν,0 ❜ H a unique solution U P Hν,0 ❜ H of the problem

• ❇0M
• ❇✁1

✟ A ✟ U ✏ f . Moreover, the solution depends continuously on the data in Hν,0 ❜ H and is causal.

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System

A modified Biot system

We shall now establish that the Biot system is of the stated form, where the skew-selfadjointness of A stems from the structure of A as a block operator matrix of the form A ✏ ✂ 0 B✝ ✁B ✡ ((H)) with B : D ♣Bq ❸ H1 Ñ H0 a closed, densely defined operator between Hilbert spaces H1 and H0. We shall specify the material law later.

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System

For simplicity and definiteness we focus here on the case of Dirichlet boundary conditions. By suitably establishing ˚ grad as the closure of the classical gradient applied to smooth functions with compact support in a non-empty, open subset Ω ❸ R3 as a mapping from the Hilbert space L2 ♣Ωq of equivalence classes (with respect to a.e.-equality as equivalence relation) of square-integrable functions to the 3-component L2-type space L2 ♣Ωq ❵ L2 ♣Ωq ❵ L2 ♣Ωq, a generalized Dirichlet boundary condition can be formulated as being in the domain of ˚

• grad. With

✁div as the adjoint of ˚ grad we obtain a skew-selfadjoint operator S ✏ ✂ div ˚ grad ✡ .

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System

Given the definition E :✏ Grad u, where u is the displacement and Grad u :✏ 1

2

✁ ❇ ❜ u ♣❇ ❜ uq❏✠ denotes the symmetric part of the Jacobi matrix ❇ ❜ u, another first

• rder dynamic equation

❇0E ✏ Grad v can be constructed, where v ✏ ❇0u describes the velocity of the

• deformation. Let Hsym denote the Hilbert space of L2 ♣Ωq-valued,

selfadjoint 3 ✂ 3-matrices, with the inner product induced by the Frobenius norm ♣Φ, Ψq ÞÑ ➺

Ω

trace

• Φ ♣xq✝ Ψ ♣xq

✟ dx .

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System

By a suitable choice of boundary condition, i.e. domain, for ✂ ✁Div ✁Grad ✡ we can enforce skew-selfadjointness in the Hilbert space H :✏

• L2 ♣Ωq ❵ L2 ♣Ωq ❵ L2 ♣Ωq

✟ ❵ Hsym. For sake of definiteness, let us again implement a Dirichlet boundary condition by considering the closure ˚ Grad of the restriction of Grad to smooth vector fields with compact support in Ω as an operator from L2 ♣Ωq ❵ L2 ♣Ωq ❵ L2 ♣Ωq to Hsym and ✁Div as its adjoint. Then E :✏ ✂ ✁Div ✁ ˚ Grad ✡ is of the form ((H)) and therefore skew-selfadjoint in H (and so also in Hν,0 ❜ H). The classical Biot system has been modified for certain types of clays by Murad and Cushman 1996. The mathematical model has been discussed by R.E. Showalter 2000. At the heart of the ✏ ♣

• ✝

❇ q ✁

✝

♣ q

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System

Then assuming that equation (4) can be solved for E, we have E ✏ ♣C trace✝λ trace ❇0q✁1 T (5) ♣C trace✝λ trace ❇0q✁1 trace✝α p, (6) and so we arrive at again at a system operator ❇0M

• ❇✁1

✟ ✁ ☎ ✝ ✝ ✝ ✝ ✆ Div ✁div

✆

Grad ✁

✆

grad ☞ ✍ ✍ ✍ ✍ ✌ .

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System

Here M

• ❇✁1

✟ is given by ☎ ✝ ✝ ✆ ̺0 0 c0 α✝trace C ✁1

λ trace✝α

α✝trace C ✁1

λ

C ✁1

λ trace✝α

C ✁1

λ

κ✁1❇✁1 ☞ ✍ ✍ ✌ with Cλ :✏ ♣C trace✝λ trace ❇0q . To arrive at (5) it is crucial to discuss the equation ♣C trace✝λ trace ❇0q U ✏ F. Noting that trace trace✝ ✏ 3, we can define the orthogonal projector P :✏ 1 3trace✝trace in C3✂3 which extends canonically to Hsym ⑨ L2 ♣Ωq3✂3.

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System

Defining P✶ :✏ 1 ✁ P, we obtain the decomposition ✂ CP 3λ ❇0 PCP✶ P✶CP CP✶ ✡ ✂ P P✶ ✡ U ✏ ✂ PF P✶F ✡ , where CP and CP✶ denote the restrictions of C to the subspaces P rHsyms and P✶ rHsyms, respectively, which clearly retain the property of strict positive-definiteness. Multiplication of this system of linear equations by ✂ 1 ✁P✶CP ♣CP 3λ ❇0q✁1 1 ✡ , i.e. performing an admissible row

• peration, we obtain the equivalent triangular system

✂ CP 3λ ❇0 PCP✶ CP✶ ✁ P✶CP ♣CP 3λ ❇0q✁1 P CP✶ ✡ ✂ P P✶ ✡ U ✏ ✏ ✂ PF ✁P✶CP ♣CP 3λ ❇0q✁1 PF P✶F ✡ .

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System

Here ♣CP 3λ ❇0q✁1 is well-defined simply as the solution operator

• f the ordinary differential equation

CPU 3λ ❇0U ✏ G in the Hilbert space P rHsyms. Since ♣C trace✝λ trace ❇0q✁1 ✏ ✏ C ✁1

P✶ P✶ ❇✁1

• 1 ✁ C ✁1

P✶ P✶C P

✟ 1

3λ✁1 P 1 3 λ✁1 C ✁1 P✶ P✶CPCP✶✟

• ❇✁2

0 Θ0

• ❇✁1

✟ with Θ0 analytic at 0 we obtain a 0-analytic material law operator

• f the form

M

• ❇✁1

✟ ✏ M0 ❇✁1

0 M1 ❇✁2 0 M2

• ❇✁1

✟ . (7)

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System

Here M0 :✏ ☎ ✝ ✝ ✆ ̺0 0 c0 α✝trace C ✁1

P✶ P✶trace✝α

α✝trace C ✁1

P✶ P✶ 0

C ✁1

P✶ P✶trace✝α

C ✁1

P✶ P✶

☞ ✍ ✍ ✌, M1 :✏ ☎ ✝ ✝ ✆ 0 α✝trace Ξ trace✝α α✝trace Ξ Ξ trace✝α Ξ κ✁1 ☞ ✍ ✍ ✌ with Ξ :✏ 1 3

• 1 ✁ P✶ C ✁1

P✶ P✶C P

✟ P λ✁1P

• 1 ✁ P C P✶ C ✁1

P✶ P✶✟

.

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System

Moreover,

M2 ✁ ❇✁1 ✠ :✏ ☎ ✝ ✝ ✝ ✆ α✝trace Θ0 ✁ ❇✁1 ✠ trace✝α α✝trace Θ0 ✁ ❇✁1 ✠ Θ0 ✁ ❇✁1 ✠ trace✝α Θ0 ✁ ❇✁1 ✠ ☞ ✍ ✍ ✍ ✌.

We observe that M0 is symmetric and also strictly positive definite

• n the range of

☎ ✝ ✝ ✆ 1 0 0 0 0 1 0 0 0 0 P✶ 0 0 0 0 0 ☞ ✍ ✍ ✌. Indeed, transforming M0 with suitable matching row and column

• perations, we obtain

☎ ✝ ✝ ✆ 1 0 0 0 0 1 0 0 0 0 P✶ 0 0 0 0 0 ☞ ✍ ✍ ✌ ☎ ✝ ✝ ✆ ̺0 0 0 c0 0 0 C ✁1

P✶

0 0 ☞ ✍ ✍ ✌ ☎ ✝ ✝ ✆ 1 0 0 0 0 1 0 0 0 0 P✶ 0 0 0 0 0 ☞ ✍ ✍ ✌.

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System

This is clearly positive definite, since ̺0, c0 and CP✶ are selfadjoint, strictly positive definite and bounded on this range. Moreover, the null space of M0 is the range of ☎ ✝ ✝ ✆ 0 0 0 0 0 0 0 0 0 0 P 0 0 0 0 1 ☞ ✍ ✍ ✌ and we have PΞP ✏ 1 3P λ✁1P.

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System

Therefore ☎ ✝ ✝ ✆ 0 0 0 0 0 0 0 0 0 0 P 0 0 0 0 1 ☞ ✍ ✍ ✌M1 ☎ ✝ ✝ ✆ 0 0 0 0 0 0 0 0 0 0 P 0 0 0 0 1 ☞ ✍ ✍ ✌✏ ✏ ☎ ✝ ✝ ✆ 0 0 0 0 0 0 0 0 0 0 P 0 0 0 0 1 ☞ ✍ ✍ ✌ ☎ ✝ ✝ ✆ 0 0 0 0 0 0 1

3 λ✁1

0 0 κ✁1 ☞ ✍ ✍ ✌ ☎ ✝ ✝ ✆ 0 0 0 0 0 0 0 0 0 0 P 0 0 0 0 1 ☞ ✍ ✍ ✌, which shows the positive definiteness of the real part of M1 on null space of M0, i.e. the range of ☎ ✝ ✝ ✆ 0 0 0 0 0 0 0 0 0 0 P 0 0 0 0 1 ☞ ✍ ✍ ✌.

A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System

Theorem 2:

Let M

• ❇✁1

✟ be given by (7) with a coupling term α continuous, linear and ̺0, C, λ, c0, κ selfadjoint, continuous and strictly positive definite in their appropriate L2 ♣Ωq-type spaces. Then for every F P Hν,0 there exists a unique solution U P Hν,0 of the problem

• ❇0M
• ❇✁1

✟ ✁ A ✟ U ✏ F, where

A :✏ ☎ ✝ ✝ ✝ ✆ Div ✁div

✆

Grad ✁

✆

grad ☞ ✍ ✍ ✍ ✌.

Moreover, the solution depends continuously on the data in Hν,0, i.e. 0 is in the resolvent set of the operator ❇0M

• ❇✁1

✟ ✁ A.

A Model in Poro-Elasticity. Summary

Summary

We have presented a solution theory for an evolutionary system describing propagation in poro-elastic media. The problem has been shown to be of the form

• ❇0M
• ❇✁1

✟ A ✟ U ✏ f with A skew-selfadjoint in a Hilbert space H and where ♣M ♣zqqzPBC♣r,rq is a uniformly bounded, holomorphic family of linear operators in H, r P R→0 small, satisfying a one-sided constraint of the form ➟

cPR→0

➞

zPBC♣r,rq

Re

• z✁1M ♣zq

✟ ➙ c. Thus it is covered by a previously developed solution scheme.

A Model in Poro-Elasticity. Literature

Literature

• M. A. Biot. General theory of three-dimensional consolidation. J.
• appl. Physics, Lancaster Pa., 12:155–164, 1941.
• M. A. Murad and J. H. Cushman. Multiscale flow and deformation

in hydrophilic swelling porous media. Int. J. Eng. Sci., 34(3):313–338, 1996.

• R. Picard. A Structural Observation for Linear Material Laws in

Classical Mathematical Physics. Technical Report Math-AN-02-2008, TU Dresden, to appear in Math. Meth. Appl.

• Sci. 2009.
• R. Picard. A Note on Poro-Elastic Media. Technical Report

Math-AN-03-2008, TU Dresden, to appear in Math. Meth. Appl. Sci. R.E. Showalter. Diffusion in poro-elastic media. J. Math. Anal. Appl., 251(1):310–340, 2000.