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 on 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 ❇ 0 V � AU ✏ f on R → 0 , V ♣ 0 �q ✏ 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 ❇ 0 V � AU ✏ f on R . (1)
A Model in Poro-Elasticity. Introduction As typical examples we may consider: ̺ 0 ❇ 2 0 u ✁ Div T ✏ f , T ✏ C Grad u , Elastic Waves: 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 H sym of L 2 ♣ Ω q -valued, self-adjoint 3 ✂ 3-matrices. This can be transformed into ❇ 0 ♣ ̺ 0 v q ✁ Div T ✏ f � ✟ C ✁ 1 T ❇ 0 ✁ Grad v ✏ 0 where v : ✏ ❇ 0 u denotes the velocity field of the displacement field u .
A Model in Poro-Elasticity. Introduction Electro-Magnetic Waves: � ✟ D � ❇ ✁ 1 ❇ 0 ✁ curl H � J ✏ 0 , ❇ 0 B � curl E ✏ 0 . 0 σ E 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 ✏ ε E , B ✏ µ H .
A Model in Poro-Elasticity. Introduction Following the lead of these examples, the abstract evolutionary problem ❇ 0 V � 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 � ✟ ❇ ✁ 1 V ✏ M U , (2) 0 where z ÞÑ M ♣ z q is bounded-operator-valued and analytic in an open ball B C ♣ r , r q 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 operator in the space L 2 ♣ R q of equivalence classes of square-integrable complex-valued functions on R . The space ˚ C ✽ ♣ R q of smooth complex-valued functions with compact support is densely embedded in the domain. Indeed, this case is occasionally 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 ✽ ♣ R q ❸ L 2 ♣ R q Ñ L 2 ♣ R q ˚ ϕ ÞÑ ♣ ϕ with ➺ 1 ϕ ♣ x q ✏ ♣ ❵ exp ♣✁ i x t q ϕ ♣ t q dt , x P R . 2 π R
As a spectral representation the Fourier transform makes 1 i ❇ 0 unitarily equivalent to the multiplication by the argument operator m given by ♣ m ϕ q ♣ x q ✏ x ϕ ♣ x q for x P R and ϕ P ˚ C ✽ ♣ R q : ✛ ✘ 1 i ❇ 0 ✏ F ✝ m F . ✚ ✙
We introduce an exponential weight function t ÞÑ exp ♣✁ ν t q , ν P R , and consider the weighted L 2 -space H ν, 0 generated by completion of ˚ C ✽ ♣ R q with respect to the inner product ① ☎ ⑤ ☎ ② ν, 0 ➺ ϕ ♣ t q ✝ ψ ♣ t q exp ♣✁ 2 ν t q dt . ♣ ϕ, ψ q ÞÑ R The associated norm will be denoted by ⑤ ☎ ⑤ ν, 0 . The multiplication operator C ✽ ♣ R q ❸ H 0 , 0 ✏ L 2 ♣ R q C ✽ ♣ R q ❸ H ν, 0 Ñ ˚ ˚ ϕ ÞÑ exp ♣✁ ν m q ϕ with ♣ exp ♣✁ ν m q ϕ q ♣ x q ✏ exp ♣✁ ν x q ϕ ♣ x q , x P R , clearly has a unitary extension, which we shall denote by exp ♣✁ ν m q , where the m serves as a reminder for ’multiplication’.
Its inverse will be denoted by exp ♣ ν m q . Thus, the operator 1 i ❇ ν : ✏ exp ♣ ν m q 1 i ❇ 0 exp ♣✁ ν m q defines a unitarily equivalent operator 1 i ❇ ν , which is now selfadjoint in H ν, 0 . We shall use again the notation ❇ 0 for the normal operator ❇ ν � ν , which is justified since indeed ♣❇ ν � ν q ϕ ✏ ❇ 0 ϕ for ϕ P ˚ C ✽ ♣ R q . Obviously we have that the spectrum of ❇ ν is purely imaginary. In fact, the spectrum σ ♣❇ ν q is also purely continuous spectrum: ✗ ✔ σ ♣❇ ν q ✏ σ c ♣❇ ν q ✏ i R . ✖ ✕
Thus, in particular for ν P R ③ t 0 ✉ we have the bounded invertibility of ❇ 0 ✏ ❇ ν � ν . With L ν : ✏ F exp ♣✁ ν m q ✛ ✘ 1 i ❇ ν ✏ L ✝ ν m L ν . ✚ ✙
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 H ν, 0 ❜ H by interpreting A as 1 H ν, 0 ❜ A with 1 H ν, 0 : H ν, 0 Ñ H ν, 0 as the identity operator in H ν, 0 and the time derivative ❇ 0 as ❇ 0 ❜ 1 H , where 1 H : H Ñ H denotes the identity operator in H . In this sense, our aim is to be able to find U P H ν, 0 ❜ H such that for a given f P H ν, 0 ❜ H we have � � ✟ ✟ ❇ ✁ 1 ❇ 0 M � A U ✏ f . (3) 0 � ✟ ❇ ✁ 1 The operator M will be referred to as the material operator. 0
Here ♣ M ♣ z qq z P B C ♣ r , r q is a uniformly bounded, holomorphic family of 1 linear operators in H with r ↕ 2 ν . It is ✤ ✜ ✂ ✡ � ✟ 1 ❇ ✁ 1 : ✏ L ✝ M ν M L ν . 0 i m � ν ✣ ✢ 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 ✛ ✘ � ✟ z ✁ 1 M ♣ z q ➙ c → 0 . (posdef) Re ✚ ✙ for all z P B C ♣ r , r q .
A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws Solution Theory Solution Theory Theorem 1: Let ♣ M ♣ z qq z P B C ♣ r , r q be a holomorphic family of 1 uniformly bounded linear operators on H , ν ➙ 2 r , satisfying our definiteness condition (posdef) and A skew-selfadjoint in H , then we have for every f P H ν, 0 ❜ H a unique solution U P H ν, 0 ❜ H of the problem � � ✟ ✟ ❇ ✁ 1 ❇ 0 M � A U ✏ f . 0 Moreover, the solution depends continuously on the data in H ν, 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 ✂ 0 ✡ B ✝ A ✏ ((H)) ✁ B 0 with B : D ♣ B q ❸ H 1 Ñ H 0 a closed, densely defined operator between Hilbert spaces H 1 and H 0 . 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 R 3 as a mapping from the Hilbert space L 2 ♣ Ω q of equivalence classes (with respect to a.e.-equality as equivalence relation) of square-integrable functions to the 3-component L 2 -type space L 2 ♣ Ω q ❵ L 2 ♣ Ω q ❵ L 2 ♣ Ω q , 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 ✂ ✡ 0 div S ✏ . ˚ 0 grad
A Model in Poro-Elasticity. Evolutionary Dynamics and Material Laws A Modified Biot System Given the definition E : ✏ Grad u , ✁ ❇ ❜ u � ♣❇ ❜ u q ❏ ✠ where u is the displacement and Grad u : ✏ 1 2 denotes the symmetric part of the Jacobi matrix ❇ ❜ u , another first order dynamic equation ❇ 0 E ✏ Grad v can be constructed, where v ✏ ❇ 0 u describes the velocity of the deformation. Let H sym denote the Hilbert space of L 2 ♣ Ω q -valued, selfadjoint 3 ✂ 3-matrices, with the inner product induced by the Frobenius norm ➺ � ✟ Φ ♣ x q ✝ Ψ ♣ x q ♣ Φ , Ψ q ÞÑ trace dx . Ω
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