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Hamiltonian and Lagrangian theory of viscoelasticity

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Abstract

The viscoelastic relaxation modulus is a positive-definite function of time. This property alone allows the definition of a conserved energy which is a positive-definite quadratic functional of the stress and strain fields. Using the conserved energy concept a Hamiltonian and a Lagrangian functional are constructed for dynamic viscoelasticity. The Hamiltonian represents an elastic medium interacting with a continuum of oscillators. By allowing for multiphase displacement and introducing memory effects in the kinetic terms of the equations of motion a Hamiltonian is constructed for the visco-poroelasticity.

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Abbreviations

]a, b[:

the open interval with ends at a, b;

\(\dot{u}\) :

 = ∂u/∂t;

\(f^\prime\) :

distributional derivative of f;

δ a :

Dirac delta measure with support at a;

\({\fancyscript{L}}^2({\mathbb{R}}_+;\Sigma;m), \Upsigma\) :

: page 9;

\({\langle}{\bf a}, {\bf b} {\rangle}=a_{kl} {\,}b_{kl}\) :

scalar product on S;

|a|:

\(:={\langle}{\bf a},{\bf a} \rangle^{1/2}\) ;

\({\bf A} \geq 0\,({\bf A} \in \Sigma )\) :

is equivalent to \(\langle{\bf v}, {\bf A}\, {\bf v} {\rangle}\geq 0\) for all \({\bf v} \in S\) ;

λ :

the Lebesgue measure on Ω \(\subset{\mathbb{R}}^d\) ;

S :

space of real symmetric tensors of rank 2;

\(S_{\mathbb{C}} = S \oplus {\rm i} \, S\) :

space of complex symmetric rank 2 tensors;

Σ, Σ \(_{\mathbb{C}}\) :

space of symmetric operators on S, \(S_{\mathbb{C}}\) ;

\({\fancyscript{I}}\) :

: page 9;

\(\langle {\bf a}, {\bf b} \rangle_{{\mathbb{C}}} = \overline{a_{kl}} \, b_{kl}\) :

scalar product on \(S_{\mathbb{C}}\) ;

da :

surface area in \({\mathbb{R}}^d\) .

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Authors and Affiliations

  1. Department of Earth Sciences, University of Bergen, Allégaten 41, 5007, Bergen, Norway

    A. Hanyga

  2. Institute of Fundamental Technological Research, Polish Academy of Sciences, ul. Świętokrzyska 21, 00-049, Warszawa, Poland

    M. Seredyńska

Authors
  1. A. Hanyga
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  2. M. Seredyńska
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Correspondence to A. Hanyga.

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Communicated by V. Berdichevsky.

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Hanyga, A., Seredyńska, M. Hamiltonian and Lagrangian theory of viscoelasticity. Continuum Mech. Thermodyn. 19, 475–492 (2008). https://doi.org/10.1007/s00161-007-0065-6

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