Discrete Spectral Modelling of Continuous Structures With Fractional Derivative Viscoelastic Behaviour

2007 ◽  
Author(s):  
Giuseppe Catania ◽  
Silvio Sorrentino
Keyword(s):  
Fractional Derivative ◽  
State Space ◽  
Linear Models ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Computational Effort ◽  
Viscoelastic Behaviour ◽  
Rheological Models ◽  
Standard Linear Solid ◽  
Standard Linear

Fractional derivative rheological models were recognised to be very effective in describing the viscoelastic behaviour of materials, especially of polymers, and when applied to dynamic problems the resulting equations of motion, after a fractional state-space expansion, can still be studied in terms of modal analysis. But the growth in matrix dimensions brought about by this expansion is in general so fast as to make the calculations too cumbersome. In this paper a discretization method for continuous structures is presented, based on the Rayleigh-Ritz method, aimed at reducing the computational effort. The solution of the equation of motion is approximated by a linear combination of shape-functions selected among the analytical eigenfunctions of standard known structures. The resulting condensed eigen-problem is then expanded in a low dimension fractional state-space. The Fractional Standard Linear Solid is the adopted rheological model, but the same methodology could be applied to problems involving different fractional derivative linear models. Examples regarding two different continuous structures are proposed and discussed in detail.

Fractional Derivative Viscoelastic Model for the Analysis of Two Colliding Spherical Shells

2012 ◽  
Author(s):  
Yury A. Rossikhin ◽  
Marina V. Shitikova
Keyword(s):  
Fractional Derivative ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Wave Fronts ◽  
Spherical Shells ◽  
Reflected Waves ◽  
Standard Linear Solid ◽  
Standard Linear ◽  
The Moment ◽  
Contact Domain

The collision of two isotropic spherical shells is investigated for the case when the viscoelastic features of the shells represent themselves only in the place of contact and are governed by the standard linear solid model with fractional derivatives. Thus, the problem concerns the shock interaction of two shells, wherein the generalized fractional-derivative standard linear law instead of the Hertz contact law is employed as a low of interaction. The pans of the shells beyond the contact domain are assumed to be elastic, and their behavior is described by the equations of motion which take rotary inertia and shear deformations into account. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. Due to the short duration of contact interaction, the reflected waves are not taken into account. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. To determine the desired values behind the wave fronts, one-term ray expansions are used, as well as the equations of motion of the contact domains for the both spherical shells.

Systematic simulation of multicomponent receiver coupling to the seafloor using rheological models

Geophysics ◽  
2014 ◽  
Vol 79 (6) ◽  
pp. P9-P19 ◽  
Author(s):  
Marcus Landschulze ◽  
Rolf Mjelde ◽  
Karin Landschulze
Keyword(s):  
Maxwell Model ◽  
General Description ◽  
Viscoelastic System ◽  
Rheological Models ◽  
S Waves ◽  
Linear Elastic ◽  
Seismic Receiver ◽  
Element Simulation ◽  
Standard Linear Solid ◽  
Standard Linear

This paper reports a comparison of three different rheological models used to characterize receiver coupling to the seafloor. We used a finite-element simulation tool to simulate the mechanical receiver coupling to the seafloor as a viscoelastic system with a combination of linear elastic springs and linear viscous dashpots, known as rheological models. Three models cover most of all mechanic coupling systems, the most commonly applied Kelvin-Voigt model (KVM), the Maxwell model (MM), and the standard linear solid (SLS) model. The models differ in behavior for different coupling aspects such as oscillation, creeping, stress relaxation, and their combinations. We tested these models’ ability and relevance for use in modeling seismic receiver coupling to the seafloor. For that purpose, we used an optimized mathematical approach to simulate coupling behavior under various coupling conditions. We found how receiver coupling will affect P- and S-waves for all three models and provided some insight into which model is most suitable to describe coupling under different circumstances. We found that the SLS model represents a general description of most of the coupling effects to the seafloor and should be used when the coupling acts as a viscoelastic system. The KVM and MM are applicable in extreme cases, such as for elastic waves in consolidated sediments (KVM) and dominant creeping effects, as in very soft biosediment (MM).

Generalization of Classical Modal Analysis

1996 ◽  
Author(s):  
F. Ma ◽  
W. C. Lee
Keyword(s):  
Modal Analysis ◽  
State Space ◽  
Equations Of Motion ◽  
Substantial Reduction ◽  
Computational Effort ◽  
Equivalence Transformations ◽  
State Space Approach ◽  
First Order ◽  
Physical Insight ◽  
Space Approach

Abstract The coefficients of a linear nonconservative structure are arbitrary matrices lacking the usual properties of symmetry and definiteness. Classical modal analysis is extended in this paper so as to apply to structures with nonsymmetric coefficients. The extension utilizes equivalence transformations and does not require conversion of the equations of motion to first-order forms. Compared with the state-space approach, the generalized modal analysis can offer substantial reduction in computational effort and ample physical insight.

scholarly journals Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

scholarly journals Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect

2013 ◽  
Vol 399 (2) ◽  
pp. 472-479 ◽  
Author(s):  
Margareth S. Alves ◽  
Celene Buriol ◽  
Marcio V. Ferreira ◽  
Jaime E. Muñoz Rivera ◽  
Mauricio Sepúlveda ◽  
...  
Keyword(s):  
Asymptotic Behaviour ◽  
Thermal Effect ◽  
Solid Model ◽  
Standard Linear Solid Model ◽  
Standard Linear Solid ◽  
Standard Linear

Analysis of Linear Nonconservative Vibrations

1995 ◽  
Vol 62 (3) ◽  
pp. 685-691 ◽  
Author(s):  
F. Ma ◽  
T. K. Caughey
Keyword(s):  
Modal Analysis ◽  
Equations Of Motion ◽  
Substantial Reduction ◽  
Computational Effort ◽  
Equivalence Transformations ◽  
State Space Approach ◽  
Nonconservative System ◽  
First Order ◽  
Physical Insight ◽  
Space Approach

The coefficients of a linear nonconservative system are arbitrary matrices lacking the usual properties of symmetry and definiteness. Classical modal analysis is extended in this paper so as to apply to systems with nonsymmetric coefficients. The extension utilizes equivalence transformations and does not require conversion of the equations of motion to first-order forms. Compared with the state-space approach, the generalized modal analysis can offer substantial reduction in computational effort and ample physical insight.

Sign in / Sign up

Export Citation Format

Share Document