Variational principles on static-dynamic analysis of viscoelastic thin plates with applications

Chang-jun, Cheng; Zhang, Neng-Hui

doi:10.1016/s0020-7683(97)00262-x

Cited by 15 publications

(11 citation statements)

References 7 publications

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“…As developed further in many of the theoretical mechanics papers on the subject of viscoelasticity (e.g. the Boltzmann relaxation law discussed by Cheng and Zhang, 1998), the state variables describing a long-term viscous behavior depend on the history of the short-term forcing. By embracing this important form of causality linking elastic and long-term viscous behaviors, glaciological response to cyclic forcing functions (e.g.…”

Section: Discussionmentioning

confidence: 99%

A model of viscoelastic ice-shelf flexure

2015

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ABSTRACT. We develop a formal thin-plate treatment of the viscoelastic flexure of floating ice shelves as an initial step in treating various problems relevant to ice-shelf response to sudden changes of surface loads and applied bending moments (e.g. draining supraglacial lakes, iceberg calving, surface and basal crevassing). Our analysis is based on the assumption that total deformation is the sum of elastic and viscous (or power-law creep) deformations (i.e. akin to a Maxwell model of viscoelasticity, having a spring and dashpot in series). The treatment follows the assumptions of well-known thin-plate approximation, but is presented in a manner familiar to glaciologists and with Glen's flow law. We present an analysis of the viscoelastic evolution of an ice shelf subject to a filling and draining supraglacial lake. This demonstration is motivated by the proposition that flexure in response to the filling/drainage of meltwater features on the Larsen B ice shelf, Antarctica, contributed to the fragmentation process that accompanied its collapse in 2002.

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Section: Discussionmentioning

confidence: 99%

A model of viscoelastic ice-shelf flexure

2015

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“…It is pointed out that the original form of the boundary condition (4) is also an integro-differential type, and here it has been treated with the help of Titchmarsh theorem [4].…”

Section: Mathematical Formulation Of the Problemmentioning

confidence: 99%

“…Cederbaum et al [1][2][3] used the phase diagram, Poincare section, power spectrum and the largest Lyapunov exponent to numerically study the chaos motion of viscoelastic plates. Cheng et al [4][5][6][7] revealed the nonlinear dynamic properties of viscoelastic plates by using the Galerkin technique.…”

Section: Introductionmentioning

confidence: 99%

Differential quadrature method for analyzing nonlinear dynamic characteristics of viscoelastic plates with shear effects

Chang-jun

2009

Nonlinear Dyn

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The integro-partial differential equations governing the dynamic behavior of viscoelastic plates taking account of higher-order shear effects and finite deformations are presented. From the matrix formulas of differential quadrature, the special matrix product and the domain decoupled technique presented in this work, the nonlinear governing equations are converted into an explicit matrix form in the spatial domain. The dynamic behaviors of viscoelastic plates are numerically analyzed by introducing new variables in the time domain. The methods in nonlinear dynamics are synthetically applied to reveal plenty and complex dynamical phenomena of viscoelastic plates. The numerical convergence and comparison studies are carried out to validate the present solutions. At the same time, the influences of load and material parameters on dynamic behaviors are investigated. One can see that the system will enter into the chaotic state with a paroxysm form or quasi-periodic bifurcation with changing of parameters.

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“…Now, paralleling to the derivation of the plate divergence equations above, one evaluates the variational constitutive equations of Eq. (27) in the form 6Jc = 5Jc~ + 5Jot,…”

Section: Macroscopic Constitutive Relationsmentioning

confidence: 99%

“…However, the analogy is of limited applications since it holds either in the theory of uncoupled thermoelasticity or in the special case of a steady temperature distribution which is a function of the space coordinates but not of time. By mostly use of the viscoelastic-elastic analogy, there are a number of studies dedicated to some types of vibrations of plates made of linear viscoelastic material (e.g., [23]- [27]). Recently, Bock [28] investigated the initial-boundary value problem for an orthotropic viscoelastic plate.…”

Section: Introductionmentioning

confidence: 99%

Thermo-viscoelastic analysis of high-frequency motions of thin plates

Altay

Dökmeci

2000

Acta Mechanica

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Summary.Within the frame of the 3D-theory of thermo-viscoelastodynamics, a hierarchic system of 2D-equations is consistently derived in invariant differential and variational forms for the analysis of highfrequency motions of thin plates. First, a differential type of variational principles is presented in terms of the Laplace transformed field variables for the fundamental equations of linear, non-isothermal, nonpolar and non-local, 3D-theory. Next, a generalized version of Mindlin's hypothesis for elastic plates is introduced for the displacement and temperature fields. Then, the hierarchic system of plate equations is systematically established by means of the differential variational principle. The hierarchic system of 2D-equations governs the extensional, thickness-shear, fiexural and torsional as well as coupled motions of thermo-viscoelastic thin plates of uniform thickness at both low-and high-frequencies. Lastly, certain cases involving special geometry, motions and material are indicated. Also, the uniqueness is investigated in solutions of the hierarchic system of plate equations. fixed, right-handed system of curvilinear orthogonal coordinates time, time interval a finite and bounded regular region, its boundary surface and closure at time t thickness of plate, thickness interval midsurface of plate, Jordan's curve which bounds A components of the unit vectors normal to 0s2 (or A) and C components of the stress tensor components of the traction vector mass density components of the body force vector components of the displacement vector components of the linear strain tensor components of the alternating tensor components of the heat flux vector normal component of the heat flux vector across 0Y2 reference temperature, temperature increment from O0 entropy density components of the thermal field vector relaxation functions of the plate material components of the thermal conductivity tensor complementary subsurfaces of 0Y2 one-sided Laplace transform of X prescribed value of X and X of order (n)

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Variational principles on static-dynamic analysis of viscoelastic thin plates with applications

Cited by 15 publications

References 7 publications

A model of viscoelastic ice-shelf flexure

A model of viscoelastic ice-shelf flexure

Differential quadrature method for analyzing nonlinear dynamic characteristics of viscoelastic plates with shear effects

Thermo-viscoelastic analysis of high-frequency motions of thin plates

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