Stability in Viscoelasticity


  • A.D. Drozdov, Ben-Gurion University of the Negev, Institute for Industrial Mathematics, Beer-Sheva, Israel
  • V.B. Kolmanovskii, Moscow Institute of Electronic Engineering, Faculty of Applied Mathematics, Moscow, Russia

The subject of stability problems for viscoelastic solids and elements of structures, with which this book is concerned, has been the focus of attention in the past three decades. This has been due to the wide inculcation of viscoelastic materials, especially polymers and plastics, in industry. Up-to-date studies in viscoelasticity are published partially in purely mathematical journals, partially in merely applied ones, and as a consequence, they remain unknown to many interested specialists. Stability in Viscoelasticity fills the gap between engineers and mathematicians and converges theoretical and applied directions of investigations.

All chapters contain extensive bibliographies of both purely mathematical and engineering works on stability problems. The bibliography includes a number of works in Russian which are practically inaccessible to the Western reader.

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Book information

  • Published: December 1994
  • Imprint: NORTH-HOLLAND
  • ISBN: 978-0-444-81951-2

Table of Contents

Preface. 1. Constitutive Models of Viscoelastic Materials. 1. Kinematics of motion. 2. Dynamics of continuum. 3. Small perturbations of the actual configuration. 4. Constitutive theory. 5. Constitutive equations for viscoelastic materials with infinitesimal strains. 6. Creep and relaxation kernels. 7. Thermodynamic potentials and variational principles in linear viscoelasticity. 8. Hyperelasticity theory. 9. Constitutive equations for viscoelastic materials with finite strains. 2. Linear Stability Problems. 1. Stability of viscoelastic bars. 2. Quasi-static stability of viscoelastic bars under non-conservative loading. 3. Stability of viscoelastic bars under the action of follower forces. 4. Stability of an integro-differential equation with non-commuting operator coefficients. 5. Stability of bars made of elastic materials with voids. 6. Concluding remarks. 3. Stability of Viscoelastic Structural Members under Periodic and Random Loads. 1. Stability of a viscoelastic shell under time-varying loads. 2. Stability of a linear integro-differential equation with periodic coefficients. 3. Stability of a viscoelastic shell driven by random loads. 4. Stability of a viscoelastic bar driven by random compressive loads. 5. Stability of a class of stochastic integro-differential equations. 6. Concluding remarks. 4. Nonlinear Problems of Stability for Viscoelastic Structural Members. 1. Stability of a non-homogeneous, ageing, viscoelastic bar under conditions of nonlinear creep. 2. Stability of a nonlinear operator integro-differential equation. 3. Stability of a system of nonlinear integro-differential equations with operator coefficients. 4. Stability of a class of nonlinear integro-differential equations. 5. Concluding remarks. 5. Applied Problems of Stability. 1. Stability of growing viscoelastic bars in a finite time interval. 2. Stability of viscoelastic bars with finite shear rigidity. 3. Stability of non-homogeneous, ageing, viscoelastic plates. 4. Stability of an elastic vertical casing in a viscoelastic medium subjected to ageing. 5. Stability of an elastic stiffening for a horizontal mine working in an ageing viscoelastic medium. 6. Concluding remarks. 6. Stability of Elastic and Viscoelastic Three-Dimensional Bodies. 1. Quasi-static stability of an ageing, linearly viscoelastic body with infinitesimal strains. 2. Dynamic stability of a viscoelastic body with infinitesimal strains. 3. Stability of a viscoelastic body with finite strains. 4. Stability of hyper-viscoelastic solids. 5. Stability of thin-walled structural members. 6. Concluding remarks. 7. Stability of Functional-Differential Equations. 1. Stability of linear stationary systems. 2. Stability of linear equations with time-varying coefficients. 3. Stability of nonlinear equations. 4. Stability of a chemostat. 5. Stability of a predator-prey system. Appendix 1. Theory of Tensors. 1. Definition of tensor. 2. Tensor algebra. 3. Tensor analysis. 4. Tensor functions. Appendix 2. Elements of Functional Analysis. 1. Banach and Hilbert spaces. 2. Linear functionals and operators. 3. Sobolev spaces. Index.