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Mathematics and Mechanics of Solids
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Reviews

Nonlinear Viscoelastic Solids—A Review

A. Wineman

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA

Elastomers and soft biological tissues can undergo large deformations and exhibit time dependent behavior that is characteristic of nonlinear viscoelastic solids. This article is intended to provide an overview of the subject of nonlinear viscoelastic solids for researchers who are interested in studying the mechanics of these materials. The article begins with a review of topics from linear viscoelasticity that are pertinent to the understanding of nonlinear viscoelastic behavior. It then discusses the topics that enter into the formulation of constitutive equations for isotropic, transversely isotropic and orthotropic nonlinear viscoelastic solids. A number of specific forms of constitutive equations have been proposed in the literature and these are discussed. Attention is restricted to constitutive equations that are phenomenological rather than molecular in origin. The emphasis is then on nonlinear single integral finite linear viscoelastic and Pipkin—Rogers constitutive equations, the latter containing the quasi-linear viscoelastic model used in biomechanics of soft tissue. Expressions for the Pipkin—Rogers model are provided for isotropy, transverse isotropy and orthotropy.

The constitutive equations are then applied to the description of homogeneous triaxial stretch and simple shear histories. The special case of uniaxial stretch histories is analyzed in detail. There is a discussion of the deviation from linear behavior as nonlinear effects become important. Non-homogeneous deformations are considered next. The combined tension and torsion of a solid cylinder on an incompressible, isotropic nonlinear viscoelastic solid is discussed in detail because of its importance in experiments involving viscoelastic materials. A large number of solutions to boundary value problems have appeared in the literature and many of these are summarized. The article concludes with comments about interesting topics for further research.

Key Words: Nonlinear single integral constitutive equations • Volterra integral equations • membranes • uniaxial and tension-torsion histories

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Mathematics and Mechanics of Solids, Vol. 14, No. 3, 300-366 (2009)
DOI: 10.1177/1081286509103660
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This Article
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