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Three-dimensional viscoelasticity in finite strain: Formulation of a rate-type constitutive law consistent with dissipation. (English) Zbl 0899.73148

Drew, Donald A. (ed.) et al., Particulate flows. Processing and rheology. Workshop at the IMA on the University of Minnesota, Twin Cities, January 8–12, 1996. New York, NY: Springer. IMA Vol. Math. Appl. 98, 67-87 (1998).
We introduce internal variables and present a derivation of a three-dimensional nonlinear rate-type viscoelastic constitutive law. Evolution of the internal variables is assumed to involve the first-order rates. Properties of the one-dimensional standard linear model as well as the existence of nonlinear models of soft tissues are used to motivate the constitutive assumptions and additional requirements. These requirements include symmetry of the stress, isotropy, reduction to hyperelasticity (via material parameters), and the existence of a hyperelastic equilibrium state. A class of objective rate-type constitutive laws satisfying dissipation and the additional requirements is derived. As an illustration, a compressible finite linear model is formulated. In infinitesimal strain, this model provides a three-dimensional analogy to the one-dimensional standard linear model with a set of constraints on the material parameters. The finite linear model is analyzed under simple time-dependent compression, extension and shear, and is shown to be consistent with the expected behavior.
For the entire collection see [Zbl 0883.00031].

MSC:

74D10 Nonlinear constitutive equations for materials with memory
74A20 Theory of constitutive functions in solid mechanics
74L15 Biomechanical solid mechanics
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