Creep, Recovery, and Waves in a Nonlinear Fiber-Reinforced Viscoelastic Solid

Abstract

We present a constitutive model capturing some of the experimentally observed features of soft biological tissues: nonlinear viscoelasticity, nonlinear elastic anisotropy, and nonlinear viscous anisotropy. For this model we derive the equation governing rectilinear shear motion in the plane of the fiber reinforcement; it is a nonlinear partial differential equation for the shear strain. Specializing the equation to the quasistatic processes of creep and recovery, we find that usual (exponential-like) time growth and decay exist in general, but that for certain ranges of values for the material parameters and for the angle between the shearing direction and the fiber direction, some anomalous behaviors emerge. These include persistence of a non-zero strain in the recovery experiment, strain growth in recovery, strain decay in creep, disappearance of the solution after a finite time, and similar odd comportments. For the full dynamical equation of motion, we find kink (traveling wave) solutions which cannot reach their assigned asymptotic limit.

Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

“…An anisotropic viscoelastic model involving neo-Hookean isotropic and the standard reinforcing model anisotropic energy density components, (2.9) (b¼0) and (2.14), with an additional anisotropic viscous constitutive term, was considered in [16].…”

Small amplitude waves and stability for a pre-stressed viscoelastic solid

“…For an incompressible material we now define eight independent invariants of the two tensors B and D by I 1 = tr B, I 2 = tr (B −1 ), I 5 = tr (DB), I 6 = tr (DB 2 ), I 7 = tr (D 2 ), I 8 = tr (D 2 B), I 9 = tr (D 2 B 2 ), I 10 = tr (D 3 ), (10) noting that the invariants I 3 = det B = 1 and I 4 = tr D = 0 have been omitted from the list by virtue of ( 6) and (8).…”

“…Antman and Seidman [1] provided a detailed mathematical study of large shearing motions of nonlinearly viscoelastic slabs (see also Rajagopal and Saccomandi [27] for some exact solutions in a similar framework). Recently, Hayes and Saccomandi [21,22,23] and Destrade and Saccomandi [8,9,10] obtained some results for finite amplitude motions and waves in some classes of nonlinear viscoelastic materials. Earlier, Hayes and Rivlin [16,17,18,19,20] had established some general results for the theory of small motions superposed on a large deformation in nonlinear viscoelastic solids (see also a recent note by Saccomandi [28] concerning such waves in a special class of materials).…”

On the Kelvin–Voigt model in anisotropic viscoelasticity

“…These simplifications of the general theory are not dictated by a rational methodology of investigation but only by empirical considerations or computational simplifications. A compromise between the complexity of the general theory in Merodio [7] and the specificity of the various particular models as in Anssari-Benam et al [8] or Destrade and Saccomandi [10] may be indicated by the isotropic model (1), i.e., by the mix between the classic theory of hyperelasticity and the Navier-Stokes theory.…”