Creep and relaxation contribution tensors for spheroidal pores in hereditary solids: fraction-exponential operators approach

Abstract

In this paper, we introduce creep and relaxation contribution tensors that describe the effect of individual pores on the overall viscoelastic properties of a porous material. Explicit analytical expressions for these tensors are obtained using the elastic-viscoelastic correspondence principle and the Laplace transform. This becomes possible when viscoelastic properties are expressed in terms of fraction-exponential operators of Rabotnov (J Appl Math Mech (PMM) 12: [53][54][55][56][57][58][59][60][61][62] 1948). Creep and relaxation contribution tensors can serve as the basic building blocks for the calculation of overall viscoelastic properties of porous hereditary materials.

Effective Viscoelastic Properties of One-Dimensional Composites

“…with 1<0, as kernel of the viscoelastic operators. These enable an analytical obtaining of the results under the use of the Laplace transform and at the same time, present an excellent concordance with the experimental data (see [17], [18]). Taking the creep kernel as a fraction-exponential function or Rabotnov's kernel,…”

“…This facilitated the obtaining of analytical expressions of the effective coefficient. However, the exponential function kernels or their linear combinations, do not always describe correctly the viscoelastic behavior for some determined materials (see [17], [18]). The situation is solved with the use of fraction-exponential functions (see…”

“… is relaxation time,   is the shear modulus at t,  0 is instantaneous shear modulus and  max is maximal shear strain. Then the viscoelastic behavior of the materials is described for the following four parameters 0, , (or ) and  (see [17], [18]).…”

Through-thickness stress relaxation in bacterial cellulose hydrogel

“…Alternatively, Scott Blair and Coppen [21,22] and Rabotnov [23] suggested to use fraction-exponential operators to describe viscoelastic behaviour and demonstrated their advantages of sufficient accuracy in description of experimental data of real materials together with convenience of solving analytical Laplace transformations. Thanks to that, the use of fraction-exponential operators in recent years attracted a growing interest in the field of solid mechanics, especially mechanics of heterogeneous materials [24,25,26,27].…”

Fraction-exponential representation of the viscoelastic properties of dentin

“…Detailed description of the approach is given, for example, in the books of Rabotnov (1977) and Gorenflo et al (2014). Methodology of application of fraction-exponential operators to heterogeneous materials has been recently developed by Sevostianov and Levin (2016) who introduced creep and relaxation contribution tensors that allow description of the effect of inhomogeneities on the overall viscoelastic properties in a unified way and thus, extend any of known micromechanical schemes from elastic materials to viscoelastic ones. The approach has been successfully verified by Sevostianov et al (2016) to calculate effective viscoelastic properties of fiber reinforced composites.…”