Interfacial Micromechanics Assessment of Classical Rheological Models. I: Single Interface Size and Viscosity

Shahidi, M.; Pichler, B.; Hellmich, Ch.

doi:10.1061/(ASCE)EM.1943-7889.0001012

Open access

Technical Papers

Oct 13, 2015

Interfacial Micromechanics Assessment of Classical Rheological Models. I: Single Interface Size and Viscosity

Authors: M. Shahidi [email protected], B. Pichler, Aff.M.ASCE [email protected], and Ch. Hellmich, M.ASCE [email protected]Author Affiliations

Publication: Journal of Engineering Mechanics

Volume 142, Issue 3

https://doi.org/10.1061/(ASCE)EM.1943-7889.0001012

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Abstract

Creep functions are often represented by “rheological models” consisting of springs and dashpots, while the actual microscopic origins of creep, such as micro-sliding along interfaces, has only recently been explicitly considered in a continuum mechanics framework. The question arises whether formal analogies between the former and the latter can be derived: This question is answered here for the rheological models of the Kelvin-Voigt and Maxwell type. Thereby, it appears a full analogy between shear stresses and strains acting on the rheological models, and those acting on a micromechanical representative volume element consisting of an elastic solid matrix with embedded viscous interfaces, whereby the respective viscosity arises from layered polar fluids absorbed at these interfaces. The corresponding Kelvin-Voigt parameters are much simpler and more intuitively related to the micromechanical quantities, when compared to the Maxwell parameters. More specifically, rheological spring parameters are always related to the shear stiffness of the elastic solid matrix, while they may additionally depend on the Poisson’s ratio of the elastic solid matrix, and on the interface density. On the other hand, dashpot viscosities are always related to interface viscosities, interface radii, and interface densities; and they may even depend on the Poisson’s ratio of the elastic solid matrix.

Introduction

In 1874, Boltzmann introduced the concept of creep functions as well as the associated superposition principle, and he confirmed these groundbreaking ideas by an initial experimental campaign (Boltzmann 1874). Ever since, these ideas have remained the fundament of the theory of viscoelasticity, which has been developed up to high mathematical maturity (Gurtin and Sternberg 1962; Salençon 1983; Leitman and Fisher 1973). In order to keep things simple, exponential creep functions are often considered, and they are standardly related to the simple rheolgical models composed of linear springs and dashpots. Their simplest versions exhibit a parallel arrangement of one spring and one dashpot (the so-called Kelvin-Voigt model) (Thomson 1865; Voigt 1890), or a serial arrangement of one spring and one dashpot (the so-called Maxwell model) (Maxwell 1867). Slightly more complex arrangements of spring and dashpots are normally called standard linear solid models. They can be either represented as a serial arrangement of a Kelvin-Voigt unit and an additional spring [Kelvin-Voigt representation or Zehner model (Zener 1948)], or as a parallel arrangement of a Maxwell unit and an additional spring (Maxwell representation). However, such simple models do not contain any direct information on the microstructural origin of creep, such as the flow of water along thin layers within the texture of hydrated bio- or geomaterials (Morrow et al. 2000; Stipp et al. 2006; Tullis and Yund 1991; Arnold and Venditti 2001; Bažant et al. 1997; Alizadeh et al. 2010; Kalinichev et al. 2007; Vlahinić et al. 2012; Youssef et al. 2011; Németh et al. 1998; Cordobés et al. 1997; Eberhardsteiner et al. 2014; Sasaki et al. 1993). These layers typically exhibit areal dimensions in the micrometer range, and nanoscopic thicknesses. Aiming exactly at the aforementioned direct consideration of viscous “micro-”interfaces when mathematically describing macroscopic creep behaviors, we recently developed a micromechanical formulation for matrix-interface composites consisting: (1) of a continuous isotropic linear elastic solid matrix; and (2) of embedded parallel interfaces filled by a liquid crystal exhibiting a linear viscous behavior (Shahidi et al. 2014), see also Fig. 1. The obvious question arising then is: How does such a micromechanical formulation and the microstructural quantities appearing therein relate to the classical rheological models made up of springs and of dashpots? As an answer to this question, we here aim at establishing relations between microstructural quantities, such as interface size, interface density, interface viscosity, as well as elastic properties of the solid material phase (Fig. 1), on the one hand, and spring stiffnesses and dashpot viscosities of macroscopic rheological models (as illustrated in Figs. 2 and 3), on the other hand. Therefore, it suffices to consider macroscopic pure shear strains in planes orthogonal to the viscous interfaces (with normals

{\underset{̲}{e}}_{z}

), i.e., in the

x, z

-plane or in the

y, z

-plane of Fig. 1. Namely, in the chosen base frame

{\underset{̲}{e}}_{x}

,

{\underset{̲}{e}}_{y}

,

{\underset{̲}{e}}_{z}

, only the aforementioned shear strain components are related to creep deformation, and hence, we here focus on relations between macroscopic strains

\underset{̲}{\underset{̲}{E}} = E_{x z} ({\underset{̲}{e}}_{x} \otimes {\underset{̲}{e}}_{z} + {\underset{̲}{e}}_{z} \otimes {\underset{̲}{e}}_{x})

(1)

and corresponding stresses

\underset{̲}{\underset{̲}{Σ}} = Σ_{x z} ({\underset{̲}{e}}_{x} \otimes {\underset{̲}{e}}_{z} + {\underset{̲}{e}}_{z} \otimes {\underset{̲}{e}}_{x})

(2)

When it comes to the rheological models as depicted in Figs. 2 and 3, it is noteworthy that they refer to the behavior of a unit cube of a tested material, i.e., the “force” acting on the rheological model is equal to the shear traction acting on the material unit cube, and the “elongation” of the spring-dashpot model is equal to the engineering shear strain of the material unit cube.

Fig. 1. Matrix-interface composite consisting of continuous isotropic solid matrix and parallel 2D interfaces; 2D sketch of 3D representative volume elements

Fig. 2. Kelvin-Voigt representation of standard linear solid, used to model time-dependent behavior under pure shear according to Eq. (4): (a) complete arrangement of spring and dashpot elements; (b) section through Kelvin-Voigt unit, for illustration of shear stress contributions

Fig. 3. Maxwell representation of standard linear solid, used to model time-dependent behavior under pure shear according to Eq. (4): (a) complete arrangement of spring and damper elements; (b) section through the rheological model, for illustration of shear stress contributions

In order to establish the aforementioned link between micromechanics and classical rheology, we will, in the remainder of this paper: (1) derive differential equations describing the material behavior in terms of overall stresses and strains defined on representative volume elements; and (2) carry out a dissipation analysis, where we will consider the fundamental thermodynamics definition of the dissipation

D

, which is equal to the rate of work of the external forces,

\underset{̲}{\underset{̲}{Σ}} : \underset{̲}{\underset{̲}{\dot{E}}}

, minus the rate of elastic or free “Helmholtz” energy

\dot{Ψ}

D = \underset{̲}{\underset{̲}{Σ}} : \underset{̲}{\underset{̲}{\dot{E}}} - \dot{Ψ}

(3)

This will be developed, in three consecutive sections, for (1) the Kelvin-Voigt representation of the standard linear solid model, for (2) the Maxwell representation of the standard linear solid model, and for (3) a matrix-interface composite. Corresponding analytical formulations will allow us to relate rheological spring stiffnesses and the dashpot viscosities of standard linear solid models, to microstructural features of a matrix-interface composite. This composite, as seen in Fig. 1, is not modelled up to each and every detail of the microstructure (which, in reality, is not known anyway), but follows the tradition of continuum micromechanics (Zaoui 2002), in reflecting only the mechanically most relevant microstructural features. Even more so, we introduce the very minimum of such features, which results in a highly “idealized” system as seen in Fig. 1, so as to relate standard material parameter of classical rheological models, to the nature of different objects found in many creep-active material systems such as clay, concrete, or bone, where interfaces filled with viscously behaving matter, are embedded into elastic domains. This explicitly reductionist vision includes the introduction of only one (average) interface size [extension to several such sizes is discussed in the companion paper, Part II (Shahidi et al. 2015)], of the 2D planar geometry of the interfaces, and of only one interface density per interface size, i.e., leaving aside explicit consideration of fluctuations of such densities within the considered RVE. Nevertheless, as often encountered in science (Popper 1935), such as reductionist approach will lead to surprisingly general and helpful results. In particular, it will allow for a micromechanical interpretation of the stresses and strains which are formally associated with the rheological springs and dashpots, and for a micromechanical illustration of the energy dissipating in the dashpots. Implications for preferential choices of rheological parameters, and their relation to experiments, will be covered, thereafter, in the Discussion section.

Review of Rheological “Spring-Dashpot” Models

Kelvin-Voigt Representation of Standard Linear Solid

The Kelvin-Voigt representation of the standard linear solid consists of a Kelvin-Voigt unit (i.e., parallelly set spring and dashpot elements) in series with an additional elastic spring (Fig. 2). Here, we focus on pure shear deformation, so that rheological shear stresses

τ

and strains

γ

are related to the following stress and strain tensors by analogy to Eqs. (1) and (2)

\underset{̲}{\underset{̲}{Σ}} = τ ({\underset{̲}{e}}_{x} \otimes {\underset{̲}{e}}_{z} + {\underset{̲}{e}}_{z} \otimes {\underset{̲}{e}}_{x}) and \underset{̲}{\underset{̲}{E}} = \frac{1}{2} γ ({\underset{̲}{e}}_{x} \otimes {\underset{̲}{e}}_{z} + {\underset{̲}{e}}_{z} \otimes {\underset{̲}{e}}_{x})

(4)

The overall shear deformation of the rheological model,

γ

, can be decomposed into a dissipative portion related to the Kelvin-Voigt unit (

γ_{1}

) and an elastic portion related to the single spring (

γ_{e}

)

γ = γ_{1} + γ_{e}

(5)

The overall shear stress

τ

, in turn, is transferred both through the Kelvin-Voigt unit (

τ_{1} = τ

) and through the elastic spring (

τ_{e} = τ

)

τ = τ_{1} = τ_{e}

(6)

The shear stress acting on the Kelvin-Voigt unit,

τ_{1}

, can be decomposed into the shear stress acting on the spring,

τ_{1}^{μ}

, and into the one acting on the dashpot,

τ_{1}^{η}

τ_{1} = τ_{1}^{μ} + τ_{1}^{η}

(7)

where indexes

μ

and

η

refer to the spring and to the dashpot, respectively.

Derivation of a differential equation in

τ

and

γ

describing the constitutive behavior of the rheological model requires individual constitutive laws for all three rheological devices. Denoting the stiffness of the additional elastic spring as

μ_{e}

, and the one of the Kelvin-Voigt spring as

μ_{1}

, as well as the dashpot viscosity as

η_{1}

, Fig. 2, the individual constitutive laws read as

τ_{1}^{μ} = μ_{1} γ_{1}, τ_{1}^{η} = η_{1} {\dot{γ}}_{1}, τ_{e} = μ_{e} γ_{e}

(8)

where a dot is standing for the time derivative

\dot{•} = \partial • / \partial t

. The sought differential equation is obtained by combining Eqs. (5–8), with the aim to eliminate the individual stress combinations

τ_{1}^{μ}

and

τ_{1}^{η}

as well as the individual strain combinations

γ_{1}

and

γ_{e}

. To this end, Eq. (7) is specified for (6) and (8), and the resulting expression is divided by

η_{1}

\frac{τ}{η_{1}} = \frac{μ_{1}}{η_{1}} γ_{1} + {\dot{γ}}_{1}

(9)

Next, the last Eq. (8) is specified for (6) and solved for

γ_{e}

, yielding

τ / μ_{e} = γ_{e}

. The equality is modified in two different directions: firstly, we consider its time derivative

\frac{\dot{τ}}{μ_{e}} = {\dot{γ}}_{e}

(10)

and secondly, we consider it multiplied by

μ_{1} / η_{1}

\frac{τ μ_{1}}{μ_{e} η_{1}} = \frac{μ_{1}}{η_{1}} γ_{e}

(11)

The differential equation describing the constitutive behavior of the rheological model follows from summing up Eqs. (9)–(11), as well as from consideration of (5) and of its time derivative,

\dot{γ} = {\dot{γ}}_{1} + {\dot{γ}}_{e}

, as

\frac{\dot{τ}}{μ_{e}} + τ \frac{μ_{1}}{η_{1}} (\frac{1}{μ_{1}} + \frac{1}{μ_{e}}) = \dot{γ} + \frac{μ_{1}}{η_{1}} γ

(12)

Eq. (12) is a first-order ordinary differential equation in the overall shear stress

τ

and the overall shear strain

γ

.

Specification of the dissipation

D

according to Eq. (3), for the Kelvin-Voigt representation of the standard linear solid model requires expressions for the rate of work of the external forces,

\underset{̲}{\underset{̲}{Σ}} : \underset{̲}{\underset{̲}{\dot{E}}}

, and the rate of elastic internal energy

\dot{Ψ}

. According to Eq. (4), the former is simply equal to “force”

τ

acting on the rheological model times the “elongation rate”

\dot{γ}

, i.e., the shear stress

τ

times the rate of the engineering strain,

\dot{γ}

\underset{̲}{\underset{̲}{Σ}} : \underset{̲}{\underset{̲}{\dot{E}}} = τ \dot{γ}

(13)

Elastic internal energy

Ψ

, in turn, is stored in the two springs. In both cases the energy is equal to

1 / 2

times the spring’s force multiplied with its elongation; under consideration of (8) we obtain

Ψ = \frac{1}{2} τ_{1}^{μ} γ_{1} + \frac{1}{2} τ_{e} γ_{e} = \frac{1}{2} γ_{1} μ_{1} γ_{1} + \frac{1}{2} γ_{e} μ_{e} γ_{e}

(14)

Taking the time derivative of (14), and consideration of (8), delivers the sought rate of elastic internal energy as

\dot{Ψ} = μ_{1} γ_{1} {\dot{γ}}_{1} + μ_{e} γ_{e} {\dot{γ}}_{e} = τ_{1}^{μ} {\dot{γ}}_{1} + τ_{e} {\dot{γ}}_{e}

(15)

The dissipation finally follows from specification of (3) for (13) and (15). This yields, under consideration of (6), of (5) in the form

\dot{γ} - {\dot{γ}}_{e} = {\dot{γ}}_{1}

, and of (7) in the form

τ - τ_{1}^{μ} = τ_{1}^{η}

D = τ (\dot{γ} - {\dot{γ}}_{e}) - τ_{1}^{μ} {\dot{γ}}_{1} = (τ - τ_{1}^{μ}) {\dot{γ}}_{1} = τ_{1}^{η} {\dot{γ}}_{1}

(16)

Eq. (16) underlines that energy dissipates exclusively in the dashpot.

Maxwell Representation of Standard Linear Solid

The Maxwell representation of the standard linear solid consists of a Maxwell unit (i.e., serially set spring and dashpot elements) in parallel with an additional elastic spring (see Fig. 3). Here, it is considered to represent a piece of material under pure shear, see (4). The overall shear stress of the rheological model,

τ

, can be decomposed into the shear stress of the Maxwell unit (

τ_{I}

) and into the one of the elastic spring (

τ_{E}

)

τ = τ_{I} + τ_{E}

(17)

The overall shear deformation

γ

, in turn, is equal to the deformation of the Maxwell unit (

γ_{I} = γ

) and to the deformation of the elastic spring (

γ_{E} = γ

)

γ = γ_{I} = γ_{E}

(18)

The shear deformation of the Maxwell unit,

γ_{I}

, can be decomposed into the shear deformation of the spring,

γ_{I}^{μ}

, and into the one of the dashpot,

γ_{I}^{η}

γ_{I} = γ_{I}^{μ} + γ_{I}^{η}

(19)

where indexes

μ

and

η

refer to the spring and to the dashpot, respectively.

Derivation of a differential equation in

τ

and

γ

, describing the constitutive behavior of the rheological model, requires individual constitutive laws for all three rheological devices. Denoting the stiffness of the additional elastic spring as

μ_{E}

, and the one of the Maxwell spring as

μ_{I}

, as well as the dashpot viscosity as

η_{I}

, the individual constitutive laws read by analogy to (8) as

τ_{I} = μ_{I} γ_{I}^{μ}, τ_{I} = η_{I} {\dot{γ}}_{I}^{η}, τ_{E} = μ_{E} γ_{E}

(20)

The sought differential equation is obtained by combining Eqs. (17–20), with the aim to eliminate the individual strain components

γ_{I}^{μ}

and

γ_{I}^{η}

, as well as the individual stress components

τ_{I}

and

τ_{E}

. To this end, the time derivative of (19) is specified for

{\dot{γ}}_{I}^{μ}

and

{\dot{γ}}_{I}^{η}

from (20), delivering under consideration of (18)

\frac{{\dot{τ}}_{I}}{μ_{I}} + \frac{τ_{I}}{η_{I}} = \dot{γ}

(21)

Next, the last Eq. (20) is specified for (18), and the resulting equality,

τ_{E} = μ_{E} γ

, is modified in two different directions: firstly, we consider it multiplied with

1 / η_{I}

\frac{τ_{E}}{η_{I}} = \frac{μ_{E} γ}{η_{I}}

(22)

and, secondly, we consider its time derivative, multiplied with

1 / μ_{I}

\frac{{\dot{τ}}_{E}}{μ_{I}} = \frac{μ_{E} \dot{γ}}{μ_{I}}

(23)

The differential equation describing the constitutive behavior of the rheological model follows from summing up Eqs. (21–23) and from consideration of (17) as well as of its time derivative,

\dot{τ} = {\dot{τ}}_{I} + {\dot{τ}}_{E}

, as

\frac{1}{μ_{I}} \dot{τ} + \frac{1}{η_{I}} τ = \dot{γ} (1 + \frac{μ_{E}}{μ_{I}}) + \frac{μ_{E}}{η_{I}} γ

(24)

Eq. (24) is a first-order ordinary differential equation in the overall shear stress

τ

and the overall shear strain

γ

.

Specification of the dissipation

D

according to Eq. (3), for the Maxwell representation of the standard linear solid model requires expressions for the rate of work of the external forces,

\underset{̲}{\underset{̲}{Σ}} : \underset{̲}{\underset{̲}{\dot{E}}}

, and the rate of elastic internal energy

\dot{Ψ}

. The former reads by analogy to (13) as

\underset{̲}{\underset{̲}{Σ}} : \underset{̲}{\underset{̲}{\dot{E}}} = τ \dot{γ}

(25)

Elastic internal energy

Ψ

, in turn, is stored in the two springs. In both cases the energy is equal to

1 / 2

times the spring’s force multiplied with its elongation; under consideration of (20) we obtain

Ψ = \frac{1}{2} τ_{I} γ_{I}^{μ} + \frac{1}{2} τ_{E} γ_{E} = \frac{1}{2} γ_{I}^{μ} μ_{I} γ_{I}^{μ} + \frac{1}{2} γ_{E} μ_{E} γ_{E}

(26)

Taking the time derivative of (26), and consideration of (20), delivers the sought rate of elastic internal energy as

\dot{Ψ} = μ_{I} γ_{I}^{μ} {\dot{γ}}_{I}^{μ} + μ_{E} γ_{E} {\dot{γ}}_{E} = τ_{I} {\dot{γ}}_{I}^{μ} + τ_{E} {\dot{γ}}_{E}

(27)

The dissipation finally follows from specification of (3) for (25) and (27). This yields, under consideration of (18), of (17) in the form of

τ - τ_{E} = τ_{I}

, and of (19) in the form

{\dot{γ}}_{I} - {\dot{γ}}_{I}^{μ} = {\dot{γ}}_{I}^{η}

D = {\dot{γ}}_{I} (τ - τ_{E}) - τ_{I} {\dot{γ}}_{I}^{μ} = τ_{I} ({\dot{γ}}_{I} - {\dot{γ}}_{I}^{μ}) = τ_{I} {\dot{γ}}_{I}^{η}

(28)

Eq. (28) underlines that energy dissipates exclusively in the dashpot.

Matrix-Interface Micromechanics

Materials Representation and Constitutive Relations

Consider matrix-interface composites consisting of one solid phase and of one interface phase comprising the entity of all viscous fluid layers (Fig. 1). The solid matrix phase exhibits linear elastic behavior characterized by an isotropic stiffness tensor

{\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{s}

\underset{̲}{\underset{̲}{σ}} (\underset{̲}{x}) = {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{s} : \underset{̲}{\underset{̲}{ε}} (\underset{̲}{x})

(29)

with

\underset{̲}{\underset{̲}{σ}} (\underset{̲}{x})

and

\underset{̲}{\underset{̲}{ε}} (\underset{̲}{x})

as Cauchy microstresses and linear microstrains at any position

\underset{̲}{x}

in the solid phase. As regards the interfaces, we consider that molecular ordering-related joining forces prevent the interfaces from opening, hence dislocations in interface normal direction

{\underset{̲}{e}}_{z}

vanish

{⟦ ξ ⟧}_{z} = 0

. The component of the traction vector acting in the interface plane,

T_{x}

, is related by a linear viscous law to the dislocation rate

{⟦ \dot{ξ} ⟧}_{x}

T_{x} = η^{int} {⟦ \dot{ξ} ⟧}_{x}

(30)

where

η^{int}

denotes a viscosity constant with physical dimension [

stress \times time / length

], and where index

x

stands for the in-plane tangential direction aligned with the macroscopic loading, see Eqs. (2) and (1).

In order to derive a differential equation describing the constitutive behavior of the studied composite in terms of macroscopic stress

Σ_{x z}

and macroscopic strain

E_{x z}

, we recall two fundamental interfacial micromechanics relations (Shahidi et al. 2014), derived in the framework of homogenization schemes for eigenstressed micro-heterogeneous materials (Pichler and Hellmich 2010):

•

The macroscopic state equation, expressing the macrostress

\underset{̲}{\underset{̲}{Σ}}

as a function of the macrostrain

\underset{̲}{\underset{̲}{E}}

and of the interface traction vector

\underset{̲}{T}

reads as (Shahidi et al. 2014)

\underset{̲}{\underset{̲}{Σ}} = {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{hom} : \underset{̲}{\underset{̲}{E}} + \underset{̲}{\underset{̲}{\underset{̲}{B}}} \cdot \underset{̲}{T}

(31)

where

{\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{hom}

denotes the homogenized stiffness tensor of the studied composite, and

\underset{̲}{\underset{̲}{\underset{̲}{B}}}

denotes a Biot-type tensor quantifying the influence of interface traction vector

\underset{̲}{T}

on the macrostress

\underset{̲}{\underset{̲}{Σ}}

provided that the macrostrain

\underset{̲}{\underset{̲}{E}}

is equal to zero. Specifying (31) for the matrix-interface composite shown in Fig. 1, and for loading in terms of pure macroscopic shear according to (1), allows for extracting the following scalar equation linking shear stress

Σ_{x z}

, shear strain

E_{x z}

, and the in-plane interface traction vector component

T_{x}

(Shahidi et al. 2014)

Σ_{x z} = \frac{μ_{s} 3 (2 - ν_{s})}{3 (2 - ν_{s}) + 16 d (1 - ν_{s})} 2 E_{x z} + \frac{16 d (1 - ν_{s})}{3 (2 - ν_{s}) + 16 d (1 - ν_{s})} T_{x}

(32)

In (32),

μ_{s}

and

ν_{s}

stand for the shear modulus and for Poisson’s ratio of the isotropic solid matrix, and

d

denotes the interface density parameter (Budiansky and O’Connell 1976)

d = N a^{3}

(33)

with

N

and

a

standing for the number of the interfaces per unit volume of the composite, and for the radius of the interfaces, respectively.

•

The concentration-influence relation expressing the dislocation vector

⟦ \underset{̲}{ξ} ⟧

as a function of the macrostrain

\underset{̲}{\underset{̲}{E}}

and of the interface traction vector

\underset{̲}{T}

, reads as (Shahidi et al. 2014)

⟦ \underset{̲}{ξ} ⟧ = \underset{̲}{\underset{̲}{\underset{̲}{A}}} : \underset{̲}{\underset{̲}{E}} + \underset{̲}{\underset{̲}{D}} \cdot \underset{̲}{T}

(34)

where

\underset{̲}{\underset{̲}{\underset{̲}{A}}}

denotes a concentration tensor quantifying the influence of macrostrain

\underset{̲}{\underset{̲}{E}}

on the dislocation vector

⟦ \underset{̲}{ξ} ⟧

provided that the interface traction vector

\underset{̲}{T}

vanishes, and

\underset{̲}{\underset{̲}{D}}

stands for an influence tensor quantifying the influence of the interface traction vector

\underset{̲}{T}

on the dislocation vector

⟦ \underset{̲}{ξ} ⟧

provided that macrostrain

\underset{̲}{\underset{̲}{E}}

vanishes. Specifying (34) for the matrix-interface composite shown in Fig. 1 and for loading in terms of pure macroscopic shear according to (1), allows for extracting the following scalar equation linking in-plane dislocation

{⟦ ξ ⟧}_{x}

, the shear strain

E_{x z}

, and the in-plane interface traction vector component

T_{x}

(Shahidi et al. 2014)

⟦ ξ ⟧_{x} = \frac{8 (1 - ν_{s}) a}{π [3 (2 - ν_{s}) + 16 d (1 - ν_{s})]} [2 E_{x z} - \frac{T_{x}}{μ_{s}}]

(35)

In order to derive the sought differential equation linking

Σ_{x z}

and

E_{x z}

, as well as their time derivatives

{\dot{Σ}}_{x z}

and

{\dot{E}}_{x z}

, we solve the concentration-influence relation (35) for the interface traction

T_{x}

, and we insert the resulting expression into (32), delivering

Σ_{x z} = 2 μ_{s} E_{x z} - μ_{s} \frac{2 π d}{a} {⟦ ξ ⟧}_{x}

(36)

Taking the time derivative of (36), multiplying

{⟦ \dot{ξ} ⟧}_{x}

by 1 in form of

η^{int} / η^{int}

yields, under consideration of (30)

{\dot{Σ}}_{x z} = 2 μ_{s} {\dot{E}}_{x z} - μ_{s} \frac{2 π d}{a η^{int}} T_{x}

(37)

Solving (37) for

T_{x}

and inserting the resulting expression into (32), delivers the sought differential equation describing the constitutive behavior of the studied composite as

{\dot{Σ}}_{x z} \frac{1}{μ_{s}} + Σ_{x z} \frac{π [3 (2 - ν_{s}) + 16 d (1 - ν_{s})]}{8 a η^{int} (1 - ν_{s})} = 2 {\dot{E}}_{x z} + 2 E_{x z} \frac{3 (2 - ν_{s}) π μ_{s}}{8 (1 - ν_{s}) a η^{int}}

(38)

Eq. (38) is a first-order ordinary differential equation in the overall shear stress

Σ_{x z}

and the overall shear strain

E_{x z}

.

Elastic Energy and Dissipation

In order to quantify the elastic energy stored in the elastic solid matrix, it is convenient to replace the “eigentractions”

\underset{̲}{T}

of the interface phase, by eigenstresses

{\underset{̲}{\underset{̲}{σ}}}_{i}^{E}

of a flat spheroidal phase (whereby the limit case of the spheroid’s thickness going to zero would again relate to the two-dimensional interface). For such a system, state equations analogous to (31) and (34) read as (Pichler and Dormieux 2010; Pichler and Hellmich 2010)

\underset{̲}{\underset{̲}{Σ}} = {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{hom} : \underset{̲}{\underset{̲}{E}} + {\underset{̲}{\underset{̲}{σ}}}_{i}^{E} : \underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{B}}}}

(39)

and

\underset{̲}{\underset{̲}{Φ}} = ϕ_{i} {\underset{̲}{\underset{̲}{ε}}}_{i} = \underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{B}}}} : \underset{̲}{\underset{̲}{E}} - {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{N}}}}}^{- 1} : {\underset{̲}{\underset{̲}{σ}}}_{i}^{E}

(40)

with

ϕ_{i}

and

{\underset{̲}{\underset{̲}{ε}}}_{i}

as volume fraction and the strains of the spheroidal phase, with

\underset{̲}{\underset{̲}{Φ}}

as the product of the latter two quantities, with

{\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{hom}

as the homogenized stiffness tensor,

\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{B}}}}

as the tensor of Biot coefficients, and

{\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{N}}}}}^{- 1}

as the inverse of the tensor of Biot moduli. The latter quantities are defined as (Pichler and Dormieux 2010)

{\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{hom} = {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{s} : {[\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}} + \frac{4 π d}{3} {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{T}}}}}_{i}]}^{- 1} \underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{B}}}} = \underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}} - {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{s}^{- 1} : {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{hom} {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{N}}}}}^{- 1} = \underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{B}}}} : {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{s}^{- 1}

(41)

see also the Appendix for more details, in particular as concern the morphology tensor

{\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{T}}}}}_{i}

. The portion of elastic energy

ψ_{s}

stored per time increment in the purely elastically behaving solid matrix is equal to the external power

L_{s}^{ext}

supplied to this elastic matrix. This reads mathematically as (Pichler and Dormieux 2010)

{\dot{ψ}}_{s} = L_{s}^{ext} = \int_{V_{s}} \underset{̲}{\underset{̲}{σ}} : \underset{̲}{\underset{̲}{\dot{ε}}} d V = \int_{V_{RVE}} \underset{̲}{\underset{̲}{σ}} : \underset{̲}{\underset{̲}{\dot{ε}}} d V - \int_{V_{i}} \underset{̲}{\underset{̲}{σ}} : \underset{̲}{\underset{̲}{\dot{ε}}} d V = \underset{̲}{\underset{̲}{Σ}} : \underset{̲}{\underset{̲}{\dot{E}}} - {\underset{̲}{\underset{̲}{σ}}}_{i}^{E} : \underset{̲}{\underset{̲}{\dot{Φ}}}

(42)

with

\underset{̲}{\underset{̲}{σ}}

and

\underset{̲}{\underset{̲}{ε}}

denoting the microscopic stress and strain fields within a representative volume element hosting an elastic matrix with spheroidal inclusions; and

V_{RVE}

,

V_{s}

, and

V_{i}

, respectively, denoting the volume of such an RVE, and of its subvolumes hosting solid matrix and spheroidal inclusions, respectively.

In order to quantify the dissipation in matrix-interface composites, it is noteworthy that the elastic energy stored in the composite,

Ψ

, is equal to the elastic energy stored in the solid matrix,

ψ_{s}

, because the viscous interfaces do neither exhibit any elastic properties, nor do they, due to their 2D nature, occupy any significant volume within the RVE. Specification of the dissipation inequality (3) for

\dot{Ψ} = {\dot{ψ}}_{s}

according to (42), yields the dissipation of the matrix-interface composite as

D = \underset{̲}{\underset{̲}{Σ}} : \underset{̲}{\underset{̲}{\dot{E}}} - {\dot{ψ}}_{s} = {\underset{̲}{\underset{̲}{σ}}}_{i}^{E} : \underset{̲}{\underset{̲}{\dot{Φ}}}

(43)

The aim is now to transform (43) into a function of the variables characterizing the dissipative interfaces, namely

\underset{̲}{T}

and

⟦ \underset{̲}{ξ} ⟧

. Therefore,

\underset{̲}{\underset{̲}{\dot{Φ}}}

follows from temporal derivation of (40):

\underset{̲}{\underset{̲}{\dot{Φ}}} = \underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{B}}}} : \underset{̲}{\underset{̲}{\dot{E}}} - {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{N}}}}}^{- 1} : {\underset{̲}{\underset{̲}{\dot{σ}}}}_{i}^{E}

(44)

When additionally considering the actual two-dimensional nature of the interface phase, the eigenstress tensor degenerates to

{\underset{̲}{\underset{̲}{σ}}}_{i}^{E} = T_{x} ({\underset{̲}{e}}_{x} \otimes {\underset{̲}{e}}_{z} + {\underset{̲}{e}}_{z} \otimes {\underset{̲}{e}}_{x})

(45)

and

\underset{̲}{\underset{̲}{\dot{Φ}}}

simplifies to

\underset{̲}{\underset{̲}{\dot{Φ}}} = {\dot{Φ}}_{x z} ({\underset{̲}{e}}_{x} \otimes {\underset{̲}{e}}_{z} + {\underset{̲}{e}}_{z} \otimes {\underset{̲}{e}}_{x})

(46)

{\dot{Φ}}_{x z}

in (46) follows from specification of (44) for (1) and (45), resulting in:

{\dot{Φ}}_{x z} = \frac{16 d (1 - ν_{s})}{3 (2 - ν_{s}) + 16 d (1 - ν_{s})} {\dot{E}}_{x z} - \frac{8 d (1 - ν_{s})}{[3 (2 - ν_{s}) + 16 d (1 - ν_{s})] μ_{s}} {\dot{T}}_{x}

(47)

whereby

{\dot{T}}_{x}

, through consideration of (35), can be expressed in terms of

{\dot{E}}_{x z}

and

⟦ {\dot{ξ} ⟧}_{x}

as

{\dot{T}}_{x} = 2 μ_{s} {\dot{E}}_{x z} - \frac{μ_{s} π [3 (2 - ν_{s}) + 16 d (1 - ν_{s})]}{8 a (1 - ν_{s})} ⟦ \dot{ξ} ⟧_{x}

(48)

Accordingly, insertion of (48) into (47) yields

\underset{̲}{\underset{̲}{\dot{Φ}}} = \frac{{⟦ \dot{ξ} ⟧}_{x} d π}{a} ({\underset{̲}{e}}_{x} \otimes {\underset{̲}{e}}_{z} + {\underset{̲}{e}}_{z} \otimes {\underset{̲}{e}}_{x})

(49)

Finally, use of (49) and of (45) for transforming the expression (43), allows us to identify the dissipation as

D = \frac{2 d π}{a} {⟦ \dot{ξ} ⟧}_{x} T_{x}

(50)

Eq. (50) expresses explicitly that the energy dissipates exclusively in the viscous interfaces.

Interfacial Micromechanics Assessment of Classical Rheological Models

Micromechanical Explanation of Rheological Parameters

In the preceding sections we have derived equations describing, in terms of the overall shear stress and the overall shear strain, the constitutive behavior of the Kelvin-Voigt representation of the standard linear solid model, see (12), of the Maxwell representation of the standard linear solid model, see (24), and of a matrix-interface composite, see (38). These differential equations are valid for any type of prescribed loading history, including the special cases of a creep scenario (vanishing shear stress rate) and of a relaxation scenario (vanishing shear strain rate), see also the Appendix. Notably, the structure of the three differential equations is the same, i.e., an analogy can be established through comparison of (2), (1), and (4), by setting equal

τ = Σ_{x z} and γ = 2 E_{x z}

(51)

In order to complete the analogy, also the factors of the stress and strain variables, as appearing in (12), (24), and (38), need to be set equal. In other words, comparison of these coefficients delivers the sought relations between spring stiffnesses and dashpot viscosities, on the one hand, and micromechanical quantities, such as interface density, size, and viscosity, as well as shear modulus and Poisson’s ratio of the solid matrix, on the other hand; see Table 1 for the result of this comparison and Figs. 4–6 for further illustration of relations between viscosities and stiffnesses.

Fig. 4. Dimensionless dashpot viscosities of Kelvin-Voigt and Maxwell models, $η_{1}$ and $η_{I}$ , over the dimensionless interface density parameter $d$ , see also Table 1

Fig. 5. Dimensionless representation of interface-micromechanical origins of Kelvin-Voigt springs: instantaneous elasticity $μ_{e}$ and delayed elasticity $μ_{1}$ (both normalized with respect to the shear modulus of the solid elastic matrix $μ_{s}$ ), as functions of the interface density parameter $d$ , depending on the magnitude of the Poisson’s ratio $ν_{s}$ of the solid matrix, see graph labels; $μ_{e}$ and $μ_{1}$ are defined in Table 1

Fig. 6. Dimensionless representation of interface-micromechanical origins of Maxwell springs: stiffness $μ_{E}$ of spring parallel to dashpot and stiffness $μ_{I}$ in series with dashpot (both normalized with respect to the shear modulus of the solid elastic matrix $μ_{s}$ ), as functions of the interface density $d$ , depending on the magnitude of the Poisson’s ratio $ν_{s}$ of the solid matrix, see graph labels; $μ_{E}$ and $μ_{I}$ are defined in Table 1

Table 1. Relations between Spring Stiffnesses and Dashpot Viscosities of the Kelvin-Voigt and Maxwell Representations of the Standard Linear Solid Model and Micromechanical Quantities of a Matrix-Interface Composite, for Illustrations See Figs. 4–6

Models	Maxwell	Kelvin-Voigt	Micromechanics
Models	According to Fig. 3	According to Fig. 2	According to Fig. 1
Maxwell	$μ_{E} = μ_{E}$	$μ_{E} = μ_{1} {[1 + (μ_{1}) / (μ_{e})]}^{- 1}$	$μ_{E} = μ_{s} [3 (2 - ν_{s})] / [3 (2 - ν_{s}) + 16 d (1 - ν_{s})]$
According to Fig. 3	$μ_{I} = μ_{I}$	$μ_{I} = μ_{e} {[1 + (μ_{1}) / (μ_{e})]}^{- 1}$	$μ_{I} = μ_{s} [16 d (1 - ν_{s})] / [3 (2 - ν_{s}) + 16 d (1 - ν_{s})]$
According to Fig. 3	$η_{I} = η_{I}$	$η_{I} = η_{1} {[1 + (μ_{1}) / (μ_{e})]}^{- 2}$	$η_{I} = [a η^{int} 128 d (1 - ν_{s})^{2}] / {π [3 (2 - ν_{s}) + 16 d (1 - ν_{s})]^{2}}$
Kelvin-Voigt	$μ_{e} = μ_{I} + μ_{E}$	$μ_{e} = μ_{e}$	$μ_{e} = μ_{s}$
According to Fig. 2	$μ_{1} = [(μ_{E}) / (μ_{I})] (μ_{I} + μ_{E})$	$μ_{1} = μ_{1}$	$μ_{1} = μ_{s} [3 (2 - ν_{s})] / [16 d (1 - ν_{s})]$
According to Fig. 2	$η_{1} = η_{I} {[(μ_{I} + μ_{E}) / (μ_{I})]}^{2}$	$η_{1} = η_{1}$	$η_{1} = (a η^{int}) / (2 π d)$

Micromechanical Interpretation of Stress and Strain Contributions of Individual Springs and Dashpots of the Rheological Models

The established relations between the rheological models and the micromechanical description of the matrix-interface composite, see (51) and Table 1, open the door to a novel, micromechanics-based interpretation of the rheological models. Accordingly, we will now express the stresses and strains acting in individual springs and dashpots, as functions of micromechanical quantities.

As for the Kelvin-Voigt representation of the standard linear solid model, such expressions concern the strain in the Kelvin-Voigt unit,

γ_{1}

, the strain in the additional spring,

γ_{e}

, the stress acting on the Kelvin-Voigt spring,

τ_{1}^{μ}

, and the stress acting on the dashpot,

τ_{1}^{η}

. As for the expression for

γ_{1}

and

γ_{e}

, we specify the constitutive law of the additional spring,

τ = μ_{e} γ_{e}

, see (8), for

γ_{e} = γ - γ_{1}

, see (5), yielding

τ = μ_{e} (γ - γ_{e})

, and we compare this relation with (36). Under consideration of (51) and the relations listed in Table 1, we then obtain

γ_{1} = \frac{2 π d}{a} {⟦ ξ ⟧}_{x} and γ_{e} = 2 E_{x z} - \frac{2 π d}{a} {⟦ ξ ⟧}_{x} = \frac{Σ_{x z}}{μ_{s}}

(52)

Next, we relate

τ_{1}^{μ} = μ_{1} γ_{1}

to micromechanical quantities, by specifying this equality for

μ_{1}

according to Table 1, and for

γ_{1}

from (52)

τ_{1}^{μ} = \frac{3 μ_{s} π (2 - ν_{s})}{8 a (1 - ν_{s})} {⟦ ξ ⟧}_{x}

(53)

Finally, the micromechanical analogon for

τ_{1}^{η}

follows from first solving both (32) and (35) with respect to

E_{x z}

, and from setting equal the corresponding results, yielding

Σ_{x z} = T_{x} + \frac{3 μ_{s} π (2 - ν_{s})}{8 a (1 - ν_{s})} {⟦ ξ ⟧}_{x}

(54)

Then this relation (54) is compared to

τ = τ_{1}^{η} + τ_{1}^{μ}

. Under consideration of (51) and (53), it follows that:

τ_{1}^{η} = T_{x}

(55)

As for the Maxwell representation of the standard linear solid model, we seek micromechanical analogons for the stress acting on the Maxwell unit,

τ_{I}

, the stress acting on the additional spring,

τ_{E}

, the strain in the Maxwell spring,

γ_{I}^{μ}

, and the strain in the dashpot,

γ_{I}^{η}

. As for a micromechanics-based expression for

τ_{E}

, we first specify the spring law

τ_{E} = μ_{E} γ

for

μ_{E} = μ_{E} (μ_{s}, ν_{s}, d)

according to Table 1, and for

γ = 2 E_{x z}

, see (51)

τ_{E} = μ_{E} γ = \frac{μ_{s} 3 (2 - ν_{s})}{3 (2 - ν_{s}) + 16 d (1 - ν_{s})} 2 E_{x z}

(56)

As for expressing

τ_{I}

, we first solve (32) for

E_{x z}

, and we then insert the resulting expression into (56); this yields

τ_{E} = Σ_{x z} - \frac{16 d (1 - ν_{s})}{3 (2 - ν_{s}) + 16 d (1 - ν_{s})} T_{x}

(57)

Comparing this relation with

τ_{E} = τ - τ_{I}

, yields, under consideration of (51)

τ_{I} = \frac{16 d (1 - ν_{s})}{3 (2 - ν_{s}) + 16 d (1 - ν_{s})} T_{x}

(58)

In order to provide a micromechanical analogon for

γ_{I}^{μ}

, we specify the spring law

γ_{I}^{μ} = τ_{I} / μ_{I}

, see (20), for

τ_{I}

from (58), and for

μ_{I}

from Table 1, yielding

γ_{I}^{μ} = \frac{T_{x}}{μ_{s}}

(59)

Similarly, we specify the dashpot law

{\dot{γ}}_{I}^{η} = τ_{I} / η_{I}

, see (20), for

τ_{I}

from (58), and for

η_{I}

from Table 1, leading to

{\dot{γ}}_{I}^{η} = \frac{π [3 (2 - ν_{s}) + 16 d (1 - ν_{s})]}{8 a (1 - ν_{s})} \frac{T_{x}}{η^{int}}

(60)

Specifying (60) for

T_{x} / η^{int} = ⟦ \dot{ξ} ⟧_{x}

, see (30), and omitting the dots indicating time derivatives, i.e. integrating over time, finally yields

γ_{I}^{η} = \frac{π [3 (2 - ν_{s}) + 16 d (1 - ν_{s})]}{8 a (1 - ν_{s})} ⟦ ξ ⟧_{x}

(61)

Micromechanical Illustration of Energy Dissipating in Dashpots of the Rheological Models

The relation between rheological parameters and micromechanical quantities, as derived in the previous two subsections, allow us to illustrate the dashpot-associated dissipation in terms of micromechanical quantities. Specification of the Kelvin-Voigt-related dissipation according to (16), for

τ_{1}^{η}

according to (55), and for

{\dot{γ}}_{1}

according to the time derivative of (52), delivers

D = τ_{1}^{η} {\dot{γ}}_{1} = \frac{2 d π}{a} {⟦ \dot{ξ} ⟧}_{x} T_{x}

(62)

Similarly, specification of the Maxwell-related dissipation according to (28), for

τ_{I}

according to (58), and for

{\dot{γ}}_{I}^{η}

according to the time derivative of (61), delivers

D = τ_{I} {\dot{γ}}_{I}^{η} = \frac{2 d π}{a} {⟦ \dot{ξ} ⟧}_{x} T_{x}

(63)

Notably, both (62) and (63) result in the same expression for the dissipation of the matrix-interface composite. This clearly illustrates that the dissipation happening in the daspots of the two rheological models is related to the dissipation happening in the viscous interfaces of the micromechanical representation. In addition, the analogy-derived micromechanical expressions (62) and (63) are identical to the directly derived dissipation relation (50). This proves that all our derivations are correct, because the analogy-defining formulas for

τ_{1}^{η}

,

{\dot{γ}}_{1}

,

τ_{I}

and

{\dot{γ}}_{I}^{η}

were obtained from comparison of the differential equations (12), (24), as well as (38), and hence their derivation was independent of the derivation of dissipation relations (16), (28), and (50).

Discussion

The standard linear solid models, both in the Kelvin-Voigt and in the Maxwell representations, can perfectly describe the rheological behavior of a matrix-interface composite whose time-dependent material properties result from the linear viscosity of the interfaces behavior, i.e., they are compatible with traction-dislocation relations originating from liquid crystal interface physics, and multiscale homogenization of eigenstressed interfaces. The relations listed in Table 1 establish the links between spring stiffnesses as well as dashpot viscosities, on the one hand, and micromechanical quantities of the matrix-interface composite, on the other hand, the latter ones being interface size, density, and viscosity, as well as shear modulus and Poisson’s ratio of the isotropic solid matrix material.

When subjecting the matrix-interface composite to sudden loading, either in terms of stresses or of strains, the effective stiffness of the material is equal to the stiffness of the solid matrix, because, according to (30), dislocations would need time to develop, so that, without time passing, the interfaces are locked, which results in an uniform microscopic stress field (Shahidi et al. 2014). The same is true for the dashpot in the rheological models, being blocked upon sudden loading. Accordingly, the then observed effective stiffness of the Kelvin-Voigt model is equal to the additional spring stiffness, and the effective stiffness of the Maxwell model is equal to the sum of the spring stiffnesses. These effective stiffnesses are directly related to how the stiffness of the solid matrix manifests itself in the two rheological models: In the Kelvin-Voigt representation, the additional elastic spring simply represents the solid matrix material, while in the Maxwell representation, the sum of the two spring stiffnesses is equal to the one of the solid matrix, see Table 1.

The time-dependent evolution of interfacial dislocations increases the effective compliance of the matrix-interface composite. Given a constant (time-independent) loading, the stiffness-reducing effect is fully developed, once the asymptotic state of the material is reached, i.e., once the interface tractions have decayed to zero and the interface dislocations have grown to their maximum (Shahidi et al. 2014). The additional time-dependent material compliance is the larger, the larger the interface density. As for the Kelvin-Voigt model, the additional compliance is simply equal to the compliance of the Kelvin-Voigt spring, see Table 1. In the Maxwell representation, in turn, the interface density appears in the expressions of both spring constants, and the compliance of the additional spring is equal to the compliance of the solid matrix plus the additional compliance stemming from the interfacial dislocations, see Table 1.

The characteristic time of the rheological processes inherent to a matrix-interface composite are triggered, in general, by interface size

a

, viscosity

η^{int}

, and density

d

. This becomes apparent from the fact that the dashpot viscosities of both rheological models are directly proportional to interface size and viscosity, but the role of interface density deserves special attention, see Table 1. The Kelvin-Voigt dashpot viscosity is simply indirectly proportional to the interface density, see Table 1. Due to a more intricate mode of spring-dashpot interaction, however, the Maxwell dashpot viscosity depends in a non-trivial fashion on the interface density, and the dashpot viscosity is the smaller, the larger the Poisson’s ratio of the solid matrix, see Table 1 and Fig. 4.

When it comes to discussing the rheological counterparts of interface tractions and interface dislocations, the Kelvin-Voigt model again allows for a more straightforward mode of interpretation of the underlying interface-micromechanics: the interface traction is simply equal to the stress acting on the Kelvin-Voigt dashpot, see (55), and the strain in the Kelvin-Voigt unit is directly proportional to both the interfacial dislocations and the interface density, while it is indirectly proportional to the interface size, see (52). In the Maxwell model, in turn, the stress acting on the Maxwell unit is also proportional to the interface tractions, but the proportionality factor depends, again in a non-trivial fashion, on both the interface density and on the Poisson’s ratio of the solid matrix, see (58). Also the relation between interfacial dislocations and the strain of the Maxwell dashpot is of increased mathematical complexity, with the interface density not any more exhibiting the nature of a simple proportionality factor, and also the Poisson’s ratio of the solid entering the picture, see (61).

The Kelvin Voigt model also reveals a surprising feature of the investigated material system. As the characteristic creep time of a Kelvin-Voigt model reads as

τ_{KV} = η_{1} / μ_{1}

, it follows from Table 2 that the characteristic creep time reads as

τ_{creep} = \frac{8 a η^{int} (1 - ν_{s})}{3 π μ_{s} (2 - ν_{s})}

(64)

Table 2. Shear Stress and Strain in Rheological Models, As Functions of Interface Tractions in Relation to Interface Traction and Dislocation in Micromechanical Representation of Matrix-Interface Composite

Models	Micromechanics
Models	According to Fig. 1
Kelvin-Voigt	$τ_{1}^{η} = T_{x}$
According to Fig. 2	$γ_{1} = [(2 π d) / (a)] {⟦ ξ ⟧}_{x}$
Maxwell	$τ_{I} = [16 d (1 - ν_{s})] / [3 (2 - ν_{s}) + 16 d (1 - ν_{s})] T_{x}$
According to Fig. 3	$γ_{I}^{η} = π [3 (2 - ν_{s}) + 16 d (1 - ν_{s})] / [8 a (1 - ν_{s})] ⟦ ξ ⟧_{x}$

It becomes obvious that the characteristic creep time is independent of the interface density, and only depends on interface viscosity and radius, as well as on the elastic properties of the solid matrix. This confirms our statements given in (Shahidi et al. 2014). On the other hand, the relaxation time of the studied micromechanical material system does depend on the interface density as well, see again (Shahidi et al. 2014) for further details.

When it comes to applications of the developed formulations to real-life material systems, particular attention needs be paid to the identification of the interface viscosity, in light of different experimental or computational sources for material creep and interface physics, respectively. More precisely, the comparison of top-down and bottom-up approaches has proven as quite instructive recently in the context of bone (Qu et al. 2015): Downscaling of creep experiments on bone (Sasaki et al. 1993; Iyo et al. 2004) by means of a multiscale continuum micromechanics model (Eberhardsteiner et al. 2014) delivered Kelvin-Voigt-type creep properties at the level of hydrated extrafibrillar crystal clusters, which when used in the relations of Table 2, provided top-down-access to the interface viscosity. Alternatively, explicit molecular modeling (Dubey and Tomar 2009) of the shear forces acting between two mutually sliding mineral crystals, with water or organic molecules filling the 1 nm thick gap between them, provided a bottom-up approach. It turned out that the huge difference in velocity magnitudes between these two approaches (the molecular approach entailing more than ten orders of magnitude larger velocities than those experienced under quasi-physiological conditions) resulted in bottom-up interface viscosities exceeding their top-down counterparts by orders of magnitudes. This is fully consistent with the well-known phenomenon of thixotropy or shear thinning (Barnes 1997), i.e. the strong dependence of viscous properties of the investigated material system, on the applied shear rates, in contrast to rate-independent elastic properties (Salençon 2001). Similar phenomena are also expected in the ongoing research work on multiscale creep mechanics of concrete, based on the aging viscoelasticity formulation of Scheiner and Hellmich (2009).

Conclusion

We conclude that the Kelvin-Voigt representation of the standard linear solid model, when compared to its Maxwell counterpart, is a more natural and elegant choice for modeling a hydrated matrix-interface composite containing viscous interfaces of identical size and viscosity, because the relations between spring-dashpot properties and micromechanical quantities turn out as very simple. Nonetheless, exactly the same model features can be achieved through a Maxwell-type representation, based on an appropriate choice of the spring stiffnesses and dashpot viscosities; their relations to micromechanics quantities, however, are significantly less intuitive when compared to the Kelvin-Voigt model. It is interesting to generalize this conclusion towards consideration of multiple interface sizes and viscosities, and corresponding chain models of the Kelvin-Voigt or Maxwell type. This is the topic of the companion paper (Shahidi et al. 2015).

Notation

The following symbols are used in this paper:

$a$: radius of the interfaces;
$\underset{̲}{\underset{̲}{\underset{̲}{A}}}$: third-order concentration tensor;
$\underset{̲}{\underset{̲}{\underset{̲}{B}}}$: third-order Biot-type influence tensor;
$\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{B}}}}$: fourth-order Biot tensor;
$C$: integration constant;
${\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{s}$: fourth-order stiffness tensor of solid;
${\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{s}^{- 1}$: inverse of fourth-order stiffness tensor of solid;
${\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{hom}$: fourth-order homogenized stiffness tensor;
$d$: interface density parameter;
$\underset{̲}{\underset{̲}{D}}$: second-order influence tensor;
$D$: dissipation;
${\underset{̲}{e}}_{x}$ , ${\underset{̲}{e}}_{y}$ , ${\underset{̲}{e}}_{z}$: unit base vectors of Cartesian coordinate system;
$\underset{̲}{\underset{̲}{E}}$: second-order tensor of macroscopic strain;
$E_{x z}$: shear component of macroscopic strain;
$E_{x z, h}$: solution of homogeneous differential equation for $E_{x z}$ ;
$E_{x z, p}$: particular solution for $E_{x z}$ ;
$i$: index for interface phase;
$\underset{̲}{\underset{̲}{1}}$: second-order identity tensor;
$\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}}$: symmetric fourth-order identity tensor;
${\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}}}_{dev}$: deviatoric part of $\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}}$ ;
${\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}}}_{vol}$: volumetric part of $\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}}$ ;
$k_{s}$: bulk modulus of solid phase;
$ℓ$: characteristic size of RVE;
$ℒ_{s}^{ext}$: external power supplied to elastic matrix;
${\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{N}}}}}^{- 1}$: inverse of fourth-order tensor of Biot moduli;
$N$: number of interfaces per unit volume of the composite;
RVE: representative volume element;
$s$: index for solid phase;
$t$: time;
$\underset{̲}{T}$: interface traction vector;
${\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{T}}}}}_{i}$: fourth-order morphology tensor for 2D interface inclusion (shape of a “sharp crack”);
$T_{x}$: shear component of interface traction vector;
$V_{i}$: volume of inclusion;
$V_{s}$: volume of solid;
$V_{RVE}$: volume of RVE;
$x$ , $y$ , $z$: Cartesian coordinates;
$\underset{̲}{x}$: position vector;
$γ$: shear strain;
$γ_{e}$: shear strain of elastic spring of Kelvin-Voigt representation of standard linear solid (Zener model);
$γ_{E}$: shear strain of elastic spring of Maxwell representation of standard linear solid;
$γ_{h}$: solution of homogeneous differential equation for $γ$ ;
$γ_{I}$: shear strain of Maxwell unit;
$γ_{I}^{μ}$: shear strain of spring in Maxwell unit;
$γ_{I}^{η}$: shear strain of dashpot in Maxwell unit;
$γ_{p}$: particulate solution for $γ$ ;
$γ_{1}$: shear strain of Kelvin-Voigt unit;
$δ$: Kronecker delta;
$\underset{̲}{\underset{̲}{ε}}$: second-order tensor of microscopic linear strain;
${\underset{̲}{\underset{̲}{ε}}}_{i}$: second-order tensor of microscopic linear strain in spheroidal inclusion;
$η^{int}$: viscosity of the interfaces;
$η_{1}$: viscosity constant of dashpot in Kelvin-Voigt unit;
$η_{I}$: viscosity constant of dashpot in Maxwell unit;
$μ_{e}$: shear modulus of elastic spring in Kelvin-Voigt representation of standard linear solid (Zener model);
$μ_{E}$: shear modulus of spring in Maxwell representation of standard linear solid;
$μ_{1}$: shear modulus of spring in Kelvin-Voigt unit;
$μ_{I}$: shear modulus of spring in Maxwell unit;
$μ_{s}$: shear modulus of isotropic solid matrix;
$ν_{s}$: Poisson’s ratio of isotropic solid matrix;
$\underset{̲}{ξ}$: displacement vector;
$⟦ \underset{̲}{ξ} ⟧$: interfacial dislocation vector;
${⟦ ξ ⟧}_{x}$: in-plane component of interfacial dislocation vector;
${⟦ ξ ⟧}_{z}$: out-of-plane component of interfacial dislocation vector;
$\underset{̲}{\underset{̲}{σ}}$: second-order tensor of microscopic stress;
${\underset{̲}{\underset{̲}{σ}}}_{i}^{E}$: second-order tensor of interface eigenstress;
$\underset{̲}{\underset{̲}{Σ}}$: second-order tensor of macroscopic stress;
$Σ_{x z}$: shear component of macroscopic stress;
$Σ_{x z, h}$: solution of homogeneous differential equation for $Σ_{x z}$ ;
$Σ_{x z, p}$: particular solution for $Σ_{x z}$ ;
$τ$: shear stress;
$τ_{e}$: shear stress in elastic spring in Kelvin-Voigt representation of standard linear solid (Zener model);
$τ_{E}$: shear stress in elastic spring in Maxwell representation of standard linear solid;
$τ_{h}$: solution of homogeneous differential equation for shear stress;
$τ_{I}$: shear stress acting on Maxwell unit;
$τ_{p}$: particulate integral;
$τ_{1}$: shear stress acting on Kelvin-Voigt unit;
$τ_{1}^{μ}$: shear stress in spring of Kelvin-Voigt unit;
$τ_{1}^{η}$: shear stress in dashpot of Kelvin-Voigt unit;
$ϕ_{i}$: volume fraction of spheroidal inclusions;
$\underset{̲}{\underset{̲}{Φ}}$: second-order tensor accounting for interfacial geometry changes;
$Φ_{x z}$: shear component of $\underset{̲}{\underset{̲}{Φ}}$ ;
$Ψ$: elastic energy of matrix-interface composites;
$ψ_{s}$: elastic internal energy of solid;
$\partial$: partial derivative;
$:$: second-order tensor contraction;
$\dot{•}$: partial derivative with respect to time (“rate”) of quantity “ $•$ ”; and
$\otimes$: dyadic product.

Appendix. Tensor Components, As Well As Details on Creep and Relaxation Studies

Defintion of Elastic Stiffness of Solid, Identity Tensor, and Interface Morphology Tensor

The isotropic elastic stiffness of solid in matrix-interface composite read as

{\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}}}_{s} = 3 k_{s} {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}}}_{vol} + 2 μ_{s} {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}}}_{dev}

(65)

where

{\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}}}_{vol} = (1) / (3) \underset{̲}{\underset{̲}{1}} \otimes \underset{̲}{\underset{̲}{1}}

and

{\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}}}_{dev} = \underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}} - {\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}}}_{vol}

, respectively, denote the volumetric and the deviatoric part of the symmetric fourth-order identity tensor

\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{I}}}}

, with components

I_{i j k l} = (1) / (2) (δ_{i k} δ_{j l} + δ_{i l} δ_{k j})

, and with

δ_{i j}

denoting the Kronecker delta being equal to 1 if

i = j

and 0 otherwise. In addition,

\underset{̲}{\underset{̲}{1}}

is the second-order identity tensor with components being equal to the Kronecker delta. In (65),

k_{s}

and

μ_{s}

denote the bulk modulus and the shear modulus of the solid phase, respectively. Bulk modulus

k_{s}

is related to Poisson’s ratio

ν_{s}

and the shear modulus

μ_{s}

as

k_{s} = \frac{2 μ_{s} (1 + ν_{s})}{3 (1 - 2 ν_{s})}

(66)

The non-vanishing components of interface morphology tensor

{\underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{T}}}}}_{i}

are functions of Poisson’s ratio of the solid, and they read as (Dormieux and Kondo 2004)

\begin{array}{l} T_{i, x z x z} = T_{i, z x x z} = T_{i, x z z x} = T_{i, z x z x} \\ T_{i, y z y z} = T_{i, z y y z} = T_{i, y z z y} = T_{i, z y z y} \end{array}} = \frac{2 (1 - ν_{s})}{π (2 - ν_{s})}

(67)

\begin{array}{l} T_{i, z z x x} \\ T_{i, z z y y} \end{array}} = \frac{4 ν_{s} (1 - ν_{s})}{π (1 - 2 ν_{s})}

(68)

T_{i, z z z z} = \frac{4 {(1 - ν_{s})}^{2}}{π (1 - 2 ν_{s})}

(69)

Kelvin-Voigt Formulation: Creep and Relaxation Study

Creep Test

In a creep study, the system is suddenly subjected to a force

τ

which is kept constant afterwards:

τ (t) = constant = τ, \dot{τ} = 0

(70)

Specification of (12) for (70) delivers the governing differential equation for creep as

τ (\frac{1}{μ_{e}} + \frac{1}{μ_{1}}) \frac{μ_{1}}{η_{1}} = \frac{μ_{1}}{η_{1}} γ (t) + \dot{γ} (t)

(71)

The solution of (71) contains two parts: the solution of the homogeneous differential equation, called complementary function

γ_{h}

, and one particulate integral

γ_{p}

γ (t) = γ_{h} (t) + γ_{p}

(72)

The particulate solution turns out to be constant and reads as

γ_{p} = τ (\frac{1}{μ_{e}} + \frac{1}{μ_{1}})

(73)

The sought complementary function follows as:

γ_{h} (t) = C \exp (- \frac{μ_{1}}{η_{1}} t)

(74)

The total solution is obtained from specification of (72) for (73) and for (74), reading as

γ (t) = C \exp (- \frac{μ_{1}}{η_{1}} t) + \frac{τ}{μ_{e}} + \frac{τ}{μ_{1}}

(75)

The integration constant

C

follows from the initial condition

γ (t = 0) = (τ) / (μ_{e})

as

C = - \frac{τ}{μ_{1}}

(76)

Backsubstitution of (76) into (75), and simplifying the corresponding result, finally yields

γ (t) = \frac{τ}{μ_{e}} + \frac{τ}{μ_{1}} [1 - \exp (- \frac{μ_{1}}{η_{1}} t)]

(77)

Relaxation Test

In a relaxation study, the system is suddenly subjected to an elongation

γ

which is kept constant afterwards

γ (t) = constant = γ, \dot{γ} = 0

(78)

Specifying Eq. (12) for (78) delivers the governing differential equation for relaxation as

\dot{τ} (t) + τ (t) \frac{μ_{e}}{η_{1}} (1 + \frac{μ_{1}}{μ_{e}}) = γ (\frac{μ_{e} μ_{1}}{η_{1}})

(79)

The solution of (79) contains two parts: the solution of the homogeneous differential equation, called complementary function

τ_{h} (t)

, and one particulate integral

τ_{p}

τ (t) = τ_{h} (t) + τ_{p}

(80)

The particulate solution turns out to be time-independent, and reads as

τ_{p} = γ \frac{μ_{1}}{(1 + \frac{μ_{1}}{μ_{e}})} = \frac{γ}{(\frac{1}{μ_{1}} + \frac{1}{μ_{e}})}

(81)

The sought complementary function reads as

τ_{h} (t) = C \exp [- \frac{μ_{e}}{η_{1}} (1 + \frac{μ_{1}}{μ_{e}}) t]

(82)

Specifying (80) for (81) and (82), yields the force history

τ (t) = C \exp [- \frac{μ_{e}}{η_{1}} (1 + \frac{μ_{1}}{μ_{e}}) t] + \frac{γ}{(\frac{1}{μ_{1}} + \frac{1}{μ_{e}})}

(83)

The integration constant

C

is identified from the initial condition

τ (t = 0) = γ μ_{e}

as

C = γ \frac{μ_{e}}{(1 + \frac{μ_{1}}{μ_{e}})}

(84)

Backsubstitution of (84) into (83) gives access to the final solution which reads as

τ (t) = γ \frac{μ_{e}}{(1 + \frac{μ_{1}}{μ_{e}})} \exp [- \frac{μ_{e}}{η_{1}} (1 + \frac{μ_{1}}{μ_{e}}) t] + \frac{γ}{(\frac{1}{μ_{1}} + \frac{1}{μ_{e}})}

(85)

Maxwell Formulation: Creep and Relaxation Study

Creep Study

In a creep study, the system is suddenly subjected to a force

τ

which is kept constant afterwards: see (70). Specification of (24) for (70) delivers the governing differential equation for creep as

\dot{γ} (t) + \frac{μ_{I} μ_{E}}{(μ_{I} + μ_{E}) η_{I}} γ (t) = \frac{μ_{I}}{(μ_{I} + μ_{E}) η_{I}} τ

(86)

The solution of (86) contains two parts: the solution of the homogeneous differential equation, called complementary function

γ_{h}

, and one particulate integral

γ_{p}

γ (t) = γ_{h} (t) + γ_{p}

(87)

The particulate solution turns out to be constant and reads as

γ_{p} = \frac{τ}{μ_{E}}

(88)

The sought complementary function follows as:

γ_{h} (t) = C \exp (- \frac{μ_{I} μ_{E}}{μ_{I} + μ_{E}} \frac{t}{η_{I}})

(89)

The total solution is obtained from specification of (87) for (88) and for (89), as

γ (t) = C \exp (- \frac{μ_{I} μ_{E}}{μ_{I} + μ_{E}} \frac{t}{η_{I}}) + \frac{τ}{μ_{E}}

(90)

In order to solve for the integration constant

C

, we consider the initial condition

γ (t = 0) = (τ) / (μ_{I} + μ_{E})

, yielding

C = - \frac{μ_{I} τ}{μ_{E} (μ_{I} + μ_{E})}

(91)

Specifying (90) for (91), delivers the deformation history of the system as

γ (t) = \frac{τ}{μ_{E}} - \frac{μ_{I} τ}{μ_{E} (μ_{I} + μ_{E})} \exp (- \frac{μ_{I} μ_{E}}{μ_{I} + μ_{E}} \frac{t}{η_{I}})

(92)

Relaxation Study

In a relaxation study, the system is suddenly subjected to an elongation

γ

which is kept constant afterwards, see (78). Specification of (24) for (78) delivers the governing differential equation for relaxation as

\dot{τ} (t) + \frac{μ_{I}}{η_{I}} τ (t) = \frac{μ_{I} μ_{E}}{η_{I}} γ

(93)

The solution of (93) contains two parts: the solution of the homogeneous differential equation, called complementary function

τ_{h} (t)

, and one particulate integral

τ_{p}

τ (t) = τ_{h} (t) + τ_{p}

(94)

where the particulate solution turns out to be time-independent, and reads as

τ_{p} = γ μ_{E}

(95)

The sought complementary function follows as:

τ_{h} (t) = C \exp [- \frac{μ_{I}}{η_{I}} t]

(96)

Specification of (94) for (95) and for (96), yields

τ (t) = C \exp [- \frac{μ_{I}}{η_{I}} t] + γ μ_{E}

(97)

In order to solve for the integration constant

C

, we apply the initial condition

τ (t = 0) = γ (μ_{E} + μ_{I})

, which yields

C = γ μ_{I}

(98)

Specification of (97) for (98) delivers the stress history of the system as

τ (t) = γ μ_{E} + γ μ_{I} \exp [- \frac{μ_{I}}{η_{I}} t]

(99)

Micromechanics Formulation: Creep and Relation Study

Creep Test

In a creep study, the RVE is suddenly subjected to a shear stress

Σ_{x z}

which is kept constant afterwards

Σ_{x z} (t) = constant = Σ_{x z}, {\dot{Σ}}_{x z} = 0

(100)

Specification of the governing differential equation of interface micromechanics (38) for (100), yields the governing equation for the macrostrain reads as

{\dot{E}}_{x z} + μ_{s} \frac{3 (2 - ν_{s}) π}{8 (1 - ν_{s}) a η^{int}} E_{x z} = \frac{π [3 (2 - ν_{s}) + 16 d (1 - ν_{s})]}{16 a η^{int} (1 - ν_{s})} Σ_{x z}

(101)

The solution of (101) contains two parts: the solution of homogeneous differential equation, called complementary function

E_{x z, h} (t)

, and one particular integral

E_{x z, p}

E_{x z} (t) = E_{x z, h} (t) + E_{x z, p}

(102)

The particular solution turns out to be time-independent, and reads as

E_{x z, p} = \frac{3 (2 - ν_{s}) + 16 d (1 - ν_{s})}{2 μ_{s} 3 (2 - ν_{s})} Σ_{x z}

(103)

The sought complementary function reads as

E_{x z, h} (t) = C \exp [- \frac{μ_{s} 3 (2 - ν_{s}) π}{8 (1 - ν_{s})} \frac{t}{a η^{int}}]

(104)

The total solution is obtained from specification of (102) for (103) and for (104), yielding

E_{x z} (t) = C [\exp (- \frac{μ_{s} 3 (2 - ν_{s}) π}{8 (1 - ν_{s})} \frac{t}{a η^{int}}] + \frac{3 (2 - ν_{s}) + 16 d (1 - ν_{s})}{2 μ_{s} 3 (2 - ν_{s})} Σ_{x z}

(105)

In order to solve for the integration constant

C

, we consider the initial condition

E_{x z} (t = 0) = (1) / (2 μ_{s}) Σ_{x z}

, yielding

C = - \frac{16 d (1 - ν_{s})}{2 μ_{s} 3 (2 - ν_{s})} Σ_{x z}

(106)

Specifying (105) for the integration constant (106) allows us to write the time evolution of the macroscopic strain as

E_{x z} (t) = \frac{Σ_{x z}}{2 μ_{s}} \frac{3 (2 - ν_{s}) + 16 d (1 - ν_{s}) [1 - \exp (- \frac{3 π (2 - ν_{s})}{8 (1 - ν_{s})} \frac{μ_{s} t}{a η^{int}})]}{3 (2 - ν_{s})}

(107)

Relaxation Test

In a relaxation study, the RVE is suddenly subjected to a shear deformation

E_{x z}

which is kept constant afterwards

E_{x z} (t) = constant = E_{x z}, {\dot{E}}_{x z} = 0

(108)

Specification of the governing differential equation of interface micromechanics (38) for (108), yields the following differential equation in

Σ_{x z}

:

{\dot{Σ}}_{x z} + μ_{s} \frac{π [3 (2 - ν_{s}) + 16 d (1 - ν_{s})]}{8 a η^{int} (1 - ν_{s})} Σ_{x z} = 2 μ_{s}^{2} \frac{3 (2 - ν_{s}) π}{8 a η^{int} (1 - ν_{s})} E_{x z}

(109)

The solution of (109) contains two parts: the solution of homogeneous differential equation, called complementary function

Σ_{x z, h} (t)

, and one particular integral

Σ_{x z, p}

Σ_{x z} (t) = Σ_{x z, h} (t) + Σ_{x z, p}

(110)

The particular solution turns out to be time-independent, and reads as

Σ_{x z, p} = \frac{2 μ_{s} 3 (2 - ν_{s})}{3 (2 - ν_{s}) + 16 d (1 - ν_{s})} E_{x z}

(111)

The sought complementary function reads as

Σ_{x z, h} (t) = C \exp (- \frac{μ_{s} π [3 (2 - ν_{s}) + 16 d (1 - ν_{s})]}{8 (1 - ν_{s})} \frac{t}{a η^{int}})

(112)

Specification of (110) for the particular solution (111) and for the complementary function (112), yields

Σ_{x z} (t) = C \exp (- \frac{μ_{s} π [3 (2 - ν_{s}) + 16 d (1 - ν_{s})]}{8 (1 - ν_{s})} \frac{t}{a η^{int}}) + \frac{2 μ_{s} 3 (2 - ν_{s})}{3 (2 - ν_{s}) + 16 d (1 - ν_{s})} E_{x z}

(113)

In order to solve for the integration constant

C

, we consider the initial condition [

Σ_{x z} (t = 0) = 2 μ_{s} E_{x z}

], delivering

C = 2 μ_{s} \frac{16 d (1 - ν_{s})}{3 (2 - ν_{s}) + 16 d (1 - ν_{s})} E_{x z}

(114)

Specification of (113) for (114), yields the stress history as

Σ_{x z} (t) = 2 μ_{s} E_{x z} \frac{3 (2 - ν_{s}) + 16 d (1 - ν_{s}) \exp [- \frac{π [3 (2 - ν_{s}) + 16 d (1 - ν_{s})]}{8 (1 - ν_{s})} \frac{μ_{s} t}{a η^{int}}]}{3 (2 - ν_{s}) + 16 d (1 - ν_{s})}

(115)

Acknowledgments

Financial support of this work through the European Research Council (ERC) under project ERC-2010-StG-257032-MICROBONE is gratefully acknowledged.

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Information & Authors

Information

Published In

Journal of Engineering Mechanics

Volume 142 • Issue 3 • March 2016

Copyright

This work is made available under the terms of the Creative Commons Attribution 4.0 International license, http://creativecommons.org/licenses/by/4.0/.

History

Received: Dec 24, 2014

Accepted: Aug 11, 2015

Published online: Oct 13, 2015

Published in print: Mar 1, 2016

Discussion open until: Mar 13, 2016

Authors

Affiliations

M. Shahidi [email protected]

Research Assistant, Institute for Mechanics of Materials and Structures, TU Wien-Vienna Univ. of Technology, Karlsplatz 13/202, A-1040 Vienna, Austria. E-mail: [email protected]

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B. Pichler, Aff.M.ASCE [email protected]

Associate Professor, Institute for Mechanics of Materials and Structures, TU Wien-Vienna Univ. of Technology, Karlsplatz 13/202, A-1040 Vienna, Austria. E-mail: [email protected]

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Ch. Hellmich, M.ASCE [email protected]

Full Professor, Institute for Mechanics of Materials and Structures, TU Wien-Vienna Univ. of Technology, Karlsplatz 13/202, A-1040 Vienna, Austria (corresponding author). E-mail: [email protected]

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Abstract

Introduction

Review of Rheological “Spring-Dashpot” Models

Kelvin-Voigt Representation of Standard Linear Solid

Maxwell Representation of Standard Linear Solid

Matrix-Interface Micromechanics

Materials Representation and Constitutive Relations

Elastic Energy and Dissipation

Interfacial Micromechanics Assessment of Classical Rheological Models

Micromechanical Explanation of Rheological Parameters

Micromechanical Interpretation of Stress and Strain Contributions of Individual Springs and Dashpots of the Rheological Models

Micromechanical Illustration of Energy Dissipating in Dashpots of the Rheological Models

Discussion

Conclusion

Notation

Appendix. Tensor Components, As Well As Details on Creep and Relaxation Studies

Defintion of Elastic Stiffness of Solid, Identity Tensor, and Interface Morphology Tensor

Kelvin-Voigt Formulation: Creep and Relaxation Study

Creep Test

Relaxation Test

Maxwell Formulation: Creep and Relaxation Study

Creep Study

Relaxation Study

Micromechanics Formulation: Creep and Relation Study

Creep Test

Relaxation Test

Acknowledgments

References

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