# How interface size, density, and viscosity affect creep and relaxation functions of matrix-interface composites: a micromechanical study

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## Abstract

Matrix-inclusion composites are known to exhibit interaction among the inclusions. When it comes to the special case of inclusions in form of flat interfaces, interaction among interfaces would be clearly expected, but the two-dimensional nature of interfaces implies quite surprising interaction properties. This is the motivation to analyze how interaction among two different classes of microscopic interfaces manifests itself in macroscopic creep and relaxation functions of matrix-interface composites. To this end, we analyze composites made of a linear elastic solid matrix hosting parallel interfaces, and we consider that creep and relaxation of such composites result from micro-sliding within adsorbed fluid layers filling the interfaces. The latter idea was recently elaborated in the framework of continuum micromechanics, exploiting eigenstress homogenization schemes, see Shahidi et al. (Eur J Mech A Solids 45:41–58, 2014). After a rather simple mathematical exercise, it becomes obvious that creep functions do not reflect any interface interaction. Mathematical derivation of relaxation functions, however, turns out to be much more challenging because of pronounced interface interaction. Based on a careful selection of solution methods, including Laplace transforms and the method of non-dimensionalization, we analytically derive a closed-form expression of the relaxation functions, which provides the sought insight into interface interaction. The seeming paradox that no interface interaction can be identified from creep functions, while interface interaction manifests itself very clearly in the relaxation functions of matrix-interface materials, is finally resolved based on stress and strain average rules for interfaced composites. They clarify that uniform stress boundary conditions lead to a direct external control of average stress and strain states in the solid matrix, and this prevents interaction among interfaces. Under uniform strain boundary conditions, in turn, interfacial dislocations do influence the average stress and strain states in the solid matrix, and this results in pronounced interface interaction.

## Keywords

Interface Interaction Solid Matrix Relaxation Function Characteristic Relaxation Time Interfacial Dislocation## List of symbols

*a*Coefficient of ordinary differential equation in γ

*a*_{1}Radius of first interface family

*a*_{2}Radius of second interface family

*A*_{I}Relaxation capacity associated with characteristic time τ

_{relax},*I**A*_{II}Relaxation capacity associated with characteristic time τ

_{relax},*II*- Open image in new window
Third-order strain-to-dislocation downscaling tensor, quantifying the concentration of macroscopic strain into the average dislocation encountered across the interfaces;

*j*= 1, 2 refers to the two interface families- Open image in new window
Third-order influence tensor describing the influence of macroscopic stress on the dislocation of interfaces;

*j*= 1, 2 refers to the two interface families*b*Coefficient of ordinary differential equation in γ

- Open image in new window
Third-order influence tensor describing the influence of interface traction on the macroscopic stress;

*j*= 1, 2 refers to the two interface families- Open image in new window
Third-order influence tensor describing the influence of interface traction on the macroscopic strain;

*j*= 1, 2 refers to the two interface families*c*_{j}Constant used in Laplace transformation method, related to dislocation of interface family

*j*= 1, 2*C*_{j}Integration constant used in the elimination-non-dimensionalization method, related to the dislocation of interface family

*j*= 1, 2*C*̄_{1}Constant appearing in relaxation function

*C*̄_{2}Constant appearing in relaxation function

- \({\mathcal{C}_j}\)
Domain of interface belonging to interface family

*j*= 1, 2- Open image in new window
Fourth-order elastic stiffness tensor of solid

- Open image in new window
Inverse of Open image in new window

- Open image in new window
Fourth-order homogenized stiffness tensor of matrix-interface composite

- Open image in new window
Inverse of Open image in new window

*d*_{j}Interface density parameter of interface family

*j*= 1, 2*D*_{j}Integration constant used in the elimination-non-dimensionalization method, related to dislocation of interface family

*j*= 1, 2*D*̄_{1}Constant appearing in relaxation function

*D*̄_{2}Constant appearing in relaxation function

*e*_{j}Constant used in Laplace transformation method, related to the dislocation of interface family

*j*= 1, 2- Open image in new window
Unit base vectors of Cartesian coordinate system

- Open image in new window
Second-order tensor of macroscopic strain

*E*_{xz}Shear component of Open image in new window

*E*_{s}Young’s modulus of the solid

*j*Index for interface phase (family)

- Open image in new window
Symmetric fourth-order identity tensor

- Open image in new window
Deviatoric part of Open image in new window

- Open image in new window
volumetric part of Open image in new window

*J*_{xzxz}Creep function

- \({\mathcal{L}}\)
Laplace transformation operator

- \({\mathcal{L}^{-1}}\)
Back transformation operator from Laplace space to time domain

- \({ \mathcal{N}_j }\)
Number of interfaces per unit volume of a matrix-interface composite;

*j*= 1, 2 refers to the two interface families- Open image in new window
Outward unit normal at any point of the boundary of the volume element representing the composite material

- Open image in new window
Unit vector oriented orthogonal to the interface family

*j*= 1, 2*R*_{xzxz}Relaxation function

*s*As subscript: index for solid phase

*s*As variable: Laplace space parameter

- Open image in new window
Interface traction vector acting on interface phase

*j*= 1, 2- Open image in new window
Fourth-order morphology tensor for 2D interface inclusion (“sharp crack” morphology)

*T*_{j,x}Shear component in

*x*-direction, of traction vector Open image in new window*T*_{j,y}Shear component in

*y*-direction, of traction vector Open image in new window*x*,*y*,*z*Cartesian coordinates

- Open image in new window
Position vector

- γ
Macroscopic engineering strain (related to

*E*_{ xz })- γ
_{j} Strain-like variable related to dislocation of interface family

*j*= 1, 2- γ
_{j,h} Solution of homogeneous differential equation in γ

_{ j }- γ
_{j,p} Particular solution of non-homogeneous differential equation in γ

_{ j }- γ
_{j} Laplace transform of function γ

_{ j }- δ
Kronecker delta

*Δμ*Loss of effective stiffness during relaxation test

- Open image in new window
Second-order tensor of microscopic linear strain

*η*_{i,j}Viscosity of interface family

*j*= 1, 2*η*_{j}Differential equation-related “viscosity constant” related to interface family

*j*= 1, 2*μ*_{s}Shear modulus of isotropic solid matrix

*μ*_{j}Differential equation-related “stiffness constant” related to interface family

*j*= 1, 2*ν*_{s}Poisson’s ratio of isotropic solid matrix

- Open image in new window
Displacement vector

- Open image in new window
(average) Dislocation vector of interface phase

*j*= 1, 2- [[
*ξ*]]_{j},_{x} Shear component of Open image in new window

- [[
*ξ*]]_{j},_{z} Normal component of Open image in new window

- Open image in new window
Second-order tensor of microscopic Cauchy stresses

- Open image in new window
Second-order tensor of macroscopic Cauchy stresses

*Σ*_{xz}Shear component of Open image in new window

- τ
_{creep},*j* Characteristic creep time related to dislocation in interface family

*j*= 1, 2- τ
_{relax},*I* First characteristic relaxation time of material system comprising two interface families

- τ
_{relax},*II* Second characteristic relaxation time of material systme comprising two interface families

*Ω*Volume of the matrix-interface composite

*Ω*_{s}Volume occupied by the solid phase

- ∂
Partial derivative

- :
Second-order tensor contraction

- \({\dot{\bullet}}\)
Partial derivative with respect to time (“rate”), of quantity “•”

- \({\otimes }\)
Dyadic product

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