Calculation of the effective properties of thermo-viscoelastic composites using asymptotic homogenization in parametric space

Abstract

In this paper we use the method of asymptotic homogenization in parametric space to determine the effective properties of thermo-viscoelastic composite materials. These materials are composed of multilayered spherical inclusions imbedded in the matrix. In comparison with the traditional method of asymptotic homogenization, our approach allows for regular non-periodic distributions of inhomogeneities as well as dependences of the material characteristics on temperature. We start with the Laplace transform of the governing equations together with their boundary and initial conditions. To do so, we treat temperature and spatial coordinates responsible for non-periodic distribution of inclusions in the material as parameters (along with the parameter of Laplace transform itself). Then we define and implement a two-level scheme of asymptotic homogenization of the resulting equations in parametric space. At the first step, we solve the problem on the microscale level (a cell problem). At the second step, for the images of Laplace transform, we derive the macroscopic equation with effective coefficients. Finally, we perform the inverse Laplace transform to compute relaxation functions and determine thermo-viscoelastic properties of the composite material. The obtained results provide an information on how the change in properties and concentration of the inclusions affect the rheological characteristics and stress relaxation patterns for the thermo-viscoelastic composites.

Appendix

The harmonic potentials in the representation Eq. (34) can be approximated with the system of complex functions $\Phi (P)$, determined by a formal series with two arbitrary analytical functions: a complex function $\psi _{0}(w)$, $w = x + i y$, and a real valued function $U_{0}(z)$:

$$ \Phi (P) = \sum _{p} \frac{( - 1)^{p}\bar{w}^{p}}{4^{p} p!}\psi _{0}^{( - p)}(w) U_{0}^{(2p)}(z),\qquad \nabla ^{2}\Phi = 0, $$

(48)

where the $\psi _{0}^{( - p - 1)}$ denote the antiderivatives $\int \psi _{0}^{( - p)} dw$ and $\psi _{0}^{(0)} = \psi _{0}(w)$.

Complex function $\Phi (P)$ in its serial representation Eq. (48) satisfies the harmonic equation $\nabla ^{2}\Phi = 0$. In the case when $U_{0}(z)$ has a polynomial form, the series (48) has a finite representation. If we choose $\psi _{0}(w) = w^{m}$ and $U_{0}(z) = z^{n - m}$, then the series (48) becomes a system of homogeneous harmonic polynomials $\Phi _{n}^{m} (P)$ used as basis functions for the generalized Eshelby problem approximating our solution. Here $n$ is the general degree of the polynomial and $m$ is its “angular” degree, i.e. the number of oscillations along the coordinate $\varphi = \arctan (y / x)$. The system of functions $\left \{ \Phi _{n}^{m} (P) \right \}$, $m \le n$ forms a complete system needed to approximate an arbitrary regular harmonic function. Along with the regular system $\left \{ \Phi _{n}^{m} (P) \right \}$, we define an additional singular system of harmonic functions $\left \{ \hat{\Phi }_{n}^{m}(P) \right \}$ corresponding to functions $\psi _{0}(w) = w^{m}$ and $U_{0}(z) = z^{ - n - 1 - m}$. Defining the system $\left \{ \hat{\Phi }_{n}^{m}(P) \right \}$ in serial form (48) can raise concerns about the convergence of the series, however, it can be defined in an equivalent form with singular multiplier $r^{ - n - 1}$ and harmonic polynomial $\Phi _{n}^{m}$ if we use a Kelvin transformation (Sobolev and Vaskevich 1997) for the spherical functions: $\hat{\Phi }_{n}^{m} (P) = r^{ - n - 1} \Phi _{n}^{m} (P)$.

The form of auxiliary potential $\boldsymbol{f}$ that ensures the solution of contact equations for quantities in Eq. (34), Eq. (35) in algebraic form is established by representing the fundamental solutions of Laplace equation in the form of products of radial multipliers $h_{n}(r)$ and homogeneous harmonic polynomials $\boldsymbol{\psi }_{n}(P)$, as well as using the Gauss theorem on the expansion for homogeneous polynomials $\boldsymbol{\phi }_{n}(P)$ in a series in harmonic polynomials (Sobolev and Vaskevich 1997):

$$ \boldsymbol{\phi }_{n}(P) = \sum _{k = 0}^{\left [ n / 2 \right ]} r^{2k}\boldsymbol{\psi }_{n - 2k}(P),\qquad \nabla ^{2}\boldsymbol{\psi }_{n - 2k}(P) = 0. $$

(49)

For a biharmonic function, such as a displacement or a stress, series (49) contains two terms. The function at the free term $\boldsymbol{\psi }_{n}$ can be calculated explicitly:

$$ \Psi _{n} = \sum _{k} \beta _{k} r^{2k} \nabla ^{2k} \phi _{n},\quad \beta _{k} =- {\beta _{k-1}} / {\left ( 2k \left ( 2n+1-2k \right ) \right ),\ \beta _{0} =1}. $$

(50)

Let $G_{I}$ represent an inclusion, $G_{M}$ represent matrix, and $G_{j}$ represent the $j$th intermediate layer (there can be arbitrary number of intermediate layers). We determine the potentials in the subdomains $G_{I}$, $G_{j}$, $G_{M}$ as products of radial multipliers and harmonic polynomial functions $\boldsymbol{f}_{n}^{(0)}$. When we substitute the potentials written in this form into Eq. (34), Eq. (35), the products of radial multipliers and harmonic polynomials take the form of the sum of two terms, one of which contains a homogeneous polynomial. To satisfy the contact conditions on the interface, we introduce the function $h(r)$ and obtain the differential relations

$$ \operatorname{div}\left ( h \boldsymbol{f} \right ) = h\operatorname{div} \boldsymbol{f} + \frac{h\,\prime}{r}\left ( \boldsymbol{r} \boldsymbol{f} \right ),\qquad \nabla \left ( h \boldsymbol{r} \boldsymbol{f} \right ) = h\nabla \left ( \boldsymbol{r} \boldsymbol{f} \right ) + \frac{h\,\prime}{r}\boldsymbol{r}\left ( \boldsymbol{r} \boldsymbol{f} \right ). $$

(51)

Using Eq. (49), Eq. (50) to analyze the homogeneous polynomials obtained in Eq. (51), we find that only five basic harmonic polynomials appear in the contact equations (see also Vlasov and Volkov-Bogorodsky 2020):

$$ \boldsymbol{\psi }_{0} = \nabla \operatorname{div} \boldsymbol{f}_{\boldsymbol{n}}^{(\boldsymbol{0})},\qquad \boldsymbol{\psi }_{1} = \boldsymbol{f}_{\boldsymbol{n}}^{(\boldsymbol{0})},\qquad \boldsymbol{\psi }_{2} =- \left ( h_{n} (r) \right )^{-1} \nabla \left ( h_{n-1} \left ( r \right ) \operatorname{div} \boldsymbol{f}_{n}^{(0)} \right ), $$

$$ \boldsymbol{\psi }_{3} = \left ( h_{n} (r) \right )^{-1} \left [ \nabla \left ( \boldsymbol{r} ( h_{n} \left ( r \right ) \boldsymbol{f}_{n}^{\left ( 0 \right )} ) \right ) - \boldsymbol{r} \operatorname{div} \left ( h_{n} (r) \boldsymbol{f}_{\boldsymbol{n}}^{(\boldsymbol{0})} \right ) \right ], $$

$$ \boldsymbol{\psi }_{4} = \left ( h_{n+2} \left ( r \right ) \right )^{-1} \nabla \operatorname{div} \left ( h_{n} \left ( r \right ) \boldsymbol{f}_{n}^{(0)} \right ), $$

where $h_{n} \left ( r \right ) = r^{-2n-1}$ is a radial multiplier in three-dimensional space.

Thus, we have five harmonic polynomials and ten potentials, which are formed of these polynomials using singular radial multipliers $h_{n}$. As a result, any of Eq. (34), Eq. (35) (which we denote $\boldsymbol{F}$) can be represented in the form of linear combinations of these polynomials with radial multipliers $h_{0}$, $h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$, which become constants on the contact surface having the shape of a sphere:

$$ \boldsymbol{F} = h_{0} r^{2} \nabla \operatorname{div} \boldsymbol{f}_{n}^{(0)} + h_{1} \boldsymbol{f}_{n}^{(0)} + h_{2} \nabla \left ( \boldsymbol{r f}_{n}^{(0)} \right ) + h_{3} \boldsymbol{r} \operatorname{div} \boldsymbol{f}_{n}^{(0)} + h_{4} r^{-2} \boldsymbol{r} \left ( \boldsymbol{r f}_{n}^{(0)} \right ). $$

(36)

In accordance with the structure of polynomials $\boldsymbol{\psi }_{\boldsymbol{0}}$, $\boldsymbol{\psi }_{\boldsymbol{1}}$, $\boldsymbol{\psi }_{\boldsymbol{2}}$, $\boldsymbol{\psi }_{\boldsymbol{3}}$, $\boldsymbol{\psi }_{\boldsymbol{4}}$, defined using the harmonic polynomial $\boldsymbol{f}_{n}^{(0)}$, the general form of the potential in representation (34) has the form

$$ \textstyle\begin{array}{l} \boldsymbol{f} = \left ( A_{j} + \hat{A}_{j}r^{ - 2n - 1} \right )\boldsymbol{f}_{n}^{(0)} + \left ( B_{j}r^{2n + 1} + \hat{B}_{j} \right )\nabla \left ( r^{ - 2n + 1}\operatorname{div} \boldsymbol{f}_{n}^{(0)} \right ) + \\ \quad \quad \quad + \left ( C_{j}r^{2n + 1} + \hat{C}_{j} \right )\left [ \nabla \left ( r^{ - 2n - 1}(\boldsymbol{r} \boldsymbol{f}_{n}^{(0)}) \right ) - \boldsymbol{r}\operatorname{div} \left ( r^{ - 2n - 1}\boldsymbol{f}_{n}^{(0)} \right ) \right ] + \\ \quad \quad \quad \quad \quad \quad + \left ( D_{j}r^{2n + 5} + \hat{D}_{j} \right )\nabla \operatorname{div} \left ( r^{ - 2n - 1}\boldsymbol{f}_{n}^{(0)} \right ) + \left ( E_{j} + \hat{E}_{j}r^{ - 2n + 3} \right )\nabla \operatorname{div} \boldsymbol{f}_{n}^{(0)}. \end{array} $$

(37)

Here $A_{j}$, $\hat{A}_{j}$, $B_{j}$, $\hat{B}_{j}$, $C_{j}$, $\hat{C}_{j}$, $D_{j}$, $\hat{D}_{j}$, $E_{j}$, $\hat{E}_{j}$, are free degrees of freedom, that is, the coefficients we need to define for each component of the composite material.

Since any homogeneous harmonic polynomial $\boldsymbol{f}_{n}^{(0)}$ can be represented as a linear combination of functions $\Phi _{n}^{m} (P)$ defined in Eq. (48), we can consider Eq. (37) as an exact approximation of the potential from representation (34) by regular and singular systems of harmonic functions $\left \{ \Phi _{n - 2}^{m}, \Phi _{n}^{m}, \Phi _{n + 2}^{m} \right \}$ and $\left \{ \hat{\Phi }_{n - 2}^{m}, \hat{\Phi }_{n}^{m}, \hat{\Phi }_{n + 2}^{m} \right \}$ of degree $n - 2$, $n$ and $n + 2$ (the degree $n - 2$ does not exist if $n = 0,1$).

$x$::

“Slow” coordinate

$\xi$::

“Fast” coordinate

$\varepsilon$::

Characteristic distance between inclusions

$L$::

Characteristic spatial size

$T$::

Temperature

$t$::

Time

$r$::

Spherical radial coordinate

$s$::

Parameter of Laplace transform

$\boldsymbol{p} = \left \{ x,T,s \right \}$::

Set of parameters

$\boldsymbol{u}$::

Displacement

$\boldsymbol{u}^{*}$::

Laplace image of displacement

$I$::

Identity matrix

$A_{ij} \left ( \xi ,x,T,t \right ) = \left \Vert c_{ikjl} \right \Vert$::

Time-dependent relaxation matrix functions

$A_{ij}^{o} = A_{ij} (\xi ,x,T,0)$::

Time-dependent relaxation matrix functions at initial moment of time

$c_{ijkl}^{*}$::

Laplace image of $c_{ikjl}$

$\tilde{c}_{ijkl} = sc_{ijkl}^{*}$::

See Eq. (13) in the text of the article

$\hat{A}_{ij} = \left \Vert \hat{c}_{ikjl} \right \Vert = \left \Vert s \hat{c}_{ikjl}^{*} \right \Vert$::

An effective matrix

$\boldsymbol{\alpha }_{i} = \left \Vert \alpha _{ij} \right \Vert$::

Vectors representing the thermal expansion tensor

$\boldsymbol{\alpha }_{i}^{*}$::

Laplace image of $\boldsymbol{\alpha }_{i}$

$\hat{\boldsymbol{\alpha }}_{i}^{*} = \left \Vert \hat{\alpha }_{ij}^{*} \right \Vert$::

Vectors determined by the components of the effective thermal expansion tensor

$\hat{\boldsymbol{b}}_{\boldsymbol{i}} = \hat{A}_{ij} \hat{\boldsymbol{\alpha }}_{j}^{*}$::

See Eq. (27) in the text of the article

$\boldsymbol{n}$::

Normal vector

$\sigma _{ij}$::

Stress tensor

$\sigma _{ij}^{*}$::

Laplace image of the stress tensor

$\varepsilon _{ij}$::

Tensor of deformations

$\varepsilon _{ij}^{*}$::

Laplace image of the tensor of deformations

$\mu$ and $\lambda$::

Relaxation functions analogous to Lame parameters

$\mu ^{*}$::

Laplace image of $\mu$

$\tilde{\mu } \left ( \xi ,x,T,s \right ) =s \mu ^{*} \left ( \xi ,x,T,s \right )$::

See Eq. (15) in the text of the article

$K_{0}$::

Elastic modulus of the volumetric expansion

$\tilde{K}$::

Complex modulus of the volumetric expansion

$\alpha$::

Linear coefficient of thermal expansion

$\alpha ^{*}$::

Laplace image of $\alpha$

$\delta _{ij}$::

Kronecker delta

$\sigma _{0}$::

First invariant of the stress tensor

$\sigma _{0}^{*}$::

Laplace image of $\sigma _{0}$

$\theta$::

Volumetric deformation

$E \left ( \xi ,x,T,t \right )$::

An analog for Young’s modulus for viscoelastic material

$E_{0}^{(i)} \left ( T \right )$ and $\mu _{0}^{(i )} \left ( T \right )$::

Instant elasticity and shear moduli

$E_{\infty }^{(i)} \left ( T \right ) \ a \mathrm{nd} \ \mu _{\infty }^{(i )} \left ( T \right )$::

Long-term elasticity and shear moduli

$\gamma _{p}^{(i)}$::

Inverse relaxation times

$\nu _{ij}^{*}$::

Viscoelastic analogs of Poisson’s ratios

$\hat{G}_{ij}^{*}$::

Viscoelastic analog of shear modulus

$\hat{E}_{i}^{*}$::

Viscoelastic analog of Young’s modulus

$\hat{C}_{ij}^{*} = \hat{c}_{iijj}^{*}$::

The image of stiffness matrix

$\hat{c}_{ijkl}^{*} (x,T,s)= \hat{c}_{ijkl} (x,T,s) /s$::

See the text of the article

$\hat{R}_{ij}^{*} = \hat{r}_{iijj}^{*}$::

Compliance matrix

$\Delta _{i}$::

Distance between the opposite faces $S_{i}^{+}$ and $S_{i}^{-}$ of a periodic cell ($i=1,2,3$)