Micromechanical analysis of pores on damping properties of particulate filled elastomer

Abstract

This study aims to analyse the effect of different shapes of pores on the damping properties of spherical particle-filled composites. A two-step stochastic homogenization approach is developed to evaluate the effective elastic properties within the micromechanical framework. Modified Mori-Tanaka homogenization approach considering the differential scheme for iteratively addition of pores as well as fillers is incorporated in the present model. From the results, the saturated ratio of the stiffness of fillers to the matrix is found to exist beyond which further increase in the stiffness of the fillers does not add in the increase in the effective elastic properties. It is further noted that the presence of voids decreases the elastic properties significantly. The results obtained from the proposed model are found to be in good agreement with that obtained from the computational study.

Graphical Abstract

Keywords

Micromechanics pores two-step homogenization particulates loss factor

Introduction

Composite materials are popularly used in many applications due to their ability to tailor their inherent mechanical properties. One of the techniques to improve properties is through dispersing fillers in the polymer matrix, denoted as particulate filled composites. However, the degree of improvement is affected by the manufacturing process adopted to synthesize materials to a great extent. The manufacturing process is generally selected according to the final product’s complexity and the matrix’s curing behavior. One of the greatest problems in composites is formation of pores during the manufacturing process, which are often unavoidable. The pores can be formed in the matrix due to (i) moisture absorption, (ii) presence of dissolved gases,¹ (iii) air/foreign particle entrapment, (iv) vigorous mixing, and (v) the presence of chemical volatiles during the curing reaction.^2,3 The pores are formed in fabric-reinforced composites due to the lack of wetting of fibers with the matrix⁴ and gaps between the fibers or inside the tows.¹ In particulate filled composites, pores are formed due to particle-matrix debonding.⁵ The degree of pores’ formation differs from one manufacturing process to another due to variability in the matrix’s fabrication path and curing/rheological behavior. Therefore, a mathematical model should be developed to account for the variations in the manufacturing process for estimating the effective inherent properties of composite materials.

Several micromechanical models have been proposed to estimate composites’ effective elastic properties consisting of elastic inclusions embedded into the elastic matrix. The Voigt⁶ and Reuss⁷ models are the earliest models for homogenization of two-phase composites that predict the upper and lower bounds of the effective properties. In Voigt⁶ model, the effective properties have been estimated as the volume-averaged stiffness of the individual phases assuming the uniform stress state in the Representative Volumetric Element (RVE). In the Reuss model,⁷ the effective properties have been estimated as the volume-averaged compliance of the individual constitutes in the RVE, assuming the iso-strain condition in the RVE. Further, Hashin and Shtrikman⁸ model provides the improved bounds for the effective properties using minimal potential energy and the concept of polarization. The limited use of these models is due to the indirect estimation of the composites’ stiffness tensor. On the other hand, other models such as Mori-Tanaka,⁹ Self-Consistent,^10–13 and Differential Scheme¹⁴ models provide the direct estimation of the stiffness tensor based on the continuum mechanics’ approach. Mori-Tanka model⁹ proposed the mean-field homogenization approach relating the averaged stress and averaged strain in the matrix’s inclusion. Benveniste¹⁵ modified the approach using Eshelby’s Equivalent inclusion theory¹⁶ and introduced a Mori-Tanaka tensor, from which the effective properties of composites can be estimated. In several literature, Mori-Tanaka, combined with Eshelby’s theory, has been adopted for the porous material¹⁷ and composites embedded with aligned and randomly oriented inclusions.¹⁸ Tian et al.¹⁹ evaluated short-fiber reinforced composites’ effective properties considering a two-step mean filed homogenization approach. Mori-Tanka scheme is a very popular approach, though not always reliable in multi-phase composites and non-dilute dispersion of inclusions. On the other hand, the Self-consistent model^10–13 accounts for the multi-phase dispersion of inclusions as well as interaction in composites. This scheme assumes that the inclusion is embedded in an infinite elastic matrix, which is considered an effective medium with the same properties as overall composites.

To estimate the effective properties of particulate-filled composites, researchers have employed several foundational models. Peng et al.²⁰ combined the Mori-Tanaka and Self-consistent schemes, while McLaughlin¹⁴ and Norris²¹ applied the differential scheme to two-phase and multi-phase composites, respectively. A key observation is that a composite’s properties are functions of the volume fraction and characteristics of its inclusions. A significant advancement was presented by Chawla et al.,²² who modified the differential scheme by introducing an iterative, self-consistent methodology. Their technique involves the stepwise addition of a specific volume fraction of pores, where the effective medium from the preceding step serves as the background matrix for the current one, calculated via the Mori-Tanaka method. This process is repeated until the target porosity is met and has been shown to successfully predict effective elastic properties. The present analysis leverages this established framework to model voids within particulate-filled composites, with the specific goal of analyzing their impact on damping characteristics.

A stochastic two-step homogenization approach for analyzing the effects of pores on the particulate-filled composites damping properties in present study. In the first step of homogenization, the effective properties are evaluated using the modified Mori-Tanaka model along with differential scheme of adding pores as inclusions in the matrix in a step-by-step manner until the specified volume fraction of pores is obtained. The matrix obtained from this step is denoted as a fictitious matrix. The effective properties of the fictitious matrix are used as a background property for the next step of homogenization. In the second step of homogenization, the same procedure is repeated for the stepwise addition of fillers as inclusions in the fictitious matrix up to a certain volume fraction. The model performance’s efficiency is verified by performing finite element (FE) simulations on RVE containing randomly distributed spherical pores and fillers of different volume fractions.

Mathematical model

Due to the intrinsically hierarchical structure of filler-reinforced composites, a multiscale strategy is required to forecast their overall effective properties. This modelling approach typically recognizes three distinct scales: the micro-scale, which includes elements like fibers, pores, and inclusions; the meso-scale, which relates to the composite laminate; and the macro-scale, which encompasses the entire structural component. A homogenization-based micromechanical modelling framework is particularly well-suited for incorporating the influence of microstructural process parameters on the mechanical behaviour of these composites.

Matrix material model

In the present study, the matrix is modelled as a linear viscoelastic material under small strain harmonic loading conditions. The constitutive response is described using a frequency-dependent complex modulus representation. The viscoelastic matrix parameters are taken from,²³ where a Prony series representation was employed for the modelling. The behaviour of matrix material is described by constitutive law as

σ_{i j} (t) = \int_{0}^{t} 2 G (t - τ) \frac{d ε_{i j}^{d e v}}{d τ} d τ + \int_{0}^{t} 3 K (t - τ) \frac{d ε_{i j}^{v o l}}{d τ} d τ

(1)

where

σ_{i j}

is stress tensor,

ε_{i j}^{d e v}

and

ε_{i j}^{v o l}

are deviatoric and volumetric strain, respectively, G(t) and K(t) are shear and bulk relaxation modulus, respectively. The complex Young’s modulus of viscoelastic materials is sum of storage modulus

E^{'}

and loss modulus

E^{″}

E *^{(m)} = E^{'} + i E^{″}

(2)

The loss factor $\tan δ$ provides a measure of damping of viscoelastic material and defined as ratio of loss modulus to storage modulus as

\tan δ = \frac{E^{″}}{E^{'}}

(3)

The complex modulus in equation (2) can be obtained using

E *^{(m)} = \frac{9 K *^{(m)} G *^{(m)}}{3 K *^{(m)} + G *^{(m)}}

(4)

Where

K *^{(m)}

and

G *^{(m)}

are complex shear and bulk modulus which can be represented by a Prony series for a viscoelastic material as

G *^{(m)} = G_{\infty} + i ω \sum_{i = 1}^{N} \frac{G_{i} τ_{i}}{(1 + i ω τ_{i})}

(5)

K *^{(m)} = K_{\infty} + i ω \sum_{i = 1}^{N} \frac{K_{i} τ_{i}}{(1 + i ω τ_{i})}

(6)

The complex Poisson’s ratio $ν *^{(m)}$ can be obtained from complex shear and bulk modulus as

ν *^{(m)} = \frac{3 K *^{(m)} - 2 G *^{(m)}}{2 (3 K *^{(m)} + G *^{(m)})}

(7)

It is clarified that the nonlinear viscoelasticity and large-strain hysteresis effects typical of rubber-like materials under high deformation amplitudes are not considered in this study. The analysis is restricted to small strain oscillatory loading, where the linear viscoelastic approximation is valid.

Modified Mori-Tanaka approach

Mori-Tanaka approach is popular for determination of effective elastic properties of composites, developed originally by Mori and Tanaka¹ and then reformulated by Benveniste.² The approach along with Eshelby’s inclusion theory³ accommodates possibilities of composites with different shapes of inclusions such as spherical, cylindrical, oblate and penny shaped as well as different orientations. The Mori-Tanaka approach was derived based on the assumption of dilute dispersion i.e., single inclusion embedded in an infinite matrix. It was noted earlier that the effective properties of composites not only depend on the volume fraction of phases but also on the process of adding inclusions in the matrix.⁴ Later, a different scheme was introduced for iterative addition of inclusion in the matrix. Based on the differential scheme, a modified Mori-Tanaka approach¹ is proposed which is also valid for non-dilute dispersion of inclusions in the matrix. In current work, a two-step procedure based on the modified Mori-Tanaka model is adopted for evaluating the damping properties of particulate filled composites containing pores of different shapes as shown in Figure 1. In the first step, the effective properties of the matrix containing pores (denoted as fictitious matrix) are obtained by using the Mori-Tanaka model along with the differential scheme of adding pores one by one. In next step, the pores of specified volume fractions are assumed to be randomly distributed in the fictitious matrix. Using the fictitious matrix properties (from the first step) as background, the effective properties can be obtained for a complete RVE containing pores as well as particulates. The analysis assumes isothermal mechanical loading and does not explicitly consider residual stresses generated by coefficient of thermal expansion (CTE) mismatch between filler and matrix. Thermal history and processing-induced stresses are therefore not included in the present formulation. A coupled thermo-viscoelastic framework would be required to account for such effects. In the present formulation, pores are modeled as void inclusions with traction-free boundaries and negligible internal stiffness. The potential effects of compressible gas within the pores, including pressure evolution under deformation, are not explicitly considered. Under the small-amplitude harmonic loading conditions investigated, gas compressibility effects are assumed to be negligible compared to the viscoelastic response of the matrix.

Figure 1.

Two-step homogenization procedure for polymer matrix containing randomly distributed voids and fillers.

Let us analyse the first step by considering RVE containing a matrix of volume $M^{(m)}$ and pores of the total volume $M^{(p)}$ . In the present approach, the pores are added iteratively. For example, for (i+1) ^th iteration, the pore of volume ${Δ M}_{(i)}^{(p)}$ is added into the matrix of volume $M^{(m)}$ , and the updated volume fraction is obtained as (Figure 2(a))

V_{(i + 1)}^{(p)} = \frac{V_{(i)}^{(p)} M^{(m)} + {Δ M}_{(i)}^{(p)}}{M^{(m)} + {Δ M}_{(i)}^{(p)}}

(8)

Figure 2.

Two-step Differential scheme adopted for modeling pores in particulate filled composites: (a) addition of pores one by one in the matrix up to prescribed volume fraction of voids of $V^{(p)}$ , denoted as the fictitious matrix in the final step of addition and (b) addition of fillers iteratively in the fictitious matrix up to prescribed volume fraction of fillers of $V^{(f)}$ .

This iterative process is continued until the final desired volume fraction of voids $V^{(p)}$ is achieved. From the Mori-Tanaka model based on the differential scheme, the effective stiffness tensor ${\bar{C}}^{* (m)}$ for the (i+1)^th iteration is obtained as²²:

{\bar{C}}^{* (m)} (i + 1) = [{\bar{C}}^{* (m)} (i) + ({C^{(p)} - {\bar{C}}^{* (m)} (i)) V}_{(i + 1)}^{(p)} \bar{T} (i + 1)] {[V_{(i + 1)}^{(p)} \bar{T} (i + 1) + (1 - V_{(i + 1)}^{(p)}) I]}^{- 1}

(9)

In equation (9), $I$ is the unity matrix, ${\bar{C}}^{* (m)}$ and $C^{(p)}$ are the stiffness tensors of the matrix and pore, respectively. The components of the stiffness tensors of the matrix and pore can be expressed as

{\bar{C}}_{i j k l}^{* (m)} = \frac{E^{* (m)} ν^{* (m)}}{(1 + ν^{* (m)}) (1 - 2 ν^{* (m)})} δ_{i j} δ_{k l} + \frac{E^{* (m)}}{2 (1 + ν^{* (m)})} (δ_{i k} δ_{j l} + δ_{i l} δ_{j k})

(10)

C_{i j k l}^{(p)} = K^{(p)} δ_{i j} δ_{k l}

(11)

where

E^{* (m)}

and

ν^{* (m)}

are the elastic modulus and Poisson’s ratio of the matrix, respectively,

K^{(p)}

is the bulk modulus of the pore containing air and

δ

is the Kronecker delta. In equation (9), T relates the elastic properties of the pore and the matrix, which can be expressed using the idea of equivalent Eshelby inclusion theory as³

{\bar{T} (i + 1) = [I - \bar{P} {\bar{C}}^{* (m)}^{- 1} ({\bar{C}}^{* (m)} - C^{(p)})]}^{- 1}

(12)

where

\bar{P}

is the fourth order Eshelby’s tensor, which updates with each iterative step. In other words, for each iteration, it updates depending upon the shape of the pore as well as Poisson’s ratio of the effective medium. For cylindrical shape of inclusion (i.e., long elliptic cylinder,

a_{3} = \infty

as shown in Figure 1), the components of

\bar{P}

are expressed as⁶

{\bar{P}}_{1111} = \frac{1}{2 (1 - {ν^{*}}^{(m)})} {\frac{a_{2}^{2} + 2 a_{1} a_{2}}{{(a_{1} + a_{2})}^{2}} + (1 - 2 {ν^{*}}^{(m)}) \frac{a_{2}}{a_{1} + a_{2}}}, {\bar{P}}_{2222} = \frac{1}{2 (1 - {ν^{*}}^{(m)})} {\frac{a_{1}^{2} + 2 a_{1} a_{2}}{{(a_{1} + a_{2})}^{2}} + (1 - 2 {ν^{*}}^{(m)}) \frac{a_{1}}{a_{1} + a_{2}}}, {\bar{P}}_{3333} = 0

{\bar{P}}_{1122} = \frac{1}{2 (1 - {ν^{*}}^{(m)})} {\frac{a_{2}^{2}}{{(a_{1} + a_{2})}^{2}} - (1 - 2 {ν^{*}}^{(m)}) \frac{a_{2}}{a_{1} + a_{2}}}, {\bar{P}}_{2233} = \frac{1}{2 (1 - {ν^{*}}^{(m)})} \frac{2 {ν^{*}}^{(m)} a_{1}}{a_{1} + a_{2}}

(13)

{\bar{P}}_{1133} = \frac{1}{2 (1 - {ν^{*}}^{(m)})} \frac{2 {ν^{*}}^{(m)} a_{2}}{a_{1} + a_{2}}, {\bar{P}}_{3311} = 0, {\bar{P}}_{2211} = \frac{1}{2 (1 - {ν^{*}}^{(m)})} {\frac{a_{1}^{2}}{{(a_{1} + a_{2})}^{2}} - (1 - 2 {ν^{*}}^{(m)}) \frac{a_{1}}{a_{1} + a_{2}}}

{\bar{P}}_{1212} = \frac{1}{2 (1 - {ν^{*}}^{(m)})} {\frac{a_{1}^{2} + a_{2}^{2}}{2 {(a_{1} + a_{2})}^{2}} + \frac{(1 - 2 {ν^{*}}^{(m)})}{2}}, {\bar{P}}_{3322} = 0, {\bar{P}}_{2323} = \frac{a_{1}}{2 (a_{1} + a_{2})}, {\bar{P}}_{3131} = \frac{a_{2}}{2 (a_{1} + a_{2})}

For penny shape of inclusion (i.e., $a_{1} = a_{2} ≫ a_{3}$ as shown in Figure 3), the components of $\bar{P}$ are expressed as⁶

{\bar{P}}_{1111} = {\bar{P}}_{2222} = \frac{13 - 8 {ν^{*}}^{(m)}}{32 (1 - {ν^{*}}^{(m)})} π \frac{a_{3}}{a_{1}}, {\bar{P}}_{3333} = 1 - \frac{1 - 2 {ν^{*}}^{(m)}}{1 - {ν^{*}}^{(m)}} \frac{π}{4} \frac{a_{3}}{a_{1}}

{\bar{P}}_{1122} = {\bar{P}}_{2211} = \frac{8 {ν^{*}}^{(m)} - 1}{32 (1 - {ν^{*}}^{(m)})} π \frac{a_{3}}{a_{1}}, {\bar{P}}_{1133} = {\bar{P}}_{2233} = \frac{2 {ν^{*}}^{(m)} - 1}{8 (1 - {ν^{*}}^{(m)})} π \frac{a_{3}}{a_{1}}

(14)

{\bar{P}}_{3311} = {\bar{P}}_{3322} = \frac{{ν^{*}}^{(m)}}{(1 - {ν^{*}}^{(m)})} (1 - \frac{4 {ν^{*}}^{(m)} + 1}{8 {ν^{*}}^{(m)}} π \frac{a_{3}}{a_{1}})

{\bar{P}}_{1212} = \frac{7 - 8 {ν^{*}}^{(m)}}{32 (1 - {ν^{*}}^{(m)})} π \frac{a_{3}}{a_{1}}, {\bar{P}}_{1313} = {\bar{P}}_{2323} = \frac{1}{2} (1 + \frac{{ν^{*}}^{(m)} - 2}{1 - {ν^{*}}^{(m)}} \frac{π}{4} \frac{a_{3}}{a_{1}})

Figure 3.

An ellipsoid pore with radii a₁, a₂ and a₃ along the principal axes x₁, x₂ and x₃, respectively.

For a spherical shape of the inclusion (i.e., $a_{1} = a_{2} = a_{3}$ as shown in Figure 3), the components of $\bar{P}$ are expressed as

{\bar{P}}_{i j k l} = \frac{5 {ν^{*}}^{(m)} - 1}{15 (1 - {ν^{*}}^{(m)})} δ_{i j} δ_{k l} + \frac{4 - 5 {ν^{*}}^{(m)}}{15 (1 - {ν^{*}}^{(m)})} (δ_{i k} δ_{j l} + δ_{i l} δ_{j k})

(15)

From equation (10), the stiffness tensor ${\bar{C}}^{* (m)}$ correspond to the final stage of achieving the prescribed volume faction of pores $V^{(p)}$ is denoted as the fictitious stiffness tensor. For the next step, spherical filler particles are added iteratively in the fictitious matrix in a similar manner (as obtained from previous step). The effective stiffness tensor $\bar{C}$ for the (j+1)^th iteration of adding filler particle is expressed as

{\bar{C}}^{*} (j + 1) = [{\bar{C}}^{* (m)} (j) + ({C^{(f)} - {\bar{C}}^{* (m)} (j)) V}_{(j + 1)}^{(f)} \bar{T} (j + 1)] {[V_{(j + 1)}^{(f)} \bar{T} (j + 1) + (1 - V_{(j + 1)}^{(f)}) I]}^{- 1}

(16)

Where

C^{(f)}

is the stiffness tensors of the filler particle, the components of which can be expressed as

{\bar{C}}_{i j k l}^{(f)} = \frac{E^{(f)} ν^{(f)}}{(1 + ν^{(f)}) (1 - 2 ν^{(f)})} δ_{i j} δ_{k l} + \frac{E^{(f)}}{2 (1 + ν^{(f)})} (δ_{i k} δ_{j l} + δ_{i l} δ_{j k})

(17)

where

E^{(f)}

and

ν^{(f)}

are elastic modulus and Poisson’s ratio of the filler particle, respectively. From equation (7), effective compliance tensor

\bar{S}

can be expressed as

{\bar{S}}^{*} (j + 1) = {{\bar{C}}^{*} (j + 1)}^{- 1} = [\begin{array}{c} S_{11}^{*} \\ S_{12}^{*} \\ \begin{array}{c} S_{12}^{*} \\ 0 \\ \begin{array}{c} 0 \\ 0 \end{array} \end{array} \end{array} \begin{array}{c} S_{12}^{*} \\ S_{11}^{*} \\ \begin{array}{c} S_{12}^{*} \\ 0 \\ \begin{array}{c} 0 \\ 0 \end{array} \end{array} \end{array} \begin{array}{l} S_{12}^{*} \\ S_{12}^{*} \\ \begin{array}{l} S_{11}^{*} \\ 0 \\ \begin{array}{c} 0 \\ 0 \end{array} \end{array} \end{array} \begin{array}{l} 0 \\ 0 \\ \begin{array}{l} 0 \\ 2 (S_{11}^{*} - S_{12}^{*}) \\ \begin{array}{c} 0 \\ 0 \end{array} \end{array} \end{array} \begin{array}{c} 0 \\ 0 \\ \begin{array}{c} 0 \\ 0 \\ \begin{array}{c} 2 (S_{11}^{*} - S_{12}^{*}) \\ 0 \end{array} \end{array} \end{array} \begin{array}{c} 0 \\ 0 \\ \begin{array}{c} 0 \\ 0 \\ \begin{array}{c} 0 \\ 2 (S_{11}^{*} - S_{12}^{*}) \end{array} \end{array} \end{array}]

(18)

The effective elastic properties for the (i+1)^th steps can be obtained as

E^{*} (i + 1) = \frac{1}{{\bar{S}}_{11}^{*} (i + 1)}, ν^{*} (i + 1) = - \frac{{\bar{S}}_{12} (i + 1)}{{\bar{S}}_{11}^{*} (i + 1)}

(19)

From the proposed method, the effective elastic properties of three-phase composites containing pores and filler particles can be computed.

It is highlighted that, in the present formulation, the pores are idealized as equivalent geometric inclusions to enable analytical homogenization. While real manufacturing-induced pores may exhibit irregular, jagged, or interconnected morphologies, micromechanical models commonly approximate such defects by equivalent spherical or ellipsoidal voids that reproduce the average stress concentration effect. It is noted that highly irregular or interconnected pore networks, which may induce percolation and strong local stress localization, are beyond the scope of mean-field homogenization and would require full-field numerical modeling for rigorous characterization. Also, the model assumes a perfect interfacial bonding between filler and matrix. Interfacial debonding, frictional sliding, or damage-induced dissipation mechanisms are not explicitly considered. The predicted damping response therefore arises solely from the viscoelastic behavior of the matrix phase. Additional dissipation due to interfacial damage would require an extended micromechanical framework incorporating cohesive or frictional interface modelling.

Model performance comparison

The proposed model is implemented in MATLAB for analysing the effects of pores on damping properties. Different pore shapes such as spherical, penny and long cylinderical of varying sizes with different volume fractions (5% and 10%) are added iteratively in the RVE. Fillers are also added iteratively in the RVE with varying volume fractions up to 30%. Particularly, four different RVEs are generated containing (i) only spherical pores, (ii) only mixed (hybrid) shape of pores, (iii) spherical pores as well as fillers and, (iv) hybrid pores as well as fillers. Figure 4 shows the four different RVEs consodered for present study. For non-spherical pore geometries, a statistically isotropic three-dimensional random orientation distribution is assumed to maintain macroscopic isotropy of the composite. Preferential alignment effects are not considered in the present formulation. The efficiency of the proposed model is assesed for RVE containing spherical fillers with available data provided in Ref. 23. The homogenized Young’s modulus as a function of frequency is obtained from equations (4)–(6) and loss factor $\tan δ$ is obtained from equations (2) and (3). The relaxation time, shear and bulk modulii of the matrix are considerd similar as provided in Table 1.²³ The fillers are considered to have $E^{(f)}$ = 64.89 GPa and ν = 0.25. The predicted damping loss factor represents the effective macroscopic response of the composite obtained from the homogenized complex modulus. The micromechanical framework inherently provides a volume-averaged measure of viscoelastic dissipation within the representative volume. While micro-scale dissipation may vary spatially, the objective of homogenization is to determine an equivalent engineering-scale damping parameter.

Figure 4.

RVE with randomly distributed (a) spherical pores, (b) different pore shapes of different sizes, (c) spherical pores as well as fillers and (d) different pores shapes as well as fillers.

Table 1.

Relaxation time, shear and bulk modulii of the matrix.²³

τ _j	G _j	τ _j	G _j	τ _j	G _j
0.032	2.512	10.000	199.526	3162.278	1.698
0.100	10.000	31.623	50.119	10000.000	1.202
0.316	56.234	100.000	19.953	31622.777	1.148
1.000	316.228	316.228	12.589	100000.000	1.096
3.162	1000.000	1000.000	2.512	—	—

To further illustrate the frequency-dependent viscoelastic response of the matrix material, cyclic stress–strain curves were reconstructed under harmonic loading. Figure 5 presents the stress–strain loops at representative low, intermediate, and high frequencies. The elliptical shape of the loops is characteristic of linear viscoelastic behaviour. The enclosed loop area, which corresponds to the energy dissipated per cycle, increases near the frequency associated with the peak loss factor and decreases at very low and very high frequencies. This behaviour is consistent with the predicted variation of damping with frequency.

Figure 5.

Frequency-dependent cyclic stress–strain curves reconstructed from the homogenized complex modulus. The loop area, representing energy dissipation per cycle, is largest near the peak loss factor frequency.

The values of $\tan δ$ as a function of frequency for varying volume fractions of spherical fillers are plotted in Figure 6(a) and (b) shows the zoomed portion of peak $\tan δ$ in Figure 6(a). The peak value of $\tan δ$ decreases with an increase in the filler’s volume fraction and the frequency at which peak $\tan δ$ occurs reduces. The peak values of $\tan δ$ at varying volume fractions obtained from proposed model are compared with values provided in Ref. 23. One can observe a good agreement between the proposed values and the literature data with a small average difference of about 3%. This discrepancy between the present model predictions and those reported in Ref. 23 can be attributed to the fundamental differences in modeling approaches. Shank et al.²³ employs stochastic finite element simulations on representative volume elements with randomly distributed inclusions, thereby explicitly capturing local stress field interactions among inclusions. In contrast, the present work utilizes a modified Mori–Tanaka homogenization approach combined with a differential scheme, which is based on a mean-field approximation. While this framework efficiently accounts for phase interactions in an average sense, it does not explicitly resolve local stress concentrations or strong inclusion–inclusion interactions, particularly at higher filler volume fractions. Furthermore, the present study incorporates pores through a two-step homogenization process, which modifies the stress redistribution and effective compliance of the composite, contributing to quantitative differences in damping predictions. Therefore, the observed deviations are consistent with the inherent assumptions of mean-field micromechanical models compared to full-field stochastic FEM simulations. This indicates that the proposed model is realible for predicting the damping properties.

Figure 6.

(a) Variation of loss factor (tan δ) with frequency for different filler volume fractions. (b) Enlarged view of the peak region highlighting the shift and reduction in peak damping with increasing filler content. (c) Variation of peak loss factor with filler volume fraction predicted by the proposed model and comparison with reference results.²³

Figure 7 illustrates the variation of effective storage modulus E′ with filler volume fraction at different loading frequencies. As expected, the storage modulus increases monotonically with filler content due to the stiffening contribution of the filler phase. Furthermore, the modulus exhibits strong frequency dependence, with higher frequencies yielding larger storage modulus values. This behaviour reflects the viscoelastic nature of the matrix material, which behaves more elastically at higher loading frequencies and undergoes relaxation at lower frequencies. The near-flat response observed at very low frequency corresponds to the fully relaxed state of the matrix.

Figure 7.

Effective storage modulus E′ as a function of filler volume fraction at different loading frequencies.

It is equally important to note that experimental datasets simultaneously reporting controlled filler content, quantified pore morphology, and frequency-dependent damping properties of particulate-filled elastomers are limited in the open literature. In most studies, porosity is either not systematically characterized or is treated as a manufacturing artifact without detailed microstructural quantification. This restricts direct one-to-one validation of micromechanical models incorporating both fillers and pores. The present validation is therefore conducted using the most comprehensive datasets available, while acknowledging that further experimental investigations with controlled pore morphology would strengthen model verification.

As a next step, the elastic modulus and loss factor are obtained for different filler shapes with and without presence of pores in RVE. Figure 8 illustrates the influence of pore shape on the complex modulus (E*) and the corresponding peak loss factor (represented by η) for a representative volume element (RVE) reinforced with spherical fillers. The results are presented for two different pore volume fractions: 5% and 10%. It is observed that, for all cases, the complex modulus E* decreases with an increase in the damping parameter η, indicating the typical stiffness–damping trade-off behaviour in particle-filled composites. The inclusion of pores, irrespective of their geometry, leads to a reduction in E* compared to the pore-free composite. However, the extent of this reduction strongly depends on the pore morphology. Among the different pore shapes considered, spherical pores cause a moderate reduction in stiffness, whereas elliptic cylindrical and penny-shaped pores result in a more pronounced decrease due to their higher aspect ratios and stress concentration effects. The hybrid-shaped pores show an intermediate behaviour, combining the effects of multiple geometries.

Figure 8.

Effect of different shape of (a) 5% and (b) 10 % volume fraction of pores on complex modulus (E*) versus peak loss factor (peak tan δ) for a RVE reinforced with spherical fillers having volume fraction ranges from 0 to 30 %.

When the pore volume fraction increases from 5% to 10%, the overall modulus decreases significantly for all pore types, reaffirming the detrimental influence of porosity on stiffness. Concurrently, the damping capacity (peak tanδ) is enhanced, suggesting that higher porosity contributes to improved energy dissipation. Overall, it demonstrates that both the shape and volume fraction of pores play a critical role in governing the stiffness–damping balance of particle-reinforced composites. These findings emphasize the need for careful optimization of pore morphology to achieve desired mechanical and damping performance.

It is noted that the matrix–pore interaction represents a limiting case of the present two-step homogenization framework when the filler volume fraction is set to zero. Under this condition, the formulation reduces to a porous viscoelastic matrix without filler contribution. Therefore, the isolated effect of pore morphology on the effective response is inherently contained within the proposed model. The combined filler–pore results discussed subsequently build upon this baseline porous matrix response.

Conclusions

In this study, the influence of pore morphology on the damping and effective elastic properties of spherical particle-filled composites has been systematically investigated through a two-step stochastic homogenization framework. The developed model, based on a modified Mori–Tanaka approach coupled with a differential scheme, effectively captures the progressive inclusion of both pores and fillers within the composite microstructure. The results reveal that there exists a saturation threshold in the stiffness ratio of fillers to the matrix, beyond which further enhancement in filler stiffness yields negligible improvement in the overall elastic response. Furthermore, the presence of pores is shown to substantially degrade the elastic stiffness, emphasizing the critical role of porosity control in composite design. The close agreement between the analytical predictions and computational simulations validates the robustness and accuracy of the proposed model. Overall, the presented framework provides a reliable and efficient tool for predicting and optimizing the mechanical and damping behavior of porous particle-reinforced composites, offering valuable insights for the design of advanced materials with tailored dynamic properties.

Footnotes

Author contributions

Sonal Nirwal: Drafting Original Manuscript, Methodology.

Neeraj Kumar Shukla: Drafting Original manuscript; Methodology; Funding.

Geeta Devi: Drafting Original Manuscript; Data Analysis.

Rahul Kumar: Review and editing.

Ajay Bhardwaj: Review and editing; Data Analysis.

Aman Garg: Review and editing.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through the Large Research Project under grant number RGP2/274/46.

ORCID iD

Geeta Devi

Data Availability Statement

MATLAB codes generated or used during the study are available from the corresponding author by request.*

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