TY - JOUR

T1 - The effective shear modulus of a random isotropic suspension of monodisperse liquid n-spheres

T2 - from the dilute limit to the percolation threshold

AU - Ghosh, Kamalendu

AU - Lefèvre, Victor

AU - Lopez-Pamies, Oscar

N1 - Funding Information:
Support for this work by the National Science Foundation through the Grant DMREF-1922371 is gratefully acknowledged.
Publisher Copyright:
© 2023 The Royal Society of Chemistry.

PY - 2022/11/23

Y1 - 2022/11/23

N2 - A numerical and analytical study is made of the macroscopic or homogenized mechanical response of a random isotropic suspension of liquid n-spherical inclusions (n = 2, 3), each having identical initial radius A, in an elastomer subjected to small quasistatic deformations. Attention is restricted to the basic case when the elastomer is an isotropic incompressible linear elastic solid, the liquid making up the inclusions is an incompressible linear elastic fluid, and the interfaces separating the solid elastomer from the liquid inclusions feature a constant initial surface tension γ. For such a class of suspensions, it has been recently established that the homogenized mechanical response is that of a standard linear elastic solid and hence, for the specific type of isotropic incompressible suspension of interest here, one that can be characterized solely by an effective shear modulus = n in terms of the shear modulus μ of the elastomer, the initial elasto-capillary number eCa = γ/2μA, the volume fraction c of inclusions, and the space dimension n. This paper presents numerical solutions—generated by means of a recently introduced finite-element scheme—for n over a wide range of elasto-capillary numbers eCa and volume fractions of inclusions c. Complementary to these, a formula is also introduced for n that is in quantitative agreement with all the numerical solutions, as well as with the asymptotic results for n in the limit of dilute volume fraction of inclusions and at percolation . The proposed formula has the added theoretical merit of being an iterated-homogenization solution.

AB - A numerical and analytical study is made of the macroscopic or homogenized mechanical response of a random isotropic suspension of liquid n-spherical inclusions (n = 2, 3), each having identical initial radius A, in an elastomer subjected to small quasistatic deformations. Attention is restricted to the basic case when the elastomer is an isotropic incompressible linear elastic solid, the liquid making up the inclusions is an incompressible linear elastic fluid, and the interfaces separating the solid elastomer from the liquid inclusions feature a constant initial surface tension γ. For such a class of suspensions, it has been recently established that the homogenized mechanical response is that of a standard linear elastic solid and hence, for the specific type of isotropic incompressible suspension of interest here, one that can be characterized solely by an effective shear modulus = n in terms of the shear modulus μ of the elastomer, the initial elasto-capillary number eCa = γ/2μA, the volume fraction c of inclusions, and the space dimension n. This paper presents numerical solutions—generated by means of a recently introduced finite-element scheme—for n over a wide range of elasto-capillary numbers eCa and volume fractions of inclusions c. Complementary to these, a formula is also introduced for n that is in quantitative agreement with all the numerical solutions, as well as with the asymptotic results for n in the limit of dilute volume fraction of inclusions and at percolation . The proposed formula has the added theoretical merit of being an iterated-homogenization solution.

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U2 - 10.1039/d2sm01219g

DO - 10.1039/d2sm01219g

M3 - Article

C2 - 36507911

AN - SCOPUS:85144689798

SN - 1744-683X

VL - 19

SP - 208

EP - 224

JO - Soft Matter

JF - Soft Matter

IS - 2

ER -