Surface tension and a self-consistent theory of soft composite solids with elastic inclusions

Abstract

The importance of surface tension effects is being recognized in the context of soft composite solids, where it is found to significantly affect the mechanical properties, such as the elastic response to an external stress. It has recently been discovered that Eshelby's inclusion theory breaks down when the inclusion size approaches the elastocapillary length L ≡ γ/E, where γ is the inclusion/host surface tension and E is the host Young's modulus. Extending our recent results for liquid inclusions, here we model the elastic behavior of a non-dilute distribution of isotropic elastic spherical inclusions in a soft isotropic elastic matrix, subject to a prescribed infinitesimal far-field loading. Within our framework, the composite stiffness is uniquely determined by the elastocapillary length L, the spherical inclusion radius R, and the stiffness contrast parameter C, which is the ratio of the inclusion to the matrix stiffness. We compare the results with those from the case of liquid inclusions, and we derive an analytical expression for elastic cloaking of the composite by the inclusions. Remarkably, we find that the composite stiffness is influenced significantly by surface tension even for inclusions two orders of magnitude more stiff than the host matrix. Finally, we show how to simultaneously determine the surface tension and the inclusion stiffness using two independent constraints provided by global and local measurements.

Capillary control of collapse in soft composite columns

“…Stiffer materials with E = O(MPa), like elastomers, may also exhibit enhanced stiffness due to capillarity when liquid inclusion size is O(0.1µm). These results hold in the dilute and non-dilute limits [42,43].…”

“…( 7), reduces to Eshelby's result for liquid droplets in an elastic solid; E c = E/(1 − φ). This differs from the stiffening case in how the dilute theory (DT) [22,23] in the softening case deviates from nondilute approaches as φ increases [24,42,43].…”

“…We take this approach because the expression is simpler than the quantitatively equivalent one obtained using the three-phase theory of soft composite solids [42,43].…”

“…( 16) and (17), for φ constant and γ′ linear, capture the behavior in the dilute limit. Moreover, the two non-dilute theories accounting for interfacial tension [42,43] apply both for stiffening and softening regardless of the volume fraction. Unfortunately, however, the expression for the effective Young's modulus in the strongly nondilute regime is so complex that Eq.…”

The mechanical equilibrium of soft solids with surface elasticity

“…This gives a basis to theoretical work that has long predicted how surface stresses might influence physical phenomena (e.g. [18][19][20][21][22][23][24][25][26][27][28]).…”