Length scale dependence in elastomers – comparison of indentation experiments with numerical simulations

Abstract

Probing depth dependent deformation at nano-and micrometer length scales has been observed in indentation experiments of polymers. Unlike in metals, where size effects are observed in plastic deformation and are attributed to geometrically necessary dislocations, the origin of size dependence in polymers is not well understood. As classical continuum theories are unable to describe such phenomena, higher order gradient theories have been developed to capture such size dependent deformation behavior. The present study adopts the penalty finite element approach for a couple stress elasticity theory under axisymmetric conditions to numerically simulate and analyze the probing depth dependent deformation. Polydimethylsiloxane (PDMS) and natural rubber have been used as model materials to analyze the depth dependent deformation at different probing depths. Simulations were performed on PDMS using spherical indenter tips of different radii to show the influence of strain/rotation gradients on elastic modulus. To capture the experimentally observed increase in hardness with decreasing probing depth, simulations applying a conical indenter tip were performed and compared with experimental data.

The Hertz contact problem in couple-stress elasticity

“…Nevertheless, the spherical indenter may be advantageous in cases where limited or no damage at the surface is in need. Furthermore, from the same set of results it is inferred that for decreasing R the effective Young's modulus E ef f increases in accordance with the results of Garg et al (2016).…”

“…In particular, size effects in polymers have been attributed to surface effects (Zhang and Xu, 2002;Zhang et al, 2004), A C C E P T E D M A N U S C R I P T friction (Lim and Chaudhri, 2006), tip imperfections (Flores and Calleja, 1998), material inhomogeneities (Briscoe et al, 1998), surface adhesion (Johnson et al, 1971), as well as multi-physics phenomena such as piezoelectricity (Giannakopoulos amd Suresh, 1999), magnetoelasticity (Giannakopoulos end Parmaklis, 2007), and flexoelectricity (Mao et al, 2016). It seems, however, that the contact related size phenomenon is mainly the result of the imposed higher order displacement gradients emerging in the vicinity of the tip of the indentor (Briscoe et al, 1998;Chong and Lam, 1999;Alisafaei et al, 2014;Lam and Chong, 1999;Lam and Chong, 2000;Lam and Chong, 2001;Swaddiwudhipong et al, 2005;Nikolov et al, 2007;Han, 2010;Alisafaei et al, 2013;Voyiadjis et al, 2014;Garg et al, 2016). It has been further shown that size effects in elastomers are considerably more pronounced than in thermoset or thermoplastic polymers in which the size effect is attributed to their molecular structure.…”

Elastic modulus of thermoplastic elastomers investigated with the instantaneous volumetric strain method

“…Previous reports have shown that a spherical indentation depth [ h ( t )] of less than 15%, as compared to the thickness of an elastoplastic material, results in K ( t ) that is close to the actual value . It has been emphasized that the sphere radius plays a significant role in K ( t ) measurements . The instantaneous strain ratio corresponding to the spherical indentation test is the ratio of h ( t ) to the sample thickness [ H ( t )].…”

“…22 It has been emphasized that the sphere radius plays a significant role in K(t) measurements. [23][24][25] The instantaneous strain ratio corresponding to the spherical indentation test is the ratio of h(t) to the sample thickness [H(t)]. However, the TPE surfaces produce spherical coronal deformation [V g (t)] under external conditions; this results in three-dimensional volumetric changes instead of two-dimensional longitudinal changes.…”