The influence of void size on the micropolar constitutive properties of model heterogeneous materials

Abstract

In this paper the mechanical behaviour of model heterogeneous materials consisting of regular periodic arrays of circular voids within a polymeric matrix is investigated. Circular ring samples of the materials were fabricated by machining the voids into commercially available polymer sheet. Ring samples of differing sizes but similar geometries were loaded using mechanical testing equipment. Sample stiffness was found to depend on sample size with stiffness increasing as size reduced. The periodic nature of the void arrays also facilitated detailed finite element analysis of each sample. The results obtained by analysis substantiate the observed dependence of stiffness on size. Classical elasticity theory does not acknowledge this size effect but more generalized elasticity theories do predict it. Micropolar elasticity theory has therefore been used to interpret the sample stiffness data and identify constitutive properties. Modulus values for the model materials have been quantified. Values of two additional constitutive properties, the characteristic length and the coupling number, which are present within micropolar elasticity but absent from its classic counterpart have also been determined. The dependence of these additional properties on void size has been investigated and characteristic length values compared to the length scales inherent within the structure of the model materials.

On the coupling number and characteristic length of micropolar media of differing topology

“…Beveridge et al [21] argue that the deliberate creation, testing and analysis of such a material is more than simply didactic; the consistency of the constitutive property data obtained through independent testing and finite-element analysis reinforces the credibility of the size effect methodology in the more challenging case of materials with genuinely threedimensional heterogeneity where only an experimental approach is viable, because analysis is presently unrealizable in practice for the reasons already outlined. Acknowledging this, Waseem et al [22] employed a similar approach albeit in a circular ring sample geometry rather than a straight beam to identify size effects in a material comprising an acrylic polymer matrix with the same topology. For consistency, micropolar elasticity theory was again used to translate the observed effects into the relevant constitutive properties; the characteristic length and coupling number.…”

A higher order control volume based finite element method to predict the deformation of heterogeneous materials

“…While the method generally shows equivalent predictive performance for the stress concentration problem when compared to the equivalent finite element based procedure, this performance varies slightly less as one of the additional constitutive parameters, the coupling factor, is altered. This supports the preferential use of the method in quantifying this parameter from experimental data via an inverse iterative approach [21,22]. Using the CV-MPLST in an inverse method the characteristic length and coupling number have been successfully quantified for a model two phase aluminium composite.…”

“…Micropolar constitutive properties are difficult to identify experimentally and few properties have been published. The higher order element, set out here, has been used in an iterative inverse procedure to identify the micropolar constitutive properties of model two phase metal composite beam [21] and polymer ring [22] samples. This inverse procedure has produced a simple method to determine constitutive properties.…”

Paradoxical Size Effects in Composite Laminates and Other Heterogeneous Materials

“…Characteristic length values were determined from the gradient of the stiffness variation while modulus values were identified from the interception of this variation with the stiffness axis. Subsequently, comparable behaviour was observed in slender ring samples with similar heterogeneity that were loaded diametrically [Waseem et al, 2013] and in ring samples in which the topology of the void array constituting the heterogeneity was varied [McGregor & Wheel, 2014].…”

“…The generalized continuum theories already mentioned predict size effects in which size scale reduction results in an apparent increase in stiffness. Behaviour like this has been observed in both polymeric foams [Lakes, 1983[Lakes, , 1986Anderson & Lakes, 1994] and materials comprised of a two dimensional homogeneous matrix perforated by a regular array of circular voids Waseem et al, 2013;McGregor & Wheel, 2014]. However, there are materials such as cortical bone, for which both increasing and decreasing stiffness with reducing size have been reported [Yang & Lakes, 1982;Choi et al, 1990].…”