Micro-structured materials: Inhomogeneities and imperfect interfaces in plane micropolar elasticity, a boundary element approach

Abstract

Abstract. In this paper we tackle the simulation of microstructured materials modelled as heterogeneous Cosserat media with both perfect and imperfect interfaces. We formulate a boundary value problem for an inclusion of one plane strain micropolar phase into another micropolar phase and reduce the problem to a system of boundary integral equations, which is subsequently solved by the boundary element method. The inclusion interface condition is assumed to be imperfect, which permits jumps in both displacements/microrotations and tractions/couple tractions, as well as a linear dependence of jumps in displacements/microrotations on continuous across the interface tractions/couple traction (model known in elasticity as homogeneously imperfect interface). These features can be directly incorporated into the boundary element formulation.The BEM-results for a circular inclusion in an infinite plate are shown to be in an excellent agreement with the analytical solutions. The BEM-results for inclusions in finite plates are compared with the FEM-results obtained with FEniCS.

‘Explicit’ and ‘implicit’ non-local continuous descriptions for a plate with circular inclusion in tension

“…The above-mentioned results are in accordance with the existing literature [32,48]. In the following subsections, the comments about each model with different scale ratios will be detailed with focusing on the conducted non-local theory.…”

“…It is important to notice that boundary effects may be of importance for domains that have dimensions comparable with the influence zone of nonlocal parameter: In order to compensate the decreasing stresses, close to boundaries, the stress at remaining regions needs to be higher due to balance requirement. Indeed, for this particular scale ratio, SCFs of 'explicit' non-local model turns out to be higher than expected for Mats As for L/a = 20.0, the plate can be assumed as an infinite medium with a small inclusion, SCFs tabulated in Table 3 are compared with the analytical values reported in the literature (Atroshchenko et al [48]), by calculating their relative difference (i.e. at point A):…”

“…From a different point of view, Dong, Wang and Rubin [47], treated the interface as a finite thickness Cosserat shell/membrane medium and compared their results with explicit solutions. More recently, Atroshchenko et al [48], modelled the matrix-inclusion structure as Cosserat media considering various material properties and showed the effect of imperfect and perfect interfaces via employing boundary element method (BEM), while Fantuzzi et al, [32] investigated the classical matrix-inclusion problem considering micropolar theory using both strong and standard forms of the finite element method.…”

Some Novel Numerical Applications of Cosserat Continua

“…Moreover, due to the symmetry of the problem only one-fourth of the total geometry has been modeled numerically. The present SFEM and FEM numerical solutions are compared to an analytical and a BEM solutions found in the literature [Atroshchenko et al (2016)]. Table 3.…”

Vibration of cracked functionally graded microplates by the strain gradient theory and extended isogeometric analysis

“…Besides, the gradient theory and Cosserat theory are not similar due to the fact that the latter considers independent rotation components in addition to the displacement components which are used in the former [7]. There are many works have been done based on the couple stress theory [8][9][10] and Cosserat theory [11][12][13] to predict the behaviours of small-scale structures. Meanwhile, the generalised strain gradient theory by Mindlin which is also called, for the lowest-order theory, dipolar gradient theory or grade-two theory is considered as one of the most effective theory.…”